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Physical mechanisms of intercalation batteries McKinnon, W. Ross 1980

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PHYSICAL MECHANISMS OF INTERCALATION BATTERIES by W. ROSS MCKINNON B . S c , Dalhousie University, 1975 M . S c , Dalhousie University, 1976 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n 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 th is thes is in p a r t i a l fu l f i lment of the requirements fo an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree ly ava i l ab le for reference and study. I fur ther agree that permission for extensive copying of th is thes is for scho la r ly purposes may be granted by the Head of my Department or by h is representat ives . It is understood that copying or p u b l i c a t i of th is thes is fo r f i nanc ia l gain sha l l not be allowed without my wri t ten permission. r i on Department of ^~V^S>\c£, The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date ~V A4p *VA \ ° \ ' 5 r C ^ i i ABSTRACT 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 f ie ld 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 e las t ic interactions; i t 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 lat t ice . Experimental studies of the voltage behaviour of three types of lithium intercalation ce l l s , Li Ti S_ , Li MoO. , and Li MoS„ , are presented, x 2 x 2 x 2 which i l lustrate the variety of voltage behaviour found in intercalation ce1 Is. TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i i i 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 8 CHAPTER 2 LAYERED COMPOUNDS AND RUTILES 9 2.1 Introduction 9 2.2 Layered Transition Metal Dicha1cogenides -Structure and Properties 10 2.3 Intercalation of Transition Metal Dicha1cogenides 16 l.k Metal Dioxides with Rutile-Re1ated Structure -Structure and Properties 19 2.5 Intercalation of Rutiles 2 5 CHAPTER 3 FURTHER PROPERTIES OF INTERCALATION AND RELATED PHENOMENA 26 3- 1 I nt roduct i on 26 3.2 Methods of Intercalation 26 3.3 Intercal at ion of Graphite 28 3.h Hydrogen in Metals 29 3.5 Interst it ial Compounds of the Transition Metal Di chal cogen i des 32 3.6 Oxide Bronzes 33 3.7 Superionic Conductors 3^ PART B: Thermodynamics of Intercalation Batteries 36 CHAPTER k LATTICE GAS THEORY OF INTERCALATION 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 Field Solution of the Problem of Ordering 52 h.S One Dimensional Lattice Gas 64 k.l Interacting Lattice Gas with Different Site Energies.. ....7^ i v Page 4.8 Inclusion of Three Body Forces 77 4.9 Changes in the Host 80 CHAPTER 5 ELECTRONIC INTERACTIONS BETWEEN INTERCALATED ATOMS 83 5-1 Introduction 83 5.2 Screened Coulomb Interaction 84 5.3 Metal - I nsul ator Transitions 95 CHAPTER 6 ELASTIC INTERACTIONS BETWEEN INTERCALATED ATOMS 99 6. 1 Introduction 99 6.2 Infinite Medium Interaction W°° 104 6.3 The Image Interaction W-1- 112 6.4 Lattice Gas Models and Elastic Interactions 117 6.5 Chemical Potential in Non-homogeneously Intercalated Hosts. 122 6.6 Limitations of the Theory 125 PART C: Kinetics of Intercalation Batteries 129 CHAPTER 7 KINETICS OF ELECTROCHEMICAL CELLS 130 7.1 Introduction 130 7.2 Electrochemistry of Intercalation Cells 131 7>3 Losses Due to Transport Across the Interfaces 136 7.4 Transport Through the Electrolyte 138 7.5 Diffusion in the Host 141 CHAPTER 8 DIFFUSION IN INTERCALATION COMPOUNDS 143 8.1 Introduction 143 8.2 Behaviour of D(x) 1 45 8.3 Model Calculation of Diffusion in a One Dimensional Lattice 148 CHAPTER 9 DIFFUSION 0VERV0LTAGES IN INTERCALATION CELLS 160 9. 1 Introduction 160 9.2 Diffusion for Constant D 161 9.3 Motion of a Phase Boundary 171 CHAPTER.10 POROUS ELECTRODES 1 80 10.1 Introduction 180 10.2 Ohmic Models 182 10.3 Electrolyte Depletion T85 PART D: Experimental Procedure and Results 189 CHAPTER 11 EXPERIMENTAL PROCEDURE 190 11.1 Introduction 190 11.2 Materials Used 190 .11.3 Cathode Preparation and Cell Assembly 192 Page 11.4 Techniques Used to Study Intercalation Cells 1 gif 11.5 Effect of Series Resistance and Diffusion on Current-Voltage Curves 196 CHAPTER 12 EXPERIMENTAL RESULTS 203 12.1 Introduction 203 12.2 Excess Capacity and Kinetic Limitations of the Cells 204 12.3 L i / L i x T i S 2 Intercalation Cells 207 12.4 L i / L i x M o 0 2 Intercalation Cells 217 12.5 Li/Li xMoS2 Intercalation Cells 221 CONCLUSION 236 CHAPTER 13 SUMMARY AND FUTURE WORK 237 13.1 Summary of the Thesis ...237 13.2 Suggestions for Future Work 240 APPENDICES 242 A. . Equivalence of Lattice Gas and Ising Models 242 B. One Dimensional Ising Model 243 C. One Dimensional Lattice Gas of Hard Spheres 248 D. One Dimensional Lattice Gas With Two Site Energies 250 E. Effects of Weak Coupling Between Lattice Gas Chains 253 BIBLIOGRAPHY 259 vi LIST OF TABLES Table Page I Transition Metals Which Form Layered D i cha 1 cogen i des 10 II Metals Whixh Form Rutile-Related Oxides 19 III Data for Cells Discussed in Chapter 12 235 VI I LIST OF FIGURES Figure Page 1. Schematic L i / L i Ti S_ Intercalation Cell 2 x 2 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 Di cha 1 cogen I des 15 6. Voltage and Inverse Derivative -8x/9V for Li TiS , 0 _< x <_ 1 . . . . 18 7. Ruti le Structure. . 20 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. Schematic Intercalation Cell 38 12. Voltage and Inverse Derivative -8x/8V for Non-interacting Lattice Gas with One Site Energy kS 13. Voltage and Inverse Derivative -8x/8V for Non-interacting Lattice Gas with Two Site Energies kj 14. Voltage and Free Energy for Lattice Gas with Attractive Interactions in Mean Field Theory 50 15. Decomposition of a Triangular Lattice into Three Sublatt ices . . . .52 16. u - 9Ux for Triangular Lattice Gas with Nearest Neighbour Interactions U = 4kT in Mean Field Theory 55 17. Voltage and Free energy near the First 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 19. Voltage and Inverse Derivative -3x/9V for Triangular Lattice Gas with Nearest Neighbour Interactions U = kkl in Three Sublattice Mean Field Theory 60 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 22. Sublattice Occupation and Voltage for Triangular Lattice Gas with Nearest Neighbour Interactions U = kT/0.72 in Three Sublattice Mean Field Theory 65 23. Voltage for One Dimensional Lattice Gas with Attractive arid Repulsive Nearest Neighbour Interactions 67 2k. One Dimensional Lattice Gas with Repulsive Nearest Neighbour Interactions Showing a F i l l i n g Mistake 68 25. Voltage and Inverse Derivative - 9 x / 3 V for One Dimensional -. Lattice Gas with Repulsive Hard Sphere Interactions 70 - 2 6 . Two Dimensional Lattice of One Dimensional Chains 71 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 78 29. Voltage for Triangular Lattice Gas with Nearest Neighbour Two and Three Body Interactions 79 30. 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 103 1J 33. Polar Plot Showing Angular Variation of Strain Induced Interaction W°°(_r) Between Two Intercalated Atoms in Layered Compounds 10 8 34. As in Fig . 33, for Rutile Structures 109 35. Schematic Summary of the Nature of the Strain Induced Interaction W°°(_r).;. 110 36. Schematic Discharge Curve of an Intercalation Cell Showing Hysteresis 119 i x Figure Page 37- Elastic Equivalence of a Plane of Intercalated Atoms to a Dislocation Loop 120 38. Interaction Between Two Dislocation Loops 121 33- Schematic Intercalation Cell 131 hO. Mobility and Enhancement Factor for One Dimensional Lattice Gas with Repulsive Nearest Neighbour Interactions U = 5kT 15/4 41. Diffusion Coefficient and "Conductivity" xM corresponding to Fig. 40 155 42. Mobility and Enhancement Factor for One Dimensional Lattice Gas with Attractive Nearest Neighbour Interactions U = -2 .5 kT 158 43. Diffusion Coefficient and "Conductivity" xM corresponding to Fig. 42 159 44. Three Geometries Considered in the Discussion of Diffusion of Intercalated Atoms.. 161 45. Surface Composition for Intercalation at Constant Current for a Constant Diffusion Coefficient 165 46. Fractional Capacity for Intercalation at Constant Current for a Constant Diffusion Coefficient 167 47. Surface Composition for Intercalation at Constant Current in the Case of Phase Boundary Motion 175 48. Effects of Diffusion on a Voltage Plateau in an Intercalation Cell 1 76 49. Fractional Capacity for Intercalation at Constant Current in the Case of Phase Boundary Motion 178 50. Planar Porous Intercalation Electrode 180 51. Resistor Chain Used to Model Porous Electrodes 183 52. Resistor-Capacitor Network Used to Model Intercalation of Porous Electrodes '....]8k 53- Two Types of Pressed Cells used for Lithium Intercalation 193 54. RC Circuit used to Discuss Effects of Cell Resistance on Inverse Derivative Curves 196 55. Current-Voltage Curves Calculated for RC Circuit of Fig . 54 199 56. Current-Voltage Curves Calculated for Large and Small Diffusion Coefficients 200 Figure Page 57. Charge/Discharge Cycles for Li TiS to 1.0 V 208 58. Current-Voltage Curves for L i x T i S 2 to 1.0 V 210 59. Current-Voltage Curves for Li TiS to 1.8 V 211 X 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 LixMo02 219 64. Voltage Behaviour of Li x MoS 2 222 65. Charge/Discharge Cycles for Li MoS„, Showing Conversion from Phase II to Phase I 224 66. Charge/Discharge Cycles for Li MoS , Showing Conversion from Phase I I I to Phase I 225 67. Current-Voltage Curves for Li MoS , Phase I 226 X £-68a. 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 229 69a. Current-Voltage Curves for Li MoS , Phase III, from 1.6 V to 2.6 V * . 230 69b. As in a, Except from 0.9 V to 2.6 V 231 69c. As in a, Except from 0.1 V to : 2 . 7 V 231 70. Inverse Derivative -8x/9V for Lattice Gas of Weakly Interacting Chains 257 71. Phase Diagram for Lattice Gas of Weakly Interacting Chains 258 XI LIST OF SYMBOLS A area A;'- apparent area of porous electrode a lattice constant a, Bohr radius b ag effective Bohr radius B magnetic f ie ld b parameter used in Chapter 11 b^,b creation and annihilation operators for lattice gas C capaci tance C^ heat capacity at constant chemical potential c lattice constant c. .. . element of e last ic stiffness tensor i j k £ c. . reduced (matrix) notation for c. .. . ij ijkJo D diffusion coefficient V .denominator in Appendices D and E d diameter of hard sphere in one dimensional lattice gas E energy E , E. , E, s i te ene rgi es a o' 1 3 E m magnetic energy of I sing model e magnitude of electronic charge base of natural logarithms &q kinetic energy of free electron state q &^  kinetic energy of highest f i l l e d kinetic energy state in the absence of perturbing ions (unperturbed Fermi energy) <5&^. change in the kinetic energy of the highest f i l l e d state caused by the perturbing ions (change in the Fermi: energy) xi i F free energy F^ magnetic free energy F Q a d d i t i v e constant t o the free energy, or n o n - e l a s t i c p o r t i o n of F f. component of body force f? component of surface force G conductance of 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. . element of e l a s t i c Green's f u n c t i o n U H Hami lton i an fi Planck's constant 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 density 3 number current density J number current density at the surface of i n t e r c a l a t i o n host s ' Bessel function of order 1 K i n t e r a c t i o n energy in Isi n g model L t r a n s f e r matrix L . transport c o e f f i c i e n t aa £ length of pore in porous e l e c t r o d e I- thickness of porous e l e c t r o d e M mob i 1i ty magnetization (Chapter k only) m average magnetization per spin in Isi n g model e l e c t r o n mass (Chapter 5 only) ground s t a t e degeneracy (Section 4.3 only) ni" e f f e c t i v e e l e c t r o n mass N number of s i t e s or number of host atoms n number of intercalated atoms n unit vector normal to the surface n a occupation number for the site a P. . i J element of e last ic dipole tensor P. diagonal element of P . . p p re s s u re Q charge 0_o charge required to change x by 1 Q charge flow in intercalation cel l to change voltage to some cut-off va 1 ue Q x charge flow in intercalation cell when voltage is cycled between 2 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 resistance ° c r pos i t i on r location of phase boundary S entropy S. sublattice entropy s Ising spin variable for site a S i j k £ e l e m e r | t of e last ic compliance tensor T temperature T r room temperature, 2S°C (kT r = 25-7 meV) t t i me t time to f i l l host to x = 1 at current -I for uniform intercalation o t time to reach cutoff voltage at current I t, half-cycle time ^aa' interaction energy between particles on sites a and a' U , U o , U 1 , U l , U " special choices of U , _u displacement f i e ld V voltage V sweep rate of voltage v volume W.|2 e last ic interaction energy between particles 1 and 2 e last ic interaction energy which is affected by boundary conditions CO W elast ic interaction energy between two particles in an inf inite host e last ic interaction energy between two particles due to the presence of the surface w j ump probab i1 i ty X porosity x composition of an intercalation compound; composition or fractional occupation of a lattice gas x. sublattice occupation x g composition at the surface Ax change in x due to a phase transition Y Young's modulus y distance along pore or in electrolyte difference in sublattice populations (Appendix E only) partition function z charge in units of e a site label a coefficient in solution of diffusion problem n 3 width of V(Q_) in Chapter 11 T density of states for free electron gas Y number of nearest neighbour sites (coordination number) structural parameter in layered compounds (Chapter 6 only) XV e . . element of strain tensor i J e strain at x = 1 o £ parameter (1 , 2, or 3) used in discussing diffusion r| overpotenti al 6 polar angle K d ie lectr ic constant \fj isothermal compressibility X Thomas-Fermi screening length decay length in porous electrodes y chemical potential y s chemical potential at the surface y g e last ic contribution to y which is sensitive to boundary conditions y electrochemical potential V Poisson's ratio 5 coherence length in one dimensional lattice gas II dipole moment operator TT 3.14159... p number density of intercalated atoms number density of electrons (Chapter 5 only) Z(t) sum in solution of diffusion problem a conductivity a., element of stress tensor i J T , T ' time constants $ tortuosity (J) e l ec tr ic potential X surface potential Xy magnetic susceptibi l i ty ¥ current-current correlation function angular frequency hopping frequency relaxation frequency ACKNOWLEDGMENTS It is a pleasure to thank my supervisor, Rudi Haering, for his advice and encouragement throughout this project. His physical insight proved indispensible in our attempts to understand this complicated subject. A large number of people have worked in Rudi Haering's group on intercalate batteries, and I have benefited from working with each one. In particular, I would like to thank fellow thesis writers Dave Wainwright, Ul ri ch Sacken , and Jeff Dahn, for their discussions and encouragement; Jeff Dahn also did some of the mean f ie ld calculations in Fig. 20 and 22. I have also benefited from discussions with John Berlinsky and B i l l Unruh. I would like to thank Peter Haas for his expert work on the diagrams. Final ly , 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 sc ient i f i ca l ly to describe the insertion of various types of guest atoms or molecules between the atomic planes of graphite," In this process, the host graphite structure changes, very l i t t l e , apart from an increase 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 inter-calation process could be used to make rechargeable high energy density batteries (Whittingham, 1976), 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 well . In keeping with this newly expanded definit ion, we wil l use the term intercalation compound to refer to any solid which has this distinction between guest and host atoms, i f the guest atoms can be reversibly added (intercalated) into the host without signif icantly altering the host struc-ture, at least over some range of composition and temperature. This necessarily requires that the guest atoms have a significant mobility in the host structure in this temperature and composition range. This definition encompasses systems not tradit ional ly included, such as the metal-hydrogen systems. It does not include those compounds whose structure can be regarded as a host latt ice with additional atoms in in ters t i t ia l s i tes , i f these inters t i t ia l atoms cannot be removed. Many such in ters t i t ia l compounds with 2 layered structures, prepared by combining the constituent elements at high temperatures, have unfortunately been Widely referred to as intercalation compounds; however, since we are most-interested in the very properties that these systems lack, namely those associated with the reversible addition of guest atoms, we wil l not include them in the definition used here. Studies of intercalation have dramatically increased since its use *n < battery systems was f i r s t suggested. Batteries employing intercalation compounds are conceptually very simple. A diagram of the best known inter-calation battery system, L i / L i ^ T i S ^ , is shown in Fig. 1. The cel l consists of an intercalation cathode, Li TiS , and an anode, Li metal, immersed in an x 2 electrolyte, a solution of some Li-bearing salt in an organic l iquid (for example, lithium perchlorate dissolved in propylene carbonate). Discharge of the cel l causes a transfer of Li from the anode to the cathode, with L i + ions migrating through the electrolyte solution and electrons through the external c i r c u i t . The process is reversed during recharge. (The terms anode„and cathode actually refer to the direction of charge transfer at the interface F i g . 1 - Schematic view of Li/LT X T IS2 intercalation cel l showing direction of flow of ions and electrons during discharge. 3 between the electrode and the solution, and so s t r i c t ly speaking the terms should be interchanged in the diagram during recharge. We shall ignore this , and apply the terms as in Fig . 1 independently of the direction 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 (of charge -e), where V is the battery voltage. Since this work is the d i f fer -ence in chemical potential, y, of Li in the cathode (c) and anode (a), we have eV = - ( V[. - y^ .,) . (D Thus, in addition to their possible technological importance, intercalation batteries provide a tool to study the process of intercalation i t se l f , through measurement of the chemical potential of the guest atom. Although this thesis wil l be most concerned with this latter use of intercalation batteries, we wil l consider briefly their practical aspects. The most demanding application of these batteries is the e lectr ic vehicle. Reviews of various competing battery systems being considered are given by Birk .e t 'a l (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, too much energy is needed just to propel the batteries. 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 lectr ic 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; accel-erations of 0 to 50 km/hr in M0 seconds could be expected. About the only available e lectr ic vehicle battery today is the lead acid battery. Although its power density and cost are adequate, its cycle l i f e 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 a l , 1976). Although this theoretical energy density is somewhat lower than for many other high energy density battery systems currently under study, the. simplicity 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 lat t ice . At present, the cost of these cel ls 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 l i f e . The intercalation portion of the battery (the cathode) appears to cycle very wel l - - guest atoms can be added and removed many times without any appreci-able degradation of the host , lat t ice . 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 reactivity 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 (the voltage of a L i / L i TiS ce l l averages about 2 volts over the range 0 ^_ x <_ 1). Because X 2 of this high react ivity , Li metal is thermodynamically unstable in elec-trolyte solutions; however, in some electrolytes, the Li is protected by the formation of a passivating surface layer. As an example, Li reacts with propylene carbonate (PC), the most common solvent used, to form lithium carbonate and propene.(Eichinger, 1976). From the free energy of formation of PC, .:105 kcal/mole, (estimated from the value for a related material, ethylene carbonate) 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 layer can completely..enclose some of the den-d r i t i c growth which is produced as Li is plated; this enclosed material becomes inactive, leading to a loss of several percent of the lithium plated in each cycle. 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 sacrif ice 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, in terms of the physical mechanisms involved in the intercalation process. Hence the thesis gives a detailed discussion of the mechanisms of interca-lation, and i l lustrates some of the wide variety of the behaviour expected with experimental results on several systems. The main body of the thesis^is divided into four parts. In Part A , a review of intercalation systems is given. Chapter 2 discusses the structure of the two types of host lattices studied, namely the layered transition metal dichalcogenides and the ruti le-related metal oxides, and reviews some of the existing l iterature on intercalation of these host structures. A review of other related systems is given in Chapter 3-In Part B, the thermodynamics of intercalation is discussed. Chapter k describes the latt ice gas model and its application to intercalation systems, stressing the simplest (mean field) solutions to the latt ice gas problem. The two major types of interaction between intercalated atoms, electronic and e last ic , which determine the parameters of the latt ice gas models, are discussed 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, pointing out how they apply to intercalation ce l l s . The effects of interactions between intercalated atoms on the diffusion of the atoms in the host is discussed in Chapter 8. Chapter 9 discusses the effects of this diffusion on the voltage of intercalation cel ls being discharged at f in i te currents, and Chapter 10 discusses the problems encountered in using porous 7 intercalation cathodes. In Part D, the experimental results are discussed. Experimental procedure is outlined in Chapter 11. Chapter 12 gives experimental results for intercalation of lithium into TiS^, MoO ,^ and MoS^, and discusses these results in the light of the theory presented in Parts B and C. Final ly , Chapter 13 summarizes the results of the thesis, and offers some suggestions for future work. 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 latt ices , 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 lustrate those properties which wil l be important in deter-mining how the voltage of an intercalation cell varies with the composition of the intercalation compound. Some of the points we wi l l look for are as follows; the effect they have on the cel l voltage wil l be discussed in subsequent chapters. (1) Type of s ite occupied by the intercalated atoms (2) Ordering of the intercalated atoms The intercalated atoms may be randomly distributed over a l l the sites available, or they may form an ordered array (a super 1attice) (3) Phase separation A host latt ice intercalated to an average composition, x, may consist of two coexisting regions of composition xi and x 2 . In discussing the - . experimental results, the phrase "intercalation compound of composition ; x" wil l 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 . coe f f i c i ent , 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 in the host 2.2 Layered Transition Metal Dicha1cogenides - Structure and Properties The layered transition metal dicha1cogenides have the chemical symbol MX ,^ where M is a transition metal from group IVB, VB, or VIB of the periodic table (Table I), and X is one of the chalcogens (sulfur, selenium, or te l lur -ium) from group VIIA. The crystal structure consists of sandwiches of close packed cha 1 cogen-metal:-cha 1 cogen planes stacked along the crystal lo-graphic 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 Shel.l > ^ IVB VB VIB 3d Ti V Cr hd Zr Nb Mo 5d Hf Ta W 11 (a) General form van der Waals gap (b) Coordination units for MX2 layer structures AbA trigonal prisrh AbC octahedron F i g . 2 - S t r u c t u r e o f l a y e r e d t r a n s i t i o n metal d i c h a 1 c o g e n i d e s , MX 2 (a) G e n e r a l form o f X-M-X san d w i c h e s . (b) C o o r d i n a t i o n u n i t s , (c) The t h r e e most common p o l y t y p e s . Two types of c o o r d i n a t i o n of the metal atom by adjacent chalcogens i s observed, namely octahedral and t r i g o n a l p r i s m a t i c , ( F i g . 2b). Because the s t r u c t u r e s are based on c l o s e packed atomic planes, i t i s convenient to describe them using the usual ABC notation shown in F i g . 3- We denote the chalcogen p o s i t i o n s with c a p i t a l l e t t e r s (ABC), the metal atom p o s i t i o n s with small l e t t e r s (abc), and the s i t e s f o r the i n t e r c a l a t e d atoms (to be described s h o r t l y ) with Greek l e t t e r s (a3y)- ' n t h i s n o t a t i o n , octahedral c o o r d i n a t i o n of the metal atoms i s represented by AbC, and t r i g o n a l p r i s m a t i c c o o r d i n a t i o n by AbA. The various s t r u c t u r e s c o n s i s t of sequences of one (or sometimes both) of these two basic sandwiches. F i g . 3 - ABC n o t a t i o n f o r close-packed spheres. The three most common s t r u c t u r e s , or polytypes, are shown in F i g . k. The 1T polytype c o n s i s t s of o c t a h e d r a l l y coordinated sandwiches, and i s found in group IVB and VB compounds. The u n i t c e l l i s one layer high (hence the " 1 " in "IT") and the s t r u c t u r e has t r i g o n a l symmetry ("T"). Two types of 2H s t r u c t u r e s (2 layer u n i t c e l l s , hexagonal symmetry "H").'a re shown in F i g . k. Both have t r i g o n a l p r i s m a t i c c o o r d i n a t i o n of the metal atoms, but they d i f f e r in the s t a c k ing sequence of the sandwiches. In the 2H-NbS2 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 ie one above the other along the c-axis; in the MoS2 structure, seen for group VIB metal atoms, they do not. Other;polytypes are discussed in Wilson and Yoffe ( 1 9 6 9 ) . Two types of sites are available for intercalated atoms, in the van der Waals gap of these materia 1s , (F ig . 4 a ) . The octahedral sites (ABC) are coordinated by 6 chalcogen atoms which l ie on the corners of a s l ightly elongated octahedron; these form a triangular latt ice of latt ice 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 s l ightly below and above the plane of the octahedral s i tes , and are coordinated by h chalcogen atoms. Each type of tetrahedral site forms a triangular latt ice of latt ice constant a; taken together, a l l the tetrahedral sites form a honeycomb lattice with nearest neighbour separation a / / J . There are two tetrahedral sites and one octahedral site 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. The composition of an intercalation compound of guest atom A in MX2 is usually given by the quantity x, as in AxMX2 ; f i l l i n g a l l the sites would give x=3. In some intercalation compounds, the sandwiches s l i p , bringing adjacent chalcogen atoms in line along c; this leads to a honeycomb lattice of two trigonal prismatic sites (ABA, AyA) per transition metal atom (Fig. 4 b ) . Schematic band structures of the host transition metal dicha1cogenides are shown in Fig. 5 . Calculations show (e.g. Mattheiss, 1 9 7 3 ) that the band structures can be roughly classif ied into two groups, according to the coordination of the transition metal by the chalcogens. In both cases, the upper and lower bands shown in Fig. 5 are derived from bonding and antibonding "V M - a t o m -j- — X s p - b a n d x M - a t o m d - b a n d X - a t o m p - b a n d 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 orbitals of the metal and chalcogen atoms, with the lower bands primarily from the chalcogens, while the two central bands are derived from the d orbitals of the transition metal. The primary difference between the two cases is in the spl i t t ing 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^ ' s a borderline case, and there is s t i l l contro-versy over whether it is a semiconductor or a semimetal. In group VB compounds, one state in the lower d band is f u l l , so both 1T and 2H polytypes are metallic. 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 iS 2 , • especially over the range 0 <_ x <_ 1 , where single phase behaviour is seen. Neutron studies for this composition range indicate that the lithium lies 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 inearly with x by about \% over this range (Bichon et aU 1 9 7 3 , Chianelli et al 1 9 7 8 ) . Knight shift measurements indicate that the density of conduction electrons near the lithium is small (Silbernagel and Whittingham 1 9 7 6 ) . The hopping time of the lithium near x = 1 . T n ~ 0 . 2 3 ys 2 - 9 2 suggests a tracer diffusion constant of order a /x^ - 5 x 10 cm /sec; the hopping is activated (D^ <* e ^ ^ T j with an activation energy E of about 0 . 3 eV (Sibernagel 1 9 7 5 ) . 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 Fig. 6 a . Detailed examination of the curve shows fine structure, as indicated in the inverse derivative -Ax/AV versus x in Fig. 6 b . 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 ) , then x = 3 (Dahn and Haering 1 9 7 9 ) , were reported. Exper-imental results are showh^in Section 1 2 . 3 . Two phase behaviour is seen both between x = 1 and 2 , and between x = 2 and 3 - The transition from x = 1. to x = 2 i s quite reversible, while the transition from x = 2 to x = 3 completely changes the charge/discharge characteristics of subsequent cycles over the range 0 <_ x <^  3 -In contrast to Li T iS„ , Na TiS„ shows several structural changes in x 2 x 2 the range 0 <_ x <_ 1 . In one of these structures, the sulfur atoms shift to produce trigonal prismatic sites for the sodium atom; in this phase, the c latt ice parameter decreases with x (Rouxel et al 1 9 7 1 ) . Li VS9 shows two narrow monoclinic phases, extending from 0 . 2 5 < x < O .38 and 0 . 5 < x < 0 . 6 . These phases disappear at higher temperatures (T < 85°C) or i f iron subst i tut ional^ replaces vanadium in the host structure (Murphy et al 1 9 7 8 a ) . 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 • ' • i I I 1 1 1 1 0 0.2 0.4 0.6 0.8 1.0 X IN L i x T i S g Fig. 6 - (a) Voltage V and (b) inverse derivative -Ax/AV versus x for L ' x T ' S 2 ' P o i n t s a r e experimental data from Thompson (1978). The sol id curve is a mean f ie ld 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 1978). 2.k Metal Dioxides With Rutile-Related Structure - Structure and Properties Metals which form metal dioxides M02 having a ruti1e-related structure are shown in Table II. As in many metal oxides, the basic building block of the rut i le oxides is an M05 octahedron, shown in two views in Fig. 7a and 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 Fig. 7c and 7d. This leads to the tetragonal structure in Fig. Ie, viewed along the c axis. The crystal lo-graphic a axis is the distance between the two nearest metal atoms in the same plane normal to c. The octahedra in Fig. 1 have been drawn as perfect TABLE I I METALS WHICH FORM RUT ILE-RELATED OXIDES ^ \ Group Shei 1 \ IVB VB VIB VI IB VIII IVA 3d n Cr Mn 4d ! Nb<2> Mo Tc Ru Rh Sn 5d Ta ' W Re Os 1 r Pt Pb Dotted line encloses atoms which form distorted rut i le structures at room temperature. (1) V converts to pure rut i le structure for T > T c = 3^0K (2) Nb converts to pure rut i le structure for T > T = 1070K 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 direction. (e) Top view of ruti le structure. octahedra; in fact, there is some distort ion. Two measures of this dis-tortion are the c/a ratio, 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 this , the materials are referred to as undistorted to distinguish them from several of the rutiles where the metal atoms dimerize, leading to a further distortion of the octahedra. These distorted ruti les have a monoclinic unit c e l l , which unfortunately is indexed with the monoclinic a axis, a^, parallel to the chains (a^ / / c_, where c_ is the c axis vector in the tetragonal unit cell of the undistorted rut i l es ) . In what follows, we shall refer to the a and c axes as those of the tetragonal unit cel l unless otherwise indicated. Possible sites available for intercalated atoms are indicated in Fig. 8 . Along the tunnels between the M0G octahedra, there are two types of sites: tetrahedral s ites , coordinated by 4 0 atoms, and octahedral s ites , coordin-ated by 6 0 atoms. It turns out that of the 6 oxygen atoms coordinating the octahedral s i te , 2 are closer to the center of the site 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 tetra-hedral s i te . It therefore seems l ikely that any atoms in the tetrahedral sites wil l have a lower energy than in the octahedral sites; the octahedral sites may even represent saddle points in the energy of an intercalated atom as it migrates along the tunnel. This is the case in calculations carried out on T i 0 2 (Ajayi et al 1 9 7 6 ; Kingsbury et al 1 9 6 8 ) . In addition to these sites along the tunnels, there are two other types of tetrahedrally coordinated sites , also shown in Fig. 8 . These two sites would be involved in diffusion of atoms normal to the tunnels. Calculations for T i 0 2 suggest that both of these sites have a considerably higher site 22 Fig. 8 - (a) Top and (b) side views of chains of octahedra in rut i le structure. 0 atoms coordinating various types of sites are indicated, o: octahedral site along tunnel. • : tetrahedral site along tunnel. A and V:two types of tetrahedral sites off the tunnel axis. energy than the tunnel sites (Kingsbury et al 1 9 6 8 ) ; this is consistent with the poor diffusion of lithium normal to the tunnels (Johnson 1 9 6 4 a ) . In the distorted rut i les , the dimerization of the metal atoms causes further distortion of the (already distorted) sites, and makes some of the sites inequivalent. Thus, the octahedral sites are separated into three types 1, 2 , 3 , in the sequence: 1 2 3 2 1 2 3 2 1 ! . . along the tunnel. Similarly, the tetrahedral sites on the tunnel axis are separated into two types in the sequence 1 1 2 2 1 1 2 2 . . . . Energies of these sites have not been calculated, and no estimation of site energies in the metallic compounds, for either distorted or undistorted structures, is available. The schematic band structure of the rut i le hosts shown in Fig. 9 a is very similar to that for the 1T transition metal dicha1cogenides (Fig. 5 a ) , where the metal atoms are also octahedrally coordinated. The dvbands,spl i t . into 6 lower and 4 upper states per metal atom. Thus, T i 0 2 is am. insulator, with the Fermi level lying .i n. the .gap be] ow the d bands, and rut i le structure V0 2 is metall ic. In Pb0 2 and Sn0 2, the d bands are f u l l , and these two materials are semiconducting; however, in this case, it is l ikely that the d bands l ie below the top of the lower sp bands, contrary to the figure. The distortion in the distorted rutiles 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 2 and Nb02 become semiconducting. In Mo02, the Fermi level lies one state above this gap, so Mo02 is metall ic. The lower d band in ferromagnetic Cr0 2 is believed to be spin sp l i t as shown in ; ' ' , Fig.. 9 c ; two states in the lowest d band are occupied, so Cr0 2 shows metallic behaviour. M - a t o m s p - b a n d M - a t o m s p - b a n d M - a t o m d - b a n d s M - a t o m d - b a n d s O - a t o m p - b a n d 1 0 O - a t o m p - b a n d 0 ^ M - a t o m A — -~ )~ " s p - b a n d \ M - a t o m 3 / M _ j x d - b a n d s. O - a t o m p - b a n d > Density of States S c h e m a t i c b a n d s t r u c t u r e o f r u t i l e - r e l a t e d m e t a l o x i d e s . T h e n u m b e r o f e l e c t r o n s t a t e s p e r m e t a l a t o m 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 2 . ( 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 2 . ( c ) F e r r o m a g n e t i c Cr02. ( A f t e r G o o d e n o u g h 1971). 2 . 5 Intercalation of Rutiles Considerably less information is available on intercalated ruti les than for the transition metal dichalcogenides. Lithium has been intercalated into Ti02 by placing the T i 0 2 in contact with metallic Li at temperatures between 200°C;.a'nd 3 0 0 ° C , but only small concentrations7 of intercalated lithium were obtained (x < 8 x 10 )^ . i n the bulk of the crystals; however, s l ightly higher concentrations were observed near dis1ocations.(Johnson 1 9 6 4 b ) . 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 2 room temperature was 6 x 10 cm /sec, with an activation energy of 0 . 3 3 eV (Johnson 1 9 6 4 a ) . Intercalation of Li into several other ruti les up to x = 1 has been reported by Murphy et al ( 1 9 7 8 b ) . 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. We then discuss some of the properties of some inters t i t ia l layered compounds, which gives further insight into intercalated layered compounds, and some properties of some tungsten bronzes, a class of materials which includes intercalated rutiles 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 solid 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 classif ied into one of the following three groups: intercalation from the vapour, inter-calation from a liquid solvent, and intercalation in an electrochemical c e l l . 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. Also, a wide variety of large organic molecules, 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 original separation (Gamble et al 1 9 7 1 ) . If the weight of the host latt ice is monitored as a function of the vapour pressure of the^interca1 ate, the chemical potential of the intercalate in the host latt ice as a function of composition can be calculated. 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 liquid solvent, the host structure is brought into contact with a solution containing the i nterca 1 ate ...;For example, alkal i metals dissolved in l iquid ammonia, and alkal i metal hydroxides dissolved in water, have been intercalated in this way.into layered compounds. Use of such small, highly polar solvents can lead to co-inter-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 lithium (Murphy and Carides 1 9 7 9 ) - Moreover, since the solvent (hexane) is non-polar, no solvent co-interca1 at ion occurs. Intercalation is done in ah electrochemical cel l by making the host one of the electrodes in the c e l l , as in Fig. 1, and passing current through the external c i r c u i t . The cel l may involve a simple mass transfer from one electrode to the other, as in Fig. 1 , or a chemical reaction. An example of the latter case is an electrolysis c e l l , where passing a current decom-poses water, giving hydrogen at one electrode and oxygen at the other. If a host is used at the hydrogen side, hydrogen may intercalate rather than bubbling off as hydrogen gas. Co-intercalat ion of the solvent is also a problem in electrochemical ce l l s . 3-3 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 lat t ice . Since a honeycomb latt ice can be obtained by placing atoms in two of the three close packed sphere positions ABC in Fig . 3, the stacking sequence for the layers can be described by giving the unfil led positions in each layer. Thus, in the common hexagonal form of graphite, denoted ABABAB.. . , .hal 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 alkal i metals have been reported (Rudorff and Schultz 1954). The carbon planes adjacent to inter-calated atoms generally shift 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 alkal 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 until klh°Z. (El lenson et al 1977). A study of intercalation of bromine into a graphite cylinder, with the cyl indrical 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 inter-calation occurs (Hooley 1 9 7 7 ) . Hence, in this case at least, adsorption of the intercalate on the surface of the graphite is essential for inter-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 definition of inter-calation compounds given in Chapter 1, but they are generally not referred to as such in the l i terature, since they were investigated independently of any other intercalation system. A wide variety of metals intercalate hydrogen; we wil l describe a couple in some detail to i l lustrate the observed behaviour. A schematic phase diagram of H^ Nb is shown in Fig. 10 for temperatures above 250K. Niobium is a body-centered cubic metal; hydrogen intercalates into the tetrahedral sites,between Nb atoms. These tetrahedral sites are distorted along the x, y, or z directions, and there are two of each of the three types of sites per Nb atom, for a total of six. Phases a and a 1 have the same structure; the a latt ice parameter of the Nb host increases l inearly with x in both phases such that Aa/a - 0.14 x. In the 3 phase, the hydrogens order, occupying one of the 6 tetrahedral sites, and the Nb host expands s l ightly in one direction, forming an orthorhombic (almost tetragonal) latt ice with c/a - 1.001. In the 6 phase, which.occurs near x = 2, "the Nb atoms form a face-centered cubic la t t i ce , with the H atoms occupying the tetrahedral sites between the close packed Nb (111) planes (the so-called f luorite structure). 500 T(K) a •+<5 300 400 0 B+6 x F i g . 10 - Schematic phase d i a g r a m o f H Nb f o r T > 250K (Schober and Wenzl 1979) i See t e x t f o r d e t a i l s . The f i r s t o r d e r t r a n s i t i o n between a and a 1 has been a t t r i b u t e d t o a t t r a c t i v e e l a s t i c i n t e r a c t i o n s between the hydrogen atoms ( f o r a d i s c u s s i o n o f the e l a s t i c i n t e r a c t i o n , see C h a p t e r 6). C a l c u l a t i o n s o f the a-a1 phase boundary u s i n g an i n t e r a c t i o n whose magnitude i s i n f e r r e d from the o b s e r v e d l a t t i c e 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 the o b s e r v e d phase diagram (Horner and Wagner 1974). F u r t h e r e v i d e n c e f o r t h i s e x p l a n -a t i o n o f the t r a n s i t i o n i s found i n s t u d i e s o f the a n e l a s t i c r e l a x a t i o n o f the hydrogen (the d i f f u s i o n o f t h e hydrogen atoms i n response t o a s t r e s s ) . Above the c r i t i c a l t e m p e r a t u r e T , the magnitude o f the s t r a i n caused by the m i g r a t i o n o f the hydrogen obeys a C u r i e - W e i s s law (the s t r a i n a m p l i t u d e p r o p o r t i o n a l t o (T - T ) 1 ) , and the v a l u e o f T c o b t a i n e d depends on the sample shape, as e x p e c t e d from the t h e o r y o f the e l a s t i c i n t e r a c t i o n ( T r e t k o w s k i e t a l 1977). In a d d i t i o n , the d i f f u s i o n c o n s t a n t i n f e r r e d from t h e s e measurements a l s o depends on the sample shape. The tracer diffusion constant in the a phase decreases approximately linearly with x, and is reduced by a factor of 3 as x varies from 0 to 0.14. Over this composition range, the activation energy rises from 0.13 to 0.18 eV. This variation in is larger than expected from a simple blocking _ i of s ites , which would give a reduction of D of (1 - .4) =1 .8 over this range of x (see the discussion in Section 8.2). At room temperature, and -6 2 near x = 0, = 3 x 10 cm /sec; in the ordered 3 phase, is considerably -8 2 lower, 5 x 10 cm /sec, but with a s l ightly 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 last ic 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. In Pd, the Fermi energy lies near the top of the metal d bands. His tor ica l ly , the effects of added H atoms were interpreted in one of two models, both based on a rigid 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. Band structure calculations indicate that neither picture is correct (Switendick 1972). The hydrogen modifies the Pd band structure, pulling states below the Fermi level; however, less than 1 state per H atom is pulled down, so the Fermi level rises with respect to the band structure. The new states below the Fermi level have been observed in photoemission studies (Eastman et al 1971). 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 o c t a h e d r a l s i t e s u p t o a m a x i m u m v a l u e x = 1, w i t h n o s t r u c t u r a l c h a n g e i n t h e h o s t a s i d e f r o m a n e x p a n s i o n o f t h e l a t t i c e . In c o n t r a s t , h y d r o g e n i n r a r e e a r t h e l e m e n t s g e n e r a l l y p r o d u c e s a f i r s t o r d e r p h a s e t r a n s i t i o n t o a f a c e c e n t e r e d c u b i c s t r u c t u r e w i t h b o t h t e t r a h e d r a l s i t e s f i l l e d ( x = 2), r e g a r d l e s s o f t h e i n i t i a l h o s t s t r u c t u r e . B e y o n d x = 2, h y d r o g e n t h e n e n t e r s o c t a h e d r a l s i t e s , f i l l i n g t h e h o s t t o x = 3, i n t h e l i g h t e r r a r e e a r t h e l e m e n t s ; i n t h e h e a v i e r e l e m e n t s , a n o t h e r f i r s t o r d e r t r a n s i t i o n o c c u r s t o a h e x a g o n a l c l o s e p a c k e d s t r u c t u r e w i t h a l l t h e s i t e s o c c u p i e d ( x = 3) . 3.5 I n t e r s t i t i a l C o m p o u n d s o f L a y e r e d T r a n s i t i o n M e t a l D i c h a l c o g e n i d e s A w i d e v a r i e t y o f t e r n a r y s y s t e m s w i t h c h e m i c a l s y m b o l h^MX^, w h i c h a r e s t r u c t u r a l l y s i m i l a r t o l a y e r e d t r a n s i t i o n m e t a l d i c h a l c o g e n i d e s M X ^ , c a n b e p r e p a r e d b y c o m b i n i n g t h e c o n s t i t u e n t e l e m e n t s a t h i g h t e m p e r a t u r e s . In t h e s e m a t e r i a l s , t h e A a t o m s o c c u p y i n t e r s t i t i a l s i t e s b e t w e e n X - M - X s a n d w i c h e s , j u s t a s i n i n t e r c a l a t i o n c o m p o u n d s ; h o w e v e r , s i n c e t h e c o m p o s i t i o n x c a n n o t b e v a r i e d o n c e t h e c o m p o u n d h a s b e e n g r o w n , we w i l l n o t c a l l t h e s e m a t e r i a l s i n t e r c a l a t i o n c o m p o u n d s h e r e . T h e s e m a t e r i a l s h a v e b e e n r e c e n t l y r e v i e w e d b y V a n d e n b e r g - V o o r h o e v e (1976). In many o f t h e s e s y s t e m s , w h e r e t h e M X 2 l a t t i c e h a s t h e same s t r u c t u r e a s 2H - N b S 2 > t h e i n t e r s t i t i a l A a t o m s f o r m o r d e r e d a r r a n g e m e n t s i n o c t a h e d r a l s i t e s a t x = i o r x = 1/3. In C u N b S „ , o n t h e o t h e r h a n d , t h e NbS„ a t o m s x 2 2 a d o p t t h e 2H-MoS 2 s t r u c t u r e , a n d t h e C u a t o m s r e s i d e i n t h e t e t r a h e d r a l s i t e s ; t h i s r e s u l t s i n a s h o r t e r C u - N b d i s t a n c e t h a n w o u l d be p o s s i b l e i n t h e 2H -NbS 2 s t r u c t u r e . In A C r X 2 c o m p o u n d s , w i t h A = A g o r C u , a n d X = S o r S e , t h e A a t o m s o c c u p y o n e o f t h e t w o t y p e s o f t e t r a h e d r a l s i t e s b e t w e e n o c t a h e d r a l l y c o o r d i n a t e d C r X 2 s a n d w i c h e s a t r o o m t e m p e r a t u r e ( t h e s t r u c t u r e is AcBy CbA3 BaCa A c B . . . ) . At higher temperatures, the A atoms disorder and randomly occupy both types of tetrahedral* sites; the order-disorder transition temperatures observed in neutron scattering experiments are 6 7 5 K for CuCrS 2 , 67O K for AgCrS 2 , and 4 7 5 K for AgCrSe2 (Engelsman et al 1 9 7 3 ) . 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 shifts 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 shift in the band edge (Yacobi et al 1 9 7 9 ) . Nuclear magnetic resonance studies on Sn TaS„ for x = 1/3 and x = 1 show considerably higher concentrations of electrons near the Sn atoms at x = 1 than at x = 1 / 3 ; i t is proposed that a Sn conduction band exists at x = 1 but not at x = 1/3,(Gossard et al 1 9 7 4 ) . 3 . 6 Oxide Bronzes Oxide bronzes are defined as solids with the chemical formula A MO x n j where MO is a transition metal oxide, and A is any element. This class of n 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 in electrochemical displays (Faughnan et al 1 9 7 5 a ) . Intercalation of H or Li into WO^  or MoO^ causes the original ly transparent host to become coloured; it is also an example of electrochemical intercalation of insulators. In this latter regard, Faughnan et al ( 1 9 7 5 b ) 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 lm, and protons from the other, the current flow is controlled by the e lectr ic f ie ld associated with this charge separation. In addition, it has been observed that H W0_ becomes metallic x 3 at x = 0.32 (Crandall and Faughnan 1977a). 3.7 Superionic Conductors Superionic conductors are ionic solids (electronically insulating) which nevertheless show a high e lectrical (ionic) conductivity, due to the high mobility of one of the constituent 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 host). A recent review of these materials is given by Boyce and Huberman (1979); as well , several collections of art ic les have recently been published (e.g. Hagenmuller and van Gool 1979). Superionic conductors are generally c lass i f ied 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 transit ion, in which the mobility increases dramatically, and in which the latt ice of immobile atoms generally undergoes a structural change. Thus, for example, Agl at low temperatures consists of Ag atoms in half the tetrahedral sites in a hexagonal close packed iodine lat t ice; at T = 420K, the iodine latt ice 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. Type II materials show no change in the immobile ion lat t ice , but the mobile ions disorder over a small temperature range, producing a peak in the specific heat and a fa i r ly rapid rise in mobility; there is controversy over whether this disordering is a second order phase transition 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 com-pounds . PART B THERMODYNAMICS OF INTERCALATION BATTERIES CHAPTER 4 LATTICE GAS THEORY OF INTERCALATION k. 1 Introduction The variation of the open c ircui t voltage, V, of an .intercalation cel l with the'state of charge of the cel l can provide a great deal of information about the intercalation process. In this chapter, we discuss how to interpret this information. We wil 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;.discussrthe application of the latt ice gas model to intercalation systems, and calculate the variation of V with the composition x of the latt ice gas for some specific cases. We stress the simplest (mean field) solutions of the latt ice gas problem, and examine their.: strengths and weaknesses in comparison with more exact (and more d i f f i cu l t ) results. A typical intercalation cel l is shown in Fig. 11. The anode is a metal composed of atoms A (usually lithium), and the cathode is some host latt ice that can intercalate A atoms. The electrolyte is some material (usually a liquid) containing mobile A ions of charge ze (where e is the magnitude of the electronic charge and z is assumed to be posit ive) , which allows A ions but not electrons to pass from the anode to the cathode. As the cell is discharged, A ions travel through the electrolyte and electrons through the external c i r c u i t , resulting in a transfer of A atoms from the anode to the cathode. To relate V to the thermodynamics of the components of the c e l l , con-sider 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 Fig. 11 - Schematic intercalation c e l l . of electrons to flow through the external c ircui 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 ) A n (2) where y and U a n o c j e are the chemical potentials of A atoms in the cathode and anode respectively. Hence the cel l voltage V is related to the chemical potential y by V = - - (u - y . ) ze anode • (3) As the cell discharges, only the composition of the cathode changes ( u a n o c | e is constant), so any variation of V is due to the variation of y caused by this composition change. The cathode composition is measured by the quantity x defined by 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 during.intercalation. 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 latt ice expands freely as the intercalate is added (the effects of stress wi l l be considered in some detail in Chapter 6). The change dF produced by infinitesimal changes in n and in the temperature, T, i s dF = -SdT + udn (1) where S is the entropy. Thermodynamic quantities are generally related to various response functions. For example, the specific heat at constant chemi-cal potential, 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 differential 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, for example, Landau and Lifschitz 1969) according to 7 =r-r kT /3x\ . . ( x " x ' m i r [ ^ ) T ' ( 3 ) Since fluctuations are greatly increased near phase transitions (see the review by Stanley 1971, for example), one expects peaks or divergences in (8x/9y)-p at compositions at which a phase transition 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) i s dF = -SdT - pdv . (k) The isothermal compressibility is defined as W ( ! F ) T <*> and so V K t for a gas is analogous to N(9x/9y)T for an intercalation system. Similarly, for a long cylinder in an external magnetic f ie ld B parallel to the cylinder axis, one has dF = -SdT - MdB (6) where M is the magnetization (the cyl indrical geometry avoids complications due to demagnetization effects). The Legendre transform of F with respect to B, F - MB, is analogous to ( 1 ) , and the isothermal magnetic suscep-t i b i l i t y x T defined by are well known (see, for example, Stanley 1 9 7 1 ) , and (3x/8y) T should show 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, especially y(x), of intercalation systems. To do this , we need some type of model of an intercalation compound. We wil l adopt here a latt ice gas model; that i s , we assume that the intercalated atoms are localized at specific sites in the host la t t ice , with no more than one atom on any s i te , and that the motion of the intercalated atoms from site to site does not affect the thermodynamics, and so can be neglected. The experimental results to date for systems intercalated with L i , although admittedly somewhat sparse, are consistent with this model. Neutron powder diffraction patterns for Li T iS 9 (0 < x < 1) can be f itted 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 1 9 8 0 ) . The measured activation energies for diffusion are about 0 . 3 eV for Li in L i x T i 0 ^ (Johnson 1 9 & 4 a ) and in Li^TiS^ (Silbernagel 1 9 7 5 ) ; in the usual picture of activated hopping, this implies that the Li atoms reside in deep potential wells (much deeper than kT). The measured diffusion coefficients at room temperature for these two systems imply hopping times (the time between hops) ( 7 ) is analogous to N(3x/3y) . Divergences in K y and x T at phase transitions 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 latt ice between hops, and the hopping should not be important in determining the thermodynamics of the system. Although the latt ice gas model appears reasonable for the Li interca-lation compounds just mentioned, it may not hold for a l l intercalation systems. X-ray diffraction 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 centers (Parry et al 1969). For these materials, a l iquid model, where the interca-lated atoms are not in registry with the host lattice atoms, may be more appropriate than a lattice gas model. Even for such cases, lattice gas models are s t i l l of interest, because a l iquid can be regarded as the limit of a lattice gas as the latt ice spacing tends to zero but the range of inter-action remains f inite (for an example, see Appendix C). According to the definition of intercalation given in Chapter 1, the host latt ice does not change appreciably during intercalation. Small changes in the host lead to effective interactions between intercalated atoms, which appear as parameters in the lattice gas model; some aspects of these host-mediated interactions are discussed in Chapters 5 and 6. Larger changes in the host, which violate the definition of intercalation and lead to a breakdown of the lattice gas model, are discussed in Section 4 .9 . Lattice gas models, or the equivalent Ising models (see Appendix A), have been studied'extensively as models which display phase transitions (see, for example, Stanley 1971 and references therein). They have been applied successfully as models of adsorbed systems (for example, Schick et al 1 9 7 7 ) . However, previous calculations were not intended to illuminate the behaviour of y(x) spec i f ica l ly , so we wil l consider calculations of y(x) i n some deta i1. To start , suppose there is no interaction between intercalated atoms. Let n^ measure the occupation of site a; since no more than one atom can be put on a given s i te , n^ = 0 or 1. In the absence of interactions, the energy E{n } of some distribution {n } of atoms over the sites is a a E{n } = Tn E (1) a where E is the energy of an atom on site a. If a l l sites have the same a 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 E{n } = E = nE (2) a o where n is the total number of intercalated atoms, as in 4 .2(4).Since 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 indis-tinguishable atoms on N sites: From (2) and ( 3 ) , using St ir l ing 's approximation for the factorials in S, the free energy F = E - TS relative to x = 0 is F = N{E Q x + kT[x£nx + ( l - x ) £ n ( l - x ) ] } . (4) (In ( 4 ) , we've neglected contributions to the entropy due to vibrations of the intercalated atoms in their sites: this wil 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 is and the response function (9x/9y)T is (dropping the subscript T) 9x _ x(1 -x) 9y kT XG) Using 4.1(3) with y a n o ( j e = 0, V and 9x/9V corresponding to (5) and (6) are plotted in Fig. 12. The "tails" which extend to V = ±°° are due to the fact that the entropy S makes it very d i f f i cu l t to completely f i l l or empty a l l the sites. 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 = . (7) + (E G -y)/kT This is just the familiar Fermi -Dirac distr ibut ion, giving the average occupation of a site (or energy level) of energy E q which can be occupied by no more than one part ic le . 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 Fig. 4. Let the site energies of the octahedral and tetrahedral sites be E q and E^ respectively, and suppose E . > E . If we let N be the number of octahedral s i tes , there are 2N 1 o ' tetrahedral sites; letting X q and x^  be the average occupation of octahedral and tet rahedra 1 s i tes respect i vel y (0 <_ X Q <_ 1 , 0 <_ x^  <_ 2) , we can ca 1 cu-late the chemical potential for each type of site 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 latt ice gas with , a single site energy, E = 0 . (8) = x + x = o 1 (E n -u)/kT + 2 (9) x ( E r y ) / k T 1 + e^o 1 + e When E Q - E^ is large compared to kT, the octahedral sites f i l l for 0 <_ x <_ 1 , then the tetrahedral si tes f i l l for 1 5 _ x < _ 3 - At x = 1, there is a sharp drop in voltage, and a minimum in 3x./3y, as shown in Fig. 13-As wil l be discussed in Section 1 2 . 3 , this simple two site energy model is a possible explanation for the large drop in voltage at x = 1 observed in Li T iS„ . shown in F iq . 6 0 . x 2 ' y At x = 1, where the sharp drop in V in Fig. 13 occurs, the f i l l e d sites form a latt ice commensurate with the total latt ice of sites. In fact, such features, a sharp drop in V and a minimum in 3x/3y, 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 latt ice (in the case shown in Fig. 1 3 , m = 1 ) . To see why this form of S leads to a drop in V, consider the energy required to take a particle from the highest energy f i l l e d site and place it on a site far enough away so that it is not affected by the vacancy created. This energy is just Ay = u(n +1) - p(n ), where = Nxc- 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. Hence Ay 0 . But 3x/3y, being non-negative, must have a minimum at low temperatures. Commensurate structures form not only because of site energy differences; Ay = ( 3 y / 3 n ) n = N - 1 ( 3 x / 3 y ) ~ 1 , "c *c so that 3x/3y = 1/NAy -»- 0 as N + OO Hence they can also be produced by repulsive interactions between intercalated atoms. We consider this problem in the following sections. k.k Lattice Gas Models With Interactions In any real system, it is unlikely that the energy of an intercalated atom on a particular site 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. This interaction wil l be very complicated, since the presence of intercalated atoms wil l distort the host la t t ice , modify the band structure, and so on. Some aspects of the interaction wil l be discussed in Chapters 5 and 6; for now, we assume the interaction can be.characterized by two body interactions U . aa 1between atoms on sites a and a 1 . In this case, the energy of the inter-calation system for a distribution {n } becomes a E{n } = YE n + i 7 U . n n . (l) a L a a L, aa 1 a a1 • v ' a aa 1 a^a" Solutions to this problem in the l iterature are often written in the language of magnetism, by introducing spin variables s^  = 2n -1, and inter-preting the system as a latt ice 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 latt ice gas models is given in Appendix A. Consider f i r s t a long range interaction, so that an atom on site a can interact with atoms on any of Y » 1 sites a 1 , with an interaction U , = U. 7 1 ' aa 1 If the range of the interaction tends to in f in i ty , so that y = N-1 N, but yU is a constant, the energy is once again independent of the details of the distr ibution, as it was for U = 0 , and depends only on n: (This can also be derived directly from the partit ion 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 solid curve in Fig . 6. However, for U < 0 , the voltage can actually rise with x, as shown in Fig . 14. This unphysical behaviour has its origins in our choice of a potential U , which depends on the size of the system (it was assumed that y l i was a constant, so that U <= 1/N). The unphysical region is avoided by arguing that for more reasonable 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 lies 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 Fig. 14b). This leads to the horizontal line in Fig. 14a, which gives the two regions between the.sol id curve and the dashed line equal areas, the so-called Maxwell construction (Huang 1963) . In fact, it has been shown (Thompson 1971) that i f one lets the range of the inter-action tend to inf inity after the thermodynamic limit (N -*-<») has been taken, the result (3) with the Maxwell construction already built in is obtained. E{n } = E = N(E x + iyUx2) . a o (2) The entropy its again given by 4 . 3 ( 3 ) , so that p ' i s given by y (3) 50 I 1 I i I i i I i I 0 0.2 0.4 0.6 0.8 1.0 x Fig. 14 - (a) Voltage V and (b) free energy F versus composition x for a latt ice gas with attractive interactions in mean f ie ld theory, with yU = -5 kT. Dotted lines are Maxwell constructions. The site energy was chosen to be E Q = 2.5 kT to make the Maxwell construction in F horizontal. 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 ites , with an energy U, then y is the number of nearest neighbour sites. In this case, we can interpret the quantity yLIx in (3) as the interaction of an atom with its nearest neighbours, of'which there are yx in a random distribution on average. For U < 0 , this is equivalent to the Weiss theory of ferromag-netism, hence the terminology "mean f ie ld theory" . Phase separation occurs for U < 0 for short range interactions as for long range interactions, and so the mean f ie ld theory is a very useful approximation, despite its errors near the c r i t i c a l region (Stanley 1971)-By contrast, for/repulsive interactions, short range forces lead to qualitatively different behaviour than that given by ( 3 ) . 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 latt ice may be lower than that of the random distribution underlying ( 3 ) . As discussed in Section h.J>, such a commensurate structure wil l produce minima in 3x/8y at the composition of the f i l l e d commensurate lat t ice . Moreover, peaks in 9x/9y wi l l occur at compositions where phase transitions between ordered and disordered arrangements of atoms occur. The random occupation result shows no such features, as shown by the sol id curve in Fig. 6; by comparison, the data for Li T iS„ , also shown in Fig. 6, shows dist inct features in 9x/9y. x 2 The .mean f ie ld result in Fig. 6 corresponds to E Q = 2 .3 volts , U - 2.5kT (with kT = 25-7 meV), and y = 6, and provides a rough f i t to the overall variation of V in Li T iS„ , indicating that only small interactions are x 2 ' 3 ' needed to explain the variation in voltage in this range of x for this particular system. Further discussion of the features in the V(x) curve for L i i is given in Section 1 3 - 1 -4 . 5 Mean Field Solution of the Problem of Ordering As an example of the ordering problem for a latt ice gas, consider the triangular latt ice with nearest neighbour repulsive interactions U. As shown in Fig. 1 5 , a triangular latt ice with latt ice constant a can be decomposed into three interpenetrating sublattices with latt ice constant / 3 a such that a l l the nearest neighbours of an atom on one sublattice wil l l ie on the other two sublattices. Because of this , at a composition x = 1/3 and a temperature T « U/k, the atoms wil l a l l l ie on one sublattice, producing an ordered structure commensurate with the overall triangular la t t ice . At high temperatures T » U / k , a l l three sublattices wil l be equally populated, and there wil l be no long range order. # # • H— a —H # Fig. 15 - Decomposition of a triangular lattice with latt ice constant a into three interpenetrating sublattices with lattice 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 solution, 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 sub-lat t ice , but interacts with any atom on either of the other two sublattices with a repulsive interaction energy U1 which is independent of the distance between the sites. For this problem, the energy F-{n^ } depends only on the overall composition x. of the three sublattices (i = 1,2,3; 0 <_ x. <_ 1) and not on the details of the distribution {n }. Since N-1 - N, we have a N 2 X 1 + X ? + ) S E - - U ' ( x 1 x 2 + x 2 x 3 + x 1 x 3 ) + N E Q 3 (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 is S = S1 + S £ + S 3 (2) where S. is the entropy of sublattice I, and Is given by (.3) S. = k&il (N/3)! (x.N/3).! ((1-x.)N/3).! This solution can be used to approximate the case of a nearest neighbour interaction U i f we use 6U = ^ (i.) which follows by equating the total interaction energy of one particle with a l l the others when the latt ice is completely ful l in the two cases. The energy E for the short range case is then E = N x„+x„+x 1 2 3 E Q ^ + U(x 1x 2+x 2x 4-x^,) (5) This expression is effectively 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 inf inite range interaction U 1 ) . Thus the system described by (3) and (5) basically has two choices: the atoms can be randomly distributed over a l l three sublattices, or randomly distributed within each of the three sublattices, but with different compositions x. . This approximation wil l be called three sublattice mean f ie ld theory (it is also known as the Bragg-Wi11iams approximation - see, for example, de Fontaine 1973)- 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 a l l values of the overall composition x, given by X..+X_+X-x = -!—^—1- , 0 < x < 1 ; 0 < x. < 1 . (6) To do this , we calculate the chemical potential y . for each sublattice and equate a l l three chemical potentials to y . This gives y = U i = U n - . , . . . i / n.,j5 * i J x. = E Q X . - 3Ux. + kT-fta-pi- + 9Ux . (7) i y-9Ux is plotted in Fig. 16a for U = kkl. 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 ie ld 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 com-positions 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 Fig. 1 6 a , and thestate of the system for some value of y-9Ux is then conveniently specified by giving the number of sublattices in the three sections as a three digit number in parentheses; thus, for example, we describe Fig. 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 poss ibi l i ty 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. The entire sequence for x going from 0 to 1 is given in Fig. 1 6 . By comparison, the completely random mean f ie ld solution, h.k(3), corresponds to the sequence ( 3 0 0 ) + ( 0 3 0 ) + ( 0 0 3 ) , and has a higher free energy than that shown in Fig. 1 6 . The onset of long range order is associated with the breaking away of one of the points on the y - 9 U x curve. Close examination of y as this occurs reveals that as the points sp l i t apart, the value of x at f i r s t decreases, then increases once again, as shown in Fig. 1 7 a . This leads to a f i r s t order phase transition over a range of composition determined by the Maxwell X F i g . 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 m e a n 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 D = 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 Q = 0.157 k T , w e r e c h o s e n t o m a k e V a n d F b o t h z e r o a t t h e p h a s e t r a n s i t i o n . T h e 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 m a x i m u m i n u -9Ux, a n d t h e l a b e l s o n ( b ) i n d i c a t e t h e s t a t e o f t h e s y s t e m c o r r e s p o n d i n g t o t h e v a r i o u s p a r t s o f t h e f r e e e n e r g y c u r v e . construction. The free energy in this region is shown in Fig. 1 7 b , where it can be seen that the states (030) , and ( 1 2 0 ) have higher free energies than ( 2 1 0 ) . By comparison, when only two points are at the maximum, such as ( 2 0 1 ) - > ( 1 1 1 ) , the chemical potential shows only a small change in slope, indicating a second order phase transit ion. For U = hkT, the behaviour of the sublattice populations x. as a function of x associated with the motion in Fig. 16 is shown in Fig . 1 8 . The arrows on each segment of the curve point in the direction corresponding to increasing/x , and the number of arrows gives the number of sublattices with that composition. The empty regions near x = 0 . 0 9 and x = 0 . 9 1 correspond to the region of phase coexistence. The voltage V corresponding to this behaviour is plotted in Fig. 1 9 . Note the large drops in V, with associated minima in 3 x / 3 y , at x = 1 /3 and x = 2 / 3 corresponding to commen-surate ordering, and the f lat regions in V, with associated inf ini t ies in 3 x / 3 y , near x = 0 . 0 9 and x = 0 . 9 1 , corresponding to phase transitions. These are 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 Fig. 1 6 . For x s l ightly less than 1 /3 only one sublattice is f i l l i n g , and the atoms added to the latt ice are able to find sites in this preferred sublattice where they feel no nearest neighbour interactions. When x reaches 1 / 3 , this sublattice is f u l l , and the atoms must then be placed on one of the other two sublattices, where they interact with three nearest neighbours on the ful l sublattice. Thus the energy gained in adding the atoms to the la t t ice , -y, drops by ^ 3 U at x = 1 / 3 . Similar considerations apply near x = 2 / 3 . The phase diagram for the triangular latt ice gas in the three sublattice mean f ie ld approximation is given approximately by curve a in Fig . 2 0 . For 59 Fig. 18 - Sublattice compositions Xj versus average composition x for three sublattice mean f i e ld theory for the triangular latt ice 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 IN L i x T i S 2 Fig. 19 - (a) Voltage V and (b) inverse derivative -Ax/AV versus x for a triangular latt ice gas with nearest neighbour interactions U = kkl in three sublattice mean f ie ld theory. The site energy E = - 2 . 3 Volts, and kT = 2 5 - 7 meV. o 61 F i g . 20 values of kT/U inside the curve, the system shows long range order; outside, a l l 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 sp l i t apart, as discussed above. The phase transit ion, as determined by the Maxwell construction, 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. In part icular, note that for U ->• 0 0 the phase transition occurs at x = 0 . This is easily understood in light of the correspondence of the mean f ie ld theory solution to an infinite range interaction U' ( 4 ) . The f i r s t atom placed on the lat t ice , on sublattice 1 say, wi l l prevent other atoms from occupying any of the sites in sublattices 2 or 3 - Hence the mean f ie ld solution for 0 <_ x <_ 1/3 as U -> 0 0 is identical to the solution for U = 0 with only N/3 sites: y = E Q + kT&\ 1/3 - x This "hard sphere" solution wil l be discussed in more detail in Section h.6. Curve b in Fig. 2 0 gives the phase boundary between ordered and dis-ordered states as determined by a renormalization group (RG) calculation (Schick et al 1 9 7 7 ) . Because the RG solution treats short range order (the mean f ie ld theory does not), it predicts that the ordered phase is confined to a smaller region of kT and x than in the mean f ie ld result. Moreover, the RG calculation predicts a continuous (higher order) phase transition to the ordered state, in contrast to the f i r s t order transition in mean f i e ld . 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 Fig. 2 1 . The voltage curve is very similar to the mean f ie ld result in Fig. 1 9 , but the fine details seen more clearly in the curve 3x/8y (Figs. 19b and 2 1 b ) ( 9 ) 63 Fig. 21 - (a) Voltage V and (b) inverse derivative -Ax/AV versus x for a triangular latt ice 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 differ considerably. It is interesting to examine the voltage curve for a value of U/kT where the mean f ie ld theory predicts a phase transition but the RG calculation does not. Fig. 22a shows the behaviour of the sublattice compositions x. versus x as predicted by the mean f ie ld theory, for U/kT = 0.72. The corresponding voltage curve, Fig. 22b, shows no noticeable features associated with the phase transit ion. These mean f ie ld calculations can straightforwardly be extended to treat other latt ices , as in the next section where we consider the one dimensional lattice' gas, or to include longer range interactions. One simply subdivides the latt ice into enough interpenetrating sublattices so .that no particles on the same sublattice interact, or until the inter-action of particles on the same sublattice is weak enough that it wi l l not produce ordering; these weak interactions can then be handled by the random mean f ie ld theory of Section k.k. A good deal of physical insight is therefore necessary in choosing an appropriate set of sublattices; such insight is also required for RG calculations. k.6 One Dimensional Lattice Gas The one dimensional latt ice gas ( lattice gas on a chain of sites) is of interest as a possible model for intercalation compounds such as the ruti le-related materials, where the intercalated atoms are located in tunnels or channels in the host lat t ice . In addition, one dimensional latt ice 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 ie ld solutions, being associated with inf inite range interactions, wi l l show long range order even in one dimension; because of this , mean f ie ld theories are 6 5 Fig. 22 - (a) Sublattice composition xj and (b) voltage V versus average composition x for the triangular latt ice gas with nearest neigh-bour interactions U = 0 . 7 2 kT in three sublattice mean f ie ld theory. Site energy E D = 2 . 3 Volts, 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 . 66 generally considered unsuccessful at treating one dimensional problems. However, in comparing the exact and mean f ie ld solutions, we shall see that the mean f ie ld results serve as useful approximations even in the one dimensional case. The one dimensional latt ice gas with nearest neighbour interactions is treated in many textbooks, usually in the magnetic language of the Ising model. A discussion of the solution is given for reference in Appendix B; from there, we find the relation between x and u is This is plotted in Fig 23a for U = -2.5kT (attractive interaction). Also shown is the result of the mean f ie ld expression 4 .3 (5) , with the Maxwell construction drawn in the two phase region. Both curves are considerably f latter than for U = 0 (Fig. 12); the mean f ie ld theory carries this flattening to the extreme of a f i r s t order phase transit ion. In Fig. 23b, (1) is plotted for U = 5kT (repulsive interaction). 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 site is occupied, so the minimum is in agreement with the discussion in Section 4 .2. If we divide the chain into two interpenetrating sublattices 1 and 2, as in Fig . 24, then at T = 0 and x = i only one of these is occupied. For T / 0, or x not precisely i , there wil l be stacking faults , or f i l l i n g mistakes, as shown in Fig. 24; at each of these mistakes the occupied sublattice.shifts from 1 to 2 or vice versa, so the long range order is destroyed. There wil l s t i l l be short range order, however, and so as x increases past i sites with two nearest neighbours begin (1) 67 Fig. 2 3 - Voltage V versus composition x for the one dimensional latt ice gas with nearest neighbour interactions U calculated using exact and mean f ie ld (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 • Fig. 2k - One dimensional latt ice 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 latt ice into two interpenetrating sublattices. • : ful l s i te , o: empty s i te . to f i l l , leading to a drop in V of order 2U/ze. Also shown in Fig. 23b is the result of a two sublattice mean f ie ld 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 latt ice because there are only 2 points on the y-4Ux curve (the n-shaped curve analogous to u-9.Ux for the triangular latt ice gas, Fig. 16). The mean f ie ld calculation, unable to handle the problems shown in Fig. 2k, predicts a second order phase transition to a long range ordered structure at x = 0.11 and x = 0.89. In spite of this , the overall shape of the voltage curve, especially the drop at x = i , provides a reasonable approximation to the exact result, unless one is interested in the fine deta i1s. To further compare mean f ie ld and exact results, we now consider a second problem in one dimension. Suppose the atoms on a one dimensional latt ice interact with a repulsive energy U = 0 0 when they are separated by less than d-1 empty sites , and do not interact otherwise, d is then essentially the diameter of the atoms in this hard sphere model, measured in units of the latt ice spacing. The occupation of the latt ice is thus restricted to the range 0 < x < 1/d. The mean f ie ld solution is given by dividing the latt ice into d sublattices; the f i r s t atom placed on the latt ice prohibits occupation of a l l sublattices except the one it is on, so the solution is the same as that for a noninteracting latt ice gas on N/d which reduces to (2) only for d = 1 (no interactions). The exact and mean f ie ld expressions are compared in Fig . 25. The voltage curve for the exact result drops much more rapidly near x = 1/d than for mean f ie ld for large d. In most situations, however, one does not expect d to be very large, so the mean f ie ld calculations again give a reasonable approximation to the exact result. These comparisons of mean f ie ld theory and the exact solutions were made to i l lustrate how well the mean f ie ld theory approximates the exact results. Of course, where the exact solutions exist and are relatively simple, they would be used rather than the mean f ie ld results to describe a one dimensional latt ice gas. However, i f we wish to use the one dimen-sional results, we must determine when it is acceptable to regard an actual intercalation compound as a one dimensional latt ice gas. To do this , consider a latt ice of chains of sites., with each chain coordinated by y nearest neighbour chains. For rut i le structures, the chains are arranged in a square la t t ice , with y = k; a two dimensional example, with y = 2, is shown in Fig . 26. Suppose that atoms interact only when on sites (cf 4.3(5)): (2) The exact solution, as shown in Appendix C, is (3) 70 N JC -8 - 1 2 a - \ \ d = 4 \ d » l \ \ 1 1 i i i . i 1 \ F i g . 25 - (a) V o l t a g e V v e r s u s x d and (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 gas 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 o v e r d l a t t i c e s i t e s . C u r v e s a r e e x a c t 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 a l1 d . o • o • o o • o o • o • o Fig. 26 - Two dimensional latt ice of one dimensional chains with nearest neighbour interactions U, L)!: indicated. Interactions such as U1 are assumed to be zero. adjacent s i tes , with an energy U when on the same chain and an energy U1 when on adjacent chains (U and U1 both positive). We have seen in the above sections that ordering occurs for interactions of order kT; hence one of the restrictions on the system to allow the chains to be considered as indepen-dent, and thus the system of chains to be regarded as a one dimensional latt ice 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 suff icient. As an example, consider the two dimensional latt ice in Fig. 26 at a composition x = i . The free energy in this particular case can be found exactly (Onsager ISkh), and the system shows long range order below a transition temperature T given by c s i n h \ 2 k T ; s i n n ( ^ r - ) = 1 ( z , ) For U 1 < K U , this can be rewritten as U " e U / 2 k T c = 1 . (5) kkT c The transition temperature T c decreases only logarithmically as U' goes to zero. Thus, for U = U 1 , kT^ = 0.567U, while for IT = 0 . 0 1 U , kT c = 0.127U, a reduction by a factor 0.22. This persistence of long range order for very small U1 is due to the fact that, at x = \ and U > kT, there is considerable order in the purely one dimensional chain ( although no long range order). Away from x = i , however, the long range order wi l l not be so persistent. As discussed above, the long range order in a purely one dimensional latt ice 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 (£ is thus a coherence length for the one dimensional chain). If there is to be at least one pair of atoms interacting on different chains, the short range order on the chains implies there wil l be of order £/2 such pairs (near x = i ) . Thus, long range order wil l be expected i f .the chains interact unless The coherence length E, can be found from the two site correlation functions <nQnr>, which gives the correlation in occupation of two sites separated by r sites. Since we wil l be interested in compositions near x = it is convenient to use the. Ising notation (Appendix A), namely s = 2n + 1 (7) a a m = <s > = 2x - 1 . (8) a As discussed in Appendix E, the two site correlation functions are given by 73 <s.s ••> = 0 r m2 + 2 [ e "U/kT ^ „ 2 + m (l-e )J 1 + .e + m (1-e ) ( -D r ( l -m 2 ) (9) For e ^ ^ » 1, (9) can be rewritten as <s.s > - m' 0 r 1 " f ( - l )r , .J « ? (10) wi th (m2 + •U/kT, (11) In this l imit , the condition (6) becomes yU , 2 . -U/kT\± < £ K l (m + e ) (12) For y = 2 and m = 0, this agrees with (5) to within a factor of 8. If U'/kT > e ^ ^ k T f the system wil l be ordered at m = 0; as m deviates from zero, however, the order wil l disappear for m ^ yU kT ' \ 2 -U/kT - e i ^ XUJ kT (13) For U1 = 0.001U, long range order is expected at m = 0 (x = i ) for T - 0.107U/k; at half this temperature, the ordered phase extends only over JmJ < 0.019, or 0.49 < x < 0.51- This small composition range is a result of the fact that although £ (11) is very large at x = j (£ - e^^^) , it fa l l s off rapidly (as 1/|m|) for |m|> e ^ ^ T ^ In Appendix E a more detailed calculation of this problem of weakly coupled chains is presented, wnich treats the interaction along the chain exactly but the interaction between chains using mean f ie ld theory. This calculation verif ies the order of magnitude estimates presented above, and indicates that the feature in 8 x /3y produced by the ordering i s very s m a l l . f o r U1 « U. It should be noted, however, that the i n t e r a c t i o n being discussed i s somewhat a r t i f i c i a l , s ince i t neglects i n t e r a c t i o n s such as Li" in F i g . 26. For widely separated chains, U" i s more l i k e l y to be of the same order as U 1, than to be zero. If U" > 0, the order which occurs f o r U" = 0 w i l l tend to be suppressed, because the energy gained by ordering w i l l be reduced. We conclude on the basis of the above d i s c u s s i o n and the r e s u l t s of Appendix E that although weak i n t e r a c t i o n s between chains may modify the c r i t i c a l behaviour, p o s s i b l y generating ordered s t a t e s in cases where the true one dimensional system would already show appreciable short range order, they have l i t t l e e f f e c t on the battery voltage curve. S i m i l a r conclusions should a l s o apply to the two dimensional l a t t i c e gas r e s u l t s discussed in Section k.5- For example, in a layered compound, a purely two dimensional ordered s t a t e would require that the occupied s u b l a t t i c e in one l a y e r be chosen randomly, and completely independently of the p o s i t i o n of the occupied s u b l a t t i c e s in the other 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 layers could cause the ordered s t a t e to c o n s i s t of a p e r i o d i c s t a c k i n g sequence of the occupied s u b l a t t i c e s from layer to l a y e r , which is a s t a t e of three dimensional order. This would not be expected to modify the o v e r a l l c e l l voltage curve V(x) from the purely two dimensional case, except near the c r i t i c a l region where ordering occurs. h.7 I n t e r a c t i n g L a t t i c e Gases with D i f f e r e n t S i t e Energies Extension of the mean f i e l d 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 s i t e energies, i s reasonably s t r a i g h t f o r w a r d . If the s i t e energy d i f f e r e n c e s are la r g e , t he 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 s i t e s . 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 of site is being f i l l e d , one can ignore the higher energy s i tes , 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 particular site of interest. In certain cases, however, more complicated behaviour can occur; we discuss one example here. Consider the sites available in a layered transition metal dichal-cogenide, shown in Fig. h. The triangular latt ice of sites consists of one sublattice of octahedral sites and two sublattices of s l ight ly smaller tetrahedral s ites. The tetrahedral sites s i t above and below the plane of the octahedral s i tes , so the distance between adjacent tetrahedral sites is larger than the nearest neighbour octahedra1-tetrahedra1 site distance. Because of this geometry, it is probable that the octahedral site energy E Q is less than the tetrahedral site energy , while the interaction energy between atoms on adjacent tetrahedral s ites , , is less than the nearest ' neighbour :octahedral-tetrahedral interaction energy U q . Thus it is possible that even though octahedral sites f i l l f i r s t , it may be favourable to have only tetrahedral sites occupied for x > 1 to avoid the larger interaction energy U . 3 7 o 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 ie ld theory, the energy of some arrangement of atoms is given by E = N [ E o x o + M V ^ + V V i + V a ' + 3 U 1 X 1 X 2J ( 1 ) neglecting a l l but nearest neighbour interactions. Fig. 27a shows the free energy as a function of x for two sequences of f i l l i n g the latt ice: curve a^  corresponds to f i l l i n g the octahedral sites for 0 <_ x 1, one type of tetrahedral site for 1 < x <2 , and the other tetrahedral sites for 76 F V 0 1 2 3 x Fig. 27 - Form of the (a) free energy F and (b) voltage V versus x for three interpenetrating sublattices of octahedral and tetrahedral sites (see text) at T = 0 , with site energies and interaction energies satisfying equation 4 .8(2) . Dotted lines indicate Maxwell construction. 2< x <_3; c u r v e c o r r e s p o n d s t o f i l l i n g o n e t y p e o f t e t r a h e d r a l s i t e f o r 0 <_ x <_ 1 , t h e o t h e r t e t r a h e d r a 1 s i t e s f o r 1 <_ x <_ 2, a n d f i n a l l y t h e o c t a h e d r a l s i t e s . A l l o t h e r s e q u e n c e s h a v e h i g h e r e n e r g i e s . F o r t h e v a l u e s o f s i t e e n e r g i e s a n d i n t e r a c t i o n s c h o s e n , t h e t w o c u r v e s c r o s s b e t w e e n x = 1 a n d x = 2. T h e M a x w e l l c o n s t r u c t i o n ( d o t t e d l i n e ) l i e s b e l o w b o t h c u r v e s , i n d i c a t i n g t h a t i n t h e r a n g e 1 < x < 2 t h e s y s t e m u n d e r g o e s a f i r s t o r d e r p h a s e t r a n s i t i o n , f r o m a p h a s e w i t h a l l t h e o c t a h e d r a l s i t e s f i l l e d ( x = 1) t o a p h a s e w i t h a l l t h e t e t r a h e d r a l s i t e s f i l l e d a n d t h e o c t a h e d r a l s i t e s e m p t y ( x = 2). T h e r e s u l t i n g v o l t a g e c u r v e i s s h o w n i n F i g . 27b. T h e c o n d i t i o n s f o r t h i s p h a s e t r a n s i t i o n t o o c c u r a r e ( a t T = 0) E ^ E i o 1 U , < : U o (2) 3 ( u o - u i > > E r E o • 4 . 8 I n c l u s i o n o f T h r e e B o d y F o r c e s T h e i n t e r a c t i o n b e t w e e n i n t e r c a l a t e d a t o m s may be m o r e c o m p l i c a t e d t h a n s i m p l e t w o b o d y f o r c e s . T h e i n c l u s i o n o f h i g h e r o r d e r f o r c e s c a n l e a d t o f u r t h e r c o m p l i c a t i o n s i n t h e b e h a v i o u r o f V ( x ) . A s a n e x a m p l e , c o n s i d e r a t r i a n g u l a r l a t t i c e o f i d e n t i c a l s i t e 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 . S u p p o s e t h e i n t e r a c t i o n b e t w e e n a n i s o l a t e d p a i r o f n e a r e s t n e i g h b o u r a t o m s i s U , b u t d u e t o t h r e e b o d y f o r c e s t h e i n t e r a c t i o n b e t w e e n t h r e e a t o m s p l a c e d o n t h e v e r t i c e s o f a n e l e m e n t a r y t r i a n g l e i s 3U + U 1 r a t h e r t h a n j u s t 31). In . . 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 , t h e e n e r g y o f some d i s t r i b u t i o n o f a t o m s i s g i v e n i n t e r m s o f t h e t h r e e s u b l a t t i c e c o m p o s i t i o n s x . ( w h e r e , a s i n S e c t Ton 4.5, 0 <_ x . <_ 1) b y E = [ E Q X + U ( x 1 x 2 + x 1 x 3 + x 2 x 3 ) + ,.2U ' x 1 x 2 x ] N (1) where x = (x^+x^+x^)/3• Once a g a i n we c o n s i d e r o n l y t h e T = 0 c a s e , so t h e f r e e e n e r g y F = E. F o r e a c h c o m b i n a t i o n o f U and U 1 , we can c o n s t r u c t t h e f r e e e n e r g y c u r v e f o r t h e c a s e where t h e t h r e e s u b l a t t i c e s f i l l s e q u e n t i a l l y (as i n S e c t i o n 4 .5 , t h i s t u r n s o u t t o be more f a v o u r a b l e t h a n f i l l i n g two o r t h r e e s u b l a t t i c e s s i m u l t a n e o u s l y ) . An e x a m p l e , f o r U > 0 and U 1 = - U , i s shown i n F i g . 28; f o r t h i s p a r t i c u l a r c h o i c e o f U and U 1 , t h e M a x w e l l c o n s t r u c t i o n i n d i c a t e s a f i r s t o r d e r t r a n s i t i o n o c c u r s between x = 1/3 and x = 1. T h i s t r a n s i t i o n i s c a u s e d by t h e f a c t t h a t a l t h o u g h t h e p a r t i c l e s r e p e l one a n o t h e r , c l u s t e r s d f p a r t i c l e s r e p e l l e s s s t r o n g l y t h a n i s o l a t e d p a i r s . The b e h a v i o u r o f t h e s y s t e m f o r a r b i t r a r y LI and U 1 can be f o u n d i n t h i s w a y , and i s s u m m a r i z e d by t h e v o l t a g e c u r v e s i n F i g . 29. These c o n d i t i o n s on U and U 1 a g r e e w i t h t h o s e o b t a i n e d i n RG t h e o r y by S c h i c k e t a l (1977). Note t h a t t h e i n c l u s i o n o f t h r e e body f o r c e s has b r o k e n t h e symmetry a b o u t x = i w h i c h o c c u r s f o r U 1 = 0 (see F i g s . 19 and 21). S i n c e e x p e r i m e n t a l v o l t a g e c u r v e s a l s o l a c k t h i s symmetry ( e . g . F i g . 6 ) , t h r e e body and h i g h e r o r d e r f o r c e s a r e p r o b a b l y p r e s e n t i n r e a l s y s t e m s . F i g . 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 gas w i t h two body and 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) and 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 e x a m p l e s h o w n , U 1 = - 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 . 79 U > 0 U' > U < 0 U' > f |u| u1 > l |u| Fig. 29 - Voltage V versus composition x for the triangular latt ice 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 far , the host has apparently been neglected in our discussion. However, small changes in the host contribute to the interaction energies and site energies in our latt ice gas models, as wi l l be discussed in the next two chapters, and so have already been implicit ly included in our latt ice gas description of intercalation systems. Large changes in the host require additional attention, which we wi l l briefly give here. A wide variety of structural forms' of a given host can be envisioned, and for each we can (in principle) calculate a free energy, F. The structure with the lowest free energy wi l l be thermodynamica11y stable, and in most cases wi l l correspond to the observed structure. As the host is interca-lated, the free energies of a l l the structures, both observed and imagined, wil l 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 wil l lead to a f i r s t order phase transit ion. In the absence of e last ic strains or surface energies, the free.energy of the coexisting phases lies on the common tangent to the two free energy curves, as shown, for example, by Hi l l er 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 last ic strains. A review of this point is given by Paterson (1973). Elastic effects in intercalation systems are discussed in Chapter 6.) In most f i r s t order phase transitions, there is some activation energy, such as the surface energy required to create a phase boundary, which must be overcome before the phase transition can occur. Thus the transition from structure 1 to 2 wi11 not occur until the common tangent condition for 1 and the activated structure 2* is sat isf ied. Hence one expects the Fig. 30 - Schematic form of free energy per s i te , F/N, versus composition x for a structural transition in the host. Maxwell construction given by dotted l ine, (a) Thermodynamic equilibrium, (b) Tran-s i t ion via some "activated" intermediate state. voltage at which the phase transition begins, which is minus the slope of the dotted line in Fig. 30a and 30b divided by ze, to be lowered as x increases in the transition 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 transition 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 iquid begins to freeze). However, in an intercalation cel l made using a powdered cathode, any given particle in the cathode is effectively in a "chemical potential bath" provided by a l l the other part ic les , and so it is more l ikely that the observed voltage wil l remain at its low value as the phase transition proceeds. In this case, the phase transition in any given particle occurs very rapidly once it begins, and the energy due to the difference between the observed and equilibrium voltages appears as heat. The voltage of the phase transition wil l be different on the charge and discharge, and the battery/curve wil l show hysteresis. (This point is discussed further in connection with the e last ic 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 LixMoS2 and other systems, as discussed in 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 of inter-calated atoms is affected by interactions between the atoms. We now wish to consider the origins of these interactions. Interactions between intercalated atoms are due to changes in the energy of one atom in the presence of the others. (In this and the following chapter, we wil l refer to Mthe 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 last ic energy; the electronic energy is associated with changes in the electron d i s t r i -butions in the atom and in the host caused by intercalation, while the elast ic energy is due to the distortions in the host latt ice caused by the intercalated atom. Changes in these energies due to the presence of other intercalated atoms give rise to electronic and elast ic interactions. The electronic interaction is discussed in this chapter; the e last ic interaction is discussed 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 specialization to a particular host material. In order to make more general comments, i t is common to adopt a rigid band model, where it is assumed that the host band structure is unchanged on intercalation, except for the addition of new electronic states associated with the intercalated atom, and a possible overall shift in the e n e r g y o f t h e b a n d s m e a s u r e d w i t h r e s p e c t t o a n e l e c t r o n a t r e s t a t a n i n f i n i t e d i s t a n c e f r o m t h e h o s t ( t h e v a c u u m l e v e l ) . I f t h e e n e r g y o f t h e s e n e w s t a t e s l i e s a b o v e e m p t y s t a t e s o f t h e h o s t , t h e r e w i l l b e a t r a n s f e r o f e l e c t r o n s f r o m t h e a t o m t o t h e h o s t , a n d t h e a t o m w i l l b e i o n i z e d . O n t h e o t h e r h a n d , t h e r e m a y b e m o r e n e w s t a t e s i n t r o d u c e d b e l o w t h e h i g h e s t f i l l e d s t a t e s o f t h e h o s t t h a n c a n b e f i l l e d b y t h e e l e c t r o n s o f t h e a t o m , s o t h e r e w i l l b e a n 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 a n a n i o n . I n 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 f r o m t h e c h a r g e d p a r t i c l e w i l l b e s c r e e n e d b y t h e e l e c t r o n s i n t h e h o s t b a n d s t r u c t u r e , a n d t h e i n t e r c a l a t e d a t o m s ( t h e i o n s p l u s t h e i r s c r e e n i n g c l o u d s ) w i l l i n t e r a c t v i a a s c r e e n e d C o u l o m b p o t e n t i a l . T h e i n t e r a c t i o n e n e r g y d e p e n d s o n t h e d e t a i l s o f t h i s s c r e e n i n g . A d i s c u s s i o n o f s c r e e n i n g i s g i v e n i n S e c t i o n 5 . 2 , l a r g e l y i n t e r m s o f t h e l i n e a r i z e d T h o m a s - F e r m i a n d 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 g a s . A f u r t h e r p o s s i b i 1 i t y i n t h e r i g i d b a n d m o d e l i s t h a t t h e r e a r e j u s t e n o u g h n e w s t a t e s i n t r o d u c e d b e l o w t h e l o w e s t e m p t y s t a t e o f t h e h o s t t o a c c o m m o d a t e t h e e l e c t r o n s o f t h e i n t e r c a l a t e d a t o m . I n t h i s c a s e , t h e i n t e r -c a l a t e d a t o m r e m a i n s n e u t r a l . I f t h e 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 o r i n s u l a t i n g , i t w i l l r e m a i n s o . T h e n , a s t h e d e n s i t y o f i n t e r c a l a t e d a t o m s i n c r e a s e s , i t i s e x p e c t e d t h a t a t r a n s i t i o n t o m e t a l l i c b e h a v i o u r w i l l o c c u r . T h i s i s d i s c u s s e d f u r t h e r i n S e c t i o n 5 - 2 . 5 . 2 S c r e e n e d C o u l o m b I n t e r a c t i o n s A n i o n i n a m e t a l i s s u r r o u n d e d b y a c l o u d o f e l e c t r o n s , w h i c h s c r e e n s t h e f i e l d o f t h e i o n i n a v e r y s h o r t d i s t a n c e . ( t h e s c r e e n i n g l e n g t h ) f r o m t h e i o n . T h u s , i f t h e r e i s a t r a n s f e r o f e l e c t r o n s t o o r f r o m t h e h o s t 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 , w e c a n s t i l l r e g a r d t h e i o n i z e d i n t e r c a l a t e p l u s i t s s c r e e n i n g e l e c t r o n c l o u d ( w h i c h i s a c t u a l l y a . d e p l e t i o n o f e l e c t r o n s f o r a n e g a t i v e i o n ) a s a n e u t r a l a t o m . T w o i n t e r c a l a t e d a t o m s 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 jel l ium 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 wil l produce a change <5p(r_) in the electron density, and the electrostatic 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. Let be the kinetic energy of electrons in the state q. 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 which is related to the electron density p by 6 f , | . W p ) 2 / 3 . 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 level) , must be independent of position, electrons move into the region near the ion (or away from i t , i f z < 0 ) , f i l l i n g additional states, as indixated"in Fig . 31. In r(s ) q Fig. 31 - Kinetic energy S q 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 shift in the Fermi energy 6&f. The potential energy of the electrons is simultaneously decreased by -e<J) (_r) the presence of the perturbing ion, the Fermi energy is conveniently given by the kinetic 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 ini te density of ions, the Fermi energy wil l rise (or fa l l i f z < 0) as wi11 be discussed shortly, so we have indicated the poss ibi l i ty of a change 6&^ . in the Fermi energy in Fig. 31, and wil 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 f ) [s&f+e<|>(r)] (2) where r(&^) is the density of kinetic energy states (number of states per unit energy and volume) for the free electron gas: r ( £ ) = _ L / 2 n i \ 3 / 2 & i ( 3 )  q 2TT2 W) q T h e n u s i n g P o i s s o n ' s e q u a t i o n , we f i n d V2cj) = A 2 <j>(_r) + | 6& f " *mze$(r) (4) w h e r e t h e l a s t t e r m , i n v o l v i n g t h e D i r a c d e l t a f u n c t i o n 6 (r) , i s d u e t o t h e i o n , a n d t h e i n v e r s e T h o m a s - F e r m i s c r e e n i n g l e n g t h , X , i s g i v e n b y X 2 = % e 2 r ( & f ) . (5) F o r a s i n g l e i o n , = 0 a n d <J> (r) 0 a s r •+ 0 0 , a n d t h e s o l u t i o n o f (4) i s < K r ) = f e - A r (6) w h i c h i s j u s t a s c r e e n e d C o u l o m b p o t e n t i a l . T h e t o t a l n u m b e r o f e l e c t r o n s l o c a l i z e d a r o u n d t h e i o n i s , a s e x p e c t e d , ( u s i n g (2) a n d (5)) /6p (r_) d v = jT(&f)ecJ>(_r) d v = z (7) ( w h e r e v i s t h e v o l u m e ) . T h e s c r e e n i n g l e n g t h X ^ c a n b e s h o r t e r t h a n a t y p i c a l a t o m i c r a d i u s , s o c l e a r l y i t i s r e a s o n a b l e t o d e s c r i b e t h e i o n p l u s i t s s c r e e n i n g c l o u d a s a n e u t r a l a t o m . T h e e n e r g y o f t h i s a t o m , E , c o r r e s p o n d s t o t h e s i t e e n e r g y o f t h e l a t t i c e g a s , E , d i s c u s s e d i n C h a p t e r h. T h i s e n e r g y i s t h e sum o f t h e k i n e t i c e n e r g y E, o f t h e e x c e s s e l e c t r o n s , <5p.(r) , a n d t h e p o t e n t i a l e n e r g y E o f t h e k — p i n t e r a c t i o n o f t h e e x c e s s e l e c t r o n s w i t h t h e i o n a n d w i t h o n e a n o t h e r . T h e i n t e r a c t i o n e n e r g i e s d u e t o t h e u n i f o r m c o n c e n t r a t i o n o f e l e c t r o n s , P, i s c a n c e l l e d b y t h e j e l l i u m b a c k g r o u n d . F r o m F i g . 31, we s e e t h a t t h e a v e r a g e k i n e t i c e n e r g y o f t h e a d d e d e l e c t r o n s a t r i s j u s t & , + [5&f+e<Mr)] / 2 , s o 2 that E. is just E k = z& f + i/(ec|>(_r)+66f)5p(£) dv . (8) The potential due to the electrons is (J) (r) - ze /r , so E is E p = - / ^ M r ) dy - f j [cj, (r) - ^ 6 p(r) I dv . (9) Adding these contributions gives E = E., + E = k p z& /^•6p (r ) dv (10) and for the case of a single ion, using (6) and 8&f = 0, which is the site energy of the intercalated atom measured with respect to the energy of the ion and z electrons separated at inf in i ty . For typical electron densities in sol ids, E ^ from (11) is negative, and several electron volts in magnitude. In fact, (11) has a minimum (most negative) value as a function of electron density, p; for z = 1, this minimum is E Q = -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 its 22 22 -3 minimum value as p varies from 0.3 x 10 to 6.5 x 10 cm . In part icular, 22 - 3 for the electron density of lithium, p = k.7 x 10 cm , E Q = -7.05 ev, in (rather fortuitous) agreement with the sum of the cohesive energy (1.7 eV) and the f i r s t ionization energy (5-4 eV) of lithium. If (11) were really the the correct expression for E q for an intercalation compound, we would con-clude that the voltage of an intercalation cel l depends only on the electron .densities in the anode and in the cathode, and for intercalation of lithium could never be greater than 0 . 5 volts . 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 and r_^, and their energy as Ij^-J^l °°- Because (k) is l inear, the total electron density from both atoms is just the sum of the electron densities from each. The interaction energy W then consists of the sum of three terms: (a) The energy of the ion of one atom in the potential cf>2(_r) of the other atom, /ze 5(_r-r^ (_r) dv (b) 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 kinetic energy change of the electrons, |/[(f)1(r_)+cf)2(L)][<Spl(i:)+6p2(r)] dv - f / ^ (r)6 P l (r.)+<|>2(L)6p2(_r)]dv . 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 ( 1 9 5 7 ) . As a result, 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\>{L]-r_2) ( 1 2 ) which becomes, using (6) for <J> (_r), 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 \ the Thomas-Fermi results are qualitatively —i —z ^ wrong. At large distances, the screening electron density osci l lates in space (Friedel osci l lat ions) rather than decaying exponentially as the Thomas-Fermi results indicate. This leads to an interaction between intercalated atoms that can be attractive at certain separations. To obtain these Friedel osc i l la t ions , we must replace (h) with the f i r s t order self-consistent equation (March and Murray 1960) for V 2cj) (r_) obtained in the Hartree approximation, or equivalently (and more commonly) by screening each Fourier component of the ionic potential ze/r with the zero frequency Lindhard die lectr ic constant rather than the Thomas-Fermi die lectr ic constant (Ziman 1 9 7 2 ) . The potential <j> (r) far from the ion tends to T r z e a , cos ( 2q ,r) *(r) - — - — s L_ d/,) q f ( l + 2 u q f a b ) 2 r 3 (Blandin 1 9 6 5 ) , where a^ is the Bohr radius "n2/me2, and q^ . is the Fermi x wavevector, (2mS^)"2/ti. 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 ) ; for q .^r < 2rr , cj> (r) rises steeply like the Thomas-Fermi solution. In calculating the interaction W(r) between two intercalated atoms using the Hartree solution, there is once again a cancellation of the change in the kinetic energy of the electrons with the interaction energy of one electron cloud with the other atom, so (12) s t i l l holds (Corless and March 1961). The interaction energy thus becomes Tr(ze) 2a, cos(2q,.r) W(r) = ze<f)(r) - — ^ — - — ^ - • (15) q f ( l + 2 T r q f a b ) 2 r 3 The f i r s t three minima where the asymptotic solution can be used occur at 22 - 3 2q f r = 2.90TT, 4 .94T T , and 6 .96TT . For an electron density of 2 x 10 cm , , . . . o o o these minima are at r = 5-4 A, 9.2 A, and 12.9 A, and correspond to values of W of -0.46kT r , -0.096kT r, and -0.035kT r (where T r = 25°C) . An energy of kT r /2 is large enough to cause clustering and/or condensation in a latt ice gas at room temperature. The discussion so far has assumed only one or two intercalated atoms are present in our inf in i te ly large free electron host. In this case, 6& .^ = 0. Eventually, however, as the number of intercalated atoms increases, we wi l l find Sp 5* 0 in regions of the host where 0 = 0, so the Fermi energy must change according to (2). We wil l discuss this change using the treatment of the Thomas-Fermi approximation given by Friedel (1954). We consider a uniform distribution of n intercalated atoms (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 atom1 at the center, then replacing each polyhedron with a sphere of radius R given by i r R 3 = ^ . (16) 3 n 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) = 6s f , A XR-1 X(R-r) A XR+1 -X(R-r)" •1 + e + e 2Xr 2Xr e (20) ze2X 6.^f = XRcoshXR - sinhTXR ' ^ (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 the energy corresponding to R °° (11). Using (10), we find W = n<5&f|-(XR+l)e~AR (23) which for large R, using (22), becomes W - n z 2 e 2 X e " 2 X R « n | o & f | . (2k) This relation, that the interaction energy per particle is much smaller than the shift in the Fermi energy, is a consequence of the fact that confining the electrons to a sphere of radius R, which increases their kinetic 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 kinetic and potential energies, resulting in (2k). Note that W in (2k) -2XR fa l l s off as e as the separation between the ions increases, as con-trasted with the interaction between a pair of atoms (13), which fa l l s off as e 1^—1 —2 I /1r — r | . Also note once again that the Thomas-Fermi calcu-lation predicts a purely repulsive interaction; i f the more accurate Hartree equation were used instead, Friedel osci l lat ions should result. At this point, i t is useful to connect these Thomas-Fermi calculations with the rigid band concept. In the Thomas-Fermi approximation, the total energy of an electron in the kinetic energy state q (measured from the vacuum level) at the position r is & - ed)(r), where & is the kinetic - q - q energy and -e'<j)(_r) the potential energy. Each electron state is shifted downward in energy by -e cf)(j^ ) - 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 ( i .e . r i g id ly ) . To apply the Thomas-Fermi results to real metals (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 "rigid band theory". It does not mean that the electron states are held r igidly in place ( i .e . 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 their screening clouds occurs. Moreover, when the Fermi energy f inal ly does begin to change, it changes faster than the interaction energy per part ic le , W/n, which means that the measured cel l voltage in an intercalation battery does not change in proportion to the change in the Fermi energy of the host, contrary to statements in the l i terature. 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 (see Lang 1973). For intercalation of a neutral atom (the ion plus its electrons), 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 site 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 ina l ly , we should point out that exchange and correlation effects, 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 ( 5 ) (Friedel 1 9 5 4 ) . Application of these calculations to real intercalation systems is complicated by the fact that the calculations assume the electron density in the sol id is uniform prior to intercalation, whereas in real systems this is definitely not true (as shown, for example, in the electron density contours calculated for Ti by Krusius et al 1 9 7 5 ) . In most intercalation systems of interest, such as the transition metal dioxides or dichalcogen-ides, the conduction bands are largely derived from the d orbitals of the transition metals, and so the conduction electrons are expected to be concentrated near the_transition metal nuclei . On the other hand, the intercalated atoms s i t on sites near the oxygen or chalcogen atoms, and are several angstroms from the transition metal atoms. As a result, much of the screening of ionized intercalated atoms wil l be done by the polarizable oxygen or chalcogen atoms. 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 shift is not in direct proportion to the intercalation cel l voltage, is expected to be va l id . F inal ly , consider what happens i f . the intercalated atom continues to bind a l l of its electrons when in the host lat t ice . Although this may seem like a different case than that discussed so far, where charge transfer to or from the host latt ice 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 interaction, and the language we would use to describe i t . The interaction is qual i -tatively similar to that expected between atoms in free space - it can be attractive or repulsive, depending on the nature of the outer electronic states. For example, an attractive interaction corresponds to the.tendency to form a molecule in free space; large attractive interactions cause clustering, which corresponds to the formation of a l iquid or a so l id . The. details of the interaction clearly require an in-depth consideration of the particular guest-host system of interest. 5-3 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 transit ion. 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 transition 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 resistive loss), it is appropriate to review some of the relevant points of meta1 -insu1ator transitions here. For s implici ty , 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 static d ie lectr ic constant of the host. This case is relevant to shallow traps in semi-conductors, a case where metal-insulator transitions have received consid-erable experimental attention.(Mott 1974) . The energy of the electron state lies below the conduction band edge by an energy V " ' - ^ • <2) Because of electron repulsions, a second electron of opposite spin added to this state wil l have an energy higher than (2) by some Coulomb repulsion energy U . Thus as x increases, and the electron clouds of the intercalated c atoms begin to overlap, broadening the electron states into bands, the band due to the state (2) wi l l be f u l l , and no metallic conduction wil l occur. However, as x increases s t i l l further, several mechanisms may bring about a transition to a metallic state: (a) As proposed by Hubbard (1964), the bands derived from the state (2) and the state U , above it wi l l eventually merge into one band. This occurs approximately when the bandwidth equals U c . In terms of the density of intercalated atoms p, this condition of band overlap and resulting transition to metallic behaviour (since the merged band i s only half ful1) i s P 1 / 3 a g ^ 1 . (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 transition from the metallic side, where the guest atom is singly ionized and screened by the conduction electrons - provided by the guest (so the density of the electrons is also p ) . -A r The resulting screened Coulomb potential e / r develops a bound state for A ^ ^ ag as p decreases, leading to a transition to an insulating state. From the expression 5 . 2 ( 5 ) 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 ) (see also Berggren 1 9 7 4 ) , based on the Claussius-Mosotti equation for the die lectr ic constant K due to an assembly of atoms of polarizabi1 i ty 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 * ) 3 / 2 for a hydrogenic o r b i t a l , this once again leads to the condition (.3) for the transition density. Since the above mechanisms a l l lead to the same condition on p, ( 3 ) , for the density at which the transition occurs, controversy s t i l l exists over the exact mechanism that drives the observed transition 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 localized, 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 transition wil 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 transition generally requires studies of the conductivity at low temperature. F ina l ly , the expression (.3) applies to the overall composition of the host only i f the intercalated atoms are homogeneously distributed throughout the host. Attractive interactions 9 8 between the guest atoms can lead to clustering, and ( 3 ) then gives the composition of the clusters at which the clusters become metall ic. Bulk metallic conduction in the host wil l then be observed at the so-called percolation transit ion, when a metallic cluster f i r s t extends throughout the host. A meta1-insulator transition has been reported for the intercalation compound H W0_ by Crandall and Faughnan ( 1 9 7 7 a ) at x = 0 . 3 2 . The voltage x 5 curve of this material is described quite well by the mean f ie ld expression h.k(3) with a repulsive interaction (yU = 0 . 5 3 eV) (Crandall et al 1 9 7 6 ) ; This has been cited as evidence against clustering of the intercalated hydrogen and a percolation transition (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 ld . This strain f ie ld can then act. on other intercalated atoms, producing a strain-mediated interaction between pairs of intercalated atoms. This interaction turns out to be attractive in a latt ice with free surfaces, and is large enough to produce condensation of the intercalated atoms at room temperature and above. An attractive elast ic interaction has been proposed as an explanation for the phase transitions in some metal-hydrogen systems (see Section 3-4). In what follows, we wil l discuss both the long range and short range aspects of the elast ic interaction between intercalated atoms. We describe the host latt ice as an elast ic continuum; a latt ice 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 . First we briefly review continuum e las t ic i ty theory (see, for example, Sokolnikoff 1 9 5 6 ) . Consider a system consisting of a linear e last ic medium together with sources of body forces _f_(_r) and surface forces _fS (r). When the body and surface forces act on the medium, the total energy of the system (medium plus forces) is changed by E = i / c . i k £ £ ; j(r)e k £ (j:) dv - / f j ( r ) u . ( r ) 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 repeated (Cartesian) indices are summed from 1 to 3- The tensor C j j ^ ' s the e last ic stiffness tensor. The displacement of the medium in the direction i at position r_, u.(r_), is related to the strain e. .(r) by U -/3u. 3u.\ with x. the Cartesian components of r_. The actual value of u^  produced by the forces is that which minimizes E; for this displacement f i e l d , (1) becomes E " - * / c i j k £ £ i j ( r ) £ k £ ( ^ d v = -i/f.(r_)uy(r_) dv - ± / f;(r>.(_r) dA . (3) I I A The stress a., is qiven by I J a : : ( n = ~ ij v-' v Be.. ' J c i j k £ £ k J l f=fS=0 and satisfies 3a!. L L = - f : ( r ) (5) 3x. i J within the medium, and a. . n . = f? (6) I J j 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 1 9 7 8 ) . Point defects exert 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 ie ld 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. . = Jx . f . dv ( 7 ) i j j i J i j This is analogous in electrostatics to the f ie ld from a charge distribution of zero net charge, which can be characterized by the dipole moment of the charge (Jackson 1 9 7 5 ) ; because of this analogy, P.^ is often referred to as the elast ic dipole tensor. 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 i j k £ P k £ P where s . . . „ is the elast ic compliance tensor (inverse of c . . , „ ) . In most i J kx, r i j k£ cases, the shear strains are small (e.. - 0 , i 4 j ) . In addition, many systems can be c lass i f ied into one of the following three cases: I. Extension in one direction only £^ 4 0 , e . j = 0 otherwise. This case applies to most layered compounds, I I. Equal extension in two directions z = e 4 0 , e . . = 0 otherwise. This case applies to most channeled 11 22 I J R R structures, including the rut i les . III. Equal extension in a l l three directions = £ „ _ = ^ 0 , e . . = 0 otherwise. This is the case of the 11 22 33 I J 102 " d i l a t i o n s p h e r e " , a n d a p p l i e s t o m e t a l - h y d r o g e n s y s t e m s s u c h a s H N b . We w i l l d i s c u s s t h e s e t h r e e c a s e s i n s o m e d e t a i l . B e c a u s e P . . c o n t a i n s a l l t h e i n f o r m a t i o n we w i l l n e e d a b o u t t h e e l a s t i c U p r o p e r t i e s o f t h e i n t e r c a l a t e d a t o m s , we w i l l h e n c e f o r t h a s s u m e t h a t t h e f o r c e s f_ h a v e o n l y a f i r s t m o m e n t . T h e f o r c e s a r e t h e n o p p o s e d ' p a i r s o f d e l t a f u n c t i o n s . A n e x a m p l e i s s h o w n i n F i g . 3 2 , f o r t h e c a s e w h e r e P . ^ i s d i a g o n a l : P = p r 0 0 \ 0 2 J 0 0 3/ (9) T h e f o r c e s t h a t p r o d u c e t h i s d i p o l e t e n s o r a r e o f t h e f o r m f 1 = Um K1iS(x2) S(x3) [fiCx^b) - 6( X l+b)] ( 1 0 ) b'+ 0 2K.b= P , i 1 w i t h s i m i l a r e x p r e s s i o n s f o r an<^ fy T h e d i a g o n a l n a t u r e o f P i n (9) i n d i c a t e s t h a t t h e i n t e r c a l a t e d a t o m e x e r t s n o s h e a r f o r c e s o n t h e h o s t l a t t i c e . We now c o n s i d e r t h e i n t e r a c t i o n b e t w e e n t w o i n t e r c a l a t e d a t o m s l o c a t e d a t r_ a n d r_2 . F o r s i m p l i c i t y , we c o n s i d e r o n l y t h e c a s e w h e r e t h e i n t e r -c a l a t e d a t o m s a r e d e s c r i b e d by t h e s a m e d i p o l e t e n s o r P . . . I f o n e u s e s ( 3 ) I J t o c a l c u l a t e t h e e n e r g y o f t h e t w o a t o m s , o n e f i n d s t w o s e l f e n e r g y t e r m s , t w o t e r m s d e s c r i b i n g t h e i n t e r a c t i o n w i t h s u r f a c e f o r c e s f_ S, a n d a n i n t e r -a c t i o n e n e r g y W ^ g i v e n b y W ^ l W = - / f i 2 ) < I ' u ! ( i l ) d v ' = - P i j e J j ^ 2 ) 103 (2) 1 where f_ is the force due to the second atom, u_ the displacement f i e ld of the f i r s t atom, and e!.(r_) the strain f ie ld 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 ld u_ (and the strain £ . ^ ) into two parts (Eshelby 1956): u_ , the displacement which would be produced in an inf inite medium; and u 1 , the additional displacement needed to satisfy the boundary 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, 0 0 2 u_ fa l l s off as 1/r , diverging at the position of the atom (in continuum theory), while u 1 gives rise to a strain which is slowly varying over the volume of the host and is proportional to 1/v. In terms of these two contributions to the total s tra in , can be written as ^^LrL2) = w °°( j l ) + w I ( L r L 2 ) ( 1 2 ) where r_ = r_^  - r_^. We now discuss these two terms separately in the following two sections. - oo » 6.2 Infinite Medium Interaction W The displacements produced in an inf inite 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 (as in 6.1(10)), the displacement uJ (r) due to an intercalated atom with dipole tensor P.. at r = Q is jk 105 u i ^ = - p m k ^ - M from which the strain is / V G . , (r) 8 2 G., (r) \ e ? . ( r ) = - i P ' k - + J k - . (2) ij — mk\tfx.9x dx.dx / \ j m i m / Hence the interaction energy between an intercalated atom at _r and one at _r = 0 i s , from 6.1 (11) 3 2 G . . (r) J m 00 The interaction W (rj is repulsive or attractive depending on the direction of r_, and averages to zero over the sphere of any non-zero radi us, indepen-dently of the anisotropy of the medium or the form of P . . (Liebfried and Breuer 1978). Along a given direction, the magnitude of W (r) fa l l s off as 1 / r 3 . oo To i l lustrate the angular variation of W ( r ) , we consider an isotropic medium, for which the Green's function can be found expl ic i t ly (Love 19^4): r (r) - JL 1 + v  G i j ^ - 8TTY x. x. 6 . . _ L ± + (3-4v-)--U-r 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 i j 3 1 discuss the three cases given in Section 6.1, corresponding to extension in one, two, or three directions respectively in uniformly intercalated materials. 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 find: i . = p2 = i V 3 ( 5 ) - = (1-2v)(1+v) - - = - = ( 6 ) 33 ( 1 - V ) Y H 3 P ' £11 £22 0 [ b ) M . P 1 = P 2 = ^ P 3 (7) - = - = (1-2V)(1 +V) - -£11 e22 Y 1p ' 33 I I I . P1 - P 2 = P 3 . (9) 1^1 = £22 = ^33 = ^"y2V^ PiP" ' ^ 1 0 ^ The expressions for W (r_) which follow from (5) - (10) are most conveniently written in terms of the average strain per unit concentration £ 0 / P 0 > = £/P> where p Q is the concentration corresponding to x = 1, and £ q is the non-zero component of the strain 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 from the z-axis: I. w°°(r) = - ( ^ , ( 1 5 0 0 5*9 - 6 c o s 20 - 1) (11) 87i(l-v 2 )r 3 \ p o / II. vT(r) = ( —) (iScos^e - 6(3+2v)cos20 + 3+4v) (12) 8^(1-V 2)r 3 \ p o / III. W°°(r) = 0 (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 the value of Poisson's ratio v while for case II the shape differs for different v . It is clear for guest atoms in layered compounds (ease l) that the interaction is attractive in the layers and repulsive normal to the layers, while in channeled hosts (case II) the interaction is attractive along the chains and repulsive normal to the chains. This is summarized schematically in Fig. 35. For di lat ion centers (case III) the interaction is identically zero. 0 0 . . In an anisotropic medium, W (r_) wil l have qualitatively similar features as for the isotropic case; the expl ic i t calculation of W (r) i s , however, much more d i f f i c u l t . One case which has been treated is that of an inter-calated atom with dipole tensor P.^ . = . ^6 j ^ in a very anisotropic hexagonal medium.(s__ » s . . . where s.. is the elast ic compliance tensor in the 33 11 i j abbreviated two subscript notation). This case was discussed by.Safran and Hamann (1979) in their treatment of intercalation of graphite; they found: e2Y_ 1 - ( l+2a: 1 )cos 29 W (r) = 3 - T L T =TF7T • 0*0 torpVa 1 Q - ( l - a 3 1 ) c o s 2 e J 5 / ^ In relating P to 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 is so, then a^ = c ^ /c^ = s^/s^y The Young's modulus Y^ = \/s^y 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 sol id angle over which the interaction is repulsive, and decreases the attraction in the xy plane. However, the qualitative features of Fig. 33 remain unaltered. 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 ratios. 109 F i g . 34 - P o l a r p l o t , s i m i l a r t o F i g . 33, f o r r u t i l e - r e l a t e d c o m p o u n d s . T h e 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 i n t e r c a l a t i o n c h a n n e l (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 e c h a n n e l (9 = 90°). T h r e e c u r v e s a r e s h o w n c o r r e s p o n d i n g t o d i f f e r e n t P o i s s o n r a t i o s , (a) v = i , 6_ % 60.3°; ( b ) v = 1/3, 6 C % 61.0°; ( c ) v = 0, 6 C ^ 63.4°. Layered Compounds Rutile Related Compounds X o Fig. 35 - Schematic summary of the nature of the strain-induced interaction W°°(-r) between two intercalated atoms in layered and rut i le -related compounds. To estimate the magnitude of the energies involved, consider a specific example: the intercalation .compound Li^MoO^. The sites in MoO^  available for the intercalation of lithium are presumably the tetrahedral sites lying in the tunnels along the pseudotetragonal c axis of the monoclinic o MoO^  crysta l , which are spaced by a distance of c/2, where c = 2.81 A (see Fig. 8). The chains of sites are arranged in a square lat t ice , ' _ o separated by a distance a//2, where a = 4.86 A is the pseudotetragonal a latt ice parameter. During intercalation, c stays nearly constant, while a increases by Aa = 0.34 A as x varies from 0 to 1 (Sacken 1980). Hence MoO^  is a good example of case II discussed iabove. The reference density _ o p = 1/33 A , and the reference strain is e = Aa/a = 0.069. Since there o o are no; ipublished e last ic constants for MoO ,^ we assume it is reasonably isotropic, and assign it e last ic constants Y = 10 ergs/cm , v = 1/3. Then from (12) 5.6kT W°°(r) = - (IScos^e - 22cos29 + 13/3) (15) r 3 o where kT = 25-7meV and r is measured in A. Along the chains the interaction r is attractive 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 k T ^ . . . The interaction is appreciable only for a small number of latt ice 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 last ic attraction is not overwhelmed by electronic repulsions. Lattice calculations give similar conclusions: Fisher (1958) calculated that attractive interactions of order 0.1 eV (4kT^) are expected between in ters t i t ia l atoms along certain directions in iron. 6.3 The Image Interaction vfl-Since is due to the image displacement, u^  , i t depends on the shape: of the host, boundary conditions, and the position of the intercalated atoms in the host. In general, it depends only weakly on the relative positions of the intercalated atoms, and its magnitude is inversely propor-tional to the volume of the host. Unlike W°°, its angular average is nonzero, leading to a net attraction or repulsion (depending on the boundary con-ditions) when averaged over a l l directions. The shape dependent behaviour of W1 makes calculations complicated i f we are interested in the details of the interaction for arbitrary positions of the guest atoms in the host. Hence we consider the simplest example: an isotropic sphere of radius R with two dilation centers (case III above), one at the center of the sphere (r = 0) and the other at an arbitrary position r. The inf inite medium component of displacement due to the atom at r = 0 is(Liebfried and Breuer 1978) £ { R ) = P (1*V)(1-2V) ± _ . ( 1 ) - 4TTY(1-V) r 3 For a f in i te sphere, (1) alone does not satisfy the boundary conditions in general, so we need the other solution of the equations of e las t ic i ty in this case, which is linear in r (the image term). For free and clamped surfaces, the image term is (Liebfried and Breuer 1978) iv- p,sTTRJF-=r ( f r e e ) < 2 ) K ^ ( 1 ) - - ^ ^ ^ ^ ( c W e „ . (3) K In these two cases, W1 (r) is independent of the position of the second atom. In terms of e and p , we can write W1 = U, where U is given by o o 7 e 2 U = - l-^-— (free) (k) N 1 - v p o .. . 1 (1+V)Y o . . . . . H ( l - v ) ( l - 2 v ) p~J (clamped) . (5) We see that W^" is attractive for a free surface and repulsive for a clamped one. Moreover, even though W1 is inversely proportional to the sample size through N in the denominator of (4) and (5), the atom at the center wi l l 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 distribution of intercalated atoms, most of the complications average out. In fact, we can obtain those terms which are dependent on the boundary conditions quite simply. Consider 6.1(1) once again, written expl ic i t ly in terms of the displacements u_a(r), the strains and the forces _f (r) of atoms on the sites labelled by a in the host: E = J n n l i f e . .. 0 e ? . ( r)ef! (r) dv - Jf?(r)u?'' (r) dv L, a a ' ijk£ I J — kl — J i — i — aa1 L -> •> (6) - J n fff(r)u?(r) dA L a\ i — i — a A In what follows, we wil l be interested in two types of boundary conditions: a free surface, where f S = 0, and a clamped surface, where the total displacement £n u (r) = 0 on the surface. For both of these cases the last a a term in (6) is zero. In evaluating the sums in the f i r s t two terms in (6), we find self energy terms ( a = a 1 ) , which contribute to the site energy in the lattice gas models of Chapter 4, and local f ie ld correction terms similar to those found in the theory of dielectrics (Kittel 1971), which arise from the short range correlation of the occupation of the sites (Siems 1970; Alefeld 1971). These terms involve the strain of an atom on a given site near the site i t se 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 i j k a e u ( ^ ) e k £ ( ^ Pjj-e. . (j_)p(_r) dv (7) where £j.(_f_) is the total strain at _r, given by £ . . ( r ) = T n (r) 1 1 — L a 11 - (8) The last term in (7) follows from the delta function nature of the forces. The stresses a (r) are defined by "J ~ , v 1 / 3 F a . . (r) = — h— 1 j — v de (9) ' J / T,p ; f S = 0 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 i j £ . . are those from W , we have 1 j e' auW = 'uufuU - p i j p ( ^ ( 1 0 ) w h e r e a., ( r ) s a t i s f i e s ' J -do. U _ 9x. J 0 (11) ( a s s u m i n g t h e r e a r e n o o t h e r s o u r c e s o f b o d y f o r c e s a s i d e f r o m i n t e r c a l a t e d a t o m s ) w i t h i n t h e m e d i u m , a n d a . . n . = f ! • (12) ' J J i o n t h e s u r f a c e w i t h u n i t n o r m a l ft ( c f . ( 4 ) - ( 6 ) ) . F o r t h e s e s t r a i n s w h i c h m i n i m i z e W , W b e c o m e s e e W e = - i / p i j e i j (£.)p(jr_) d v ( 1 3 ) T h e e f f e c t s o f s t r a i n o n W c a n c l e a r l y be s e e n f r o m ( 1 3 ) . F o r a c l a m p e d e s u r f a c e a n d a u n i f o r m d i s t r i b u t i o n o f i n t e r c a l a t e d a t o m s (p(r_) = p ) , e . . ( r ) = 0, a n d s o W =0. F o r a f r e e s u r f a c e a n d a u n i f o r m d i s t r i b u t i o n , I J — e o " . j ( r _ ) = 0, s o t h e s t r a i n i s g i v e n b y e. . ( r ) = i " . . = s . . . „ P . „p" (14) i j — i j i j kx, k£ v ' ( w h i c h we q u o t e d e a r l i e r w i t h o u t p r o o f i n 6 . 1 ( 8 ) ) , a n d W g b e c o m e s W e = - i s i j k £ P i j P k £ p 2 v (15) I n t r o d u c i n g x = p/pQ> w h e r e p Q = N/v a s b e f o r e , we c a n r e w r i t e ( 1 5 ) a s W = i N 2 U x2 ( 1 6 ) w h e r e U i s d e f i n e d b y e V ^ P i j P k * s i J k A ( 1 7 ) We see that (16) is of the same form as the infinite range interaction discussed in connection with mean f ie ld theory in k.k(2), with an effective interaction U e between the intercalated atoms. given by (17) is the change in this interaction on going from a free to a clamped surface. For the three special cases considered in Section 6.1, becomes ' • Ue ~ N (1+V)(1-2V) ( ™ > " • Ue • - N (1+V)2(1-2V) Vo ( 1 9 ) e 2 III. U = - 1-3Y -2. (20) e N 1-2v p o Note that (20) agrees with the difference between (h) and (5) for two dilation 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. As an example, consider once aqain the case of Li Mo0o. Usinq the values of strain e and 3 x 2 3 o the estimated e last ic constants given in Section 6.2, we find NU = -17-7kT , e r which is the effective total interaction energy of one atom due to a l l the others, and hence the parameter which would be used in latt ice 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 Elastic Interactions The e last ic interaction energies discussed in the previous two sections contribute to the interaction energies U , in the latt ice gas models of Chapter 4 (see 4 . 4 ( 1 ) ) . The infinite medium terms W produce f a i r l y short ranged interactions, and because of their 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. They are large enough to lead to staging in layered compounds, where fu l l layers alternate with empty ones, or similar ordering effects in channeled structures, where fu l l channels are surrounded by empty ones. The image terms produce long range interactions; in fact, 6 . 3 ( 1 6 ) is precisely of the form of the inf inite range interaction y l ) with y = N introduced in Section 4 . 4 . It should be pointed out that the local f ie ld terms mentioned in Section 6 . 3 tend to reduce the attractive interaction U e from the value given in 6 . 3 ( 1 7 ) , but they do not overwhelm it in the calculations which have been done to date (Wagner 1 9 7 8 ) . As a result, the image terms are expected to lead to unphysical regions in the free energy and chemical potential , as in Fig. 1 4 . Phase separation according to the Maxwell construction wi l l occur i f a low energy interface can be formed between the two phases. Such a low energy interface wil l be formed in a sol id i f the two phases actually break apart, or i f an incoherent interface is produced by dislocations. However, i f the interface remains completely coherent (the crystal latt ice remains continuous across the interface), the e last ic energies associated with the so-called coherency stresses required to hold the two phases, of different latt ice parameters, together becomes too large for this simple phase separation to have the lowest free energy. In these cases more complicated distributions p(_r_) of the intercalated atoms, called density modes, are produced (Wagner and Horner 1 9 7 4 ) . Real systems are expected to be intermediate between the completely coherent and completely incoherent cases, involving both coherency stresses and plastic deformation. Because of this , the chemical potential of the intercalated atoms as measured by the voltage in an intercalation cel l wi l l show hysteresis over a charge-discharge cycle as indicated schematically in Fig. 3 6 . Such hysteresis is produced in two ways: (a) Energy is lost as interfaces between phases move and the crystal is plast ical ly deformed, in analogy with the losses associated with the motion of domain walls in ferromagnets. (b) As discussed in Section h.3, the particles in an intercalation cathode act as a "chemical potential bath", preventing the voltage of a particular particle from relaxing to the equilibrium value once the phase separation begins and the coherency stresses relax by plast ic deformation. Such hysteresis effects are to be expected in a l l condensation phenomena (f irst order phase transitions) which involve two coexisting phases of different latt ice parameters, whether or not the phase transition is actually produced by the e last ic interaction. There i s , however, a special case where the coherency stresses associated with the phase transition 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 strain, which occurs i f the strains produced for uniform intercalation of a host with free surfaces ( 6 . 3 0 * 0 ) are such that they leave a l l planes perpendicular to some direction (with a unit vector ft, say) undistorted. For example, i f e i 1 ' £ 2 2 ' a n c ' £12 a r e a ^ z e r o > then fi = (0,0,1) is such a direction. In this case, the two phases-can separate into thin plates normal to ft without 119 Fig. 36 Schematic discharge curve of an intercalation cel 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 this , consider the following. Cut the unintercalated host into thin plates whose normal is along n. Inter-calate some of these plates to a composition and the rest to x^ (where x.| and x^ are the compositions of the two phases), allowing the plates to expand freely. Then f i t the plates back together again. Since planes perpendicular to n remain unchanged, the latt ice f i ts 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 restriction that determines their thickness. In a real system, however, there wil l be regions where the plates of phase 1 terminate in a region of phase 2 and vice versa, as in Fig. 37- The elast ic properties of such a region are those of a dislocation loop surrounding the plate (in fact, the standard textbook example of an edge dislocation is essentially that in Fig. 37a - see Kittel 1971)-Since parallel dislocations repel, plates of one phase wi l l repel one another i f placed directly above one another along fi, and attract one another i f placed side by side, as indicated in Fig. 38,. These considerations imply that Fig. 37 - Elast ic equivalence of (a) a plane of intercalated atoms, the f i l l e d c irc les , to (b) a dislocation loop. 121 att ract i ve Fig. 38 - Interaction between two dislocation loops the plates wil l tend to be as thin as possible, one atomic layer, and as far apart as possible, which is equally spaced in the direction n. Such an explanation has. been recently proposed for staging 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 ) . This suggests that there is an additional driving force for condensation in the layers, beyond the e last ic energy considered above, namely the free energy decrease due to this structural rearrangement. However, as long as the two phases have different lat t ice 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 intercalation, and so there is no invariant plane strain. 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 of the intercalated atoms at position _r, y (_r) , in a host where the composition x varies with r_, because the voltage V depends on the chemical potential at the surface of the host (see Chapter 7 ) , and because the variation of y(r) with r_ determines the diffusion of the guest atoms in the host (Chapter 8 ) . Normally, one expects the chemical potential to depend on r_only through the local composition x(r_); that i s , y (r_) = y(x(r_)). However, when e las t ic effects are important, the term Wg ( 6 . 3 ( 1 3 ) ) wi l l give rise to a contri -bution y g t o y which depends on the details of the total distribution x(_r), or p (_r) , throughout the sample; that i s , y depends non locally on P (_r) . To discuss this , we divide the free energy F of the intercalation compound into two parts, Wg and the remainder F Q : F = F + W . (1) o e The term F leads to a contribution y to y which depends locally on x: o o 1 8 F / \ 1 o y (r) = — - K — o — N ox (2) The e last ic term W gives rise to a term y in y; y is found by calculating e e e the variation 6W caused by a variation 6p(r) over some infinitesimal e — volume v about the point r: r — 6W 1 e ( 3 ) r Because the change <5p(_r) produces long range strains ^ e j j » the variation <$w calculated from 6 . 3 ( 1 3 ) wi l l depend on the total distribution p, not just the value at _r. As a specific example, consider a distribution of dilation centers (case III, P. . = P6. .) in an i sotropi c medi um with a free surface. The 'J i J relation between stress, s train, and density of intercalated atoms in this case is (from 6 . 3 ( 1 0 ) ) a i j ( ^ " c i j k £ £ k £ ( l ' " PS\f{L> This is identical in form to the relation between stress, s train, and temper-ature T in thermoelasticity (Landau and Lifschitz 1 9 7 0 ) a i j ( ^ = c i j k A e k A ( ^ " ^ i j 1 ^ (5) where a is the thermal expansion coefficient and K = Y / 3 ( 1 - V ) the bulk modulus. Thus we can use the solutions to thermoelastic problems to discuss intercalation systems i f we make the substitutions Ka -> P arid T •> p . In particular, for an isotropic sphere of radius R with a spherically symmetric distribution p(r), the radial displacement u^(r) is (Landau and Lifschitz 1 9 7 0 pg. 2 2 ) : I \ P 1+V - L f p(r) r 2 dr + 2 ( 1 " 2 V > - r , r 2 0 1+v r R / P(r) L o r 2 dr ( 6 ) This can be used to evaluate W in 6 . 3 ( 1 3 ) , giving e (l+v)Y ( 1 - 2 v ) ( 1 - v ) (7) where we have eliminated P in terms of e and p . The contribution u to.the o o e chemical potential is then found from ( 7 ) and ( 3 ) to be c\ (l+v)Y e o / f v 2(1-2V) -\ y e ( r ) = - (l-2v)(l-v) IT [ x i r ) + V n * ) 2 (8) W l i th the average composition x = v /p(r) dv. The chemical potential at the radius r depends not only on x(r) but also on the total amount of intercalate through x. For a uniform distribution x(r) = x, y = NU x"with U qiven e 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 ). This can be s important in transient experiments intended to study diffusion in the host (see Chapter 9) where a small composition change at the surface is produced 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 (8) s ^ . - ^ - f e V V s i r 1 — ^ • <9) dx e For short times, however, when only the composition near the surface has changed, so that x = 0, (8) gives ^ s - U i T - 3 A y s ( 1 0 ) + NU.' dx e where Ue 1 +v u 3TT^y ' e The quantity U V U varies from 1/3 to 1 for v varying from 0 to i , and is equal to 2/3 for v = 1/3- Since U E can be many times k T r , as we saw in Section 6.3, this can be a significant effect, especially at compositions where 8U q/8X + NUe is small (recall is negative, while 3u/8x is always positive); 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. Also, since diffusion is driven by gradients in u, the diffusion coefficient wi l l depend on the details of the distribution of the intercalated atoms and, as it turns out, on the shape of the host (Janssen 1976). This dependence of the diffusion coefficient 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 last ic interaction is responsible for the phase transitions seen in these systems. 6.6 Limitations of the Theory The above theory of the e last ic interaction is based on several assumptions, which we wil l now discuss. (a) Infinitesimal strains The relation between displacement and strain, 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 10% of the terms linear in e. (b) Hooke's law The relation between the elastic energy and the strains that we used, 6 . 1 ( 1 ) , neglects cubic and higher order terms in £ . The expression for the elast ic energy E in a volume v is actually of the form v 2 C i j k £ £ i j £ k £ + 3~ C ij l<£mn e ij £ : k£ e mn + 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 " * C 11 £ 11 I1 + 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 wil l be of the form (written per unit volume) - = — - — (3) v r i o r -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 usual e last ic energy, i ^ e ^ • I f we assume that the challcogen-metal-cha 1 cogen 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 Y 3 V 2 V , 1 + T £ 3 3 / \ 1 + Y £ 3 3 , (A) where y is the ratio of the c latt ice parameter (per layer) to the distance between adjacent chalcogen atoms (y % 2 typically) 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 is actually larger than the quadratic term at e = 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 latt ice around i t , independent of the strain at the atom's position, . :!t':<i'S :pr.obable, however, that the atom wil l exert less force as the strain increases due to the presence of the other intercalated atoms. This wil l effectively cause P. . to decrease as x increases. Note that this 1 J may compensate to some degree the softening of the lattice at large strains due to anharmonic effects, 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) Elastic constants of the host independent of x This point must be checked experimentally. It is unlikely that interca-lation compounds with x ^ 1 or greater wil l have the same elast ic constants as the unintercalated host. However, the effects we have discussed above should s t i l l be valid even i f the elast ic constants vary, although the actual values of the interaction energies wil l involve some effective elast ic constants different from those of the pure host. (e) Elastic isotropy Some hosts are quite anisotropic e las t i ca l ly . The effects of anisotropy was estimated in connection with 6.2(14); for that case, the qualitative features of the elast ic interaction were in aqreement with the corresponding isotropic result. Care must be taken, however, in .classifying an anisotropic material as one of the three cases discussed in Section 6 .1 ; one should compare the elastic.energies required to produce the observed strains rather than the magnitudes of the strains themselves. For example, i f Y^ and Y^  are respectively the Young's modu1L fd<n.extensions normal and parallel to the basal plane of a layered compound, the condition for the compound to be case I is Y ^ e ^ » Y ^ e 2 ^ rather than simply e » e ^ . For graphite, Y ^ Y = 28 (Blakslee et al 1970); for MoS2, Yj/Y - c 1 1 / c 3 3 = * k 6 ' e s t i m a t e d from the neutron data of Wakabayashi et al (1975). (f) Use of continuum e last ic i ty The expressions given in Section 6.3 for are exact (subject, of course, to assumptions (a), through (f)). By contrast, the short range interaction results of section 6.2 are valid for intercalated atoms separated by several oo latt ice spacings. The qualitative features of W , that i t is attractive in some directions and repulsive in others, are expected to be true in a latt ice calculation, but some of the details may di f fer . Calculations by Bui lough and Hardy (1968) on vacancies in hypothetical isotropic aluminum (an example of case III) indicate..a nonzero interaction which varies as l/r"' and oscil lates in sign along a given direction; 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 latt ice wil l make to the continuum results, but it could be appreciable for intercalated atoms separated by one or two latt ice 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 cel l due to changes in the composition, x, of the interca-lation cathode. This discussion assumed that the cel l was in equilibrium throughout the intercalation process. In practice, intercalation occurs at a f ini te rate, leading to changes in the cel 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 cell current and on time. F i r s t , we look at an intercalation cell in de ta i l , to see where over-potentials occur. This also allows the considerations of Chapter k to be related to the conventional picture of an electrochemical c e l l . We briefly discuss the loss mechanisms that an intercalation cel l has in common with other types of electrochemical ce l l s : losses due to current flow through the electrolyte, charge transfer across the electrolyte-electrode inter-face, and possible rate-1imiting surface reactions. We then discuss . diffusion of the intercalated atoms, which causes an additional over-potential not present in most electrochemical ce l l s . The details of this diffusion overpotential are presented in Chapter 9, following a discussion in Chapter 8 of the variation of the diffusion coefficient with the com-position of the intercalation compound. Final ly , since intercalation electrodes generally consist of powdered host.materia 1 f i l l e d with electro-lyte, we consider how such porous electrodes modify the details of the relationship between overpotential and current. 7.2 Electrochemistry of Intercalation Cells In order to discuss the various types of loss, or overpotential, in an intercalation c e l l , we must f i r s t look in detail at the cel l to see where such loss can occur. A schematic view of an electrochemical: eel 1. i s shown in Fig. 39 ( c f . Fig . 1 ) . We immediately specialize to a case appropriate to an intercalation cel l by taking the "reaction" which provides the cel l voltage to be a simple transfer of an atom A from the anode, a, to the cathode, c The electrolyte is assumed to be a binary electrolyte, con-taining ions of the atom A, which we wi l l denote by A and which carry a charge z^e, ions B of charge zge (with zg < 0 ) , and solvent molecules. The contacts d and d1 are made from identical materials, since only then can a potential difference be measured. The open c ircui t voltage V of the cel l is just the potential difference between d and d 1 . We wi11 use the symbol $ to denote the e lectr ic potential , with a superscript to refer to the d c b a d' t t y = 0 y = £ Fig. 39 - Schematic view of intercalation cel l.c a: anode, b: electrolyte, c: cathode, d and d 1 : contacts made from the same material. 132 particular material in the c e l l . In this notation V = cf>d - (j) d (D Just as in Chapter k, V is given in terms of the difference in the chemical potentials y of the atom A in the cathode and in the anode as V = cj)d - <J)d' = - J L ( y c _ y a) (2  v T . z-e V M A * K A This use of subscripts to denote species of particle and superscripts to distinguish different parts of the cel l is standard electrochemical notation (see any text in electrochemistry, such as Bockris and Reddy 1970); the use of t i ldas to distinguish ions (A) from neutral atoms (A) is not. Although an electrochemical cel l can contain several different species of charged part ic les , the open c ircui t (equilibrium) cel l voltage can always be expressed in terms of thermodynamic quantities of neutral ent i t ies . For example, (2) involves the chemical potential of the neutral atom, A, even though in the transfer of atoms from a to c electrons flow through the external c ircui t (a d ^ . d 1 -> c) and ions A flow through the electrolyte (a -> b -> c) . On the other hand, in order to discuss the kinetics , or losses, in an electrochemical c e l l , we must consider the details of the motion of the charged part ic les . This requires the use of the electro-chemical potential , which we denote as y. The electrochemical potential of some charged particle a, y~, is1 defined as the change in the free energy F of a system when the number of a par-t ic les changes at constant temperature T and volume v: (3) y~ is thus the work required to add an a particle to the system'.atoconstant T,v. 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 lectr ic potential , done against the e lectr ic fields 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 particles has a dipole layer at the surface; in a metal, this is produced by the sp i l l ing over of the electrons into the vacuum. This dipole layer is modified by adsorbed atoms or molecules. (c) The work y~ (the chemical potential) resulting from the forces which bind a into the body. Although Coulombic in or ig in , these are local forces, since the system is charge neutral when viewed over several atomic spacings,(except for the small amount of excess charge which produces (j)). y~ can be altered by changing the chemical composition of the system. Thus we have y~ - y~ + Y ~ + zecj) (k) This is only an approximate relationship, since the three terms are not completely independent. For example, addition of excess charge to change cp wil l also affect I1-,' however, for values of cf) of interest, the chemical and surface changes produced by adding the small amounts of charge needed to give these changes in cj> are completely negligible. We are not interested in x^ > so it can be absorbed into y~ or <j>, giving (5) Although not necessary, it is useful to use electrochemica1. potentia 1s in discussing the equilibrium voltage of the cell in Fig. 39, because it simplifies our discussion of the kinetics. Al l of the parts of the cel l in Fig . 39 are assumed to be conductors, so that in equilibrium e lectr ic f ields exist only at the interfaces. These fields are produced when the inter-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 particle 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 solution, b, and A atoms and electrons, e, in the cathode, c. This equilibrium requires y A " P A + Z A P S ( 6 ) which follows from the condition that there be no free energy change in equilibrium on transferring A from b to c and combining with electrons e ~b ~c to form the neutral A atom. Using (5) for y^ and y~ gives * c = - ^ ( y A - A - z^t] • (7) Note that cj)C - cj)'3 is defined only up to an additive constant, due:to_the ambiguity in separating y into y and zetj) and dropping x ' n (5). Similar equations can be written for each of the other interfaces in Fig . 39-These are: <$>d - c|,c = 1 (yg -ug) (8) ,b ,a 1 / .a a b \ * " * = ^e" ( Y A " Z A % " V (9) * d ' = i " " ^ ' : y ' ( 1 0 ) Combining (7) to (10) gives (2) once again, since y~ = LU 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 in these equations are to be evaluated at these interfaces. In equilibrium this is an irrelevant point, because the chemical potentials must be constant throughout the bulk of each material. It becomes important, however, when a current flows, since the concentrations, and hence the chemical potentials, of the various particles can then vary, throughout the materials. Thus, for example, when the cell is discharged and A ions are neutralized by electrons and trans-ferred into the cathode at the c-b interface, 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 cel l voltage. This change in the cel l voltage due to gradients in concentration is called the concentration or diffusion overpotential. There wi l l also be a reduction in the cel l voltage when a current flows (an increase on recharge) due to resistive losses in the bulk materials (sometimes called 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 briefly 136 reviewe'd" in Section 7-3. The flow of current in the e l e c t r o l y t e i s compli-cated by the f a c t that when the concentration in the e l e c t r o l y t e v a r i e s , so does the c o n d u c t i v i t y , so the concentration 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 i s discussed in Section J.k. The e f f e c t s of d i f f u s i o n in the host l a t t i c e are b r i e f l y considered in Section 7-5, and then in more d e t a i l in Chapter 9- The r e s i s t i v e losses due to current flow in the e l e c t r o n i c conductors are assumed to be described by Ohm's law, and so are not discussed f u r t h e r . 7.3 Losses Due to Transport Across the Interfaces The e l e c t r i c f i e l d at 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 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 in Fig- 39, i s considered to occur across one or two layers of solvent molecules adsorbed on the e l e c t r o d e surface (see, f o r example, Bockris and Reddy 1970). When an ion A in the e l e c t r o l y t e s o l u t i o n (b in 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 bulk of the i n t e r c a l a t i o n e l e c t r o d e ( c ) , i t must pass through t h i s s o - c a l l e d "Helmholtz l a y e r " . The t r a n s f e r of charge through the Helmholtz layer i s g e n e r a l l y regarded as an a c t i v a t e d process, with 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 with the p o t e n t i a l drop c ,b ,c ,b across the i n t e r f a c e , <P' - <P . If 1 i s the change in 9 - <P with current (the overpotentia 1) , such an a c t i v a t e d process gives the Butler-Volmer equation f o r the dependence of the current density i on n. , which i s where i i s known as the exchange current d e n s i t y . In ( l ) , the t r a n s f e r c o e f f i c i e n t s a and c i _ s a t i s f y . : . / a+en/kT - i I = I (e ^ - e o •ct_erj'/kT (D a + + a (2) w h e r e z ^ e i s t h e c h a r g e o f t h e i o n b e i n g n e u t r a l i z e d ; f o r z ^ = 1 , o n e u s u a l l y h a s a + - a - \. N o t e t h a t b y c o n v e n t i o n t h e c u r r e n t i s p o s i t i v e w h e n i t f l o w s f r o m t h e e l e c t r o d e t o t h e e l e c t r o l y t e . (1) c a n a l s o b e s h o w n t o h o l d i f t h e n e u t r a l i z a t i o n o c c u r s b y t h e t u n n e l i n g o f e l e c t r o n s a c r o s s t h e H e l m h o l t z l a y e r ( G e r i s c h e r 1 9 6 1 ) . F o r n » k T / e , t h e c u r r e n t v a r i e s e x p o n e n t i a l l y w i t h o v e r p o t e n t i a l , w h i c h i s k n o w n a s t h e T a f e l r e l a t i o n . E q u a t i o n ( 1 ) g i v e s a c u r r e n t d e n s i t y ; t o c o n v e r t t h i s t o a c u r r e n t , we n e e d t o k n o w t h e a r e a i n v o l v e d i n t h e c h a r g e t r a n s f e r . F o r a n i n t e r c a -l a t i o n e l e c t r o d e , i f t h e n e u t r a l i z e d i o n i n t e r c a l a t e s i m m e d i a t e l y , t h i s a r e a i s t h a t f r a c t i o n o f t h e t o t a l s u r f a c e a r e a t h r o u g h w h i c h i n t e r c a l a t i o n c a n o c c u r . F o r e x a m p l e , c r y s t a l s o f l a y e r e d c o m p o u n d s a r e u s u a l l y t h i n p l a t e l e t s , w h o s e f a c e s a r e p a r a l l e l t o t h e c l o s e p a c k e d a t o m i c l a y e r s ; a t o m s i n t e r c a l a t e o n l y . t h r o u g h c r a c k s o r s t e p s i n t h e f a c e s , o r t h r o u g h t h e e d g e s o f t h e p l a t e l e t s . I t i s p o s s i b l e , h o w e v e r , t h a t t h e n e u t r a l i z e d i o n s a r e f i r s t a d s o r b e d o n t o t h e s u r f a c e , a n d t h e n d i f f u s e o v e r t h e s u r f a c e t o a r e a s w h e r e t h e y c a n i n t e r c a l a t e . S u r f a c e d i f f u s i o n a l s o o c c u r s i n t h e e l e c t r o p l a t i n g o f m e t a l s ; i n t h i s c a s e , a d s o r b e d m e t a l a t o m s d i f f u s e t o g r o w t h s i t e s , w h e r e t h e y a r e i n c o r p o r a t e d i n t o t h e m e t a l l a t t i c e . A s d i s c u s s e d b y V e t t e r ( 1 9 6 7 ) . , s u r f a c e d i f f u s i o n m a k e s t h e e f f e c t i v e s u r f a c e a r e a a f u n c t i o n o f t h e c u r r e n t . A t l o w c u r r e n t d e n s i t i e s , c u r r e n t f l o w s f r o m t h e e l e c t r o l y t e o v e r t h e e n t i r e a r e a o f t h e e l e c t r o d e ; a t h i g h c u r r e n t s , a d s o r b e d a t o m s c a n n o t d i f f u s e f a s t e n o u g h f r o m a d s o r p t i o n s i t e s f a r f r o m t h e e n t r y p o i n t s ( o r g r o w t h s i t e s i n t h e c a s e o f e l e c t r o p l a t i n g ) a n d s o t h e c u r r e n t f r o m t h e e l e c t r o l y t e f l o w s o n l y n e a r t h e e n t r y p o i n t s , d e c r e a s i n g t h e e f f e c t i v e a r e a . A d d i t i o n a l c o m p l i c a t i o n s a r i s e i f t h e a d s o r b e d a t o m m u s t b e a c t i v a t e d i n s o m e w a y b e f o r e i t c a n i n t e r c a l a t e , p e r h a p s b y b r e a k i n g f r e e o f a t t a c h e d s o l v e n t m o l e c u l e s ; i f t h e a c t i v a t i o n p r o c e s s i s s l o w , t h e c u r r e n t s t h a t c a n f l o w may b e l i m i t e d t o s o m e m a x i m u m v a l u e . ( A c t i v a t e d 138 s t a t e s o f a d s o r b e d atoms have been p o s t u l a t e d i n t h e i n t e r c a l a t i o n o f h y d r a z i n e f r o m t h e v a p o u r i n t o NbSe^ by B e a l and A c r i v o s , 1978,) 1.k T r a n s p o r t T h r o u g h t h e E l e c t r o l y t e In e q u i l i b r i u m , a l l t h e w o r k r e q u i r e d t o t r a n s f e r an A atom f r o m a t o c i n t h e c e l l i n F i g . 39 i s done on t h e e l e c t r o n s , s i n c e t h e i o n s A a r e i n e q u i l i b r i u m t h r o u g h o u t t h e s o l u t i o n , b . When f i n i t e c u r r e n t s f l o w , a g r a d i e n t i s p r o d u c e d i n y ^ , t h e e l e c t r o c h e m i c a l p o t e n t i a l o f t h e A i o n s i n t h e s o l u t i o n ; as a r e s u l t , i t t a k e s work A y ^ = y ^ ( y = £ ) - y^(y=0) ( w i t h y t h e d i s t a n c e d e f i n e d i n F i g . 39) t o t r a n s p o r t t h e i o n s t h r o u g h t h e s o l u t i o n . S i n c e t h e work t o move t h e e n t i r e atom A i s u n c h a n g e d , t h e c e l l v o l t a g e V i s r e d u c e d f r o m i t s open c i r c u i t v a l u e V q t o V = V - — A y £ = V - n : (1) o z^e A o ( I n t h i s s e c t i o n , we n e g l e c t any o t h e r s o u r c e s o f l o s s e x c e p t t r a n s p o r t t h r o u g h t h e e l e c t r o l y t e s o l u t i o n . ) To f i n d t h e r e l a t i o n s h i p between t h e c u r r e n t and t h e o v e r p o t e n t i a 1 n , we must see how n depends on t h e c u r r e n t . In d i s c u s s i n g c u r r e n t f l o w i n t h e e l e c t r o l y t e , we must c o n s i d e r t h e e f f e c t s o f h a v i n g more t h a n one s p e c i e s o f m o b i l e c h a r g e . In s u c h a c a s e , t h e c u r r e n t d e n s i t i e s J ~ o f t h e v a r i o u s s p e c i e s o f m o b i l e c h a r g e a a r e r e l a t e d t o t h e e l e c t r o c h e m i c a l p o t e n t i a l g r a d i e n t s by t h e f o l l o w i n g c o u p l e d e q u a t i o n s ( s e e , f o r e x a m p l e , B o c k r i s and Reddy 1970): J ~ = VfL, (2) a aa' or a-;1-where I , = L ~ . ~ . F o r s i m p l i c i t y , we w i l l c o n s i d e r an i d e a l . ( i n f i n i t e l y aa' a'a d i l u t e ) s o l u t i o n where t h e o f f - d i a g o n a l t e r m s L~~, , a 4 a', a r e z e r o , and 139 w h e r e we c a n n e g l e c t d i f f u s i o n o f t h e s o l v e n t m o l e c u l e s . I f we a l s o n e g l e c t c o n v e c t i o n , t h e e q u a t i o n s g o v e r n i n g t h e f l o w o f i o n s ft a n d B a r e a A °n J „ = - -4- V * - — — V u * (3) " * Z A e - ( z ^ e ) 2 —b z^e /_ . v2- ts "B ( z B e ) ' : ( w h e r e we h a v e d r o p p e d t h e s u p e r s c r i p t b ) . In (3) a n d ( 4 ) , z ~ e a n d z ~ e a r e A D t h e c h a r g e s o n A a n d B r e s p e c t i v e l y , a n d a~ a n d o~ a r e t h e c o n d u c t i v i t i e s o f t h e t w o i o n s . In t h e s t e a d y s t a t e , c u r r e n t i s c a r r i e d o n l y b y t h e A i o n s , s o J ~ = 0, a n d (k) g i v e s —B V U R Vcb = - — 2- . (5) z B e S u b s t i t u t i n g (5) i n t o (3) g i v e s ( z ^ e ) B F o r a n i d e a l d i l u t e s o l u t i o n , t h e c h e m i c a l p o t e n t i a l u o f one o f the s o l u t e a t o m s i s r e l a t e d t o t h e c o n c e n t r a t i o n p ( a t c o n s t a n t T ) b y u = '.:klZnp + c o n s t a n t (7) a n d t h e d i f f u s i o n c o e f f i c i e n t o f t h e A i o n s , . , D ~ , i s g i v e n b y ' aA 3y f i . k T - a A ( M * ( z ^ e ) 2 8 p A ( z A e ) 2 PA ' S i n c e t h e c o n d u c t i v i t y ^ i s p r o p o r t i o n a l t o p^, i s i n d e p e n d e n t o f p^. S u b s t i t u t i n g (7) a n d (8) i n t o (6) g i v e s ( s i n c e p^ * p-g b y c h a r g e n e u t r a l i t y ) ^ = - D A ( + ] | | - ) V P A - (9) We s e e t h a t t h e p r e s e n c e o f t h e s e c o n d i o n e f f e c t i v e l y i n c r e a s e s t h e d i f f u s i o n c o e f f i c i e n t n A t o ( 1 + z ^ / | | ) . E q u a t i o n (9) l e a d s t o l i m i t i n g c u r r e n t s , w h i c h i s m o s t e a s i l y s e e n b y c o n s i d e r i n g a p l a n a r c e l l a s i n F i g . 39 ( L e v i c h 1 9 6 2 ) . S i n c e t h e c u r r e n t i s i n d e p e n d e n t o f t h e p o s i t i o n , y , b e t w e e n t h e a n o d e a n d c a t h o d e , t h e c o n c e n -t r a t i o n . 'p^ ( y ) . . m u s t v a r y l i n e a r l y i n t h i s r e g i o n . T h u s J A " D A 1 + , z Z A B p 2 ( 0 ) - p 2(£) \ p « A (10) w h e r e p^ . i s r t h e c o n c e n t r a t i o n o f A i o n s f o r = 0 . We a l s o h a v e , u s i n g (5) a n d ( 7 ) : A u A = U A U ) - y R ( 0 ) - k T ^ ^ j - J (11) k T / P A ( £ ) \ A + ^ ( A , - * ( 0 ) - - ^ W ^ 5 r J (12) s o t h a t t h e o v e r p o t e n t i a l n i s n = -A * A k T / . Z A E L / P A ( I ° 1 + A - . ^ A . . (13) z A e z A e y - | z B e | / \o^0) F i n a l l y , w i t h t h e f a c t t h a t t h e t o t a l n u m b e r o f s o l u t e i o n s i n t h e e l e c t r o -l y t e i s f i x e d , i . e . / P R M d y = p^£ we o b t a i n 2D, J A " 1 + u l P K tanhlr^ F A kt 1 + z R / | z B | 2 (15) We s e e t h a t t h e c u r r e n t b e c o m e s i n d e p e n d e n t o f n f o r |n | » k T / e . T h i s o c c u r s a s p(0) o r p ( £ ) a p p r o a c h e s z e r o . I t i s c o m m o n t o s e p a r a t e t h e o v e r p o t e n t i a l n i n t o t w o p a r t s ( V e t t e r 1967).: t h e 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 A y - v / z - e . (11) , a n d t h e r e s i s t a n c e p o l a r i z a t i o n Acj) (12). T h e e f f e c t s o f c o n v e c t i o n i n t h e e l e c t r o l y t e c a n b e c o n s i d e r e d q u a l i t a t i v e l y b y r e p l a c i n g £ b y t h e t h i c k n e s s o f t h e u n s t i r r e d e l e c t r o l y t e l a y e r n e x t t o t h e e l e c t r o d e . 7.5 D i f f u s i o n i n t h e H o s t C o n v e n t i o n a l e l e c t r o c h e m i c a l c e l l s , w h i c h i n v o l v e s u r f a c e r e a c t i o n s a t t h e e l e c t r o d e s , s h o w t h e t y p e s o f o v e r p o t e n t i a l s d i s c u s s e d i n S e c t i o n s 7-3 a n d 7-4, n a m e l y l o s s e s d u e t o t h e t r a n s f e r o f c h a r g e a c r o s s i n t e r f a c e s a n d t h r o u g h t h e e l e c t r o l y t e . T h e r e a c t i o n i n a n i n t e r c a l a t i o n c e l l i s i n a s e n s e a b u l k r e a c t i o n , r a t h e r t h a n a s u r f a c e r e a c t i o n , s i n c e t h e r e a c t i o n p r o d u c t , . ( t h e ' i n t e r c a l a t e d c o m p o u n d ) c o n s i s t s o f i n t e r c a l a t e d a t o m s s p r e a d t h r o u g h o u t t h e h o s t l a t t i c e . T h e f i n i t e r a t e o f d i f f u s i o n o f t h e i n t e r c a l a t e d a t o m s i n t h e h o s t p r o v i d e s a n a d d i t i o n a l l o s s , o r o v e r p o t e n t i a l , i n a d d i t i o n t o t h o s e a l r e a d y d i s c u s s e d . A s w a s s e e n i n t h e d i s c u s s i o n i n S e c t i o n 7-2, t h e v o l t a g e o f a n i n t e r c a l a t i o n c e l l d e p e n d s o n ' t h e c h e m i c a l p o t e n t i a l u f o f ' t h e i n t e r c a l a t e d a t o m s a t t h e s u r f a c e o f t h e h o s t ; t h i s i n 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 w e c a n n e g l e c t n o n l o c a l e f f e c t s i n y ( x ) o f t h e t y p e d i s c u s s e d i n S e c t i o n 6 .5 . ( N o t e t h a t when we speak of atoms at the surface here and in the following chapters, we mean intercalated atoms just inside the host rather than atoms adsorbed on the surface.) During intercalation, a gradient in the composition develops in the host,-so that x g varies more rapidly than the average com-position. This produces a difference between the observed cell voltage and the voltage which would be measured i f no such gradients existed; this difference is referred to as a diffusion overvoltage. In Chapter :S, we wil l discuss solutions of the diffusion problem and arrive at relations between this overvoltage and the cel l current I. These solutions wil l assume two idealized forms of the dependence of the diffusion coefficient D on the composition; to just ify these idealizations, we wi l l f i r s t discuss the expected composition dependence of D in Chapter 8. C H A P T E R 8 D I F F U S I O N IN I N T E R C A L A T I O N COMPOUNDS 8 . 1 I n t r o d u c t i o n In t h i s c h a p t e r , we d i s c u s s t h e e f f e c t s o f i n t e r a c t i o n s b e t w e e n i n t e r -c a l a t e d a t o m s o n t h e d i f f u s i o n c o e f f i c i e n t , D , o f t h e a t o m s . We w i l l o n c e a g a i n u s e a l a t t i c e g a s d e s c r i p t i o n o f t h e i n t e r c a l a t i o n s y s t e m , a n d n e g l e c t a n y c h a n g e s i n t h e m o b i l i t y o f t h e a t o m s d u e t o c h a n g e s i n t h e h o s t . T o i l l u s t r a t e t h e g e n e r a l a r g u m e n t s , we e x a m i n e t h e v a r i a t i o n o f D w i t h t h e c o m p o s i t i o n x i n a s i m p l e m o d e l o r i g i n a l l y p r o p o s e d b y M a h a n ( 1 9 7 6 ) w h i c h d e s c r i b e s h o p p i n g o f a t o m s o n a o n e d i m e n s i o n a l l a t t i c e 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 . N e u t r a l p a r t i c l e s m o v e i n r e s p o n s e t o g r a d i e n t s i n t h e i r c h e m i c a l p o t e n t i a l ]i; t h e n u m b e r c u r r e n t d e n s i t y o f t h e p a r t i c l e s i s l i n e a r l y r e l a t e d t o V u b y J_ = - M p V y ( 1 ) w h e r e p i s t h e n u m b e r d e n s i t y o f t h e p a r t i c l e s a n d M i s t h e i r m o b i l i t y . ( N o t e t h a t M d e f i n e d h e r e i s e t i m e s t h e m o b i l i t y u s e d i n s e m i c o n d u c t o r p h y s i c s . ) ( 1 ) d e s c r i b e s d i f f u s i o n o f t h e p a r t i c l e s , a s c a n b e s e e n by w r i t i n g J_ = - DVp (2) w i t h t h e d i f f u s i o n c o n s t a n t D d e f i n e d b y D - N p j £ . . (3) D d e f i n e d b y (3) i s s o m e t i m e s c a l l e d t h e c h e m i c a l d i f f u s i o n c o e f f i c i e n t . If the particles 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 which is the familiar Einstein relation. Experimentally, diffusion is often studied by measuring^ the rate of mixing of labelled (e.g. radioactive isotopes) and unlabelled particles which are chemically identical . Such a procedure measuresVthe tracer diffusion coefficient D^. In general, differs from MkT by a factor of order unity (see, for example, Flynn 1972); however, in the case of diffusion in a one dimensional la t t i ce , = 0 since the particles cannot get around one another, whereas M and D are nonzero. Equation (1) can be generalized to the case of charged particles by introducing the electrochemical potential y, which gives J_ = - MpVy = - MpVy - zeMpVcf) (6) where ze is the charge of the part ic les . The f i r s t term in (6) describes diffusion, while the second describes e lectr ical conductivity. The conduc-t iv i ty a is defined by y = kT^np + constant so the diffusion coefficient becomes D = MkT (5) zeJ = -.:'aVcf> (7) so we can identify a by comparing (6) and (7) as a = (ze)2Mp . (8) Because of (8), the discussion in this chapter on the effects of the : interaction on M for neutral particles can be applied to charged particles as well , such as in the case of superionic conductors. Also in reference to charged part ic les , we should note that it is possible to regard an intercalated atom as an ion A ofucharge ze together with z electrons (see Chapter 5 ) . If this 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 remain in equilibrium (Vu~ = 0) even in the presence of the ionic motion. 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 part ic les . 8.2 Behaviour of D(x) Consider a latt ice gas of particles as in Chapter 4 , where x measures the fraction of occupied sites. If there are no interactions between the particles (except for the hard core repulsion that prevents more than one particle from occupying any site) one expects the mobility M for a simple hopping motion of the particles from site to site to decrease as 1 - x as x increases, due to the blocking of s ites. However, in a non-interacting latt ice gas, the variation of 8y/9p calculated from 4 .3(6) exactly cancels this factor of 1 - x in the expression (3) for D, so D is a constant (independent of x). For simple hopping between adjacent s i tes , D can be related to w, the probability per unit time that a hop wil l occur between a a fu l l and an empty s i te . For example, on a one dimensional latt ice with nearest neighbour site separation c, (Flynn 1972) D = wc2 . (1) Near x = 0 we can speak of the Independent hopping of particles with the mobility M = wc 2 /kT, and near x = 1, the independent hopping of holes or vacancies with the same mobility. Repulsions between intercalated atoms keep atoms apart. For some values of x and of the interaction, this may increase M over the case U = 0, since adjacent sites are less l ikely to be occupied. However, a reduction in M is expected near compositions corresponding to ordered arrangements of the part ic les , since the repul s i on.-respons i ble for the ordering wil l prevent the particles from jumping out of the ordered superlattice. On the other hand, we saw in Chapter k that the factor 9 y / 9 p should become very large ( 9 x / 9 y small) at such compositions, which wi 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 . Because the factor (p /kT ) 9 y / 9 p = (x/kT)9y/9x increases D over the value predicted by the Einstein relation (5), it has been referred to as the '.'enhancement factor" (Weppner and Huggins 1977). Attractive interactions between intercalated atoms wi l l also reduce M, because of the clustering of the atoms produced by the attraction. However, in this case ,9y/9p is also reduced over the non-interacting case (Section h.k). Hence D may be considerably smaller at intermediate x values than near x = 0 (1) where particles (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 latt ice to x = 1, large concentration gradients wil l form in regions of intermediate values of x (since Vp °c 1/D). This wi l l result in.a sharp boundary separating the empty region (x = 0) from the fu l l one (x = 1); this boundary wil l then move through the latt ice as intercalation proceeds. In fact, 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 qual 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 if 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 attractive interact" t i ons. In Chapter 9 we wi11 discuss diffusion for these two cases. F i r s t , however, we wi l l consider a simple model calculation describing diffusion on a one dimensional latt ice which i l lustrates the conclusions reached above. 8.3 Model Calculation of Diffusion on a One Dimensional Lattice Consider particles localized on sites in a one dimensional latt ice with 1a t t i c e „ c o n s t a n t c which are described by the Hamiltonian H = E I n + U l n n + fiO),Y(b + b _,,:+. b') (1) o £ a L a a+1 h^ a a+1 a+1 a . a a . .a ¥ In (1), b and b are creation and annihilation operators for particles CX CX on the site a, and are related to the number operator n^ by n =.b+b . (2) a a a 'Because no more than one particle can reside on a single s i te , b operators for the same site obey anticommutation relations b tb + b b + = 1 (3) a a a a b b +: b b = b + b + + b + b + = 0 (4) a a a a a a a a while b operators for different 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 wil l use it to discuss the variation of the mobility M and the diffusion coefficient D as a function of x = <n > at fixed T in intercalation systems. If the final term a in (1) (the-hopping term) is absent, H describes a one dimensional latt ice gas with nearest neighbour interactions U (see Section 4.6). Following Mahan (1976) we wi l l assume fico, « kT, so that we can use the latt ice gas n results to evaluate any thermal averages. The hopping term should more correctly be called a tunneling term, since it describes the overlap of the wavefunction of a particle on one site with the adjacent s ites. The problem is thus analogous to a tight binding problem in sol id state physics: the overlap of the single particle wavefunctions on separate sites means that the true wavefunctions are Bioch states, which for U = 0 would produce an energy band of width 2ficon; note, however, that in contrast with the usual solid state problem the bandwidth is much less... than kT here. This leads to an inf inite mobility in a perfect la t t ice , so we wil l 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 J r 6 ( r - r )b+b — L -a a a a a (5) where is the position of the site a . The number current density operator cl is .related to II (_r) by (Mahan 1976) v ' ze dt = 1 f-L V j if. n(r) ze H (6) 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 latt ice of noninteracting chains, which occupy a volume v. Then J becomes, using (1) CO, c j = J 2 _ . y ( b + b - b + b J . iv L a+1 a a a+1 a (7) The conductivity at frequency co is then evaluated using the Kubo formula (Kubo 1957) a (co) _ tanh (fico/2kt) v J . e " i a J t S ( t ) dt :(ze): fico/2 (8) where the correlation function y(t) is given by Y(t) = ± < J ( t ) J(0) + J(0)J(t)> ( 9 ) and then a is related to M using 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 . ( 1 0 ) To evaluate Y(t) we need the results (which follow from ( 1 ) and the commu-tation relations for the b operators) e X V b A , e X H = e"XU(na-1-na+2)btb a a+1 a a+1 v e X H b + J . 1 b e"XH = e"X U ( na-1-n a+2)bt b ( 1 2 ) a+1 a a+1 a 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] cos ^ ( n XJ > ( 1 3 ) \ v / L la a+1 a+1 a J -ft a-1 a+2 \ / a: The terms in ( 1 3 ) have an obvious interpretation. The factors n a ^ " n a + i ) and n a + i ( l " n a ) describe a hop from a to a+1 and from a+1 to a respectively. The cosine term is unity i f site a-1 and a+2 are both fu 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 site a-1 or a+2 is fu 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^  = 2n -1; i f 1 , 2 , 3 , 4 are any four adjacent s i tes , we have Y(t) = (1 -<s 1 s 2 >+<s 1 s i t >-<s 1 s 2 s 3 s i t >) + (1 - < s 1 s 2 > - < s 1 s / ( > + < s 1 s 2 s 3 s i ( > ) c o s . (14) Substituting (14) into (8) and using 8 .1(8) we find M x = Wr^' ( 1 " < S 1 S 2 > + < S 1 S 4 > " < S 1 S 2 S 3 S 4 > ) 6 " & ) ) TT tanh(U/2kT) : 2 z /, „ ^ „ ^ ^  ^  ^\ .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 inf inite at GO = 0 and OJ = ± U/h. To make them f in i t e , we assume some scattering exists , so that the current-current correlation function decays exponentially in time; that i s , we multiply (13) by e l^r 1 -!^ where oo^ is some scattering rate. This causes the delta functions in (15) to become Lorentzians: 6 (co) - 1 7 3 - 4 — • (16) ?r ai2+co2 r 1 W 6 (aj±U/h) -> • (17) 7 7 co2 + (oj±U/fi)2 We f i r s t consider (15) for the case of no interactions, U = 0. Then the two Lorentzians (delta functions in (15)) merge into one, and in : add i t i on 1 - <s 1s 2> = 4X(1-X) (18) so we obtain 2 CO M(co) = f r < P (1-x) (19) k T h oo2+co2 CO D(co) = c2co2 — - (20) h w 2 +(o 2 r Note that M <* 1-x and D is independent of x, in agreement with the qual i ta-tive discussion of Section 8.1 of the noninteracting latt ice gas. Com-paring (20) at co = 0 with (1), we see that the jump probability w is given by w = co2/co r. To apply this result to true hopping, weishould have w % cor (the scattering rate of order the hopping rate), which in turn implies co ^ co,. I f we apply (20) at co = 0 to di ff us ion of Li ; in Li Ti 0 for x « 1 , r h x 2 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 calculations, where the motion of the particles 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 ly 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) increasing with frequency while others, like Agl , show M(co) decreasing with frequency (Kimbal 1 and Adams 1978). Now we return to co = 0 (setting w = cojVco )^ and consider repulsive interactions. The:expression (15) for M is evaluated using the correlation functions for the one dimensional latt ice gas model, given for reference in Appendix B; only the f i r s t term in (15) is used, the second one being negligible i f fico^ « U ^ kT, which we assume here. The mobility at zero frequency for U = 5kT (which corresponds to the voltage curve in Fig. 23b) is shown in Fig. 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. In Fig. kOb the "enhancement factor" tfy/8x for the same interaction, 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 Fig. 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 Fig . 41b, to show the ionic conductivity expected i f the intercalated atoms were charged part ic les . It is interesting to compare these results with the predictions of mean f ie ld theory, which are also shown in Fig . kO and 41 . The mean f ie ld results were calculated using a two sublattice decomposition of the one dimensional latt ice (see Section 4.6), so that the required correlation functions are F i g . kO - (a) M o b i l i t y M a n d (b) e n h a n c e m e n t f a c t o r ( x / k T)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 a n d 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 s h o w n . 155 F i g . 41 - (a) D i f f u s i o n c o e f f i c i e n t D and (b) " c o n d u c t i v i t y " xM c o r r e s p o n -d i n g t o F i g . 40. < S l s 2 > = <s1s i (> = ( 2 x ^ : 1 . ) ( 2 x 2 - 1 ) ( 2 1 ) <s 1 s 2 s 3 s 4 > = [ ( 2 X 1 - 1 ) ( 2 X 2 - 1 ) | 2 ( 2 2 ) where and x 2 are the fractional occupations of the two sublattices (note x^  + x 2 = 2 x ) . We see that the mean f ie ld theory gives semi-quanti-tative agreement with the exact results, except near x = i , where mean f ie ld strongly underestimates M,and overestimates x 9 y / 9 x . The discontinuity in D in the mean f ie ld results is a consequence of the kink in the voltage curve in Fig. 2 3 b produced by the second order phase transition predicted in mean f i e l d . For very strong repulsions (U » kT), the expressions for M and D simplify considerably. We quote the expressions for x < i.and co = 0; the actual solutions for xM and D are symmetric about x = i . For the exact solution to the latt ice gas (with w = co,2/co ) h r M = w c ^ 1 ^ 2 x ( } kT 1-x while for the mean f ie ld solution M = ^ ( 1 - 2 X ) ( 2 5 ) D = wc2:' ( 2 6 ) D varies by a factor of 4 in ( 2 4 ) , while in the mean f ie ld result ( 2 6 ) , D is independent of x. The concentration dependence of M in ( 2 3 ) is identical to that found by Dietrich et al ( 1 9 7 7 ) in their master equation solution of hopping in a one dimensional la t t ice . Note that the mean f i e ld results for U -»• o o , (25) and (26), may be obtained from the U = 0 case by replacing x in (19) and (20) by 2x; a similar simplification of the voltage curves in the inf inite interaction limit in mean f ie ld was noted in Chapter 4 (see especially 4.6(2)) . F ina l ly , consider attractive interactions. Fig. 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 Fig . 23a. 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 Fig. 4 3 a , is also decreased. The "conductivity" xM is plotted for reference in Fig. 43b. Also shown are the same quantities calculated using the simple (random) mean f ie ld theory of Chapter 4 (4 .4(3)) which predicts a f i r s t order phase transition for 0.14 < x < 0 .86. The mean f ie ld results for M and xM in the two phase region represent the effective quantities for:' the entire lat t ice , calculated on the basis that the resistance of the chain which would be measured i f the particles had a charge ze (from the conduc-t iv i ty (ze)2Mp) 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 .86, . 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 latt ice as particles are added for 0.14 < x < 0 :86. 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 latt ice gas solution. For large attractive interactions in both the mean ' f ie ld and exact results, M = 0 for a l l x except x = 0, and D = 0 except at x = 0 and x = 1,(where D = wc 2); this behaviour in the mean f ie ld case 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 latt ice gas with nearest neighbour interactions U = -2.5 kT. Results are shown for the exact and mean f ie ld solutions to the latt ice gas problem. The curve for U = 0 is also shown. Fig. hi - (a) Diffusion coefficient D and (b) "conductivity" Mx corres-ponding to Fig. kl. 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 ini te currents causes overvoltages in interca 1 at ionjcel1s. The types of behaviour expected can be understood at least qualitatively in terms of one of the following assumptions about the concentration dependence of the diffusion coefficient 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 effectively equivalent to the case where D = 0 over some range of x. We discuss the behaviour of cel l current and cel l voltage resulting from these two cases in the following sections. The f i r s t case has also been discussed by Atlung et al ( 1 9 7 9 ) . We assume for simplicity in what follows that a l l other overvoltages except those associated with diffusion in the host are negligible. 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 results, we shall give their equation numbers preceded by the letters CJ, as in CJ 7 - 5 ( 1 ) . We wil l discuss the three geometries shown in Fig. kk: (a) an. infinite slab of halfwidth R (b) an infinite c ircular cylinder of radius R (c) a sphere of radius R. It wi l l be useful to define a parameter £ = 1 , 2 , 3 respectively for the three cases. The distance in each case wil 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 . Fig. kk - The three geometries considered in discussing the effects of diffusion of the intercalated atoms on the behaviour of inter-calation eel 1s. The symmetry of cases (a) and (b) allows them to be applied to materials with very anisotropic diffusion. . This enables us to establish a corres-pondence 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 ruti le related materials-; (b) to a host with diffusion in two dimensional layers, as in the layered compounds; and (c) to a host where the diffusion is isotropic, as in the metal-hydrogen systems. Note that here the correspondence of the cases to:-the rutiles 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 last ic strains produced by intercalation; this is because in rutiles the intercalated atoms move in one direction while expanding the latt ice in two directions, 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 cel l current flowing in the negative sense according to the conventions of Chapter 7, and wil l lead to a negative overpotential (reduction of the cel l voltage). The number density of intercalated atoms at the surface at time t , P s ( t ) , found by solving the diffusion equation iis^ . (CJ .3.8(3), CJ 7.8(1), CJ 9.7(D) J t J R / . \ where Z(t) is defined by z ( t ) , 2 j - i ^ e - » n < > t / K \ ( 2 ) n=1 n In (2), a is a coefficient given by the solution of one of the following: a = n T T , n 5 = 1 (3a) J ^ ) = 0, ? = 2 (3b) a cot a = 1, £ = 3 (3c) n n where is the Bessel function of order 1 . We can rewrite (1) in terms of the composition x of the intercalation compound. We define 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 lectr ic current which must flow through.the cel l i f intercalation is occuring from a solution where the intercalate is ionized to a charge ze. Then we have p = p Q X (k) I = zeJ s A (5) A = £ f - (6) In addition, it 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 cel l) i f intercalation proceeded uniformly throughout the host: zevp Q, Introducing these variables into (l) gives the following result for x s ( t ) , the value of x at the surface: x (t) = f r + T T " - TT(2+S)X(t) (8) o o o where x' is defined by ~ D • ( 9 ) For t « T ' , (8) reduces to the case of a semi - i nf i n i te sol id (CJ 2 .9 (8 ) ) : x (t) = / - ^ (10) ' o where T is defined by 1 R2 £+2 , / , , \ After the current has been flowing for a long time, so t » x , £ ( t ) ->• 0, and **M~jr + -r (12) o o Equation (8) is plotted in Fig. 45. We.'ve also plotted a useful interpolation formula given by x s ( t ) =1^- coth/Y (13) o which also shows the limiting behaviour (10) and (11), with x = T T X ' / 4 . For t > x, the surface composition,: , increases linearly with t, as expected for a constant current, but there is an extra contribution, x ' / 3 t , to x o s which would not be present i f intercalation were uniform throughout the host. This corresponds to an overpotential ( x ' / 3 t ) ( 3 V / 8 x ) for t •> x. o x=xs ^ This nonuniform intercalation at f ini te currents reduces the capacity of an intercalation cel l below its theoretical value. Suppose the cel l is considered discharged (its voltage too low to be useful) at a cutoff voltage g. kS - Surface composition x s versus time t for intercalation of the three geometries shown in Fig. 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. corresponding to a composition x^ _. Since the surface composition x^  (t) exceeds the average composition t / t during discharge of the c e l l , the time t needed to reach the cutoff voltage wil l be less than the corresponding time i f intercalation were uniform, x t . Let Q be the maximum capacity c o m (charge) available above the cutoff voltage, given by Qm = ltr,xr- 0 4 )  o c and Q the charge passed in time t , c c Q. = It . (15) c c The relation between x^  and t^ is given by ( 8 ) , namely x = x (t ) (16) C S C ' ' 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 Fig . 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 , 1 I T m m At high currents, the capacity is inversely proportional to i ' 0_ Q. c _ TT_ _m 0. 4 IT m (17) 0 8 ) Equations (14) to (16) give the capacity for a single discharge of an intercalation c e l l , starting from an unintercalated host. In laboratory tests, cel ls are often cycled continuously between fixed voltage limits at a current ±i , corresponding to a variation of the surface composition g. 46 - Fractional capacity QQ/Q,^  versus current I for intercalation of the three geometries shown in Fig. 44 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 half-cycle time t i , say. The capacity over each half cycle, Q, , varies with current in 2 2 a fashion similar to the single discharge, Fig. 46. For small currents (t,. » T ) * = 1 - i I T ' m 3 Q„ (19) For large currents, we can use the solution for the steady state change in surface concentration in response to a sinusoidal current JssinO)t (CJ 2.9(13)): Ap (t) = —— sin (cot -S /coD (20) together with the Fourier expansion of the surface current appropriate for a square wave: 4J J s ( t ) = TT L 2 n-1 n=1 s i n ( 2 n - l h r | (21) to obtain the variation in surface composi:tion, Ax,(t) Ax.(t) = - W - f r - I TT 3 /V l o n=1 ( 2 n - D 3 / 2 s i n / „ 1 \ T T t TT ( 2 n - l ) - - 7 (22) The peak-to-peak variation in Ax g (t) is x^, given by 172 Hr s ! n ^ _i — — 3 7 2 • n=1 (2n-l)' (23) The sum in ( 2 3 ) is 1 . 6 8 8 7 6 1 . . . so that Q2-= 0 . 3 4 0 ^ • ( 2 4 ) 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 cel l on current to estimate the diffusion constant, D. Alternatively, D can be measured using one of the following transient techniques. If the cel l is in equilibrium at t = 0 , and the cel l voltage is changed by V, which causes the surface concentration to change by 6p = P Q S X J a current l ( t ) flows. At short times, the number current density at the surface of the host is (CJ 3 - 3 ( 9 ) , CJ 1 3 - 3 ( 3 ) , CJ 9 - 3 ( 5 ) ) : v o - j e K - ^ p , • <«> The current, l ( t ) , is then given by I ( T ) = ^ | ^ - | 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 wi l l give a straight l ine, whose slope gives T , and hence D i f the particle size, R, is known. Note that the quantity 9x/8V must also be known i f D is to be found. 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 ircui t voltage as a function of x; this point is discussed in Section 6 . 5 for the e last ic interaction. This problem is most important when the magnitude of 9x/9V is large. 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 cel l voltage change is ™ ( t l = | 5 7 w f ^  • (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 2 . Also, i f a sinusoidal current is applied, the diffusion coefficient can be found from the expression (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. The main problem in applying the above results to real intercalation cells is in specifying the particle s ize, R. In single crystals , intercalation can begin at cracks in the surface, and so the effective particle size is smaller than the measured crystal dimensions. In any practical intercalation c e l l , the host wil l be used in powder form, so there is a large distribution in R, rather than a single value as assumed until now. The restriction t «: x used in some of the results wi l l then refer to the smallest part ic les , 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 particles via the electrolyte. There are also further complications in pressed powders introduced by the f in i te conductivity of the electrolyte, which are discussed in Chapter 10. 9.3 Motion of a Phase Boundary As discussed in Section 8.2, intercalation of a host latt ice 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 wi l l also form if the host undergoes a structural phase transit ion. As intercalation proceeds, the boundary moves through the host crystal . In this section, we discuss the motion of thi.s boundary. We treat the same geometries as in Fig. hh. We model the system as follows. The boundary, located at position r = r(t) at time t, is assumed to be inf ini te ly sharp, separating a region of composition (phase 1) from a region of composition x^  (phase 2), with x^  < x^. To be specif ic , we assume that atoms are flowing into the host lat t ice , so phase 2 lies outside the boundary (closer to the surface), which corresponds to a negative e lectr ic current flowing through the cel 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 crysta 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 = x2 " x i = i r - ( 1 ) o We wil 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 (r = R) . s These steady state concentration profiles are (CJ 3 - 2 (1 ) , CJ 7 - 2 ( 4 ) , CJ 9 - 2 ( 7 ) : ) ( r - r ) , p (r , t)-p„ = < J R In J R' C = 1 ? = 2 C = 3 ( 2 a ) ( 2 b ) ( 2 c ) with p 2 = P Q X 2 the concentration just outside the boundary. The rate of motion of the boundary, dr/dt , is determined by the number 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 dr dt J s Ap Ct Ax o ( 3 ) The factor (f/R) 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 distribution from that appropriate when the phase boundary is at r and that needed when the phase boundary is at r + dr; that i s JJp ( r , t + d t ) -p(r,t)] \ d r « A p A ^ j 1 (-dr) . ( 4 ) Referring to the steady state equations, we see that we can write C-1 (-dr) (5) Using (5) the integral in (4) can be performed, and we obtain the following condition on Ax for the val idity of the steady state approximation: concentration = PQX^ is established from r = 0 to r =:"?; we wil l assume that this uniform concentration already exists at t = 0. First we consider constant current. Integrating (3) with the condition r(t=0) = R gives The phase boundary reaches r = 0 at t = t Q Ax, which from 9,2(7) is just the time needed to homogeneously change the composition by Ax at the surface current density J ^ ; the steady state approximation clearly neglects the additional current which must flow to change the composition outside the phase boundary. Substituting (7) into (2) gives for the surface composition x (t) the following: (6) In addition to (6), the approximation wil l not be valid until the uniform (7) s 174 Tt FAX o x s ( t ) - x 2 J - i ^ l - ^ ) t (1 - t / t A x ) 1 / 3 - 1 5 = 1 ? = 2 5 = 3 (8a) (8b) (8c) This is plotted in Fig. 47 ( c f . the case of constant D, 9 . 2 (8) and Fig . 45). The diffusion overvoltage associated with x^  - is (x^-x^) (9V/9x) X s ; the magnitude of this overvoltage increases/with time as the phase boundary moves, in contrast to other types, of overvoltage (such as activation over-voltage or resistive losses), which would be constant (see, however, the discussion of porous electrodes in Section 1 0 . 3 , where similar effects occur). Diffusion overvoltages can clearly wash out the plateau in the voltage curve expected at a f i r s t order phase transit ion, as shown schematic-al ly in^ Fig. 48. Note that i f the current is interrupted, the steady state approximation predicts that when the current resumes, the voltage wi l l drop to the same value it had just before the interruption occurs (df:course, i t wi l l take a time of order x to re-establish the steady state concentration prof i le ) . The val idi ty of the expressions (8) ns given by the condition (6). For 5 = 1 , the solutions are valid for a l l time t < tQAx provided t Ax » x o 5 = 1 . (9a) For 5 = 2 , 3 , the phase boundary moves more rapidly as r decreases, and the solution eventually breaks down; thus (8) is valid for t such that Fig. kl - Surface composition x s versus time t for intercalation at constant current of the three geometries shown in Fig . kk in the case of the motion of a phase boundary in the host. ) tQAx - t » T , x, = 2 (9b) t QAx - t » ( t ( A x ) 1 / V A , ? = 3 (9c) If the cell 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 . In this case, the maximum capacity Q_m is independent of x^, and is given by Qm = QQAx . ( 1 0 ) For £ = 1 , we find = < 1 , Q G (x c -x 2 ) I < Q O(X C-X 2)/T1 I > Q (x -x )/T o c 2 C = 1 (11a) The lower term in (11 a) has the same form as 9.2(18) except TTQ Ik has been replaced by 0- o (x c -x 2 ) . T n e lower expression in (11a) also holds for r, = 2,3, but only for I large enough that Q /Q < 0.1. For arbitrary I , c m _ £ = 1 - e " ( x s - x 2 > V l T , m ? = 2 (11b) = 1 -m 1 + V V X 2 > 3lx S = 3 (11c) Equations (11) are plotted in Fig. kS ( c f . Fig . 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 (or t ) in (3) s o is time varying, so it must be eliminated using (2); then (3) can be integrated to give V X 2 Ax ? = 1 (12a) i £ An £ t Xs"X2 ~k~ kx Ax ? = 2 (12b) W r y 1 / r y . 1 3\RJ 21R/ F t X s _ X 2 9T Ax ? = 3 (12c) In each case, the phase boundary reaches r = 0 at t = t , given by Fig. hS The current l ( t ) can be found by eliminating r from (12) using (2); however, a simple expression results only for 5 = 1 : 179 K t ) 1 f(x - x J A x ' s I 2-rt (14) For 5 = 2,3, the limiting behaviour at small t is given by i(0 - _ L - l ( W x 2Tt 5 - 1 2 Xs"X2 5 3 x ( 1 5 ) (which also reduces to (14) for 5 = 1 ) . (15) is an accurate approximation to better than 2% until the host is 50% ful l for 5 = 2,3. Note that,just as in the case of constant D, the current varies as t 2 ( c f . 9.2(26)); in this case, however, the current varies as the square root of the applied voltage V-V^. = (x^-x^)(3V/3x). The limitations of (12) are given by (6). Again for 5 = 1 the solutions are valid for a l l t, in thi.s case i f Ax V X 2 >> 1 5 = 1 ( 1 6 a ) For 5 = 2,3 the conditions are Ax V X 2 F O 2 y t In (R/r) 1JL 3 y 5 = 2 5 = 3 (16b) (16c) For example, i f L\x/(x^-x^) = 100, these conditions for 5 = 2 and 3 in terms of t are t « 96x = 0.96t f for 5 = 2 and t « 142X = 0.94t f for 5 = 3 , with t f given by ( 1 3 ) . CHAPTER 10 POROUS ELECTRODES 10.1 Introduction As is evident from the previous chapter, losses in intercalation cathodes due to diffusion in the host latt ice are decreased by decreasing the size of the host latt ice crystals , R. In general, practical intercalation electrodes consist of finely powdered host material. The pores in the powder are f i l l e d with electrolyte to allow the ions in the solution to reach particles throughout the electrode. This structural arrangement, however, can also cause problems, which we now discuss. We shall consider a planar intercalation electrode, as shown in Fig . 50. The electrode is a slab of thickness 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 electrolyte; the quantity X is known as the porosity. The particles in the powder are e lectr ical contact porous electrode elect rolyte e A Fig. 50 - Planar porous intercalation electrode, showing a pore of length H. 181 assumed small enough so that any effects due to diffusion in the host can be neglected. In Fig . 50, electrons arrive from the anode via the external c i rcu i t and enter the cathode from the left , and ions arrive from the right. The resistance to electron flow is determined both by the bulk res i s t iv i ty of the host material, and by the contact resistance between the part ic les . In the fa i r ly porous materials used in electrodes, the contact resistance can be larger than the bulk resistance, so the electronic resistance generally depends on the particle size and the procedure used to prepare the electrode. The resistance to ion flow is determined by the bulk electrolyte 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 particular depth in the electrode, and we have a one dimensional system with some effective electrolyte conductivity. Tp relate the effective electrolyte conductivity of the electrode to the bulk electrolyte conductivity, we assume that the pores have a length, SL, given by where $ is.the tortuosity. Then i f a l l of the electrolyte volume in the pores is accessible from the surface, the total cross section of a l l the pores has area A , given by and the total ionic resistance, R, of the electrolyte in the pores is related to R*, the resistance which would be measured for a slab of bulk electrolyte with dimensions SL* and A*, by SL = $£* (D £A = £*A*X (2) This ratio, R/R*, is referred to as the formation factor. Empirically, it varies with porosity roughly as JL ^ _L n ^ 2 R * k n > (Archie, 19^2), so that, from (3), the tortuosity should vary as One particular measurement of <£> from the transit time of ions through porous material gave :(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 electrolyte in the pores, intercalation does not proceed uniformly through the electrode. Such spatial ly nonuniform reactions in porous electrodes were f i r s t discussed by Euler and Nonemacher (1960) in terms of a simple resistive chain as shown in Fiq . 51. R, and R are the total resistances of the electrolyte in the 3 b c pores and of the powder respectively; G is the conductivity of the interface between the particles and the electrolyte, given, for example, by l inearizing the Butler-Volmer equation 7 - 3 (1) - Current flowing from y = SL to y = 0 in Fig. 51 wi l l be distributed to give equal potential drops in the upper and (5) (6) y = £ y = 0 Fig. 51 - Resistive chain used to model porous electrodes. R^  and R c are the total resistances of the electrolyte and of the host matrix respectively. G is the total conductance of the interface. I indicates the direction of positive current flow. lower chains; i f G is large (corresponding to a low impedance:interface), some of the current wil 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 . Generalization of these arguments to non-c b linear behaviour of G and R, , with the results of numerical calculations, b is given by Newman and Tobias ( 1 9 6 2 ) and Grens and Tobias ( 1 9 6 4 ) . The resistor chain of F ig . 51 is useful in discussing the current distribution just as intercalation begins (t = 0 ) . At later times, as the front or back of the electrode is intercalated, current begins to flow deeper in the electrode. This can be discussed in terms of a resistor-capacitor network as in Fig . 5 2 ; for s implici ty , we neglect the resistance of the particles and of the interface (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 Fig. 52 - Resistor-capacitor network used to model the intercalation of porous electrodes. R is the total resistance of the electrolyte, and C the differential capacity of the entire host. I indicates the direction of positive current flow, and V is the measured cell 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) DRC = W • (3) Hence we can apply the results of Section 9 -2 , relating the current density and number density, to this case, i f we make the substitutions V, J i i / C , and D •> D , where V is the measured voltage shown in Fig . 52. s R L Similarly, i f a f i r s t order transition occurs in the host lat t ice , so that the open c ircui t cel l voltage shows a plateau over some range Ax , a boundary separating the two phases wil l move through the electrolyte, and we can use the results of Section 9 -3; in part icular, the cell voltage wil l f a l l l inearly in time at constant current (9-3(8a)) , and the current wi l l vary as t 2 i f the cel l is held at constant voltage (9-3(14)). 10.3 Electrolyte Depletion So far, i t has been assumed that the electrolyte shows ohmic behaviour. However, as discussed in Section l.k, this may not be true - as the voltage drop in the electrolyte increases, the current eventually saturates at some limiting value due to depletion of the ions in the solution. The same behaviour occurs in porous electrodes, but is complicated by the fact that the limiting current depends on the depth in the electrode that the current reaches. We wil l discuss only the case where a f i r s t order transition 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 (see Fig. 50). 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. Final ly , since in most cel ls of interest the anode and cathode are separated only by a thin e lec tro lyte - f i l l ed membrane, we assume that the amount of electrolyte outside the pores is negligible. As a result, 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 electrolyte must remain constant due to charge neutra1ity (we are only removing A ions from the solution). Too large an increase in concentration at y = £ wi11 produce precipitat ion, which wi l l block the pores and limit the current that can flow. Even i f precipitation does not occur, the current can s t i l l be limited 186 i f the ion concentration at the position of the phase boundary, y, becomes zero. In the steady state approximation, the concentration in the electro-lyte wi l l vary linearly between y = y and y = £ , and wil l be constant from y = 0 to y = y. (see the discussion of the case 5 = 1 of Section 9.3). Note that it wi l l take a time of order £ 2 / D to establish such a prof i l e , a where is the ambipolar diffusion coefficient of the e lectrolyte , which describes diffusion with no e lectr ic current flow. For ideal solutions, the equations relating overpotential, n , ar>d the number current density of A ions in the solution, J ^ , are (using the notation of Section 7-4) J A - D A ^ 1 + l ^ T J £-y J i - D i h + T i f t r (2) where J~ > 0 corresponds to current flow to the right in Fig. 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. Equations (1) to (3) can be solved for n, (t) or J^(t) for constant current or constant voltage conditions. 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, i f | l | > I L > the cell wil l appear ful ly discharged or recharged (|n| 00) at time t L given by Q Axl, t L = - 2 ^ . (5) At this time, only a fraction t^/tQAx of the electrode wil l have been converted in phase. Thus the apparent fractional capacity 0-c/n-m is given by m 1 > | I | < I L ( 6 ) jrp | I | > X L (assuming that the cutoff voltage corresponds to [n | » k T / z ^ e ) . ( 6 ) should be compared to the corresponding expressions for the case of diffusion in the host, 9 . 2 ( 1 8 ) and 9 - 3(1 l a ) . We can rewrite the expression for I , ( 4 ) , in a somewhat more trans-parent form. Using 7 - 4 ( 8 ) to relate the conductivity due to the 'A- ions, a^, to D^, and defining the resistance in the pores due to the A ions, R^, by Rs • ^  <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 electrolyte, and regard the electrolyte as a fixed (ohmic) resistance. 188 This wi l l be valid only i f the voltage drop in the electrolyte 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 / z A e • ( 9 ) In the other l imit , the current flow when the boundary is at y is indepen-dent of the overpotential n, a n d ' s given by 1 = • Ml > > k T / z A e • ( 1 0 ) If the voltage of the cel l is held at a value much larger than kT/z^e below the open c ircui t voltage of the coexisting phases., so that the limiting current appropriate to y, (10), always flows, y(t) is given by and the measured current as a function of time is K t ) - ! ^ . (12) -2/3 - i The current varies as t , in contrast to the variation t expected if the resistance of the electrolyte were constant. PART D EXPERIMENTAL PROCEDURE AND RESULTS CHAPTER 11 EXPERIMENTAL PROCEDURE 11.1 Introduction In order to i l lustrate the wide variety of behaviour found in inter-calation systems, several types of lithium intercalation cel ls were studied experimentally. In this chapter, we describe the methods used to prepare and test these ce 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. For most of the samples, 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 (1978c); later samples of T i S 2 indicated, however, that keeping the entire tube at 550°C (ho temperature gradient) gives identical results. The reactants were heated slowly (over a period of several 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 particle size (a diameter of order 10 ym, and a thickness of order 1 ym), and below 550°C, which resulted in the formation of appreciable amounts of TiS^. For each material, the samples were X-rayed to confirm their structure. Natural MoS^  was also used in the experiments, in the form of single crystals (> 1 mm diameter) and powder. Two grades of powder were obtained from Molybond Laboratories: very fine powder (^0.1 ym particle size) suspended in o i l , and free flowing powder with a platelet diameter of order 1 ym. The M0O2 used was prepared by passing hydrogen over MoO^  at 475°C, or by reacting stoichiometric quantities of Mo and MoO^  in sealed quartz tubes at 750°C. These two methods gave particle sizes of M ym and MO ym respectively. Lithium fo i l (0.38 mm thick, 99-95% pure) was used as received from Alfa Ventron. Single crystal TiS^ was prepared using standard vapour transport techniques.(Balchin 1976). The electrolyte used in the cel ls 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 disti1 led twice, then passed through three columns of activated alumina and stored under argon. Gas chromatography showed that this purification procedure for the PC reduced the concentration of the principle impurity, propylene g lycol , to ^3 ppm. (Subsequent study has shown that equally good results can be obtained without the columns, by optimizing the d i s t i l l a t i on 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 lycol , spreading the slurry over a nickel or aluminum f o i l , and baking the propylene glycol away in a stream of nitrogen gas at 200°C. For the Molybond MoS^  suspended in o i l , the o i l suspension was applied directly to a nickel substrate and baked at 750°C. to remove the heaviest tars from the M 0 S 2 part ic les . 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 disc; 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 cel ls which used baked cathodes were beaker ce l l s , where the substrate of the cathode was soldered or spotwelded to a wire and suspended in electrolyte, together with a lithium anode, in a beaker sealed with a neoprene stopper. These cel ls showed rapid capacity loss i f the powder did not adhere well to the substrate. This problem was largely avoided by using pressed ce l l s . In these ce l l s , a polypropylene Celgard #2500 or #3501 microporous film (the separator) was placed between the cathode and the lithium anode, and the resulting sandwich pressed tightly together. Two types of pressed cel ls were used, and are shown in Fig . 53. In "flange cel ls", a cathode-separator-anode sandwich was held between steel flanges coated with si l icone grease and screwed together; an 0-ring seal was used to keep the cell a ir t ight . In "button cells", the sandwich Fig. 53 - The two types of pressed cells used for intercalation: (a) flange ce l l s , (b) teflon button cel ls . 194 was held between teflon plugs screwed into a teflon barrel; this type of cel l was also used for pressed cathodes. In the button ce l l s , e lectr ical 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 , the steel flanges themselves provided the contact. 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 cel ls to be dis-cussed in the following chapter was intended to establish the behaviour of the cel l voltage V as a function of the composition x of the intercalation cathode. In the simplest test used, the cel 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 irect ly , 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 electrolyte 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 reactions i:s..approximately one half of the difference in the length of the discharge and recharge voltage curves between the same voltage l imits . In general, the cel l was cycled repeatedly between fixed voltage l imits , which allows changes in cell capacity to be easily seen. 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 built by the UBC Physics electronics shop. The inverse derivative of these voltage curves gives the quantity 3x/3V, which, as discussed in Chapter h, can reveal ordering processes in inter-calation systems. 3x/3V can also be obtained directly by recording the cel l current I versus V as V is changed at a constant rate, V. This follows from I = d Q = d Q d V = - 9 x ( )  1 dt dV dt V 3 V v " where Q, is the total charge needed to change x by Ax = 1. Hence I is directly proportional to 3x/3V. Current-voltage curves generated in this way are called linear sweep voltammograms by electrochemists; we wil l refer to them as inverse derivative curves or current-voltage curves. Scan rates • V used in our experiments were typically 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 ce 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 constant current has already been discussed in Section 9-2; the apparent cel l capacity fa l l s l inearly with I at low currents, and varies as I ^ at high currents. The effect of a resistance R in the cell is to add a constant voltage IR to the cel l during recharge, and subtract IR during discharge. This is effectively equivalent to lowering the upper voltage limit and raising the lower l imit , so the change in capacity with I depends oh the details of V(x) at the voltage l imits . The variation of apparent cell capacity with current can be used quantitatively, but is more often used as a qualitative 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 wil 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 coefficient, were also done using the PAR equipment. (A general review of such transient techniques 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 easily discussed by representing the cell as a series RC c i r c u i t , as in Fig. 54. We wil l consider R independent of current; for nonohmic behaviour, the discussion wil l be more complicated, but no qualitative changes are expected. The capacitance C= (dV(0j/d0) ^repre-sents the open c ircui 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 cel l is not l inear, we expect C to be a function of "the voltage V(0_) across the capacitor. In the calculation to be discussed, we wil l take the following F i g . 54 - RC c i r c u i t used 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 c u r v e s . V(Q.) i s t he v o l t a g e a c r o s s the c a p a c i t o r , and I the c u r r e n t which f l o w s as th e t o t a l v o l t a g e a c r o s s t he c i r c u i t v a r i e s as V t . form of C(V) Ci(V) = R T T - ^ 17577- (1 ) which corresponds to the following form of V(Q): V ( QW* n(v*)' ( 2 ) Except for a minus sign due to the convention of current flow in Fig. 54> this corresponds to the non-interacting lattice gas result, 4.3(5), shown in Fig. 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 ful l width at half maximum of C(V) is 3-53 3 ) . The current l ( t ) which flows when the voltage across the cel l (or across the RC c ircui t in Fig. 54) is swept at a constant rate V is found by solving the loop equation for the RC c i rcu i t : R -^ j-jr + V(Q) = Vt (3) where I = dQ/dt. With the particular form of V(Q) in (2), we can rewrite (3) in dimensionless form as d(Q/Q ) / Q./Q \ b - — ^ 2 - + &i hpT7§- = 3Vt (4) where the parameter b is defined as b = 3 2QQVR . (5) b measures the change in shape of the curve Q/QQ or d(Q/Q q ) /d (3Vt) versus 198 ( 3V 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 solved numerically f o r several values of b, and the r e s u l t s are shown in F i g . 55. For small b, the curve i s s h i f t e d only s l i g h t l y from the case b = 0 ; the s h i f t of the peak i s given by A($Vt) = b / 2, which corresponds to a s h i f t in voltage of A(C't) = 21 ,R, where I , = V/C , = V A 3 ( t h i s 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) to f i r s t order ini'b). For large b, the voltage across the c a p a c i t o r changes much more slowly than the voltage across the r e s i s t o r except near Q = 0 or Q = 0_ , so the current begins to look l i k e a ramp, expected f o r a l i n e a r change in the voltage across the r e s i s t o r . The peak in the current i s s h i f t e d to much higher voltages than f o r smal1 b. We now turn to the e f f e c t s of d i f f u s i o n , in the host. A l l s e r i e s resistances are assumed to be zero, so the a p p l i e d voltage Vt c o n t r o l s the composition x^ at the cathode s u r f a c e , as discussed in Chapter 9 . The d i f f u s i o n problem i s considerably more d i f f i c u l t to 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 the s o l u t i o n f o r a simpler form of the voltage-composition r e l a t i o n V'(x) than ( 2 ) , namely V.(x) = j - ^ ( l - x ) . (6) The inverse d e r i v a t i v e of t h i s i s f O , V < 0 dx d\l - e 3 V , V > 0 (7) V(x) and 8x/3V are p l o t t e d in F i g . 56a and 56b r e s p e c t i v e l y . The d i f f u s i o n problem was t r e a t e d in d e t a i l in Chapter 9 , and we use the notation developed there. We assume that the i n t e r c a l a t i o n host i s i n i t i a l l y empty (x = 0 at t = 0 ). At t = 0 , we begin d i s c h a r g i n g the c e l l 199 Fig. 55 - Current-voltage curves given by sweeping the voltage of the RC circui t of Fig. 5k at a constant rate V, with the capacity C(V) given by 11.5(1), for several values of the parameter b. 200 1 I 1 1 a x 1 1 1 > 1 1 r Fig. 56 - (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. s t a r t i n g f r o m V = 0, s o V = - | V 11, a n d t h e s u r f a c e c o m p o s i t i o n v a r i e s a s x.(t) - l - . - e l * ! * (8) w h i c h f o l l o w s f r o m (6). I f |v| i s l a r g e , we c a n u s e t h e s o l u t i o n f o r a s e m i - i n f i n i t e m e d i u m , e q u a t i o n s 2.5(2) a n d 2.5(9) i n C a r s l a w a n d J a e g e r (1959), t o f i n d t h e n u m b e r c u r r e n t d e n s i t y f l o w i n g i n t o t h e s u r f a c e o f t h e h o s t d u e t o t h e s e i n i t i a l a n d b o u n d a r y c o n d i t i o n s : i p / $ | V | D fiM«i. ' J s ( t ) = — e ^ l ^ [ e r f c ( i / B 7 v T 0 - e r f c ( - \ S W M t ) ] (9) w h e r e p Q i s t h e c o n c e n t r a t i o n o f i n t e r c a l a t e d a t o m s a t x = 1. T h e c o m p l e m e n -t a r y e r r o r f u n c t i o n o f i m a g i n a r y a r g u m e n t i s t a b u l a t e d i n C a r s l a w a n d J a e g e r (1959), a n d a p l o t o f (9) i s g i v e n i n F i g . 56. N o t e t h a t t h e p o s i t i o n o f t h e p e a k i s s h i f t e d v e r y l i t t l e f r o m t h e s l o w d i s c h a r g e c a s e (V -> 0 ) , a n d i s i n f a c t i n d e p e n d e n t o f D f o r l a r g e • | V | ; : - t h i s i s i n c o n t r a s t t o t h e c a s e o f r e s i s t i v e l o s s e s , w h e r e t h e p e a k c u r r e n t p o s i t i o n d e p e n d s o n R a n d c a n b e s h i f t e d b y a l a r g e a m o u n t . A f t e r t h e p e a k , J g ( t ) ( o r l ( t ) ) d e c a y s a s l / / t ~ : t h i s d e c a y w i l l c o n t i n u e u n t i l t ^ R 2 / D , w h e r e R i s t h e 0 0 p a r t i c l e s i z e i n t h e c a t h o d e . A s w a s s e e n i n t h e d i s c u s s i o n s o f C h a p t e r 9 , t h e f i n i t e p a r t i c l e s i z e b e c o m e s i m p o r t a n t f o r t > R 2 / D , a n d t h e c u r r e n t w i l l f a l l t o z e r o m o r e r a p i d l y t h a n 1 / /F i n t h i s l i m i t . A s |\/| d e c r e a s e s , t h e f i n i t e p a r t i c l e s i z e w i l l b e c o m e i m p o r t a n t a t a s m a l l e r v a l u e o f | V | t , a n d e v e n t u a l l y we w i l l r e g a i n t h e V -»- 0 r e s u l t . In s u m m a r y , we c o n c l u d e t h a t r e s i s t a n c e a n d d i f f u s i o n h a v e a l m o s t o p p o s i t e e f f e c t s o n p e a k s i n t h e i n v e r s e d e r i v a t i v e c u r v e . R e s i s t a n c e c a u s e s p e a k s t o s h i f t t o l a t e r t i m e s ( h i g h e r v o l t a g e o n r e c h a r g e , l o w e r v o l t a g e o n d i s c h a r g e ) a n d p r o d u c e s a l o n g r i s i n g e d g e b e f o r e t h e p e a k . D i f f u s i o n d o e s n o t s h i f t t h e p e a k p o s i t i o n v e r y m u c h , b u t p r o d u c e s c o n s i d e r a b l e b r o a d e n i n g over a time t ^ ^Q^> where RQ ' s the particle size, giving rise to a long ta i l after the peak i f D is small. CHAPTER 12 EXPERIMENTAL RESULTS 12.1 Introduction In this chapter, we present and discuss experimental results obtained in studies of lithium intercalation ce l l s . In the course of these experi-mental studies, the author personally prepared and tested over 60 ce l l s , and in addition had access to data from several hundred other cel ls prepared by cd-workers in the laboratory. Cell 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 ce l l s . In discussing the data, we wi l l refer to each cel l by a cell number (e.g. RM12); Table III at the end of the chapter l i s ts a l l of the cel ls discussed, together with relevant information on each one. In what follows, we f i r s t characterize the cel ls used, and discuss results obtained with cel ls with no intercalation host. Then we present data for three systems: Li TiS Li MoCL , and Li MoS„. The Li Ti S_ results X A X ^ X ^ X £-are presented f i r s t , 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 participation of the solvent in the reaction in our ce l l s . Results for Li Mo0„ are given x 2 next, showing hysteresis associated with f i r s t order transitions. F ina l ly , L i^oS^ is discussed, i l lustrat ing the effects of a large structural change in the host which leads to large changes in the variation of the cel l voltage V with composition x. 20 k 12.2 Excess Capacity and Kinetic Limitations of the Cells In this section, we discuss some of the properties of the cel ls 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 cel l to measure..the composition of the intercalation com-pound. We then examine the kinetic limitations of the c e l l s , especially the problems associated with transport through the electrolyte and with the lithium metal anodes. If a l l of the charge which flows through a lithium intercalation cel l a-, results in uniform intercalation of the cathode with lithium, then the Li composition of the host, x, can be found directly from the charge flow i f the weight of the host is known. It is therefore important to see i f any reactions aside from intercalation occur. Such extra sources of cel l capacity are generally referred to col lect ively as side reactions. As .a check for such side reactions, cel ls were constructed which were identical in a l l respects to those used for intercalation, except the cathode consisted only of a cleaned nickel disc. In one such cel l with a nickel cathode of 2 area 2 cm (R26) , 100 mC of charge flowed through the cel 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 cel ls tested had a capacity greater than 1 C within these same voltage l imits , the background reactions due to the other components of the cel 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 electrolyte, or there may be a reaction of the electro-lyte with the intercalated L i . Such side reactions can be identified by a persistent difference between the amount of charge which flows on charge and 205 d i s c h a r g e o f t h e c e l l . H o w e v e r , t h e r e s t i l l r e m a i n s a p r o b l e m i n i d e n t i -f y i n g s i d e r e a c t i o n s d u r i n g t h e f i r s t d i s c h a r g e o f t h e c e l l , w h i c h i n s o m e c a s e s i s q u i t e d i f f e r e n t f r o m t h e d i s c h a r g e o n s u b s e q u e n t c y c l e s . M o r e o v e r , c h e m i c a l m e a s u r e m e n t o f t h e L i c o n t e n t o f t h e i n t e r c a l a t e d h o s t i s c o m p l i -c a t e d b y t h e s m a l l s a m p l e s n e c e s s a r y f o r r e a s o n a b l e d i s c h a r g e r a t e s ( s e e b e l o w ) a n d b y t h e p r e s e n c e o f t h e L i s a l t i n t h e e l e c t r o l y t e . A s a r e s u l t , t h e a b s o l u t e L i c o n t e n t a t s o m e c e l l v o l t a g e i s l e s s a c c u r a t e l y k n o w n t h a n c h a n g e s i n t h e L i c o n t e n t b e t w e e n t w o v a l u e s o f t h e c e l l v o l t a g e . A f u r t h e r c o m p l i c a t i o n a r i s e s i f L i + ! i o n s i n t h e e l e c t r o l y t e c a n n o t r e a c h a l l o f t h e p a r t i c l e s i n t h e c a t h o d e , l e a d i n g t o t h e r e d u c t i o n i n t h e a p p a r e n t c a p a c i t y o f t h e c a t h o d e b y t h e s o - c a l l e d " c a t h o d e u t i l i z a t i o n f a c t o r " ; t h i s d o e s n o t a p p e a r t o b e a p r o b l e m i n t h e b a k e d c a t h o d e s d e s c r i b e d i n C h a p t e r 11, b u t may b e i m p o r t a n t i n p r e s s e d c a t h o d e s . We now d i s c u s s t h e k i n e t i c s o f t h e c e l l s u s e d . T h e t o t a l c o n d u c t i v i t y o f t h e e l e c t r o l y t e u s e d i n m o s t o f t h e c e l l s , a o n e m o l a r s o l u t i o n o f - 3 -1 L i C l O ^ i n p r o p y l e n e c a r b o n a t e ( P C ) , h a s b e e n r e p o r t e d a s 5 x 1 0 ( f i - c m ) , w i t h t h e L i + i o n s c a r r y i n g 20% o f t h e c u r r e n t ( J a s i n s k i 1971). H e n c e , t h e c o n d u c t i v i t y o f t h e L i + i o n s , a. , i s 1 x 10~ 3 ( f i - c m ) " ^ . T h e C e l g a r d Li s e p a r a t o r s u s e d a r e 25 ym 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 t o r t u o s i t y n e a r u n i t y . I f we 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 c h a r a c t e r i s t i c s o f c u r r e n t f l o w t h r o u g h 1 cm b f a s i n g l e s e p a r a t o r ( s e t t i n g ~ 2 y = 0 i n 10.3(10)) ,we f i n d a l i m i t i n g c u r r e n t o f 21 m A / c m , a n d a r e s i s t a n c e f o r |n |« k T / e o f o r d e r 5^ i n t h e s t e a d y s t a t e . In t h e s a m e w a y , we c a n e s t i m a t e t h e p r o b l e m s a s s o c i a t e d w i t h t h e p o r o u s e l e c t r o d e s . T h e p o r o s i t i e s a n d t h i c k n e s s e s o f t y p i c a l b a k e d c a t h o d e s a r e o f t h e s a m e o r d e r a s f o r t h e s e p a r a t o r s , s o t h e l i m i t i n g c u r r e n t s s h o u l d b e o f t h e o r d e r o f t e n s o f mA f o r i n t e r c a l a t i o n t h r o u g h o u t t h e c a t h o d e . T h e s e c u r r e n t s a r e m o r e t h a n a d e q u a t e f o r s t u d y i n g t h e c e l l s ; f o r e x a m p l e , a 5 mg c a t h o d e 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 porosities. 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 cel l to x = 3-At higher currents, the cathode can not be completely 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 10.3-To check the magnitude of the losses in the electrolyte and at the Li interface, L i / L i cel ls were constructed by using Li fo i l for both anode and cathode in pressed ce l l s . Such cel ls typically showed an impedance of order 2 100 9, for 2 cm area of Li at each electrode, considerably higher than that expected from the electrolyte. Moreover, reasonably ohmic behaviour was seen - there was no exponential variation of current with voltage expected from the Butler-Volmer equation (7-3(1)), and no limiting currents as would be produced by electrolyte depletion. The resistance appears to be due to a surface layer on the Li f o i l . The resistance could be lowered by a factor of two after passing current through the cel l and exposing fresh metal surface; further, the resistance dropped above frequencies of order 1 kHz, implying a capacitance in parallel with the resistance of order 10 yF and hence a surface layer which is a few angstroms thick. These high resis-tances were s t i l l low enough so that a third (reference) electrode was 207 u n n e c e s s a r y i n t h e l i t h i u m i n t e r c a l a t i o n c e l l s . When a l a r g e amount o f c h a r g e p a s s e s i n one d i r e c t i o n t h r o u g h a L i / L i c e l l , p e n e t r a t i o n o f t h e s e p a r a t o r by d e n d r i t e s i s somet imes o b s e r v e d . Such d e n d r i t e p e n e t r a t i o n c a u s e s t h e c e l l impedance t o d r o p t o v a l u e s o f o r d e r V/Q, and a l l o w s a c h a r g e t o f l o w t h r o u g h t h e c e l l w h i c h i s f a r l a r g e r t h a n t h e e q u i v a l e n t amount o f L i i n t h e c e l l . S e p a r a t o r s f r o m s u c h d e n d r i t i c c e l l s a r e s o m e t i m e s c o m p l e t e l y f u l l o f L i , w i t h a d u l l g r e y a p p e a r a n c e ; when washed w i t h m e t h a n o l , t h e s e p a r a t o r s r e g a i n t h e i r o r i g i n a l w h i t e c o l o u r and show no s i g n s o f p u n c t u r e , i m p l y i n g t h a t t h e L i has grown t h r o u g h t h e p o r e s . S e p a r a t o r 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 i n t e r c a l a t i o n c e l l s , c a u s i n g t h e v o l t a g e t o r e m a i n r e a s o n a b l y s t e a d y w h i l e 1 a r g e „ a m o u n t s o f c h a r g e f l o w t h r o u g h t h e c e l l . An e x a m p l e o f s u s p e c t e d d e n d r i t e p e n e t r a t i o n i s shown i n F i g . 57-12.3 L i / L i T i S I n t e r c a l a t i o n C e l l s X Z-The s t u d y o f L i x T i S 2 w a s i n t e n d e d i n i t i a l l y as a c h e c k o f t h e e x p e r i -m e n t a l t e c h n i q u e s u s e d , s i n c e a d e t a i l e d s t u d y o f t h e v o l t a g e v e r s u s x has been r e p o r t e d by Thompson (1978), whose r e s u l t s have a l r e a d y been shown i.n F i g . 1. I t was f o u n d h o w e v e r , t h a t t h e i n t e r c a l a t i o n o f T i S ^ i s more com-p l i c a t e d t h a n had been b e l i e v e d i n i t i a l l y . F i g . 57 shows a p o r t i o n o f t h e c h a r g e / d i s c h a r g e b e h a v i o u r o f a L i / L i C 1 0 ^ , P C / L i T i c e l l made f r o m T i S ^ p o w d e r , between 2.3 V. and 1.0 V . I f t h e c e l l v o l t a g e i s k e p t above 1.8 V , t h e v o l t a g e c u r v e i s v e r y s i m i l a r t o T h o m p s o n ' s , e x c e p t f o r t h e e x t r a c a p a c i t y a t 2 .8 V on t h e f i r s t d i s c h a r g e , and t h e t o t a l : c e l l c a p a c i t y o b s e r v e d , t y p i c a l l y x ^ 0.7 on t h e f i r s t d i s c h a r g e t o 1.8 V , and x - 0.5 on s u b s e q u e n t c y c l e s , a l t h o u g h somewhat 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 used i n F i g . 57- D i s c h a r g e o f t h e c e l l t o l o w e r v o l t a g e s p r o d u c e s a l o n g p l a t e a u i n t h e range 1.4 V t o 1.6 V , w h i c h c a n be as l o n g as x = 2, and w h i c h t (hours) Fig. 57 " Charge-discharge cycles for Li TiS^, cel 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 cel l is s t i l l recharging during this time. considerably changes the subsequent cycles of the c e l l . The deviations from Thompson's results have been associated with intercalation of PC. Powder X-ray diffraction 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 their size and weight, absorbing a l l of the PC in the c e l l , before the cel 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 intercalation, while the s imilarity 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 is inactive above 1 .8 V after the f i r s t discharge. The plateau near 1.4 V may be associated with a structural transition involving the PC intercalated T i $ 2 ; it does not seem to produce any change in the Ti which contributes to the observed capacity above 1 .8 V. This is most clearly seen in the inverse derivative curves shown in Fig. 5 8 , taken before and after the cell 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 Fig . 2 are similar 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 Fig. 5 9 ; Thompson's data is also reproduced there, plotted against voltage rather than x, and normalized to correspond to the same total 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 F i g . 5 8 - 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 S2 , 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 a n d (b) a f t e r t h e c e l l was d i s c h a r g e d t h r o u g h t h e p l a t e a u a t 1 . 4 V . 211 F i g . 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, a t a s w e e p r a t e V = 17-1 y V / s ( s o l i d c u r v e s ) . P o i n t s a r e d a t a f r o m T h o m p s o n (1978) n o r m a l i z e d t o t h e s a m e c e l l c a p a c i t y . 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 cell of 50 £2. Thompson avoided this shift by incrementing the cel l voltage by AV = 10 mV, and measuring the charge AQ which flowed until the current had dropped to some small l imit; in his data, AQ./AV coincided on recharge and discharge. Except for this deta i l , our method appears to give equally good results. 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 clear. The poorer resolution of the features seen in Fig. 58 is attributed to diffusion in the TiS^ host. If the TiS^ particles have radius R, diffusion effects wil l smear the features over a time ^ R 2 /D , where D is the diffusion coefficient; this corresponds to smearing over a voltage ^ VR /D at a sweep rate V. The TiS^ used"in Fig. 58 consisted of particles with R ^ 10 ym, so that VR 2/D ^ 20 mV, assuming D ^  10 9 cm 2/sec; for the data in Fig. 59, R ^ 2 ym, and VR 2/D ^ 0.8 mV. The peak at 2.33 V was studied in more detail at slower sweep rates, to see i f the top of the peak was being rounded by the f in i te rates. If the peak is actually a divergence in 9 x / 8 V , as might be expected i f i t is produced by a phase transit ion, 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 clearly in cel ls 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 particle s ize, as discussed by Dahn (1980). F ig . 60 shows the voltage curve for the f i r s t discharge of a L i /L i^TiS^ cel l to 0 .2 volts and its subsequent recharge. The plateau near 0.5 V suggests a f i r s t order phase transit ion, 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 similar to those in Fig. 6jc. However, when the cell is discharged to 0.05 V, a second plateau is seen, and subsequent cycles of the cell show considerably different voltage behaviour. Cells made with powdered T\S^ grown directly 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 ignifies an ordered structure of the Li in the Ti$2 host at this composition. Neutron diffraction studies by Dahn et al -:(1980)> indicate that throughout the range 0 < x < 1, the Li atoms occupy predominantly (and perhaps entirely) octahedral sites 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 difference in site energy of octahedral and tetrahedral s i tes , as discussed in connec-tion with Fig. 13, and with nearest neighbour octahedral-tetrahedral site i nteract i ons. It is l ikely that both the plateau for 1 < x < 2 and that for 2 < x < 3 correspond to f i r s t order phase transitions. It is clear from Fig. 60 and 61 that these transitions are quite different. The transition from x - 1 to x - 2 is quite reversible; the difference in the plateau voltage on charge and discharge in Fig. 60 is i< 0.2 V. The transition 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 x T i S 2 , ce l l JDty?, at a current of 75 yA. Note that the x scale applies only to the discharge. X in L i x T i S 2 (discharge only) 1.0 2.0 3.0 TIME (hrs) 61 - Charge/discharge cyc les for L i x T i S 2 , c e l l JD49, at a current of 75 yA. Note that the x sca le appl ies only to the d ischarge. 216 subsequent recharge; moreover, the voltage characteristics are completely changed on subsequent cycles. It is l ikely that the distinction between these two transitions is in the degree of change of the host latt ice as the transition occurs. Unfortunately, detailed structural information is not available, and it is impossible to quantify this statement. If the change in the host structure in the transition from x - 1 to x - 2 is small, then this transition might-be understandable in terms of a latt ice gas model for the Li in the host; in part icular , it is tempting to speculate that the transition is caused by the particular range of site energies and interaction energies discussed in Section 4.7, which produce a phase transition from a composition x = 1 where a l l the ";octahedral sites are f u l l , to a phase at x = 2 where a l l the tetrahedral sites are f u l l . The large change in the voltage behaviour following the transition from x - 2 to x - 3 makes it unlikely that this transition can be understood in a latt ice gas model; the situation is more like that^out 1ined in 4.9- A latt ice 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 wil l be discussed further in x 2' Section 12.5. 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 particular, the sharp peak near x = 0.25 in Fig. 6, which stimulated much of the work in this thesis, 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 dimen-sional order in the neutron diffraction 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 well , that the value of U = 2 . 5 kT for the interaction energy of Li atoms on neareat neigh-bour octahedral sites inferred from the simple mean f ie ld f i t in Fig. 6 lies above the phase boundary predicted by RG calculations (Fig.. 2 0 ) , which also argues against an ordered Li arrangement for x < 1 . 1 2 . 4 L i / L i Mo0„ Intercalation Cells x 2 Typical cycles of L i x M o 0 2 are shown in Fig. 62. The cel 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 negative direction. The recharge is seen to consist of two peaks, each corresponding to a change in Li composition of x - 0 . 5 ; the discharge is somewhat more complicated, but also consists predominantly of two peaks. The width of the peaks in Fig. 63 is smaller than the width of the non-interacting latt ice gas curve.:in Fig. 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 transitions. X-ray studies by Sacken (1980) confirm two phase behaviour between 0 <• x < i and i < x < 1. The curve in Fig . 63 indicates hysteresis in the Li^MoO^ system. 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 .37 V and 1.67 V, and the two largest peaks on the discharge at 1 .30 V and 1 . 5 8 V (Sacken 1 9 8 0 ) . The voltage behaviour of Li^MoO^ also shows interesting history dependence; for example, i f the cel 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 .36 V in Fig . 63 , but rather the curve proceeds along the dotted line indicated in Fig. 63 . h—Ax = I * t (hours) Fig. 62 - Charge/discharge cycles for Li xMo02, cel l U08, at a current I = 500 yA. The time corresponding to x = 1 is indicated. I I I I I _ l 0 1.2 1.4 1.6 1.8 2.0 V (volts) . 63 - Current-voltage curves for LixMoC>2, ce l l U04, at a sweep rate V = 9-6 yV/s . Note that the discharge current is plotted in the negative direction. The dotted line indicates the curve obtained on discharge i f the previous recharge was stopped at 1.6 V. The drops in voltage near x = 0.5 and x = 1 indicate ordered structures of the Li at these compositions. In the absence of information on the position of the Li atoms in the MoO^  host, we can only speculate on the nature of these ordered states. They could be ordered occupations of octahedral or tetrahedral sites in every tunnel, produced by repul-sive interactions in the tunnel direction. On the other hand, as mentioned in Section 2.h, the distortions of the MoO^  structure from the pure r u t i l e structure d i s t o r t the octahedral and tetrahedral sites along the tunnels, and lead to two types of octahedral sites which can each account for a composition of x = £, and one type of octahedral s i t e and two types of tetrahedral sites each capable of accomodating Li atoms up to x = 1. It is therefore possible that s i t e energy differences produce either or both of the voJtage drops near x = 0.5 and x = 1. Recalling the angular variation of the e l a s t i c interaction appropriate for Li^MoO^ in Section 6.2 (Fig. 3*0, we can also propose that either or both of the ordered states involve ordered arrays of occupied and unoccupied tunnels, to minimize the e l a s t i c interaction energy. The small amount of Li in the MoO^  host when the t r a n s i t i o n from x - 0 to x - 0.5 begins makes i t unlikely that this t r ansition is produced by some peculiar combination of s i t e energies and repulsive interaction energies, in analogy with the discussion in Section k.J in connection with the triangular l a t t i c e . The transitions from x - 0 to x - 0.5 and from x - 0.5 to x - 1 are probably produced by attractive interactions between the intercalated Li atoms, and the magnitude of the observed stra i n indicates that e l a s t i c interactions contribute td these attractions. ; l t i s l i k e l y that e l a s t i c effects also contribute to the hysteresis in the voltage behaviour. 12.5 L i / L i MoS„ Intercalation Cells x 2 The voltage behaviour seen in L i / L i MoS„ cel ls is summarized in F iq . 6k. x 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 cel l voltage is kept above 1.1 V, the cel l cycles over the curve labelled I. If the cel l is discharged through the f i r s t plateau but not the second, the cel l cycles over the curve labelled II. F inal ly , 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 latt ice occurs during the 1.1 V plateau. This change in structure appears to involve a shift in the planes of Mo atoms, so that Mo atoms in adjacent sandwiches l ie 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 latt ice 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 clearly 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 x MoS 2 , cel l RM11. The different curves labelled I, II, III are discussed in the text. and phase III at a slow rate at high voltages. This conversion can be seen by cycling the cel ls at high voltages, as shown in Fig . 65 and 66. The high voltage capacity disappears (note that phase I has very l i t t l e capacity), but discharge to a lower voltage 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 cell 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 Fig. 6k. We now look in more detail at the voltage curves for the three phases by examining the inverse derivative curves. Fig. 67 shows the curve obtained for phase I. 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 Fig. 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 Fig . 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 lowered, and the features near 1.8 V on the discharge increase correspondingly. Fig . 69 shows the inverse derivative curves obtained in phase III. These show the considerable hysteresis in phase III which was also seen in the voltage curve of Fig. 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 evident, independently of the depth of the previous discharge. The curves in Fig. 69 are complicated by a loss*in the capacity of the cell between each curve, Fig. 65 - Charge/discharge cycles f o r Li xMoS 2, c e l l R37, at a current I = 340 yA. The c e l l had been previously discharged to 0.7 V through the plateau at 1.1 V. Fig. 66 - Charge/discharge cycles for Li xMoS2, cel l RM13, at a current I = 50 uA. The cel l had been previously discharged to 0.25 V through the two plateaus at 1.1 V and 0.6 V. 3 0 < 2 0 4 . 1 0 0 1 i i 1 \ Discharge v Charge — — Ii i 1 > 1.2 1 . 6 1 . 8 2 . 0 2 . 2 V (volts) Fig. 67 _ Current-voltage curve for LixMoS2> ce l l RM16, at a sweep rate V = 15-2 uV/s. The cell had never been discharged below 1.4 V before this data was taken, and hence was in phase I. h o ON 1.5 _L_ 2.0 2.5 V (volts) Current-voltage curves for Li xMoS2, cel l RM11, at a sweep rate V = 30.1 yV/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. V (volts) Fig. 68b - Current-voltage curve on cell RM11 following Fig . 68a at the same sweep rate. Discharge curve taken immediately after recharge. ho ro oo Charge l O O h < Fig. 68c - Current-voltage curve f o n c e l l RM11 following Fig. 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 LixMoS2 , ce l l RM11, at a sweep rate V = 28.3 uV/s. The discharge curve was taken immediately before the recharge curve. The cell had been previously dis-charged to 0.3 V, so it was in phase III before the discharge began. ro o V (volts) Fig. 69b - Current-voltage curve for cell RM11 following Fig. 69a at the same sweep rate. The discharge curve was taken before the recharge curve. K > Fig. 6 9 c - Current-voltage curves for cel l RM11 following Fig . 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 clear, 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: r x 2 structural change in the host, and so l ie outside our latt ice gas descrip-tion, as discussed in Section 4 . 9 - The latt ice gas model may s t i l l apply separately to phases I, II, or III, with different site energies and interaction energies due to the changes in the host. At the present time, however, we are unable to explain the features in the inverse derivative curves with a simple latt ice gas model. 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 is far more complicated than a simple lattice gas; in view of this large hysteresis, it is l ikely :that large structural changes in the host occur within phase III, which would invalidate the latt ice gas model except possibly over narrow composition ranges. Within phase. II , where.the structural changes are not too dramatic, a latt ice gas model might be more successfully used. The voltage of the plateaus associated with the transitions between the phases is very l ikely considerably lower than the voltage corresponding to thermodynamic equilibrium of the phases., This is suggested in the inverse derivative curve in Fig. 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 ) . Moreover, the plateau voltage is lower on the f i r s t transition from phase I to I I than for subsequent transitions, as seen in Fig. 6 5 . The energy corresponding to the difference between the observed and thermodynamic values of the plateau voltage wil l be released as heat as the transition 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 1 2 Cell # Cathode material Cathode Cell Case electrolyte Mass (mg) ' 3 C3 MoS2 natura1 832 tefIon 1M L ClO^/PC JD49 T i S 2 vapour transport 1.8 f1ange 1M L ClO^/PC Zi JD61 T i S 2 550°C 3-9 f1ange 1M 1 ClO^/PC JD685 T i S 2 800°C 7.4 flange 1M L ClO^/PC R26 Ni - flange 1M L" C10^/PC R 3 7 MoS2 Molybond, in o i l 2.2 f1ange 1M L Br/PC RM11 MoS2 550°C 5.2 tef1 on 1M L ClO^/PC RM12 T i S 2 550°C 4.9 tef1 on 1M L' C10^/PC RM13 MoS2 Molybond, i n oi1 1. teflon 1M L ClO^/PC RM16 MoS2 550°C 5.0 teflon 1M L ClO^/PC U046 Mo02 reduction of MoO^ in H 2 4.5 flange 1M L Br/PC U086 Mo02 reduction of MoO^  in H 2 5. flange 1M L Br/PC Notes: Al 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 directly from the elements. 2 One of the two.types of cel ls shown in Fig. 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 intercalation, and to provide a conceptual framework in which to discuss intercalation systems. The definition of an interca-lation compound in Chapter 1 led naturally in Chapter k to the application of the lattice gas model to describe intercalation systems. We saw in our discussion of the latt ice gas model that phase transitions in the latt ice gas lead to f lat regions in the voltage V as a function of the composition x of the intercalation compound, corresponding to peaks or divergences in -3x/3V. ^Moreover, at compositions corresponding to a f i l l e d latt ice of particles commensurate with the total la t t ice , a drop in voltage, and a corresponding minimum in -3x/3V, occurs. These commensurate structures can be due to different site energies in the la t t ice , 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. Attractive 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 site energy differences and repul-sive interactions (Section 4.7), or by three body forces between intercalated atoms (Section 4 .8 ) . In our discussions of the latt ice gas model, we emphasized the mean f ie ld 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 satisfactory results for the derivatives of the voltage, 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 last ic . In the discussion of the electronic inter-action, we saw in a specific example, where the host is regarded as a free electron gas in a jel l ium 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) is not proportional to the variation of the Fermi level of the electrons. In discussing the elast ic interaction, we saw that e last ic 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 kinetic properties of intercalation ce l l s , with particular 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 lustrate these effects; this model calculation also provided another example of the degree of success of the mean f ie ld approximation in treating lattice gas problems. We extended, the recent treatment of the effects of a constant diffusion coefficient on the discharge characteristics of an intercalation cell by Atlung et al ( 1 9 7 9 ) to the case of a f i r s t order phase transition in the host. We saw in particular how diffusion problems can smear the voltage plateau produced by a f i r s t order transtion. The complications produced by trying to avoid diffusion problems by using powdered cathodes in 239 intercalation cel ls was also discussed. Experimental results were presented for three intercalation systems, Li T i S „ , Li Mo0_, and Li MoS„. These systems are complex, and none corres-X £- X X pond exactly to the simple model systems discussed. Li^TiS^ for 0 < x < 1 is probably describable as a latt ice gas with fa ir ly weak nearest neighbour interactions (of order 2 . 5 kT). No ordering is observed in this range of x; the peak in -9x/8V near x = 0 .25 is s t i l l a puzzle. If this peak does indicate a phase transit ion, we do not know as yet what kind of state is produced by the transit ion. L i i S ^  also shows two very different f i r s t order phase transitions. The f i r s t involves only a small amount of hysteresis, and may be explainable in terms of a latt ice gas model, such as that discussed in Section k.7. The second transit ion, from x - 2 to x - 3 , produces a large change in the voltage behaviour of the ce l l s , and probably involves a considerably larger change in the host structure than the f i r s t transit ion, from x - 1 to x - 2. Li Mo0„ shows f i r s t order ' x 2 transitions with some hysteresis, and is probably an example of a latt ice gas with attractive interactions, with e last ic effects presumably of some importance. Li M 0 S 2 i l lustrates the effects of a large change in the host structure. Here the latt ice 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 intercalation. However, our understanding of individual systems is not complete enough at present to allow quantitative calculations. Further work is needed in both theory and experiment before a complete understanding of intercalation systems is achieved. On the theoretical side, detailed calculations of e last ic and electronic interactions in specific 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 to find out how large a deviation from the inf inite medium elast ic interaction, 00 W (_r) , can be expected for atoms separated by one or two lattice spacings. More work is needed on lattice gas models, to increase our understanding of the ways that interactions between the atoms modify the voltage curves of intercalation systems. In part icular, calculations of latt ice gas models with interactions of the form found in Fig . 33 and 34, appropriate for the interaction of e last ic dipoles, would be useful. Further work is also needed on the experimental side, to explore in more detail the mechanisms responsible for specific features in the voltage curves of intercalation ce l l s . In this regard, neutron diffraction studies, and more careful dynamic X-ray diffraction 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 particle size on the cel l voltage characteristics; the e last ic interactions between intercalated atoms, i f important, should lead to observable effects. Very l i t t l e work has been done to study the changes in features in the voltage curve with temperature, and such work is needed. It could be useful 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 ina l ly , 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 practical 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 has not yet been discovered. A considerable effort in materials research is needed to explore new intercalation systems. Further work is also needed to solve the problems associated with lithium cycling and electrolyte decomposition, which we only briefly mentioned. Final ly , the considerations of Chapter 9 and 10 make it clear that there is an optimum combination of particle 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 ie ld on a simple cubic latt ice 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 lattice 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 lattice gas and Ising models for a lattice of N sites , where each site has y nearest neighbours. We assume nearest neighbour interactions only. In the latt ice gas model, atoms occupy sites on the latt ice . Each site a is assigned an occupation number, n^ = 0,1; n^ is unity i f the site is occupied, zero i f it is empty. If the energy of an isolated atom is E Q (the site energy), and the interaction energy of adjacent atoms is U, then the energy E{n }^ of some configuration ^ n a ^ of atoms is E{n) = E y n + U y n n . (l) a o L a ^ . a a 1 • a <aa'> In (1), the f irs t sum is over a l l s i tes , and the second sum is over a l l pairs of sites. From (1), the grand partition function is -(F-yn)/kT Y / E o" y Y U v \ , . in^i \ a <aa'> / using the abbreviated notation 1 1 1 •I = • I I • • • I (3) { n a } n 1 = 0 n 2 = 0 n N = 0 * In (2), F is the free energy, y the chemical potential, and n the total number of occupied sites , given by n - I n a . (4) a In the Ising model, each latt ice site a is assigned a spin, s^  = ± 1 , which can point either up (s^ = +1) or down (s = -1). If parallel (anti-parallel) 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 ie ld B is E (s } = - B J s - K J s s , + E (5) m a ^ a ^ ^ . ^ a a ' m o v / a <aa1> where E m Q is some additive constant. From ( 5 ) , the partition function is "Fm/kT = f t a y e X p J T ^ r y S + -pjr y S S - - ; -=- ) (6) r L \ \kTL a kT L.^ a a 1 kT / v ' ts J- \ a <aa'> / where F is the magnetic free energy, m We relate the latt ice 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 latt ice gas to the magnetic free energy, F , of the Ising model. We replace s in (6) by n using s = 2n - 1 . (7) a a w ' Then, introducing the average occupation, x, of the lattice gas model x = <n > = AI n = xr (8) a N L a N v ' and the magnetization per spin, m, of the Ising model 1 r m = <s > = TT ) S (q) a WL a K Z" a we obtain the following relations: Fm = F " ^ n (10) m = 2x- 1 ( n ) K = " ^ (12) B =i(y-Eo-f) (13) Emo=!^o>-T • (1*) A P P E N D I X B ONE D I M E N S I O N A L I S I N G MODEL H e r e we g i v e f o r r e f e r e n c e some o f t h e i m p o r t a n t f o r m u l a s n e e d e d i n t r e a t i n g t h e o n e d i m e n s i o n a l l a t t i c e g a s m o d e l . T h e f o r m u l a s a r e m o r e c o m p a c t l y w r i t t e n i n t e r m s o f t h e I s i n g m o d e l ( s e e A p p e n d i x A ) . A n e x c e l l e n t r e f e r e n c e f o r t h e o n e d i m e n s i o n a l I s i n g m o d e l i s T h o m p s o n (1972); t h e r e t h e c o m p l e t e d e t a i l s o f t h e t r a n s f e r m a t r i x s o l u t i o n a r e g i v e n . We a s s u m e a o n e d i m e n s i o n a l l a t t i c e o f N s p i n s , w i t h p e r i o d i c b o u n d a r y c o n d i t i o n s ( s o s = s^ ) . T h e p a r t i t i o n f u n c t i o n 2^  i s ( i n t h e n o t a t i o n o f A p p e n d i x A ) In t h e t r a n s f e r m a t r i x s o l u t i o n , o n e i n t r o d u c e s a m a t r i x L w i t h e l e m e n t s (D L J s 1 ) g i v e n b y (2) T h i s a l l o w s Z. . t o b e w r i t t e n a s N z = 7 n ( s ILIS _,_,) N ; s a 1 'a+1 (3) is . a a n d t h e n e x p r e s s e d s i m p l y i n t e r m s o f t h e e i g e n v a l u e s A . . , A 9 ( A . > A „ ) o f L a s N a s N ->• 0 0 T h e s e e i g e n f u n c t i o n s a r e g i v e n b y A1,2 = 6 K/kT cosh B/kT ± s inh 2 B/kT + e -4K/kTj ( 5 ) where the + (-) sign refers to X (X 2). From (k) and ( 5 ) we can obtain the thermodynamics, such as the magnetization per spin, m, which is m = sinh B/kT sinh B/kT + e -WkTjJ (6) As shown by Thompson (1972), the correlation functions can be found quite easily in terms of the eigenvalues, X^  and X 2 , and the (real) eigen-f/unotions .".'<)>.(s); j = 1,2, of L. The resulting expressions involve the "inner product" ( i | s | j ) = I s<J>.(s)<J>.(s) s=±1 1 J (7) The various terms from (7) can be written in terms of m (6) very simply: 0 | s | l ) = m (8) (2|s|2) = -m (1 |S |2) = (21s | 1) = (1-m 2) 2 (9) (10) For two spin c o r r e l a t i o n functions, t h e r e s u l t is = j 1 f t - ) r ( , l s , J ) 2 - m a + ( ^ ) r ° - m 2 ) (11) In evaluating the transport properties of the hopping model of Chapter 8, we wil l need the following k spin correlation function: 247 X. X. X, <s s s s,> = I -1-1 >i ( i | s | i ) ( i | s | j ) ( j | s | k ) ( k | s | l ) 1 L * 4 ljk A1 A1 A1 •4 # a " (12) 248 APPENDIX C ONE DIMENSIONAL LATTICE GAS OF HARD SPHERES We wish to calculate the chemical potential of a one dimensional latt ice gas of hard spheres of diameter d; that i s , a lattice gas where the interaction between two atoms is infinite 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 sites; this number is e , where k is Boltzmann's constant. We assume periodic boundary conditions, so the latt ice 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 ring, which gives For d = 2, we use the following construction (Rao and Rao 1978). Consider two rings of s i tes , one consisting of n f i l l e d s i tes , the other consisting of N-n empty sites. Cut n of the bonds in the empty ring, and place the n segments between the n sites in the fu l l ring. The number of ways to do this is This construction works only for N ;large, since it does not distinguish between arrangements which differ by a cycl ic permutation of the site labels. When the two rings have been f i tted together, a l l pairs of f i l l e d sites are separated by at least one empty s i te . (D (2) 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 (2) with N replaced by N-n: g S / k = /(N-n) - nj = /N - 2n (3) The continuation to arbitrary d is obvious. We thus have = Nk{ [l - x ( d - l ) ] £ w [1 - x(d-l)] - x£nx - (l-xd)^i (1-xd)} . (4) The chemical potential is p . T | s . k m / » 6 - « ( d - i i ^ ( 5 ) This reduces to the non-interacting result, 4.3(5), for d = 1, and to the solution for nearest neighbour interactions, U, in the infinite U l imit , 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 S = nk [l +&i (x"1-d)] (6) which agrees with the continuum result, 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 latt ice gas, where the site energy alternates between two values, E^  and E^, as we move along the latt ice . 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 latt ice 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 latt ice with alternating site energies may be described as two interpenetrating sublattices, i = 1 and i = 2, of s ite energy E^  and E^. We label the sites along each sublattice by a, so..that each site is identified by the two labels a and i , in the sequence ai = 11, 12, 21, 22, 32 , . . . If nearest neighbour atoms interact with an energy U, then the energy Efn^.}of some configuration of atoms on the latt ice is E { n a i } = I'E1 na1 + £ E 2 n a 2 + U ^ ( n a1 n a2 + W l , 1 > • ( l ) a a a ' It is convenient to solve the problem in the Ising notation (see Appendix A) , so we introduce the following Ising variables: (2) Then the partition function (grand partition function of the lattice gas) becomes ( J . } E X P a i ! l 7 B, k T ^ s a 1 + kT ^ s a 2 + kT ^ ( s a 1 s a 2 + s a 2 sa+ 1 ,1 * a a a ' (5) If we now perform the sum over sublattice 2 , and introduce the fields B and B1 defined by B = V B 2 (6) B1 = V B 2 (7) we can easily show that (5) can be written in terms of a transfer matrix L as *"-(,*} 5(*«I|LL+I*«H.I> a l (8) where the matrix elements of L are (s j L | s ' ) = exp _K_ __, . J _ (s+s') BJ_ (s-s1) kT s s kT 2 kT 2 (9) and Lt is the transpose of L. For B' = 0 , L reduces to the transfer matrix given in appendix B. Z N is given in terms of the eigenvalues, X^ and X^, of the matrix LLt by _ N/2 N/2 N/2 . . ^ N 1 2 1 a S (10) where X^ and X^ are given by , 2K/kT . 2B ^ -2K/kT . 2B' A 1 = e cosh 7-=- + e cosh - r = -1,2 kT kT , 2 K / k T ,2B A -2K/kT . 2 B , x 2 . . 0 , , 4K (e cosh-j^jr + e cosh-j^j + 2 - 2cosh (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 2K/kT . . 2-B e s i nh kT V (12) where the denominator, V, is given by V2 = e 2K/kT . 2B , -2K/kT , 2B1 cosh T-=- + e kT cosh kT + 2 - 2 coshf^ kT (13) m is related to the average occupation of the two sublattices, x^  and x^, in the latt ice gas problem by m = x.j + - 1 = 2x - 1 (14) where x is the average occupation of the overall lat t ice . We can also calculate the difference in occupation of the two sublattices: -2K/kT . . 2B e sinh X1 " X2 = kT V (15) with V given by (13). (12) and (15) are needed in Appendix E. 252 2 5 3 APPENDIX E EFFECTS OF WEAK COUPLING BETWEEN LATTICE GAS CHAINS The effects of weak interchain interactions in a latt ice gas on a latt ice of chains was discussed qualitatively in Section k.G. Here we present a more detailed solution of this problem, using the exact one dimensional latt ice gas results to describe intrachain interactions, and mean f ie ld theory to introduce the interactions between chains. This problem was treated previously by Stout and Chisholm (1962 ) , and applied to antiferromagnetic ordering in linear chain crystals of CuCl 0 . We assume a latt ice of chains of s i tes , with each chain coordinated by y neighbouring chains, as in the ruti le structure, where y = k, or in the two dimensional example shown in Fig. 2 6 , where y = 2. We exclude a triangular lattice of chains, or cases where alternate chains are shifted by half a latt ice spacing along the chain direction. The occupation of some given site is denoted by n . , where 3 labels the different chains, and a 7 ai ' ' 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 U1 are assumed positive (repulsive interactions), and a l l other inter-actions are assumed to be zero. Assuming a l l sites have the same site energy, E^, the energy of some configuration of atoms over the sites is where <33'> indicates a sum over pairs of nearest neighbour chains. We treat the interaction U 1 in mean f ie ld by replacing one of the occupation ai <33 '> ai ai (0 numbers in the last term in (l) by an average value. Anticipating an ordered 3' structure as shown in Fig. 26, we replace the value of n . for a given sublattice on one chain 6 ' by the average occuptation of the other sub-lattice on the adjacent chains, 3 . This means that in (1), we make the subst i tut i on 3' 3 n , •+ <n _> = x. al a2 2 ( 2 a ) OI D n p „ •> <nP-> = x. . a2 al 1 (2b) Now (1) can be rewritten as E(n .} ai ( ^ ^ . • M ^ - v * x 'a x 'a + U v / 3 3 3 3 H n a l V + na2na+l,1 a x (3) We see from (1) of Appendix D that each term.in the sum over 3 in (3) is the energy of the one dimensional latt ice gas with two site energies, and which are given by E l = Eo + V X2 (ha) (hb) Thus, (12) and (15) of Appendix D can be used immediately (note K = -kii) , i f we make the following identif ication: U - E - U o ^'(x + x h U 2 x1 ( 5 a ) B ' = f ( x 2 - X l ) (5b) We thus 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 e q u a t i o n s f o r m = x £ + x-| " 1 and y = x 2 - x 1 m = e " U / 2 k T s i n h 2 (B -yU' m/8)/kT V e U / 2 k T s i n h ( y U ' y A k T ) V where , o - i ( M - E 0 - U - ^ ) - B + I ^ and the d e n o m i n a t o r , V, i s g i v e n by V1 = | e " U / 2 k T cosh[2(B o-YU'm/8)/kT] + e U / 2 k T c o s h (yU 1 y A k T ) + 2 - 2cosh(U/kT) I f y = 0, then x^ = x^, and no long range o r d e r e x i s t s . We see t h a t y = i s always a s o l u t i o n o f ( 7 ) • O r d e r i n g o c c u r s when (7) i s a l s o s a t i s f i e d by some nonzero v a l u e o f y. For U 1 <K kT and U 1 « U , 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 the above e q u a t i o n s o c c u r s . F o r y 4 0, (6) and (7) become m = 4 e " U / k T s i n h ( 2 B /kT) o yU 1/kT Y = 6 4 e " U / k T c o s h 2 ( B /kT)" o ( y U ' / k T ) 2 O r d e r i n g o c c u r s 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 (11) T h i s g i v e s For B Q = 0 and Y = 2, the condition (11) is identical to the Onsager result k.G(k) , for U 1 « U . For T > T , y = 0 and m is given by the usual one dimensional Ising result, (6) of Appendix B. To see what sort of effect the ordering has on the chemical potential, we show in Fig. 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 transit ion, only a very small feature is produced in 8x/3u. For U 1 > kT, the solutions (6), (7), suffer the same problem encountered in the mean f ie ld solutions discussed in Chapter k: they predict that the ordered phase extends over' the entire range 0 < x < 1. A phase diagram for Y U ' / 2 = 0.01 U is shown in Fig . 71. 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. 2 5 7 X Fig. 70 - (a) Inverse derivative -3x/8V versus composition x for a lattice gas of weakly interacting one dimensional chains,' calculated using the mean f ie ld approximation discussed in the text. The intrachain interaction is U = 10 kT, while the interchain interaction is U 1 = 0 . 0 2 M/y. (b) Enlargement of (a) near x = 0 . 5 , also showing the long-range order parameter y = x 2 -x-|. The dotted curve is for U 1 = 0 . 258 O.I4| ' 1 1 1 1 — | i 1 i 1 x Fig. 71 - Phase diagram of latt ice chains for U1 = 0.02 U/y imation discussed in the gas of weakly interacting one dimensional , calculated using the mean f ie ld approx-text. BIBLIOGRAPHY Ajayi , O.B. , Nagel, L . E . , Raistrick, l . D . , and Huggins, R.A. 1976. J . Phys. Chem. Solids 3 7 , 167-Alefeld, G. 1971. In Cr i t i ca l Phenomena in Al loys , Magnets, and Supercon- ductors (eds. R.E. M i l l s , E. Ascher, and R.I . Jaffee) , McGraw-Hill, New York. Alefeld, G . , and Volk l , J . (eds.). 1978. Hydrogen in Metals (2 v o l . ) , Springer-Verlag, Berl in. Alfred, L . C . R . , and March, N.H. 1957. Phil Mag. 2, 9 8 5 . Archie, G.E. 1942. Trans. A . I . M . E . T46, 5 4 . Atlung,...S. , Weat, K. , and Jacobsen, T. 1979. J . Electrochem. Soc. 126, 1311. Balchin, A.A. 1976. In Crystallography and Crystal Chemistry of Materials  with Layered Structures (ed. F. Levy), Reidel, Dordrecht. Beal, A . 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