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. (b) Enlargement of (a) near x = 0 . 5 , also showing the long-range order parameter y = x -x-|. 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Physical mechanisms of intercalation batteries McKinnon, W. Ross 1980
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Title | Physical mechanisms of intercalation batteries |
Creator |
McKinnon, W. Ross |
Date Issued | 1980 |
Description | This thesis identifies and discusses physical mechanisms in intercalation batteries. The effects of interactions and ordering of intercalated atoms on the voltage behaviour of intercalation cells is described, largely in terms of the lattice gas model of intercalation. Particular emphasis is given to the mean field solutions of the lattice gas model, which are compared to more exact solutions for several cases. Two types of interaction between intercalated atoms are discussed, namely electronic and elastic interactions; it is found that both can be important in intercalation compounds. The kinetics of intercalation batteries is also discussed, with emphasis on overpotentials due to diffusion of the intercalated atoms in the host lattice. Experimental studies of the voltage behaviour of three types of lithium intercalation cells, Li[sub=x]TiS₂, Li[sub=x]MoO₂, and Li[sub=x]MoS₂, are presented, which illustrate the variety of voltage behaviour found in intercalation cells. |
Subject |
Clathrate compounds Storage batteries |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-03-26 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0085598 |
URI | http://hdl.handle.net/2429/22626 |
Degree |
Doctor of Philosophy - PhD |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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