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Physical mechanisms of intercalation batteries McKinnon, W. Ross 1980

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PHYSICAL MECHANISMS OF INTERCALATION BATTERIES  by  W. ROSS MCKINNON B.Sc, M.Sc,  Dalhousie University, 1975 Dalhousie University, 1976  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  in THE FACULTY OF GRADUATE STUDIES (Department of Physics)  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA Apri1 , 1980  0  W. Ross McKinnon, 1980  In  presenting this  thesis  an advanced degree at  in p a r t i a l  further  for  the  freely  available  for  that  r e f e r e n c e and study.  t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f  representatives.  this  thesis for  It  i s understood that  financial  gain s h a l l  written permission.  Department The  of  ^~V^S>\c£,  U n i v e r s i t y o f B r i t i s h Columbia  2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5  Date  ~V  A4p*VA  \°\'5rC^  not  fo r  requirements  this  thesis  s c h o l a r l y purposes may be granted by the Head of my Department  by h i s of  agree  of  the U n i v e r s i t y of B r i t i s h Columbia, I agree  the L i b r a r y s h a l l make i t I  fulfilment  copying or  or  publicati  be allowed without my  i on  ii  ABSTRACT  This thesis i d e n t i f i e s batteries.  and discusses physical mechanisms in intercalation  The effects of interactions  and ordering of intercalated atoms  on the voltage behaviour of intercalation c e l l s is described, terms of the l a t t i c e  largely in  gas model of i n t e r c a l a t i o n .  P a r t i c u l a r emphasis  given to the mean f i e l d solutions of the l a t t i c e  gas model, which are  compared to more exact solutions  for several  between intercalated atoms are discussed, interactions; compounds.  namely e l e c t r o n i c and e l a s t i c  The kinetics of intercalation batteries  host l a t t i c e .  Two types of interaction  i t is found that both can be important in intercalation  emphasis on overpotentials  of  cases.  is  is also discussed, with  due to diffusion of the intercalated atoms in the  Experimental studies of the voltage behaviour of three types  lithium intercalation c e l l s ,  Li Ti S_ , Li MoO. , and Li MoS„ , are presented, x 2 x 2 x 2  which i l l u s t r a t e the variety of voltage behaviour found in intercalation ce1 Is.  TABLE OF CONTENTS Page  ABSTRACT  i i  TABLE OF CONTENTS  iii  LIST OF TABLES  vi  LIST OF FIGURES  vi i  LIST OF SYMBOLS  xi  ACKNOWLEDGEMENTS  xvi i  CHAPTER 1  INTRODUCTION  1  1 .1 Intercalation and Batteries  1  1.2 Contributions of This Thesis  6  PART A: Review of  Intercalation Systems  CHAPTER 2 LAYERED COMPOUNDS AND RUTILES 2.1 Introduction 2.2 Layered Transition Metal Dicha1cogenides Structure and Properties 2.3 Intercalation of Transition Metal Dicha1cogenides l.k Metal Dioxides with Rutile-Re1ated Structure Structure and Properties 2.5 Intercalation of Rutiles CHAPTER 3  FURTHER PROPERTIES OF INTERCALATION AND RELATED PHENOMENA  3- 1 I nt roduct i on 3.2 Methods of Intercalation 3.3 Intercal at ion of Graphite 3.h Hydrogen in Metals 3.5 I n t e r s t i t i a l Compounds of the Transition Metal Di chal cogen i des 3.6 Oxide Bronzes 3.7 Superionic Conductors  PART B: Thermodynamics of CHAPTER k  Intercalation Batteries  LATTICE GAS THEORY OF INTERCALATION  8 9 9 10 16 19 5  2  26 26 26 28 29 32 33 3^  36 37  h. 1 Introduct ion 37 h.2 Thermodynamics 39 4.3 Lattice Gas Models Applied to Intercalation Systems *t1 k.k Lattice Gas Models with Interactions ^8 h.5 Mean F i e l d Solution of the Problem of Ordering 52 h.S One Dimensional Lattice Gas 64 k.l Interacting Lattice Gas with Different Site Energies.. ....7^  iv  Page 4.8 Inclusion of Three Body Forces 4.9 Changes in the Host CHAPTER 5  77 80  ELECTRONIC INTERACTIONS BETWEEN INTERCALATED ATOMS  5-1 Introduction 5.2 Screened Coulomb Interaction 5.3 Metal - I nsul ator Transitions CHAPTER 6  83 84 95  ELASTIC INTERACTIONS BETWEEN INTERCALATED ATOMS  6. 1 6.2 6.3 6.4 6.5  Introduction Infinite Medium Interaction W°° The Image Interaction W-Lattice Gas Models and E l a s t i c Interactions Chemical Potential in Non-homogeneously Intercalated Hosts. 6.6 Limitations of the Theory 1  PART C: Kinetics of CHAPTER 7 7.1 7.2 7>3 7.4 7.5  Intercalation  Batteries  Introduction Electrochemistry of Intercalation Cells Losses Due to Transport Across the Interfaces Transport Through the Electrolyte Diffusion in the Host DIFFUSION IN INTERCALATION COMPOUNDS  8.1 Introduction 8.2 Behaviour of D(x) 8.3 Model Calculation of Diffusion Lattice CHAPTER 9  DIFFUSION 0VERV0LTAGES  122 125  130 130 131 136 138 141 143  in a One Dimensional  IN INTERCALATION CELLS  POROUS ELECTRODES  10.1 Introduction 10.2 Ohmic Models 10.3 Electrolyte Depletion PART D: Experimental Procedure and Results CHAPTER 11  99 104 112 117  143 1 45  9. 1 Introduction 9.2 Diffusion for Constant D 9.3 Motion of a Phase Boundary CHAPTER.10  99  129  KINETICS OF ELECTROCHEMICAL CELLS  CHAPTER 8  83  EXPERIMENTAL PROCEDURE  11.1 Introduction 11.2 Materials Used .11.3 Cathode Preparation and Cell Assembly  148 160 160 161 171 1 80 180 182 T85 189 190 190 190 192  Page 11.4 Techniques Used to Study Intercalation Cells 11.5 Effect of Series Resistance and Diffusion on Current-Voltage Curves CHAPTER 12 12.1 12.2 12.3 12.4 12.5  EXPERIMENTAL RESULTS  Introduction Excess Capacity and Kinetic Limitations of the Cells Li/Li TiS Intercalation Cells Li/Li Mo0 Intercalation Cells Li/Li MoS2 Intercalation Cells x  x  2  2  x  CONCLUSION CHAPTER 13  196 203 203 204 207 217 221 236  SUMMARY AND FUTURE WORK  13.1 Summary of the Thesis 13.2 Suggestions for Future Work APPENDICES A. . Equivalence of Lattice Gas and Ising Models B. One Dimensional Ising Model C. One Dimensional Lattice Gas of Hard Spheres D. One Dimensional Lattice Gas With Two Site Energies E. Effects of Weak Coupling Between Lattice Gas Chains  BIBLIOGRAPHY  1 gif  237 ...237 240 242 242 243 248 250 253  259  vi  LIST OF TABLES  Table  Page  I  Transition Metals Which Form Layered D i cha 1 cogen i des  II  Metals Whixh Form Rutile-Related  III  Data for Cells Discussed in Chapter 12  Oxides  10 19 235  VI I  LIST OF FIGURES Figure  Page  1.  Schematic L i / L i  2.  Structure of the Layered Transition Metal Dichalcogenides  11  3.  ABC Notation for Close-Packed Spheres  12  k.  Sites Available for Intercalated Atoms in Layered Compounds..... 13  5.  Schematic Band Structures of the Transition Metal  Ti S_ Intercalation Cell x 2  Di cha 1 cogen I des  2  15  6.  Voltage and Inverse Derivative -8x/9V for Li T i S , 0 _< x <_ 1 . . . . 1 8  7.  Ruti le Structure. .  8.  Sites Available for Intercalated Atoms in the Rutile Structure..22  9.  Schematic Band Structure of the Ruti1e-Related Metal Oxides  2k  10.  Phase Diagram of h^Nb  30  11. 12.  Schematic Intercalation Cell Voltage and Inverse Derivative -8x/8V for Non-interacting Lattice Gas with One Site Energy Voltage and Inverse Derivative -8x/8V for Non-interacting Lattice Gas with Two Site Energies  38  13.  14.  Voltage and Free Energy for Lattice Gas with A t t r a c t i v e Interactions in Mean Field Theory  20  kS kj 50  15.  Decomposition of a Triangular Lattice into Three S u b l a t t i c e s . . . .52  16.  u - 9Ux for Triangular Lattice Gas with Nearest Neighbour Interactions U = 4kT in Mean F i e l d Theory  55  17.  Voltage and Free energy near the F i r s t Order Phase Transition in a Triangular Lattice Gas with Nearest Neighbour Interactions U = 4kT in Three Sublattice Mean Field Theory 57  18.  Sublattice Occupations for Triangular Lattice Gas with Nearest Neighbour Interactions U = 4kT in Three Sublattice Mean Fi^eld Theory  59  Voltage and Inverse Derivative -3x/9V for Triangular Lattice Gas with Nearest Neighbour Interactions U = kkl in Three Sublattice Mean F i e l d Theory  60  19.  vi i i  Figure  Page  20.  Phase Diagram for Triangular Lattice Gas with Repulsive Nearest Neighbour Interactions Calculated Using Three Sublattice Mean Field Theory and Renormalization Group Techniques 61  21.  Voltage and Inverse Derivative - 3 x / 3 V for Triangular Lattice Gas with Nearest Neighbour Interactions U = kkT calculated using Renormalization Group Techniques  63  Sublattice Occupation and Voltage for Triangular Lattice Gas with Nearest Neighbour Interactions U = kT/0.72 in Three Sublattice Mean F i e l d Theory  65  Voltage for One Dimensional Lattice Gas with A t t r a c t i v e arid Repulsive Nearest Neighbour Interactions  67  One Dimensional Lattice Gas with Repulsive Nearest Interactions Showing a F i l l i n g Mistake  68  22.  23.  2k.  25.  -26.  Neighbour  Voltage and Inverse Derivative - 9 x / 3 V for One Dimensional Lattice Gas with Repulsive Hard Sphere Interactions Two Dimensional  Lattice of One Dimensional  -. 70 71  Chains  27.  Free Energy and Voltage of Triangular Lattice Gas with Two Site Energies and Repulsive Nearest Neighbour Interactions 76  28.  Free Energy for Triangular Lattice Gas with Two and Three Body I nte ract i ons  29.  30.  Voltage for Triangular Lattice Gas with Nearest Two and Three Body Interactions  78  Neighbour 79  Schematic Form of the Free Energy of an Intercalation Compound with a Structural Transformation in the Host  80  31.  Kinetic Energy versus Density of States for a Free Electron Gas.83  32.  Force Density of Intercalated Atoms Giving Rise to a Diagonal Dipole Tensor P 1J Polar Plot Showing Angular Variation of Strain Induced Interaction W°°(_r) Between Two Intercalated Atoms in Layered Compounds  33.  103  10 8  34.  As in F i g . 33, for Rutile Structures  109  35.  Schematic Summary of the Nature of the Strain Induced Interaction W°°(_r).;.  110  Schematic Discharge Curve of an Intercalation Hysteresis  119  36.  Cell  Showing  ix  Figure  37-  Page  E l a s t i c Equivalence of a Plane of  Intercalated Atoms to a  Dislocation Loop  120  38.  Interaction  121  33-  Schematic  hO.  Mobility and Enhancement Factor for One Dimensional Lattice Gas with Repulsive Nearest Neighbour Interactions U = 5kT 15/4 Diffusion Coefficient and "Conductivity" xM corresponding to F i g . 40 155  41.  Between Two Dislocation Loops  Intercalation  131  Cell  42.  Mobility and Enhancement Factor for One Dimensional Lattice Gas with A t t r a c t i v e Nearest Neighbour Interactions U = - 2 . 5 kT 158  43.  Diffusion Coefficient to F i g . 42  44. 45.  46. 47.  48. 49.  and "Conductivity" xM corresponding 159  Three Geometries Considered in the Discussion of of Intercalated Atoms..  Diffusion 161  Surface Composition for Intercalation at Constant Current for a Constant Diffusion Coefficient  165  Fractional Capacity for Intercalation at Constant Current for a Constant Diffusion Coefficient  167  Surface Composition for Intercalation at Constant Current in the Case of Phase Boundary Motion  175  Effects of Diffusion on a Voltage Plateau in an Intercalation Cell  1 76  Fractional Capacity for Intercalation at Constant Current in the Case of Phase Boundary Motion  178  50.  Planar Porous Intercalation  180  51.  Resistor Chain Used to Model Porous Electrodes  52.  Resistor-Capacitor Network Used to Model  Electrode  183  Intercalation  of Porous Electrodes  '....]8k  53-  Two Types of Pressed Cells used for Lithium Intercalation  54.  RC C i r c u i t used to Discuss  Effects of Cell  193  Resistance  on Inverse Derivative Curves  196  55.  Current-Voltage Curves Calculated for RC Circuit of F i g . 54  199  56.  Current-Voltage Curves Calculated for Large and Small Diffusion Coefficients  200  Figure  Page  57.  Charge/Discharge Cycles for Li TiS  58.  Current-Voltage Curves for L i T i S  59.  Current-Voltage Curves for Li TiS  x  X  2  to 1.0 V  208  to 1.0 V  210  to 1.8 V  211  L-  60.  Charge/Discharge Cycles for Li^TiS^ to x = 2:  214  61.  Charge/Discharge Cycles for Li^TiS^ to x = 3  215  62.  Charge/Discharge Cycles for Li^MoO^  218  63.  Current-Voltage Curves for Li Mo02  219  64.  Voltage Behaviour of Li MoS  222  65.  Charge/Discharge Cycles for Li MoS„, Showing Conversion from Phase II to Phase I Charge/Discharge Cycles for Li MoS , Showing Conversion from Phase I I I to Phase I  225  Current-Voltage Curves for Li MoS , Phase I  226  66. 67.  x  x  2  X  68a.  224  £-  Current-Voltage Curves for Li MoS , Phase II, from 1.0 V to 2 . 8 V  227  68b.  As in a; Except from 1.0 V to 2,2 V  228  68c.  As in a, Except from 1.0 V to 1.95 V  69a.  Current-Voltage Curves for Li MoS , Phase from 1.6 V to 2.6 V  229 III,  *.  230  69b.  As in a, Except from 0.9 V to 2.6 V  231  69c.  As in a, Except from 0.1 V t o 2 . 7 V  231  70.  Inverse Derivative -8x/9V for Lattice Gas of Weakly Interacting Chains Phase Diagram for Lattice Gas of Weakly Interacting Chains  257 258  71.  :  XI  LIST OF SYMBOLS  A  area  A'-  apparent area of porous electrode  a  lattice  a, b  Bohr radius  ag  effective  B  magnetic f i e l d  b  parameter used in Chapter 11  ;  b^,b  constant  Bohr radius  creation and annihilation operators for l a t t i c e  C  capaci tance  C^  heat capacity at constant chemical potential  c  lattice  gas  constant  c. .. . element of e l a s t i c stiffness i jk£  tensor  c. . reduced (matrix) notation for c. .. . ij ijkJo D  diffusion  coefficient  .denominator in Appendices D and E  V d  diameter of hard sphere in one dimensional l a t t i c e  E  energy  E , E. , E, a o' 1 E  m  e  gas  s i te ene rgi es 3  magnetic energy of I sing model magnitude of e l e c t r o n i c charge base of natural  logarithms  &q  k i n e t i c energy of free electron state q  &^  k i n e t i c energy of highest f i l l e d k i n e t i c energy state in the absence of perturbing ions (unperturbed Fermi energy)  <5&^. change in the k i n e t i c energy of the highest f i l l e d state caused by the perturbing ions (change in the Fermi: energy)  xi i  F  f r e e energy  F^  magnetic f r e e energy  F  a d d i t i v e c o n s t a n t t o the f r e e energy, o r n o n - e l a s t i c p o r t i o n o f F  Q  f.  component o f body  f?  component o f s u r f a c e  G  conductance o f e l e c t r o d e - e l e c t r o l y t e i n t e r f a c e  G. . U  force force  element o f e l a s t i c Green's  function  H  Hami l t o n i an  fi  P l a n c k ' s c o n s t a n t d i v i d e d by 2TT  I  current  1^  l i m i t i n g current of e l e c t r o l y t e  i  current  3  number c u r r e n t  density  number c u r r e n t  d e n s i t y at the s u r f a c e '  J  density  s Bessel  function of order 1  K  i n t e r a c t i o n energy i n I s i n g model  L  transfer  L  .  o f i n t e r c a l a t i o n host  matrix  transport c o e f f i c i e n t  aa £  length o f pore i n porous  I-  thickness  M  mob i 1i t y  o f porous  magnetization m  electrode  electrode  (Chapter k o n l y )  average m a g n e t i z a t i o n p e r s p i n  i n I s i n g model  e l e c t r o n mass (Chapter 5 o n l y ) ground s t a t e degeneracy ni" N  ( S e c t i o n 4.3 o n l y )  e f f e c t i v e e l e c t r o n mass number o f s i t e s o r number o f host atoms  n  number of intercalated atoms  n  unit vector normal to the surface  n  a  occupation number for the s i t e a  P. .  element of e l a s t i c dipole tensor  i J  P.  diagonal element of P . .  p  p re s s u re  Q  charge  0_  charge required to change x by 1  Q  charge flow in intercalation c e l l to change voltage to some cut-off va 1 ue  o  Q  x  2  charge flow in intercalation c e l l when voltage is cycled between two limits  Q maximum value of Q m c q.p  magnitude of Fermi wave vector in free electron gas  R  radius of sphere of cylinder or halfwidth of slab  R,R, ,R ° c  resistance  r  pos i t i on  r  location of phase boundary  S  entropy  S.  sublattice entropy  s  Ising spin variable for s i t e a  S  ijk£  T T  l  e m e r |  t  of e l a s t i c compliance tensor  temperature r  t t  e  room temperature, 2S°C  (kT = 25-7 meV) r  t i me o  time to f i l l  host to x = 1 at current -I for uniform intercalation  t  time to reach cutoff voltage at current I  t,  h a l f - c y c l e time  interaction energy between p a r t i c l e s on sites a and a '  ^aa'  U,U ,U ,U ,U"  special choices of U ,  l  o  1  _u  displacement  V  voltage  V  sweep rate of  v  volume  W.|2  field  voltage  e l a s t i c interaction energy between p a r t i c l e s  1 and 2  e l a s t i c interaction energy which is affected by boundary conditions CO  W  e l a s t i c interaction energy between two p a r t i c l e s  in an i n f i n i t e host  e l a s t i c interaction energy between two p a r t i c l e s due to the of the surface  presence  w  j ump probab i 1 i ty  X  porosity  x  composition of an intercalation compound; composition or fractional occupation of a l a t t i c e gas  x.  sublattice  x  composition at the surface  g  occupation  Ax  change in x due to a phase t r a n s i t i o n  Y  Young's modulus  y  distance along pore or in e l e c t r o l y t e difference partition  in sublattice  charge in units of e  a  s i t e label n  (Appendix E only)  function  z  a  populations  coefficient  in solution of diffusion problem  3  width of V(Q_) in Chapter 11  T  density of states for free electron  Y  number of nearest neighbour sites  gas  (coordination number)  structural parameter in layered compounds (Chapter 6 only)  XV  e.. iJ e  o  element of strain tensor s t r a i n at x = 1  £  parameter (1 , 2, or 3) used in discussing diffusion  r|  overpotenti al  6  polar angle  K  d i e l e c t r i c constant  \fj  isothermal compressibility  X  Thomas-Fermi screening length decay length in porous electrodes chemical potential  y y  s  chemical potential at the surface  y  g  elastic  contribution to y which is sensitive to boundary conditions  y  electrochemical  potential  V  Poisson's  ratio  5  coherence  length in one dimensional l a t t i c e  II  dipole moment operator  TT  3.14159...  p  number density of intercalated atoms number density of electrons  Z(t) a a., iJ T,T'  (Chapter 5 only)  sum in solution of diffusion problem conductivity element of stress tensor time  constants  $  tortuosity  (J)  e l e c t r i c potential  X  surface potential  Xy  magnetic s u s c e p t i b i l i t y  ¥  current-current correlation function  gas  angular frequency hopping frequency relaxation  frequency  ACKNOWLEDGMENTS  It is a pleasure to thank my supervisor, Rudi Haering, for his and encouragement indispensible  throughout this project.  His physical  insight proved  in our attempts to understand this complicated  subject.  A large number of people have worked in Rudi Haering's group on batteries,  and I have benefited  advice  from working with each one.  intercalate  In p a r t i c u l a r ,  I would like to thank fellow thesis writers Dave Wainwright, Ul ri ch Sacken , and Jeff Dahn, for their discussions and encouragement; did some of the mean f i e l d calculations benefited  Jeff Dahn also  in F i g . 20 and 22.  from discussions with John Berlinsky and B i l l  I have also  Unruh.  like to thank Peter Haas for his expert work on the diagrams.  I would Finally,  I thank the National Research Council of Canada for financial support.  1  CHAPTER 1  INTRODUCTION  1.1  Intercalation and Batteries  The term "intercalation" was f i r s t used s c i e n t i f i c a l l y insertion of various types of guest atoms or molecules planes of graphite," very l i t t l e ,  In this process,  apart from an increase  to describe  the  between the atomic  the host graphite structure changes,  in the separation of the planes and a.  possible change in their stacking arrangement.  The same term was later  extended to describe similar processes in other layered compounds, notably the transition metal dica1cogenides.  With the recognition that the  calation process could be used to make rechargeable high energy batteries  (Whittingham, 1976),  inter-  density  a search began for the optimum host materials  for. use in such battery systems.  These new materials do not necessarily  have  layered structures, but the term intercalation has been carried over to these other systems as w e l l .  In keeping with this newly expanded d e f i n i t i o n , we w i l l intercalation compound to refer to any s o l i d which has this between guest and host atoms, (intercalated)  use the term distinction  i f the guest atoms can be reversibly added  into the host without s i g n i f i c a n t l y a l t e r i n g the host struc-  ture, at least over some range of composition and temperature. necessarily  This  requires that the guest atoms have a s i g n i f i c a n t mobility in the  host structure in this temperature and composition range.  This d e f i n i t i o n  encompasses systems not t r a d i t i o n a l l y included, such as the metal-hydrogen systems.  It does not include those compounds whose structure can be regarded  as a host l a t t i c e with additional atoms in i n t e r s t i t i a l i n t e r s t i t i a l atoms cannot be removed.  sites,  i f these  Many such i n t e r s t i t i a l compounds with  2  layered structures, prepared by combining the constituent  elements at high  temperatures, have unfortunately been Widely referred to as compounds; however, since we are most-interested  intercalation  in the very properties  that  these systems lack, namely those associated with the reversible addition of guest atoms, we w i l l  not include them in the d e f i n i t i o n used here.  Studies of intercalation have dramatically increased since battery systems was f i r s t suggested.  Batteries employing intercalation  compounds are conceptually very simple. calation battery system,  Li/Li^TiS^,  of an intercalation cathode, electrolyte, example,  its use *n <  A diagram of the best known inter-  is shown in F i g . 1.  The c e l l  Li TiS , and an anode, Li metal, x 2  consists  immersed in an  a solution of some Li-bearing s a l t in an organic liquid  lithium perchlorate dissolved  in propylene carbonate).  (for  Discharge of  the c e l l causes a transfer of Li from the anode to the cathode, with L i migrating through the e l e c t r o l y t e c i r c u i t . The process  solution and electrons  is reversed during recharge.  1 -  ions  through the external  (The terms  anode„and  cathode actually refer to the direction of charge transfer at the  Fig.  +  interface  Schematic view of Li/LT T I S 2 intercalation c e l l showing direction of flow of ions and electrons during discharge. X  3  between the electrode and the s o l u t i o n , and so s t r i c t l y speaking the terms should be interchanged in the diagram during recharge.  We shall  ignore t h i s ,  and apply the terms as in F i g . 1 independently of the d i r e c t i o n of current flow.)  Since the L i  +  ions are in equilibrium throughout the transfer of  atoms from anode to cathode.(for  infinitesimal current flow), the only work  done is that done by the electrons, which is just eV for each electron charge -e), where V is the battery voltage.  (of  Since this work is the d i f f e r -  ence in chemical p o t e n t i a l , y, of Li in the cathode (c) and anode (a), we have  eV = - (  [. - y^.,)  V  .  (D  Thus, in addition to their possible technological  importance, intercalation  batteries provide a tool to study the process of intercalation  itself,  through measurement of the chemical potential of the guest atom. Although this thesis w i l l  be most concerned with this l a t t e r use of  intercalation batteries, we w i l l  consider b r i e f l y their practical  The most demanding application of these batteries  aspects.  is the e l e c t r i c v e h i c l e .  Reviews of various competing battery systems being considered are given by Birk.et'al  (1979) and McCoy (1977).  Although many different parameters must  be considered, the four most important are energy density, peak power density, cycle l i f e , and cost.  The energy density, or energy available per unit mass,  is generally quoted in watt-hours per kilogram (Wh/kg), and must be at least 100 Wh/kg for .thelbattery to be viable in an urban vehicle; otherwise, much energy is needed just to propel the batteries.  too  The peak power density  is the power (per unit mass) that the battery can supply over a brief period (usually taken to be 15 seconds) and must be greater than 100 W/kg for adequate acceleration.  The battery should cost less'than $50/kWh, and should  provide about 500 deep cycles,  A typical e l e c t r i c vehicle would  4  then contain 400 kg of batteries, with the batteries  i n i t i a l l y costing ^$2000  and capable of providing M00,000 km of service before replacement; a c c e l erations of 0 to 50 km/hr in M 0 seconds could be expected.  About the only available e l e c t r i c vehicle battery today is the lead acid battery.  Although its power density and cost are adequate,  its cycle  life  is limited ( 300 deep cycles) and its energy density is too low ( 40 Wh/kg). The achieved energy density is substantially less than the theoretical energy density of 175 Wh/kg for the active materials alone.  The Li T"iS  system has  a theoretical energy density of ^500 Wh/kg, and the actual energy density expected in a commercial battery has been estimated as 134 Wh/kg (Gaines et al,  1976).  Although this theoretical energy density is somewhat lower than  for many other high energy density battery systems currently under study,  the.  s i m p l i c i t y of the intercalation battery allows 1ight weight: cases to.be:used, with a considerable saving in mass over the cases needed for the competing systems.  For example, the sodium-sulphur battery has a theoretical energy  density of 793 Wh/kg, but projected total energy density of about 150 Wh/kg, due to the problems in confining the molten constituents at the battery's operating temperature of 350°C; the intercalation battery operates at room temperature.  The power density of an intercalation battery is acceptable  (MIO W/kg for Li TiS ), due to the high mobility of the guest atoms in the x 2 host l a t t i c e .  At present, the cost of these c e l l s would probably exceed  $50/kWh, but the wide variety of intercalation systems under investigation should eventually produce a battery at this cost. The greatest remaining problem with intercalation batteries  is cycle  life.  The intercalation portion of the battery (the cathode) appears to cycle very w e l l - - guest atoms can be added and removed many times without any appreciable degradation of the h o s t , l a t t i c e .  The lithium, however, presents a  5  problem.  The high energy density of a lithium intercalation battery is due  to the small atomic mass and high r e a c t i v i t y of the lithium; the  latter  property leads to a large energy difference between Li as an atom in the Li metal anode and as a guest in the intercalation cathode L i / L i TiS X  (the voltage of a  c e l l averages about 2 volts over the range 0 ^_ x <_ 1).  Because  2  of this high r e a c t i v i t y , Li metal trolyte solutions;  however,  is thermodynamically unstable  in some e l e c t r o l y t e s ,  formation of a passivating surface layer.  of PC, .:105 kcal/mole, (estimated ethylene carbonate)  the Li is protected by the  As an example, Li reacts with  propylene carbonate (PC), the most common solvent carbonate and propene.(Eichinger, 1976).  in elec-  used, to form lithium  From the free energy of formation  from the value for a related material,  this reaction should be favourable by a free energy of  3 eV per L i . However, the reaction quickly forms a protective  layer of  insoluble lithium carbonate over the lithium metal surface.  During cycling  of the battery, the passivating  some of the den-  layer can completely..enclose  d r i t i c growth which is produced as Li is plated; this enclosed material becomes inactive, in each c y c l e .  leading to a loss of several  percent of the lithium plated  Not only does this require a large excess of lithium in the  battery, but the dendritic lithium can lead to internal shorting of the c e l l . A review of some of the work being done to solve this problem is given in Besenhard and Eichinger (1976).  An alternative solution is to replace the  lithium metal anode with another.1ithiurn intercalation compound whose voltage versus lithium is almost zero; the s a c r i f i c e  in weight that this  implies  is  too high at the present time, but further research may lead to more suitable systems.  6  1.2  Contributions of This Thesis The aim of this thesis is to understand how the voltage of an interca- :  lation battery varies with the composition of the intercalation cathode, terms of the physical mechanisms involved in the intercalation  in  process.  Hence the thesis gives a detailed discussion of the mechanisms of intercal a t i o n , and i l l u s t r a t e s with experimental  some of the wide variety of the behaviour expected  results on several  systems.  The main body of the thesis^is divided into four parts. review of intercalation systems is given.  In Part A , a  Chapter 2 discusses the structure  of the two types of host l a t t i c e s studied, namely the layered transition metal dichalcogenides and the r u t i l e - r e l a t e d metal and reviews some of the existing structures. In Part B,  oxides,  l i t e r a t u r e on intercalation of these host  A review of other related systems is given in Chapter 3Chapter k  the thermodynamics of intercalation is discussed.  describes the l a t t i c e gas model and its application to intercalation systems, stressing  the simplest  (mean f i e l d )  solutions  to the l a t t i c e gas problem.  The two major types of interaction between intercalated atoms, and e l a s t i c , discussed  electronic  which determine the parameters of the l a t t i c e gas models, are  in Chapters 5 and 6 .  In Part C, the kinetics of intercalation batteries  are discussed.  Chapter 7 reviews the types of losses in electrochemicaliccel 1 s, out how they apply to intercalation c e l l s .  The effects of  pointing  interactions  between intercalated atoms on the diffusion of the atoms in the host discussed  in Chapter 8.  Chapter 9 discusses the effects of this  is  diffusion  on the voltage of intercalation c e l l s being discharged at f i n i t e currents, and Chapter 10 discusses the problems encountered in using porous  7  intercalation cathodes.  In Part D, the experimental results are discussed. procedure is outlined in Chapter 11.  Experimental  Chapter 12 gives experimental  results  for intercalation of lithium into T i S ^ , MoO^, and MoS^, and discusses  these  results in the light of the theory presented in Parts B and C. F i n a l l y , Chapter 13 summarizes the results of the thesis, some suggestions  for future work.  and offers  8  PART  A  REVIEW OF INTERCALATION SYSTEMS  \  9  CHAPTER 2 LAYERED COMPOUNDS AND RUTILES 2.1  Introduction  This chapter reviews some of the relevant properties of two types of host lattices,  the layered transition metal dichalcogenides  and the  ruti1e-related  metal dioxides, and the intercalation of these hosts.  The results  presented  are intended to i l l u s t r a t e those properties which w i l l  be important in deter-  mining how the voltage of an intercalation c e l l varies with the composition of  the intercalation compound.  follows;  Some of the points we w i l l  the effect they have on the c e l l voltage w i l l  subsequent  look for are as  be discussed  in  chapters.  (1) Type of s i t e occupied by the intercalated atoms  (2) Ordering of the intercalated atoms The intercalated atoms may be randomly distributed over a l l the sites a v a i l a b l e , or they may form an ordered array (a super 1attice) (3) Phase separation A host l a t t i c e of -. ;  intercalated to an average composition, x, may consist  two coexisting  experimental x" w i l l  regions of composition xi and x .  results,  2  In discussing  the  the phrase "intercalation compound of composition  be used only i f a,homogenous  (one phase) region in the host  latti.ce can be prepared.  {k) Mobility of the intercalated atoms Most of the results quoted are in terms of the tracer diffusion .coefficient,  D^, which is approximately equal to MkT, where M is the  . ".mob i 1 i ty :of the intercalated atom, T is the absolute temperature, and  k is Boltzmann's constant.  To provide a feeling for the scale of D^,  3 2 we note that MkT is 2 x 10 cm /sec for electrons in the semiconductor 2 -7 2 InSb, 1.1 cm /sec for electrons in copper, and 3 x 10 cm /sec for lithium ions in propylene carbonate, a l l at room temperature.  (5) Changes in the electronic properties of the host due to intercalation (6) Structural changes  2.2  in the host  Layered Transition Metal Dicha1cogenides - Structure and Properties  The layered t r a n s i t i o n metal dicha1cogenides have the chemical symbol MX^, where M is a t r a n s i t i o n metal from group IVB, VB, or VIB of the periodic table (Table I), and X is one of the chalcogens ium) from group VIIA.  (sulfur, selenium, or t e l l u r -  The crystal structure consists of sandwiches of  close packed cha 1 cogen-metal:-cha 1 cogen planes stacked along the crystal lographic c-axis,  as shown in Fig 2a.  Because of the weak van der Waals bonds  between adjacent chalcogen planes, the layers are easily separated, and a wide variety of atoms or even large organic molecules can be intercalated into the van der Waals gap.  TABLE I Transition Metals Which Form Layered Dichalcogenides  Group IVB  VB  VIB  3d  Ti  V  Cr  hd  Zr  Nb  Mo  5d  Hf  Ta  W  Shel.l  >^  11  (a) General form  van der Waals gap  (b) Coordination units for  MX2 layer structures  AbA trigonal prisrh  Fig.  2 - Structure of layered t r a n s i t i o n  metal  (a) G e n e r a l f o r m o f X-M-X s a n d w i c h e s . ( c ) T h e t h r e e m o s t common p o l y t y p e s .  AbC octahedron  dicha1cogenides, (b) C o o r d i n a t i o n  MX  2  units,  Two  types o f c o o r d i n a t i o n o f the metal atom by a d j a c e n t  o b s e r v e d , namely o c t a h e d r a l  and  trigonal  p r i s m a t i c , ( F i g . 2b).  s t r u c t u r e s a r e based on c l o s e packed atomic p l a n e s , d e s c r i b e them u s i n g  the usual ABC  chalcogen p o s i t i o n s w i t h c a p i t a l w i t h small described  letters  ( a b c ) , and  letters  the s i t e s f o r the  o f one  (a3y)-  The  denote  the  i n t e r c a l a t e d atoms (to be  '  n  this notation,  by AbC,  and  octahedral  trigonal  v a r i o u s s t r u c t u r e s c o n s i s t of sequences  (or sometimes both) o f these two  F i g . 3 - ABC  The  The  We  (ABC), the metal atom p o s i t i o n s  c o o r d i n a t i o n of the metal atoms i s r e p r e s e n t e d p r i s m a t i c c o o r d i n a t i o n by AbA.  Because the  i t i s convenient to  n o t a t i o n shown i n F i g . 3-  s h o r t l y ) w i t h Greek l e t t e r s  chalcogens i s  notation f o r close-packed  b a s i c sandwiches.  spheres.  t h r e e most common s t r u c t u r e s , or p o l y t y p e s , are shown i n F i g . k.  1T p o l y t y p e  c o n s i s t s of o c t a h e d r a l l y coordinated  found in group IVB and  VB compounds.  unit cell  i s one  is  l a y e r high  and  o f 2H s t r u c t u r e s  (2 l a y e r u n i t c e l l s , hexagonal symmetry "H").'a re shown i n  Both have t r i g o n a l  they d i f f e r  t r i g o n a l symmetry ("T").  Two  (hence  the " 1 " i n "IT")  F i g . k.  the s t r u c t u r e has  The  sandwiches, and  types  p r i s m a t i c c o o r d i n a t i o n o f the metal atoms, but  i n the s t a c k i n g sequence of the sandwiches.  In the 2H-NbS  2  Sites available for intercalated atoms in layered compounds, (a) before and (b) after slipping of adjacent chalcogen planes.  structure, observed for group VB metal atoms, the metal atoms l i e one above the other along the c-axis; atoms, they do not.  in the MoS structure, seen for group VIB metal 2  Other;polytypes are discussed  in Wilson and Yoffe  (1969).  Two types of sites are available for intercalated atoms, in the van der Waals gap of these materia 1 s , ( F i g . 4 a ) . The octahedral sites  (ABC) are  coordinated by 6 chalcogen atoms which l i e on the corners of a s l i g h t l y elongated octahedron; these form a triangular l a t t i c e of l a t t i c e constant a, where a is the distance between adjacent chalcogen atoms in the close packed atomic plane.  The two types of tetrahedral sites  (AaB, ABB) s i t  slightly  below and above the plane of the octahedral s i t e s , and are coordinated by h chalcogen atoms. of  Each type of tetrahedral s i t e forms a triangular l a t t i c e  l a t t i c e constant a; taken together,  a l l the tetrahedral sites form a  honeycomb l a t t i c e with nearest neighbour separation a / / J . There are two tetrahedral  sites and one octahedral s i t e per transition metal atom M in MX . 2  Note that these sites are not unique to layered compounds, but occur between any pair of close packed atomic planes; hence planes of these sites occur in both hexagonal close packed and face centered cubic metals. of  The composition  an intercalation compound of guest atom A in MX is usually given by the 2  quantity x, as in A MX ; f i l l i n g a l l the sites would give x = 3 . x  2  In some  intercalation compounds, the sandwiches s l i p , bringing adjacent chalcogen atoms in line along c; this prismatic sites  leads to a honeycomb l a t t i c e of two trigonal  (ABA, AyA) per transition metal atom (Fig. 4 b ) .  Schematic band structures of the host transition metal dicha1cogenides are shown in F i g . 5 .  Calculations show (e.g.  structures can be roughly c l a s s i f i e d  Mattheiss, 1 9 7 3 ) that the band  into two groups, according to the  coordination of the t r a n s i t i o n metal by the chalcogens.  In both cases,  the  upper and lower bands shown in F i g . 5 are derived from bonding and antibonding  "V  M - atom Xsp-band  -j- — x  M - atom d - band  X-  atom  p-  band  Density  of  States  I T  2H  Schematic band structures of the transition metal dicha1cogenides. The number of electron states per metal atom M in each band is indicated. (a) IT polytype, such as TiS (b) 2H polytype, such as MoS„. z  combinations of the s and p o r b i t a l s of the metal and chalcogen atoms, with the lower bands primarily from the chalcogens, while the two central bands are derived from the d o r b i t a l s of the transition metal.  The primary  difference between the two cases is in the s p l i t t i n g of the d bands; for octahedral coordination, the lower d band contains 6 states and the upper d band k states per transition metal atom, whereas for trigonal prismatic coordination the lower band has 2 states and the upper band 8 states.  For  group IVB metal atoms, which are a l l octahedrally coordinated, the Fermi level  lies at the bottom of the d band, so the group IVB compounds are  semimetals or semiconductors.  The width of the gap between the s-p bands  and the d bands decreases with increasing metal atom mass or decreasing chalcogen atom mass.  ^iS^ '  versy over whether it  is a semiconductor or a semimetal.  s  a  borderline case, and there is s t i l l  compounds, one state in the lower d band is f u l l , are m e t a l l i c .  contro-  In group VB  so both 1T and 2H polytypes  In group VIB compounds, only the trigonal prismatic coordin-  ation is seen, so the Fermi level  lies between the two d bands, leading to  semiconducting behaviour.  2.3  Intercalation of Transition Metal  Dicha1cogenides  A detailed review of this subject has been given by Whittingham ( 1 9 7 8 a ) . The most extensively studied system has been Li T i S , • especially over the 2  range 0 <_ x <_ 1 , where single phase behaviour is seen.  Neutron studies for  this composition range indicate that the lithium l i e s predominantly in octahedral sites  (Dahn et al 1 9 8 0 ) .  The c axis increases  by 10% from  x = 0 to x = 1, with most of the increase at small x; the a axis  increases  approximately l i n e a r l y with x by about \% over this range (Bichon et aU 1 9 7 3 , Chianelli et al 1 9 7 8 ) .  Knight shift measurements  of conduction electrons near the lithium is small  indicate that the density (Silbernagel and  Whittingham 1 9 7 6 ) .  The hopping time of the lithium near x = 1 .  suggests a tracer diffusion constant the hopping is activated 0.3  eV (Sibernagel  T n  ~ 0.23  ys  of order a 2 / x ^ - 5 x 1 0- 9 cm2 /sec;  (D^ <* e ^ ^ T j with an activation energy E of about  1975).  The voltage of a L i / L i ^ T i S ^ battery varies from 2.k to 1 . 8 volts as x increases from 0 to 1, as shown in F i g . 6 a .  Detailed examination of  curve shows fine structure, as indicated in the inverse -Ax/AV versus x in F i g . 6 b .  the  derivative  Over thi.s range, the intercalation  is highly  reversible, with charge and discharge giving the same voltage ;to within 10 mV at any value of x at low currents.  It has only been recently  learned that greater amounts of lithium than  x = 1 can be incorporated into the TiS^ host; f i r s t values of x = 2 (Murphy and Carides 1 9 7 9 ) , imental  results  then x = 3 (Dahn and Haering 1 9 7 9 ) , were reported.  are showh^in Section 1 2 . 3 .  Exper-  Two phase behaviour is seen  both between x = 1 and 2 , and between x = 2 and 3 -  The t r a n s i t i o n from  x = 1. to x = 2 i s quite reversible, while the t r a n s i t i o n from x = 2 to x = 3 completely changes the charge/discharge c h a r a c t e r i s t i c s of subsequent cycles over the range 0 <_ x <^ 3 -  In contrast to Li T i S „ , x 2 the range 0 <_ x <_ 1 .  Na TiS„ shows several x 2  In one of these structures,  structural changes in the sulfur atoms shift  to produce trigonal prismatic sites for the sodium atom; in this phase, c lattice  parameter decreases with x (Rouxel et al  1971).  the  Li VS shows two 9  narrow monoclinic phases, extending from 0 . 2 5 < x < O . 3 8 and 0 . 5 < x < 0 . 6 . These phases disappear at higher temperatures substitutional^ 1978a).  (T < 85°C) or i f  replaces vanadium in the host structure  iron  (Murphy et al  VS^ is unique in that the host structure has only been prepared to  date by de-i nterca 1 at i ng the compound L i ^ S ^ grown at high temperatures.  |T  •  0  '  0.2  •  i  0.4  X IN  Fig.  I  I  1  0.6  Li TiS x  1  0.8  1  1  1.0  g  6 - (a) Voltage V and (b) inverse derivative -Ax/AV versus x for 'x ' 2' experimental data from Thompson (1978). L  T  S  P  o  i  n  t  s  a  r  e  The s o l i d curve is a mean f i e l d f i t to the data, discussed in Section k.k, with U = 2.5 kT, kT = 25.7 meV, E = -2.3 eV, and  Further intercalation of Li^VS^ produces two phase behaviour to x = 2 (Murphy et al 1979)-  In VSe2, two coexisting phases are seen between x = 0  and x = 1, and between x = 1 and x = 2 (Whittingham 1 9 7 8 ) .  2.k  Metal Dioxides With Rutile-Related Structure - Structure and Properties  Metals which form metal dioxides M0 having a r u t i 1 e - r e l a t e d structure 2  are shown in Table II. of and  As in many metal oxides,  the basic building block  the r u t i l e oxides is an M0 octahedron, shown in two views in F i g . 7a 5  7b.  The octahedra share edges ( i . e .  two 0 atoms) and form chains along  the crystallographic c axis; points of the octahedra in adjacent chains are shared to connect the chains as in F i g . 7 c and 7 d .  This leads to the  tetragonal structure in F i g . Ie, viewed along the c axis. graphic a axis  The c r y s t a l l o -  is the distance between the two nearest metal atoms in the  same plane normal to c.  The octahedra in F i g . 1 have been drawn as perfect  TABLE I I METALS WHICH FORM RUT ILE-RELATED OXIDES  ^ \  Group IVB  Shei 3d 4d 5d  VB  VIB  VI IB  VIII  IVA  1 \ n  Cr  Mn  ! Nb< > Mo  Tc  Ru  Rh  Re  Os  1r  2  Ta  '  W  Sn Pt  Pb  Dotted line encloses atoms which form distorted r u t i l e structures at room temperature. (1)  V converts to pure r u t i l e structure for T > T  (2)  Nb converts to pure r u t i l e structure for T > T = 1070K  c  = 3^0K  20  Fig.  7 - Rutile structure. (a) Top and (b) side views of MOg octahedron, (c) Top and (d) side views of chains of octahedra joined by sharing points (0 atoms). Note that the width of the lines in (c) distinguishes the positions of the octahedra along the chain d i r e c t i o n . (e) Top view of r u t i l e structure.  octahedra; in fact,  there is some d i s t o r t i o n .  Two measures of this  dis-  tortion are the c/a r a t i o , and the fractional distance u indicated in Fig.  7 e ; for ideal octahedra, these are c/a = (1 + 1 / / 2 )  1  - O . 5 8 6 , and  u = 1 / ( 2 +/I") - 0 . 2 9 2 9 , whereas in actual materials, c/a varies from 0.64 to 0 . 6 6 , while u varies from 0 . 3 0 0 to 0 . 3 0 7 -  In spite of t h i s ,  the materials  are referred to as undistorted to distinguish them from several of the r u t i l e s where the metal atoms dimerize, leading to a further distortion of the octahedra.  These distorted r u t i l e s have a monoclinic unit c e l l , which  unfortunately is indexed with the monoclinic a a x i s , a^, p a r a l l e l to the chains (a^ of  //  c_, where c_ is the c axis vector in the tetragonal  the undistorted r u t i l e s ) .  In what follows, we shall  axes as those of the tetragonal  unit c e l l  refer to the a and c  unit c e l l unless otherwise  indicated.  Possible s i t e s available for intercalated atoms are indicated in F i g . 8 . Along the tunnels between the M0G octahedra, there are two types of s i t e s : tetrahedral s i t e s , coordinated by 4 0 atoms, and octahedral s i t e s , coordinated by 6 0 atoms.  It turns out that of the 6 oxygen atoms coordinating  the octahedral s i t e , 2 are closer to the center of the s i t e than the other 4, and in fact the oxygen-site distance for these two atoms is shorter than the oxygen-site distance for any of the 4 0 atoms coordinating the t e t r a hedral s i t e . sites w i l l  It therefore seems l i k e l y that any atoms in the tetrahedral  have a lower energy than in the octahedral s i t e s ; the octahedral  sites may even represent saddle points as it migrates along the tunnel. out on T i 0  2  in the energy of an intercalated atom  This is the case in calculations  (Ajayi et al 1 9 7 6 ; Kingsbury et al  1968).  In addition to these s i t e s along the tunnels, of  there are two other types  tetrahedrally coordinated s i t e s , also shown in F i g . 8 .  These two sites  would be involved in diffusion of atoms normal to the tunnels. for  Ti0  2  carried  Calculations  suggest that both of these sites have a considerably higher s i t e  22  Fig. 8 -  (a) Top and (b) side views of chains of octahedra in r u t i l e structure. 0 atoms coordinating various types of sites are indicated, o: octahedral s i t e along tunnel. • : tetrahedral s i t e along tunnel. A and V:two types of tetrahedral sites off the tunnel axis.  energy than the tunnel sites  (Kingsbury et al  this is consistent with  1968);  the poor diffusion of lithium normal to the tunnels  In the distorted r u t i l e s ,  s i t e s , and makes some of  Thus, the octahedral sites are separated  types 1, 2 , 3 , in the sequence: 1 2 3 2 1 2 3 2 1 ! . .  sequence 1 1 2 2 1 1 2 2 . . .  .  Similarly,  into two types in the  Energies of these sites have not been calculated,  no estimation of s i t e energies in the metallic compounds, for either  distorted or undistorted structures,  The  is a v a i l a b l e .  schematic band structure of the r u t i l e hosts shown in F i g . 9 a is  very similar to that for the 1T t r a n s i t i o n metal dicha1cogenides where the metal atoms are also octahedrally coordinated. into 6 lower and 4 upper states per metal atom. with the Fermi V0  2  the  into three  along the tunnel.  the tetrahedral sites on the tunnel axis are separated  and  1964a).  the dimerization of the metal atoms causes  further d i s t o r t i o n of the (already distorted) sites inequivalent.  (Johnson  level  is m e t a l l i c .  (Fig. 5 a ) ,  The d v b a n d s , s p l i t .  Thus, T i 0  2  is am. insulator,  lying .i n. the .gap be] ow the d bands, and r u t i l e structure  In P b 0 a n d Sn0 , the d bands are f u l l , 2  2  materials are semiconducting; however,  in this case,  it  is  and these two l i k e l y that  the  d bands l i e below the top of the lower sp bands, contrary to the figure. The  d i s t o r t i o n in the distorted r u t i l e s causes a gap to open between one  state in the lower d band and the rest of the d states (Fig. 9 b ) . When this occurs, the group VB oxides V0 and Nb0 become semiconducting.  In Mo0 ,  Fermi  The lower  2  level  l i e s one state above this gap, so Mo0  d band in ferromagnetic Cr0 Fig..  2  2  is m e t a l l i c .  is believed to be spin s p l i t as shown in ' ' , ;  2  9 c ; two states in the lowest d band are occupied, so Cr0  metallic behaviour.  2  2  shows  the  M- atom sp-band  M-atom d-bands M-atom sp-band  O-atom  M-atom  p-band  d-bands  A — -~ 1 0) ~ "  ^  M-atom sp-band  O-atom p-band  \  M-atom d - band  3 / M _ j x  s.  O-atom p -  band  >  0  Density of States  Schematic band s t r u c t u r e o f r u t i l e - r e l a t e d metal o x i d e s . The number o f e l e c t r o n s t a t e s p e r metal atom M i s i n d i c a t e d , (a) U n d i s t o r t e d r u t i l e s t r u c t u r e , s u c h a s T i 0 . (b) D i s t o r t e d r u t i l e s t r u c t u r e , a s i n Mo0 . ( c ) F e r r o m a g n e t i c Cr02. (After G o o d e n o u g h 1971). 2  2  2.5  Intercalation of Rutiles  Considerably less information is available on intercalated r u t i l e s than for  the t r a n s i t i o n metal dichalcogenides.  into Ti02 by placing the T i 0  2  Lithium has been  in contact with metallic Li at  between 200°C;.a'nd 3 0 0 ° C , but only small concentrations of 7  lithium  were obtained  intercalated temperatures  intercalated  (x < 8 x 10 ^) . i n the bulk of the c r y s t a l s ;  however,  s l i g h t l y higher concentrations were observed near dis1ocations.(Johnson 1964b).  The discoloration of the transparent TiU2 caused by the Li was used  to measure the diffusion constant of the Li atoms; the value obtained at -7  room temperature was 6 x 10 (Johnson 1 9 6 4 a ) .  2  cm /sec,  with an activation energy of 0 . 3 3 eV  Intercalation of Li into several other r u t i l e s up to  x = 1 has been reported by Murphy et al  (1978b).  CHAPTER 3 FURTHER PROPERTIES OF INTERCALATION AND RELATED PHENOMENA  3.-1  Introduction In this chapter, we continue our review of intercalation and related  phenomena.  F i r s t , the methods of intercalation are discussed.  Then-, a  review is given of two more intercalation systems, namely graphite and the metal-hydrogen compounds. some i n t e r s t i t i a l  We then discuss some of the properties of  layered compounds, which gives further insight  into  intercalated layered compounds, and some properties of some tungsten a class of materials which includes  bronzes,  intercalated r u t i l e s as a special  case.  The chapter ends with a discussion of superionic conductors, a group of solids  in which one of the constituents of the s o l i d shows a high mobility,  which is also a property of intercalation compounds. 3.2  Methods of  Intercalation  The various methods of intercalation can in general be c l a s s i f i e d  into  one of the following three groups: intercalation from the vapour, intercalation from a liquid solvent,  and intercalation in an electrochemical  cell.  To intercalate from the vapour, the host material is exposed to the vapour phase of the substance to be intercalated  (the  intercalate).  Intercalation of hydrogen into metals is generally done in this way. a wide variety of large organic molecules,  Also,  such as pyridine, have been ',.  intercalated into layered compounds using this technique, producing an expansion of the layer spacing of up to 10 times the o r i g i n a l separation (Gamble et al 1 9 7 1 ) .  If the weight of the host l a t t i c e  is monitored as a  function of the vapour pressure of the^interca1 ate, of the intercalate calculated.  the chemical  potential  in the host l a t t i c e as a function of composition can be  In some cases, such as the intercalation of hydrazine into  2H NbSe^, the rate limiting step in the intercalation process appears to be the absorption of the vapour molecules on the surface of the crystal (Beal and Acrivos 1 9 7 8 ) .  In intercalation from a l i q u i d solvent,  the host structure is brought  into contact with a solution containing the i nterca 1 ate ...;For example, a l k a l i metals dissolved dissolved  in l i q u i d ammonia, and a l k a l i metal  in water, have been intercalated  compounds.  in this way.into  hydroxides layered  Use of such small, highly polar solvents can lead to c o - i n t e r -  calation of the solvent molecules;  the solvent molecules  can often be  removed by heating the sample under vacuum (Whittingham 197*0 .  In some  cases, the intercalated atom is produced at the surface of the host in a chemical reaction; an example is n-buty11ithium (C^H^Li), which reacts with the host to form intercalated  lithium and octane.  The reaction  of some host with n-buty11ithium produces an intercalation compound with a composition corresponding to a voltage of about one volt against (Murphy and Carides 1 9 7 9 ) -  Moreover, since the solvent  (hexane)  lithium  is  non-polar, no solvent co-interca1 at ion occurs.  Intercalation  is done in ah electrochemical  cell  by making the host  one of the electrodes in the c e l l , as in F i g . 1, and passing current through the external electrode  circuit.  The c e l l may involve a simple mass transfer from one  to the other, as in F i g . 1 , or a chemical reaction.  of the l a t t e r case is an e l e c t r o l y s i s  An example  c e l l , where passing a current decom-  poses water, giving hydrogen at one electrode and oxygen at the other. a host is used at the hydrogen side,  hydrogen may intercalate  If  rather than  bubbling off as hydrogen gas. problem in electrochemical  3-3  Co-intercalat ion of the solvent is also a  cells.  Intercalation of Graphite This is the oldest known intercalation system, and detailed reviews are  available, such as those by Ebert (1976), Fischer and Thompson (1978), and Gamble and Geballe (1976).  Graphite is a layered crystal form of carbon,  where the carbon atoms in each layer form a honeycomb l a t t i c e .  Since a  honeycomb l a t t i c e can be obtained by placing atoms in two of the three close packed sphere positions ABC in F i g . 3, the stacking sequence for the can be described by giving the u n f i l l e d positions the common hexagonal  form of graphite, denoted  in each layer.  layers  Thus, in  A B A B A B . . . , . h a l f the carbons  in one layer are above carbons in the layer below, and half are above empty s i tes. In many graphite intercalation compounds, a phenomenon known as staging is observed.  A stage n compound is one where only every nth layer  is  intercalated; stage 5 intercalation compounds of the a l k a l i metals have been reported (Rudorff and Schultz 1954). calated atoms generally shift  The carbon planes adjacent to inter-  to l i e one above the other so the sites  occupied are trigonal prismatic.rather than tetrahedral.  Thus, for example,  a stage k compound would have a structure -ABAB-BCBC-CACA-ABAB-, where the dashes indicate the layers occupied by intercalated atoms.  Disordering of  intercalated atoms in the layers has been observed for the a l k a l i  metals;  stage 2 compounds disorder in the range -150°C to -50°C (Parry et al 1969), whereas stage 1 Rb^gC remains ordered u n t i l klh°Z. (El lenson et al  1977).  A study of intercalation of bromine into a graphite c y l i n d e r , with the c y l i n d r i c a l axis normal to the carbon layers, showed that the material near  the end of the cylinder intercalated before that in the center.  Moreover,  i f the ends are capped so the bromine gas cannot contact them, no i n t e r calation occurs (Hooley  1977).  Hence, in this case at least, adsorption of  the intercalate on the surface of the graphite is essential  for i n t e r -  calation to occur.  3-4  Hydrogen in Metals  A detailed review of hydrogen in metals has recently been published (Alefeld and Volkl  1978).  These systems satisfy  the d e f i n i t i o n of  inter-  calation compounds given in Chapter 1, but they are generally not referred to as such in the l i t e r a t u r e , since they were investigated any other intercalation system.  A wide variety of metals  independently of intercalate  hydrogen; we w i l l describe a couple in some detail to i l l u s t r a t e the observed behaviour. A schematic phase diagram of H^Nb is shown in F i g . 10 for temperatures above 250K.  Niobium is a body-centered cubic metal; hydrogen  into the tetrahedral sites,between Nb atoms.  intercalates  These tetrahedral s i t e s are  distorted along the x, y, or z d i r e c t i o n s , and there are two of each of the three types of sites per Nb atom, for a total of s i x .  Phases a and a 1 have  the same structure; the a l a t t i c e parameter of the Nb host increases with x in both phases such that Aa/a - 0.14 x.  linearly  In the 3 phase, the hydrogens  order, occupying one of the 6 tetrahedral s i t e s , and the Nb host expands slightly  in one d i r e c t i o n , forming an orthorhombic (almost  l a t t i c e with c/a - 1.001.  tetragonal)  In the 6 phase, which.occurs near x = 2, "the Nb  atoms form a face-centered cubic l a t t i c e , with the H atoms occupying the tetrahedral s i t e s between the close packed Nb (111) f l u o r i t e structure).  planes  (the  so-called  500 T(K)  a •+<5 B+6 400  300  0 x  Fig.  10 - S c h e m a t i c p h a s e 1979)i  The  first  See  d i a g r a m o f H Nb  text  o r d e r t r a n s i t i o n b e t w e e n a and a  the e l a s t i c i n t e r a c t i o n ,  boundary lattice  u s i n g an  ation  of the t r a n s i t i o n  hydrogen  i s found  is inferred  temperature T  proportional  t o (T - T )  1  atoms  e t a l 1977).  , the magnitude  from t h e s e measurements a l s o  the observed  the value of T  c  d e p e n d s on  relaxation  the sample  by  (the s t r a i n amplitude  o b t a i n e d d e p e n d s on  the d i f f u s i o n  of  stress).  of the s t r a i n caused  a C u r i e - W e i s s law  In a d d i t i o n ,  phase  1  from the observed  in response to a  shape, as e x p e c t e d from t h e t h e o r y o f t h e e l a s t i c  (Tretkowski  discussion  o f t h e a-a  i n s t u d i e s of the a n e l a s t i c  obeys  ) , and  (for a  to  Further evidence f o r t h i s explan-  (the d i f f u s i o n o f the hydrogen  m i g r a t i o n o f the hydrogen  sample  atoms  Calculations  whose m a g n i t u d e  ( H o r n e r and Wagner 1 9 7 4 ) .  Above the c r i t i c a l the  been a t t r i b u t e d  e x p a n s i o n w i t h x a r e i n r e a s o n a b l e agreement w i t h  diagram  the  has  1  between the hydrogen  see C h a p t e r 6).  interaction  phase  Wenzl  for details.  attractive e l a s t i c interactions of  f o r T > 2 5 0 K ( S c h o b e r and  interaction  constant  shape.  the  inferred  The tracer diffusion constant  in the a phase decreases approximately  l i n e a r l y with x, and is reduced by a factor of 3 as x varies from 0 to Over this composition range, the activation energy rises from 0.13 0.18 eV.  This variation in  the discussion -6  near x = 0,  = 3 x 10 -8  lower, 5 x 10  to  is larger than expected from a simple blocking  of s i t e s , which would give a reduction of D of range of x (see  0.14.  in Section 8.2).  (1 - .4)  _i  =1.8  over  this  At room temperature, and  2 cm /sec;  in the ordered 3 phase,  is considerably  2 cm /sec,  but with a s l i g h t l y  lower activation energy,  0.11 eV. Intercalation of hydrogen into palladium shows a similar f i r s t order transition between two phases, a and a ' , presumably also due to e l a s t i c interactions.  In addition, considerable study has been done on H Pd to x  learn how the addition of H atoms modifies the Pd band structure. the Fermi energy l i e s near the top of the metal d bands.  In Pd,  H i s t o r i c a l l y , the  effects of added H atoms were interpreted in one of two models, both based on a r i g i d band picture of the Pd host: the anion model, where the H atom removes an electron from the Pd band structure, forming H ; and the proton model, where the H atom donates its electron to the Pd band structure. structure calculations  indicate that neither picture is correct  Band  (Switendick  1972). The hydrogen modifies the Pd band structure, pulling states below the Fermi l e v e l ; however, Fermi level  rises with respect to the band structure.  the Fermi level 1971).  less than 1 state per H atom is pulled down, so the The new states below  have been observed in photoemission studies (Eastman et al  Moreover, the sudden drop in the density of states expected when the  Fermi level  rises above the metal d band is seen in both magnetic  suscep-  t i b i l i t y and specific heat measurements. Palladium has a face-centered cubic structure, and hydrogen atoms occupy  octahedral the  host  sites  aside  rare  earth  face  centered  from an expansion  elements  regardless enters  up t o a maximum v a l u e  cubic  octahedral  earth  elements;  occurs  produces  structure  with  host  sites,  i n the heavier  t o a hexagonal  close  a first  both  order  elements,  phase sites  change  t o x = 3, another  transition filled  structure  with  order  to a  ( x = 2), then  in the lighter  first  in  hydrogen i n  B e y o n d x = 2, h y d r o g e n  t h e host  packed  In c o n t r a s t ,  tetrahedral  structure.  filling  no s t r u c t u r a l  of the lattice.  generally  of the initial  x = 1, w i t h  rare  transition  a l l the sites  occupied  (x = 3).  Interstitial  3.5  A wide  variety  are  structurally  can  be p r e p a r e d  In  these  Compounds  of ternary  similar  materials,  composition  x cannot  have  these  been  recently  In m a n y o f t h e s e as  2H-NbS  sites  2 >  be v a r i e d  once  where  t h e i n t e r s t i t i a l A atoms  sites;  t h e 2H-MoS this  2H-NbS  Se,  t h e A atoms  octahedrally  structure,  results  the  2  2  occupy  In A C r X  2  form ordered  CrX  2  between X - M - X since the grown,  These  (1976).  h a s t h e same  hand,  structure  in octahedral  t h e NbS„  atoms  2 reside  Cu-Nb d i s t a n c e compounds,  sandwiches  we w i l l  materials  arrangements  on t h e o t h e r  MX^,  temperatures.  2  one o f t h e two types  coordinated  sites  here.  lattice  a n d t h e Cu atoms  in a shorter  structure.  2  at high  h a s been  compounds  the MX  In Cu N b S „ , x  adopt  t h e compound  h^MX^, w h i c h  dichalcogenides  however,  by V a n d e n b e r g - V o o r h o e v e  systems,  a t x = i o r x = 1/3.  interstitial  Dichalcogenides  symbol  elements  compounds;  intercalation  reviewed  chemical  t r a n s i t i o n metal  occupy  as i n i n t e r c a l a t i o n  materials  with  the constituent  t h e A atoms  just  call  systems  to layered  by c o m b i n i n g  sandwiches,  not  o f Layered T r a n s i t i o n Metal  with  in the tetrahedral  than would  be p o s s i b l e i n  A = Ag o r C u , and X = S o r  of tetrahedral  sites  a t room t e m p e r a t u r e  between  (the structure  is AcBy CbA3 BaCa A c B . . . ) .  At higher temperatures,  the A atoms disorder  and randomly occupy both types of tetrahedral* s i t e s ; the order-disorder transition temperatures observed in neutron scattering experiments 6 7 5 K for CuCrS , 6 7 O K for AgCrS , and 4 7 5 K for AgCrSe 2  2  are  (Engelsman et al  2  1973).  Optical studies oh A MS , with A = Ni , Cu, or Fe, and M = Zr or Hf, X  z.  indicate that these materials are semiconducting, but with a band edge which s h i f t s to lower energies as x increases.  This is  interpreted as  evidence that the A atoms produce states in the band gap of the MS^ host; electrons excited from these states into the conduction band would then account for the s h i f t  in the band edge (Yacobi et al  1979).  Nuclear  magnetic resonance studies on Sn TaS„ for x = 1 / 3 and x = 1 show considerably higher concentrations of electrons x = 1 / 3 ; it at  is proposed that a Sn conduction band exists at x = 1 but not  x = 1/3,(Gossard et  3.6  near the Sn atoms at x = 1 than at  al  1974).  Oxide Bronzes  Oxide bronzes are defined as solids with the chemical formula A MO x nj where MO is a t r a n s i t i o n metal oxide, and A is any element. n  This class of  materials thus includes intercalation of atoms into metal oxides as a special case.  For a review, see Dickens and Wiseman (1975)•  An interesting application of these materials is displays  (Faughnan et al 1 9 7 5 a ) .  in electrochemical  Intercalation of H or Li into WO^ or MoO^  causes the o r i g i n a l l y transparent host to become coloured; i t example of electrochemical regard, Faughnan et al  intercalation of insulators.  (1975b)  is also an  In this  latter  have shown that on de-interca1 ation the flow  34  of  current is space charge limited in films of H^WO^; that i s , when electrons  are removed from one face of the f i l m , and protons from the other,  the  current flow is controlled by the e l e c t r i c f i e l d associated with this charge separation. at x = 0.32  3.7  In addition, it has been observed that H W0_ becomes metallic x 3 (Crandall and Faughnan 1977a).  Superionic Conductors Superionic conductors are ionic solids  nevertheless  show a high e l e c t r i c a l  mobility of one of the constituent  (electronically  insulating) which  (ionic) conductivity, due to the high ions.  The composition of the mobile ion  is usually fixed, so that these are not intercalation compounds.  However,  they are similar to intercalation compounds because of the rapid motion of one of the constituents  of the structure, and we can regard the mobile atom  as a guest within the structure provided by the immobile atoms (the A recent review of these materials is given by Boyce and Huberman  host). (1979);  as w e l l , several collections of a r t i c l e s have recently been published (e.g.  Hagenmuller and van Gool  1979).  Superionic conductors are generally c l a s s i f i e d according to the way that the mobility of the mobile atom increases as T is increased.  In Type I  materials, there is a f i r s t order t r a n s i t i o n , in which the mobility  increases  dramatically, and in which the l a t t i c e of immobile atoms generally undergoes a structural change. Thus, for example, Agl at low temperatures consists of Ag atoms in half the tetrahedral s i t e s in a hexagonal close packed iodine l a t t i c e ; at T = 420K, the iodine l a t t i c e becomes body centered cubic, with •Ag atoms randomly occupying tet rahedral ' s i tes-, and the mobility of the Ag increases by four orders of magnitude. in the immobile ion l a t t i c e ,  Type II materials show no change  but the mobile ions disorder over a small  temperature range, producing a peak in the s p e c i f i c heat and a f a i r l y rapid rise in mobility; there is controversy over whether this disordering is a second order phase t r a n s i t i o n or not.  In Type III materials, the mobility  of the mobile ions increases very gradually with temperature; these material are generally non-stoichiometric, and some of them are intercalation compounds .  PART  B  THERMODYNAMICS OF INTERCALATION BATTERIES  CHAPTER  4  LATTICE GAS THEORY OF INTERCALATION k. 1  Introduction  The variation of the open c i r c u i t voltage, V, of an .intercalation c e l l with the'state of charge of the c e l l about the intercalation process. interpret this information.  can provide a great deal of  information  In this chapter, we discuss how to  We w i l l f i r s t outline the thermodynamics of an  intercalation c e l l , and draw analogies with more familiar thermodynamic systems.  We then;.discuss the application of the l a t t i c e gas model to r  intercalation systems, and calculate the variation of V with the composition x of the l a t t i c e gas for some s p e c i f i c (mean f i e l d )  cases.  We stress the  simplest  solutions of the l a t t i c e gas problem, and examine  their.:  strengths and weaknesses in comparison with more exact (and more d i f f i c u l t ) results.  A typical  intercalation c e l l  is shown in F i g . 11. The anode is a  metal composed of atoms A (usually lattice  lithium), and the cathode  that can intercalate A atoms.  The e l e c t r o l y t e  is some host  is some material  (usually a liquid) containing mobile A ions of charge ze (where e is magnitude of the electronic charge and z is assumed to be p o s i t i v e ) ,  the which  allows A ions but not electrons to pass from the anode to the cathode. the c e l l  is discharged, A ions travel through the e l e c t r o l y t e  through the external anode to the  c i r c u i t , resulting  As  and electrons  in a transfer of A atoms from the  cathode.  To relate V to the thermodynamics of the components of the c e l l , consider the free energy change of the c e l l , AF, when An A atoms are transferred from the anode to the cathode.  This transfer causes a charge -zeAn  11 - Schematic intercalation c e l l .  Fig.  of electrons  to flow through the external c i r c u i t and do work zeAnV, so that  AF, which is minus the work done,  is  AF = -zeAnV  (D  But AF is also given by  A  F  " ^ " ^anode (  where y and U  a n o c  j  e  anode respectively.  (2)  )An  are the chemical potentials Hence the c e l l  of A atoms in the cathode and  voltage V is related to the chemical  potential y by  V = - - (u - y . ) ze anode • As the c e l l discharges, is constant),  (  u a n o c  |  e  so any variation of V is due to the variation of y caused by  this composition change. x defined by  only the composition of the cathode changes  (3)  The cathode composition is measured by the quantity  where n is the number of intercalated A atoms, and N is some reference number, usually taken to be the number of one species of host atoms in the cathode  (as  in Li^TiS^).  Hence the aim of this chapter is to understand  how u varies with x, and how this variation reflects  the underlying physical  processes occuring d u r i n g . i n t e r c a l a t i o n .  4.2 Thermodynamics  It is useful  to compare the thermodynamic relations for an intercalation  compound with analogous  relations for other more familiar systems.  Let F  denote the free energy of an intercalation compound of composition x, where x is given by 4.1(4) (we refer to equations  in other sections of the thesis  by giving the number of the section and the equation in this way). We assume that the number of host atoms  (which is proportional to N) is constant, and  also that the host l a t t i c e expands freely as the intercalate effects of stress w i l l  be considered in some detail  is added  (the  in Chapter 6). The  change dF produced by infinitesimal changes in n and in the temperature, T, is  dF = -SdT + udn  where S is the entropy.  (1)  Thermodynamic quantities are generally related to  various response functions. cal  For example, the s p e c i f i c heat at constant chemi-  p o t e n t i a l , C^ = T(3S/3T)^, measures the heat absorbed by a change in T.  Similarly,  the response  in composition to a change in u is given by  (3x/9u)-|..  This quantity is proportional to the d i f f e r e n t i a l  capacity  AQ/AV of an intercalation c e l l , the amount of charge AQ passed per change in voltage AV; in fact  In addition, (8x/9y)y also measures the fluctuation in composition of an intercalation system held at constant y (see, 1969) according to  Lifschitz  7 (  =r-r x  "  x  for example, Landau and  '  m  kT / 3 x \ i r [ ^ ) T  . . '  ( 3 )  Since fluctuations are greatly increased near phase transitions  (see  the  review by Stanley 1971, f o r example), one expects peaks or divergences in (8x/9y)-p at compositions at which a phase t r a n s i t i o n occurs in the intercalation system.  As an example of a more familiar system, consider a gas of volume v at a pressure p.  The relation for the free energy change analogous to (1)  is  (k)  dF = -SdT - pdv .  The isothermal compressibility  W(!F) and so V K  t  is defined as  <*>  T  for a gas is analogous to N(9x/9y)  T  for an intercalation system.  S i m i l a r l y , for a long cylinder in an external magnetic f i e l d B p a r a l l e l  to  the cylinder a x i s , one has  dF = -SdT - MdB  (6)  where M is the magnetization (the c y l i n d r i c a l due to demagnetization e f f e c t s ) .  geometry avoids complications  The Legendre transform of F with respect  to B, F - MB, is analogous to ( 1 ) , and the isothermal magnetic tibility x  T  suscep-  defined by  (7)  is analogous to N(3x/3y) . are well known (see,  Divergences  in K  for example, Stanley  and x  y  1971),  at phase  transitions  and ( 3 x / 8 y )  should show  T  T  similar behaviour, as mentioned above.  4.3  Lattice Gas Models Applied to Intercalation Systems We would like to explore now what kinds of behaviour might be expected  in the thermodynamic properties, e s p e c i a l l y y ( x ) ,  of intercalation systems.  To do t h i s , we need some type of model of an intercalation compound. We w i l l adopt here a l a t t i c e gas model; that i s , we assume that intercalated atoms are localized at s p e c i f i c  sites in the host l a t t i c e ,  no more than one atom on any s i t e , and that the motion of the atoms from s i t e to s i t e does not affect neglected.  The experimental  the with  intercalated  the thermodynamics, and so can be  results to date for systems intercalated with  L i , although admittedly somewhat sparse, are consistent with this model. Neutron powder d i f f r a c t i o n patterns for Li T i S  9  ( 0 < x < 1) can be f i t t e d  well by assuming that the Li atoms are localized in octahedral sites in the van  der Waals gap of the T i S  2  host (Dahn et al  1980).  The measured  activation energies for diffusion are about 0 . 3 eV for Li in L i T i 0 ^ x  (Johnson of  19&4a)  and in Li^TiS^ (Silbernagel  activated hopping, this  wells  1975);  in the usual picture  implies that the Li atoms reside  (much deeper than kT).  in deep potential  The measured diffusion coefficients  at room  temperature for these two systems imply hopping times (the time between hops)  42  of about 10 ^ s,  several orders of magnitude larger than the periods of  typical optical phonons; hence the intercalated atoms should be well equilibrated with the l a t t i c e between hops, and the hopping should not be important in determining the thermodynamics of the system.  Although the l a t t i c e  gas model appears reasonable for the Li interca-  lation compounds just mentioned, it may not hold for a l l systems.  intercalation  X-ray d i f f r a c t i o n intensity measurements of the graphite interca-  lation compounds Rb^^^C  and Cs^^^C  are inconsistent with a structural model  where a l l of the Rb or Cs atoms are located over carbon hexagon (Parry et al 1969).  centers  For these materials, a l i q u i d model, where the interca-  lated atoms are not in registry with the host l a t t i c e atoms, may be more appropriate than a l a t t i c e gas model. models are s t i l l of interest,  Even for such cases,  lattice  gas  because a liquid can be regarded as the limit  of a l a t t i c e gas as the l a t t i c e spacing tends to zero but the range of i n t e r action remains f i n i t e  (for an example, see Appendix C).  According to the definition of intercalation given in Chapter 1, the host l a t t i c e  does not change appreciably during i n t e r c a l a t i o n .  in the host lead to effective  Small  changes  interactions between intercalated atoms, which  appear as parameters in the l a t t i c e gas model; some aspects of these host-mediated interactions are discussed in Chapters 5 and 6. changes  Larger  in the host, which violate the d e f i n i t i o n of intercalation and  lead to a breakdown of the l a t t i c e gas model, are discussed in Section 4 . 9 .  Lattice gas models, or the equivalent have been studied'extensively (see,  Ising models  Appendix A),  as models which display phase transitions  for example, Stanley 1971 and references  applied successfully  (see  therein).  They have been  as models of adsorbed systems (for example, Schick  et al  However, previous calculations were not intended to illuminate  1977).  the behaviour of y(x) s p e c i f i c a l l y , so we w i l l consider calculations of y(x) i n some deta i1.  To s t a r t , suppose there is no interaction between intercalated atoms. Let n^ measure the occupation of s i t e a ; since no more than one atom can be put on a given s i t e , n^ = 0 or 1.  In the absence of interactions,  energy E{n } of some d i s t r i b u t i o n {n } of atoms over the sites a  the  is  a  (1)  E{n } = Tn E a  where E  a  is the energy of an atom on s i t e a .  If a l l sites have the same  energy, E = E , then the total energy E{n } is independent of the d i s t r i ct o a bution {n }: a  (2)  E{n } = E = nE a o  where n is the total number of intercalated atoms, as in 4 . 2 ( 4 ) . S i n c e a l l distributions {n^} with the same value of n give the same energy, the entropy S is just k times the logarithm of the number of ways to place n indistinguishable atoms on N s i t e s :  From (2) and ( 3 ) , using S t i r l i n g ' s approximation for the f a c t o r i a l s in S, the free energy F = E - TS r e l a t i v e to x = 0 is (4)  F = N{E x + kT[x£nx + ( l - x ) £ n ( l - x ) ] } . Q  (In ( 4 ) , we've neglected contributions to the entropy due to vibrations of the intercalated atoms in their s i t e s : this w i l l  introduce another term  proportional to x, which can be incorporated into E  q  i f desired.)  From ( 4 ) ,  the chemical potential y ' = ' (9F/9n)  T  and the response function (9x/9y)  is  T  is  (dropping the subscript T)  9x _ x(1 -x) 9y kT  XG)  Using 4.1(3) with y  j  a n o (  plotted in F i g . 12.  = 0, V and 9x/9V corresponding to (5) and (6) are  e  The " t a i l s " which extend to V = ±°° are due to the  fact that the entropy S makes it very d i f f i c u l t to completely f i l l all  the s i t e s .  or empty  The half width of 9x/9V versus x is 3-53kT/ze; at room  temperature, and for z = 1, this  is 90.7 mV.  (5) can be rewritten to give  x in terms of y as  x = +  (E -y)/kT  .  (7)  G  This is just the familiar Fermi -Dirac d i s t r i b u t i o n , giving the average occupation of a s i t e (or energy level) of energy E  q  which can be occupied  by no more than one p a r t i c l e .  This analysis is easily extended to the case where not a l l of the sites have the same energy.  As an example, consider the sites available in a  layered compound such as TiS^, shown in F i g . 4. the octahedral and tetrahedral sites be E E. > E . 1 o  and E^  q  Let the s i t e energies of respectively, and suppose  If we let N be the number of octahedral s i t e s , there are 2N '  tetrahedral s i t e s ;  letting X  and tet rahedra 1 s i tes  q  and x^ be the average occupation of octahedral  respect i vel y (0 <_ X <_ 1 , 0 <_ x^ <_ 2 ) , we can ca 1 cuQ  late the chemical potential for each type of s i t e and equate the two chemical potentials to satisfy the;>condi t ions of equilibrium.  The result  is  Fig.  1 2 - (a) Voltage V and (b) inverse derivative -3x/3V versus x, and (c). -3x/8V versus V for a non-interacting l a t t i c e gas with , a single s i t e energy, E = 0 .  (8)  x = xo + x 1 =  2 1 +  (E -u)/kT  e^o n  +  1  + e  (9)  (E y)/kT r  When E - E^ is large compared to kT, the octahedral sites f i l l Q  0 <_ x <_ 1 , then the tetrahedral si tes f i l l  for 1 5 _ x < _ 3 -  for  At x = 1, there  is a sharp drop in voltage, and a minimum in 3x./3y, as shown in F i g . 13As w i l l  be discussed in Section 1 2 . 3 , this simple two s i t e energy model  is a possible explanation for the large drop in voltage at x = 1 observed in Li T i S „ . shown in F i q . 6 0 . x 2' y  At x = 1, where the sharp drop in V in F i g . 1 3 occurs, the f i l l e d s i t e s form a l a t t i c e commensurate with the total  l a t t i c e of s i t e s .  In fact, such  features, a sharp drop in V and a minimum in 3 x / 3 y , are expected at any composition x  c  which corresponds to a stable commensurate structure.  This  follows from the fact that any such structure has a ground state entropy (the entropy at T = 0 ) of S = k£nm, where m is the number of ways to f i t the structure onto the total  lattice  (in the case shown in F i g . 1 3 , m = 1 ) .  To see why this form of S leads to a drop in V, consider the energy required to take a p a r t i c l e from the highest energy f i l l e d s i t e and place it on a s i t e far enough away so that it is not affected by the vacancy created. This energy is just Ay = u(n + 1 ) - p(n ), where  = Nx c  If Ay were zero,  this excited state would be degenerate with the commensurate structure, and so the entropy would contain an additional term of order kJlnN, contrary to the above form of S for commensurate structures. Ay = ( 3 y / 3 n )  = N ( 3 x / 3 y ) ~ , so - 1  "c n  1  *c  that  Hence Ay  0.  3x/3y = 1/NAy -»- 0 as  N +  But OO  Hence  3 x / 3 y , being non-negative, must have a minimum at low temperatures.  Commensurate structures form not only because of s i t e energy differences;  they can also be produced by repulsive interactions between intercalated atoms.  k.k  We consider this problem in the following sections.  Lattice Gas Models With  In any real system,  it  Interactions  is unlikely that the energy of an intercalated  atom on a p a r t i c u l a r s i t e is independent of the occupation of the other sites.  Any change in the energy of one atom due to the presence of the  others can be regarded as an interaction between the atoms. will  This  interaction  be very complicated, since the presence of intercalated atoms w i l l  d i s t o r t the host l a t t i c e ,  modify the band structure, and so on.  aspects of the interaction w i l l  be discussed  Some  in Chapters 5 and 6; for now,  we assume the interaction can be.characterized by two body interactions U  .  aa 1  between atoms on sites a and a .  In this case, the energy of the i n t e r -  1  calation system for a d i s t r i b u t i o n {n } becomes E{n } = L YE n  a  a  a a  + i 7 U .n n .  , aa a a aa a^a" 1  L  1  1  a  •  v  '  (l)  Solutions to this problem in the l i t e r a t u r e are often written in the language of magnetism, by introducing spin variables s^ = 2n - 1 , and interpreting the system as a l a t t i c e of spins which can point only up (s^ = +1) or down (s  = -1).  This interpretation, known as the Ising model, further  strengthens the analogy between intercalation and magnetic systems mentioned in Section h.2. The translation between the Ising and l a t t i c e gas models is given in Appendix A.  Consider f i r s t a long range interaction, so that an atom on s i t e a can interact with atoms on any of Y » 7  1  1 sites a , with an interaction U  , = U.  1  '  If the range of the interaction tends to i n f i n i t y , so that y = N-1  aa  1  N, but  yU is a constant,  the energy is once again independent of the details of the  d i s t r i b u t i o n , as it was for U = 0 , and depends only on n:  E{n } = E = N(E x + iyUx ) 2  a  o  .  (2)  The entropy its again given by 4 . 3 ( 3 ) , so that p ' i s given by  y  (3)  (This can also be derived d i r e c t l y from the p a r t i t i o n function - see Kac 1968.)  For U > 0 (a repulsive interaction)  the voltage drops more rapidly  with x than for U = 0 , as shown in the s o l i d curve in F i g . 6 .  However, for  U < 0 , the voltage can actually rise with x, as shown in F i g . 14. unphysical behaviour has its origins in our choice of a potential  This U  , which  depends on the size of the system (it was assumed that y l i was a constant, so that U <= 1/N). reasonable  The unphysical region is avoided by arguing that for more  long-range interactions,  the system can form two separate phases,  with compositions x^ and x^ say, such that the energy of interaction between the two phases is negligible compared with the interaction energy within . each phase (note that this  is not true i f U  , is independent of a , a ) . 1  The free energy of such a mixture l i e s on a chord joining the free energies of compositions x^ and  in the free energy diagram, so that the lowest free  energy is given for x^ and x^-determined by the common tangent to the curve (the dashed line in F i g . 14b). Fig.  This leads to the horizontal line in  14a, which gives the two regions between the.sol id curve and the dashed  line equal areas,  the so-called Maxwell construction  it has been shown (Thompson 1971)  (Huang 1963) .  that i f one lets the range of the  In f a c t , inter-  action tend to i n f i n i t y after the thermodynamic limit (N -*-<») has been taken, the result  (3) with the Maxwell construction already b u i l t in is obtained.  50  I  0  1  I  0.2  i  I  i  0.4  0.6  i  I  0.8  i  I 1.0  x  Fig.  14 - (a) Voltage V and (b) free energy F versus composition x for a l a t t i c e gas with a t t r a c t i v e interactions in mean f i e l d theory, with yU = -5 kT. Dotted lines are Maxwell constructions. The s i t e energy was chosen to be E = 2.5 kT to make the Maxwell construction in F h o r i z o n t a l . Q  The solution  (3) with the Maxwell construction can be used as an  approximation for short range interactions;  this  is equivalent to assuming  that the atoms remain randomly distributed in spite of the interaction. As an example,  i f atoms interact only when they are on adjacent s i t e s , with an  energy U , then y is the number of nearest neighbour s i t e s . can  In this case, we  interpret the quantity yLIx in (3) as the interaction of an atom with  its  nearest neighbours, of'which there are yx in a random d i s t r i b u t i o n on average.  For U < 0 , this  is equivalent to the Weiss theory of ferromag-  netism, hence the terminology "mean f i e l d theory" . for  Phase separation occurs  U < 0 for short range interactions as for long range interactions, and so  the mean f i e l d theory is a very useful approximation, despite its errors near the c r i t i c a l  region (Stanley  By contrast,  1971)-  for/repulsive interactions,  short range forces  q u a l i t a t i v e l y different behaviour than that given by ( 3 ) .  lead to  For U > kT, the  free energy of a state where the atoms avoid one another as much as  possible  by forming an ordered structure commensurate with the total  l a t t i c e may be  lower than that of the random d i s t r i b u t i o n underlying ( 3 ) .  As discussed  Section h.J>, such a commensurate structure w i l l  in  produce minima in 3x/8y at  the composition of the f i l l e d commensurate l a t t i c e .  Moreover, peaks in  9x/9y w i l l occur at compositions where phase transitions between ordered and disordered arrangements of atoms occur. no such features,  The random occupation result shows  as shown by the s o l i d curve in F i g . 6 ; by comparison, the  data for Li T i S „ , also shown in F i g . 6 , shows d i s t i n c t features x 2 The .mean f i e l d result  in F i g . 6 corresponds to E  Q  = 2.3 volts,  (with kT = 25-7 meV), and y = 6 , and provides a rough f i t  in 9x/9y. U - 2.5kT  to the overall  variation of V in Li T i S „ , indicating that only small interactions are x 2' ' 3  needed to explain the variation in voltage particular system.  in this  range of x for this  Further discussion of the features  in the V(x) curve  for L i  4.5  i  is given in Section  13-1-  Mean F i e l d Solution of the Problem of Ordering  As an example of the ordering problem for a l a t t i c e gas, consider the triangular l a t t i c e with nearest neighbour repulsive interactions U.  As shown  in F i g . 1 5 , a triangular l a t t i c e with l a t t i c e constant a can be decomposed into three interpenetrating sublattices with l a t t i c e constant / 3 a such that all  the nearest neighbours of an atom on one sublattice w i l l  other two s u b l a t t i c e s .  l i e on the  Because of t h i s , at a composition x = 1 / 3 and a  temperature T « U/k, the atoms w i l l  all  l i e on one s u b l a t t i c e ,  producing an  ordered structure commensurate with the overall triangular l a t t i c e . temperatures T » U / k , a l l three sublattices w i l l there w i l l  At high  be equally populated, and  be no long range order.  #  #• H—  a  #  Fig.  —H  15 - Decomposition of a triangular l a t t i c e with l a t t i c e constant a into three interpenetrating sublattices with l a t t i c e constant / J a .  We wish to discuss  the onset of ordering as the composition x rather  than the temperature T varies.  To find an approximate s o l u t i o n , we look for  another ordering problem, but one with long range interactions.  Suppose an  atom on one sublattice does not interact with other atoms on the same sublattice,  but interacts with any atom on either of the other two  sublattices  with a repulsive interaction energy U which is independent of the 1  between the s i t e s .  For this problem, the energy F-{n^} depends only on the  overall composition x. of the three sublattices not on the d e t a i l s of the d i s t r i b u t i o n {n }. a E -  N  2  X  -U'(x x 1  2  distance  +  x x 2  3  +  x x ) 1  3  + N E  1  Q  + X  ?  + )  3  (i = 1,2,3; 0 <_ x. <_ 1) and  Since N-1 - N, we have  S  (D  Because of the long range interaction, the atoms are randomly distributed over the three sublattices with a fraction x. of the sites on sublattice i i  occupied, so the entropy  S = S  1  + S  £  + S  is  (2)  3  where S. is the entropy of sublattice  S. = k&il  I, and Is given by  (N/3)! (x.N/3).! ((1-x.)N/3).!  (.3)  This solution can be used to approximate the case of a nearest neighbour interaction U i f we use  6U  = ^  which follows all  (i.)  by equating the total  the others when the l a t t i c e  interaction energy of one p a r t i c l e with  is completely f u l l  energy E for the short range case is then  in the two cases.  The  E =N E  Q  x„+x„+x 1 2 3 ^  (5)  + U(x x +x x 4-x^,) 1  2  This expression is e f f e c t i v e l y  2  that obtained by neglecting the short range  order of the atoms, that i s , by ignoring the fact that an atom can avoid interactions with other atoms without going into a state of long range order (which is not true for the i n f i n i t e range interaction U ) . 1  Thus the system  described by (3) and (5) b a s i c a l l y has two choices: the atoms can be randomly distributed over a l l three s u b l a t t i c e s , or randomly distributed within each of the three s u b l a t t i c e s , This approximation w i l l  but with different compositions x. .  be called three sublattice mean f i e l d theory  is also known as the Bragg-Wi11iams approximation - see, de Fontaine 1973)-  (it  for example,  The extension to a different number of sublattices  is  obvious; a general expression is given in de Fontaine (1973)-  The solution of the ordering problem defined by (3) and (5) involves finding those values of x. which minimize the free energy F = E-TS for all  values of the overall composition x, given by  X..+X_+Xx = -!—^—1- ,  0 < x < 1 ;  0 < x. < 1 .  (6)  To do t h i s , we calculate the chemical potential y . for each sublattice and equate a l l three chemical potentials to y .  y  =  U  i Un-., =  This gives  ...  i/n.,j5*i J  x.  = E X . - 3Ux. + kT-fta-pi- + 9Ux Q  i  .  y-9Ux is plotted in F i g . 16a for U = kkl.  (7)  It is apparent that for this  value of U, different values of x. for the three sublattices can satisfy  55  Fig.  16 - u~9Ux versus sublattice population Xj for three sublattice mean f i e l d theory discussed in the text, with U = kkl, (a) Separation of x; into three regions, (b) -»- (h) Sequence of motion of the three roots representing the three sublattice populations. Arrows indicate motion of the points as x increases. In the notation explained in the text, this sequence is (300) -v (210) -»• (201) + (111) -»• (102) + (012) + (003). Other sequences have a higher free energy.  (7)  for a given value of y - 9 U x ;  thus a state of long range order is  possible.  By comparison, y-9Ux for U = 0 is identical to the curve y(x);shown in Fig.  1 2 a , and no ordering occurs.  To follow the state of the system as x increases,  i t is most convenient  to observe the motion of the three points representing the sublattice compositions x. on the curve y-9Ux.  For U..= .4kT,  this curve can be divided into  three sections by the position of the maximum and minimum as in F i g . 1 6 a , and thestate of the system for some value of y-9Ux is then conveniently specified by giving the number of sublattices three d i g i t number in parentheses;  in the three sections as a  thus, for example, we describe F i g . 1 6 b ,  whixh is appropriate for small x, as ( 3 0 0 ) .  As x increases from zero, the  three points move up the f i r s t leg of the curve until  they reach the maximum  value, at x given by the solution of  i  I  I  Here one, two, or three of the points can move onto the second section of the curve.  Examination of the free energy for each p o s s i b i l i t y shows that only  one point breaks off and moves into the second section, and the other two points remain in the f i r s t section. 0 to 1 is given in F i g . 1 6 . s o l u t i o n , h.k(3),  The entire sequence for x going from  By comparison, the completely random mean f i e l d  corresponds to the sequence  (300)+(030)+(003)  , and has  a higher free energy than that shown in F i g . 1 6 .  The of  onset of long range order is associated with the breaking away of one  the points on the y - 9 U x curve.  Close examination of y as this occurs  reveals that as the points s p l i t apart, the value of x at f i r s t  decreases,  then increases once again, as shown in F i g . 1 7 a . This leads to a f i r s t order phase t r a n s i t i o n over a range of composition determined by the Maxwell  X Fig.  17  -  (a) V o l t a g e V a n d (b) f r e e e n e r g y F v e r s u s x n e a r t h e t r a n s i t i o n (300) -> (210) i n t h r e e s u b l a t t i c e mean f i e l d t h e o r y f o r U = 4kT, k T = 25.7 m e V . T h e s i t e e n e r g y , E = 0.560 k T , a n d t h e f r e e e n e r g y a t x = 0, F = 0.157 k T , w e r e c h o s e n t o make V a n d F b o t h z e r o at t h e p h a s e t r a n s i t i o n . The p o i n t i n (a) a n d (b) c o r r e s p o n d s t o t h e maximum i n u 9Ux, a n d t h e l a b e l s on (b) i n d i c a t e the state o f the system corresponding to the various p a r t s o f the free energy c u r v e . D  Q  -  construction. can  The free energy in this region is shown in F i g . 1 7 b , where it  be seen that the states ( 0 3 0 ) , and ( 1 2 0 ) have higher free energies than  (210).  By comparison, when only two points are at the maximum, such as  (201)->(111),  the chemical potential shows only a small change in slope,  indicating a second order phase t r a n s i t i o n . For U = hkT, the behaviour of the sublattice populations x. as a function of x associated with the motion in F i g . 16 is shown in F i g . 1 8 . The arrows on each segment of the curve point in the d i r e c t i o n corresponding to i n c r e a s i n g / x , and the number of arrows gives the number of with that composition.  The empty regions near x = 0 . 0 9 and x = 0 . 9 1  correspond to the region of phase coexistence. to this behaviour is plotted in F i g . 1 9 . associated minima in 3 x / 3 y ,  The voltage V corresponding  Note the large drops in V, with  at x = 1 / 3 and x = 2 / 3 corresponding to commen-  surate ordering, and the f l a t regions in V, with associated 3x/3y  are  sublattices  i n f i n i t i e s in  , near x = 0 . 0 9 and x = 0 . 9 1 , corresponding to phase t r a n s i t i o n s .  These  in agreement with the general discussion of Section h.3-  The drops in V at x = 1/3 and 2/3 can also be understood quite easily in terms of the f i l l i n g  behaviour shown in F i g . 1 6 .  than 1 / 3 only one sublattice is f i l l i n g , are  For x s l i g h t l y  and the atoms added to the  less lattice  able to find sites in this preferred sublattice where they feel no  nearest neighbour interactions. full,  When x reaches 1 / 3 , this sublattice  is  and the atoms must then be placed on one of the other two s u b l a t t i c e s ,  where they interact with three nearest neighbours on the f u l l Thus the energy gained in adding the atoms to the l a t t i c e ,  sublattice.  -y, drops by ^ 3 U  at x = 1 / 3 . Similar considerations apply near x = 2 / 3 .  The phase diagram for the triangular l a t t i c e gas in the three sublattice mean f i e l d approximation is given approximately by curve a in F i g . 2 0 . For  59  Fig.  18 - Sublattice compositions Xj versus average composition x for three sublattice mean f i e l d theory for the triangular l a t t i c e with U = kkT. The arrows point in the direction of increasing x, and the number of arrows on each segment of the curve indicates the number of sublattices of that composition. The breaks in the curve near x = 0.1 and x = 0.9 correspond to phase coexistence.  60  x  Fig.  IN L i T i S x  2  19 - (a) Voltage V and (b) inverse derivative -Ax/AV versus x for a triangular l a t t i c e gas with nearest neighbour interactions U = kkl in three sublattice mean f i e l d theory. The s i t e energy E = - 2 . 3 Volts, and kT = 2 5 - 7 meV. o  61  Fig.  20  values of kT/U inside the curve, the system shows long range order; outside, all  three sublattices  are equally populated.  The curve is the solution of  ( 8 ) which gives the value of x. where the three points on the curve y-3Ux f i r s t s p l i t apart, as discussed above. by the Maxwell construction,  The phase t r a n s i t i o n , as determined  lies on or outside this curve; calculations  have been done for several values of U/kT, and the positions and widths of the regionsiof phase coexistence are indicated. for U ->•  the phase transition occurs at x = 0 .  0 0  In p a r t i c u l a r , note that This is easily  understood  in light of the correspondence of the mean f i e l d theory solution to an i n f i n i t e range interaction U' ( 4 ) . The f i r s t atom placed on the on sublattice  1 say, w i l l prevent other atoms from occupying any of the sites  in sublattices as U ->  0 0  lattice,  2 or 3 -  Hence the mean f i e l d solution for 0 <_ x <_ 1/3  is identical to the solution for U = 0 with only N/3 s i t e s :  y = E  Q  +  kT&\  1/3  (9)  - x  This "hard sphere" solution w i l l be discussed  in more detail  in Section h.6.  Curve b in F i g . 2 0 gives the phase boundary between ordered and d i s ordered states as determined by a renormalization group (RG) calculation (Schick et al  1977).  Because the RG solution treats short range order  (the mean f i e l d theory does not),  it predicts that the ordered phase  confined to a smaller region of kT and x than in the mean f i e l d Moreover, the RG calculation predicts a continuous transition to the ordered state, in mean f i e l d .  result.  (higher order) phase  in contrast to the f i r s t order transition  RG methods have also been used to calculate V versus x  for U = kkl (Berlinsky et al 1 9 7 9 ) ; the result  is shown in F i g . 2 1 . The  voltage curve is very similar to the mean f i e l d result fine details  is  in F i g . 1 9 , but the  seen more c l e a r l y in the curve 3x/8y (Figs.  1 9 b and 2 1 b )  63  Fig. 21 - (a) Voltage V and (b) inverse derivative -Ax/AV versus x for a triangular l a t t i c e gas with the same parameters as in Fig. 1 9 , calculated using renormalization group techniques. The points in (a) are the results of Monte Carlo calculations, again for the same parameters. From Berlinsky et al ( 1 9 7 9 ) .  6k  d i f f e r considerably.  It is interesting to examine the voltage curve for a value of U/kT where the mean f i e l d theory predicts a phase t r a n s i t i o n but the RG calculation does not.  F i g . 22a shows the behaviour of the sublattice compositions x. versus x  as predicted by the mean f i e l d theory, for U/kT = 0.72.  The corresponding  voltage curve, F i g . 22b, shows no noticeable features associated with the phase t r a n s i t i o n .  These mean f i e l d calculations can straightforwardly be extended to treat other l a t t i c e s , dimensional  as in the next section where we consider the one  lattice' gas, or to include longer range interactions.  simply subdivides the l a t t i c e  into enough interpenetrating  so .that no p a r t i c l e s on the same sublattice action of p a r t i c l e s on the same sublattice  One  sublattices  interact, or until  the i n t e r -  is weak enough that i t w i l l  not  produce ordering; these weak interactions can then be handled by the random mean f i e l d theory of Section k.k. therefore necessary insight  k.6  A good deal of physical insight  in choosing an appropriate set of sublattices;  is such  is also required for RG c a l c u l a t i o n s .  One Dimensional Lattice Gas  The one dimensional l a t t i c e gas  ( l a t t i c e gas on a chain of sites)  is  of interest as a possible model for intercalation compounds such as the r u t i l e - r e l a t e d materials, where the intercalated atoms are located in tunnels or channels in the host l a t t i c e .  In addition, one dimensional l a t t i c e  gas  problems can in some cases be solved exactly, and show no long range order i f the interactions are short ranged.  By contrast, the mean f i e l d  solutions,  being associated with i n f i n i t e range interactions, w i l l show long range order even in one dimension; because of t h i s , mean f i e l d theories are  65  Fig. 2 2 - (a) Sublattice composition xj and (b) voltage V versus average composition x for the triangular l a t t i c e gas with nearest neighbour interactions U = 0 . 7 2 kT in three sublattice mean f i e l d theory. Site energy E = 2 . 3 V o l t s , kT = 2 5 - 7 meV. The arrows in (a) are explained in Fig. 1 8 ; the insert in (b) shows the Maxwell construction for the f i r s t order phase transition near x = 0 . 3 9 . D  66  generally considered unsuccessful  at treating one dimensional problems.  However, in comparing the exact and mean f i e l d solutions, we shall see  that  the mean f i e l d results serve as useful approximations even in the one dimensional case.  The one dimensional l a t t i c e gas with nearest neighbour interactions is treated in many textbooks, model.  usually in the magnetic language of the  A discussion of the solution is given for reference  Ising  in Appendix B;  from there, we find the relation between x and u is  (1)  This is plotted in Fig 23a for U = -2.5kT (attractive  i n t e r a c t i o n ) . Also  shown is the result of the mean f i e l d expression 4 . 3 ( 5 ) , with the Maxwell construction drawn in the two phase region.  Both curves are considerably  f l a t t e r than for U = 0 (Fig. 12); the mean f i e l d theory carries this flattening to the extreme of a f i r s t order phase t r a n s i t i o n .  In F i g . 23b, (1) is plotted for U = 5kT (repulsive i n t e r a c t i o n ) .  Note  the drop in V at x = i , which would correspond to a minimum in 3x/Su.  This  is the result of the existence of a commensurate structure at x = i and T = 0 , where every second s i t e is occupied, so the minimum is in agreement with the discussion in Section 4 . 2 . interpenetrating sublattices  If we divide the chain into two  1 and 2 , as in F i g . 24, then at T = 0 and  x = i only one of these is occupied. For T / 0, or x not precisely i , will  there  be stacking f a u l t s , or f i l l i n g mistakes, as shown in F i g . 24; at each  of these mistakes the occupied s u b l a t t i c e . s h i f t s so the long range order is destroyed.  There w i l l  from 1 to 2 or vice versa, still  be short range order,  however, and so as x increases past i s i t e s with two nearest neighbours begin  67  F i g . 2 3 - Voltage V versus composition x for the one dimensional l a t t i c e gas with nearest neighbour interactions U calculated using exact and mean f i e l d (MFT) solutions. (a) U = - 2 . 5 k T . (b) U = 5 kT.  1  2  1  2  1  2  1  2  1  2  1  2  • O• O• O• •O •O • F i g . 2k - One dimensional l a t t i c e gas with repulsive interactions near x = i indicating the destruction of long range order by f i l l i n g mistakes. The numbers 1 and 2 give the decomposition of the l a t t i c e into two interpenetrating s u b l a t t i c e s . • : full site, o: empty s i t e .  to f i l l ,  leading to a drop in V of order 2U/ze.  Also shown in F i g . 23b is the result of a two sublattice mean f i e l d theory calculation for the same U.  This calculation was done in the same  way as described in Section k.5; the calculations are simpler here than for the triangular l a t t i c e because there are only 2 points on the y-4Ux curve (the n-shaped curve analogous to u-9.Ux for the triangular l a t t i c e gas, Fig.  16).  The mean f i e l d c a l c u l a t i o n , unable to handle the problems shown  in F i g . 2k, predicts a second order phase t r a n s i t i o n to a long range ordered structure at x = 0.11 and x = 0.89.  In spite of t h i s , the overall shape of  the voltage curve, especially the drop at x = i ,  provides a reasonable  approximation to the exact r e s u l t , unless one is interested in the fine deta i1s.  To further compare mean f i e l d and exact r e s u l t s , we now consider a second problem in one dimension. lattice  Suppose the atoms on a one dimensional  interact with a repulsive energy U =  0 0  when they are separated by  less than d-1 empty s i t e s , and do not interact otherwise, essentially  d is then  the diameter of the atoms in this hard sphere model, measured  in units of the l a t t i c e spacing.  The occupation of the l a t t i c e  restricted to the range 0 < x < 1/d.  is thus  The mean f i e l d solution is given by  dividing the l a t t i c e  into d sublattices;  prohibits occupation of a l l sublattices  the f i r s t atom placed on the except the one it  lattice  is on, so the  solution is the same as that for a noninteracting l a t t i c e gas on N/d sites  (cf 4.3(5)):  (2)  The exact solution, as shown in Appendix C, is  (3)  which reduces to (2) only for d = 1 (no interactions). f i e l d expressions  The exact and mean  are compared in F i g . 25. The voltage curve for the exact  result drops much more rapidly near x = 1/d than for mean f i e l d for large d. In most s i t u a t i o n s ,  however, one does not expect d to be very large, so  the mean f i e l d calculations again give a reasonable approximation to the exact  result.  These comparisons of mean f i e l d theory and the exact solutions were made to i l l u s t r a t e how well the mean f i e l d theory approximates the exact results.  Of course, where the exact solutions exist and are r e l a t i v e l y  simple, they would be used rather than the mean f i e l d results a one dimensional sional  l a t t i c e gas.  to describe  However, i f we wish to use the one dimen-  results, we must determine when i t is acceptable to regard an actual  intercalation compound as a one dimensional  lattice  gas.  To do t h i s , consider a l a t t i c e of chains of sites., with each chain coordinated by y nearest neighbour chains.  For r u t i l e structures, the chains  are arranged in a square l a t t i c e , with y = k; a two dimensional example, with y = 2, is shown in F i g . 26.  Suppose that atoms interact only when on  70  a  \ \ d =4  -  \  N JC  d»l\  \  -8  -12  Fig.  25 -  1  1  i  i  i  .  i  1  \  (a) V o l t a g e V v e r s u s x d a n d (b) i n v e r s e d e r i v a t i v e - 9 x / 3 V v e r s u s V f o r a one d i m e n s i o n a l l a t t i c e g a s w i t h r e p u l s i v e h a r d s p h e r e i n t e r a c t i o n s over d l a t t i c e s i t e s . Curves are exact r e s u l t s ; mean f i e l d t h e o r y c a l c u l a t i o n c o i n c i d e s w i t h t h e c u r v e d = 1 f o r al1 d .  o • o •o o•o o • o •o F i g . 26 - Two dimensional l a t t i c e of one dimensional chains with nearest neighbour interactions U, L)!: indicated. Interactions such as U are assumed to be zero.  1  adjacent s i t e s , with an energy U when on the same chain and an energy U when on adjacent chains (U and U both p o s i t i v e ) . 1  1  We have seen in the above  sections that ordering occurs for interactions of order kT; hence one of the r e s t r i c t i o n s on the system to allow the chains to be considered as independent, and thus the system of chains to be regarded as a one dimensional l a t t i c e gas,  is  U' < kT.  However, for the system being considered, where  an atom on one chain can interact with only one atom on an adjacent chain, this  is not s u f f i c i e n t .  As an example, consider the two dimensional  in F i g . 26 at a composition x = i . can be found exactly  (Onsager ISkh),  lattice  The free energy in this particular case and the system shows long range order  below a t r a n s i t i o n temperature T given by c  s i n h  For U  1  \2kT; <KU,  s  i  n  n  ( ^ r - )  =  1  this can be rewritten as  ( z , )  U  "  kkT  e  c = 1 .  U / 2 k T  The transition temperature T zero.  (5)  c  Thus, for  c  decreases only logarithmically as U' goes to  U = U , kT^ = 0 . 5 6 7 U , while for 1  a reduction by a factor 0 . 2 2 . This persistence very small U  IT = 0 . 0 1 U , k T  c  of long range order for  is due to the fact that, at x = \ and U > kT, there  1  = 0.127U,  is  considerable order in the purely one dimensional chain ( although no long range order). Away from x = i ,  however, the long range order w i l l not be so  persistent.  As discussed above, the long range order in a purely one dimensional gas is destroyed by f i l l i n g mistakes  (Fig. 2k). Suppose one such mistake  occurs on average once out of every E, sites for  the one dimensional chain).  lattice  (£ is thus a coherence  length  If there is to be at least one pair of  atoms interacting on different chains, the short range order on the chains implies there w i l l be of order £/2 such pairs (near x = i ) . order w i l l be expected  The coherence  Thus, long range  i f .the chains interact unless  length E, can be found from the two s i t e correlation functions  <nQn >, which gives the correlation in occupation of two sites separated by r  r sites.  Since we w i l l be interested  convenient  s a  to use the. Ising notation  = 2n  a  +  it  is  (Appendix A ) , namely  (7)  1  m = <s > = 2x -  in compositions near x =  1  .  (8)  a As discussed  in Appendix E, the two s i t e correlation functions are given by  73  2  <s.s ••> = 0 r  For e ^ ^  m +  » 1,  2  [ "U/kT ^ „ + m (l-e e  )J  2  1 + .e  + m (1-e  (-D (l-m ) r  2  (9)  )  (9) can be rewritten as  <s.s > - m' 0 r  1  "f(-l)  r  , .J «  (10)  ?  wi th  (m + 2  In this  (11)  •U/kT,  l i m i t , the condition (6) becomes  yU . -U/kT\± < £ + e )  , (m 2  (12)  K l  For y = 2 and m = 0, this agrees with (5) If U'/kT > e ^ ^ k T the system w i l l f  from zero, however, the order w i l l  m  For  U  ^  1  yU ' \ kT  2  = 0.001U,  T - 0.107U/k;  -U/kT - e  i  disappear for  (13)  long range order is expected at m = 0 (x = i ) for  at half this temperature, the ordered phase extends only over  the fact that although £ (11)  f a l l s off  be ordered at m = 0; as m deviates  ^ XUJ kT  JmJ < 0.019, or 0.49 < x < 0.51of  to within a factor of 8.  rapidly  This small composition range is a result is very large at x = j  (as 1/|m|) for |m|>  e ^^T^  (£ - e ^ ^ ^ ) ,  In Appendix E a more  detailed calculation of this problem of weakly coupled chains is wnich treats the interaction along the chain exactly but the between chains using mean f i e l d theory.  it  presented,  interaction  This c a l c u l a t i o n v e r i f i e s  of magnitude estimates presented above, and indicates  the order  that the feature in  8x/3y produced  by the o r d e r i n g i s v e r y s m a l l . f o r U  «  1  U.  It s h o u l d be n o t e d , however, t h a t the i n t e r a c t i o n b e i n g d i s c u s s e d i s somewhat a r t i f i c i a l , s i n c e i t n e g l e c t s i n t e r a c t i o n s such as L i " i n F i g . 26. For w i d e l y s e p a r a t e d c h a i n s , U" U, 1  to  than t o be z e r o .  i s more l i k e l y t o be o f the same o r d e r as  I f U" > 0, the o r d e r which o c c u r s f o r U" = 0 w i l l  be s u p p r e s s e d , because the energy  We  gained by o r d e r i n g w i l l  be  tend  reduced.  conclude on the b a s i s o f the above d i s c u s s i o n and the r e s u l t s o f  Appendix E t h a t a l t h o u g h weak i n t e r a c t i o n s between c h a i n s may critical  modify  the  b e h a v i o u r , p o s s i b l y g e n e r a t i n g o r d e r e d s t a t e s i n cases where the  t r u e one d i m e n s i o n a l system would a l r e a d y show a p p r e c i a b l e s h o r t range o r d e r , they have l i t t l e e f f e c t on the b a t t e r y v o l t a g e c u r v e . should a l s o a p p l y t o the two d i m e n s i o n a l in S e c t i o n k.5-  l a t t i c e gas  Similar conclusions results discussed  For example, i n a l a y e r e d compound, a p u r e l y two  o r d e r e d s t a t e would r e q u i r e t h a t the o c c u p i e d s u b l a t t i c e i n one  dimensional  layer  be  chosen randomly, and c o m p l e t e l y i n d e p e n d e n t l y o f the p o s i t i o n o f the o c c u p i e d s u b l a t t i c e s i n the o t h e r l a y e r s .  Weak i n t e r a c t i o n s between atoms  in d i f f e r e n t l a y e r s c o u l d cause the o r d e r e d s t a t e t o c o n s i s t o f a p e r i o d i c s t a c k i n g sequence of the o c c u p i e d s u b l a t t i c e s from l a y e r t o l a y e r ,  which  is a s t a t e of three dimensional order.  to  modify  the o v e r a l l  cell  v o l t a g e curve V(x) from the p u r e l y two  c a s e , except near the c r i t i c a l  h.7  T h i s would not be expected  r e g i o n where o r d e r i n g o c c u r s .  I n t e r a c t i n g L a t t i c e Gases w i t h D i f f e r e n t S i t e  E x t e n s i o n o f the mean f i e l d  dimensional  Energies  c a l c u l a t i o n s to l a t t i c e s w i t h d i f f e r i n g  energies, i s reasonably s t r a i g h t f o r w a r d .  I f the s i t e energy  l a r g e , t h e d i f f e r e n t types of s i t e s f i l l  s e q u e n t i a l l y , s t a r t i n g w i t h the  lowest energy  sites.  site  d i f f e r e n c e s are  In d i s c u s s i n g a range of x where a p a r t i c u l a r type o f  s i t e is being f i l l e d , one can ignore the higher energy s i t e s , and treat  the  interaction with atoms on the f i l l e d lower energy sites as a modification of the site energy of the p a r t i c u l a r s i t e of interest.  In certain cases,  however, more complicated behaviour can occur; we discuss one example here.  Consider the sites available in a layered t r a n s i t i o n metal shown in F i g . h.  cogenide,  The triangular l a t t i c e of sites consists of one  sublattice of octahedral sites and two sublattices tetrahedral s i t e s .  dichal-  of s l i g h t l y  smaller  The tetrahedral sites s i t above and below the plane of  the octahedral s i t e s , so the distance between adjacent tetrahedral is larger than the nearest neighbour octahedra1-tetrahedra1 s i t e Because of this geometry,  sites  distance.  it is probable that the octahedral s i t e energy E  is less than the tetrahedral s i t e energy  , while the interaction energy  between atoms on adjacent tetrahedral s i t e s , neighbour :octahedral-tetrahedral  , is less than the nearest '  interaction energy U . q  that even though octahedral sites f i l l  Q  first,  Thus i t  is  possible  it may be favourable to have  only tetrahedral sites occupied for x > 1 to avoid the larger interaction energy U . o 3 7  The fundamental aspects of this problem can be obtained from the zero temperature case.  Let x^, x^, and x^ be the fractional occupation of the  octahedral and the two tetrahedral sites respectively.  In mean f i e l d  theory,  the energy of some arrangement of atoms is given by  E  =  N  [ o o E  x  +  MV^  neglecting a l l but nearest  +  VVi+Va'  +  3U  1 1 2J X  X  neighbour interactions.  (  F i g . 27a shows the  energy as a function of x for two sequences of f i l l i n g the l a t t i c e : corresponds to f i l l i n g the octahedral sites for 0 <_ x tetrahedral s i t e for 1 < x < 2 ,  )  free  curve a^  1, one type of  and the other tetrahedral sites for  1  76  F  V  0  1  2  3  x  F i g . 27 - Form of the (a) free energy F and (b) voltage V versus x for three interpenetrating sublattices of octahedral and tetrahedral s i t e s (see text) at T = 0 , with s i t e energies and interaction energies satisfying equation 4 . 8 ( 2 ) . Dotted lines indicate Maxwell construction.  2<  x<_3;  curve  corresponds  0 <_ x <_ 1 ,  the other  octahedral  sites.  of  site  and  x = 2.  phase to  that  empty  with  ( x = 2).  conditions  o  ^ i1  U,  <:U  E  3  4.8  (  u  a  >  u  E  r o  voltage  transition  filled  curve  below  both  a first  sites  filled  curves, order (x =  and the octahedral  i s shown  to occur  x = 1  are  i n F i g . 27b.  (at T =  1)  sites The  0)  •  E  interaction  simple  Suppose  Body  between  two body  complications lattice  between  is given  as  i n S e c t Ton 4.5,  mean  i n terms  may b e m o r e  of V(x).  with  an i s o l a t e d  forces  field  nearest pair  triangle  theory,  complicated  order  forces  can lead  As an e x a m p l e ,  consider  neighbour  of nearest  the interaction  of the three  the energy  sublattice  neighbour  between  i s 3U + U  interactions.  three 1  atoms  rather  o f some  compositions  atoms  than  just  distribution x.  (where,  0 <_ x . <_ 1) b y  E =[E X + U(x x +x x +x x ) Q  sites  o f an elementary  sublattice  atoms  atoms  The i n c l u s i o n o f h i g h e r  identical  body  on the v e r t i c e s  of  intercalated  in the behaviour  of  the interaction  In ..three  Forces  forces.  U , but due t o three  placed  sites  between  cross  undergoes  a l l the octahedral  values  (2)  triangular  31).  phase  lies  For the  o  o- i>  further  is  The r e s u l t i n g  energies.  line)  for  and f i nal l y the  the two curves  (dotted  site  E  The  to  with  a l l the tetrahedral  I n c l u s i o n o f Three  than  of tetrahedral  higher  1 < x < 2 the system  from a phase  for this  have  chosen,  construction  i n t h e range  transition,  a phase  sequences  and i n t e r a c t i o n s  The Maxwell  indicating  one type  t e t r a h e d r a 1 s i t e s f o r 1 <_ x <_ 2,  A l l other  energies  to f i l l i n g  1  2  1  3  2  3  + .2U'x x x ] N ,  1  2  (1)  where  x = (x^+x^+x^)/3•  free  energy  F = E.  free  energy  curve  (as  in Section  or  three  shown  Once a g a i n we c o n s i d e r o n l y  F o r each  c o m b i n a t i o n o f U and U  f o r t h e c a s e where  4.5, t h i s  sublattices  simultaneously).  indicates  x = 1.  t r a n s i t i o n i s caused  one a n o t h e r ,  pairs. in  clusters  order  t r a n s i t i o n occurs  df particles  al  symmetry a b o u t  voltage  curves forces  also  1  curves  obtained  for U lack  1  between  1  isolated  c a n be f o u n d  i n F i g . 29.  body f o r c e s  are probably present  x = 1/3 a n d  than  i n RG t h e o r y  symmetry  = - U , is  the p a r t i c l e s  strongly  = 0 (see F i g s .  this  1  , t h e Maxwell  although  less  the i n c l u s i o n of three  x = i which occurs  body a n d h i g h e r o r d e r  Fig.  agree w i t h those  Note t h a t  (1977).  experimental  1  sequentially  t h a n f i l l i n g two  f o r a r b i t r a r y LI a n d U  t h i s w a y , a n d i s s u m m a r i z e d by t h e v o l t a g e on U a n d U  that  repel  fill  f o r U > 0 and U  o f U and U  by t h e f a c t  The b e h a v i o u r o f t h e s y s t e m  conditions et  a first  so the  , we c a n c o n s t r u c t t h e  sublattices  An e x a m p l e ,  p a r t i c u l a r choice  construction  repel  1  t u r n s o u t t o be more f a v o u r a b l e  i n F i g . 28; f o r t h i s  This  the three  the T = 0 case,  These  by S c h i c k  has broken t h e  19 a n d 2 1 ) .  ( e . g . F i g . 6),  in real  Since three  systems.  28 - F r e e e n e r g y F v e r s u s c o m p o s i t i o n x for-..-.the t r i a n g u l a r l a t t i c e g a s w i t h t w o body a n d t h r e e body n e a r e s t n e i g h b o u r i n t e r a c t i o n s , L) a n d U ' r e s p e c t i v e l y . F o r the p a r t i c u l a r example shown, U = - U . The d o t t e d l i n e i s t h e M a x w e l l c o n s t r u c t i o n . 1  79  U' >  U > 0  U' >  f |u|  U <0  u > l |u| 1  Fig. 29 - Voltage V versus composition x for the triangular l a t t i c e gas with nearest neighbour two body interactions U and three body interactions IT , for a l l ranges of U ' / U , at T = 0. Dotted portions of the curve indicate two phase coexistence.  4.9  Changes in the Host  So f a r , the host has apparently been neglected  in our discussion.  However, small changes in the host contribute to the interaction energies and s i t e energies in our l a t t i c e gas models, as w i l l be discussed  in the next  two chapters, and so have already been i m p l i c i t l y included in our  lattice  gas description of intercalation systems.  Large changes in the host require  additional attention, which we w i l l b r i e f l y give here.  A wide variety of structural forms' of a given host can be envisioned, and for each we can (in p r i n c i p l e ) calculate a free energy, F. with the lowest free energy w i l l be thermodynamica11y stable, cases w i l l correspond to the observed structure.  The structure and in most  As the host is  interca-  lated, the free energies of a l l the structures, both observed and imagined, will  vary.  It may happen that the free energy of one of the conceptual  structures crosses F for the observed structure at some value of x, as indicated in Fig. 30a. This w i l l  lead to a f i r s t order phase t r a n s i t i o n .  In the absence of e l a s t i c strains or surface energies, the free.energy coexisting  of the  phases l i e s on the common tangent to the two free energy curves,  as shown, for example, by H i l l e r t  (1974).  (Note that this and most other  proofs of this use the chemical potential of the host, which is a well defined quantity only in the absence of e l a s t i c s t r a i n s . point is given by Paterson (1973). are discussed  E l a s t i c effects  A review of  this  in intercalation systems  in Chapter 6.)  In most f i r s t order phase t r a n s i t i o n s ,  there is some activation energy,  such as the surface energy required to create a phase boundary, which must be overcome before the phase t r a n s i t i o n can occur.  Thus the t r a n s i t i o n  from structure 1 to 2 wi11 not occur until the common tangent condition for 1 and the activated structure 2* is s a t i s f i e d .  Hence one expects the  Fig. 30 - Schematic form of free energy per s i t e , F / N , versus composition x for a structural t r a n s i t i o n in the host. Maxwell construction given by dotted l i n e , (a) Thermodynamic equilibrium, (b) Trans i t i o n via some "activated" intermediate state.  voltage at which the phase t r a n s i t i o n begins, which is minus the slope of the dotted line in F i g . 30a and 30b divided by ze, to be lowered as x increases  in the t r a n s i t i o n from 1 to 2 (or raised as x decreases in going  back from 2 to 1) over what is expected from equilibrium thermodynamics. As the phase t r a n s i t i o n proceeds, one might expect the voltage to rise again as the new phase grows and surface energies  become less important (in  analogy with the rise in temperature as a supercooled l i q u i d begins freeze).  to  However, in an intercalation c e l l made using a powdered cathode,  any given p a r t i c l e in the cathode is e f f e c t i v e l y  in a "chemical potential  bath" provided by a l l the other p a r t i c l e s , and so i t is more l i k e l y that the observed voltage w i l l proceeds.  remain at its  low value as the phase t r a n s i t i o n  In this case, the phase t r a n s i t i o n in any given p a r t i c l e occurs  very rapidly once i t begins, and the energy due to the difference the observed and equilibrium voltages appears as heat. phase t r a n s i t i o n w i l l battery/curve w i l l  between  The voltage of the  be different on the charge and discharge, and the  show hysteresis.  connection with the e l a s t i c  (This point is discussed further in  interaction in.Chapter 6.)  If the activation  energy is high, the host might remain in a given phase over a much wider range of x and V than expected thermodynamically; this phenomenon of metastable phases occurs in Li MoS2 and other systems, as discussed in x  Chapter 13-  CHAPTER 5  ELECTRONIC INTERACTIONS BETWEEN INTERCALATED ATOMS  5.1  Int roduct i on  In the previous chapter, we saw how the chemical potential calated atoms is affected  by interactions between the atoms.  to consider the origins of these interactions.  of  inter-  We now wish  Interactions between  intercalated atoms are due to changes in the energy of one atom in the presence of the others. to  M  (In this and the following chapter, we w i l l  refer  the energy of an intercalated atom" rather than using the more correct  phrase, "the change in the energy of the atom and of the host when the atom is  intercalated".)  The energy of an intercalated atom can be  conveniently divided into two parts, electronic energy and e l a s t i c energy; the electronic energy is associated with changes in the electron  distri-  butions in the atom and in the host caused by i n t e r c a l a t i o n , while e l a s t i c energy is due to the distortions intercalated atom.  the  in the host l a t t i c e caused by the  Changes in these energies due to the presence of other  intercalated atoms give rise to electronic and e l a s t i c interactions. electronic  interaction is discussed  is discussed  in this chapter; the e l a s t i c  The  interaction  in Chapter 6.  A complete discussion of the electronic  interaction would require a  comparison of the band structures of the intercalated and uninterca1ated host.  Such an ambitious project requires s p e c i a l i z a t i o n  material.  In order to make more general  r i g i d band model, where it  comments,  it  to a p a r t i c u l a r host  is common to adopt a  is assumed that the host band structure  is  unchanged on i n t e r c a l a t i o n , except for the addition of new electronic associated with the intercalated atom, and a possible overall shift  states  in the  energy of the bands measured w i t h respect infinite distance new s t a t e s  from the host  to an e l e c t r o n at  (the vacuum l e v e l ) .  l i e s above empty s t a t e s o f the h o s t ,  electrons  rest at  an  If t h e e n e r g y o f  t h e r e w i l l be a t r a n s f e r  from t h e atom t o t h e h o s t , and t h e atom w i l l be i o n i z e d .  o t h e r h a n d , t h e r e may b e m o r e new s t a t e s  these  On t h e  i n t r o d u c e d below the highest  filled  s t a t e s o f t h e h o s t t h a n c a n be f i l l e d by t h e e l e c t r o n s o f t h e a t o m , s o will  there  be an e l e c t r o n t r a n s f e r f r o m t h e h o s t t o t h e a t o m , c r e a t i n g an a n i o n .  In b o t h o f t h e s e c a s e s , t h e f i e l d s by t h e e l e c t r o n s (the  from the charged p a r t i c l e w i l l  in the host band s t r u c t u r e , and the  ions plus t h e i r screening clouds) w i l l  potential.  is given  be  intercalated  i n S e c t i o n 5.2,  of this  largely  e n o u g h new s t a t e s  in terms o f  introduced below the  c a l a t e d atom remains n e u t r a l .  increases, occur.  it will it  remain so.  is expected  An i o n i n a m e t a l the f i e l d of the the ion.  interor  atoms  that a t r a n s i t i o n to m e t a l l i c behaviour w i l l 5-2.  Interactions i s s u r r o u n d e d by a c l o u d o f e l e c t r o n s , w h i c h  ion in a very short d i s t a n c e . ( t h e  b a n d s t r u c t u r e d u r i n g . i n t e r c a 1 a t i o n , we c a n s t i l l  of electrons  In t h i s c a s e , t h e  T h e n , as the d e n s i t y o f i n t e r c a l a t e d  screening  Thus, i f there is a transfer of electrons  intercalate plus  to  If the h o s t w a s . o r i g i n a l l y s e m i c o n d u c t i n g  This is d i s c u s s e d f u r t h e r in Section  Screened Coulomb  just  lowest empty s t a t e o f the host  i n t e r c a l a t e d atom.  the  gas.  in the r i g i d band model is t h a t t h e r e a r e  accommodate the e l e c t r o n s of the  insulating,  Coulomb screening.  l i n e a r i z e d Thomas-Fermi and H a r t r e e s o l u t i o n s f o r a f r e e e l e c t r o n A further possibi1ity  screened  atoms  interact via a screened  The i n t e r a c t i o n e n e r g y depends on the d e t a i l s  A discussion of screening  5.2  of  length)  to or from the  regard the  from host  ionized  its s c r e e n i n g e l e c t r o n c l o u d (which is a c t u a l l y  f o r a n e g a t i v e i o n ) as a n e u t r a l atom.  screens  a.depletion  Two i n t e r c a l a t e d  atoms  will  interact i f they are within a screening length of each other, so that  their screening clouds overlap; this leads to a screened Coulomb interaction between them.  This interaction is most easily discussed in terms of the linearized Thomas-Fermi equation (see,  for.example, Ziman 1972). Consider a single  intercalated atom, consisting of an ion of charge ze at r = 0 and z electrons, added to an i n i t i a l l y uniform free electron gas of number density p with a positive j e l l i u m background.  (We use the symbol p without  a subscript to denote the electron density only in this section - elsewhere it is used to denote the density of intercalated atoms.)  The ion w i l l  produce a change <5p(r_) in the electron density, and the e l e c t r o s t a t i c potential < J > (r_) at any point r_ wi 1 1 be the sum of the potential of the ion (ze/r, where r = |rj) and of these electrons.  The relationship between <5p(r_)  and < J ) (r_) is given by the following semi c 1 ass i ca 1 argument. kinetic energy of electrons  in the state q.  Let  be the  In the absence of the perturbing  ion, the kinetic energy states q of the free electron gas are f i l l e d to the unperturbed Fermi energy  6  f  , | . W p )  2  /  3  which is related to the electron density p by  .  ti)  The Thomas-Fermi approximation consists of assuming the potential <f>(r_) '  s  slowly varying in space, so that on the scale of variation of (J) (j_) we can regard the electrons as localized at r_.  Then the total energy of an electron  at _r (measured with respect to the vacuum level) in the state q is just &q - ecb (r_).  Since the Fermi energy, which is the total energy of the most  energetic electron (again, measured from the vacuum l e v e l ) , must be independent of p o s i t i o n , electrons move into the region near the ion (or away from i t ,  if z < 0),  f i l l i n g additional states, as indixated"in F i g . 31. In  r(s ) q  Fig.  31 - Kinetic energy S versus density of states r ( £ q ) for states q of a free electron gas, showing increase in the number of f i l l e d states at r^ in the presence of a potential (J) (r) and a possible s h i f t in the Fermi energy 6&f. The potential energy of the electrons is simultaneously decreased by -e<J) (_r) q  the presence of the perturbing ion, the Fermi energy is conveniently given by the k i n e t i c energy of the highest f i l l e d state at a position where (J)(r)  =  0.  Clearly for a single  ion, the Fermi energy  is unchanged.  However, for a f i n i t e density of ions, the Fermi energy w i l l  rise (or f a l l  i f z < 0) as wi11 be discussed s h o r t l y , so we have indicated the p o s s i b i l i t y of a change 6&^. in the Fermi energy in F i g . 31, and w i l l  carry  through  our discussion.  If (SSy + e<M_r) is small, the change in electron density 6p(_r)  is  proportional to 6&^- + e<f>(_T_), according to  6p(_r) = r ( 6 ) [s& +e<|>(r)] f  f  where r ( & ^ ) is the density of k i n e t i c energy states (number of states per unit energy and volume) for the free electron gas:  (2)  r  (  )  £  _L/2ni\  =  2TT  q  Then  using  3 / 2 &  W)  2  equation,  2 2  + |  2  where ion,  last  and the  X  For  the  term,  inverse  = %e r(& )  2  f  is just  localized  (where typical plus  length,  6(r),  i s due t o t h e  is given  X,  by  (5)  0 as  r •+  0 0  ,  and the s o l u t i o n  of  (4) i s  (6)  the  Coulomb  potential.  i o n i s , as e x p e c t e d ,  The t o t a l (using  (2)  number  and  of  electrons  (5))  (7)  f  the volume).  atomic  radius,  energy  of  E, o f k  The s c r e e n i n g  so c l e a r l y  cloud  this  the excess of  interaction  energies  it  atom,  h.  This  of  to the site  energy  with  is  the added  electrons  energy  t h e sum o f  the  and the p o t e n t i a l the  i o n and w i t h  due t o the u n i f o r m c o n c e n t r a t i o n  by t h e j e l l i u m b a c k g r o u n d .  energy  to describe  than  a  the ion  atom.  <5p.(r), —  electrons  X ^ c a n be s h o r t e r  reasonable  E, corresponds  electrons,  the excess  length  is  as a n e u t r a l  in Chapter  interaction  kinetic  function  d v = jT(& )ecJ>(_r) d v = z  is  cancelled  delta  screening  = 0 a n d <J> (r)  E , discussed  energy  the Dirac  Thomas-Fermi  a screened  i t s screening  The gas,  v  (4)  A r  around  /6p(r_)  find  " *mze$(r)  f  involving  ion,  <Kr)=fe-  which  6&  we  .  2  a single  ( 3 )  q  Poisson's  V cj) = A <j>(_r)  i  From  F i g . 31,  at  is just  r  of  the  kinetic  energy  &, +  E of p  one a n o t h e r .  of electrons,  we s e e  that  lattice  the  P,  The is  average  [5& +e<Mr)] / 2 , s o f  the  that E. is just  E  k  = z& + i/(ec|>(_r)+66 )5p(£) dv . f  The potential due to the electrons  E  p  (8)  f  = - / ^ M r )  dy - f j [cj, (r) - ^  is (J) (r) - z e / r , so E  6 p(r) I dv  is  (9)  .  Adding these contributions gives  E = E., + E = z& k p  / ^ • 6 p ( r ) dv  and for the case of a single  (10)  ion, using (6) and 8&  f  = 0,  which is the s i t e energy of the intercalated atom measured with respect the energy of the ion and z electrons  separated at i n f i n i t y .  to  For typical  electron densities in s o l i d s ,  E ^ from (11)  volts  (11) has a minimum (most negative) value as a  in magnitude.  In f a c t ,  is negative, and several  function of electron density, p; for z = 1, this minimum is E  Q  electron  = -7-5 eV,  22 -3 which occurs at p = 1.8 x 10 cm (approximately 1/5 the electron density . - 1 o in copper), which corresponds to = 2.5 eV and \ = 0.71 A. This is a very shallow minimum; the value of E  q  for z = 1 remains within 1 eV of  minimum value as p varies from 0.3 x 10  22  to 6.5 x 10 22  for the electron density of lithium, p = k.7 x 10  22 -  cm  cm  -3  .  In p a r t i c u l a r ,  3 , E  Q  = -7.05 ev, in  (rather fortuitous) agreement with the sum of the cohesive energy and the f i r s t  ionization energy (5-4 eV) of lithium.  its  (1.7 eV)  If (11) were really the  the correct expression for E for an intercalation compound, we would conq  clude that the voltage of an intercalation c e l l depends only on the  electron .densities of  in the anode and in the cathode, and for intercalation  lithium could never be greater than 0 . 5 v o l t s .  Of course, actual  intercalation compounds are far from being free electron gases, and the fact that ( 1 1 ) gives the correct order of magnitude for E  q  is as much as we could  expect.  We can also use ( 1 0 ) to estimate the interaction energy between two intercalated atoms.  This is the difference between the energy when the two  atoms are located at (k)  and r_^, and their energy as Ij^-J^l  Because  is l i n e a r , the total electron density from both atoms is just the sum  of the electron densities of  °°-  from each.  The interaction energy W then consists  the sum of three terms: (a) The energy of the ion of one atom in the potential cf>(_r) of the 2  other atom, /ze 5(_r r^ -  (b)  (_r) dv  The energy of the electron cloud 6p^(_r_) of one atom in the potential of the other atom, -ej^^{r)6p^ (r) dv  (c) The k i n e t i c energy change of the electrons, |/[(f) (r_)+cf) (L)][<Sp (i:)+6p (r)] d v - f / ^ ( r ) 6 1  2  l  2  Pl  (r.)+<|> ( )6p (_r)]dv . 2  L  2  We see that the potential energy (b) is exactly cancelled by the change in kinetic energy (c) , a fact f i r s t noted by Alfred and March  (1957).  As a  r e s u l t , the interaction energy W is just the energy of the ion of one atom in the total potential of the other atom, (a), which is  W = zec\>{ -r_ ) L]  2  which becomes, using (6) for < J > (_r),  (12)  a  screened Coulomb repulsion.  The intercalated atoms strongly repel when  they are closer than a distance of order X \ However, for I r _ — r _ 1-' > X \ —i —z ^ wrong.  At large distances,  results are q u a l i t a t i v e l y  the screening electron density o s c i l l a t e s in  space (Friedel o s c i l l a t i o n s ) Thomas-Fermi results  the Thomas-Fermi  rather than decaying exponentially as the  indicate.  This leads to an interaction between  intercalated atoms that can be a t t r a c t i v e at certain separations.  To obtain  these Friedel o s c i l l a t i o n s , we must replace (h) with the f i r s t order s e l f consistent  equation (March and Murray 1960)  Hartree approximation, or equivalently  for  constant  (Ziman 1 9 7 2 ) .  Trzea,  cos  f  f  (2q  r  2  b  (r_) obtained in the  z e / r with the zero frequency  rather than the Thomas-Fermi  The potential  *(r) - — - — s q (l+2uq a )  2  (and more commonly) by screening  each Fourier component of the ionic potential Lindhard d i e l e c t r i c constant  V cj)  <j> ( r )  dielectric  far from the ion tends to  ,r)  d/,)  L_  3  (Blandin 1 9 6 5 ) , where a^ is the Bohr radius "n /me , and q^. is the Fermi 2  2  x wavevector,  (2mS^)" /ti. 2  For electron densities of typical metals,  (14)  provides a reasonable approximation to the complete Hartree solution for q^r > 2TT, as can be seen in Blandin ( 1 9 6 5 ) ; steeply like the Thomas-Fermi  for q^.r < 2 r r ,  cj>  (r)  rises  solution.  In calculating the interaction W(r) between two intercalated atoms using the Hartree s o l u t i o n , there is once again a cancellation of the change in the k i n e t i c energy of the electrons with the interaction energy of one  electron cloud with the other atom, so (12) March 1961).  still  holds  (Corless and  The interaction energy thus becomes  Tr(ze) a,  cos(2q,.r)  2  W(r) = ze<f)(r) - — ^ — q (l+2Trq a ) 2  f  f  b  -  (15)  — ^ - •  r  3  The f i r s t three minima where the asymptotic solution can be used occur at 22  2 q r = 2.90TT, 4 . 9 4 T T , and 6 . 9 6 T T . , . . . o f  For an electron density of 2 x 10 o o  -3  cm  ,  these minima are at r = 5-4 A, 9.2 A, and 12.9 A, and correspond to values of W of - 0 . 4 6 k T , -0.096kT , and -0.035kT r  r  r  (where T = 25°C) . r  k T / 2 is large enough to cause clustering and/or condensation r  An energy of in a l a t t i c e  gas at room temperature. The discussion so far has assumed only one or two intercalated atoms are present  in our i n f i n i t e l y large free electron host.  In this case, 6&^. = 0.  Eventually, however, as the number of intercalated atoms increases, we w i l l find Sp 5* 0 in regions of the host where 0 = 0, so the Fermi energy must change according to (2).  We w i l l  discuss  this change using the treatment of  the Thomas-Fermi approximation given by Friedel uniform d i s t r i b u t i o n of n intercalated atoms  (1954).  We consider a  (ionized to a charge ze)  in a  volume-V, and focusj.'bur attention on a. single one by dividing the volume v into n Wigner-Seitz polyhedra of volume v / n , each with an intercalated atom at the center,  1  then replacing each polyhedron with a sphere of radius R  given by  irR 3  3  =^ . n  (16)  Each sphere contains one ion and an excess of z electrons over the uniform background, so that'.<j>'(R) = 0 by Gauss' law and dcj)(R)/dr = 0 by symmetry. We can solve (4) subject to these boundary conditions, to obtain  <Hr) = 6 s  ,  A  •1 +  f  XR-1 X(R-r) 2Xr  A  e  ze X XRcoshXR - sinhTXR  +  XR+1 2Xr  -X(R-r)"  (20)  e e  2  6  .^f  =  (21) follows  '  ^  from (20) at r = R, using cb (R) = 0.  For large R, we find  -XR' 6&  F  = 2 z e  2  ^ -  (22) -•2  which goes to zero as R -> °° faster  than R  , and cb(r) reduces to  (6).  The interaction energy of the atoms, W, is n times the difference between the energy per atom at a concentration of atoms correspnding to R and to °° (11).  the energy corresponding to R  W = n<5& |-(XR+l)e~  Using (10), we find  (23)  AR  f  which for large R, using (22), becomes  W - nz e Xe" 2  2  2 X R  «n|o& | f  (2k)  .  This r e l a t i o n , that the interaction energy per p a r t i c l e is much smaller than the s h i f t  in the Fermi energy,  the electrons  is a consequence of the fact that confining  to a sphere of radius R, which increases  their k i n e t i c energy  over the case R -* °°, also pulls them closer to the ion than for R -> °°, so the screening  is more complete.  Thus, there is a large cancellation between  the k i n e t i c and potential energies,  resulting in (2k).  Note that W in (2k)  -2XR  f a l l s off as e  as the separation between the ions increases,  trasted with the interaction between a pair of atoms as e ^1—1 —2 I / 1 r — r | .  as con-  (13), which f a l l s  off  Also note once again that the Thomas-Fermi c a l c u -  lation predicts a purely repulsive interaction; i f the more accurate Hartree  equation were used instead, Friedel o s c i l l a t i o n s  At this point,  i t is useful  with the r i g i d band concept. energy of an electron  should r e s u l t .  to connect these Thomas-Fermi  calculations  In the Thomas-Fermi approximation, the total  in the k i n e t i c energy state q (measured from the  vacuum level) at the position r is & - ed)(r), where & is the k i n e t i c q q energy and -e'<j)(_r) the potential energy. downward in energy by -e cf)(j^) -  Each electron state is  shifted  If we regard the states q as forming a band,  we can say that the entire band is shifted downward in energy by -ecf)(r) without changing its shape to real metals  (i.e.  rigidly).  To apply the Thomas-Fermi  results  (within the limitations of the theory, of course), we simply  regard the states q as the electron states for the metal, and a l l of the above results apply.  This is the meaning of " r i g i d band theory".  not mean that the electron states are held r i g i d l y in place  (i.e.  It does unchanged  in energy) as atoms are intercalated, which would cause the Fermi energy to rise in direct proportion to the number of electrons added to the host band structure; as we have seen, there is no shift  in the Fermi energy at a l l  until enough atoms are added so that overlap of t h e i r screening clouds occurs.  Moreover, when the Fermi energy f i n a l l y does begin to change,  it  changes faster than the interaction energy per p a r t i c l e , W/n, which means that the measured c e l l voltage  in an intercalation battery does not change  in proportion to the change in the Fermi energy of the host, statements in the l i t e r a t u r e .  contrary to  We should also note that the energy of the  electron states in a metal measured from the vacuum level contains a large contribution from the dipole layer at the metal surface For intercalation of a neutral atom (the  (see  Lang 1973).  ion plus its e l e c t r o n s ) ,  the  effects of this dipole layer on the energy required to add the ion cancels the effects on the electrons.  Hence i f we wished to calculate the s i t e  energy of an intercalated atom, we would have to subtract this dipole energy  from the work function of the electrons  in the host before we could use the  work function in place of &^ in the formulas above.  F i n a l l y , we should point  out that exchange and correlation e f f e c t s , which have not been considered, are quite important in an electron gas of metallic densities;  as a f i r s t  approximation, their effect  is to shorten the screening length A ^ over the  value given by  1954).  (5)  (Friedel  Application of these calculations to real  intercalation systems is  complicated by the fact that the calculations assume the electron density in the s o l i d is uniform prior to i n t e r c a l a t i o n , whereas  in real systems this  is d e f i n i t e l y not true (as shown, for example, in the electron density contours calculated for Ti systems of interest,  by Krusius et al  1975).  In most intercalation  such as the t r a n s i t i o n metal dioxides or dichalcogen-  ides, the conduction bands are largely derived from the d o r b i t a l s of the transition metals, and so the conduction electrons are expected to be concentrated near the_transition metal n u c l e i .  On the other hand, the  intercalated atoms s i t on sites near the oxygen or chalcogen atoms, and are several angstroms from the t r a n s i t i o n metal atoms. screening of ionized intercalated atoms w i l l oxygen or chalcogen atoms.  As a r e s u l t , much of the  be done by the polarizable  However, the interaction between intercalated  atoms is s t i l l expected to be short ranged, and the conclusion reached above, that the Fermi energy s h i f t intercalation c e l l voltage,  is not in direct proportion to the  is expected to be v a l i d .  F i n a l l y , consider what happens i f . t h e  intercalated atom continues  bind a l l of its electrons when in the host l a t t i c e .  to  Although this may seem  like a different case than that discussed so f a r , where charge transfer to or from the host l a t t i c e occurs, the electronic interaction between intercalated atoms occurs.in the same way - through the overlap of the atoms'  electron clouds.  The only differences  are in the details of the i n t e r a c t i o n ,  and the language we would use to describe i t .  The interaction is q u a l i -  t a t i v e l y similar to that expected between atoms in free space - it can be a t t r a c t i v e or repulsive, depending on the nature of the outer e l e c t r o n i c states.  For example, an a t t r a c t i v e interaction corresponds to  to form a molecule in free space;  the.tendency  large a t t r a c t i v e interactions cause  c l u s t e r i n g , which corresponds to the formation of a l i q u i d or a s o l i d .  The.  details of the interaction c l e a r l y require an in-depth consideration of the particular guest-host system of  5-3  interest.  Metal - Insulator Transitions Although guest atoms may bind a l l of their electrons at low values of  the composition x, some of the electrons may become itinerant as x increases. In an i n i t i a l l y insulating host,  this  leads to a meta1 -insu1ator t r a n s i t i o n .  The subject of meta1 -insu1ator transitions has been extensively  studied in  recent years, and detailed reviews are available (for example, Mott 197^0 . Since such a t r a n s i t i o n can be generated by the electronic interactions of intercalated atoms, and since a large change in the conductivity of the host might be seen in the charging and discharging of an intercalation battery (through a modification of the r e s i s t i v e  loss),  it is appropriate to review  some of the relevant points of meta1 -insu1ator transitions here.  For s i m p l i c i t y , suppose the outermost electron bound by the guest atom lies in a hydrogenic s-state, with Bohr radius ag given by  . m"e  where m- is the effective  mass of the electron, and K the s t a t i c  dielectric  constant of the host.  This case is relevant to shallow traps in semi-  conductors, a case where metal-insulator transitions have received considerable experimental attention.(Mott l i e s below the conduction band edge  V " ' - ^  1974).  The energy of the electron state  by an energy  •  <2)  Because of electron repulsions, a second electron of opposite spin added to this state w i l l have an energy higher than (2) by some Coulomb repulsion energy U . c  Thus as x increases,  and the electron clouds of the  intercalated  atoms begin to overlap, broadening the electron states into bands, the band due to the state (2) w i l l be f u l l , However, as x increases  still  and no metallic conduction w i l l occur.  further, several mechanisms may bring about  a transition to a metallic state: (a) As proposed by Hubbard ( 1 9 6 4 ) , the bands derived from the state (2) and the state U , above it w i l l eventually merge into one band. occurs approximately when the bandwidth equals U . c  This  In terms of the  density of intercalated atoms p, this condition of band overlap and resulting t r a n s i t i o n to metallic behaviour (since the merged band i s only half ful1)  P  1  /  3  a g  ^  1  is  .  (3)  (b) The band from the state (2) may merge with the conduction band. This also-occurs at p given by (3) (Mattsubara and Toyozawa 1961).  (c) Mott's (1949) proposal for the metal-insulator transition  involves  approaching the t r a n s i t i o n from the metallic side, where the guest atom is singly -  ionized and screened by the conduction  provided by the guest (so the density of the electrons  electrons is also p ) .  The resulting screened Coulomb potential state for A ^ ^ ag as p decreases, insulating state.  e  -A r / r develops a bound  leading to a t r a n s i t i o n  From the expression  5.2(5)  to an  for X, this gives  ( 3 ) once again as the condition on p.  (d) The f i r s t proposal for a mechanism for a metal-insulator transition  appears to have been made by Herzfeld ( 1 9 2 7 )  Berggren 1 9 7 4 ) ,  : . (see  also  based on the Claussius-Mosotti equation for the  d i e l e c t r i c constant K due to an assembly of atoms of p o l a r i z a b i 1 i t y at a density p:  1-4Trpa/3  v  '  This expression diverges at 4?rpa/3 = 1 , leading to metallic behaviour (Berggren 1 9 7 4 ) .  Since a = 9 ( a * ) / 2 for a hydrogenic o r b i t a l , 3  once again leads to the condition  Since the above mechanisms a l l  (.3) for the t r a n s i t i o n  this  density.  lead to the same condition on p, ( 3 ) , for  the density at which the t r a n s i t i o n occurs, controversy s t i l l  exists over the  exact mechanism that drives the observed t r a n s i t i o n in semiconductors.  The  above arguments are complicated i f the atoms are randomly placed in the host, since this. randomness  is expected to cause the upper and lower states in  the band derived from ( 2 ) to be l o c a l i z e d , and in cases (a), or Cb) hopping conductivity  is expected due to electrons in these states when band  merging f i r s t occurs.  At room temperature, the t r a n s i t i o n w i l l  be smeared  out by conduction due to electrons thermally excited out of the s t a t e . ( 2 ) ~ so the observation of a sharp t r a n s i t i o n generally conductivity at low temperature.  Finally,  overall composition of the host only i f the homogeneously  requires studies of  the expression  (.3) applies to the  intercalated atoms are  distributed throughout the host.  Attractive  the  interactions  98  between the guest atoms can lead to c l u s t e r i n g , and ( 3 ) then gives the composition of the clusters  at which the clusters  metallic conduction in the host w i l l  become m e t a l l i c .  then be observed at the  Bulk  so-called  percolation t r a n s i t i o n , when a metallic cluster f i r s t extends throughout  the  host.  A meta1-insulator t r a n s i t i o n has been reported for the compound H W0_ by Crandall and Faughnan ( 1 9 7 7 a ) x 5 curve of this material h.k(3) with a repulsive  is described quite well interaction  intercalation  at x = 0 . 3 2 .  The voltage  by the mean f i e l d  expression  (yU = 0 . 5 3 eV) (Crandall et al  This has been cited as evidence against clustering of the hydrogen and a percolation t r a n s i t i o n  1976);  intercalated  (Crandall and Faughnan 1 9 7 7 b ) .  CHAPTER 6  ELASTIC INTERACTIONS BETWEEN INTERCALATED ATOMS 6.1  Introduction  When an atom is intercalated into a host structure, it pushes  aside  the neighbouring host atoms, which push on their neighbours, and so on, setting up a long range strain f i e l d .  This s t r a i n f i e l d can then act. on  other intercalated atoms, producing a strain-mediated interaction between pairs of intercalated atoms.  This interaction turns out to be a t t r a c t i v e  in a l a t t i c e with free surfaces, and is large enough to produce condensation of  the intercalated atoms at room temperature and above.  elastic  An a t t r a c t i v e  interaction has been proposed as an explanation for the phase  transitions in some metal-hydrogen systems (see In what follows, we w i l l aspects of the e l a s t i c  Section 3-4).  discuss both the long  range and short range  interaction between intercalated atoms.  We describe  the host l a t t i c e as an e l a s t i c continuum; a l a t t i c e description is  also  possible, but to be applied it must be translated into the continuum approximation for a l l but the simplest problems (Wagner and Horner 1 9 7 * 0 .  F i r s t we b r i e f l y review continuum e l a s t i c i t y Sokolnikoff  1956).  theory (see,  for example,  Consider a system consisting of a linear e l a s t i c medium  together with sources of body forces _f_(_r) and surface forces _f ( r ) . S  When  the body and surface forces act on the medium, the total energy of the system (medium plus forces)  E = i/c.  i k £  is changed by  £ ; j ( r ) e ( j : ) dv - / f j ( r ) u . ( r ) k£  dv - / f * ( r ) u . ( r )  dA  (1)  where the f i r s t two integrals run over the volume v of the system, the third  over the surface A, and a l l 1 to 3of  The tensor C j j ^  '  repeated s  (Cartesian)  the e l a s t i c  indices are summed from  s t i f f n e s s tensor.  The displacement  the medium in the direction i at position r_, u.(r_), is related to the  s t r a i n e..(r) by U /3u.  with x.  3u.\  the Cartesian components of r_.  the forces  The actual value of u^ produced by  is that which minimizes E; for this displacement f i e l d ,  (1)  becomes  E  " -*/ ijk£ ij r c  £  (  ) £  k£ ^ (  d  v  = -i/f.(r_)uy(r_) dv - ± / f ; ( r > . ( _ r ) dA . I I A  (3)  The stress a . , is qiven by IJ  a  :ij: (-n' = ~ v Be..  c  v  'J  and  ijk£ kJl £  f=f =0 S  satisfies  3a!. L L  3x.  =-f (r) i  (5)  :  J  within the medium, and  a..n. IJ  j  (6)  = f? i  on the surface,  n is a unit vector normal to the surface.  We are interested  in a special  type of body force - that produced by  an intercalated atom or some other type of point defect  (a general  discussion of point defects can be found in Liebfried and Breuer Point defects exert  1978).  local body-forces f_, called Kanzaki forces, on the host  near the defect, with the property that the net force, Jf_ dv, and "torque, Jr_ x f_ dv, vanish.  Because of these conditions on f_, the strain f i e l d  far from the defect which these forces produce can be completely charac- ' terized by the f i r s t moment of the forces,  P.^, defined by  P. . = P . . = J x . f . dv ij ji i j  (7)  J  This is analogous  in e l e c t r o s t a t i c s  to the f i e l d from a charge d i s t r i b u t i o n  of zero net charge, which can be characterized by the dipole moment of the charge (Jackson  1975);  because of this analogy, P.^ is often  as the e l a s t i c dipole tensor.  referred to  P . . can be determined from the observed 'J  strains in a host with free surfaces which is uniformly intercalated to a number density p; this strain is  - ' J  =  s  ijk£ k£P P  where s . . . „ is the e l a s t i c compliance tensor i J kx, r  cases, the shear strains are small systems can be c l a s s i f i e d  (e..  (inverse of c . . , „ ) . i j k£  - 0, i 4 j).  In most  In a d d i t i o n , many  into one of the following three cases:  I. Extension in one direction only £^  4 0 , e . j = 0 otherwise.  I I. Equal extension z  11  = e  22  IJ  = £„_ =  This case applies to most channeled R  R  including the r u t i l e s .  III. Equal extension 22  in two directions  4 0 , e . . = 0 otherwise.  structures,  11  This case applies to most layered compounds,  33  in a l l three directions ^ 0 , e . . = 0 otherwise. IJ  This is the case of the  102  "dilation  We w i l l  discuss  Because properties forces delta  sphere",  and applies  these  three  P.. contains U  cases  only  functions.  i n some  systems  a first  atoms,  moment.  An example  need  we w i l l  henceforth  The f o r c e s  a r e then  i s shown  such  as H Nb.  detail.  a l l t h e i n f o r m a t i o n we w i l l  of the intercalated  f_ h a v e  to metal-hydrogen  about assume  the elastic that  the  o p p o s e d 'pa i r s o f  i n F i g . 32, f o r t h e case where  P.^ i s  diagonal:  p P =  The  forces  f  1  0  2 0  \  0  0  r  (9) J  0  that  3/  produce  = Um  this  dipole  2  a r e o f the form  [fiCx^b) - 6( +b)]  K iS(x ) S(x ) 1  tensor  3  (10)  Xl  b'+ 0 2K.b=P, i 1  with  similar  indicates  expressions  that  for  an<  the intercalated  ^  The diagonal  fy  atom e x e r t s  nature  no shear  forces  of P in  (9)  on the host  lattice.  We n o w c o n s i d e r at  r_ a n d r_ . atoms  between  F o r s i m p l i c i t y , we c o n s i d e r  2  calated  the interaction  are described  by t h e same  two i n t e r c a l a t e d only  dipole  the case  tensor  atoms  where  P . . .  located  the inter-  I f one uses  (3)  IJ  to two  calculate terms  action  the energy  describing  energy  W ^ l W  -/ i f  one finds  the interaction with  W ^ given  =  o f t h e two atoms,  2 )  surface  by  <I' !(il) u  d  v  ' =  - ij Jj^ P  e  ) 2  two s e l f  forces  energy  f_ , a n d a n S  terms, inter-  103  where f_ of  (2)  1 is the force due to the second atom, u_ the displacement f i e l d  the f i r s t atom, and e!.(r_) the s t r a i n f i e l d of the f i r s t atom at the i j —2  position of the second atom.  The second equality in (11) follows from the  del ta funct ion form we have assumed for the forces _F_. It is convenient  to  divide the displacement f i e l d u_ (and the s t r a i n £ . ^ ) into two parts (Eshelby 1956): u_ , the displacement which would be produced in an i n f i n i t e medium; and u , the additional displacement needed to satisfy the boundary 1  conditions at the surface of the host (the superscript I is intended to imply that u_ is due to image forces).  In general, for an atom at r_= 0,  2  00  u_ f a l l s off as 1/r , diverging at the position of the atom (in continuum theory), while u  1  gives rise to a s t r a i n  which is slowly varying over  the volume of the host and is proportional to 1/v. contributions to the total s t r a i n , ^^L L ) r  2  = °°(jl) w  +  w  I  (L L r  In terms of these two  can be written as  )  (  1  2  2  where r_ = r_^ - r_^. We now discuss these two terms separately  in the  following two sections.  -  6.2  oo  »  Infinite Medium Interaction W  The displacements produced in an i n f i n i t e medium can be conveniently written in terms of the Green's function G.j (£."£.'), which gives the displacement in the direction i at position _r due to a delta function force density in the direction j at r_ . l  For the idealized defects we are  considering, which involve opposed pairs of delta function forces 6.1(10)), the displacement uJ (r) tensor P.. at r = Q is jk  (as in  due to an intercalated atom with dipole  )  105  i ^  u  = -  p m  k  ^ -  from which the strain  e  ?.( ) r  ij  —  -  =  i P  M  is  8 G . , (r) \  (r)  /VG., '  k  -  mk\tfx.9x \ j m  2  +  J k -  dx.dx i m  .  (2)  / /  Hence the interaction energy between an intercalated atom at _r and one at _r = 0 i s , from 6.1 (11) 3 G.. 2  J  (r) m  00  The of  interaction W (rj is repulsive or a t t r a c t i v e depending on the direction r_, and averages to zero over the sphere of any non-zero radi us,  dently of the anisotropy of the medium or the form of P . . Breuer 1 9 7 8 ) .  indepen-  (Liebfried and  Along a given d i r e c t i o n , the magnitude of W (r) f a l l s  off  as 1 / r . 3  oo  To  i l l u s t r a t e the angular variation of W ( r ) , we consider an isotropic  medium, for which the Green's function can be found e x p l i c i t l y  r G  (r) -  ij^  JL  - 8TTY  1 + v  6..  x. x. _L±  r  (Love 1 9 ^ 4 ) :  +  (3-4v-)--U-  3  where Y is Young's modulus and v is Poisson's  ratio for the medium.  In addition, we choose P . . to be diagonal, as in 6 . 1 ( 9 ) , with P^ = P^. This form of P . . , with suitable choice of the ratio P , / P , , allows us to ij 3 1 discuss the three cases given in Section 6.1, corresponding to extension in one, two, or three directions respectively materials.  in uniformly intercalated  The relations between P^ and P^ for the three cases are found  106  by applying the conditions on the average strain £ „ for the three cases to 6.1 (8) ; we f i n d :  =p = i V  i.  2  -  (1-2v)(1+v)  =  33  M . P  1  £  =  11  2  = ^  -  =  e  I I I . P  ^11  -  ( 1 - V ) Y  = P  -  ( 5 )  3  =  £  22  '  P 3  (1-2V)(1 V)  =  The expressions  ^33  =  £  =  22  0  -  +  = P  11  Q  6  )  [  b  )  1  -  '  p  33  (9)  .  3  ^"y ^ iP"  =  2V  P  '  ^  for W (r_) which follow from (5) -  1 0  ^  (10) are most conveniently  written in terms of the average s t r a i n per unit concentration where p  (  (7)  Y  2  £  3  22  - P  1  P  H  is the concentration corresponding to x = 1, and £  q  £ 0  /P >  =  0  £  /P>  is the non-zero  component of the s t r a i n at that concentration for each of the three cases. Using (3),  (4),  and (5) -  (10), we find the following expressions  for  CO  W (r_), written in terms of the polar angle 0 measured  I.  w°°(r) =  87i(l-v )r 2  II. vT(r) = 8^(1-V )r 2  III. W°°(r) = 0  (^,(15005*9-6cos 0 \ o/ 2  3  3  from the z - a x i s :  - 1)  (11)  p  ( —) \ o/  (iScos^e - 6(3+2v)cos 0 + 3+4v) 2  (12)  p  (13)  107  Polar plots of W (r) for cases I and II are shown in Figures 33 and 34 respectively.  Note that in case I the shape of the curve is independent of ratio v while for case II the shape d i f f e r s  the value of Poisson's different v .  It is clear for guest atoms in layered compounds (ease l)  that the interaction is a t t r a c t i v e  in the layers and repulsive normal  the layers, while in channeled hosts (case II) the interaction is along the chains and repulsive normal to the chains. schematically  for  in F i g . 35.  to  attractive  This is summarized  For d i l a t i o n centers (case III)  the  interaction  is i d e n t i c a l l y zero. 00.  .  In an anisotropic medium, W (r_) w i l l  have q u a l i t a t i v e l y similar  as for the isotropic case; the e x p l i c i t calculation of W (r) much more d i f f i c u l t .  is,  features  however,  One case which has been treated is that of an inter-  calated atom with dipole tensor P.^. = . ^6 j ^ in a very anisotropic medium.(s__ » s . . . where s.. is the e l a s t i c compliance tensor in the 33 11 ij abbreviated two subscript notation).  hexagonal  This case was discussed by.Safran and  Hamann (1979) in their treatment of intercalation of graphite; they found: 1 - (l+2a: )cos 9  e Y_ 2  W (r)  =  3  1  -  T  torpVa In relating P to  1  L  2  =TF7T  T  •  0*0  Q - (l-a )cos eJ ^ 1  2  5 /  3  it has been assumed that the Poisson's  ratio  v_ = -s._/s__ = 0 , which appears to have been assumed by Safran and 5 13 55 Hamann (1979); i f this modulus Y^ = \/s^y  is so, then a^ = c ^/c^ = s^/s^y  The Young's  If (14) is compared to the isotropic result for case I  (11), one finds that the anisotropy increases the repulsion along the z-axis while decreasing the s o l i d angle over which the interaction repulsive, and decreases the attraction in the xy plane. q u a l i t a t i v e features of F i g . 33 remain unaltered.  is  However, the  108  Fig. 33 - Polar plot showing the angular variation of the strain-induced interaction W°°(jr) , between intercalated atoms in layered compounds. The interaction is attractive within a given layer (G = 90°) and repulsive perpendicular to the layer (8 = 0 ° ) . The c r i t i c a l angle at which the interaction changes sign is about 4 3 .5° for a l l Poisson r a t i o s .  109  Fig.  34  -  Polar  plot,  similar  to  Fig.  33,  for  rutile-related  The i n t e r a c t i o n i s a t t r a c t i v e a l o n g t h e (6 = 0°) a n d r e p u l s i v e p e r p e n d i c u l a r t o T h r e e c u r v e s a r e shown c o r r e s p o n d i n g t o  compounds.  i n t e r c a l a t i o n channel t h e c h a n n e l (9 = 9 0 ° ) . different Poisson ratios,  (a) v = i , 6_ % 60.3°; ( b ) v = 1/3, 6 % 61.0°; ( c ) v = 0, 6 ^ 63.4°. C  C  Layered Compounds  Rutile Related  X  Compounds  o  F i g . 35 - Schematic summary of the nature of the strain-induced interaction W°°(-r) between two intercalated atoms in layered and r u t i l e related compounds.  To estimate  the magnitude of the energies  example: the intercalation .compound Li^MoO^. for  involved, consider a s p e c i f i c The sites in MoO^ available  the intercalation of lithium are presumably the tetrahedral sites  lying in the tunnels along the pseudotetragonal c axis of the monoclinic o MoO^ c r y s t a l , which are spaced by a distance of c/2, where c = 2.81 A (see F i g . 8).  The chains of sites are arranged in a square  lattice,  '_ o separated by a distance a//2, where a = 4.86 A is the pseudotetragonal a l a t t i c e parameter.  During i n t e r c a l a t i o n , c stays nearly constant, while  a increases by Aa = 0.34 A as x varies from 0 to 1 (Sacken 1980). MoO^ is a good example of case II discussed iabove. p  _ o  o  = 1/33 A  are  The reference  , and the reference s t r a i n is e = Aa/a = 0.069. o  no; ipublished e l a s t i c constants for MoO^, we assume i t  i s o t r o p i c , and assign i t e l a s t i c constants Y = 10  Hence density  Since there  is reasonably  ergs/cm , v = 1/3.  Then from (12) 5.6kT  W°°(r) =  - (IScos^e - 22cos 9 + 13/3)  (15)  o where kT = 25-7meV and r is measured in A. r  Along the chains the interaction  2  r  3  is a t t r a c t i v e and equal to -5.4kT , -0.7kT , -0.1kT for f i r s t , secondhand r r r third nearest neighbours respectively.  Perpendicular to the chains the  interaction is repulsive, the sequence for 8 = 90° being +0.6 kT^, +0.2 kT^, +0.1  kT^...  The interaction is appreciable only for a small number of  l a t t i c e spacings; at these short distances,  however, continuum theory can  give at best only order of magnitude estimates.  These energies a « e large  enough to cause pairing or clustering of atoms along the chains, assuming the e l a s t i c a t t r a c t i o n is not overwhelmed by electronic repulsions. calculations give similar conclusions: Fisher (1958) calculated that a t t r a c t i v e interactions of order 0.1 eV (4kT^) are expected  between  Lattice  i n t e r s t i t i a l atoms along certain directions in iron.  The Image Interaction vfl-  6.3  Since of  is due to the image displacement,  the host,  u^ , i t depends on the shape:  boundary conditions, and the position of the  atoms in the host.  intercalated  In general, it depends only weakly on the  relative  positions of the intercalated atoms, and its magnitude is inversely proportional  to the volume of the host.  Unlike W°°, its angular average is nonzero,  leading to a net attraction or repulsion (depending on the boundary conditions) when averaged over a l l d i r e c t i o n s . The shape dependent behaviour of W makes calculations complicated i f 1  we are interested of  in the details of the interaction for arbitrary  the guest atoms in the host.  Hence we consider the simplest  an isotropic sphere of radius R with two d i l a t i o n centers  positions  example:  (case III  above),  one at the center of the sphere (r = 0) and the other at an arbitrary position r.  The i n f i n i t e medium component of displacement due to the atom  at r = 0 i s ( L i e b f r i e d and Breuer 1978)  £  { R )  =  P  -  (1*V)(1-2V)  ±_  4TTY(1-V)  r  For a f i n i t e sphere,  .  ( 1 )  3  (1) alone does not satisfy  the boundary conditions in  general, so we need the other solution of the equations of e l a s t i c i t y this case, which is linear in r (the image term). surfaces,  iv-  the image term is  p  ,sTTRJF-=r K  in  For free and clamped  (Liebfried and Breuer 1978)  (free)  <2)  ^  (  1  )  -  -  ^  ^  ^  (cWe„ .  ^  (3)  K  In these two cases, W (r) 1  is  independent of the position of the second atom.  In terms of e and p , we can write W = U, where U is given by o o 1  7  e  U = - l-^-— N 1-v p  ..  2  (k)  (free) o  . 1 (1+V)Y o H ( l - v ) ( l - 2 v ) p~J  . . (clamped)  ... (5)  .  We see that W^" is a t t r a c t i v e for a free surface and repulsive for a clamped one.  Moreover, even though W is 1  inversely proportional to the sample  size through N in the denominator of will  (4) and (5), the atom at the center  interact equally with a l l the other atoms intercalated into the host.  Calculation of W" '" for other positions of the two atoms, or for other sample shapes,  is considerably more complicated than for the above problem.  However, for a reasonably uniform d i s t r i b u t i o n of intercalated atoms, most of  the complications average out.  In f a c t , we can obtain those terms which  are dependent on the boundary conditions quite simply. once again, written e x p l i c i t l y strains  Consider 6.1(1)  in terms of the displacements  u_ (r), a  the  and the forces _f (r) of atoms on the sites labelled by a in  the host:  E = J n n l i f e . .. e ? . ( r ) e f ! (r) dv - Jf?(r)u?'' (r) dv , a a L ' ijk£ i — i — aa -> I J — •> kl — J  0  L  1  - J n f f f ( r ) u ? ( r ) dA a\ i — i — a A  (6)  L  In what follows, we w i l l  be interested  a free surface, where f  = 0, and a clamped surface, where the total  S  in two types of boundary conditions:  displacement £n u (r) = 0 on the surface. a  For both of these cases the  last  a  term in (6) is zero.  In evaluating the sums in the f i r s t two terms in ( 6 ) ,  we find s e l f energy terms ( a = a 1 ) , which contribute to the s i t e energy in the l a t t i c e gas models of Chapter 4 , and local f i e l d correction terms s i m i l a r to those found in the theory of d i e l e c t r i c s  (Kittel 1971), which arise from  the short range correlation of the occupation of the sites Alefeld 1971).  (Siems 1970;  These terms involve the strain of an atom on a given s i t e  near the s i t e i t s e l f , and so do not depend on the boundary conditions (Wagner 1978).  The terms that do depend on the boundary conditions are given by  a straightforward replacement of the occupation numbers n^ by the coarse grained density p (_r) of the intercalated atoms, together with a conversion of the sums in (6) to integrals; we denote these terms by W^, given by  p  ijka u ^ e  where £j.(_f_)  £..(r) 11—  (  ) e  (  is the total  (7)  Pjj-e. . (j_)p(_r) dv  k£ ^  s t r a i n at _r, given by  (8)  = T n (r) a 11 L  The last term in (7) follows The stresses a  from the delta function nature of the  forces.  (r) are defined by "J ~  , v 1/3F a . . (r) = — h— 1j v de  —  (9)  ' T,p;f =0 S  J /  where F is- the free energy of the intercalated host.  Note that since we  are now considering the intercalated atoms in addition to the host, we do not set  the forces they exert to zero in (9) ( c f .  6.1(4)).  For a given p ,  the resulting strain e.. minimizes F; since the only terms in F which contain ij  £..  are those from W , we have 1j e'  uW  = 'uufuU  a  w h e r e a., ( r ) 'J -  do.  -  p  i j  p  (  ^  (  1  0  )  satisfies  U _ 0  (11)  9x. J  (assuming atoms)  there  within  a..n. 'J  on  surface  minimize  W  e  =  The  e..(r)  p  W  of  = 0,  so  =  —  the  i  is  ft  ( c f .  from  intercalated  (4)  -  For these  (6)).  strains  which  (13)  on W e  can  distribution  strain  For a is  clearly of  free  given  be  seen  from  i n t e r c a l a t e d atoms surface  and a  For a  (13). (p(r_) =  uniform  clamped  p),  distribution,  by  i j kx, k £  v  e a r l i e r without  ijk£ ij k£p  s  P  P  x = p/p > Q  2  e  aside  i " . . = s . .. „ P . „p"  = iN U x  U  forces  d v  so W =0. e  ij  Introducing  where  strain  uniform  and  = -  W  body  becomes  we q u o t e d  e  normal  e  = 0,  e. . ( r )  (which  unit  - i / i j i j (£.)p(jr_)  and a  i j  of  (12)  W , W e e  —  o".j(r_)  sources  medium, and  with  effects  surface  IJ  the  no o t h e r  = f! • i  J  the  are  in 6.1(8)),  and W  g  becomes  (15)  v  where  p  Q  = N/v a s b e f o r e , we c a n r e w r i t e  (15)  as  (16)  2  defined  2  proof  (14) '  by  V ^  i j  P  P  We see that discussed  k * i J k A s  (16)  e  between the intercalated atoms.  change in this  7  )  with an effective  given by (17)  is  the  interaction on going from a free to a clamped surface.  the three special  cases considered in Section 6.1,  '•  U  e  "•  U  e • - N (1 V) (1-2V) V  For  becomes  N (1+V)(1-2V)  (  2  +  ™ >  (  1  9  )  o  e  2  III. U = - 1-3Y -2. e N 1-2v p o Note that  1  is of the same form as the i n f i n i t e range interaction  in connection with mean f i e l d theory in k.k(2),  interaction U  ~  (  (20)  (20) agrees with the difference between (h) and (5) for two  d i l a t i o n centers in a sphere.  The magnitude in the change in interaction on going from a free to a clamped surface can be very large in intercalation systems. consider once aqain the case of Li Mo0 . x 2 o  3  the estimated e l a s t i c  others,  Usinq the values of strain e and o 3  constants given in Section 6.2,  which is the effective total  As an example,  we find NU = -17-7kT , e r  interaction energy of one atom due to a l l  the  and hence the parameter which would be used in l a t t i c e gas models.  This is a very large quantity, and we conclude that boundary conditions play a crucial  role in determining the magnitude of the interactions between  intercalated atoms.  6.4  Lattice Gas Models and E l a s t i c  The e l a s t i c  Interactions  interaction energies discussed  contribute to the interaction energies U Chapter 4 (see 4 . 4 ( 1 ) ) .  in the previous two sections  , in the l a t t i c e gas models of  The i n f i n i t e medium terms W produce f a i r l y short  ranged interactions, and because of t h e i r anisotropic form, they tend to lead to clustering of intercalated atoms along the channels of a channeled structure or in the layers of a layered compound. lead to staging  in layered compounds, where f u l l  ones, or s i m i l a r ordering effects  in f a c t , 6 . 3 ( 1 6 )  layers alternate with empty  in channeled structures, where f u l l  channels are surrounded by empty ones. interactions;  They are large enough to  The image terms produce long range  is precisely of the form of the i n f i n i t e range  interaction y l ) with y = N introduced in Section 4 . 4 .  It should be pointed  out that the local f i e l d terms mentioned in Section 6 . 3 tend to reduce the attractive  interaction U from the value given in 6 . 3 ( 1 7 ) , but they do not  overwhelm it  e  in the calculations which have been done to date (Wagner 1 9 7 8 ) .  As a r e s u l t , the image terms are expected to lead to unphysical regions in the free energy and chemical p o t e n t i a l , as in F i g . 1 4 . Phase separation according to the Maxwell construction w i l l occur i f a low energy can be formed between the two phases.  interface  Such a low energy interface w i l l  be  formed in a s o l i d i f the two phases actually break apart, or i f an incoherent interface is produced by d i s l o c a t i o n s . completely coherent interface),  (the crystal  lattice  However, i f the interface remains remains continuous across  the  the e l a s t i c energies associated with the so-called coherency  stresses required to hold the two phases, of different  l a t t i c e parameters,  together becomes too large for this simple phase separation to have the lowest free energy.  In these cases more complicated d i s t r i b u t i o n s p(_r_) of  the intercalated atoms, called density modes, are produced (Wagner and Horner 1974).  Real systems are expected to be intermediate between the completely coherent and completely incoherent cases, and p l a s t i c deformation.  Because of t h i s ,  involving both coherency stresses the chemical potential of the  intercalated atoms as measured by the voltage show hysteresis in F i g . 3 6 .  (a)  in an intercalation c e l l  will  over a charge-discharge cycle as indicated schematically  Such hysteresis  is produced in two ways:  Energy is lost as interfaces  between phases move and the crystal  is  p l a s t i c a l l y deformed, in analogy with the losses associated with the motion of domain walls  in ferromagnets.  in Section h.3,  (b) As discussed  the p a r t i c l e s in an intercalation cathode  act as a "chemical potential bath", preventing the voltage of a p a r t i c u l a r p a r t i c l e from relaxing to the equilibrium value once the phase separation begins and the coherency stresses relax by p l a s t i c deformation.  Such hysteresis  effects are to be expected  in a l l condensation phenomena  ( f i r s t order phase transitions) which involve two coexisting phases of different  l a t t i c e parameters, whether or not the phase t r a n s i t i o n is actually  produced by the e l a s t i c  interaction.  There i s , however, a special case where the coherency stresses  associated  with the phase t r a n s i t i o n are zero, and so the phase separation should occur with very l i t t l e hysteresis.  This is the case of invariant plane s t r a i n ,  which occurs i f the strains produced for uniform intercalation of a host (6.30*0)  with free surfaces  are such that they leave a l l planes perpendicular  to some direction (with a unit vector ft, say) e  i1' 22' £  a n c  ' 12 £  a  r  e  a  ^  z e r o  >  undistorted.  For example, i f  then fi = (0,0,1) is such a d i r e c t i o n .  case, the two phases-can separate  In this  into thin plates normal to ft without  119  Fig. 36  Schematic discharge curve of an intercalation c e l l showing condensation due to attractive interactions between the intercalated atoms; (a) Maxwell construction, (b) real system showing hysteresis.  120  producing any coherency stresses.  To see t h i s ,  consider the following.  Cut the unintercalated host into thin plates whose normal is along n. calate some of these plates to a composition  Inter-  and the rest to x^ (where  x.| and x^ are the compositions of the two phases), allowing the plates to expand f r e e l y .  Then f i t the plates back together again.  perpendicular to n remain unchanged, the l a t t i c e  fits  Since planes  into the same registry  as before intercalation with no coherency stresses required.  If the plates extend a l l the way to the boundary, there Is no r e s t r i c t i o n that determines their thickness.  In a real system, however, there w i l l  be  regions where the plates of phase 1 terminate in a region of phase 2 and vice versa, as in F i g . 37-  The e l a s t i c properties of such a region are those of a  dislocation loop surrounding the plate of an edge dislocation is essentially Since p a r a l l e l dislocations  (in f a c t ,  the standard textbook example  that in F i g . 37a - see Kittel  repel, plates of one phase w i l l  1971)-  repel one another  i f placed d i r e c t l y above one another along fi, and attract one another i f placed side by side, as indicated in F i g . 38,. These considerations  Fig.  37 - E l a s t i c equivalence of (a) a plane of intercalated atoms, f i l l e d c i r c l e s , to (b) a dislocation loop.  imply that  the  121  att ract i ve  F i g . 3 8 - Interaction between two dislocation  the plates w i l l  loops  tend to be as thin as possible, one atomic layer, and as far  apart as possible, which is equally spaced in the direction n. explanation has. been recently proposed for staging  Such an  in graphite (Safran and  Hamann, 1979).  The graphite system has an additional complication, however - the atomic layers of carbon shift  during intercalation (see  Section 3 - 3 ) .  that there is an additional driving force for condensation  This suggests  in the  layers,  beyond the e l a s t i c energy considered above, namely the free energy decrease due to this structural rearrangement. have different  However, as long as the two phases  l a t t i c e parameters, the above arguments s t i l l hold and s t i l l  provide a possible explanation of staging  in spite of this complication.  These considerations also suggest a reason for the absence of staging in other layered hosts: in these materials there is often a non-negligible expansion of the basal plane during i n t e r c a l a t i o n , and so there is no invariant plane s t r a i n .  6.5  Chemical Potential  in Nonhomogeneously  Intercalated Hosts  On calculating the behaviour of an intercalation battery at non-zero currents  (see  Part C) , it  is necessary  to know the chemical potential  the intercalated atoms at position _r, y (_r) , in a host where the  of  composition  x varies with r_, because the voltage V depends on the chemical potential the surface of the host (see with r_ determines  Chapter 7 ) , and because the variation of y ( r )  the diffusion of the guest atoms in the host (Chapter 8 ) .  Normally, one expects the chemical potential the  local composition x(r_); that  to depend on r_only through  i s , y (r_) = y(x(r_)).  effects are important, the term W ( 6 . 3 ( 1 3 ) ) w i l l g  bution y  t o y which depends on the details  g  at  or p (_r) , throughout the sample; that  However, when e l a s t i c  give rise to a c o n t r i -  of the total  d i s t r i b u t i o n x(_r),  i s , y depends non l o c a l l y on P (_r) .  To discuss t h i s , we divide the free energy F of the  intercalation  compound into two parts, W and the remainder F : g  F  =  F  o  Q  + W . e  (1)  The term F leads to a contribution y to y which depends o o  1 1  8 F  o y (r) = — - K — o — N ox /  \  l o c a l l y on x:  (2)  The e l a s t i c term W gives rise to a term y in y; y is found by c a l c u l a t i n g e e e the variation 6W caused by a variation 6p(r) over some infinitesimal e — volume v  r  about the point r: — 6W 1  (3)  e  r Because the change <5p(_r) produces  long range strains ^ j j » e  the variation <$w  calculated from  w i l l depend on the total d i s t r i b u t i o n p, not just  6.3(13)  the  value at _r. As a s p e c i f i c example, consider a d i s t r i b u t i o n of d i l a t i o n centers (case III, P . . = P6. .) 'J iJ  in an i sotropi c medi um with a free surface.  The  relation between stress, s t r a i n , and density of intercalated atoms in this case is  a  (from  ij ^ (  6.3(10))  " ijk£ k£ l' c  £  This is identical  "  (  ij ^ (  =  c  {  in form to the relation between s t r e s s , s t r a i n , and temper-  ature T in thermoelasticity  a  \f L>  PS  ijkA kA ^ e  (Landau and L i f s c h i t z 1 9 7 0 )  " ^ i j  (  1  (5)  ^  where a is the thermal expansion coefficient modulus.  Thus we can use the solutions  and K =  Y/3(1 V) -  the bulk  to thermoelastic problems to  intercalation systems i f we make the substitutions  Ka -> P arid T •> p .  discuss In  p a r t i c u l a r , for an isotropic sphere of radius R with a spherically symmetric distribution 1970  pg.  I  p(r), the radial displacement u^(r)  is  (Landau and L i f s c h i t z  22):  \  P  1+V  -Lf ,r 2  p(r) r  2  dr  2 ( 1 +  "  0  This can be used to evaluate W in e  >  1+v  6.3(13),  (l+v)Y (1-2v)(1-v)  where we have eliminated P in terms of e chemical potential  2 V  -  r  r  /  R  P(r) r  2  dr  (6)  L o giving  (7)  and p . The contribution u to.the o o e is then found from ( 7 ) and ( 3 ) to be  2  (l+v)Y o / f v - ( l - 2 v ) ( l - v ) IT [  c\ e  y  2(1-2V) - \ V n * )  e  ( r )  x  =  i  r  )  +  i th the average composition x = v  / p ( r ) dv.  Wl  (8)  The chemical potential at the  radius r depends not only on x(r) but also on the total amount of through x.  For a uniform d i s t r i b u t i o n x(r) = x, y e  intercalate  = NU x"with U qiven e e  by 6.3(20).  The nonlocal behaviour of y through y^ destroys  the one-to-one corres-  pondence between the surface composition x^ = x(R) and the voltage V (which is determined by the chemical potential at the surface y ). s important in transient experiments  This can be  intended to study diffusion  in the host  Chapter 9) where a small composition change at the surface is produced  (see  by incrementing y^ by A y . g  After a long time, when the composition has  changed uniformly throughout the host by Ax = Ax , we have from s  (8)  ^.-^-feVVsir —^ • 1  dx For short times,  x = 0,  3  A  +  e  however, when only the composition near the surface has  changed, so that  ^ s - U i T -  <9)  y  (8) gives  s  (  NU.'  dx  e  where  U  e  u  1 v +  3TT^y  '  e The quantity U V U  varies from 1/3 to 1 for v varying from 0 to i ,  equal to 2/3 for v = 1/3-  and is  Since U can be many times k T , as we saw in E  r  1  0  )  Section 6.3, this can be a significant effect, where 8U /8X + NU is small (recall  is negative, while 3u/8x  e  q  is always p o s i t i v e ) ;  especially at compositions  in such a case,  the i n i t i a l  change in  may be consid-  erably smaller than Ax^ after a long time, so that the surface composition is varying with time, despite the intentions of the experiment. diffusion  Also,  since  is driven by gradients in u , the diffusion coefficient w i l l depend  on the details of the d i s t r i b u t i o n of the intercalated atoms and, as it turns out, on the shape of the host diffusion coefficient  (Janssen 1 9 7 6 ) .  This dependence of the  on the macroscopic details of the system has been  observed in some meta1-hydrogen systems (Tretkowski et al 1977) and cited as evidence that the e l a s t i c  interaction is responsible for the phase  transitions seen in these systems.  Limitations of the Theory  6.6  The above theory of the e l a s t i c  interaction is based on several  assumptions, which we w i l l now discuss. (a)  Infinitesimal strains The relation between displacement and s t r a i n , 6.1(2), neglects terms  of order e . 2  For strains of order 0 . 1 , as seen in some intercalation com-  pounds, these neglected terms may be of order 1 0 % of the terms linear in e. (b) Hooke's law The relation between the e l a s t i c energy and the strains that we used, 6 . 1 ( 1 ) , neglects cubic and higher order terms in £ .  The expression for the  e l a s t i c energy E in a volume v is actually of the form  v  2 C  i jk£ i j k£ £  £  +  3~ ijl<£mn ij k£ mn C  e  £:  e  +  The cubic term is appreciable for strains of order 0.1 and higher.  For  example, for a s t r a i n - k e e p i n g only the quadratic and cubic terms in (1), we have (in the abbreviated index notation)  7 "  * 11 11 C  £  I  1  +  3  c  ) •  n  (  2  )  In typical metals, the second term is equal to the f i r s t for strains between 0.3 and 0 . 6 , so the cubic term can be 30% of the quadratic term at strains of 0 . 1 ; however, the quartic and higher order terms tend to compensate the cubic terms, so the actual error involved in using the quadratic terms is somewhat less (Liebfried and Breuer 1978).  Anharmonic terms are more important, however, in layered compounds such as the transition metal dichalcogenides where the forces binding the adjacent chalcogen layers together are of the weak van der Waals type.  If we assume  that the interaction energy between chalcogen atoms is of the usual 6-12 type, then the interaction between planes of atoms separated by a distance r will  be of the form (written per unit volume)  -  v  = — - — r  io  r  (3)  -  where A and B are constants.  We determine these constants by requiring that  the minimum in E lies at the observed separation of the atomic planes and that the leading term in the expansion of (3) about this minimum be the I f we assume that the challcogen-metal-cha 1 cogen  usual e l a s t i c energy, i ^ e ^ •  sandwich.thickness remains fixed as the planes pull apart, so that only the thickness of the van der Waals gap changes, we find that  E V  Y  3  2V  (A) ,  1 + T £  33/  \  1 + Y £  33,  where y is the ratio of the c l a t t i c e parameter (per layer) to the distance between adjacent chalcogen atoms (y % 2 t y p i c a l l y ) and  is the Young's  modulus associated with expansion normal to the layers.  The change in ( 4 )  fo r an expansion of 10% ( e ^ = 0 . 1 ) is only about k0% of the change in the quadratic term i\'^e33*  ^  '  s  interesting to note that the magnitude of the  cubic term in an expansion of [k) in powers of e the quadratic term at e  is actually larger than  = 0 . 1 and has the opposite sign, but is largely  cancelled by the higher order terms.)  (c) Dipole moment tensor P . . independent of strain We have assumed throughout that the intercalated atom exerts a constant force on the host l a t t i c e around i t ,  independent of the strain at the atom's  p o s i t i o n , . :!t':<i'S :pr.obable, however, that the atom w i l l exert  less force as  the strain increases due to the presence of the other intercalated atoms. This w i l l  effectively  cause P . . to decrease as x increases.  Note that this  1J  may compensate to some degree the softening of the l a t t i c e at large strains due to anharmonic e f f e c t s ,  which could lead to an approximately linear  variation of strain with x even though the linear theory presented above is breaking down, and predicting interaction energies higher than those actually present. (d) E l a s t i c constants of the host independent of x This point must be checked experimentally. lation compounds with x ^ 1 or greater w i l l as the unintercalated host.  It is unlikely that interca-  have the same e l a s t i c  constants  However, the effects we have discussed above  should s t i l l be v a l i d even i f the e l a s t i c constants vary, although the actual values of the interaction energies w i l l  involve some effective  constants different from those of the pure host.  elastic  (e)  E l a s t i c isotropy Some hosts are quite anisotropic e l a s t i c a l l y .  in connection with 6.2(14); for that case, the q u a l i t a t i v e  was estimated  features of the e l a s t i c isotropic result.  interaction were in aqreement with the corresponding  Care must be taken, however,  material as one of the three cases discussed compare the elastic.energies  in .classifying an anisotropic  in Section 6 . 1 ; one should  required to produce the observed strains rather  than the magnitudes of the strains themselves. respectively  The effects of anisotropy  For example,  i f Y^ and Y^ are  the Young's modu1L fd<n.extensions normal and p a r a l l e l to the  basal plane of a layered compound, the condition for the compound to be case I is Y ^ e ^ » Y ^ e ^ 2  Y^Y  rather than simply e  = 28 (Blakslee et al 1970); for MoS , Yj/Y 2  from the neutron data of Wakabayashi et al  (f)  » e ^ . c  For graphite, 11 33 / c  =  *  k  6  '  e  s  t  i  m  a  t  e  d  (1975).  Use of continuum e l a s t i c i t y The expressions given in Section 6.3 for  to assumptions  (a), through (f)).  are exact (subject,  of course,  By contrast, the short range interaction  results of section 6.2 are v a l i d for intercalated atoms separated by several oo  l a t t i c e spacings.  The q u a l i t a t i v e features of W , that i t is attractive in  some directions and repulsive in others, are expected to be true in a l a t t i c e c a l c u l a t i o n , but some of the details may d i f f e r . and Hardy (1968) on vacancies of case III) oscillates  in hypothetical  Calculations by Bui lough  isotropic aluminum (an example  indicate..a nonzero interaction which varies as l/r"' and  in sign along a given d i r e c t i o n ; recall our continuum result CO  indicated no interaction in case III.  For cases I or I I, where W 5* 0 , it  is not clear how large a correction the discreteness of the l a t t i c e w i l l make to the continuum results,  but it could be appreciable for intercalated  atoms separated by one or two l a t t i c e  spacings.  129  PART  C  KINETICS OF INTERCALATION BATTERIES  CHAPTER 7 KINETICS OF ELECTROCHEMICAL CELLS  7.1  Introduction In Chapter h, we discussed the changes in the voltage of an  intercalation c e l l due to changes in the composition, x, of the intercalation cathode.  This discussion assumed that the c e l l was in equilibrium  throughout the intercalation process. a finite  rate,  In p r a c t i c e , intercalation occurs at  leading to changes in the c e l l voltage  (called overpotentia 1s  and conventionally denoted by n ) which are due to various loss mechanisms in the c e l l .  In this chapter, we discuss  these various mechanisms, to see  how the overpotent i a 1 each one produces'. depends on the c e l l current and on time. F i r s t , we look at an intercalation c e l l potentials  in d e t a i l , to see where over-  This also allows the considerations of Chapter k to be  occur.  related to the conventional picture of an electrochemical discuss  We b r i e f l y  the loss mechanisms that an intercalation c e l l has in common with  other types of electrochemical the e l e c t r o l y t e , face,  cell.  cells:  losses due to current flow through  charge transfer across the electrolyte-electrode  and possible  rate-1imiting surface reactions.  inter-  We then discuss  .  diffusion of the intercalated atoms, which causes an additional overpotential not present  in most electrochemical  diffusion overpotential are presented  cells.  The details of this  in Chapter 9, following a discussion  in Chapter 8 of the variation of the diffusion coefficient position of the intercalation compound.  F i n a l l y , since  with the com-  intercalation  electrodes generally consist of powdered host.materia 1 f i l l e d with e l e c t r o l y t e , we consider how such porous electrodes modify the details of the  relationship between overpotential  7.2  Electrochemistry of  and current.  Intercalation Cells  In order to discuss the various types of loss, or overpotential, intercalation c e l l , we must f i r s t such loss can occur. in F i g . 39 ( c f .  look in detail at the c e l l  in an  to see where  A schematic view of an electrochemical: eel 1. i s shown  Fig. 1 ) .  to an intercalation c e l l  We immediately specialize  to a case appropriate  by taking the "reaction" which provides the c e l l  voltage to be a simple transfer of an atom A from the anode, a, to the cathode, c  The e l e c t r o l y t e  is assumed to be a binary e l e c t r o l y t e ,  taining ions of the atom A, which we w i l l charge z^e,  ions B of charge zge  contacts d and d a potential  1  con-  denote by A and which carry a  (with zg < 0 ) , and solvent molecules.  The  are made from identical materials, since only then can  difference  is just the potential  be measured. difference  The open c i r c u i t voltage V of the c e l l  between d and d . 1  We wi11 use the symbol  $ to denote the e l e c t r i c p o t e n t i a l , with a superscript to refer to the  d  c  b  a  d'  t  t  y = 0  y = £  F i g . 39 - Schematic view of intercalation c e l l . c a: anode, b: e l e c t r o l y t e , c: cathode, d and d : contacts made from the same material. 1  132  particular material in the c e l l .  V = cf> - (j) d  In this  notation  d  (D  Just as in Chapter k, V is given in terms of the difference potentials y  V =  of the atom A in the cathode and in the anode as  cj) - <J)' = - J L ( d  d  v  in the chemical  . z-e A  T  VM  c y  _ )  (  a  y  A  *  K  This use of subscripts to denote species of p a r t i c l e and superscripts distinguish different parts of the c e l l notation  Although an electrochemical  to  is standard electrochemical  (see any text in electrochemistry,  the use of t i l d a s to distinguish ions  2  such as Bockris and Reddy 1970);  (A) from neutral atoms  c e l l can contain several  (A) is not.  different  species  of charged p a r t i c l e s , the open c i r c u i t (equilibrium) c e l l voltage can always be expressed example,  in terms of thermodynamic quantities of neutral e n t i t i e s .  For  (2) involves the chemical potential of the neutral atom, A, even  though in the transfer of atoms from a to c electrons external c i r c u i t ( a (a -> b -> c ) .  d^.d  1  flow through the  -> c) and ions A flow through the  On the other hand, in order to discuss  losses, in an electrochemical  electrolyte  the k i n e t i c s , or  c e l l , we must consider the details of the  motion of the charged p a r t i c l e s .  This requires the use of the e l e c t r o -  chemical p o t e n t i a l , which we denote as y . The electrochemical  potential of some charged p a r t i c l e a , y~, is  1  defined  as the change in the free energy F of a system when the number of a part i c l e s changes at constant temperature T and volume v:  (3)  y~ is thus the work required to add an a p a r t i c l e to the T,v.  system'.atoconstant  This work can be considered as the sum of three contributions: (a) The work ze<J>, where ze is the charge of a and cj) is the e l e c t r i c p o t e n t i a l , done against the e l e c t r i c f i e l d s  i f the system is not  charge neutral. (b) The work x~ (the surface potential)  to take a through the surface.  In general, a system composed of charged p a r t i c l e s has a dipole layer at the surface; in a metal, this is produced by the s p i l l i n g over of the electrons  into the vacuum.  This dipole layer is  modified by adsorbed atoms or molecules. (c) The work y~ (the chemical potential) bind a into the body.  resulting from the forces which  Although Coulombic in o r i g i n , these are local  forces, since the system is charge neutral when viewed over several atomic spacings,(except produces (j)).  for the small amount of excess charge which  y~ can be altered by changing the chemical  composition of the system.  Thus we have  (k)  y~ - y~ + Y ~ + zecj)  This is only an approximate r e l a t i o n s h i p , since the three terms are not completely independent. cp w i l l also affect  For example, addition of excess charge to change I -,' 1  however, for values of cf) of i n t e r e s t ,  the  chemical and surface changes produced by adding the small amounts of charge needed to give these changes in cj> are completely n e g l i g i b l e . interested  in x^> so it can be absorbed into y~ or <j>, giving  We are not  (5)  Although not necessary,  it is useful  to use electrochemica1. potentia 1s  in discussing the equilibrium voltage of the c e l l simplifies our discussion of the k i n e t i c s .  in F i g . 39, because  A l l of the parts of the c e l l  F i g . 39 are assumed to be conductors, so that in equilibrium e l e c t r i c exist only at the interfaces.  it in  fields  These f i e l d s are produced when the i n t e r -  faces f i r s t form; charge is transferred across each interface until dynamic equilibrium is established, with the same rate of charge flow in both directions.  The potential differences  produced by this charge transfer can  be calculated using (5) for each species of p a r t i c l e involved in the charge transfer.  Although we cannot measure such potential differences, we need  to consider the changes. in them which occur when current flows.  As an  ;  example, consider the. c-b interface, where equilibrium is  established  between A ions in the s o l u t i o n , b, and A atoms and electrons, e, cathode, c.  y  A " A P  This equilibrium requires  +  Z  which follows  A S P  (  to form the neutral A atom.  =  - ^  ( y  A -  ]  (7)  3  ambiguity in separating y into y and zetj) and dropping x ' equations can be written for each of the other interfaces These are:  electrons e  A - ^t •  Note that cj) - cj)' is defined only up to an additive constant, C  )  ~b ~c Using (5) for y^ and y~ gives z  c  6  from the condition that there be no free energy change in  equilibrium on transferring A from b to c and combining with  *  in the  n  due:to_the  (5).  Similar  in F i g . 39-  <$> - c|, = 1 (yg -ug) d  ,b *  (8)  c  ,a " *  /.a  1 =  *d'=  ^e"  ( Y  A "  a b \ Z  " ^'  i"  A % "  : y  V  (9)  '  ( 1 0 )  Combining (7) to (10) gives (2) once again, since y~ = L U i f d and d' are e e made from the same material.  Equations (7) to (10) were derived by considering equi1ibrium at the various interfaces,  so that the chemical potentials  to be evaluated at these interfaces. point, because the chemical potentials of each material.  In equilibrium this must be constant  is an irrelevant  throughout the bulk  It becomes important, however, when a current  since the concentrations,  and hence the chemical potentials,  p a r t i c l e s can then vary, throughout the materials. the c e l l  in these equations are  of the various  Thus, for example, when  is discharged and A ions are neutralized by electrons  ferred into the cathode at the c-b interface,  flows,  and trans-  the concentration of A ions  may be depleted, and the concentration of A atoms increased, over their respective  values  in the bulk of b and c.  Since 9y/8p > 0, both the  depletion of A and the increase of A lead to a reduction in the total voltage.  This change in the c e l l voltage due to gradients  is called the concentration or diffusion overpotential.  cell  in concentration  There w i l l also  be a reduction in the c e l l voltage when a current flows  (an increase on  recharge) due to r e s i s t i v e  (sometimes called  losses in the bulk materials  ohmic polarization) and at the interfaces  (the activation  overpotential).  In the remainder of this chapter, we discuss the current dependence of these types of overpotential.  The activation overpotential  is b r i e f l y  136  reviewe'd" i n S e c t i o n 7-3.  The f l o w o f c u r r e n t  cated by t h e f a c t t h a t when the c o n c e n t r a t i o n  i n the e l e c t r o l y t e i s c o m p l i i n the e l e c t r o l y t e v a r i e s ,  so does t h e c o n d u c t i v i t y , so the c o n c e n t r a t i o n o v e r p o t e n t i a l and the ohmic losses are i n t i m a t e l y r e l a t e d ; t h i s e f f e c t s of d i f f u s i o n 7-5,  i n the host  and then i n more d e t a i l  i s discussed  i n S e c t i o n J.k.  l a t t i c e are b r i e f l y considered  i n Chapter 9-  The r e s i s t i v e  The in Section  l o s s e s due t o  c u r r e n t f l o w i n the e l e c t r o n i c c o n d u c t o r s a r e assumed t o be d e s c r i b e d by Ohm's law, and so a r e not d i s c u s s e d f u r t h e r .  7.3  Losses Due t o T r a n s p o r t  Across the I n t e r f a c e s  The e l e c t r i c f i e l d a t the i n t e r f a c e between a m e t a l l i c e l e c t r o d e and a concentrated Fig-  e l e c t r o l y t e s o l u t i o n , such as the c-b and a-b i n t e r f a c e s i n  39, i s c o n s i d e r e d  t o o c c u r a c r o s s one o r two l a y e r s o f s o l v e n t m o l e c u l e s  adsorbed on t h e e l e c t r o d e s u r f a c e  ( s e e , f o r example, B o c k r i s and Reddy 1970).  When an i o n A i n the e l e c t r o l y t e s o l u t i o n (b i n F i g . 39) i s n e u t r a l i z e d and  i n t e r c a l a t e d i n t o the b u l k o f t h e i n t e r c a l a t i o n e l e c t r o d e  pass through t h i s s o - c a l l e d " H e l m h o l t z l a y e r " .  ( c ) , i t must  The t r a n s f e r o f charge  through the H e l m h o l t z l a y e r i s g e n e r a l l y regarded as an a c t i v a t e d  process,  w i t h an a c t i v a t i o n energy which v a r i e s l i n e a r l y w i t h t h e p o t e n t i a l drop c ,b t h e i n t e r f a c e , <P' - <P .  across  ,c ,b I f 1 i s the change i n 9 - <P w i t h  (the o v e r p o t e n t i a 1) , such an a c t i v a t e d process equation  . I  f o r the dependence o f the c u r r e n t d e n s i t y  i s known as the exchange c u r r e n t d e n s i t y .  coefficients a  +  Butler-Volmer  i on n. , which i s  . / a+en/kT - i•ct_erj'/kT I (e ^ - e o  :  =  where i  a  gives the  current  + a  and c i _  (D In ( l ) , the t r a n s f e r  satisfy  (2)  where  z^e  is  the  usually  has  when  flows  to  it  hold  the  if  a  - a  +  Helmholtz  we  need  lation is  to  know  atoms the ions to  the  of  whose  of  a  by  to  the  the  the  of  parallel  platelets.  to  cracks  It  is  areas  where  they  intercalate.  discussed area  a  from  the  by  where Vetter  function  of  the  atoms  the  entry  points  the  current  the  effective  (or  from the area.  must  be  activated  free  of  attached that  can  the  At  fast  sites  Additional  i n some way  enough in  flows  be  case  only  before  it if  limited  can the  to  that  metal  the  activation  surface in  diffuse  perhaps is  from  and  adsorbed  maximum v a l u e .  currents,  far  points,  process  flows  high  sites  the  to  surface  current at  the  As  effective  electroplating)  if  can  neutralized  lattice.  entry  area  through  occurs  atoms  intercalate,  some  or  also  electrode;  the  this  layers;  over  densities,  arise  interca-  thin  the  diffuse  the  near  current,  intercalation  faces,  makes  of  a  F o r an  from adsorption  complications  molecules;  may  the  the  metal  the  across varies  atomic  the  low c u r r e n t of  to  shown  relation.  usually  diffusion  into  area  are  then  adsorbed  diffusion  entire  electrolyte  flow  and  be  immediately,  packed  in  also  Tafel  this  however,  Surface  case,  surface  diffuse  solvent  steps  incorporated  current.  growth  or  close  positive  current  through which  the  one  electrons  transfer.  compounds  surface,  this  are  over  cannot  the  in  (1967).,  electrolyte  adsorbed  currents  they  convert  possible,  onto  metals;  area  layered  adsorbed  sites,  to  the  intercalates  first  growth  known as  1, is  can  the  ion  only.through  of  is  »kT/e,  neutralized surface  current  tunneling of  For n  density;  are  electroplating  the  charge  are  can  by  z^ =  (1)  the  crystals  faces  the  in  total  for  electrolyte.  which  involved  the  neutralized;  convention  1961).  current  area  if  intercalate  edges  being  that  (Gerischer  For example,  platelets,  Note  electrode  gives  (1)  fraction  occur.  ion  with overpotential,  electrode,  that  the  neutralization occurs layer  exponentially  of  - \.  from the  the  Equation  charge  by  so  decreasing atom breaking  slow,  the  (Activated  138  states o f adsorbed hydrazine  Through  In e q u i l i b r i u m , in  the c e l l  gradient  throughout  defined  t h e work  V i s reduced  V = V  (In  the  this  required  - —  o  to transport  t o move t h e e n t i r e  z^e  i t s open  Ay£ = V  section,  A  o  circuit  value  related  solution.)  flow  through  ions i n  (with y the  the s o l u t i o n .  the c e l l  voltage  to  sources  To f i n d  of  loss  except  the r e l a t i o n s h i p  s e e how n  in the e l e c t r o l y t e ,  J ~ of the various  (see, f o r example,  depends  gradients  B o c k r i s a n d Reddy  between  the  on t h e c u r r e n t .  we must c o n s i d e r  species of mobile  potential  transport  the  In s u c h a c a s e , charge  a are  by t h e f o l l o w i n g  coupled  1970):  VfL,  J~ =  a-;1  dilute)  densities  of the A  a  (1)  any o t h e r  to the electrochemical  equations  I  q  flow,  - y^(y=0)  o f h a v i n g more t h a n o n e s p e c i e s o f m o b i l e c h a r g e .  current  where  currents  potential  the ions  V  the ions A a r e i n  - n :  we n e g l e c t  current  since  atom A i s unchanged,  a n d t h e o v e r p o t e n t i a 1 n , we must  In d i s c u s s i n g  1978,)  an A atom f r o m a t o c  i t t a k e s work A y ^ = y^(y=£)  current  a  to transfer  When f i n i t e  the e l e c t r o l y t e  the  and A c r i v o s ,  in y ^ , the electrochemical  i n F i g . 39)  from  b.  through  effects  i n the i n t e r c a l a t i o n of  Electrolyte  the s o l u t i o n ,  s o l u t i o n ; as a r e s u l t ,  Since  i n t o NbSe^ by B e a l  a l l t h e work  i s produced  distance  postulated  i n F i g . 39 i s done on t h e e l e c t r o n s ,  equilibrium  the  h a v e been  from the vapour  Transport  1.k  atoms  , = L~.~.  aa'  (2)  aa' or  a'a  F o r s i m p l i c i t y , we w i l l  s o l u t i o n where  the off-diagonal  consider  terms  an i d e a l  .(infinitely  L~~, , a 4 a', a r e z e r o , a n d  139  where  we  can  convection,  the  a  J„ "*  Z  diffusion  equations  A  -4- V *  = -  —b  A  we  of  the  two  so  J ~ = 0, —B  the  solvent  the  flow  molecules. of  ions  ft  If  we  and B  also  neglect  are  °n  (z^e)  -  e  Vu*  (3)  2  /_ . v2- ts (z e)' :  B  have  charges  of  governing  - — —  z^e "B  (where the  neglect  dropped  on A and ions. and  VUR Vcb = - — 2 -  B  In (k)  the  superscript  respectively,  the  steady  b).  a n d a~  state,  In  and  (3)  a n d o~  current  is  are  (4), the  carried  z~e  and  A  z~e  are  D  conductivities only  by  the  A  ions,  gives  .  z e  (5)  B  Substituting  into  (5)  gives  (3)  (z^e)  For  an  atoms  and  ideal is  B  dilute  related  to  u =  '.:klZnp +  the  diffusion  a *  (z^e)  2  8 p  the  A  fi  the  chemical  concentration  p  (at  potential constant  u T)  of  one  of  the  solute  by  (7)  constant  coefficient  3y  A  solution,  .  of  the  - A  A  ions,.,D~,  is  given  by'  a  k T  (z e) A  2  P  A '  (  M  Since  the  conductivity^  Substituting  ^ = -  We s e e  (7)  A (  D  that  increases  and  ] | | - )  +  the  the  leads  considering  a planar  independent  of  the  t r a t i o n . 'p^(y)..must  Z  A  where  "  A  D  1  , z  +  p^. i s r t h e  (5) a n d  Au  of  the  so  +  A  to  the  A  is  A  p^ * p-g  in  Fig.  position,  y,  between  vary  linearly  effectively  to  (1+z^/|  39  in  by  charge  of  p^.  neutrality)  the  | ) .  which  (Levich  this  p ( 0 ) - p (£) \ 2  ion  currents,  as  is  most  1962).  anode  region.  and  easily  Since  the  cathode,  the  seen  by  current  is  concen-  Thus  p«  2  (10)  A  B  concentration  of  A ions  for  = 0.  We a l s o  have,  -kT^^j-J  -y (0) R  /  A  n = -  Finally,  n  limiting  overpotential  kT  * A  n  /  . 1  z  A  e  with fixed ,  z  the  A  fact  i.e.  +  e y  that  A  P  ^( ,-*(0)--^W^  that  lyte  second  coefficient  cell  A  (since  independent  (9)  kT A  gives  is  using  (7):  = U U)  A  (6)  t o p^,  P A -  V  presence  (9)  proportional  into  diffusion  Equation  J  (8)  is  5  (  r  £  (11)  \  )  (12)  J  is  Z  AA  - | z  B  E  -L . e | /  the  ^ / AA . ° \o^0)  total  P  (  I  number  (13)  .  of  solute  ions  in  the  electro-  dy  PRM  /  =  p^£  we o b t a i n 2D, J  A  1 + u  "  tanhlr^F kt 1 + z /|z |  the c u r r e n t becomes  o c c u r s a s p(0)  o r p(£)  overpotential n  i n t o two p a r t s  in the e l e c t r o l y t e  £ by t h e t h i c k n e s s  Diffusion  (15)  2  B  independent of n for  approaches  zero.  (Vetter  . (11) , a n d t h e r e s i s t a n c e  convection  7.5  A  PK  R  We s e e t h a t  Ay-v/z-e  l  It  polarization  Acj)  a n d 7-4,  losses  layer next  discussed  The r e a c t i o n  r e a c t i o n product,. (the' i n t e r c a l a t e d  replacing electrode.  the host  lattice.  reactions  The f i n i t e  of  As was s e e n  t h e v o l t a g e o f an i n t e r c a l a t i o n c e l l  7-3  interfaces is  the  intercalated  atoms  rate of diffusion of  i n t h e h o s t p r o v i d e s an a d d i t i o n a l  at  in Sections  reaction, since  compound) c o n s i s t s  in a d d i t i o n to those a l r e a d y d i s c u s s e d .  f  of  i n an i n t e r c a l a t i o n c e l l  in a sense a bulk r e a c t i o n , r a t h e r than a s u r f a c e  potential u of'the  by  to the  due t o the t r a n s f e r o f c h a r g e a c r o s s  and t h r o u g h the e l e c t r o l y t e .  i n S e c t i o n 7-2,  The e f f e c t s  (12).  c e l l s , which involve surface  show the t y p e s o f o v e r p o t e n t i a l s  i n t e r c a l a t e d atoms  overpotential  Host  the e l e c t r o d e s ,  spread throughout  the  1967).: t h e c o n c e n t r a t i o n  c a n be c o n s i d e r e d q u a l i t a t i v e l y  Conventional electrochemical  namely  This  i s common t o s e p a r a t e  of the u n s t i r r e d e l e c t r o l y t e  in the  |n | » k T / e .  loss, or in the  the  overpotential,  discussion  depends o n ' t h e  i n t e r c a l a t e d atoms at the s u r f a c e o f the h o s t ;  chemical  this  in  t u r n d e p e n d s u n i q u e l y o n t h e s u r f a c e c o m p o s i t i o n , x^ , p r o v i d e d we c a n  neglect  nonlocal effects  that  in y(x)  of the type d i s c u s s e d  i n S e c t i o n 6.5.  (Note  when we speak of atoms at the surface here and in the following chapters, we mean intercalated atoms just on the surface.) develops  During i n t e r c a l a t i o n , a gradient in the composition  in the host,-so that x  position.  inside the host rather than atoms adsorbed  g  varies more rapidly than the average com-  This produces a difference between the observed c e l l voltage and  the voltage which would be measured i f no such gradients existed; difference will  is referred to as a diffusion overvoltage.  In Chapter S, we  discuss solutions of the diffusion problem and arrive at  between this overvoltage and the c e l l current I.  :  relations  These solutions  assume two idealized forms of the dependence of the diffusion D on the composition; to j u s t i f y these i d e a l i z a t i o n s , we w i l l the expected composition dependence of D in Chapter 8.  this  will  coefficient first  discuss  CHAPTER  DIFFUSION  INTERCALATION  COMPOUNDS  Introduction  8.1  In  this  calated again any  chapter,  atoms  use  a  in the  composition describes  we  the  the  x  in  discuss  diffusion gas  a  effects  of  interactions  coefficient,  D,  of  the  arguments,  simple of  the  description of  mobility of  general  hopping  neighbour  model  atoms  on  the  atoms we  one  atoms.  intercalation due  examine  originally a  the  to  the  variation  proposed  dimensional  in  by  of  Mahan  lattice  inter-  We w i l l  system,  changes  the  between  and  neglect  host.  To  D with  the  (1976)  with  once  which  nearest  interactions.  Neutral potential Vu  on  lattice  changes  illustrate  to  IN  8  particles the  ]i;  move  number  in  response  current  to  density  gradients  of  the  in  their  particles  is  chemical linearly  related  by  J_ = -  MpVy  where  p  is  (Note  that  physics.)  the  (1)  number  M defined (1)  density here  describes  is  of e  the  times  diffusion  of  particles the the  and M is  mobility  used  particles,  as  their in  mobility.  semiconductor  can  be  seen  by  w r i t i ng  J_ = -  with  the  D -  (2)  DVp  diffusion  Npj£.  D defined  by  constant  D defined  by  .  (3)  (3)  is  sometimes  called  the  chemical  diffusion  coefficient.  If the p a r t i c l e s form an ideal gas, or are solute atoms in a d i l u t e  solution,  the concentration dependence of y is of the form y = kT^np + constant  so the diffusion coefficient  becomes  D = MkT  (5)  which is the familiar Einstein r e l a t i o n .  Experimentally, diffusion is  studied by measuring^ the rate of mixing of labelled (e.g.  radioactive  isotopes) and unlabelled p a r t i c l e s which are chemically i d e n t i c a l . procedure measuresVthe tracer diffusion coefficient differs from MkT by a factor of order unity (see, however,  D^.  often  Such a  In general,  for example, Flynn 1972);  in the case of diffusion in a one dimensional l a t t i c e ,  = 0  since the p a r t i c l e s cannot get around one another, whereas M and D are nonzero. Equation (1) can be generalized to the case of charged p a r t i c l e s by introducing the electrochemical  potential y, which gives (6)  J_ = - MpVy = - MpVy - zeMpVcf) where ze is the charge of the p a r t i c l e s .  The f i r s t term in (6) describes  d i f f u s i o n , while the second describes e l e c t r i c a l  conductivity.  The conduc-  t i v i t y a is defined by  zeJ = -.:'aVcf>  (7)  so we can identify a by comparing (6) and (7) as a = (ze) Mp 2  Because of  .  (8), the discussion  (8) in this chapter on the effects of the :  interaction on M for neutral p a r t i c l e s can be applied to charged p a r t i c l e s as w e l l , such as in the case of superionic conductors.  Also in reference  to charged p a r t i c l e s , we should note that i t is possible to regard an intercalated atom as an ion A ofucharge ze together with z electrons Chapter 5 ) .  If this  (see  is done, we can write two coupled equations of the form  7 . 4 ( 2 ) to describe the motion of the ions and electrons,  in which case we  can speak of the ionic conductivity of an intercalation compound as.-.defined by applying (8) to the mobility of the ions.(Weppner and Huggins 1977). However, in the intercalation systems we wish to describe, the electron mobility is much larger than the ion mobility, so the electrons equilibrium (Vu~ = 0) even in the presence of the ionic motion.  remain in This causes  the coupled equations to reduce to an equation of the form (1) involving only the electrochemical potential gradient of the neutral species A = A + ze, and we are back to our discussion of neutral p a r t i c l e s .  8.2  Behaviour of D(x) Consider a l a t t i c e gas of p a r t i c l e s as in Chapter 4 , where x measures  the fraction of occupied s i t e s . particles  If there are no interactions between the  (except for the hard core repulsion that prevents more than one  p a r t i c l e from occupying any site) one expects the mobility M for a simple hopping motion of the p a r t i c l e s from s i t e to s i t e to decrease as 1 - x as x increases,  due to the blocking of s i t e s .  l a t t i c e gas,  the variation of 8y/9p calculated from 4 . 3 ( 6 ) exactly  this factor of 1 - x in the expression (independent of x).  However, in a non-interacting cancels  (3) for D, so D is a constant  For simple hopping between adjacent s i t e s , D can be  related to w, the probability per unit time that a hop w i l l occur between a a full  and an empty s i t e .  For example, on a one dimensional  nearest neighbour s i t e separation c,  (Flynn 1972)  l a t t i c e with  D = wc  2  .  (1)  Near x = 0 we can speak of the Independent hopping of p a r t i c l e s with the mobility M = w c / k T , and near x = 1, the independent hopping of holes or 2  vacancies with the same mobility. Repulsions between intercalated atoms keep atoms apart. of  For some values  x and of the i n t e r a c t i o n , this may increase M over the case U = 0,  since adjacent sites are less l i k e l y to be occupied.  However, a reduction  in M is expected near compositions corresponding to ordered arrangements of  the p a r t i c l e s , since the repul s i on.-respons i ble for the ordering w i l l  prevent the p a r t i c l e s from jumping out of the ordered s u p e r l a t t i c e . other hand, we saw in Chapter k that the factor 9 y / 9 p large ( 9 x / 9 y  On the  should become very  small) at such compositions, which w i l l compensate this  :  reduction in M, just as in the noninteracting case just discussed where the variation 1 - x in M is exactly cancelled by 9 y / 9 p . (p/kT)9y/9p  = (x/kT)9y/9x  Einstein relation (5),  Because the factor  increases D over the value predicted by the it has been referred to as the '.'enhancement factor"  (Weppner and Huggins 1977).  Attractive interactions between intercalated atoms w i l l also reduce M, because of the clustering of the atoms produced by the a t t r a c t i o n . in this c a s e , 9 y / 9 p h.k).  is also reduced over the non-interacting case  However,  (Section  Hence D may be considerably smaller at intermediate x values than  near x = 0 (1) where p a r t i c l e s (vacancies)  move independently.  As a  consequence of such a concentration dependence of D, i f we try to intercalate an  i n i t i a l l y empty l a t t i c e to x = 1, large concentration gradients w i l l  in  regions of intermediate values of x (since Vp °c 1/D).  in.a  This w i l l  sharp boundary separating the empty region (x = 0) from the f u l l  (x = 1); this boundary w i l l  form  result one  then move through the l a t t i c e as intercalation  proceeds.  In f a c t ,  i f the attraction is large enough to produce phase  separation, this boundary is just the surface separating the two coexisting phases.-of composition x^ and x^ , say.  We can thus think of a phase  boundary as being caused by the vanishing of D for compositions. x where x^ < x < x^, due to the fact that ou/9p = 0 for these compositions.  These considerations suggest that we can understand (at least q u a l i tatively)  diffusion in intercalation systems in terms of one of the following  assumptions of the form of D(x):  (a) D is  independent of x, which is appropriate for systems with  repulsive interactions between intercalated atoms  (or for any system  i f only small variations in x are considered)  (b) D is zero over some range Ax, so that motion of a phase boundary occurs, which is appropriate for systems with a t t r a c t i v e interact" t i ons.  In Chapter 9 we wi11 discuss diffusion for these two cases.  First,  however,  we w i l l consider a simple model c a l c u l a t i o n describing diffusion on a one dimensional  l a t t i c e which i l l u s t r a t e s the conclusions reached above.  8.3  Model Calculation of Diffusion on a One Dimensional Lattice  Consider p a r t i c l e s  localized on sites in a one dimensional  lattice  with 1 a t t i c e „ c o n s t a n t c which are described by the Hamiltonian H = E In + U l nn o£ a a a+1  + fiO),Y(b b _,,:+. b') h^ a a+1 a+1 a .  a  a  ¥ In (1), b and b CX  CX  (1)  +  L  . .a  are creation and annihilation operators for  particles  on the s i t e a, and are related to the number operator n^ by n  =.b b .  (2)  +  a  aa  'Because no more than one p a r t i c l e can reside on a single s i t e , b operators for  the same s i t e obey anticommutation  bb t  + b b  a a  aa  b b +: b b  a a  +  = 1  = b b +  a a  relations  (3)  +  + b b +  a a  +  a a  = 0  while b operators for different  (4)  sites commute.  This Hamiltonian was used  by Mahan (1976) to discuss the variation of the conductivity as a function of  temperature T in superionic conductors; here we w i l l use it to discuss  the variation of the mobility M and the diffusion coefficient function of x = <n > at fixed T in intercalation systems.  D as a  If the final term  a in  (1) (the-hopping  term)  is absent, H describes a one dimensional  gas with nearest neighbour interactions  U (see  lattice  Section 4.6). Following  Mahan (1976) we w i l l assume fico, « kT, so that we can use the l a t t i c e n results to evaluate any thermal averages.  gas  The hopping term should more correctly be called a tunneling term, since it describes  the overlap of the wavefunction of a p a r t i c l e on one s i t e with  the adjacent s i t e s .  The problem is thus analogous  to a tight binding problem  in s o l i d state physics: the overlap of the single p a r t i c l e wavefunctions on separate sites means that the true wavefunctions  are Bioch states, which  for U = 0 would produce an energy band of width 2fico ; note, however, n  that  in contrast with the usual s o l i d state problem the bandwidth is much less... than kT here. we w i l l  This leads to an i n f i n i t e mobility in a perfect  lattice,  so  introduce scattering phenomenologically with a relaxation rate  which describes the time decay of current fluctuations. Calculation of M and D proceeds as follows.  We introduce the dipole  moment operator II (r_) as  n(r) = ze —  where  J-a r 6(r-r a  L  a  )b b a a  (5)  +  is the position of the s i t e a .  The number current density operator  cl is .related to II (_r) by (Mahan 1976)  v ' ze  =  dt  1 f - L n(r) ze V if.  (6)  H  j  where the last term in (6) involves the commutator of the Hamiltonian ( 1 ) . In evaluating the volume integral in ( 6 ) , we assume a three dimensional l a t t i c e of noninteracting chains, which occupy a volume v.  Then J becomes,  using (1)  CO, c j = J2_.y(  iv  + b  -b b +  b  a+1 a  L  J .  (7)  a a+1  a  The conductivity at frequency co is then evaluated using the Kubo formula (Kubo 1957) a (co) :(ze)  :  _ tanh (fico/2kt) v J.e" fico/2  i a J t  S ( t ) dt  (8)  where the correlation function y ( t ) is given by  Y(t)  = ± < J ( t ) J ( 0 ) + J(0)J(t)>  and then a is related to M using  (9)  8.1(8).  In  (9),  J ( t ) is the current .  operator in the Heisenberg representation, related to J in ( 6 ) by  J(t) = e  i H t /  ^e-  i H t / f i  .  (10)  To evaluate Y(t) we need the results  (which follow from ( 1 ) and the commu-  tation relations for the b operators)  eXVb e  XH  A  ,e  X  J  = " a-1- a 2) t a a+1 = " a-1-n +2) t a+1 a XU(n  n  e  a a+1 b . b e" a+1 a +  H  XH  +  b  v  XU(n  e  1  b  a  b  (12)  b  where A is a constant  (these are the corrected forms of equations 2 . 3 a and  2 . 3 b in Mahan 1 9 7 6 ) .  Using ( 1 1 ) and ( 1 2 ) Y(t) becomes  Y(f)--.=P^) J<fn ( 1 - n ^ ) + n _ ( 1 - n j] \ v / la a+1 a+1 a J \ / a: L  The terms in ( 1 3 ) have an obvious interpretation. and  n a +  i(l"  n a  )  J>  cos ^ ( n X -ft a-1 a+2  The factors  (13)  n a  ^"  n a +  i)  describe a hop from a to a+1 and from a+1 to a respectively.  The cosine term is unity i f s i t e a 1 -  and a+2 are both f u l l  or both empty,  so that the hop does:,not change the number of nearest neighbour, pai rs in the chain, and cos(Ut/fi)  i f either s i t e a-1 or a+2 is f u l l  but not both, in  which case the number of nearest neighbour pairs changes by one.  We can  write y ( t ) more simply in terms of the "spin" operators s^ = 2 n -1; i f 1,2,3,4  are any four adjacent s i t e s , we have  = (1 - < s s > + < s s > - < s s s s > )  Y(t)  1  2  1  i t  1  2  3  i t  + (1 - < s s > - < s s > + < s s s s > ) c o s . 1  =  1  / (  1  2  3  (14)  i (  (14) into (8) and using 8.1(8) we find  Substituting  M x  2  Wr^'  "  ( 1  < S  1 2 S  > +  < S  TT tanh(U/2kT) :  1 4 S  z  2  >  "  /,  < S  1 2 3 4 S  „  S  S  ^  > )6  "  & ) )  „  ^^^  ^\  .6(co-U/fi) + 6 (oo+U/fi)  D can be found from M using 8.1(3) and 4.6(1) for u (x) . We see that M and D are i n f i n i t e at GO = 0 and OJ = ± U/h. To make them f i n i t e , we assume some scattering e x i s t s , so that the current-current correlation function decays exponentially (13)  in time; that i s , we multiply  by e l^r -!^ where oo^ is some scattering 1  functions  rate.  This causes the delta  in (15) to become Lorentzians:  6 (co) - 1 7 3 - 4 — • ?r ai +co r 2  (16)  2  1  W  6 (aj±U/h) ->  • 77  co + (oj±U/fi) 2  (17)  2  We f i r s t consider (15) for the case of no interactions, U = 0. the two Lorentzians (delta functions  Then  in (15)) merge into one, and in  :  add i t i on  1 - <s s > = 4X(1-X) 1  2  (18)  so we obtain  2  CO  P  M(co) = f r < k T  oo +co  h  2  (1-x)  (19)  2  CO  D(co) = c co 2  2  h  —  w +(o r 2  2  (20)  -  Note that M <* 1-x and D is independent of x,  in agreement with the q u a l i t a -  tive discussion of Section 8.1 of the noninteracting l a t t i c e gas.  Com-  paring (20) at co = 0 with (1), we see that the jump probability w is given by w = co /co . 2  r  To apply this result to true hopping, weishould have w % co  r  (the scattering rate of order the hopping rate), which in turn implies co ^ co,. r h  I f we apply (20) at co = 0 to di ff us ion of Li ; in Li Ti 0 x 2  for x « 1 ,  where D = 6 x 10 ^ cm /sec at T = 25°C (Johnson 1964a) and the jump distance o g -_i _c c = 1.5 A, we find co^ = 2.7 x 10 sec , so fico^ = 7 x 10 kT. Thus, we expect that when we discuss  interactions U ^ kT, the contribution of the  Lorentzian at ±U/fi can be completely neglected  in calculating D and M at  co = 0, and we need only worry about the f i r s t term in  (15).  While we are not concerned with the frequency dependence of M and D in app:lyii ng„thi s model, it is nonetheless interesting to contrast the frequency dependence of (15) with the results of true hopping c a l c u l a t i o n s , where the motion of the p a r t i c l e s is described by a master equation rather than being incorporated into the Hamiltonian, (Dietrich et al 1977)-  For U = 0, the current-current correlation function  ¥ ( t ) becomes a delta function in time rather than a constant or an o s c i l l a tory function as in (14),  so the mobility is then independent of frequency  rather than consisting of delta functions in frequency as in (15).  In  the presence of : interactions, the hopping calculations predict that M(co) is  a s t r i c t l y increasing function of frequency.  In superionic conductors,  neither a tunneling type model such as we have discussed here or a true hopping model is capable of describing the frequency dependence of a l l systems; some systems, such as 3~alumina, show M(co) while others,  increasing with frequency  like A g l , show M(co) decreasing with frequency (Kimbal 1 and  Adams 1978). Now we return to co = 0 (setting w = cojVco^) and consider interactions.  The:expression  (15)  functions for the one dimensional  for M is evaluated using the correlation l a t t i c e gas model, given for  in Appendix B; only the f i r s t term in (15) negligible  repulsive  reference  is used, the second one being  i f fico^ « U ^ kT, which we assume here.  The mobility at zero  frequency for U = 5kT (which corresponds to the voltage curve in F i g . 23b) is shown in F i g . kOa. The mobility is decreased from the noninteracting case, and shows a minimum at x = 5 due to the large amount of short range order at this composition produced by the repulsive interaction. Fig.  kOb the "enhancement factor"  tfy/8x  In  for the same i n t e r a c t i o n , U = 5kT,  is presented, showing an increase over the noninteracting case, and a maximum at x = i.  The diffusion coef f i cieht  . ca 1 cu.l ated from M and x3y/9x  is shown in F i g . 41a; it is larger than for the case 11 = 0 and shows a maximum at x = i , but on the whole variesmuch less rapidly than either M or x3y/3x.  For reference, xM is plotted in F i g . 41b, to show the  conductivity expected  ionic  i f the intercalated atoms were charged p a r t i c l e s .  It is interesting to compare these results with the predictions of mean f i e l d theory, which are also shown in F i g . kO and 41 .  The mean f i e l d  results were calculated using a two sublattice decomposition of the one dimensional  lattice  functions are  (see  Section 4.6), so that the required correlation  Fig.  kO - (a) M o b i l i t y M and (b) e n h a n c e m e n t f a c t o r (x/kT)8y/8x v e r s u s c o m p o s i t i o n x f o r a one d i m e n s i o n a l l a t t i c e g a s w i t h n e a r e s t n e i g h b o u r i n t e r a c t i o n s U = 5kT. R e s u l t s a r e shown f o r t h e e x a c t and mean f i e l d s o l u t i o n s t o t h e l a t t i c e g a s p r o b l e m . The c u r v e f o r U = 0 i s a l s o shown.  155  Fig.  41 -  (a) D i f f u s i o n c o e f f i c i e n t d i n g t o F i g . 40.  D and  (b)  "conductivity"  xM  correspon-  < s > = <s s > = ( 2 x ^ : 1 . ) ( 2 x - 1 )  (21)  <s s s s > =  (22)  S l  2  1  1  2  where  3  4  and x  (note x^ + x  i(  2  2  [(2X -1)(2X -1)|  2  2  are the fractional occupations of the two sublattices  = 2x).  2  1  We see that the mean f i e l d theory gives semi-quanti-  tative agreement with the exact r e s u l t s , except near x = i , where mean f i e l d strongly underestimates M,and overestimates in D in the mean f i e l d results  x9y/9x.  The discontinuity  is a consequence of the kink in the voltage  curve in F i g . 2 3 b produced by the second order phase t r a n s i t i o n predicted in mean f i e l d . For very strong repulsions considerably. solutions  (U » kT), the expressions  We quote the expressions  for M and D simplify  for x < i.and co = 0 ; the actual  for xM and D are symmetric about x = i .  For the exact solution  to the l a t t i c e gas (with w = co, /co ) h r 2  M  =  wc^1^2x  kT  (  }  1-x  while for the mean f i e l d  solution  M = ^ ( 1 - 2 X )  (25)  D = wc :' 2  (26)  D varies by a factor of 4 in ( 2 4 ) , while in the mean f i e l d result is independent of x.  (26), D  The concentration dependence of M in ( 2 3 ) is identical  to that found by Dietrich et al hopping in a one dimensional  (1977)  lattice.  in their master equation solution of Note that the mean f i e l d  results  for U -»• o o ,  (25)  and (26),  x in (19) and (20)  may be obtained from the U = 0 case by replacing  by 2x; a s i m i l a r s i m p l i f i c a t i o n of the voltage  curves  in the i n f i n i t e  interaction limit in mean f i e l d was noted in Chapter 4  (see especially  4.6(2)).  F i n a l l y , consider a t t r a c t i v e  interactions.  F i g . 42a and 42b show the  variation of M and x9u/9x with x at co = 0 for U = -2.5 kT, which corresponds to the voltage curve shown in F i g . 2 3 a .  Again the correlation functions  in Appendix B were used and only the f i r s t term in (15) was retained.  In  this case, both M and X'3u/3x are decreased from the case U = 0, and so D, shown in F i g . 4 a , is also decreased. 3  reference  in F i g . 43b.  The "conductivity" xM is plotted for  Also shown are the same quantities calculated using  the simple (random) mean f i e l d theory of Chapter 4 ( 4 . 4 ( 3 ) ) which predicts a f i r s t order phase t r a n s i t i o n for 0.14 < x < 0 . 8 6 .  The mean f i e l d  results  for M and xM in the two phase region represent the e f f e c t i v e quantities the entire  lattice,  for:'  calculated on the basis that the resistance of the chain  which would be measured i f the p a r t i c l e s had a charge ze (from the conductivity  (ze) Mp) 2  is just the series combination of the resistance of the two  phases; such a resistance would be independent of x for 0.14 < x < 0 . 8 6 , . since both.-.phases have the same value of xM.  The diffusion constant  is  zero in the coexistence region since 3y/9x is zero there, which, as was argued in Section 8 . 2 , is consistent  with the idea that a phase boundary  moves through the l a t t i c e as p a r t i c l e s are added for 0.14 < x < 0 : 8 6 . Note that taking D = 0 and xM = constant over this  range of x is a reason-  able f i r s t order approximation to the results obtained using the exact l a t t i c e gas solution.  For large a t t r a c t i v e  interactions  in both the mean  f i e l d and exact r e s u l t s , M = 0 for a l l x except x = 0, and D = 0 except at x = 0 and x = 1,(where D = wc ); this behaviour in the mean f i e l d case 2  corresponds to phase coexistence over the entire range 0 < x < 1.  '  158  Fig.  Ul -  (a) Mobility M and (b) enhancement factor (x/kT)9y/9x versus composition x for a one dimensional l a t t i c e gas with nearest neighbour interactions U = -2.5 kT. Results are shown for the exact and mean f i e l d solutions to the l a t t i c e gas problem. The curve for U = 0 is also shown.  F i g . hi - (a) Diffusion coefficient ponding to F i g . kl.  D and (b) "conductivity" Mx corres-  CHAPTER 9  DIFFUSION OVERVOLTAGES IN INTERCALATION CELLS  9.1  Introduction  As discussed  in Chapter 7 , the nonuniform concentration in interca-  lation compounds produced by f i n i t e currents causes overvoltages  in interca  1 at i o n j c e l 1 s .  The types of behaviour expected can be understood at  qualitatively  in terms of one of the following assumptions about the  concentration dependence of the diffusion coefficient  least  D on the composition  of the intercalation compound:  (a) D is  independent of x  (b) A phase boundary moves through the intercalation compound as intercalation proceeds, which is e f f e c t i v e l y  equivalent to the  case where D = 0 over some range of x. We discuss the behaviour of c e l l current and c e l l these two cases in the following sections. discussed by Atlung et al that a l l other overvoltages host are n e g l i g i b l e .  (1979).  voltage resulting from  The f i r s t case has also been  We assume for s i m p l i c i t y in what  except those associated with diffusion  follows in the  9.2  Diffusion for a Constant D  In this section, we discuss diffusion in an intercalation compound for constant D.  The standard reference for the solutions of the diffusion  equation in this case is Carslaw and Jaeger ( 1 9 5 9 ) , and we shall use their results extensively in what follows.  In referring to their r e s u l t s , we  shall give their equation numbers preceded by the letters C J , as in CJ 7 - 5 ( 1 ) .  We w i l l discuss  the three geometries shown in F i g . kk:  (a) an. i n f i n i t e slab of halfwidth R (b) an i n f i n i t e c i r c u l a r cylinder of radius R (c) a sphere of radius R. It w i l l be useful cases.  to define a parameter £ = 1 , 2 , 3 respectively  The distance  for the three  in each case w i l l be measured by r, with r = 0 corres-  ponding to the plane midway between the surfaces  for £ = 1 , the axis of the  cylinder for £ = 2 , and the center of the sphere for £ = 3 .  F i g . kk - The three geometries considered in discussing the effects of diffusion of the intercalated atoms on the behaviour of intercalation eel 1s.  The  symmetry of cases (a) and (b) allows them to be applied to materials  with very anisotropic d i f f u s i o n . . This enables us to establish a correspondence between the three cases here and the three types of  intercalation  compounds distingiushed in Section 6.1: (a) corresponds to a host where diffusion  is along one dimensional tunnels, as in the r u t i l e related  materials-; (b) to a host with diffusion in two dimensional the layered compounds; and (c)  layers, as in  to a host where the diffusion is  as in the metal-hydrogen systems.  isotropic,  Note that here the correspondence of  the cases to: the r u t i l e s and to the layered compounds is opposite to that -  discussed  in Section 6.1; where the correspondence was made on the basis of  the e l a s t i c  strains produced by i n t e r c a l a t i o n ;  this  is because in r u t i l e s  the intercalated atoms move in one direction while expanding the  lattice  in two d i r e c t i o n s , and vice versa for the layered compounds.  We f i r s t consider intercalation into an i n i t i a l l y empty host (x = 0 at t = 0) at a constant number current density at the surface,  .  Note that  this corresponds to a c e l l current flowing in the negative sense according to the conventions of Chapter 7, and w i l l (reduction of the c e l l voltage).  lead to a negative  overpotential  The number density of intercalated atoms  at the surface at time t , P ( t ) , found by solving the diffusion equation iis^. s  (CJ .3.8(3),  CJ 7.8(1), CJ 9.7(D) J t  where Z(t)  z ( t  )  J R/ .  \  is defined by  , j-i^ -»n<>t/K\ e  2  (  n=1 n  In  (2), a  is a coefficient  given by the solution of one of the following:  2  )  a  n  =  nTT,  J ^ )  a  n  where  = 0,  cot a = 1, n  5 = 1  (3a)  ? = 2  (3b)  £ = 3  (3c)  is the Bessel  function of order 1 .  We can rewrite (1)  the composition x of the intercalation compound.  We define  in terms of as the density  corresponding to x = 1, A and v as the surface area and volume of the host, and I as the magnitude of the e l e c t r i c current which must flow through.the cell  i f intercalation is occuring from a solution where the  is ionized to a charge ze.  intercalate  Then we have  (k)  p = p X Q  I = zeJ A  (5)  A = £f  (6)  s  -  In a d d i t i o n , i t is useful  to define t  as the time which would be required o  to f i l l  the host to x = 1 (a total  charge Q  q  passed through the c e l l )  if  intercalation proceeded uniformly throughout the host: zevp  Q,  Introducing these variables into (l)  gives the following result for x ( t ) , s  the value of x at the surface:  x (t) = f r o  +  TT" o  - TT(2+S)X(t) o  (8)  where x' is defined by  ~  D •  For t « T ' ,  (  (8) reduces to the case of a semi - i nf i n i te s o l i d  9  )  (CJ 2 . 9 ( 8 ) ) :  x (t) = / - ^ ' o  (10)  where T is defined by  1  R  £+2  2  ,  /,,\  After the current has been flowing for a long time, so t » x , £ ( t ) ->• 0, and  **M~jr  + -r o  (12)  o  Equation (8) is plotted  in F i g . 45.  We.'ve also plotted a useful  interpolation formula given by  x (t) s  =1^- c o t h / Y o  (13)  which also shows the limiting behaviour (10) and ( 1 1 ) , with x = T T X ' / 4 . t > x , the surface composition,:  For  , increases l i n e a r l y with t, as expected  is an extra contribution, x ' / 3 t , to x o s i f intercalation were uniform throughout the host.  for a constant current, but there which would not be present  This corresponds to an overpotential  (x'/3t  This nonuniform intercalation at f i n i t e of an intercalation c e l l below its considered discharged (its  ) (3V/8x) o  x=x  s  for t •> x . ^  currents reduces the capacity  theoretical  value.  Suppose the c e l l  voltage too low to be useful)  is  at a cutoff voltage  g. kS - Surface composition x versus time t for intercalation of the three geometries shown in F i g . kh, for the case of a constant diffusion coefficient D. The dashed curve gives the results for the interpolation formula, (13). The straight line is the asymptotic solution for large t. s  corresponding to a composition x^_. exceeds the average composition t / t t  Since the surface composition x^ (t) during discharge of the c e l l , the time  needed to reach the cutoff voltage w i l l be less than the corresponding  time i f intercalation were uniform, x t . Let Q be the maximum capacity co m (charge) available above the cutoff voltage, given by  Qmm  =  r, r-o c  lt  0 4 )  x  and Q the charge passed in time t , c c Q. = It  c  c .  (15)  The relation between x^ and t^ is given by ( 8 ) , x  C  namely  = x (t ) S  C  (16) '  '  Equations (14)  to (16)  can be solved numerically to give Q /Q. as a function c ,m of the current, i ; the result of this calculation is plotted in F i g . 46. At low currents, the fractional capacity Q /Q. f a 11 s 1. inea r 1 y: wi th „ i : c m Q  c  ,  ,  1IT  m  (17)  m  At high currents, the capacity is inversely proportional to i ' 0_ Q. c _ TT_ _m 0. 4 IT m  Equations (14)  0 8 )  to (16) give the capacity for a single discharge of an  intercalation c e l l , starting from an unintercalated host.  In laboratory  t e s t s , c e l l s are often cycled continuously between fixed voltage  limits  at a current ± i , corresponding to a variation of the surface composition  g. 4 6 - Fractional capacity QQ/Q,^ versus current I for intercalation of the three geometries shown in F i g . 4 4 for the case of a constant diffusion coefficient D. The dashed curve is the result for the interpolation formula (13)-  over a range x^.  If there are no other reactions  in the c e l l , which would  make the time of the discharge longer or shorter than the time of the recharge, this procedure is equivalent  (once a steady state has been  reached) to applying a square wave current to the c e l l , with a h a l f - c y c l e time t , i  say.  The capacity over each half cycle, Q, , varies with current in  2  2  a fashion similar to the single discharge, F i g . 46. (t,. »  For small currents  T)  *  =  1  m  - i ' 3 Q„ I T  (19)  For large currents, we can use the solution for the steady state change in surface concentration in response to a sinusoidal current J sinO)t s  (CJ 2.9(13)):  Ap (t) =  —— sin (cot  (20)  -  /coD  S  together with the Fourier expansion of the surface current appropriate for a square wave: 4J J (t) = s  TT  L -1 n=1 2n  s i n (2n-lhr|  (21)  to obtain the variation in surface composi:tion, Ax,(t)  Ax.(t) = - W - f r -  TT V 3/  l  /„  I  n=1 ( 2 n - D  o  The peak-to-peak variation in A x ( t ) g  172 H r  s  !  n  ^ _i  1  \TTt  TT  sin (2n-l)--  3 / 2  7  (22)  is x^, given by  ——372  n=1 (2n-l)'  •  (23)  The sum in ( 2 3 ) is 1 . 6 8 8 7 6 1 . . .  Q2-=  0.340  ^  so that  •  (24)  m  This is smaller than ( 1 8 ) for a single discharge by a factor of 0 . 4 3 3 -  Given the expressions  above, one can use the dependence of the capacity  of an intercalation c e l l on current to estimate  the diffusion constant, D.  A l t e r n a t i v e l y , D can be measured using one of the following transient techniques.  If the c e l l  is changed by 6p  =  PQSXJ  a  is in equilibrium at t = 0 , and the c e l l  voltage  V, which causes the surface concentration to change by  current l ( t )  flows.  at the surface of the host is  At short times, the number current density  (CJ 3 - 3 ( 9 ) ,  CJ 1 3 - 3 ( 3 ) ,  CJ 9 - 3 ( 5 ) ) :  vo-jeK-^p, • The current, l ( t ) ,  I ( T  )  =  <«>  is then given by  ^ | ^ - | J . 3 x  6  v .  m  (  2  6  )  The f i r s t term.; in ( 2 5 ) and ( 2 6 ) corresponds to the result for a semi - i nf i n i te _x  medium (CJ 2 . 4 ( 6 ) ) .  A plot of l ( t )  versus t  2  will  give a straight  whose slope gives T , and hence D i f the p a r t i c l e s i z e ,  line,  R, is known.  that the quantity 9x/8V must also be known i f D is to be found.  Note  Also note  that i f the chemical potential of the intercalated atoms depends nonlocally on x for nonuniform compositions,  8x/8V needed in ( 2 6 ) is not the same as  8x/3V found by measuring the open c i r c u i t voltage as a function of x; point is discussed in Section 6 . 5 for the e l a s t i c  interaction.  is most important when the magnitude of 9x/9V is large.  this  This problem  An alternate transient technique to measure D involves applying a brief pulse of current which causes n atoms to intercalate at the host surface. Using the semi-infinite medium result (CJ 2 . 2 ( 1 ) ) , the surface concentration varies as follows after the pulse has been applied:  ' • P ^ O - T —  (27)  so the measured c e l l voltage change is  ™  ( t l  =|57wf ^  •  >  (28  (The constant terms in (28) analogous to those in (26) have not been evaluated.)  In this case, D can be found.by plotting  V versus t  i f a sinusoidal current is applied, the diffusion coefficient from the expression  2  .  Also,  can be found  (20) for the resulting surface concentration change.  As discussed in Section 8.2, the approximation of D independent of x should be a reasonable one in many intercalation systems. in applying the above results to real the p a r t i c l e s i z e , R.  is in specifying  In single c r y s t a l s , intercalation can begin at cracks  in the surface, and so the effective measured crystal dimensions. will  intercalation c e l l s  The main problem  p a r t i c l e size is smaller than the  In any practical  intercalation c e l l , the host  be used in powder form, so there is a large d i s t r i b u t i o n in R, rather  than a single value as assumed until now. some of the results w i l l  The r e s t r i c t i o n t «: x used in  then refer to the smallest p a r t i c l e s , which may  limit the small t results to a range of time which is too small to be useful; beyond the small t regime the results described above must then be generalized, because there are current paths between the p a r t i c l e s via the e l e c t r o l y t e .  There are also further complications in pressed powders  introduced by the f i n i t e conductivity of the e l e c t r o l y t e ,  which are discussed  in Chapter 10.  9.3  Motion of a Phase Boundary As discussed  in Section 8.2, intercalation of a host l a t t i c e where the  intercalated atoms attract one another leads to the formation of a boundary separating a region of low concentration from a region of high concentration. Similar boundaries w i l l also form i f the host undergoes a structural phase transition. crystal.  As intercalation proceeds, the boundary moves through the host  In this section, we discuss  the same geometries as in F i g . hh.  the motion of thi.s boundary.  We model the system as follows.  boundary, located at position r = r(t) at time t, sharp, separating a region of composition composition x^ (phase 2 ) , with x^ < x^. are flowing into the host l a t t i c e ,  We treat The  is assumed to be i n f i n i t e l y  (phase 1) from a region of To be s p e c i f i c , we assume that atoms  so phase 2 lies outside the boundary  (closer to the surface), which corresponds to a negative e l e c t r i c current flowing through the c e l l  in the conventions of Chapter 7-  The results  obtained then depend on the values of D and 8x/9V in phase 2 only; when atoms flow out of the c r y s t a l , the values for phase 1 should be used in the formulas to be derived.  We assume that intercalation begins at t = 0, with r(0) = R (the boundary at the surface).  As the boundary moves past any point in the host,  the composition there jumps by  A  x  = 2 " x  x  i  =  i r -  (  1  o We w i l l assume that the current is small enough that we can use the steady  )  172  state approximation (Crank 1 9 7 5 ) ;  that i s , we assume that at any time t the  composition for r < r i s constant and equal to x^ , while for r > r , the composition is  identical to that obtained in steady state with a current  density flowing between r = R and r = r , equal, to J at the_surface s  (r = R) .  These steady state concentration p r o f i l e s are (CJ 3 - 2 ( 1 ) , CJ 7 - 2 ( 4 ) , CJ  9-2(7):)  (r-r),  C  =  1  (2a)  ?  = 2  (2b)  C = 3  (2c)  J R  In  p(r,t)-p„ = <  J R'  with p  2  = P  X Q  2 the concentration just outside the boundary.  motion of the boundary, d r / d t , is determined by the number of  The rate of intercalated  atoms needed to increase the composition from x^ to x^ in the region r to r+dr (note dr < 0 ) :  r f  1  The factor  dr dt  (f/R)  J s  (3)  Ct Ax o  Ap  arises from the fact that in a cylinder and sphere  the area through which current must flow decreases as r decreases.  This  approximation assumes that we can neglect the number of intercalated atoms needed to change the steady state d i s t r i b u t i o n from that appropriate when the phase boundary is at r and that needed when the phase boundary is at r + dr; that i s  J J p ( r , t + d t ) -p(r,t)]  \  dr«ApA^j  1  (-dr)  .  (4)  Referring to the steady state equations, we see that we can write  C-1  (-dr)  Using (5) the integral  (5)  in (4) can be performed, and we obtain the  following  condition on Ax for the v a l i d i t y of the steady state approximation:  (6)  In addition to (6), the approximation w i l l concentration  = PQX^  is established  not be v a l i d until the uniform  from r = 0 to r =:"?; we w i l l assume  that this uniform concentration already exists at t = 0.  F i r s t we consider constant current.  Integrating  (3) with the condition  r(t=0) = R gives  (7)  The phase boundary reaches  r = 0 at t = t A x , which from Q  the time needed to homogeneously  9,2(7)  is just  change the composition by Ax at the surface  current density J ^ ; the steady state approximation c l e a r l y neglects the additional  current which must flow to change the composition outside  phase boundary. x (t) s  Substituting  the following:  (7)  the  into (2) gives for the surface composition  174  Tt  FAX  5  = 1  (8a)  ?  = 2  (8b)  5  = 3  (8c)  o  x (t)-x s  2  J - i ^ l - ^ )  t  -  (1 - t / t  Ax)  This is plotted in F i g . 47 ( c f . The  1  1 / 3  the case of constant D, 9 . 2 ( 8 ) and F i g . 4 5 ) .  diffusion overvoltage associated with x^ -  is  (x^-x^) ( 9 V / 9 x ) ; Xs  the magnitude of this overvoltage increases/with time as the phase boundary moves, in contrast to other types, of overvoltage (such as activation overvoltage or r e s i s t i v e  losses), which would be constant  discussion of porous electrodes occur).  (see,  however, the  in Section 1 0 . 3 , where similar effects  Diffusion overvoltages can c l e a r l y wash out the plateau in the  voltage curve expected at a f i r s t order phase t r a n s i t i o n , as shown schematica l l y in^ F i g . 48.  Note that i f the current is interrupted, the steady state  approximation predicts that when the current resumes, the voltage w i l l drop to the same value it had just before the interruption occurs (df:course, will  it  take a time of order x to re-establish the steady state concentration  profile).  The For  v a l i d i t y of the expressions  5 = 1 , the solutions are v a l i d for a l l time t < t Ax provided Q  t Ax » o  For  (8) ns given by the condition ( 6 ) .  5 = 2,3,  x  5 = 1 .  (9a)  the phase boundary moves more rapidly as r decreases, and the  solution eventually breaks down; thus (8) is v a l i d for t such that  F i g . kl - Surface composition x versus time t for intercalation at constant current of the three geometries shown in F i g . kk in the case of the motion of a phase boundary in the host. s  )  t Ax - t » Q  T ,  t Ax - t » ( Q  If the c e l l  t  (  A x )  1  /  V  A  ,  x, = 2  (9b)  ? = 3  (9c)  is considered discharged at x^ = x^, the equations  (8) can  be used to calculate the variation in the apparent capacity, Q , with current, just as in Section 9 - 2 . is  In this case, the maximum capacity Q_  m  independent of x^, and is given by  Q  m  = Q Ax . Q  For £ = 1 , we find  ( ) 10  I < Q (X -X )/T1  1 ,  O  C  2  (11a)  C =1  = <  Q (x -x ) G  c  2  I > Q (x -x )/T o c 2 The  lower term in (11 a) has the same form as 9.2(18) except TTQ Ik has been  replaced by 0 - ( x - x ) . o  but  c  T  n  lower expression in (11a) also holds for r, = 2,3,  e  2  only for I large enough that Q /Q < 0.1. cm _£ = 1 m  e  "  ( x  s- 2>V x  ,  l T  VV 2>  For arbitrary I ,  ? = 2  (11b)  S = 3  (11c)  X  = 1 -  1 +  3lx  m  Equations (11) are plotted in F i g . kS ( c f .  F i g . 46 for the case D'=~con'st.) .  Now consider the variation of current i f the surface c o m p o s i t i o n , ^ , is held at a constant value after t = 0.  In this case, J  s  (or t ) in (3) o  is time varying, so i t must be eliminated using (2); then (3) can be integrated to give  V 2 Ax X  i £  Wry  3\RJ  An £  t  s" 2 ~k~ kx Ax  1/ry . 1 t s 2 21R/ F 9T Ax X  X  ? = 1  (12a)  ? = 2  (12b)  ? = 3  (12c)  X  _ X  In each case, the phase boundary reaches r = 0 at t = t , given by  Fig. hS  179  The current l ( t )  can be found by eliminating r from (12)  using (2);  however,  a simple expression results only for 5 = 1 :  f(x - x J A x ' s I 2-rt  1  Kt)  For 5 = 2,3,  (14)  the limiting behaviour at small t is given by  i(0 - _ L - l  (  W  x  5 - 1 2 s" 2 5 3 x X  2Tt  (which also reduces to (14) for 5 = 1 ) . to better than 2% until  the host  X  (15)  (15)  is 50% f u l l  is an accurate approximation for 5 = 2,3.  Note t h a t , j u s t as in the case of constant D, the current varies as t (cf.  2  9.2(26)); in this case, however, the current varies as the square  root of the applied voltage V-V^. = (x^-x^)(3V/3x). (12) are given by (6).  The limitations of  Again for 5 = 1 the solutions are v a l i d for a l l  t,  in thi.s case i f  Ax  >> 1  V 2  5  = 1  (16a)  5  = 2  (16b)  5  = 3  (16c)  X  For 5 = 2,3 the conditions are  FO  y  Ax  2  t  In (R/r)  V 2 X  1JL 3  y  For example, i f L\x/(x^-x^) = 100, these conditions for 5 = 2 and 3 in terms of  t are t « 96x = 0 . 9 6 t  t  given by ( 1 3 ) .  f  f  for 5 = 2  and t « 142X = 0 . 9 4 t  f  for 5 = 3 , with  CHAPTER 10 POROUS ELECTRODES  10.1  Introduction  As is evident from the previous chapter, losses in intercalation cathodes due to diffusion in the host l a t t i c e are decreased by decreasing the size of  the host l a t t i c e c r y s t a l s , R.  In general, practical  electrodes consist of finely powdered host material. powder are f i l l e d with e l e c t r o l y t e  intercalation  The pores in the  to allow the ions in the solution  reach p a r t i c l e s throughout the electrode.  to  This structural arrangement,  however, can also cause problems, which we now discuss. We shall consider a planar intercalation electrode, The electrode  is a slab of thickness  as shown in F i g . 50.  and front surface area A * , consisting  of a fine powder of host crystals occupying a fraction 1 - X of the total volume H'*.l\*. The rest of the volume is filledi!with e l e c t r o l y t e ; quantity X is known as the porosity.  electrical  contact  The p a r t i c l e s  porous  electrode  the  in the powder are  elect rolyte  A e  Fig.  50 - Planar porous intercalation electrode, length H.  showing a pore of  181  assumed small enough so that any effects due to diffusion in the host can be neglected. In F i g . 50, electrons arrive from the anode via the external c i r c u i t and enter the cathode from the l e f t ,  and ions arrive from the right.  The  resistance to electron flow is determined both by the bulk r e s i s t i v i t y of the host material, and by the contact resistance between the p a r t i c l e s . In the f a i r l y porous materials used in electrodes,  the contact resistance can  be larger than the bulk resistance, so the e l e c t r o n i c resistance generally depends on the p a r t i c l e size and the procedure used to prepare the electrode. The resistance to ion flow is determined by the bulk e l e c t r o l y t e properties, and by the paths the ions follow through the pores of the electrode.  In  general, these pores are well cross-1inked, so the voltage is constant at a p a r t i c u l a r depth in the electrode, and we have a one dimensional system with some effective e l e c t r o l y t e conductivity.  Tp relate the effective e l e c t r o l y t e conductivity of the electrode to the bulk e l e c t r o l y t e conductivity, we assume that the pores have a length, SL, given by  (D  SL = $£* where $ i s . t h e t o r t u o s i t y .  Then i f a l l of the e l e c t r o l y t e volume in the  pores is accessible from the surface, the total cross section of a l l the pores has area A , given by  £A = £*A*X and the total  (2)  ionic resistance, R, of the e l e c t r o l y t e in the pores  is  related to R*, the resistance which would be measured for a slab of bulk e l e c t r o l y t e with dimensions SL* and A*, by  This r a t i o , R/R*, is referred to as the formation factor.  Empirically,  it varies with porosity roughly as  J L ^ _L R*  k  n >  n  ^  2  (Archie, 19^2), so that, from (3),  the tortuosity should vary as  (5)  One p a r t i c u l a r measurement of <£> from the transit time of ions through porous material gave  (6)  :(Wi:nsauer et al 1952)  rather than (5),  suggesting that some of the pores  in that measurement were isolated from the surface.  10.2  Ohmic Models  Because of the resistance of the powder and of the e l e c t r o l y t e pores,  intercalation does not proceed uniformly through the electrode.  s p a t i a l l y nonuniform reactions  in porous electrodes were f i r s t  by Euler and Nonemacher (1960) in terms of a simple r e s i s t i v e in F i q . 51. 3  in the  R, and R are the total b c  pores and of the powder respectively;  the Butler-Volmer equation 7 - 3 (1) -  discussed chain as shown  resistances of the e l e c t r o l y t e G is the conductivity of the  between the p a r t i c l e s and the e l e c t r o l y t e ,  Such  in the interface  given, for example, by l i n e a r i z i n g  Current flowing from y = SL to y = 0 in  F i g . 51 w i l l be distributed to give equal potential  drops in the upper and  y=£  y  F i g . 51  -  =0  Resistive chain used to model porous electrodes. R^ and R are the total resistances of the e l e c t r o l y t e and of the host matrix respectively. G is the total conductance of the interface. I indicates the direction of positive current flow. c  lower chains;  i f G is large (corresponding to a low impedance:interface),  some of the current w i l l cross the interface near y = I, and the rest near y = 0.  The current crossing the interface decays exponentially with  distance, y, from each end of the electrode, with a decay length X ^ given by (R.+R )G  Clearly,  i f R, » R , most of the current crosses the interface near y = I, b c  and for R » R, , near y = 0 . c b  Generalization of these arguments to non-  linear behaviour of G and R, , with the results of numerical c a l c u l a t i o n s , b  is given by Newman and Tobias  (1962)  and Grens and Tobias  The resistor chain of F i g . 51 is useful d i s t r i b u t i o n just as intercalation begins front or back of the electrode deeper in the electrode.  (1964).  in discussing the current  (t = 0 ) .  At later times, as the  is intercalated, current begins to flow  This can be discussed  in terms of a r e s i s t o r -  capacitor network as in F i g . 5 2 ; for s i m p l i c i t y , we neglect the of the p a r t i c l e s and of the interface  resistance  (R = G ^ = 0 ) . The total capacitance c of the network is related to the total capacity of the intercalation host,  I  b-  y =£  1  1  1  1  1  y =0 I  F i g . 52 - Resistor-capacitor network used to model the intercalation of porous electrodes. R is the total resistance of the e l e c t r o l y t e , and C the d i f f e r e n t i a l capacity of the entire host. I indicates the direction of positive current flow, and V is the measured c e l l voltage (up to an additive constant) i f there are no other losses in the eel 1.  Q. , which was defined in 9 . 2 ( 7 ) , as  for small changes in x .  The voltage and current along such a capacitive  transmission line obey the diffusion equation, with a diffusion  coefficient  D__ given by (de Levie 1967)  D  RC  =  W•  (3)  Hence we can apply the results of Section 9 - 2 , relating the current density and number density, J  s  ii/C,  Similarly,  to this case, i f we make the substitutions  and D •> D  RL  V,  , where V is the measured voltage shown in F i g . 52.  i f a f i r s t order t r a n s i t i o n occurs in the host l a t t i c e ,  so that  the open c i r c u i t c e l l voltage shows a plateau over some range A x , a boundary separating the two phases w i l l move through the e l e c t r o l y t e ,  and we can use  the results of Section 9 - 3 ; in p a r t i c u l a r , the c e l l voltage w i l l  fall  l i n e a r l y in time at constant current ( 9 - 3 ( 8 a ) ) , and the current w i l l vary as t  2  i f the c e l l  is held at constant voltage  (9-3(14)).  10.3  Electrolyte Depletion  So f a r , i t has been assumed that the e l e c t r o l y t e shows ohmic behaviour. However, as discussed in Section l.k,  this may not be true - as the voltage  drop in the e l e c t r o l y t e increases, the current eventually saturates at some limiting value due to depletion of the ions in the solution. behaviour occurs in porous electrodes,  The same  but is complicated by the fact that  the limiting current depends on the depth in the electrode that the current reaches.  We w i l l discuss only the case where a f i r s t order t r a n s i t i o n  occurs in the host, so we can use the steady state approximation discussed in Section 9.3 to describe the variation of the current and voltage with time.  The pores are assumed to have a small diameter, so any variation  of concentration across the width of a pore is negligible - we need only consider the variation in concentration with depth in the pore, y F i g . 50).  (see  We also assume that the total charge Q Ax needed to complete the o  phase transition in the host is much greater than the total  ionic charge  contained in the pores, so that the ions in the pores must be replenished by ion flow from the surface, y = £ , of the electrode.  F i n a l l y , since in  most c e l l s of interest the anode and cathode are separated only by a thin e l e c t r o l y t e - f i l l e d membrane, we assume that the amount of e l e c t r o l y t e outside the pores is n e g l i g i b l e .  As a r e s u l t , i f the concentrations p ~ ,  p~ , of the ions within the pores drops, the concentration at the surface D  of the electrode  (y = £ ) must increase, since the total number of ions in  the e l e c t r o l y t e must remain constant due to charge neutra1ity (we are only removing A ions from the s o l u t i o n ) .  Too large an increase in concentration  at y = £ wi11 produce p r e c i p i t a t i o n , which w i l l block the pores and limit the current that can flow. Even i f p r e c i p i t a t i o n does not occur, the current can s t i l l  be limited  186  if  the ion concentration at the position of the phase boundary, y, becomes  zero.  In the steady state approximation, the concentration in the e l e c t r o -  lyte w i l l vary l i n e a r l y between y = y and y = £ , and w i l l  be constant from  y = 0 to y = y. (see the discussion of the case 5 = 1 of Section 9.3). that i t w i l l where  take a time of order £ / D to establish such a p r o f i l e , a 2  is the ambipolar diffusion coefficient  of the e l e c t r o l y t e , which  describes diffusion with no e l e c t r i c current flow.  For ideal  solutions,  relating overpotential, n , >d the number current density  the equations of  Note  ar  A ions in the s o l u t i o n , J ^ , are (using the notation of Section 7-4)  J iA -- D iA h^+ T il f^ tT rJ J  D  1  (2)  £-y  +  where J~ > 0 corresponds to current flow to the right in F i g . 50 (from y = 0 to y = £ ) , which in turn corresponds to recharging the c e l l .  (3) follows  from the conservation of the total number of ions in the pores. (1) to (3) can be solved for n, (t) or J^(t) voltage conditions.  Equations  for constant current or constant  Note from (2) that the magnitude of the maximum current  which can flow (corresponding to p^(y) = 0 or p^(£) = 0) varies  inversely  with £ - y , and for y = 0 is given by  /  Z  A  \ P A  (4)  Thus, under constant current conditions, f u l l y discharged or recharged (|n|  00  if | l | > I  ) at time t  L  L >  the c e l l w i l l appear  given by  Q Axl,  t  L  = - 2 ^  .  (5)  At this time, only a fraction t^/t Ax of the electrode w i l l have been Q  converted in phase.  1  >  Thus the apparent fractional capacity 0- / - is given by n  c  |I| < I  m  L  (6)  m jrp  |  I  |  >  X  L  (assuming that the cutoff voltage corresponds to [n | » k T / z ^ e ) . be compared to the corresponding expressions the host,  9.2(18)  and  ( 6 ) should  for the case of diffusion in  9-3(1 l a ) .  We can rewrite the expression for I , ( 4 ) , in a somewhat more transparent form.  Using 7 - 4 ( 8 )  to relate the conductivity due to the 'A ions, a^,  to D^, and defining the resistance  R  -  in the pores due to the A ions, R^, by  s•^  <> 7  we find  which to within a numerical factor of order unity is just the current flow through a resistance  R~ due to a potential  drop kT/z^e = 2 5 . 7 / z ^ mV.  For  currents much less than I , we can neglect the concentration variations in the e l e c t r o l y t e ,  and regard the e l e c t r o l y t e  as a fixed  (ohmic)  resistance.  188  This w i l l  be v a l i d only i f the voltage drop in the e l e c t r o l y t e is small  compared with kT/z^e.  In this case, the current flowing when the boundary  is at y is  i  -iV^  '  h  l  <  <  k  T  /  A  z  •  e  (  9  )  In the other l i m i t , the current flow when the boundary is at y is independent of the overpotential n,  1  If  =  •  Ml  >  >  k  T  /  z  the voltage of the c e l l  A  a n  e  d '  s  given by  •  (  1  0  )  is held at a value much larger than kT/z^e  below the open c i r c u i t voltage of the coexisting phases., so that the limiting current appropriate to y,  and  if  is given by  the measured current as a function of time is  K t ) - ! ^  The  (10), always flows, y(t)  current varies as t  . -2/3  (12) - i  , in contrast to the variation t  the resistance of the e l e c t r o l y t e were constant.  expected  PART  D  EXPERIMENTAL PROCEDURE AND RESULTS  CHAPTER 11 EXPERIMENTAL PROCEDURE  11.1  Introduction  In order to i l l u s t r a t e the wide variety of behaviour found in i n t e r calation systems, several experimentally. and  types of lithium intercalation c e l l s were studied  In this chapter, we describe the methods used to prepare  test these c e l l s .  The experimental  results are presented and discussed  in Chapter 12.  11.2  Materials Used  Free flowing powders of the layered compounds IT-TiS^ and 2H-M0S2 were prepared by reacting stoichiometric quantities of powders of the elements (purity > 99.3%) in quartz tubes at 550°C for two or three days. of  the samples,  For most  the quartz tube was placed in a temperature gradient, with  the metal powder at the warmer end (T ^ 550°C) , and the molten sulfur condensed at the cooler end (k50°C).  The temperature gradient was intended  to allow simultaneous control of the sulfur vapour pressure (determined by the temperature of the cooler end) and of the reaction temperature  (the  warmer end) as suggested by Whittingham ( 1 9 7 8 c ) ; later samples of T i S  2  indicated, however, that keeping the entire tube at 550°C (ho temperature gradient) gives identical results. a period of several  The reactants were heated slowly  (over  hours) to the reaction temperature to avoid rapid  reaction of the sulfur with the unreacted metal surface.  When the reaction  was complete, the products were cooled down to room temperature over a period of several hours.  The products were fine powders, consisting of  platelets with a diameter of one or two ym, and thicknesses  considerably less than 1 ym.  TiS^ was also prepared at 800 C, which gave  a larger p a r t i c l e size (a diameter of order 10 ym, and a thickness of order 1 ym), and below 550°C, which resulted amounts of T i S ^ .  in the formation of appreciable  For each material, the samples were X-rayed to confirm  their structure.  Natural MoS^ was also used in the experiments, crystals  (> 1 mm diameter)  and powder.  in the form of single  Two grades of powder were obtained  from Molybond Laboratories: very fine powder (^0.1 ym p a r t i c l e size) suspended  in o i l , and free flowing powder with a p l a t e l e t diameter of  order 1 ym.  The M0O2 used was prepared by passing hydrogen over MoO^ at 475°C, or by reacting stoichiometric quantities tubes at 750°C. respectively.  of Mo and MoO^ in sealed quartz  These two methods gave p a r t i c l e sizes of M ym and MO ym Lithium f o i l  received from Alfa Ventron.  (0.38 mm t h i c k , 99-95% pure) was used as Single crystal TiS^ was prepared using standard  vapour transport techniques.(Balchin  The e l e c t r o l y t e  1976).  used in the c e l l s was 1 molar LiClO^ or Li B r dissolved  in propylene carbonate  (PC).  The LiClO^ (obtained from Alfa Ventron) and  LiBr (from MCB) were vacuum dried for one week at 150°C, and then stored under argon.  The PC (Eastman Kodak) was.vacuum d i s t i 1 led twice, then passed  through three columns of activated alumina and stored under argon.  Gas  chromatography showed that this p u r i f i c a t i o n procedure for the PC reduced the concentration of the p r i n c i p l e impurity, propylene g l y c o l , to ^3 ppm. (Subsequent  study has shown that equally good results  without the columns, by optimizing the d i s t i l l a t i o n  can be obtained  procedure.)  11 > 3 Cathode Preparation and Cell Assembly Cathodes from powdered materials were generally prepared by making a slurry with the powder in propylene g l y c o l , spreading the slurry over a nickel or aluminum f o i l , of nitrogen gas at 200°C.  and baking the propylene glycol away in a stream For the Molybond MoS^ suspended in o i l , the o i l  suspension was applied d i r e c t l y to a nickel substrate and baked at 750°C. to remove the heaviest  tars from the  M0S2  particles.  The powder coatings  applied using both of these methods adhered well to the substrate for 2 thicknesses  corresponding to a few mg per cm  or less:  The coatings were  quite porous, with porosity X, as defined in 10.1(2), of order 0.7 seen in some cases.  Thicker cathodes were occasionally made by pressing the  powders into a d i s c ; this method worked well for the layered compounds but  not for MoO^.  The powders in pressed cathodes were considerably more  densely packed than in baked cathodes, and porosities as low as 0.2 were seen. The simplest c e l l s which used baked cathodes were beaker c e l l s , where the substrate of the cathode was soldered or spotwelded to a wire and suspended in e l e c t r o l y t e ,  together with a lithium anode, in a beaker  sealed with a neoprene stopper. if  These c e l l s showed rapid capacity loss  the powder did not adhere well to the substrate.  avoided by using pressed c e l l s .  In these c e l l s ,  This problem was largely  a polypropylene Celgard  #2500 or #3501 microporous film (the separator) was placed between the cathode and the lithium anode, and the resulting sandwich pressed t i g h t l y together.  Two types of pressed c e l l s were used, and are shown in F i g . 53.  In "flange c e l l s " , a cathode-separator-anode sandwich was held between steel  flanges  coated with s i l i c o n e grease and screwed together;  seal was used to keep the c e l l a i r t i g h t .  In "button c e l l s " ,  an 0-ring  the sandwich  Fig. 53 - The two types of pressed cells used for i n t e r c a l a t i o n : (a) c e l l s , (b) teflon button c e l l s .  flange  194  was held between teflon plugs screwed into a teflon b a r r e l ; this type of c e l l was also used for pressed cathodes.  In the button c e l l s ,  electrical  contact to the electrodes was made with a wire soldered to a metal, disc behind the electrodes and fed through the teflon plugs; in flange c e l l s , steel flanges themselves provided the contact.  the  Cells were assembled under  an argon atmosphere in a Vacuum Atmospheres glovebox.  11.4 Techniques Used to Study Intercalation Cells Most of the experimental study of the intercalation c e l l s to be d i s cussed in the following chapter was intended to establish  the behaviour of  the c e l l voltage V as a function of the composition x of the cathode.  intercalation  In the simplest test used, the c e l l was charged or discharged  at a constant current I , and the voltage V recorded as a function of time. If the weight of the intercalation host is known, this gives a curve of V versus x d i r e c t l y , provided no reactions other than intercalation occur in the c e l l .  Such extra reactions could be due to the reaction of the  lithium with the e l e c t r o l y t e or with impurities in the c e l l .  Since most  of these reactions are expected to produce a current that would tend to recharge the c e l l , they should decrease the recharge time and increase the discharge time; i f this  is the case, the capacity due to side  i:s..approximately one half of the difference  in the length of the discharge  and recharge voltage curves between the same voltage c e l l was cycled repeatedly between fixed voltage changes in c e l l capacity to be e a s i l y seen.  reactions  limits.  In general,  l i m i t s , which allows  Such cycling was done with a  Princeton Applied Research .(PAR) 174 Galvanostat/Potentiostat  (which is  basically an elaborate current/voltage source) equipped with a PAR 175 Programmer and a Par 179 Digital Coulometer, or with a cycling system designed and b u i l t by the UBC Physics electronics  shop.  the  The inverse derivative of these voltage curves gives the quantity 3x/3V, which, as discussed  in Chapter h, can reveal ordering processes in i n t e r -  calation systems.  3x/3V can also be obtained d i r e c t l y by recording the c e l l  current I versus V as V is changed at a constant  I  =  1  dQ dQdV dt dV dt =  =  rate, V.  This follows  from  -9x  (  V 3V  v  where Q, is the total charge needed to change x by Ax = 1. d i r e c t l y proportional to 3x/3V.  Hence I  )  "  is  Current-voltage curves generated in this  way are called linear sweep voltammograms by electrochemists;  we w i l l  to them as inverse derivative curves or current-voltage curves.  refer  Scan rates  •  V used in our experiments were t y p i c a l l y a few yV/sec, as compared with values of several mV/sec used in most applications of this method.  The  current-voltage curves to be presented in Chapter 12 were produced with the PAR equipment described above. Since the cycling and derivative techniques just described are used to infer the equilibrium properties of intercalation c e l l s ,  it is important  to know how the curves produced are affected by the various loss mechanisms in the c e l l .  The effects of diffusion on charge-discharge cycles at  current has already been discussed in Section 9-2; falls  the apparent c e l l  constant capacity  l i n e a r l y with I at low currents, and varies as I ^ at high currents.  The effect of a resistance R in the c e l l  is to add a constant voltage I R  to the c e l l during recharge, and subtract I R during discharge. effectively  equivalent to lowering the upper voltage  This  is  limit and raising the  lower l i m i t , so the change in capacity with I depends oh the details of V(x) at the voltage  limits.  The variation of apparent c e l l capacity with current  can be used q u a n t i t a t i v e l y , but is more often used as a q u a l i t a t i v e measure of the losses in the c e l l .  The effect of series resistance and diffusion on  the inverse derivative curves is somewhat more complicated, and w i l l  be  discussed in the next section. In addition to these measurements, simple transient experiments, such as those described in Section 9-2 to measure the diffusion c o e f f i c i e n t , also done using the PAR equipment. techniques  were  (A general review of such transient  in electrochemistry is given by Yeager and Kuta 1970.)  11.5 Effect of Series Resistance and Diffusion on Current-Voltage Curves The  effects of a series resistance R on current-voltage curves is  considered f i r s t .  This is most e a s i l y discussed by representing the c e l l  as a series RC c i r c u i t , as in F i g . 54.  We w i l l  consider R independent of  current; for nonohmic behaviour, the discussion w i l l but  no q u a l i t a t i v e changes are expected.  be more complicated,  The capacitance C= (dV(0j/d0) ^repre-  sents the open c i r c u i t capacity of the c e l l .  A constant C would correspond  to a linear voltage-composition curve; since V(0_) for an intercalation c e l l is not l i n e a r , we expect C to be a function of "the voltage V(0_) across the capacitor.  Fig.  In the calculation to be discussed, we w i l l  take the following  54 - RC c i r c u i t u s e d t o d i s c u s s e f f e c t s o f c e l l r e s i s t a n c e on i n v e r s e d e r i v a t i v e curves. V(Q.) i s t h e v o l t a g e a c r o s s t h e c a p a c i t o r , a n d I t h e c u r r e n t w h i c h f l o w s as t h e t o t a l v o l t a g e a c r o s s t h e c i r c u i t v a r i e s as V t .  form of C(V)  Ci(V)  =  RTT-^  (1 )  17577-  which corresponds to the following form of V(Q):  V(Q  W* (v*)' n  (2)  Except for a minus sign due to the convention of current flow in F i g . 54> this corresponds to the non-interacting l a t t i c e gas r e s u l t , 4 . 3 ( 5 ) , shown in F i g . 12, i f 3 ^ = kT/ze.  Here we regard (2) as a convenient  represen-  tation of a voltage curve leading to a simple peak in C, so 3 is just a parameter measuring the width of the peak of C(V) in (1) (the f u l l half maximum of C(V) is 3-53 3  width at  ) .  The current l ( t ) which flows when the voltage across the c e l l across the RC c i r c u i t in F i g . 54) is swept at a constant  (or  rate V is found  by solving the loop equation for the RC c i r c u i t :  R -^j-jr + V(Q) = Vt  where I = dQ/dt.  With the p a r t i c u l a r form of V(Q) in ( 2 ) , we can rewrite  (3) in dimensionless  d(Q/Q  )  b -—^2-  (3)  form as  / Q./Q  \  + &i hpT7§- = 3Vt  (4)  where the parameter b is defined as  b = 3 Q VR .  (5)  2  Q  b measures the change in shape of the curve Q/Q or d ( Q / Q ) / d ( 3 V t ) Q  q  versus  198  ( 3 V t ; c l e a r l y i n c r e a s i n g R has the same e f f e c t as i n c r e a s i n g V.  (k)  was  s o l v e d n u m e r i c a l l y f o r s e v e r a l v a l u e s of b, and the r e s u l t s are shown i n F i g . 55. b = 0;  For small b, the curve  i s s h i f t e d o n l y s l i g h t l y from the case  the s h i f t of the peak i s g i v e n by A ( $ V t ) = b/2,  a shift  i n v o l t a g e of  A(C't) =  21  which corresponds  ,R, where I , = V/C , = VA3 peak peak peak  r e s u l t can a l s o be seen by s o l v i n g (k)  t o f i r s t o r d e r ini'b).  to  (this  For l a r g e b,  the v o l t a g e a c r o s s the c a p a c i t o r changes much more s l o w l y than the v o l t a g e a c r o s s the r e s i s t o r except  near Q = 0 o r Q = 0_ , so the c u r r e n t begins  look l i k e a ramp, expected  f o r a l i n e a r change i n the v o l t a g e a c r o s s  resistor.  The  f o r smal1 We  to  the  peak i n the c u r r e n t i s s h i f t e d t o much h i g h e r v o l t a g e s  than  b.  now  t u r n t o the e f f e c t s of d i f f u s i o n , i n the h o s t .  A l l series  r e s i s t a n c e s are assumed t o be z e r o , so the a p p l i e d v o l t a g e Vt c o n t r o l s the x^ at the cathode s u r f a c e , as d i s c u s s e d i n Chapter 9 .  composition  The  d i f f u s i o n problem i s c o n s i d e r a b l y more d i f f i c u l t t o t r e a t than the case of a s e r i e s r e s i s t a n c e , and we present voltage-composition  r e l a t i o n V'(x) than  V.(x) = j - ^ ( l - x )  The  the s o l u t i o n f o r a s i m p l e r form o f (2),  the  namely  .  (6)  i n v e r s e d e r i v a t i v e of t h i s i s  dx  fO  V < 0  ,  (7)  d\l  -e V(x) and The  3 V  ,  V > 0  8x/3V are p l o t t e d in F i g . 56a d i f f u s i o n problem was  the n o t a t i o n developed t h e r e .  and 56b r e s p e c t i v e l y .  treated in d e t a i l We  i n i t i a l l y empty (x = 0 at t = 0 ) .  i n Chapter 9 ,  and we  use  assume t h a t the i n t e r c a l a t i o n host i s At t = 0 ,  we  begin d i s c h a r g i n g the  cell  199  Fig. 55 - Current-voltage curves given by sweeping the voltage of the RC c i r c u i t of F i g . 5k at a constant rate V, with the capacity C(V) given by 11.5(1), for several values of the parameter b.  200  I  1  1  1  a  x 1  Fig. 56 -  1  1  >  1  1  r  (a) Voltage versus composition curve used to discuss the effects of diffusion on inverse derivative curves. (b) Current-voltage curves corresponding to the voltage curve in (a); curves bj and b£ are for large and small values of the diffusion coefficient respecti vely.  starting  f r o m V = 0,  s o V = - | V 11, a n d t h e s u r f a c e  composition varies  as  x.(t) - l - . - e l * ! *  which  follows  semi-infinite  t h e number  due t o these  ip  J (t)  where tary  p  and a plot  (1959),  o f t h e peak  ( V -> 0 ) ,  and i s i n fact  of  (9)  is shifted  the surface  - erfc(-\SWMt)]  atoms  (9)  a t x = 1.  The complemen-  is tabulated  i n Carslaw and  is given  in Fig.  Note  little  56.  from the slow  that the  discharge  o f D f o r l a r g e • | V |;:-thi s  where  t h e peak  current  position  by a l a r g e  amount.  After  the peak,  decays  decay  continue  until  t ^ R / D , where  this  will  J (t)  2  particle  size  the  finite  will  fall  the  finite  and  eventually In  to zero  to shift  shift  size  more  particle  rapidly will  regain  we c o n c l u d e on peaks  to later  and produces t h e peak  As was seen  becomes  size  we w i l l  effects  discharge) not  particle  summary,  opposite peaks  i n the cathode.  important  than  1//F  become  that  a long  position  rising  very  much,  o f Chapter  9,  f o r t > R /D, and the current 2  in this  resistance  (higher  is the  limit.  A s |\/|  at a smaller  decreases,  value  of  | V | t ,  result.  i n the inverse  times  R  0  in the discussions  important  t h e V -»- 0  depends on  (or l ( t ) )  g  0  case  is in contrast  R a n d c a n be s h i f t e d as l//t~:  of the  argument  very  losses,  and Jaeger  '  M«i.  independent  of resistive  into  for a  conditions:  of intercalated  o f imaginary  i n Carslaw  flowing  e ^ l ^ [erfc(i/B7vT0  function  the case  density  and boundary  fi  we c a n u s e t h e s o l u t i o n  a n d 2.5(9)  2.5(2)  /$|V|D  position  to  is large,  current  is the concentration  Q  error  Jaeger  initial  = —  s  I f |v|  (6).  medium, equations  to find  (1959), host  from  (8)  and diffusion  derivative  voltage edge  have  curve.  on recharge,  before  t h e peak.  but produces  almost  Resistance lower  v o l t a g e on  Diffusion  considerable  causes  does  broadening  over a time t ^ ^ Q ^ > where R tail  Q  '  s  after the peak i f D is small.  the p a r t i c l e s i z e ,  giving rise to a long  CHAPTER 12  EXPERIMENTAL RESULTS  12.1  Introduction  In this chapter, we present and discuss experimental in studies of lithium intercalation c e l l s .  results obtained  In the course of these experi-  mental studies, the author personally prepared and tested over 60 c e l l s , in addition had access to data from several by cd-workers in the laboratory.  Cell  and  hundred other c e l l s prepared  testing involved using cycling and  derivative techniques outlined in the previous chapter.  The data presented  here is a sample of the data collected on these c e l l s .  In discussing  data, we w i l l  RM12); Table III at  refer to each c e l l  by a c e l l  number (e.g.  the end of the chapter l i s t s a l l of the c e l l s discussed, relevant  the  together with  information on each one.  In what follows,  we f i r s t characterize the c e l l s used, and discuss  results obtained with c e l l s with no intercalation host. data for three systems: Li TiS X  are presented f i r s t ,  A  Li MoCL , and Li MoS„. X  ^  X  ^  Then we present The Li Ti S_ results X  £-  to allow comparison with similar studies previously  reported by Thompson (1978) for 0 < x < 1; our results for this range of x are more complicated than Thompson's, due to the p a r t i c i p a t i o n of the solvent in the reaction in our c e l l s .  Results for Li Mo0„ are given x 2  next, showing hysteresis associated with f i r s t order t r a n s i t i o n s . L i ^ o S ^ is discussed,  Finally,  i l l u s t r a t i n g the effects of a large structural  in the host which leads to large changes in the variation of the c e l l voltage V with composition x.  change  20 k  12.2  Excess Capacity and Kinetic Limitations of the Cells  In this section, we discuss some of the properties of the c e l l s used to obtain the results to be presented in the following sections.  We f i r s t  discuss the problems encountered in using the net amount of charge which flows through a c e l l to measure..the composition of the intercalation compound.  We then examine the k i n e t i c limitations of the c e l l s ,  especially  the problems associated with transport through the e l e c t r o l y t e and with the lithium metal  anodes.  If a l l of the charge which flows through a lithium intercalation c e l l a-, results  in uniform intercalation of the cathode with lithium, then the Li  composition of the host, x, can be found d i r e c t l y from the charge flow i f the weight of the host is known.  It is therefore important to see  reactions aside from intercalation occur.  i f any  Such extra sources of c e l l  capacity are generally referred to c o l l e c t i v e l y as side reactions.  As .a  check for such side reactions, c e l l s were constructed which were identical in a l l  respects to those used for i n t e r c a l a t i o n , except the cathode  only of a cleaned nickel d i s c .  consisted  In one such c e l l with a nickel cathode of  2  area 2 cm  (R26) , 100 mC of charge flowed through the c e l l on the f i r s t  discharge, while on subsequent charge and discharge cycles, only 10 mC flowed between voltage  limits of 2.8 V and 0.3 V.  Since most intercalation  c e l l s tested had a capacity greater than 1 C within these same voltage l i m i t s , the background reactions due to the other components of the c e l l can be neglected except on the f i r s t discharge.  Of course, the host  materials themselves may lead to side reactions; the host may catalyze decomposition of the e l e c t r o l y t e , lyte with the intercalated L i . persistent  or there may be a reaction of the e l e c t r o -  Such side reactions can be identified by a  difference between the amount of charge which flows on charge and  205  discharge  of the c e l l .  fying  side  cases  is quite  chemical cated  measurement  absolute  arises  i f Li !ions  t o be a p r o b l e m  leading  We n o w d i s c u s s the electrolyte  on subsequent  f o r reasonable  voltage  is  host  Moreover, is  compli-  rates  (see  As a  result,  known  voltage.  cannot  reach  factor";  described  than  A further  a l l of the  i n the apparent  utilization  cathodes  i n some  cycles.  accurately  of the cell  identi-  which  discharge  less  to the reduction  "cathode  in  in the electrolyte.  two values  i n t h e baked  in pressed  of the c e l l ,  in the electrolyte  +  by t h e s o - c a l l e d  may b e i m p o r t a n t  of  between  a problem  of the intercalated  of the Li salt cell  remains  discharge  necessary  a t some  i n the cathode,  the cathode  appear  samples  i n the Li content  particles  the f i r s t  still  o f the Li content  Li content  complication  there  from the discharge  a n d by t h e p r e s e n c e  changes  of  during  different  by t h e s m a l l  below) the  reactions  However,  capacity  this  in Chapter  does n o t 11, b u t  cathodes.  the kinetics  of the cells  used  of the cells,  i n most  used.  The t o t a l  a one molar  conductivity  solution of  - 3 LiClO^ with the  in propylene  the L i  +  ions  carbonate  carrying  (PC) , h a s been  reported  20% o f t h e c u r r e n t  conductivity of the L i  +  i o n s , a.  ,  as 5x10  (Jasinski  i s 1 x 10~  3  (fi-cm)  ,  Hence,  1971).  (fi-cm)"^.  -1  The Celgard  Li separators tortuosity  used near  a r e 25 y m t h i c k , a n d h a v e a p o r o s i t y o f r o u g h l y 50%, w i t h a unity. I f w e u s e t h e r e s u l t s d f S e c t i o n 9-3 t o e s t i m a t e t h e  2 characteristics  of current  flow  through  1 cm  bf a single  ~ find  for  5^  |n|« kT/e  estimate  of order  the problems  thicknesses  separators, for  (setting  2  y = 0 i n 10.3(10)),we  and  separator  a  associated  of typical  baked  so the l i m i t i n g  f o r studying  current  i n the steady  i n t e r c a l a t i o n throughout  adequate  limiting  with  state.  electrodes.  a r e o f t h e same  These  f o r example,  currents  The p o r o s i t i e s as f o r t h e  o f tens  a r e more  a 5 mg c a t h o d e  resistance  w a y , we c a n  order  s h o u l d be o f t h e o r d e r  the cathode.  the cells;  In t h e same  the porous  cathodes  currents  o f 21 m A / c m , a n d a  o f mA than  o f MoS_ c a n  206  be discharged to x = 3 in one hour (which is faster than usual) at a current of only 2.5 mA.  On the other hand, pressed cathodes are considerably thicker  and show lower p o r o s i t i e s .  For example, consider a disc of MoS^ , 1 mm thick,  2 1 cm  in area, with a porosity of 20%. The limiting current for interca-  lation throughout the cathode  (y = 0 in 10.3(10)) is 200 yA; using such a  current would require 1000 hours - 6 weeks to discharge the c e l l At higher currents, the cathode can not be completely  to x = 3-  intercalated.  Such  incomplete intercalation was observed in an actual MoS^ cathode (cell C3) 2 of this thickness and porosity, and area 2.5 cm , where at a current of 2 400 yA/cm  the cathode could be intercalated only in the top ^6.1 mm  (compared to 0.7 mm predicted by 10.3(10), using 10.2(5) with n = 2 for the tortuosity).  This incomplete  intercalation might also have been caused  by precipitation of LiClO^ in the large concentration gradients  expected  at these currents, as discussed in Section 10.3To check the magnitude of the losses in the e l e c t r o l y t e and at the Li interface, L i / L i c e l l s were constructed by using Li f o i l for both anode and cathode in pressed c e l l s .  Such c e l l s t y p i c a l l y showed an impedance of order  2 100 9, for 2 cm  area of Li at each electrode,  that expected from the e l e c t r o l y t e . was seen - there was no exponential  Moreover, reasonably ohmic behaviour variation of current with  expected from the Butler-Volmer equation as would be produced by e l e c t r o l y t e  considerably higher than  (7-3(1)), and no limiting currents  depletion.  due to a surface layer on the Li f o i l .  voltage  The resistance appears to be  The resistance  could be lowered by a  factor of two after passing current through the c e l l and exposing metal surface; further, the resistance 1 kHz, implying a capacitance  dropped above frequencies of order  in p a r a l l e l with the resistance of order 10 yF  and hence a surface layer which is a few angstroms thick. tances were s t i l l  fresh  low enough so that a third  (reference)  These high electrode was  resis-  207 unnecessary  in the  When a l a r g e cell,  lithium  amount o f  penetration  of  the  than  V/Q, a n d a l l o w s the e q u i v a l e n t  cells  are  Separator causing charge is  causes  Li  the  sometimes  impedance  to  of  Separators  L i , with a dull  regain  implying that  L i has  steady  An e x a m p l e  is  from such  grey  of  dendritic  the  pores.  cells,  1 arge„amounts dendrite  when  c o l o u r and  intercalation  suspected  larger  appearance;  grown t h r o u g h  while  of  far  t h e i r o r i g i n a l white  the  remain r e a s o n a b l y  Intercalation  Li TiS2was  of  penetration  Cells  used,  since  been  r e p o r t e d by Thompson  Fig.  1.  I t was  charge/discharge  ( 1 9 7 8 ) , whose  the v o l t a g e  curve  capacity  2.8  V. a n d  is  very  V on t h e  of 1.0  the  study of results  If the  cell  first  discharge,  on t h e  although  first  in  F i g . 57-  Discharge of  the c e l l  in  the  1.4  V , w h i c h c a n be as  1.6  somewhat to  x  has  shown i.n  a p o r t i o n of  for  1.8  the  made f r o m T i S ^ above  the  total: c e l l to  experi-  versus been  is kept  except  discharge  subsequent  cell  voltage  the  T i S ^ i s more com-  shows  Ti  and t h e  on  V to  have a l r e a d y  F i g . 57  a Li/LiC10^,PC/Li  of  the v o l t a g e  i n t e r c a l a t i o n of  initially.  V.  as a c h e c k  s i m i l a r to Thompson's,  t y p i c a l l y x ^ 0.7 cycles,  that  believed  behaviour 2.3  initially  a detailed  found however,  t h a n had been  between  intended  x  techniques  range  to drop to v a l u e s  Z-  The s t u d y o f  at  observed.  through the c e l l which  in the c e l l .  Li/Li  57-  TiS X  observed,  cell  is  p e n e t r a t i o n may a l s o o c c u r on r e c h a r g i n g  Li/Li  powder,  the  separators  puncture,  in F i g .  plicated  by d e n d r i t e s  full  flow through the c e l l .  mental  passes  to flow  completely  the v o l t a g e  shown  12.3  i n one d i r e c t i o n t h r o u g h a  a charge  washed w i t h m e t h a n o l , of  charge  amount o f  sometimes  show no s i g n s  cells.  separator  Such d e n d r i t e p e n e t r a t i o n order  intercalation  1.8  extra  capacity  V , and x -  s h o r t e r on t h e p a r t i c u l a r c e l l  lower v o l t a g e s produces l o n g as x = 2,  V,  a long  and w h i c h  0.5 used plateau  t  (hours)  F i g . 57 " Charge-discharge cycles for Li T i S ^ , c e l l JD61 , at a current of 310 uA. The time interval corresponding to x = 1 is shown. A case of suspected separator penetration by Li dendrites is also indicated; the c e l l is s t i l l recharging during this time.  considerably changes the subsequent  cycles of the c e l l .  The deviations  Thompson's results have been associated with intercalation of PC.  from  Powder  X-ray d i f f r a c t i o n patterns of the Li TiS~ cathode following a discharge to 1 . 8 V indicate two structures, one of which is identical to that reported by Whittingham and Gamble ( 1 9 7 5 )  for Li^TiS^.  Absorption of PC has been  seen in pressed T l c a t h o d e s , which swelled to twice t h e i r size and weight, absorbing a l l of the PC in the c e l l , before the c e l l voltage had reached 2 . 0 V.  This indicates that the extra capacity at 2 . 3 V on the f i r s t  ".  discharge is associated with PC i n t e r c a l a t i o n , while the s i m i l a r i t y of the subsequent  cycles above 1 . 8 V with Thompson's results  (which were obtained  with a different solvent) suggests that the PC intercalated material inactive above 1 . 8 V after the f i r s t  is  discharge.  The plateau near 1.4 V may be associated with a structural t r a n s i t i o n involving the PC intercalated T i $ 2 ; the Ti  i t does not seem to produce any change in  which contributes to the observed capacity above 1 . 8 V.  This  is  most c l e a r l y seen in the inverse derivative curves shown in F i g . 5 8 , taken before and after the c e l l  has been discharged through the 1 . 4 V plateau.  After the discharge through the plateau, the portion of the curve above 1 . 8 V is unchanged in shape, but is shifted upward because of additional capacity resulting from the low discharge.  The curves in F i g . 2 are s i m i l a r to those given by Thompson ( F i g . 6 ) , but the small peak near x - 1 / 9 ( 2 . 4 V) is not resolved.  Better resolution  can be seen in the curves of F i g . 5 9 ; Thompson's data is also reproduced there, plotted against voltage to the same total  rather than x, and normalized to correspond  capacity as our experimental curves.  Agreement between  our results and those of Thompson is..good, except below 2 V, where our data is distorted by the beginning of the plateau which is eventually seen near  Fig.  58 -  C u r r e n t - v o l t a g e c u r v e f o r L i x T i S 2 , c e l l JD68, a t a sweep r a t e V = 1 6 . 3 u V / s (a) b e f o r e and (b) a f t e r t h e c e l l was d i s c h a r g e d through the p l a t e a u at 1.4 V.  211  Fig.  59  _  C u r r e n t - v o l t a g e c u r v e s f o r L i ^ T i S ^ , c e l l RM12, V = 17-1 y V / s ( s o l i d c u r v e s ) . Points are data (1978) n o r m a l i z e d t o t h e same c e l l capacity.  a t a sweep r a t e f r o m Thompson  1.4 V.  The shift  in voltage of the largest peak in our results from charge  to discharge is consistent with a resistance  in the c e l l of 50 £2. Thompson  avoided this shift by incrementing the c e l l voltage by AV = 10 mV, and measuring the charge AQ which flowed until  the  current had dropped to some small l i m i t ; in his data, AQ./AV coincided on recharge and discharge. equally good results.  Except for this d e t a i l , our method appears to give The reason for the discrepancy in the voltage of the  large peak in his data (2.30 V) and its average position for charge and discharge in ours (2.33 V) is not c l e a r .  The poorer resolution of the  features seen in F i g . 58 is attributed to diffusion in the TiS^ host. the TiS^ p a r t i c l e s have radius R, diffusion effects w i l l  smear the  over a time ^ R / D , where D is the diffusion c o e f f i c i e n t ;  Fig.  features  this corresponds  2  to smearing over a voltage ^ VR /D at a sweep rate V.  If  The TiS^ used"in  58 consisted of p a r t i c l e s with R ^ 10 ym, so that VR /D ^ 20 mV, 2  assuming D ^ 10  9  cm /sec; for the data in F i g . 59, R ^ 2 ym, and 2  VR /D ^ 0.8 mV. 2  The peak at 2.33 V was studied in more detail at slower sweep rates, see  i f the top of the peak was being rounded by the f i n i t e  peak is actually a divergence in  9x/8V,  rates.  as might be expected  if it  to  If the is  produced by a phase t r a n s i t i o n , then the peak should become sharper as the sweep rate decreases.  It was found that the height of the peak (in  current) was proport i ona 1 to the sweep rate, V , within experi menta 1 error for  a reduction in V by a factor of 75, and so there appears to be no  rounding, or at least no change in the rounding over this range of V.  L i ^ T i . c e l l s show additional capacity below 1.0 V.  This capacity  is  seen most c l e a r l y in c e l l s made from crushed TiS^ crystals grown by iodine vapour transport methods, where very l i t t l e  intercalation of PC is seen.  This absence of PC intercalation appears to be related to the larger p a r t i c l e s i z e , as discussed by Dahn (1980). f i r s t discharge of a L i / L i ^ T i S ^ c e l l recharge.  F i g . 60 shows the voltage curve for the to 0 . 2 volts and its  subsequent  The plateau near 0 . 5 V suggests a f i r s t order phase t r a n s i t i o n ,  in analogy with that seen, in Li VSe  for 1 < x < 2 (Murphy and Carides 1979).  Further cycling between 2 . 8 V and 0 . 2 V gives curves s i m i l a r to those in Fig. 6jc. However, when the c e l l  is discharged to 0 . 0 5 V, a second plateau  is seen, and subsequent cycles of the c e l l voltage behaviour.  show considerably  different  Cells made with powdered T\S^ grown d i r e c t l y from the  elements also show evidence of these plateaus near 0 . 5 V and 0.1 V, but the voltage curves are complicated by the intercalation of PC.  In the light of the discussion of Chapter k, the sharp drop in the voltage curve at x = 1 s i g n i f i e s Ti$2 host at this composition.  an ordered structure of the Li in the Neutron d i f f r a c t i o n studies by Dahn et al -  :(1980)> indicate that throughout the range 0 < x < 1, the Li atoms occupy predominantly (and perhaps e n t i r e l y ) octahedral s i t e s in the van der Waals gap of the Ti  host.  Since the octahedral sites' can accomodate Li only  up to x = 1, the drop in voltage  is therefore associated with the  in s i t e energy of octahedral and tetrahedral s i t e s , as discussed  difference in connec-  tion with F i g . 13, and with nearest neighbour octahedral-tetrahedral  site  i nteract i ons. It is l i k e l y that both the plateau for 1 < x < 2 and that for 2 < x < 3 correspond to f i r s t order phase t r a n s i t i o n s .  It is clear from F i g . 60 and 61  that these transitions are quite d i f f e r e n t .  The t r a n s i t i o n from x - 1  to x - 2 is quite reversible; the difference  in the plateau voltage on charge  and discharge in F i g . 60 is i< 0 . 2 V.  The t r a n s i t i o n from x - 2 to x - 3  produces a plateau on the discharge, but no corresponding plateau on the  21k  Fig. 60 - Charge/dis charge cycles for L i T i S , c e l l JDty?, at a current of 75 yA. Note that the x scale applies only to the discharge. x  2  X in L i T i S x  2  (discharge  1.0  2.0  TIME  61 -  only) 3.0  (hrs)  C h a r g e / d i s c h a r g e c y c l e s f o r L i T i S , c e l l JD49, at a c u r r e n t 75 yA. Note that the x s c a l e a p p l i e s only t o the d i s c h a r g e . x  2  of  216  subsequent  recharge; moreover, the voltage c h a r a c t e r i s t i c s are completely  changed on subsequent  cycles.  these two transitions  is in the degree of change of the host l a t t i c e as the  transition occurs.  It is l i k e l y that the d i s t i n c t i o n between  Unfortunately, detailed structural information is not  a v a i l a b l e , and it is impossible to quantify this statement.  If the change  in the host structure in the t r a n s i t i o n from x - 1 to x - 2 is small, then this t r a n s i t i o n might-be understandable in terms of a l a t t i c e gas model for the Li in the host; in p a r t i c u l a r , it is tempting to speculate  that the  transition is caused by the p a r t i c u l a r range of s i t e energies and interaction energies discussed  in Section 4.7, which produce a phase t r a n s i t i o n from a  composition x = 1 where a l l the ";octahedral sites are f u l l , x = 2 where a l l the tetrahedral s i t e s are f u l l .  to a phase at  The large change in the  voltage behaviour following the t r a n s i t i o n from x - 2 to x - 3 makes  it  unlikely that this t r a n s i t i o n can be understood in a l a t t i c e gas model; the situation  is more like that^out 1ined in 4.9-  A l a t t i c e gas description  may, however, be appropriate for the new phase produced after the transition occurs.  This type of change of voltage behaviour after a plateau in the  voltage curve is also seen in Li MoS., and w i l l x 2' Section 12.5.  be discussed further in  At present, we are unable to account for the detailed features  in the  inverse derivative voltage curves for Li^TiS^ in the range 0 < x < 1.  In  p a r t i c u l a r , the sharp peak near x = 0.25 in F i g . 6, which stimulated much of the work in this t h e s i s , remains a mystery.  It seems unlikely that  it  is associated with spatial ordering of the Li at fractional x values, because there are no sharp drops in the voltage  in the range 0 < x < 1.  Further evidence against spatial ordering is the absence of three dimensional order in the neutron d i f f r a c t i o n studies of Dahn et al (1980), although it should be noted that these neutron studies were unable to  exclude purely two dimensional ordering.  We should note, as w e l l , that the  value of U = 2 . 5 kT for the interaction energy of Li atoms on neareat neighin F i g . 6  bour octahedral s i t e s inferred from the simple mean f i e l d f i t l i e s above the phase boundary predicted by RG calculations  (Fig.. 2 0 ) ,  which also argues against an ordered Li arrangement for x < 1 .  12.4  Li/Li  x  Mo0„  2  Intercalation Cells  Typical cycles of L i M o 0 2 are shown in F i g . 6 2 . x  The c e l l cycles over  the range 0 < x < 1 , with a drop in voltage near x = 0 . 5 .  A more detailed  view of the voltage behaviour is given by the inverse derivative curves shown in F i g . 6 3 ;  here the current on discharge is plotted in the  direction.  The recharge is seen to consist  to a change in Li composition of  negative  of two peaks, each corresponding  x - 0 . 5 ; the discharge is somewhat more  complicated, but also consists predominantly of two peaks.  The width of the  peaks in F i g . 63 is smaller than the width of the non-interacting  lattice  gas curve.:in F i g . 12c,(which has a halfwidth of 3-53 kT = 91 mV), and becomes sharper at lower sweep rates, strongly indicating f i r s t order t r a n s i t i o n s . X-ray studies by Sacken (1980) confirm two phase behaviour between 0 <• x < i and i < x < 1. system.  The curve in F i g . 63 indicates hysteresis in the Li^MoO^  Moreover, this hysteresis persists  at much lower discharge  rates;  very slow cycles taking up: to a month place the two peaks on the recharge at 1 . 3 7 V and 1.67 V, and the two largest peaks on the discharge at 1 . 3 0 V and  1.58  V (Sacken 1 9 8 0 ) .  The voltage behaviour of Li^MoO^ also shows  interesting history dependence;  for example,  i f the c e l l  is  recharged from  x = 1 to x = 1 only, the inverse derivative of the subsequent  discharge does  not contain the small peak at 1 . 3 6 V in F i g . 6 3 , but rather the curve proceeds along the dotted line indicated in F i g . 6 3 .  h—Ax = I  *  t (hours)  Fig. 62 - Charge/discharge cycles for Li Mo02, c e l l U08, at a current I = 500 yA. The time corresponding to x = 1 is indicated. x  I  I  I  0  1.2  1.4  I  1.6 V  I  1.8  (volts)  . 63 - Current-voltage curves for Li MoC>2, c e l l U04, at a sweep rate V = 9-6 y V / s . Note that the discharge current is plotted in the negative d i r e c t i o n . The dotted line indicates the curve obtained on discharge i f the previous recharge was stopped at 1.6 V. x  _ l  2.0  The drops in voltage near x = 0.5 of the Li at these compositions.  structures  In the absence of information on the  p o s i t i o n of the Li atoms in the MoO^ nature of these ordered s t a t e s .  and x = 1 indicate ordered  host, we can only speculate on the  They could be ordered occupations  of  octahedral or tetrahedral s i t e s in every tunnel, produced by repulsive i n t e r a c t i o n s in the tunnel d i r e c t i o n .  On the other hand, as mentioned  in Section 2.h,  s t r u c t u r e from the pure r u t i l e  the d i s t o r t i o n s of the MoO^  structure d i s t o r t the octahedral and tetrahedral s i t e s along the tunnels, and lead to two types of octahedral s i t e s which can each account f o r a composition  of x = £, and one type of octahedral s i t e and two types of  tetrahedral s i t e s each capable of accomodating Li atoms up to x = 1.  It is  therefore possible that s i t e energy differences produce e i t h e r or both of the voJtage drops near x = 0.5  and x = 1.  R e c a l l i n g the angular v a r i a t i o n  of the e l a s t i c i n t e r a c t i o n appropriate f o r Li^MoO^ in Section 6.2  (Fig.  3*0,  we can also propose that e i t h e r or both of the ordered states involve ordered arrays of occupied and unoccupied tunnels, to minimize the e l a s t i c i n t e r a c t i o n energy. The small amount of Li in the MoO^ to x - 0.5  host when the t r a n s i t i o n from x - 0  begins makes i t u n l i k e l y that t h i s t r a n s i t i o n i s produced by  some p e c u l i a r combination of s i t e energies and repulsive i n t e r a c t i o n energies, in analogy with the discussion in Section k.J in connection the t r i a n g u l a r l a t t i c e . x - 0.5  The t r a n s i t i o n s from x - 0 to x - 0.5  with  and from  to x - 1 are probably produced by a t t r a c t i v e i n t e r a c t i o n s between  the i n t e r c a l a t e d Li atoms, and the magnitude of the observed s t r a i n indicates that e l a s t i c i n t e r a c t i o n s contribute td these a t t r a c t i o n s . ; l t i s  l i k e l y that  e l a s t i c e f f e c t s a l s o contribute to the hysteresis in the voltage  behaviour.  12.5  Li/Li  x  MoS„ Intercalation Cells 2  The voltage behaviour seen in L i / L i x  MoS„ c e l l s is summarized in F i q . 6k. 2 3  The f i r s t discharge, i f taken a l l the way to 0.3 V, shows two plateaus,  at  1.1 V and 0.6 V, with the f i r s t plateau ending near x - 1 and the second near x > 3-  The voltage behaviour of charge/discharge cycles depends on  the depth of the previous discharges.  If the c e l l voltage  1.1 V, the c e l l cycles over the curve labelled I.  is kept above  If the c e l l  is discharged  through the f i r s t plateau but not the second, the c e l l cycles over the curve labelled II.  F i n a l l y , after discharge to 0.3 V-(through both plateaus),  curve 111 is obtained.  We denote the Li MoS„ associated with these three x 2  different curves as phase I,  II, and III  respectively.  X-ray studies by  Wainwright (1978) reveal that no noticeable change in the host occurs in phase I, but a structural change in the MoS^ host l a t t i c e occurs during the 1.1 V plateau.  This change in structure appears to involve a s h i f t  planes of Mo atoms, so that Mo atoms in adjacent sandwiches  in the  l i e one above  the other, as would be the case in either the 1T-T i S2 or 2H-Nb$2 structures (Fig.  2).  The small change in the layer spacing on going from phase I  to phase II (the layer spacing in phase I is 6.15 A*, while its maximum o value in phase I I, at 1.9 V, is 6.k0 A) appears to rule out intercalation of propylene carbonate.  The X-ray patterns obtained for phase III, however,  did not permit determination of the l a t t i c e parameters, and PC intercalation in phase III cannot as yet be ruled out. Fig. 6k implies that a l l three phases can coexist over a range of  voltage,  so c l e a r l y the phases must be only metastable over part of their voltage range.  The reverse transitions from phases III and II to phase I do not  occur rapidly enough to be observed as plateaus  in the voltage curves;  however, conversion back to phase I does appear to occur from both phase I I  6k - Summary of voltage behaviour of L i M o S , c e l l RM11. The different curves labelled I, II, III are discussed in the text. x  2  and phase  III at a slow rate at high voltages.  This conversion can be seen  by cycling the c e l l s at high voltages, as shown in F i g . 65 and 66. voltage capacity disappears  (note that phase I has very l i t t l e  but discharge to a lower voltage  The high  capacity),  reveals a plateau near the voltage  associ-^  ated with the original phase conversion, and regenerates the high voltage capacity.  The net charge flowing out of the c e l l as the high voltage  cycles proceed and phase conversion occurs is very small, indicating that the high voltage portion of phases I L a n d f i l l  are at very small x values,  smaller than indicated in F i g . 6k. We now look in more detail at the voltage curves for the three phases by examining the inverse derivative curves. obtained for phase  I.  F i g . 67 shows the curve  Considerable hysteresis, is evident.  The sharp rise  in current on discharge below 1.3 V is associated with the phase conversion to phase  I I.  The inverse derivative curves for phase  II are shown in F i g . 68.  There  is considerable structure on the curves, with the double peak near 1.8 V being the most pronounced on recharge.  On discharge, the large peak at  1.3 V in F i g . 68a is attributed to the phase transition to phase II of that material which converted to phase I on the previous recharge.  The size of  this peak decreases as the upper limit on the previous recharge is and the features  lowered,  near 1.8 V on the discharge increase correspondingly.  F i g . 69 shows the inverse derivative curves obtained in phase These show the considerable hysteresis in phase  III.  III which was also seen in  the voltage curve of F i g . 6k. On the discharge, 3 broad peaks are seen near 2.0 V, 1.2 V, and 0.3V; on recharge, only the peak at 2.2 V is independently of the depth of the previous discharge. are complicated by a loss*in the capacity of the c e l l  evident,  The curves in F i g . 69 between each curve,  Fig.  65 - Charge/discharge c y c l e s f o r L i M o S , c e l l R37, at a c u r r e n t I = 340 yA. The c e l l had been p r e v i o u s l y d i s c h a r g e d to 0.7 V through the p l a t e a u at 1.1 V. x  2  F i g . 66 - Charge/discharge cycles for Li MoS2, c e l l RM13, at a current I = 50 uA. The c e l l had been previously discharged to 0.25 V through the two plateaus at 1.1 V and 0.6 V. x  1  i  i  1  3 0  \  Discharge  < 2 0 4.  v 10  0  Charge  —  —  Ii  1.2  i  1  1.6  1.8 V  Fig. 67  _  >  2.0  2.2  (volts)  Current-voltage curve for Li MoS2> c e l l RM16, at a sweep rate V = 15-2 u V / s . The c e l l had never been discharged below 1.4 V before this data was taken, and hence was in phase I . x  ho  ON  _L_ 2.0  1.5  2.5  V (volts) Current-voltage curves for Li MoS2, c e l l RM11, at a sweep rate V = 30.1 y V / s . Discharge curve was taken immediately after the recharge curve. The:cell had previously been discharged to 0.7 V, and hence was in phase II just before the recharge began. x  V (volts)  F i g . 68b - Current-voltage curve on c e l l RM11 following F i g . 68a at the same sweep rate. Discharge curve taken immediately after recharge. ho ro oo  Charge  lOOh  <  Fig. 68c - Current-voltage curve f o n c e l l RM11 following F i g . 68b.at the same sweep rate. Discharge taken immediately after recharge.  ho  300  Charge Discharge ~ 200h < 4.  100  1.5  Fig. 69a - .Current-voltage curves for Li MoS2 , c e l l RM11, at a sweep rate V = 28.3 u V / s . The discharge curve was taken immediately before the recharge curve. The c e l l had been previously d i s charged to 0.3 V, so it was in phase III before the discharge began. x  ro o  V  (volts)  F i g . 69b - Current-voltage curve for c e l l RM11 following F i g . 69a at the same sweep rate. The discharge curve was taken before the recharge curve.  K>  F i g . 6 9 c - Current-voltage curves for c e l l RM11 following F i g . 6 9 b same sweep rate. Discharge taken before recharge.  as seen by the decrease  in the height of the peak at 2 . 0 V on discharge.  The reason for this capacity loss is not c l e a r , but it may be partly a kinetic effect associated with a layer of reaction products from side reactions on the Li anode or on the  M0S2  cathode.  The transitions from phase I to I I and I I to I I I in Li MoS_ involve: x 2 r  structural change in the host, and so l i e t i o n , as discussed  in Section 4 . 9 -  separately to phases I,  outside our l a t t i c e gas descrip-  The l a t t i c e gas model may s t i l l apply  II, or III, with different s i t e energies and  interaction energies due to the changes in the host. however, we are unable to explain the features curves with a simple l a t t i c e gas model.  At the present time,  in the inverse derivative  The fact that there are three peaks  in the inverse derivative curve for the discharge in phase III but only one for the recharge indicates that phase III a simple l a t t i c e gas;  is far more complicated than  in view of this large hysteresis,  large structural changes  it is l i k e l y :that  in the host occur within phase III, which would  invalidate the l a t t i c e gas model except possibly over narrow composition ranges.  Within phase. II , where.the structural changes are not too dramatic,  a l a t t i c e gas model might be more successfully  used.  The voltage of the plateaus associated with the transitions between the phases is very l i k e l y considerably lower than the voltage corresponding to thermodynamic equilibrium of the phases.,  This is suggested in the  inverse  derivative curve i n F i g . 6 7 , where the current begins to rise sharply well before the voltage of 1.1 V at which the plateau is observed at a normal discharge rate (Fig. 6 4 ) . first  Moreover, the plateau voltage  is lower on the  transition from phase I to I I than for subsequent t r a n s i t i o n s , as  seen in F i g . 6 5 .  The energy corresponding to the difference between the  observed and thermodynamic values of the plateau voltage w i l l  be released  as heat as the t r a n s i t i o n occurs, which could increase the rate of side reactions, and hence increase the length of the plateau over its value in the absence of such side reactions.  This would further support the arguments  already given that the high voltage portions of phases II and III correspond to a small value of x.  235  TABLE III DATA FOR CELLS DISCUSSED IN CHAPTER 12  Cell #  '  C3  Cathode material  3  Zi JD61 JD68  5  Cathode Mass (mg)  2 Cell Case  electrolyte  MoS  natura1  832  tefIon  1M L ClO^/PC  TiS  2  vapour transport  1.8  f1ange  1M L ClO^/PC  TiS  2  550°C  3-9  f1ange  1M 1 ClO^/PC  TiS  2  800°C  7.4  flange  1M L ClO^/PC  -  flange  1M L" C10^/PC  2  JD49  1  R26  Ni  R37  MoS  Molybond, in o i l  2.2  f1ange  1M L Br/PC  RM11  MoS  550°C  5.2  tef1 on  1M L ClO^/PC  RM12  TiS  550°C  4.9  tef1 on  1M L' C10^/PC  RM13  MoS  Molybond, i n oi1  1.  teflon  1M L ClO^/PC  RM16  MoS  550°C  5.0  teflon  1M L ClO^/PC  U04  Mo0  2  reduction of MoO^ in H  2  4.5  flange  1M L Br/PC  Mo0  2  reduction of MoO^ in H  2  5.  flange  1M L Br/PC  U08  2  2  2  2  6  6  Notes:  2  A l l cathodes baked onto a Ni substrate except U08, which was baked onto A l , and C3, which was a pressed cathode. Temperature, when given, indicates temperature used in growing the powders d i r e c t l y from the elements. 2  One of the two.types of c e l l s shown in F i g . 53 3 Prepared and tested by Chris Hodgson  4  Prepared and tested by Jeff Dahn ^Prepared by Jeff Dahn ^Prepared and tested by Ulrich Sacken  236  CONCLUSION  CHAPTER 13  SUMMARY AND FUTURE WORK  13.1  Summary of the Thesis  The purpose of this thesis has been to elucidate the physical mechanisms which may occur during i n t e r c a l a t i o n , and to provide a conceptual framework in which to discuss  intercalation systems.  The d e f i n i t i o n of an interca-  lation compound in Chapter 1 led naturally in Chapter k to the application of  the l a t t i c e gas model to describe intercalation systems.  discussion of the l a t t i c e gas model that phase transitions gas lead to f l a t of  We saw in our in the  lattice  regions in the voltage V as a function of the composition x  the intercalation compound, corresponding to peaks or divergences  in  -3x/3V. ^Moreover, at compositions corresponding to a f i l l e d l a t t i c e of p a r t i c l e s commensurate with the total  l a t t i c e , a drop in voltage,  corresponding minimum in -3x/3V, occurs.  and a  These commensurate structures can  be due to different s i t e energies in the l a t t i c e , or can be produced by repulsive interactions between the intercalated atoms; in the latter case, the onset of long range order is accompanied by a peak in -3x/3V, but this peak occurs at a value of x which is different than that corresponding to the f i l l e d commensurate structure.  A t t r a c t i v e interactions between interca-  lated atoms can lead to phase separation, and a plateau in the voltage curve over an appreciable range of x; we also saw examples of such plateaus produced by appropriate combinations of s i t e energy differences sive interactions  and repul-  (Section 4.7), or by three body forces between intercalated  atoms (Section 4 . 8 ) .  In our discussions of the l a t t i c e gas model, we  emphasized the mean f i e l d solutions, which were shown to provide a reasonable approximation to calculations of the variation of the voltage  (or chemical  potential)  with composition, although they provide less  results for the derivatives of the voltage,  satisfactory  such as 9x/3V.  Changes in the host caused by intercalation contribute to the  interaction  between intercalated atoms, and we discussed these contributions in two parts, electronic and e l a s t i c .  In the discussion of the electronic  inter-  action, we saw in a s p e c i f i c example, where the host is regarded as a free electron gas in a j e l l i u m background and the intercalated atom as a screened ion, that it can be misleading to separate the energy of the intercalate  into  an ionic and an electronic component, because the variation in energy of the total atom (the interaction energy) of the Fermi level of the electrons.  is not proportional to the variation In discussing the e l a s t i c  interaction,  we saw that e l a s t i c interaction energies can have the same order of magnitude as electronic  interaction energies,  and moreover lead to a large  dependence  of the interaction energy on the boundary conditions at the host surface.  We also discussed some of the k i n e t i c properties of intercalation  cells,  with p a r t i c u l a r emphasis on the role of diffusion of the intercalated atoms in the host.  We discussed the effect of interactions between intercalated  atoms on the variation of the diffusion coefficient  with composition, and  presented results of a simple hopping model to i l l u s t r a t e these effects; this model calculation also provided another example of the degree of success of the mean f i e l d approximation in treating l a t t i c e gas problems. We extended, the recent treatment of the effects of a constant coefficient Atlung et al host.  diffusion  on the discharge characteristics of an intercalation c e l l by (1979)  to the case of a f i r s t order phase transition in the  We saw in p a r t i c u l a r how diffusion problems can smear the  plateau produced by a f i r s t order transtion.  voltage  The complications produced  by trying to avoid diffusion problems by using powdered cathodes in  239  intercalation c e l l s was also discussed.  Experimental results were presented for three intercalation systems, Li T i S „ , X  Li Mo0_, and Li MoS„.  £-  X  These systems are complex, and none corres-  X  pond exactly to the simple model systems discussed.  Li^TiS^ for 0 < x < 1  is probably describable as a l a t t i c e gas with f a i r l y weak nearest neighbour interactions  (of order 2 . 5 kT).  No ordering is observed in this  the peak in -9x/8V near x = 0 . 2 5 is s t i l l a puzzle.  range of x;  If this peak does  indicate a phase t r a n s i t i o n , we do not know as yet what kind of state produced by the t r a n s i t i o n . order phase transitions. hysteresis,  Li  i S ^ also shows two very different  The f i r s t  is  first  involves only a small amount of  and may be explainable in terms of a l a t t i c e gas model, such  as that discussed in Section k.7. The second t r a n s i t i o n , from x - 2 to x - 3 , produces a large change in the voltage behaviour of the c e l l s , and probably involves a considerably larger change in the host structure than the f i r s t t r a n s i t i o n , from x - 1 to x - 2. ' transitions with some hysteresis,  Li Mo0„ shows f i r s t order x 2  and is probably an example of a l a t t i c e  gas with a t t r a c t i v e interactions, with e l a s t i c effects presumably of some importance. host structure.  Li M0S2 i l l u s t r a t e s the effects of a large change in the Here the l a t t i c e gas models apply in a piecewise fashion at  best, over a restricted range of composition.  Phase II appears to be a  true intercalation phase, but phase III may not be.  240  13.2  Suggestions  for Future Work  This thesis has identified and discussed the physical mechanisms underlying i n t e r c a l a t i o n .  However, our understanding of individual systems  is not complete enough at present to allow quantitative c a l c u l a t i o n s . Further work is needed in both theory and experiment before a complete understanding of intercalation systems is achieved.  On the theoretical interactions  side, detailed calculations of e l a s t i c and electronic  in s p e c i f i c systems are needed, to explore the  limitations  of the continuum approximations which were used in both Chapter 5 and Chapter 6.  For example, it would be very useful  out how large a deviation from the i n f i n i t e medium e l a s t i c  to find  interaction,  00  W (_r) , can be expected for atoms separated by one or two l a t t i c e  spacings.  More work is needed on l a t t i c e gas models, to increase our understanding of the ways that interactions between the atoms modify the voltage curves of intercalation systems.  In p a r t i c u l a r , calculations of l a t t i c e gas models  with interactions of the form found in F i g . 33 and 34, appropriate for the interaction of e l a s t i c  dipoles, would be useful.  Further work is also needed on the experimental side, to explore in more detail  the mechanisms responsible for s p e c i f i c features  of intercalation c e l l s .  In this  in the voltage curves  regard, neutron d i f f r a c t i o n studies, and  more careful dynamic X-ray d i f f r a c t i o n studies of the type reported by Chianelli et al  (1978), promise to provide a great deal of information.  It would be of value to explore the effects of p a r t i c l e size on the c e l l voltage c h a r a c t e r i s t i c s ; the e l a s t i c if  interactions between intercalated atoms,  important, should lead to observable effects.  done to study the changes  in features  and such work is needed.  It could be useful  Very l i t t l e work has been  in the voltage curve with temperature, to study the effects of .  modifying the host compound by substitutionally replacing host atoms with atoms nearby in the periodic table.  Experiments such as NMR or perturbed  angular correlation studies (of the type recently reported by Butz et al 1979) may also be of interest.  F i n a l l y , the large amount of information  which could be obtained in single crystal experiments, such as transport studies,  makes further attempts to produce large single crystals of inter-  calated hosts very worthwhile.  On the more p r a c t i c a l side,  it has become evident that a large variety  of host materials intercalate lithium, and it is conceivable that the optimum intercalation host for use in high energy density batteries not yet been discovered.  A considerable effort  needed to explore new intercalation systems.  has  in materials research is  Further work is also needed  to solve the problems associated with lithium cycling and e l e c t r o l y t e decomposition, which we only b r i e f l y mentioned.  F i n a l l y , the considerations  of Chapter 9 and 10 make it clear that there is an optimum combination of p a r t i c l e size and cathode thickness consistent with the requirement of a high capacity and high discharge rate c e l l , and further study in this area is also needed.  Note added in proof:  Calculations for an Ising model in zero f i e l d on  a simple cubic l a t t i c e with both dipolar and short range (nearest neighbour) interactions between spins have been reported recently by Kretschmer and Binder ( 1 9 7 9 ) ; this represents one step toward understanding l a t t i c e gases with dipolar interactions at arbitrary compositions, one of the suggestions for future work given above.  242  APPENDIX A EQUIVALENCE OF LATTICE GAS AND ISING MODELS  For reference, we give the formulas relating the l a t t i c e gas and Ising models for a l a t t i c e of N s i t e s , where each s i t e has y nearest neighbours. We assume nearest neighbour interactions only.  In the l a t t i c e gas model, atoms occupy sites on the l a t t i c e . is assigned an occupation number,  Each s i t e a  n^ = 0,1; n^ is unity i f the s i t e  is  If the energy of an isolated atom is E  occupied, zero i f it is empty.  Q  (the s i t e energy), and the interaction energy of adjacent atoms is U, then the energy E{n^} of some configuration ^  E{n) = E y n + U y n n . a o a ^. a a L  a  In (1),  n a  ^ of atoms is  (l)  •  1  <aa'>  the f i r s t sum is over a l l s i t e s , and the second sum is over a l l  pairs of s i t e s .  From (1),  -(F-yn)/kT  the grand p a r t i t i o n function is  /  Y  in^i  E  o"  U  y  \  Y  a  v  <aa'>  \  , . /  using the abbreviated notation 1 =• I  •I { n  a  In (2),  1  n  }  1  I = 0  n  2  = 0  1 ••• n  I  N  = 0  (3)  *  F is the free energy, y the chemical p o t e n t i a l , and n the total  number of occupied s i t e s , given by  n - I n a  a  .  (4)  In the Ising model, each l a t t i c e s i t e a is assigned a spin, s^ = ± 1 , which can point either up (s^ = +1) or down (s  = -1).  If parallel  (anti-  p a r a l l e l ) spins on adjacent sites have an interaction energy - K ( + K ) , then the energy Ei;{s } of some configuration {s } of spins in an upward magnetic m ot ot f i e l d B is  E  (s } = ma  where E  m  Q  -B  Js - K J s s , + E ^ a ^ ^ . ^ a a ' m o a <aa >  is some additive constant.  "F /kT m  (5)  v  /  1  =  r  y  \  L  eXpJT^ry  \kT  f t ts Ja  \  L  a  S  a  +  -pjr  y  From ( 5 ) , the p a r t i t i o n function is  S  S  kT .^ a a <aa'> L  1  -  -;-=-  )  kT / /  (6)  '  v  where F is the magnetic free energy, m We relate the l a t t i c e gas and Ising models by observing that the sums (2) and (6) have the same form, so we can equate the grand free energy, F-yn,of"the l a t t i c e gas to the magnetic free energy, F , of the Ising model. We replace s  s a  in (6) by n  using  = 2n - 1 . a  (7) ' w  Then, introducing the average occupation, x, of the l a t t i c e gas model  x = <na >  =  AI an  N  L  =Nxr  v  (8)  '  and the magnetization per spin, m, of the Ising model  m = <s > a  =  1  r  W  L  TT) S  a  a  we obtain the following relations:  (q) "  K Z  =  F m  m  K  =  =  F  "^  n  2x- 1  " ^  (-o-f) o=!^o>-T • B =i y E  E  m  (10)  ( ) n  (12)  (13) (1*)  APPENDIX  B  ONE D I M E N S I O N A L  H e r e we treating  give  the  for  one  dimensional  compactly  written  excellent  reference  there  complete  the  We a s s u m e conditions of  Appendix  in  terms for  s  some  of  lattice  of  the  details  a one  (so  reference  the  one of  ISING  the  gas  Ising  important  model. model  dimensional  the  transfer  dimensional  lattice  = s^ ) .  partition  The  MODEL  of  The  (see Ising  matrix  needed  formulas  are  Appendix model  is  solution  N spins, function  formulas  with 2^  is  A).  more An  Thompson are  the  (1972);  given.  periodic (in  in  boundary notation  A)  (D  In  the  transfer  L J s )  given  1  matrix  solution,  one  introduces  a matrix  L with  elements  by  (2)  This  allows  Z.. t o N  =  n  z N  and of  ;  then L  7 i ss .  be w r i t t e n  (s  a  a  expressed  ILIS _,_,)  (3)  'a+1  1  simply  in  terms  as  N  These  as  eigenfunctions  as  are  N ->•  given  0 0  by  of  the  eigenvalues  A.., A  9  (A. > A „ )  K/kT A  1,2  =  cosh B/kT ± s i n h B/kT + e 2  6  where the + (-)  -4K/kTj  (5)  ( X ) . From (k) and ( 5 ) we can obtain  sign refers to X  2  the thermodynamics, such as the magnetization per s p i n , m, which is  sinh B/kT -WkTjJ sinh B/kT + e  m=  (6)  As shown by Thompson (1972), the correlation functions can be found quite e a s i l y  in terms of the eigenvalues,  f/unotions .".'<)>.(s); j = 1,2,  of L.  X^ and X , and the (real) 2  The resulting expressions  eigen-  involve the "inner  product"  (i|s|j) =  I s<J>.(s)<J>.(s) s=±1 1  (7)  J  The various terms from (7) can be written in terms of m (6) very simply:  0|s|l) = m  (8)  (2|s|2) = -m  (9)  ( 1 | S | 2 ) = (21s | 1) = For  (1-m ) 2  (10)  2  two s p i n c o r r e l a t i o n f u n c t i o n s ,  = j  1  f t - )  r  (  ,  l  s  ,  J  )  2  -  theresult is  m  a  +  ( ^ )  r  °-  m 2 )  In evaluating the transport properties of the hopping model of Chapter 8, we w i l l  need the following k spin correlation function:  (11)  247  <s 1  s s s,> * L  4  =  X. X. X, I -1-1 >i ( i | | i ) ( i | s | j ) ( j | s | k ) ( k | s | l ) ljk 1 1 1 s  A  A  A  •4  # a"  (12)  248  APPENDIX C  ONE DIMENSIONAL LATTICE GAS OF HARD SPHERES  We wish to calculate the chemical potential of a one dimensional l a t t i c e gas of hard spheres of diameter d; that i s , a l a t t i c e gas where the interaction between two atoms is i n f i n i t e i f they are separated by less than d-1 empty sites and zero otherwise.  The entropy, S, is determined by  the number of ways to place n atoms on N sites such that a l l atoms are -S/k separated by at least d-1 empty s i t e s ; this number is e  , where k is  Boltzmann's constant. We assume periodic boundary conditions, so the l a t t i c e can be considered as a ring of N sites held together by N bonds.  For d = 1, the atoms do  not interact, and so they can be placed at random over the r i n g , which gives  (D For d = 2, we use the following construction (Rao and Rao 1978).  Consider  two rings of s i t e s , one consisting of n f i l l e d s i t e s , the other consisting of N-n empty s i t e s .  Cut n of the bonds in the empty ring, and place the  n segments between the n sites in the f u l l this  ring.  The number of ways to do  is  (2)  This construction works only for N ;large, since it does not distinguish between arrangements which d i f f e r by a c y c l i c permutation of the s i t e labels.  When the two rings have been f i t t e d together,  sites are separated by at least one empty s i t e .  a l l pairs of f i l l e d  For d = 3, we use the same construction, except that when making the cuts in the empty ring we must ensure that any pair of cut bonds is separated by at least one uncut bond.  The number of ways to make the cuts is then the  same as the number of ways to place n atoms on N-n sites with no nearest neighbours, which is just  g  S/k  =  /(N-n) - nj  =  (2) with N replaced by N-n:  /N - 2n  (3)  The continuation to arbitrary d is obvious.  = Nk{ [l - x ( d - l ) ] £ w [1 - x(d-l)]  The chemical potential  p.T|s.  k m  We thus have  - x£nx - (l-xd)^i (1-xd)}  .  (4)  is  /»6-«(d-ii^  ( 5 )  This reduces to the non-interacting r e s u l t , 4.3(5), for d = 1, and to the solution for nearest neighbour interactions, U, in the i n f i n i t e U l i m i t , for d = 2.  The limit d  0 0  corresponds to the continuum limit of a one dimen-  sional, gas of rods of length d in a box; of length N.  In this case,  x ^ 1/d «: 1, and we find  (6)  S = nk [l +&i (x" -d)] 1  which agrees with the continuum r e s u l t , the so-called Tonks gas  (see Thompson  1972) when we interpret 1/x as the average length available for each rod.  APPENDIX D  ONE DIMENSIONAL LATTICE GAS WITH TWO SITE ENERGIES  Consider a one dimensional l a t t i c e gas, where the s i t e energy alternates between two values, E^ and E^, as we move along the l a t t i c e .  The solution to  this problem is needed in Appendix E, which discusses the effects on one dimensional solutions of introducing weak three dimensional coupling between chains of a l a t t i c e gas.  This problem was considered by Stout and  Chisholm (1962) using a transfer matrix solution; here we extend their solution to derive the results we need in Appendix E .  The one dimensional l a t t i c e with alternating s i t e energies may be described as two interpenetrating s u b l a t t i c e s , energy E^ and E^.  i = 1 and i = 2, of s i t e  We label the sites along each sublattice by a, so..that  each s i t e is i d e n t i f i e d by the two labels a and i , in the sequence ai = 11, 12, 21, 22, 3 2 , . . .  If nearest neighbour atoms interact with an energy U,  then the energy Efn^.}of some configuration of atoms on the l a t t i c e  E { n  ai  }  =  I' 1 a1 E  a  n  +  £ 2 a2 E  a  n  +  U  ^  aa 1 a 2 W l , 1 >' •  (n  n  +  is  (  l  )  It is convenient to solve the problem in the Ising notation (see Appendix A ) , so we introduce the following Ising variables:  (2)  Then the p a r t i t i o n function (grand p a r t i t i o n function of the l a t t i c e gas) becomes  (J.}  B,  !l7  kT^sa1  E X P  +  a  kT ^ s a 2  +  a  kT ^  a  ( s  a1sa2  +  s  (5)  a 2 a + 1 ,1 * s  '  ai  If we now perform the sum over sublattice 2 , and introduce the f i e l d s B  1  B and  defined by  B=  B = 1  2  B  V  V  B  (6)  2  (7)  we can easily show that  *"-(,*}  5 *«I (  (5) can be written in terms of a transfer matrix L as  |LL+I  (8)  *«H.I>  al  where the matrix elements of L are  (s j L | s ' ) = exp _K_ _ _ , . J _ (s+s') kT kT 2 s  s  and Lt is the transpose of L. given in appendix B. of  Z  N  BJ_ ( s - s ) kT 2 1  (9)  For B' = 0 , L reduces to the transfer matrix  is given in terms of the eigenvalues, X^ and X^,  the matrix LLt by  _ N  N/2 1  N/2 2  N/2 1  where X^ and X^ are given by  .. ^ a S  (10)  252  , A  = e  1,2  1  2K/kT  . 2B ^ -2K/kT . 2B' cosh 7-=- + e cosh - r = kT kT  ,2K/kT ,2B -2K/kT . 2B x (e cosh-j^jr + e cosh-j^j ,  A  2  ..  ,  ,  + 2 - 2cosh 0  4K  (11)  with the positive sign referring to Xy From (10) and (11), we can derive the following expression for the magnetization per spin, m:  m=  <s , + s „> al a2  e  2K/kT . . 2-B s i nh kT  (12)  V  where the denominator, V, is given by  2K/kT . 2B , -2K/kT , 2B cosh T-=- + e e cosh kT kT  1  V  2  =  + 2-2  coshf^ kT  (13)  m is related to the average occupation of the two sublattices, x^ and x^, in the l a t t i c e gas problem by  m = x.j +  - 1 = 2x - 1  where x is the average occupation of the overall l a t t i c e .  (14)  We can also  calculate the difference in occupation of the two sublattices:  e 1 "2  X  X  =  -2K/kT . . 2B sinh kT  V  with V given by (13).  (12) and (15) are needed in Appendix E.  (15)  253  APPENDIX E  EFFECTS OF WEAK COUPLING BETWEEN LATTICE GAS CHAINS  The effects of weak interchain interactions in a l a t t i c e gas on a l a t t i c e of chains was discussed q u a l i t a t i v e l y in Section k.G.  Here we  present a more detailed solution of this problem, using the exact one dimensional  l a t t i c e gas results to describe intrachain interactions, and  mean f i e l d theory to introduce the interactions between chains.  This  problem was treated previously by Stout and Chisholm ( 1 9 6 2 ) , and applied to antiferromagnetic ordering in linear chain crystals of  CuCl . 0  We assume a l a t t i c e of chains of s i t e s , with each chain coordinated by y neighbouring chains, as in the r u t i l e structure, where y = k, or in the two dimensional example shown in F i g . 2 6 , where y = 2.  We exclude a  triangular l a t t i c e of chains, or cases where alternate chains are shifted by half a l a t t i c e spacing along the chain d i r e c t i o n .  The occupation of  some given s i t e is denoted by n . , where 3 labels the different chains, and a ai ' ' 7  the two indices a and i label sites along each chain, as discussed in Appendix D.  The intrachain nearest neighbour interaction, U, is assumed to  be much larger than the interchain nearest neighbour interaction, U ' ; both U and U are assumed positive 1  (repulsive interactions), and a l l other i n t e r -  actions are assumed to be zero.  Assuming a l l s i t e s have the same s i t e  energy, E ^ , the energy of some configuration of atoms over the sites  is  (0  ai <33 '> ai ai where <33'> indicates a sum over pairs of nearest neighbour chains. treat the interaction  U  1  We  in mean f i e l d by replacing one of the occupation  numbers in the last term in (l)  by an average value.  Anticipating an ordered 3'  structure as shown in F i g . 26, we replace the value of n . for a given sublattice on one chain 6 ' by the average occuptation of the other subl a t t i c e on the adjacent chains, 3 .  This means that in (1), we make the  subst i tut i on 3'  3  OI  D  n , •+ <n _> = x. al a2 2  (2a)  n „ •> <n -> = x. a2 al 1 p  (2b)  .  P  Now (1) can be rewritten as  E(n  ai  (^^.•M^-v*  .}  'a  x  +U  v/ 3  3  x  3  3  'a  (3)  H a l V a2 a+l,1 a n  +  n  n  x  We see from (1) of Appendix D that each term.in the sum over 3 in (3) is the energy of the one dimensional l a t t i c e gas with two s i t e and  E  l  energies,  which are given by  =  E  o  V  +  X  (ha)  2  (hb)  Thus, (12) and (15) of Appendix D can be used immediately (note K = -kii) , i f we make the following  U  - E -U o  B'=f(x 2  X l  )  ^'(x h 2 U  identification:  x 1  + x  (5a)  (5b)  o b t a i n t h e . f o l 1 owi ng s e l f - c o n s i s t e n t  We t h u s and  y = x  - x  2  e"  equations  form = £ x  +  -| " 1  x  1  sinh  U / 2 k T  2 (B -yU' m/8)/kT  m =  V  e  U / 2 k T  sinh(yU'yAkT)  V where  ,  o  - i ( M - E  0  - U - ^ )  t h e d e n o m i n a t o r , V,  and  V  1  =| "  - B  +  I ^  i s given  by  cosh[2(B -YU'm/8)/kT] + e  U / 2 k T  e  U / 2 k T  o  c o s h (yU y A k T ) 1  + 2 - 2cosh(U/kT)  I f y = 0, t h e n  x^ = x ^ , a n d no l o n g  i s always a s o l u t i o n by  range o r d e r  of (7)• Ordering  occurs  exists.  when  We s e e t h a t y =  (7) i s a l s o s a t i s f i e d  some n o n z e r o v a l u e o f y.  For U  <K kT a n d U  1  equations  occurs.  4e" m =  U / k T  o  /kT)  yU /kT 1  Y =  This  c o n s i d e r a b l e s i m p l i f i c a t i o n i n t h e above  F o r y 4 0, ( 6 ) a n d (7) become  sinh(2B  64e"  Ordering  «U,  1  c o s h ( B /kT)" o (yU'/kT)  U / k T  2  2  occurs  gives  f o r T < T , where T  c  i s d e f i n e d by s e t t i n g y = 0 i n ( 1 1 )  For B = 0 and Y = 2, the condition (11)  is identical to the Onsager result  Q  k.G(k)  , for U  dimensional  1  «U.  For T > T , y = 0 and m is given by the usual one  Ising r e s u l t ,  (6) of Appendix B.  To see what sort of effect  the ordering has on the chemical p o t e n t i a l , we show in F i g . 70 a plot of 8x/9u versus x for U = lOkT and Y U ' / 2 = 0.01 U .  It is seen that even though  the "order parameter", y, rises abruptly at the t r a n s i t i o n , only a very small feature  For U  1  is produced in 8x/3u.  > kT, the solutions  (6),  (7),  in the mean f i e l d solutions discussed  suffer the same problem encountered  in Chapter k: they predict that the  ordered phase extends over' the entire range 0 < x < 1. for Y U ' / 2 = 0.01 U is shown in F i g . 71.  A phase diagram  It is seen that for U' « kT,  the ordered phase is confined to a narrow region near x = i , but for U  1  > kT, the ordered phase extends over most of the range 0 < x < 1.  257  X  Fig.  70 - (a) Inverse derivative -3x/8V versus composition x for a l a t t i c e gas of weakly interacting one dimensional chains,' calculated using the mean f i e l d approximation discussed in the text. The intrachain interaction is U = 10 kT, while the interchain interaction is U = 0 . 0 2 M/y. 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