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Thermodynamics and structure of LixTiS₂ : theory and experiment Dahn, Jeffery Raymond 1982

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THERMODYNAMICS AND STRUCTURE OF L i TiS : THEORY AND EXPERIMENT by J e f f e r y Raymond Dahn B.S c , Dalhousie University, 1978 M.Sc. , rUniv.ersity';of B r i t i s h Columbia, 1980 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 t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1982 c Jeffery.Raymond Dahn,"1982 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 DE-6 (3/81) i i ABSTRACT This thesis describes experimental methods, including i n s i t u X-ray d i f f r a c t i o n , e s p e c i a l l y suited to the study of l i t h i u m i n t e r c a l a t i o n systems,and discusses the i n t e r p r e t a t i o n of the r e s u l t s obtained i n a study of Li xTiS2» A r i g i d p l a t e and spring model of layered i n t e r c a l a t i o n systems i s developed and i s used to investigate the r o l e of l a t t i c e expansion and e l a s t i c energy i n layered i n t e r c a l a t i o n compounds. When the e l a s t i c energy, calculated using the spring and plate model, i s included i n the Hamiltonian of a three dimensional l a t t i c e gas model f o r L i TiS„ x 2 good agreement between the experimental r e s u l t s and the t h e o r e t i c a l predictions are obtained. Staging, not l i t h i u m ordering, i s i d e n t i f i e d as the dominant physical mechanism i n L i Y T i S 0 . i i i TABLE OF CONTENTS Page ABSTRACT . i i TABLE OF CONTENTS i i i LIST OF TABLES v i LIST OF FIGURES v i i LIST OF SYMBOLS x i i i ACKNOWLEDGEMENTS x v i i PART I I N T R O D U C T I O N 1 CHAPTER 1 INTRODUCTION 2 1.1 I n t e r c a l a t i o n 2 1.2 Contributions of t h i s thesis 8 CHAPTER 2 REVIEW OF EARLY WORK ON L i TiS„ 11 x 2 2.1 Thermodynamics of i n t e r c a l a t i o n 11 2.2 Structure and properties of TiS^ 18 2.3 Ea r l y experimental r e s u l t s on L i H S 2 23 2.4 L a t t i c e gas models f o r L i T i S 2 29 2.5 Summary of early work on il^TiS^ 38 PART I I E X P E R I M E N T A L METHODS FOR S T U D Y I N G L I T H I U M I N T E R C A L A T I O N S Y S T E M S : A P P L I C A T I O N TO L i x T i S 2 39 CHAPTER 3 THE MEASUREMENT OF V(x) AND -3x/3V 40 3.1 Preparation of T i S 2 40 3.2 C e l l construction and components 44 3.3" Measurement of V(x) and -3x/3V 47 3.4 E l e c t r o l y t e c o - i n t e r c a l a t i o n 53 3.5' Results and discussion 57 CHAPTER 4 THE MEASUREMENT OF 3V/3T) 68 x 4.1 Experimental d e t a i l s 68 4.2 Results and discussion 73 iv CHAPTER 5 IN SITU X-RAY DIFFRACTION 79 5.1 Advantages of the i n s i t u technique 79 5.2 Experimental 82 5.3 S t r u c t u r a l measurements on L i TiS^ 89 5.4 Discussion 110 CHAPTER 6 THE NEUTRON DIFFRACTION EXPERIMENT 111 6.1 Sample preparation and experimental d e t a i l s 111 6.2 Results f o r sample JN-3, L i ^ T i S j ^ 124 6.3 Results f o r sample JN-7, L i " 2 5 T i S 2 142 6.4 Discussion " 145 PART I I I L A T T I C E GAS MODELS OF I N T E R C A L A T I O N S Y S T E M S 1 4 7 CHAPTER 7 THE ROLE OF LATTICE EXPANSION AND ELASTIC ENERGY IN INTERCALATION SYSTEMS 148 7.1 Introduction 148 7.2 Importance of the e l a s t i c energy 150 7.3 The spring and plate model 153 7.3.1 zero applied pressure 153 7.3.2 applied pressure 159 7.4 Inclusion of the e l a s t i c energy i n l a t t i c e gas models of i n t e r c a l a t i o n 162 7.4.1 Mean f i e l d theory 162 7.4.2 Bragg-Williams approximation 173 7.4.3 Monte Carlo simulations of L i x T i S 2 186 7.5 .' Applicationcof ::the;:theory';toi; other i i n t e r -c a l a t i o n compounds 201 7.5.1 Graphite i n t e r c a l a t i o n compounds 201 7.5.2 E f f e c t s of pressure on staged i n t e r -c a l a t i o n compounds 204 • 7.5.3 Li xNbSe 2 207 7.5.4 L i T i . ,S 0 208 x 1.1 2 PART IV C O N C L U S I O N S 2 1 3 CHAPTER 8 SUMMARY AND SUGGESTIONS FOR FUTURE WORK 214 8.1 Summary of the thesis 214 8.2 Suggestions f o r future work 217 A P P E N D I C E S 2 1 9 Al 3V/8T) for the one dimensional l a t t i c e gas 219 A2 x > 1 ?n L i x T i S 2 2 2 4 A2.1 Introduction 224 A2.2 Experimental r e s u l t s 225 A2.3 A l a t t i c e gas model f o r L i TiS , 0<x<2 235 A3 Least squares refinement of the l a t t i c e parameters of hexagonal c r y s t a l s 242 A4 I n t e n s i t y c a l c u l a t i o n s f o r X-ray and Neutron d i f f r a c t i o n 246 A4.1 X-ray d i f f r a c t i o n 246 A4.2 Neutron d i f f r a c t i o n 253 A5 Imperfect staging 256 A5.1 The H e n d r i c k s - T e l l e r s o l u t i o n to d i f f r a c t i o n from disordered l a y e r l a t t i c e s 256 A5.2 The e f f e c t of the Daumas-Herold domain model of staging 273 A6 Determining phase diagrams of l a t t i c e gas models solved w i t h the Bragg-Williams approximation 277 A7 Monte C a r l o s i m u l a t i o n s 281 B I B L I O G R A P H Y VI LIST OF TABLES Table Page 1. Analogies between the thermodynamics of the i n t e r c a l a t i o n c e l l and the gas 16 2. The growth conditions and properties of the . T i S 2 used i n the experiments described i n t h i s t h e s i s 42 3. Observed and calculated plane spacings f o r L i TiS„ i n c e l l JX-13 at 2.296 v o l t s (x = .3421.008) x 93 4. The observed plane spacings of the e l e c t r o l y t e c o - i n t e r c a l a t e d material 95 5. Relative X-ray powder d i f f r a c t i o n i n t e n s i t y c a l c u l a t i o n f o r L i ,,TiS„ 104 .16 2 8. Absorption and scattering cross-sections f o r the n u c l e i present i n sample JN-3, L i ^ T i S 2 118 Relative i n t e n s i t y c a l c u l a t i o n f o r L i ^ ^ T i S 2 measured with neutron powder d i f f r a c t i o n 120 The thermal h i s t o r y of sample JN-3, L i ^ T i S 2 125 9. " The!results o f f i t t i n g the neutron d i f f r a c t i o n data 134 A2.1 Bragg peaks of L i 2 T i S 2 230 A2.2 Values of the s i t e energies and i n t e r a c t i o n parameters used i n the model c a l c u l a t i o n to describe L i TiS. f o r 0<x<2 240 x 2 _ -A4.1 Values of the parameters needed to f i t the experimental v a r i a t i o n of the P h i l i p s PW1386/50 automatic divergence s l i t 250 v i i LIST OF FIGURES Figure Page 1. Schematic representation of a L i / L i TiS„ c e l l 5 x 2 2. V(x) and -3x/3V f o r a L i / L i TiS c e l l , Oijx^l 7 3. 3V/3T) x corresponding to an entropy minimum 15 .4. The atomic structure of TiS,, 19 5. The van der Waals gap of T i S 2 6. V(x) and -3x/3V f o r a L i / L i x T i S 2 c e l l , O^xgl 20 24 7. Dependence of the l a t t i c e parameters of L i T i S 2 on x ( a f t e r Whittingham 1976b) 26 8. Mean f i e l d theory c a l c u l a t i o n f o r V(x) and -3x/3V of L i / L i T i S 2 c e l l s using a two dimensional t r i a n g u l a r l a t t i c e gas model 33 . 9. The decomposition of a t r i a n g u l a r l a t t i c e into three interpenetrating s u b l a t t i c e s 35 10. Renormalization group c a l c u l a t i o n f o r V(x) and -3x/3V of L i x T i S 2 c e l l s using a two dimensional t r i a n g u l a r l a t t i c e gas model 36 11. "Exploded" view of an electrochemical c e l l 45 12. Several cycles of a L i / L i C 1 0 4 , PC/Ni c e l l 48 13. The e f f e c t s of e l e c t r o l y t e c o - i n t e r c a l a t i o n on the electrochemical behaviour of L i / L i C l O , , PC/Li TiS„ c e l l s 54 x 2 14. V(x) and -3x/3V f o r L i / L i x T i S 2 c e l l JD-211 58 15. -3x/3V versus V f o r L i / L i T i S 9 c e l l JD-235 59 16. -3x/3V versus x f o r L i / L i TiS„. c e l l JD-235 61 x Z 17. V(x) for L i / L i x T i S 2 c e l l JD-235 62 18. -3x/3V versus x f o r L i / L i TiS„ c e l l JD 234 63 x 2 19. .'V(x) f o r L i / L i x T i S 2 c e l l JD-234 64 20. Linear sweep voltammogram of L i / L i TiS ' c e l l JD-212 x 66 v i i i Figure Page 21. "Exploded" view of an electrochemical c e l l used f o r 3V/3T) x measurements 70 22. 3V/3T) measurements oh L i / L i TiS c e l l JD-218 72 X X c. 23. 3V/3T) versus V for L i / L i TiS„ c e l l JD-218 74 x x 2 24. 3V/3T) versus x f o r L i / L i TiS„ c e l l JD-218 75 x x 2 25. 3V/3T) versus x f o r L i / L i T i S 0 c e l l JD-156 76 x x 2 26. "Exploded" view of the electrochemical c e l l used f o r i n s i t u X-ray d i f f r a c t i o n experiments 83 27. X-ray d i f f r a c t i o n p r o f i l e obtained from a L i / L i TiS„ c e l l X 86 28. The "breathing" battery 90 29. V a r i a t i o n of the l a t t i c e parameters of L i T i S 2 as a function of x measured using i n s i t u X-ray d i f f r a c t i o n 92 30. T y p i c a l (004) Bragg peaks f o r 3p38 L i x T i S 2 97 31. The (004) Bragg peak o f - L i / L i T i S ? c e l l JX-17 at 2.332 v o l t s X 98 32. Several high rate charge-discharge cycles measured on L i / L i TiS„ c e l l JX-17 99 x 2 33. V a r i a t i o n of the (004) Bragg peak width of L i TiS„ c e l l JX-17 X 101 9 34. A search f o r the (00y) s u p e r l a t t i c e peak i n L i w T i S „ 106 .14 2 35. "Exploded" view of the flange c e l l used to prepare neutron d i f f r a c t i o n samples 113 36. The aluminum sample holders used in the neutron d i f f r a c t i o n experiment, 116 37. The A l (111) peak from the JN-3, L i 1 4 T i S 2 sample holder measured using neutron d i f f r a c t i o n 123 38. The (001) and (002) peaks of sample JN-3, L i wTiS, at 17.K v * 126 3 39. (OOj) neutron d i f f r a c t i o n scans at 100K and 300K measured on sample JN-3, L i ^ ^ T i S 2 128 3 (00^) neutron d i f f r a c t i o n scans measured on sample JN-3, L i ^ T i S 2 as a function of temperature 130 XX Figure Page 3 41. (OCb) peak as a function of temperature f o r sample JN-3, L i 1 4 T i S 2 135 3 42. (00y) peak width as a function of temperature fo r sample JN-3, L i 1 4 T i S 2 136 3 43. (00y) peak p o s i t i o n as a function of temperature f o r sample JN-3, L i 1 4 T i S 2 137 44. The c-axis of L i ^ 4 T i S 2 as a function of temperature * 140 45. The (002) and (100) peaks of. sample JN-7, L i 9 STiS„ at 300K 143 3 46. (00j) scans measured on sample JN-7, L i „_TiS 2 at I00K and 300K ' 144 47. The spring and plate model of a layered i n t e r c a l a -t i o n compound 154 48. A f i t to the c-axis expansion of L i T i S 2 using the spring and plate model 156 49. Gibbs f r e e energy per layer f o r a non-interacting l a t t i c e gas model of a layered i n t e r c a l a t i o n compound where the e l a s t i c energy has been calculated with the spring and plate model 165 50. Phase diagram f o r a non-interacting l a t t i c e gas model of a layered i n t e r c a l a t i o n compound where the e l a s t i c energy has been calculated with the spring and plate model 167 51. The v a r i a t i o n of T and x with:-.a" 168 c c 52. V(x) and -9x/3V f o r several values of j/kT for the l a t t i c e gas described by Figure 50 169 53. Phase diagram f o r an i n t e r a c t i n g l a t t i c e gas model of a layered i n t e r c a l a t i o n compound where the i n t e r a c t i o n has been treated i n mean f i e l d theory and the e l a s t i c energy has been cal c u l a t e d with the spring and plate model 171 54. F i t s to V(x) and -3x/3V of L i / L i T i S 2 c e l l s using the l a t t i c e gas model with nearest neighbor i n t e r a c t i o n treated i n mean f i e l d theory 172 55. Phase diagrams including staged phases f o r the l a t t i c e gas model with interplane i n t e r a c t i o n s calculated using the Bragg-Williams approximation 175 X Figure Page 56. V(x) and -3x/3V c a l c u l a t e d using the Bragg-W i l l i a m s approximation f o r s p e c i f i c values of the nearest neighbor i n t e r a c t i o n and e l a s t i c parameters Cu' = 23, kT/J = 1.6 and a = 0.2). 177 57. Phase diagram, i n c l u d i n g staged phases, f o r the l a t t i c e gas model w i t h interplan-e i n t e r a c t i o n s c a l c u l a t e d w i t h the Bragg-Williams approximation 178 58. Phase diagram, i n c l u d i n g staged and ordered phases, f o r the l a t t i c e gas model w i t h i n t e r -plane and i n t r a p l a n e i n t e r a c t i o n s 179 59. V(x) c a l c u l a t e d f o r s p e c i f i c values of the e l a s t i c parameters and i n t e r a c t i o n energies f o r the l a t t i c e gas described by f i g u r e 56 i n an attempt to describe V(x) f o r L i / L i T i S 0 c e l l s 181 x 2 60. -3x/3V corresponding to V(x) shown i n Fi g u r e 59 182 61. 3S/3x) c a l c u l a t e d w i t h the same parameters as V(x) i n Fi g u r e 59 183 184 62. c-axis expansion c a l c u l a t e d using the spring and p l a t e model f o r the same parameters as V(x) i n Figure 59 and compared to the L i TiS„ data x 2 Figures 63-70 are the r e s u l t s of Monte Carlo s i m u l a t i o n s , on 4x4x4 and 6x6x6 l a t t i c e s , of a l a t t i c e gas model w i t h nearest neighbor i n t e r p l a n e and i n t r a p l a n e i n t e r a c t i o n s . The e l a s t i c energy, c a l c u l a t e d using the spring and p l a t e model, has been included i n the Hamiltonian. The r e s u l t s of the si m u l a t i o n s are compared w i t h the appropriate L i ^ T i S ^ data. 63. V(x) f o r L i / L i TiS„ c e l l s 190 x I 64. -3x/3V f o r L i / L i T i S 0 c e l l s (4x4x4 l a t t i c e ) 191 x 2 65. -3x/3V f o r L i / L i TiS„ c e l l s (6x6x6 l a t t i c e ) 192 x 2 66. 3V/3T) f o r L i / L i TiS„ c e l l s (4x4x4 l a t t i c e ) 194 x x 2 67. 3V/3T) f o r L i / L i T i S n c e l l s (6x6x6 l a t t i c e ) 195 x x 2 68. c-axis expansion of L i TiS„ (4x4x4 l a t t i c e ) 196 x 2 69. The c - a x i s thermal expansion c o e f f i c i e n t , a , as a f u n c t i o n of x (6x6x6 l a t t i c e ) 197 70. The (004) Bragg peak width (6x6x6 l a t t i c e ) 199 71. Phase diagrams i n c l u d i n g higher stages t o q u a l i t a t i v e l y d e s c r i b e staging i n GICs and i n L i NbSe„ 202 x i Figure Page 72. Phase diagram i l l u s t r a t i n g the e f f e c t s of pressure on staged i n t e r c a l a t i o n compounds 205 73. V(x) and -9x/9V for L i / L i T i , ,S„ c e l l s x 1.1 I 74. Voltage curves i l l u s t r a t i n g the e f f e c t s of 209 i n t e r l a y e r titanium on L i / L i T i , , S„ c e l l s 210 x 1+y 2 75. -3x/9V corresponding to V(x) i n f i g u r e 74 211 A l . l . " The entropy, chemical p o t e n t i a l and 9V/9T) f o r the one dimensional l a t t i c e gas with nearest neighbor i n t e r a c t i o n s 222 A2.1. V(x) of L i / L i C l 0 4 , P C / L i x T i S 2 c e l l JD-49 for 0<x<2 226 A2.2. The voltage and corresponding X-ray d i f f r a c t i o n p r o f i l e s of c e l l JX-15 at several values of x between one and two 227 A2.3. The X-ray d i f f r a c t i o n p r o f i l e of L i 2 T i S 2 229 A2 . 4 . V(x) of L i / L i x T i S 2 c e l l JD-49 f o r 0<x<3 231 A2.5. Portion of the.X-ray d i f f r a c t i o n p r o f i l e measured on c e l l JX-15 at 0.025 v o l t s 233 A2.6. The octahedral and tetrahedral s i t e s within a sin g l e van der Waals gap of T i S 2 237 A2.7. Free energy and voltage of the t r i a n g u l a r l a t t i c e gas with two s i t e energies and r e p u l s i v e i n t e r a c t i o n s 239 A2.8. Voltage of the t r i a n g u l a r l a t t i c e gas with two s i t e energies and repulsive i n t e r a c t i o n s compared to V(x) f o r L i / L i T i S 0 c e l l s f o r 0<x<2 241 x 2 A4.1. The Bragg-Brentano focussing geometry 247 A4.2. D e f i n i t i o n of the quantities i n equation A4.2 247 A4.3. V a r i a t i o n of the PW1386/50 automatic divergence s l i t width as a function of Bragg angle 251 A4 . 4 . Two geometries used for neutron d i f f r a c t i o n experiments 254 A5.1. Perfect (stage 2) and imperfect staging 257 A5.2. The. incident and scattered X-rays from a'.layered c r y s t a l 259 A5.3. The stacking of layers with two c h a r a c t e r i s t i c layer spacings 261 F i g u r e A5.4. The s c a t t e r e d X-ray i n t e n s i t y from disordered l a y e r s t r u c t u r e s A5.5. The Daumas-Herold domain model of staging LIST OF SYMBOLS A area (Chapter 7 only) A parameter used to d e s c r i b e staging c o r r e l a t i o n s A(0), A'(0) Absorption f a c t o r s (Appendix 4 only) a l a t t i c e parameter b^ r e c i p r o c a l l a t t i c e v e c t o r c l a t t i c e parameter c value of c at x=0 o c average c-axis c' v a l u e of c stored i n neutron spectrometer computer c^, c^ c h a r a c t e r i s t i c l a y e r spacings c^ l a y e r spacing of the i * " * 1 l a y e r c parameter used i n chapter 7 c. . reduced n o t a t i o n f o r e l a s t i c s t i f f n e s s tensor D d i f f u s i o n constant d, d(hk£) plane spacing E e l a s t i c energy E , s i t e energies o t ° E^ e l a s t i c energy of l a y e r i e magnitude of the e l e c t r o n i c charge F Helmholtz f r e e energy F, F(hk£) geometrical s t r u c t u r e f a c t o r (Appendices 4 and 5 on f molecular s t r u c t u r e f a c t o r ft f o r c e per l a t t i c e spring i' tf/(fe(eL - C Q ) ) G Gibbs f r e e energy x i y Gibbs f r e e energy of i " " l a y e r Hamiltonian M i l l e r index when used as (hk€) Current (Chapter 3 only) s c a t t e r e d i n t e n s i t y (X-ray or neutron) e l a s t i c parameter l a t t i c e s p r i n g constant Boltzmann's constant wave v e c t o r s s p r i n g constant a s s o c i a t e d w i t h d i s t o r t i o n s near i n t e r c a l a n t atoms M i l l e r index when used as (hk£) M i l l e r index when used as.(hk£) p a r t i c l e r a d i u s (Chapter 3 only) molecular weight m u l t i p l i c i t y cathode mass (Chapter 3 only) number of s i t e s or number of host atoms number of i n t e r c a l a t e d atoms pressure parameters i n f i t s to neutron d i f f r a c t i o n data charge corresponding to Ax=l s c a t t e r i n g v e c t o r s e r i e s r e s i s t a n c e (Chapter 3 only) goniometer r a d i u s (Chapter 5 and Appendix 4 only) lengths entropy ,- .th , entropy of x l a y e r temperatures c r i t i c a l temperature time i n t e r n a l energy i n t e r n a l energy of i ^ layer i n t e r a c t i o n energies voltage layer structure f a c t o r (Appendix 5 only) volume volume of i * " * 1 layer c o r r e l a t i o n function f r a c t i o n by weight of i * " * 1 element in a compound composition of an i n t e r c a l a t i o n compound, composit f r a c t i o n a l occupation of a l a t t i c e gas average occupancy of layer k occupancy of i * " * 1 s i t e i n k*"^  layer (x-j^O o r 1) composition at the surface c o r r e l a t i o n function T 1 l + y S 2 p a r t i t i o n function : valence of metal c-axis thermal expansion c o e f f i c i e n t K/k length angular divergence (Appendix 4 only) s t r a i n (reduced notation) (c C )/(.c + c ) a b a b xvi n D diffusive overpotential n resistive oyerp©'tential R 0 Bragg angle 6 corrected angle corr K T isothermal compressibility A wavelength y chemical potential U linear absorption coefficient (Chapter/ 6 and ApPeudice.s 4 and 5 only) U* effective linear absorption coefficient v related to the Bragg angle 5 number of lattice spacings over wh£ch cgrrelatioiis.- extend p density a- scattering cross-section a standard deviation T residence time cj> monochromator angle (Appendix 4 only) . <j>, <f>^, <)>k phase shifts i> average phase shift X, X' goodness of f i t grand partition function a- absorption cross-section x v i i ACKNOWLEDGEMENTS I am indebted to my supervisor, Rudi Haering, for h i s help and guidance throughout the course of t h i s i n v e s t i g a t i o n . The i n t e r e s t shown by him, John Berlinsky and B i l l Unruh i n my work and my career i s g r e atly appreciated. Throughout the course of my work i n Rudi Haering's group I have benefitted from discussions with many co-workers i n the lab. In p a r t i c u l a r , c o l l a b o r a t i o n with my co-authors Ross McKinnon, Marcel Py and Doug Dahn, has been very u s e f u l . Geoff Johnson has, i n e f f e c t , given me a working knowledge of modern computing f a c i l i t i e s . Discussions with Alec Rivers-Bowerman about t e c h n i c a l problems were always u s e f u l . I thank U l r i c h Sacken, Jim Chiu, Peter Mulhern, Andre van Schyndel, Rod McMillan, Rick Clayton and Richard Marsolais f o r t h e i r help and encourage-ment . The assistance given to me by the s t a f f and s c i e n t i s t s of the A.E.C.L. Neutron and S o l i d State Physics branch i s appreciated. In p a r t i c u l a r , the help of B i l l Buyers, Brian Powell, Harald Nieman, Don Tennant and Jim Evans i s acknowledged. I thank Peter Haas for his expert work on the diagrams i n the thesis and Carmen deSilva f o r typing the t h e s i s . F i n a l l y , I thank the Natural Sciences and Engineering Research Council f o r f i n a n c i a l support. PART I 2 C H A P T E R O N E INTRODUCTION 1.1 I n t e r c a l a t i o n I n t e r c a l a t i o n i s the r e v e r s i b l e i n s e r t i o n of guest atoms or molecules i n t o a host s t r u c t u r e without d r a s t i c a l l y a l t e r i n g the atomic s t r u c t u r e of the host. In order f o r i n t e r c a l a t i o n to proceed there must be s i t e s i n the host a v a i l a b l e f o r occupation by the guest species ( i n t e r c a l a n t ) . For the a d d i t i o n of any s i g n i f i c a n t q u a n t i t y of i n t e r c a l a n t , these s i t e s must be a c c e s s i b l e from the surface and the i n t e r c a l a n t must be mobile w i t h i n the host. The t r a d i t i o n a l example of an i n t e r c a l a t i o n host i s g r a p h i t e , where i t i s found that many d i f f e r e n t types of atoms and molecules can be i n t e r -c a l a t e d between the g r a p h i t i c planes. Dresselhaus and Dresselhaus (1981) review much of the work performed on graphite i n t e r c a l a t i o n compounds (GICs). I n t e r c a l a t i o n has a l s o been found, to occur i n other lay e r e d hosts such as the t r a n s i t i o n metal dichalcogenides (MX,,) . The review by Whittingham (1978) describes many MX 2 i n t e r c a l a t i o n compounds. M a t e r i a l s such as the t r a n s i t i o n metal oxides (e.g. (Murphy et a l . , 1979), W02 (Murphy et a l . , 1978) and Mo0 2 (Sacken 1979)) w i t h non-layered s t r u c t u r e s have a l s o been found to be hosts f o r r e v e r s i b l e i n t e r c a l a t i o n . The metal-hydrogen systems (Alefeld and V o l k l 1978) and doped polyacetylene ( F l a n d r o i s 1981) can a l s o be in c l u d e d as i n t e r c a l a t i o n systems under t h i s d e f i n i t i o n . 3 I n t e r c a l a t i o n systems are of current i n t e r e s t for two reasons. F i r s t , the large number of i n t e r e s t i n g physical phenomena exhibited by these systems makes these materials a t t r a c t i v e from a basic science viewpoint. Secondly, the possible technological importance of many i n t e r c a l a t i o n systems renders them i n t e r e s t i n g for p r a c t i c a l and economic reasons. The most s t r i k i n g e f f e c t exhibited by GICs i s that of staging, i . e . an ordered sequence of n graphite layers and 1 i n t e r c a l a n t layer, where n defines the stage. Staging i s not l i m i t e d to GICs and has recently been observed i n Na TiS„, (Zanini et a l . , 1981), Ag TaS„ (Scholz and F r i n d t x 2 x 2 1980) and L i NbSe (Dahn and Haering 1982). The i n plane i n t e r c a l a n t X z. structure of GICs i s quite varied, including structures both commensurate and incommensurate with the host (Dresselhaus and Dresselhaus 1981). Most studies of phase t r a n s i t i o n s i n GICs have concentrated on the order-disorder t r a n s i t i o n s r e l a t e d to the occupation of s i t e s within a s i n g l e layer. Changes i n the e l e c t r i c a l conductivity of GICs of several orders of magnitude are observed upon i n t e r c a l a t i o n due to the transfer of charge from the i n t e r c a l a n t to the graphite la y e r s . Theoretical studies attempting to describe the phenomenon of staging (Safran 1980), i n plane i n t e r c a l a n t ordering (Lee et a l . , 1980) and to compute the stage dependent d i s t r i b u t i o n of electrons i n GICs (Safran and Hamann 1980, Pietronero et a l . , 1978) have been made. There i s much a c t i v i t y i n the study of i n t e r c a l a t i o n compounds and only a t i n y f r a c t i o n of publications i n the f i e l d have been referenced above. There are many ways to prepare i n t e r c a l a t e d materials. The most common method of preparing GICs and metal-hydrides i s to allow the host to come i n contact with the i n t e r c a l a n t vapor (Dresselhaus and Dresselhaus 1981) . By c o n t r o l l i n g j the vapor pressure of the i n t e r c a l a n t , i t i s possible 4 to change the i n t e r c a l a n t concentration i n the host. Chemical reactions, such as the n-butyl l i t h i u m reaction, (Dines 1975), can be used to i n t e r -calate many compounds with l i t h i u m . The i r r e v e r s i b l e reaction of n-butyl l i t h i u m , C.H„Li, with TiS„ i s as follows: 4 9, 2 xC.H.Li + TiS„ -> L i TiS„ + x C 0 L n , producing L i TiS„ and octane. 4 9 2 x 2 2 8 18 ° x 2 I n t e r c a l a t i o n can also be performed i n electrochemical c e l l s and t h i s was the p r i n c i p a l method used i n the experiments described i n the t h e s i s . Whittingham (1976a) r e a l i z e d that the t r a n s i t i o n metal dichalcogenides could be used as electrodes i n high energy density r e v e r s i b l e battery systems based on i n t e r c a l a t i o n . The operation of an i n t e r c a l a t i o n c e l l i s conceptually simple. A schematic diagram of a L i / L i TiS„ electrochemical X z. c e l l i s given i n Figure 1. The h a l f c e l l reactions are x L i % x L i + + xe 1.1 at the l i t h i u m anode and x L i + + xe" + TiS„ t L i TiS„ 1.2 2 x 2 at the L i TiS„ cathode. The electron makes the journey from anode to x 2 cathode v i a an external e l e c t r i c c i r c u i t and the L i + ion i s transported by an e l e c t r o l y t e . These c e l l s are not only i n t e r e s t i n g from a p r a c t i c a l point of view, but also a great deal of thermodynamic information i s contained i n t h e i r electrochemical behaviour. This i s because the voltage, V(x), of a L i / L i ^ T i S 9 c e l l i s given by V(x) ± I ' ( v - u (x) ) e a c 1.3 L i x T i S 2 Electrolyte Figure 1. Schematic representation of the discharge of a L i / L i T i S . c e l l , x z 6 where u and u (x) are the chemical p o t e n t i a l s of the l i t h i u m atoms i n a c the anode and cathode r e s p e c t i v e l y and e i s the e l e c t r o n i c charge. u (x) and V(x) are functions of l i t h i u m concentration because the chemical c composition of the L i ^ T i S ^ cathode changes as current flows. The V(x) behaviour f o r a L i / L i x T i S 2 c e l l i s shown i n Figure 2. One can also use the electrochemical c e l l to prepare i n t e r c a l a t e d samples for study by other means. This i s done by c o n t r o l l i n g the number of electrons which flow through the external c i r c u i t of Figure 1, which i n turn, by charge n e u t r a l i t y , controls the number of l i t h i u m atoms which i n t e r c a l a t e (equations 1.1 and 1.2). The study of i n t e r c a l a t i o n systems such as L i TiS using electrochemical c e l l s provides one with the opportunity to do e x c i t i n g physics and to a s s i s t i n the understanding of very promising battery systems. .4 .6 X in L i x T i S 2 8 Figure 2. V(x) and -3x/3V of a L i / L i x T i S 2 c e l l . A fter Dahn et a l . (1982). 8 1.2 Contributions of t h i s thesis This t h e s i s i s pri m a r i l y concerned with L i TiS„ . Experimental methods X *£ p a r t i c u l a r l y suited to the study of l i t h i u m i n t e r c a l a t i o n systems are described and applied to L i TiS-. The r e s u l t s of the experiments, when X £. combined, y i e l d the information needed to determine the dominant phys i c a l e f f e c t s present i n Li xTiS2« A simple theory i s developed to incorporate l a t t i c e expansion and e l a s t i c energy in the framework of a l a t t i c e gas model. This model, when applied to L i TiS„, produces good agreement X £. with the experimental r e s u l t s described. Part I of the the s i s i s devoted to a b r i e f discussion of i n t e r c a l a t i o n and a detai l e d review of early experimental and t h e o r e t i c a l work on L i ^ T i S ^ In Chapter 2 the thermodynamics of i n t e r c a l a t i o n c e l l s , the structure and properties of H S 2 and electrochemical and s t r u c t u r a l experiments on L i TiS- are discussed. A de t a i l e d d e s c r i p t i o n of l a t t i c e gas models for x 2 i n t e r c a l a t i o n systems i s given and various methods of solving the models are described. The status of theory and experiment for L i x T i S 2 p r i o r to t h i s work i s summarized. Part II of the thesis contains a d e s c r i p t i o n of experimental techniques which are e s p e c i a l l y suited to the study of i n t e r c a l a t i o n systems. Chapter 3 deals with the measurement of the v a r i a t i o n of c e l l voltage, V, with i n t e r c a l a n t content, x. The measurement of the inverse d e r i v a t i v e , -3x/3V, of the V(x) r e l a t i o n i s also described. The techniques are applied to L i / L i x T i S 2 c e l l s and the r e s u l t s are discussed. Chapter 4 demonstrates how measurements of the v a r i a t i o n of open c i r c u i t c e l l voltage with temperature at constant intercalant content, 3V/8T) , are made. 3V/3T) i s shown to give information about the entropy of i n t e r c a l a t i o n systems. 9 The r e s u l t s of experiments on L i / L i TiS„ c e l l s are presented. S t r u c t u r a l x 2 studies using x-ray powder diffractometry are described i n Chapter 5. A unique electrochemical c e l l which incorporates a beryllium x-ray window allows one to monitor changes in structure which occur when the intercalant concentration i s al t e r e d electrochemically. The d e t a i l e d design of the c e l l s and experimental problems which a r i s e when using t h i s in s i t u d i f f r a c t i o n technique are discussed. Results on L i TiS„ are presented and x 2 interpreted. The r e s u l t s of a neutron d i f f r a c t i o n experiment on e l e c t r o -chemically prepared L i TiS samples are presented i n Chapter 6. The d e t a i l s X z of sample preparation and design for t h i s experiment are also discussed. In Part III of the t h e s i s a simple theory i s presented which trea t s the l a t t i c e expansion and e l a s t i c energy present in L i x T i S 2 and other layered i n t e r c a l a t i o n compounds. In Chapter 7 the e f f e c t s of the i n c l u s i o n of the e l a s t i c energy i n l a t t i c e gas models f o r i n t e r c a l a t i o n systems are discussed. Sample V(x) and -8x/8V r e l a t i o n s are c a l c u l a t e d . Prompted by experimental r e s u l t s on L i TiS„ as well as on other i n t e r c a l a t i o n systems, x 2 e f f e c t s l i k e staging and in t e r c a l a n t ordering are included in the theory. Phase diagrams are c a l c u l a t e d . The theory i s applied to L i TiS and X z compared to the experimental r e s u l t s obtained in Part I I . The theory i s applied to other i n t e r c a l a t i o n systems. F i n a l l y i n Chapter 8, the experimental and t h e o r e t i c a l r e s u l t s on L i TiS are summarized. Suggestions for future experimental and t h e o r e t i c a l X £. work are given. Seven appendices, devoted to s p e c i f i c experimental and t h e o r e t i c a l points, have also been included. Appendices 1, 3, 4 and 5 deal with the i n t e r p r e t a t i o n and analysis of the experimental data reported in Part II of the t h e s i s . Appendices 6 and 7 give d e t a i l e d descriptions of the 10 t h e o r e t i c a l methods used in Chapter 7. Recent experimental and t h e o r e t i c a l r e s u l t s f o r x>l in L i TiS„ are presented i n Appendix 2. 11 CHAPTER TWO REVIEW OF EARLY WORK ON L i TiS„ x 2 2.1 Thermodynamics of I n t e r c a l a t i o n The EMF, V, of an i n t e r c a l a t i o n c e l l can be r e l a t e d to the thermo-dynamics of the c e l l components. For the discharge of a Li/Li^ T i S 2 c e l l as i n Figure 1, l i t h i u m ions t r a v e l from anode to cathode v i a the e l e c t r o l y t e and e l e c t r o n s v i a the e x t e r n a l c i r c u i t , the net e f f e c t being a t r a n s f e r of l i t h i u m atoms from the L i anode to the L i T i S . cathode. The change i n x 2 Helmholtz f r e e energy of the c e l l , AF, when An l i t h i u m atoms are t r a n s f e r r e d from anode to cathode i s AF = ( u - y ) An 2.1 c a where F and F are the Helmholtz f r e e energies of the anode and cathode, a c ° The e l e c t r o n s t r a n s f e r r e d have done work eAnV i n moving through the p o t e n t i a l d i f f e r e n c e i n the e x t e r n a l c i r c u i t . Because•energy i s conserved, AF i s 12 minus the work done, AF = -e An V which y i e l d s V = - i ( y - y ) e a c 2.2 Because the composition of the l i t h i u m anode i s f i x e d , y & = constant and a l l changes in the c e l l voltage as the c e l l charges or discharges at constant temperature are due to changes in y . The t o t a l number of l i t h i u m atoms in the cathode, n, i s usually referenced to some fixed number, N, for instance the number of titanium atoms. Then n = x N 2.3 for L i _TiS„ . S t r i c t l y speaking, the free energy considered i n deriving the EMF of the c e l l should be the Gibbs free energy, G, (Maclnnes 1939) since i t i s the pressure, P, not the volume, v, which i s controlled in the c e l l , i . e . one should write AG = -eAnV. U i s the i n t e r n a l energy and 5- i s the .entropy in equation 2.4. The change, dG, produced for i n f i n i t e s i m a l changes in T, n and P i s x 2 G = C(n,P,T) - U - TS + Pv 2.4 dG = -SdT + ydn + vdP 2.5 13 The Helmholtz f r e e energy, F(n,v,T), defined by F = U - TS 2.6 y i e l d s dF = -SdT + ydn + Pdv 2.7 f o r i n f i n i t e s i m a l changes i n T, n and v. At constant zero pressure, the Gibbs and Helmholtz f r e e energies are equ i v a l e n t . This i s the case i n the b a t t e r y when the c e l l components are allowed to expand f r e e l y . The e f f e c t s of pressures of a few atmospheres are n e g l i g i b l e compared to the other energies involved (see Chapter 7 ) . Thus F and G can be used i n t e r -changeably at low pressure. However, when the e f f e c t s of pressure on model c a l c u l a t i o n s of V(x) are considered, the Gibbs f r e e energy must be used (Chapter 7 ) . The change i n the Helmholtz f r e e energy of the cathode, dF, (F w i l l be c a l l e d F now as the anode i s u n i n t e r e s t i n g ) f o r i n f i n i t e s i m a l changes i n the i n t e r c a l a n t content and the temperature i s dF = -SdT + ydn 2.8 From equation 2.8, the Maxwell r e l a t i o n 2.9 14 can be obtained. Using equations 2.2, 2.3 and 2.9 31) = ±31) _ f ( T ) 2.10 3T Jx " Ne 3x ;T K ' because 3y /3T) = f(T) i s independent of x. Equation 2.10 r e l a t e s measurements of the temperature c o e f f i c i e n t of the c e l l voltage at constant x to the change i n entropy of the i n t e r c a l a t i o n compound with x at constant temperature. In Figure 3 the e f f e c t of a minimum in S on 3V/3T) i s shown X schematically. 3V/3T) x i s s e n s i t i v e to short range order as well as long range order. The e f f e c t of the short range order present i n the one dimensional l a t t i c e gas, on SV/ST)^ i s calculated i n Appendix One. It i s useful to make analogies between the thermodynamics of i n t e r -c a l a t i o n c e l l s and gases. Consider the change in the Helmholtz free energy, dF, of a.gas at fi x e d number of p a r t i c l e s f o r i n f i n i t e s i m a l changes i n volume and temperature. dF = - SdT - Pdv 2.11 Equations 2.8 and 2.11 are i d e n t i c a l i f the substitutions v->n and P-*-y are made. In f a c t , the Maxwell r e l a t i o n s f o r the c e l l can be obtained from those of the gas by making t h i s s u b s t i t u t i o n (Stanley 1971). In Table 1 comparable thermodynamic quantities for the gas and the c e l l are l i s t e d . y(n,T) f o r the c e l l i s analogous to P(v,T) for the gas. Thus, V(x,T) for the c e l l can be thought of as an equation of state f o r the i n t e r c a l a t i o n compound (equation 2.2 i s used to r e l a t e V and y) . The V(x) behaviour at constant T i s analogous to a P-v isotherm for the gas. The isothermal c o m p r e s s i b i l i t y K , for the gas 0 0.5 I X Figure 3. V a r i a t i o n of 3V/3T) X corresponding to the entropy, S(x), of the hypothetical i n t e r c a l a t i o n system shown. The constant s h i f t , due to the anode, (f(T) i n equation 2.10) i n 3V/3T)_ has not been included. Table 1 Analogies between thermodynamics of the i n t e r c a l a t i o n c e l l and the gas Quantity Gas Cell:. Free energy F=F(v,T) F=F(n,T) dF=-SdT-Pdv dF=-SdT+ydn Equation of state P=P(v,T) y=y(n,T) Compressibility 1 3v. K T v 3P^T 1 3n. 1 3x n 3y^T x 3y^T Maxwell r e l a t i o n 3P_, _ 3S. 3T'v 3v^T 3y.. _ -3S. 3T n 3n^T 17 1 i z \ T ~ v BP-'T i s analogous to n 3^T x 3y^T ~ ex WT 2.12 for the i n t e r c a l a t i o n system. 3x/8V) T r e l a t e s the response of int e r c a l a n t concentration to a change in c e l l voltage. (9x/9y) T also measures the fluc t u a t i o n s in composition of an i n t e r c a l a t i o n system held at constant y (Landau and L i f s h i t z 1969) where k i s Boltzmann's constant. Since f l u c t u a t i o n s are greatly increased near phase t r a n s i t i o n s , one expects peaks i n 3x/3y at values of y where phase t r a n s i t i o n s occur. It i s known that the compressibility of a gas shows divergences at phase t r a n s i t i o n s as w e l l . In summary, measurements of the electrochemical behaviour of i n t e r c a l a t i o n c e l l s y i e l d thermodynamic information about the i n t e r c a l a t i o n system. . The c e l l voltage, V(x,T) i s the equation of state for the i n t e r c a l a t i o n system, peaks or divergences i n -9x/9V)^, give information about phase t r a n s i t i o n s and measurements of 3V/9T) x y i e l d information about the entropy. 2.13 18 2.2 St r u c t u r e and P r o p e r t i e s of TIS^ The c r y s t a l s t r u c t u r e of T i S 2 i s the w e l l known cadmium i o d i d e — 3 s t r u c t u r e (Wyckof f 1963) . The space group i s P3ml - which has a t r i g o n a l p o i n t group (Henry and. Lonsdale 1952). This i s a layered s t r u c t u r e which c o n s i s t s of S-Ti-S "sandwiches" stacked upon each other and held together by r e l a t i v e l y weak van der Waals f o r c e s as depicted i n Figure 4a. The atomic s t r u c t u r e of the "sandwiches" i s shown i n Figure 4b. The o o hexagonal l a t t i c e parameters of T i S 2 are a = 3.407 A and c = 5.695 A. (Thompson et a l . 1975). The f r a c t i o n a l coordinates of the atoms i n the u n i t c e l l are t i t a n i u m at (0,0,0) and s u l f u r atoms at ± (1/3, 2/3, .250). The l o c a t i o n of s i t e s a v a i l a b l e f o r l i t h i u m occupation i n T i S 2 can be e a s i l y v i s u a l i z e d . Figure 5 shows a p r o j e c t i o n of the van der Waals gap of T i S 2 i n the (001) plane. There are three types of s i t e s , the f i r s t coordinated o c t a h e d r a l l y by three s u l f u r atoms top and bottom, the second coordinated t e t r a h e d r a l l y by three s u l f u r atoms on top and one on bottom and the t h i r d merely the i n v e r s i o n of the second. The f r a c t i o n a l coordinates of the s i t e s are (0,0,1/2) f o r the octahedral s i t e and ±(1/3, 2/3, z) w i t h z 5/8 f o r the t e t r a h e d r a l s i t e s . One observes that there are two t e t r a h e d r a l s i t e s and one octahedral s i t e per t i t a n i u m atom suggesting a l i m i t i n g composition of x=3 i n L i TiS„. One f a c t o r important i n the i n t e r c a l a t i o n process i s the p h y s i c a l s i z e of the v o i d at each s i t e . We consider a model where the s u l f u r atoms are represented by c l o s e packed hard spheres and determine the s i z e of the l a r g e s t hard sphere that can be i n s e r t e d i n t o the v o i d l o c a t e d at each type o of s i t e . A sphere of r a d i u s .72 A can be i n s e r t e d i n t o the octahedral o s i t e and a l i m i t i n g r a d i u s of .40 A i s found f o r the t e t r a h e d r a l s i t e s . 19 Figure 4b. The atomic structure of TiS2- Titanium atoms are depicted by s o l i d c i r c l e s and s u l f u r atoms by open c i r c l e s . The p r e f i x IT r e f e r s to the 1 layer unit c e l l with t r i g o n a l (T) symmetry. 20 Figure 5. A projection of the van der Waals gap of i n the 001 plane. Large open c i r c l e s represent s u l f u r atoms from the plane located at z = .75 and large dashed c i r c l e s represent s u l f u r atoms with z = .25. The small s o l i d c i r c l e s give the positions of octahedral s i t e s and the tr i a n g l e s give the positions of tetrahedral s i t e s 21 The r a d i u s of a L i i o n i s approximately equal to .60 A and that of a p L i atom approximately 1.5 A (Ashcroft and Mermin 1976). This suggests that occupation of the octahedral s i t e s by l i t h i u m i n i o n i c form i s favourable. (In the r i g i d band theory, the outer l i t h i u m e l e c t r o n i s donated to the host t i t a n i u m d band). However, simple models of t h i s type do not work as i t has been found that f o r Cu^TiS^, the copper r e s i d e s ++ i n t e t r a h e d r a l s i t e s at x = .7 (LeNagard et a l . 1975). The Cu i o n r a d i u s o i s .72 A. One must th e r e f o r e consider the T i S 2 l a t t i c e as not being completely r i g i d and a l l o w some l o c a l d i s t o r t i o n s of the l a t t i c e around the i n t e r c a l a t e d s p e c i e s . A p r i o r i assignments of l i t h i u m atom (ion) l o c a t i o n i n L i TiS„ cannot be made, x 2 I t i s of i n t e r e s t to c a l c u l a t e the va r i o u s i n t e r s i t e distances i n o TiS,,. The nearest neighbor o c t a h e d r a l - t e t r a h e d r a l d i s t a n c e i s 2.11 A and o the nearest neighbor t e t r a h e d r a l - t e t r a h e d r a l d i s t a n c e i s 2.47 A. This d i f f e r e n c e i s due to the t e t r a h e d r a l s i t e s being d i s p l a c e d by ± c/8 from the center of the gap. The octahedral-octahedral nearest neighbor distance o i s given by the a-axis of TiS,,, 3.407 A. These dista n c e s must be con-sid e r e d when one attempts to place l i t h i u m atoms (ions) on adjacent s i t e s . I f the s i t e s are too c l o s e , t h i s may be d i f f i c u l t to do because of the f i n i t e s i z e of the atoms ( i o n s ) . One would expect a l i m i t i n g composition of x=3 i n L i TiS provided that i t i s p o s s i b l e to f i l l a l l the s i t e s . X £. Much controversy e x i s t s i n the l i t e r a t u r e over the e l e c t r o n i c nature of T i S 2 . For in s t a n c e , Thompson et a l . (1975) c l a i m T i S 2 i s a semimetal. Other researchers b e l i e v e that TiS,, i s a semiconductor (e.g. Wilson 1977) and that i t s s e m i m e t a l l i c behaviour i s a t t r i b u t a b l e to the e f f e c t s of t i t a n i u m r i c h non-stoichiometry. The non—stoichiometry a r i s e s due to the 22 existence of a IT - T i ^ S 2 phase for 0<y<0.1 (Tronc and Moret. 1981). In f a c t , preparation of the stoichiometric compound i s d i f f i c u l t (Thompson et a l . 1975) and usu a l l y excess titanium atoms can be found i n the van der Waals gap. Vacancies i n the titanium layer and i n t e r l a y e r titanium are found even i n stoichiometric material i f made at temperatures above 800°C (Lieth and T e r h e l l 1977). The dependence of the e l e c t r o -chemical behaviour of L i / L i ^ T i S ^ c e l l s on TiS2 prepared i n various ways w i l l be discussed l a t e r i n the thes i s . 23 2.3 E a r l y Experimental Results oh L ^ T i S ^ E a r l y work by Whittingham (1976a) e s t a b l i s h e d that l i t h i u m could be i n t e r c a l a t e d to a composition of x=l i n L i ^ T i S ^ . Subsequently, Murphy et a l .. (1979) showed t h a t L ^ T i S ^ could i n t e r c a l a t e r e v e r s i b l y to x=2. More recent work (Dahn and Haering 1979) reported evidence f o r the existence, of L i ^ T i S ^ w i t h 0 <_ x _< 3. This t h e s i s i s s p e c i f i c a l l y concerned w i t h L i x T i S 2 f o r x <1. Appendix two i s devoted to a d i s c u s s i o n of recent r e s u l t s and t h e i r i n t e r p r e t a t i o n f o r L ^ T i S ^ w i t h x > 1. C h i a n e l l i (1976) performed o p t i c a l microscopic s t u d i e s on T i S ^ c r y s t a l s placed i n contact w i t h n-butyl l i t h i u m . He found that l i t h i u m entered the c r y s t a l a t the edges and d i f f u s e d inward p a r a l l e l to the l a y e r s . This could be observed by the appearance of a " f r o n t " of darker colour which appeared at the c r y s t a l edges and moved inward. No d i f f u s i o n perpendicular to the l a y e r s was observed. C h i a n e l l i a l s o noted that the s i n g l e c r y s t a l used developed many cracks and r i f t s as i n t e r c a l a t i o n proceeded, suggesting l a r g e s t r a i n s are present which cause the breakup of the s i n g l e c r y s t a l . In f a c t i t i s very d i f f i c u l t to prepare i n t e r c a l a t e d s i n g l e c r y s t a l s f o r t h i s reason. S i m i l a r o p t i c a l experiments performed i n our lab confirm t h i s behaviour (Chiu and Haering 1979). Thompson (1978) reported the d e t a i l e d V(x) and -3x/3V behaviour f o r 0 < x <_ 1 of L i / L i C 1 0 ^ , D i o x o l a n e / L i x T i S 2 c e l l s . Thompson's data i s shown i n Figure 6. The peaks i n -3x/8V near x = 1/9, 1/4 and 6/7 were a t t r i b u t e d by Thompson to phase t r a n s i t i o n s to s t a t e s where the i n t e r c a l a t e d l i t h i u m was ordered. Thompson c i t e d unpublished work by Jacobsen to c l a i m that the l i t h i u m atoms occupied octahedral s i t e s i n the van der Waals gap of the T i S 9 host. With reference to Figure 5, one observes that the octahedral Figure 6a). The quasi-open c i r c u i t voltage of a L i / L i x T i S 2 c e l l . b ) . The inverse d e r i v a t i v e of a. (After Thompson 1978) 25 s i t e s w i t h i n a s i n g l e l a y e r form a two dimensional t r i a n g u l a r l a t t i c e . Ordered compositions can be formed a t the compositions x = 1/3, 1/4, 1/7, 1/9 ... on the 2D t r i a n g u l a r l a t t i c e . The s t r u c t u r e at 6/7 was suggested to be due to the o r d e r i n g of unoccupied s i t e s on the 1/7 s t r u c t u r e . Other researchers (McKinnon 1980, Jacobsen et a l . 1979) have measured -9x/3V f o r L i / L i TiS c e l l s and the agreement w i t h Thompson's data i s good except X z. that the f e a t u r e near x = 6/7 i s much weaker. Thompson (1981) has a l s o measured 3V/3T f o r L i / L i x T i S 2 c e l l s . Minima i n 9V/3T were suggested to correspond to minima i n the p a r t i a l molar entropy of L i TiS„. Weak l o c a l minima were observed a t x = 1/4, 1/7 J x 2 and 1/12 as w e l l as a g l o b a l minimum near x = .3. This was taken as f u r t h e r evidence f o r the formation of ordered l i t h i u m s t r u c t u r e s . Thompson's i n t e r p r e t a t i o n of the data i s i n c o r r e c t s i n c e minima i n the entropy le a d to f e a t u r e s i n 9V/9T) x s i m i l a r to Figure 3. Minima i n 3V/9T) do not correspond to minima i n the entropy. This data was measured by s i n u s o i d a l l y o s c i l l a t i n g the c e l l temperature by ± 1 K and monitoring the corresponding v o l t a g e o s c i l l a t i o n s during a slow discharge. As w i l l be shown l a t e r (Chapter 4 ) , the presence of temperature gradients w i t h i n the c e l l leads to t h e r m o e l e c t r i c e f f e c t s which are l a r g e compared to 9V/9T) x-Thompson's r e s u l t s are suspect due to t h i s e f f e c t . Measurements of the s t r u c t u r e of L i T i S 9 by Whittingham (1976b) X /. revealed t h a t the c r y s t a l l a t t i c e expands as l i t h i u m i s i n t e r c a l a t e d . The c-axis expands by roughly 10% and the a-axis by 1.5% f o r x - 1 i n L i ^ T i S ^ . Whittingham's data i s shown i n Figure 7. The s t r u c t u r e remains IT (Figure 4b) as i n t e r c a l a t i o n proceeds. A study by C h i a n e l l i et a l . (1978) us i n g an e l e c t r o c h e m i c a l c e l l w i t h a b e r y l l i u m x-ray window found f i n e s t r u c t u r e i n the v a r i a t i o n of the spacing of the (101) planes as a 26 Figure 7. Dependence of the l a t t i c e parameters c and a on x i n L i x T i S 2 . C i r c l e s are c a x i s data points and t r i a n g l e s are a a x i s data p o i n t s . Note the d i f f e r e n c e i n the scal e s f o r a and c. ( A f t e r Whittingham 1976b) 27 f u n c t i o n of x. The data shows a plateau between .30 < x <^  .50 which i s suggested to be due to l i t h i u m o r d e r i n g . Unfortunately C h i a n e l l i et a l . (1978) only report the p o s i t i o n of t h i s s i n g l e Bragg peak as a f u n c t i o n of x, so i t i s not p o s s i b l e to a s c e r t a i n how the l a t t i c e constants are behaving independently. No s u p e r l a t t i c e peaks were detected. E l e c t r o c h e m i c a l l y i n t e r c a l a t e d " s i n g l e " c r y s t a l s were stud i e d using x-ray d i f f r a c t i o n by Hibma, (1980). Homogeneously i n t e r c a l a t e d c r y s t a l s were not obtained. Weak superstructures corresponding to 2a (x ~ 1/4) and /3a (x a 1/3) were reported a t room temperature although no data was shown. These superstructures were taken as evidence f o r l i t h i u m o r d e r i n g a t x = .25 and x = .33. X-rays give l i t t l e i n f o r m a t i o n about the l o c a t i o n of the l i t h i u m atoms due to the low x-ray s c a t t e r i n g f a c t o r of l i t h i u m . A neutron d i f f r a c t i o n experiment, however, i s s e n s i t i v e to l i t h i u m atom l o c a t i o n because the neutron s c a t t e r i n g c r o s s - s e c t i o n of l i t h i u m i s comparable to those of t i t a n i u m and s u l f u r . Dahn et a l . (1980) performed room temperature neutron d i f f r a c t i o n s t u d i e s on powder samples of T i S ^ , L i ^ T±S^, L i ^TiS^, L i ,,TiS„ and Li_TiS„. In a l l the l i t h i a t e d samples, the i n t e r c a l a t e d . D O L 1 Z l i t h i u m was found to occupy octahedral s i t e s and no evidence f o r l i t h i u m o r d ering was observed. D i f f r a c t i o n p r o f i l e s of the L i ^^Y-iS^ sample taken a t 106°K a l s o showed no evidence f o r l i t h i u m o rdering. The s t a t i s t i c s i n t h i s data were good enough to exclude the p o s s i b i l i t y of three dimensional long range l i t h i u m , order, but two dimensional order (uncorrelated s t a c k i n g of ordered i n t e r c a l a n t l a y e r s ) could not be excluded by the data. 7 L i NMR work by K l e i n b e r g et a l . (1982) on L i 3 3 T i S 2 showed that the / T a s u p e r l a t t i c e , expected i f l i t h i u m order occurs, was, not formed at temperatures above 150°K. K l e i n b e r g et a l . concluded t h a t the question of l i t h i u m o r d e r i n g could only be resolv e d w i t h b e t t e r low temperature d i f f r a c t i o n experiments. No good o p t i c a l data or r e s i s t i v i t y s t u d i e s on L i x T i S 2 have been reported to date to the author's knowledge. NMR s t u d i e s by S i l b e r n a g e l and Whittingham (1976) i n d i c a t e that the l i t h i u m donates most of i t s 2s e l e c t r o n to the TiS2 l a y e r s . Because of the s e m i m e t a l l i c (or semi-conducting) nature of TiS2> t h i s i m p l i e s an increase i n the d e n s i t y of s t a t e s a t the Fermi l e v e l as TiS2 i s i n t e r c a l a t e d w i t h l i t h i u m . T h e o r e t i c a l e f f o r t s to e x p l a i n the V(x) and -3x/8V r e s u l t s f o r L i / L i TiS_ c e l l s began s h o r t l y a f t e r Thompson's suggestion of l i t h i u m o r d e r i n g . These e a r l y t h e o r e t i c a l e f f o r t s w i l l now be discussed. 29 2 .4 L a t t i c e Gas Models for L i TiS„ x 2 L a t t i c e gas models for i n t e r c a l a t i o n systems have been proposed and studied (e.g. McKinnon 1980 , McKinnon 1981 , Berlinsky et a l . 1979 , Lee et a l . 1980 , Osorio and F a l i c o v 1982). These models assume that i n t e r c a l a t e d atoms are l o c a l i z e d at s p e c i f i c s i t e s i n the host l a t t i c e and that motion of the atoms from s i t e to s i t e does not a f f e c t the equilibrium thermodynamics. The neutron d i f f r a c t i o n r e s u l t s of Dahn et a l . (1980) can be f i t t e d w ell by assuming that the L i atoms are l o c a l i z e d i n the octahedral s i t e s of L i ^ T i S ^ . L a t t i c e gas models are expected to apply to L i TiS.. J x 2 The physical o r i g i n of the parameters i n the l a t t i c e gas model w i l l now be considered. Consider a l a t t i c e of equivalent s i t e s that i s i n i t i a l l y u n f i l l e d . The change i n i n t e r n a l energy which occurs when a s i n g l e s i t e i s f i l l e d i s termed the s i t e energy, E q . This energy re -presents the f a c t that i t i s more favourable, i n the case of L i ^ T i S ^ , for l i t h i u m atoms to reside i n the T i S 2 host than i t i s for them to reside i n l i t h i u m metal. (The i n t e r c a l a t e d l i t h i u m i s referred to as atoms due to the screening cloud that would form around an i o n ) . As the s i t e s on the l a t t i c e begin to f i l l , i t i s unreasonable to expect that f o r each a d d i t i o n a l atom added the-change i n i n t e r n a l energy i s E Q » This i s because the i n t e r c a l a t e d atoms i n t e r a c t , with the host by s t r a i n i n g the c r y s t a l and by modifying i t s e l e c t r o n i c properties and with each other through the screened Goulomb i n t e r a c t i o n . Attempts have been made to deal with these in t e r a c t i o n s . ( e . g . Safran 1980 , Safran and Hamann 198Q, McKinnon 1980 and McCanny 1979) although the r e s u l t s are not e a s i l y incorporated i n l a t t i c e gas models. What i s usually done i s that 30 the complicated i n t e r a c t i o n s , host mediated or otherwise, are replaced by two body i n t e r a c t i o n s , u „ , between intercalated atoms on s i t e s i and j which r e f l e c t the physics of the o r i g i n a l i n t e r a c t i o n s . Interactions which depend on the f i l l i n g of three or more s i t e s can a l s o be included. Thus when a s i t e i n a p a r t i a l l y occupied l a t t i c e i s f i l l e d , the change in energy i s the s i t e energy plus the sum of the i n t e r a c t i o n energies. A discussion of the o r i g i n of these i n t e r a c t i o n s i s given in the t h e s i s by McKinnon (1980) . E a r l y l a t t i c e gas models for L i TiS (Berlinsky et a l . 1979, X Zm Osorio and F a l i c o v 1982, McKinnon 1980) have assumed that the i n t e r c a l a t e d l i t h i u m atoms w i l l i nteract strongly with neighboring atoms i n the same layer and weakly with atoms in other l a y e r s . This assumption i s based on the f a c t that l i t h i u m atoms in d i f f e r e n t layers are separated by at l e a s t one S-Ti-S "sandwich". These authors neglected the i n t e r l a y e r i n t e r a c t i o n and therefore could model L i TiS„ as a two dimensional t r i a n g u l a r lattice., x 2 gas with i n t e r a c t i o n s . The e f f e c t s of c r y s t a l l a t t i c e expansion and band structure changes were not treated e x p l i c i t l y i n these models. The statistical.mechanics of the problem w i l l now be discussed. The i n t e r n a l energy, U, i s N U = T E x. + h J u..x.x. / . o x L i ] i ] i=l 1 >J ±*3 2.14 wher e x^ i s the occupation of s i t e i , x^ = 1 i f the s i t e i s f i l l e d , x. = 0 otherwise. u „ i s the two body i n t e r a c t i o n energy between atoms on s i t e s i and j and N i s the number of s i t e s on the l a t t i c e . The average occupation of the l a t t i c e , x, i s given by 31 1 N = « 1' x-N . . l 1=1 x  — > 2.15 It i s observed that t h i s Hamiltonian (equation 2.14) i s isomorphic to the Ising model for magnetic systems (Stanley 1971) . For the time being consider interactions only between nearest neighbors. Then N U = T E x . + h J ux.x. , 2.16 1=1 <1J> where <ij> denotes the sum over nearest neighbors and u i s the nearest neighbor i n t e r a c t i o n energy. For a given value of x, we must f i n d the set {x_^ } that minimizes the free energy. This problem cannot be solved exactly for the two dimensional triangular l a t t i c e gas except at x = 1/2, However, approximate solutions can be made and several w i l l now be discussed. The simplest approximation we can make i s that of "mean f i e l d " theory. In t h i s case i t i s assumed that the int e r c a l a t e d atoms remain d i s t r i b u t e d at random i n sp i t e of the i n t e r a c t i o n . The sum over nearest neighbor pairs i n equation 2.16 i s now easy; x^ and x^ . are replaced with x and one r e c a l l s that each s i t e on the tri a n g u l a r l a t t i c e has 6 nearest neighbors. Thus U = N ( E x + 3ux 2 ) . 2.17 o The entropy i s the logarithm of the number of ways of arranging xN atoms on N sites:, 32 S = k i n I NI i 2.18 where k i s Boltzmann's constant. Using 2.17, 2.18 and S t i r l i n g ' s approximation f o r l o g ( N i ) , the Helmholtz f r e e energy, F, f o r l a r g e N i s F = N { E x + 3ux 2 + kT( x l n ( x ) + ( l - x ) l n ( l - x ) ) } . 2.19a The chemical p o t e n t i a l , 9x/9y) and 9y/9T) are u = E + 6ux + kT l n ( - r ^ - ) , 2.19b o 1-x and The c e l l v o l t age i s r e l a t e d to the chemical p o t e n t i a l of the l i t h i u m atoms i n the cathode by equation 2.2. An attempt to f i t the observed V(x) and -9x/3V f o r L i / L i x T i S 2 c e l l s i s shown i n Figure 8. The parameters used are E q = -2.3 eV, u = 2.5 kT and kT = 25.7 meV. Since voltages are measured w i t h respect to the l i t h i u m anode, i t s chemical p o t e n t i a l i s set to zero i n equation 2.2. One observes a q u a l i t a t i v e agreement w i t h the V£x) data, although the -9x/9V r e s u l t shows none of the s t r u c t u r e evident i n the data. By making the mean f i e l d approximation, 33 34 i . e . s i t e s occupied randomly, we have not allowed the system to form ordered s t a t e s . A b e t t e r approximation i s needed. The t r i a n g u l a r l a t t i c e of s i t e s w i t h l a t t i c e constant a can be decomposed i n t o three i n t e r p e n e t r a t i n g s u b l a t t i c e s of l a t t i c e constant /Ta (Figure 9 ) . For a composition x = 1/3, the system can avoid the r e p u l s i v e nearest neighbor i n t e r a c t i o n e n t i r e l y i f a l l the atoms are placed on one s u b l a t t i c e . Order-disorder t r a n s i t i o n s are expected because at h i g h temperatures, kT » u, a l l three s u b l a t t i c e s w i l l be occupied e q u a l l y , w h i l e f o r kT « u the system w i l l order w i t h a l l atoms on one s u b l a t t i c e . A second neighbor r e p u l s i v e i n t e r a c t i o n i s needed before the ordered s t a t e at x = 1/4 w i t h l a t t i c e constant 2a w i l l form. Approximate s o l u t i o n s which d e a l w i t h the problem of o r d e r i n g have been discussed i n the l i t e r a t u r e . In Figure 10 the r e s u l t of a r e -n o r m a l i z a t i o n group c a l c u l a t i o n ( B e r l i n s k y et a l . 1979) f o r V(x) and -3x/3V w i t h u - 4kT i s shown. The l a r g e peak i n -3x/3V near x = 1/4 i s due to a phase t r a n s i t i o n to the ordered s t a t e w i t h one of the s u b l a t t i c e s occupied. The drop i n V(x) and the minimum i n -3x/3V at x = 1/3 come about due to the c ost i n energy, j_ 3u, to add each subsequent atom to the system when one s u b l a t t i c e i s already f i l l e d . -3x/3V i s symmetric about x = 1/2 due to the p a r t i c l e - h o l e symmetry inherent i n l a t t i c e gas models w i t h two body i n t e r a c t i o n s . For x £> .3, the r e s u l t s of the c a l c u l a t i o n resembles Thompson's L i ^ T i S ^ data (Figure 6). However the l a r g e drop i n voltage and the minimum i n -3x/3V at x = 1/3 are absent i n the data. Minima i n -3x/3V are always expected at ordered compositions ( B e r l i n s k y et a l . 1979). I t i s found t h a t e f f o r t s to decrease the magnitude of the v o l t a g e drop at x =. 1/3, by making u s m a l l e r , a l s o decrease the s i z e of the peak i n -3x/3'V at x = 1/4. 35 o o # o o o o o o 1 Figure 9. Decomposition of a triangular l a t t i c e with la t t i c e constant a into three interpenetrating sublattices with la t t i c e constant /3a. T 1 1 1 1 1 1 1 r X in LixTiS2 Figure 10. (a) Voltage V and (b) i n v e r s e d e r i v a t i v e -Ax/AV versus x f o r a t r i a n g u l a r l a t t i c e gas w i t h u=4kT c a l c u l a t e d using r e n o r m a l i z a t i o n group techniques. The p o i n t s i n (a) are the r e s u l t s of Monte Carlo c a l c u l a t i o n s , again f o r the same parameters. From B e r l i n s k y et a l . (1979) 37 The symmetry about x = 1/2 e x h i b i t e d by the c a l c u l a t i o n can be removed by the i n c l u s i o n of three body i n t e r a c t i o n s . This was done by B e r l i n s k y et a l . (1979) and Osorio and F a l i c o v (1982). The former found that Thompson's V(x) curve could be f i t q u a l i t a t i v e l y without c o n s i d e r i n g l i t h i u m o r d e r i n g by a s u i t a b l e choice of the three body i n t e r a c t i o n . However the c a l c u l a t e d -9x/9V showed none of the f i n e r f e a t u r e s of the data i n Figure 6. Osorio and F a l i c o v (1982) , u s i n g the c l u s t e r v a r i a t i o n approximation, considered the e f f e c t of the three body i n t e r a c t i o n when l i t h i u m o r d e r i n g was s t i l l allowed to occur. They found that although the symmetry i n Figure 10 could be removed, the minimum i n -9x/9V at x = 1/3 was always present provided the system formed an ordered s t a t e . 38 2.5 Summary of Early Work on Ll^TlS^ Thompson (1978) measured structure In -3x/3V for Li/Li xTiS2 c e l l s . This structure was suggested to be due to the formation of ordered states at compositions x = 1/4, 1/9 and 6/7. Other researchers cite evidence which supports (.e.g. Chianelli et a l . 1978, Thompson 1981, Hibma 1980) or disagrees with this hypothesis (e.g. Dahn et a l . 1980, Kleinberg et a l . 1982). Theoretical studies have confirmed that peaks are to be expected in -3x/3V at phase transitions and that minima are expected at ordered compositions. The lack of minima in the Thompson data at x = 1/3, 1/4, 1/7 ... suggests that lithium ordering on the two dimensional triangular la t t i c e of octahedral sites i s not occurring. A minimum near x = .16 may be present in the data, although this is not commensurate with the ordered states on the 2D triangular l a t t i c e . The inclusion of three body interactions in the calculation removes the symmetry of -3x/3V but does not improve the f i t to Thompson's data. This was roughly the status of L i TiS„ when the work about to be J x 2 described in the thesis was started. It was clear that better data on Li^TiS^ was needed and also that two dimensional triangular l a t t i c e gas models had l i t t l e hope of explaining the data. Both of these points were addressed and the progress that was made w i l l now be reported. 39 P A R T E X P E R I M E N T A L METHODS I N T E R C A L A T I O N S Y S T E M S ; I I F OR S T U D Y I N G L I T H I U M A P P L I C A T I O N TO L I X T I S 2 40 C H A P T E R ' T H R E E THE MEASUREMENT OF V(x) AND -9x/9V 3.1 Preparation of TiS^ T i S 2 i s most e a s i l y prepared by d i r e c t reaction of the elements i n sealed ampoules. The elements can either be mixed s t o i c h i o m e t r i c a l l y or some excess s u l f u r , t y p i c a l l y .5% by weight, can be added. There are no stable T i S 2 + z phases for z<l, so the addit i o n of excess s u l f u r helps to ensure the proper stoichiometry of the product (Thompson et a l . 1975, L i e t h and T e r h e l l 1977). Because TiS^ i s stable below 500°C, samples prepared with excess s u l f u r must be quenched to room temperature to avoid formation of the t r i s u l f i d e . The e f f e c t of T i S 2 synthesis conditions on the electrochemical behaviour of L i / L i TiS„ c e l l s has been reported previously (Dahn 1980). In t h i s study, the d e t a i l s of the synthesis were also found to be important. Stoichiometric mixtures of highly pure titanium (at l e a s t 99.9%) and sulf u r (at l e a s t 99.999%) powders were sealed in evacuated quartz ampoules. The sample was then heated to 250°C ( t y p i c a l l y ) and then the temperature was raised at 20°C/hr to between 550°C and 850°C (depending on the s p e c i f i c batch) where i t was held f o r approximately two days. Samples containing excess s u l f u r were quenched to room,.temperature by a d i r e c t transfer of the hot ampoule from the furnace to a water bath. It was found that higher 41 growth temperatures and excess s u l f u r helped to Increase the c r y s t a l l i t e s i z e . Many batches of TiS2 were prepared for these studies and a l i s t of relevant parameters for each batch discussed in the t h e s i s i s given in Table 2. The e f f e c t of excess titanium on the l a t t i c e parameters of-.Tl^ has been measured by Thompson et a l . (1975) who found that the c-axis increases as y increases. The c-axis expansion caused by i n t e r l a y e r titanium i s about 10 times l e s s than the expansion caused by an equal amount of in t e r c a l a t e d l i t h i u m . The i n t e r l a y e r titanium i s involved i n the bonding between the molecular "sandwiches" and helps to pin them together (Thompson et a l . 1975). L a t t i c e parameter measurements were made on our samples and were found to be consistent from sample to sample. o o In a l l cases, a = 3.408 ± .001 A and c = 5.700 + .001 A. E a r l i e r neutron d i f f r a c t i o n experiments on s i m i l a r samples (Dahn et a l . 1980) yielded o o a = 3.407 9 ± .004 A and c = 5.6989 ± .0006 A for T i S ^ Thompson et a l . O O (1975) found a = 3.4073 ± .0002 A and c = 5.6953 ± .0002 A. According to O Thompson et a l . (1975) c = .5.700 A corresponds to y * 0.02 i n T ^ S 2. However, we estimate maximum errors in stoichiometry, due to unc e r t a i n t i e s in weighing the powders and loading the ampoule, of y ~ .005 for our samples, i n c o n f l i c t with the data of Thompson et a l . . The growth temperature of 600°C used by Thompson et a l . i s s u b s t a n t i a l l y lower than the 750°C t y p i c a l l y used here. (Table 2). It i s possible that some titanium atoms have moved from the S-Ti-S "sandwich" into the van der Waals gaps as suggested by L i e t h and T e r h e l l (1977) for TIS^ prepared at high temperatures, causing a minor expansion of the c-axis. It i s also possible that the data reported by Thompson et . a l . . (197.5) could be i n error . The magnitude of the v a r i a t i o n i n,c, A c , measured from sample to sample 42 Table 2 The growth conditions and properties of the T1S2 used in the experiments described in this thesis Batch number Growth temp. (C°) Excess s u l f u r Average c r y s t a l l i t e s i z e (ym) E l e c t r o l y t e c o - i n t e r c a l a -t i o n C e l l s using t h i s material 2 P63 800 No 15 moderate JD-63 3p38 750 No 15 moderate f JD-211, JD-218 JX-12 JX-13 JX-14 JX-15 JX-17 > -4 P100 750 No 2 severe JD-212 5 P96 750 Yes 20 moderate JD-222 5 p l l l 850 Yes 20 minor JD-234 5 P151 750 Yes 20 moderate JD-235 JN-3 JX-32 . 6 P13 750 Yes 20 moderate JN-7 RM9* 800 No 15 moderate JD-156 L-038** US9*** sin g l e X - t a l transport 40. minor JD-49 Grown by Ross McKinnon C e l l prepared and tested by L. Abello Grown by U l r i c h Sacken (Ac< .001 A) i s consistent with our estimates of the errors in sample stoichiometry, y ~ .005. 44 3.2 C e l l Construction and Components The construction of c e l l s for V(x) and -9x/9V measurements has been described i n d e t a i l previously (Dahn 1980) and therefore w i l l be b r i e f l y summarized here. Cathodes were prepared by spreading a s l u r r y of the cathode material (TiS^) and propylene g l y c o l i n a th i n layer on a n i c k e l substrate. The cathode was then baked at 200°C under flowing argon u n t i l dry. By weighing the substrate before and a f t e r a p p l i c a t i o n of the cathode powder, an accurate determination of the cathode mass was obtained. 2 Cathodes were t y p i c a l l y 2 cm i n area with a density of cathode material 2 between 5 and 10 mg/cm . Nickel substrates were used due to t h e i r low capacity for a l l o y i n g with l i t h i u m as w i l l be discussed l a t e r . The e l e c t r o l y t e i n these c e l l s was either IM UiAsFg/PC or IM l i C l O ^ / P C . The propylene carbonate, PC, was p u r i f i e d as described elsewhere (Dahn 1980), and the anhydrous LiAsF^ was obtained from United States Steel A g r i -chemicals. Hydrous l i t h i u m perchlorate was vacuum dried at 130 ?C and stored under argon u n t i l use. No dif f e r e n c e s i n electrochemical behaviour were observed between c e l l s d i f f e r i n g only i n the choice of e l e c t r o l y t e anion. The separators used were Celgard #3501 or #2500 microporous f i l m s . #3501 separators were cut into one inch square sheets, given two soaks in p u r i f i e d PC (to remove a surfactant used as a wetting agent) under argon and then f i n a l l y soaked i n the e l e c t r o l y t e . #2500 separators, which lack the surfactant, were cut to s i z e , then wetted d i r e c t l y with the e l e c t r o l y t e in a s p e c i a l l y designed "bomb" under a pressure of 80 p s i . Electrochemical c e l l s were constructed as shown in Figure 11. Stainless s t e e l flanges were greased with s i l i c o n high vacuum grease which 45 Figure 11. Exploded view of an electrochemical c e l l . 46 was found to be inert with respect to the l i t h i u m and e l e c t r o l y t e . This was done to protect the s t a i n l e s s s t e e l . The cathode, two separators and a piece of l i t h i u m metal f o i l were stacked as shown i n Figure 11 and placed in the c e l l which was sealed by the 0-ring when tightened. E l e c t r i c a l connection to the l i t h i u m anode and the cathode i s made by the flanges. Pressures of approximately 100 p s i (Clayton and Haering 1982) are needed to keep the cathode p a r t i c l e s e l e c t r i c a l l y connected to the substrate and can be obtained e a s i l y in c e l l s of t h i s type. 47 3.3 Measurement of V(x) and -3x/3V When a c e l l i s discharged at constant current, the amount of charge passed i s simply the current m u l t i p l i e d by the time. By knowing the mass of the i n t e r c a l a t i o n cathode i t i s a simple matter to c a l c u l a t e x. One obtains = -• t M I X 96,500 z m ' 3 , 1 where I i s the e l e c t r o n i c current, t i s the time the current has been flowing, z i s the valence of the metal ion being transported by the e l e c t r o l y t e , m i s the cathode mass and M i s the molecular weight of the cathode,material. It i s therefore possible, in theory, to control the amount of intercalated species in the host by simply discharging the c e l l at constant current for a s p e c i f i e d time. This i s c a l l e d a coulometric t i t r a t i o n by chemists. In p r a c t i c e , equation 3.1 does not s t r i c t l y hold due to the presence of various sinks for l i t h i u m . Reactions between the l i t h i u m and the n i c k e l substrate or the e l e c t r o l y t e are two examples of these processes which are c a l l e d "side reactions". Another problem i s that of "cathode u t i l i z a t i o n " ; the e n t i r e cathode may not be e l e c t r i c a l l y connected to the substrate. However, cathodes that have been properly prepared, when cycled i n c e l l s under s u f f i c i e n t pressure, show cathode u t i l i z a t i o n s near 100%. Figure 12 shows the experimental voltage versus charge curve for a c e l l containing no cathode material, j u s t a bare n i c k e l substrate.of 2 area 2 cm . This data shows that n i c k e l substrates have capacity of 2 order 50 milliCoulombs per cm on the f i r s t discharge to 0.3 v o l t s . 48 3 TIME (hrs) Figure 12. Several cycles of a Li/LiClO^, PC/Ni electro-chemical c e l l . The constant current cycles were measured at a current of 10 uA 49 Recharge and subsequent discharge c a p a c i t y i s reduced by a f a c t o r of 6. Note that a L i TiS cathode has a c a p a c i t y of .86 G/mg f o r Ax=l. For X L. reasonable mass cathodes, the e f f e c t of the n i c k e l substrate i s small. 2 Side r e a c t i o n c u r r e n t s can u s u a l l y be maintained at l e v e l s below. ..l]ia/cm f o r v o l t a g e s between 1.5 and 2.7 v o l t s w i t h c a r e f u l e l e c t r o l y t e p u r i f i c a t i o n and c e l l assembly (Sacken 1979). In p r a c t i c e one u s u a l l y looks f o r an e a s i l y r e c o g n i z a b l e f e a t u r e on the V(x) r e l a t i o n where x i s known. Then one can normalize the constant current data to t h i s v a l u e of x to c o r r e c t f o r the cathode u t i l i z a t i o n problem. For L i / L i TiS„ c e l l s i t i s known that the r a p i d drop i n v o l t a g e X z (Figures 2, 6, A2.1), which begins near 1.8 v o l t s , occurs at x=l. This has been e s t a b l i s h e d by chemical a n a l y s i s (Dines 1975) as w e l l as by the r e s u l t of many el e c t r o c h e m i c a l c e l l t e s t s which give A x ~ l on the f i r s t discharge to 1.8 v o l t s f o r L i / L i TiS c e l l s . A p l o t of c e l l v o l t a g e as a X z. f u n c t i o n of time can be converted to a measurement of V(x) s i n c e Ax i s p r o p o r t i o n a l to At when the c e l l i s c y c l i n g at constant c u r r e n t . The v o l t a g e measured i n t h i s way i s not the e q u i l i b r i u m open c i r c u i t v o l t a g e corresponding t o the composition x. This i s because of the r e s i s t i v e l o s s e s i n the c e l l due to the anode, e l e c t r o l y t e and surface processes as w e l l as the overvoltages due to s o l i d s t a t e d i f f u s i o n of the l i t h i u m i n the host. The overvoltage, . n. , due to the r e s i s t i v e l o s s e s , modeled by a s e r i e s r e s i s t a n c e R, i s simply n„ = I R 3.2 50 The e f f e c t s of s o l i d s t a t e d i f f u s i o n have been t r e a t e d by McKinnon (1980). I t i s shown t h a t , f o r s i n g l e phase i n t e r c a l a t i o n , a f t e r the current has been f l o w i n g f o r a time, t>>£2/D, the overvoltage due to d i f f u s i o n , f| , i n a layered host i s V X s ) = * D Q "rfx=x 3 ' 3 ^o s In t h i s expression, £ i s the p a r t i c l e r a d i u s , D i s the d i f f u s i o n constant, Q q i s the charge corresponding to x=l and the d e r i v a t i v e i s evaluated at x=x , where x i s the l i t h i u m content at the surface of the p a r t i c l e s , s s This i s again a term l i n e a r i n I , so we can speak of a d i f f u s i o n " r e s i s t a n c e " . 2 A t y p i c a l value of the s e r i e s r e s i s t a n c e R f o r c e l l s w i t h a 2 cm cathode -9 2 i s 200. Q. The d i f f u s i o n " r e s i s t a n c e " , f o r £=10ym, D=10 cm /sec, Q Q=10 c o u l . and 9V/9x = 1 v o l t i s 500,. For a current of 50 uA, one expects T) = 10 mV and ri = 2.5 mV. D e t a i l e d measurements of V(x) should be performed at low current to minimize these overvoltages. However, w i t h 1=50 yA and Q =10' C, a discharge time of 2x10"* seconds i s needed. The experimenter must trade o f f time f o r accuracy. In cases l i k e L i ^ T i S 2 where there appears to be no h y s t e r e s i s between charge and discharge, the e q u i l i b r i u m V(x) r e l a t i o n can be approximated by averaging the charge and discharge. Constant current c y c l i n g t e s t s were performed on constant current c y c l e r s b u i l t i n the U.B.C. physics e l e c t r o n i c s shop. The inverse d e r i v a t i v e , -9x/3V, can be obtained from the V(x) data i n two ways. The data can be d i f f e r e n t i a t e d by hand using f i n i t e d i f f e r e n c e s and -9x/9V obtained ( t h i s procedure i s c a l l e d numeric v o l t -ammetry). A more elegant way i s to monitor the c e l l v o l t a g e as a f u n c t i o n of time w i t h a s e n s i t i v e computer c o n t r o l l e d voltmeter. Voltage 51 measurements can be taken at f i x e d time i n t e r v a l s and the d e r i v a t i v e , At/AV, c a l c u l a t e d by the computer and then output to a X-Y recorder. Since At/AV i s p r o p o r t i o n a l to Ax/AV at constant c u r r e n t , one obtains a p l o t of Ax/AV as the c e l l i s charging or d i s c h a r g i n g . A system such as t h i s has r e c e n t l y been designed and b u i l t by the U.B.C. physics e l e c t r o n i c s shop and i t s o p e r a t i o n a l d e t a i l s are given elsewhere (Johnson 1982). This u n i t w i l l be c a l l e d the U.B.C. 3t/3V u n i t . Inverse d e r i v a t i v e s taken i n t h i s way are c a l l e d constant current voltammograms. Because the A/D converter on t h i s instrument has a r e s o l u t i o n of 45 ]iV, i t was found t h a t the c e l l s had to be temperature c o n t r o l l e d f o r optimum system performance. This was done by immersing the c e l l s i n t o the f l u i d r e s e r v o i r of a Haake F3K temperature c o n t r o l l e r which contained e l e c t r i c a l l y i n s u l a t i n g S h e l l D i a l a ax transformer o i l . Another way t o obta i n -3x/3V i s t o use l i n e a r sweep voltammetry. In t h i s method the c e l l v o l t a g e i s swept l i n e a r l y i n time and the current supplied to the c e l l i s monitored. The current i s 1 " dt dV dt % dV dt ' where i s the charge corresponding to x=l. We see that f o r constant dV/dt, the current i s d i r e c t l y p r o p o r t i o n a l to dx/dV. In d e r i v i n g equation 3.4, the e f f e c t s of s e r i e s r e s i s t a n c e and d i f f u s i o n have been neglected. These e f f e c t s have been tre a t e d p r e v i o u s l y f o r the case of s i n g l e phase i n t e r c a l a t i o n (McKinnon 1980) . I t i s shown th a t as dV/dt ->• 0, the current measured approaches the value given by 3.4. The case of c o e x i s t i n g phases at a f i r s t order phase t r a n s i t i o n i s more complicated and has a l s o been treated p r e v i o u s l y (Dahn and Haering 1981a, b ) . I t i s shown that "notches" in the l i n e a r sweep voltammogram correspond to the posit i o n s of the equilibrium voltages of f i r s t order phase t r a n s i t i o n s . A Princeton Applied Research (PAR) model 173 potentiostat/galvanostat and a PAR 175 u n i v e r s a l programmer were used to perform these experiments. 3.4 E l e c t r o l y t e C o - I n t e r c a l a t i o n I n t e r c a l a t i o n of l i t h i u m ions along w i t h t h e i r PC s o l v a t i o n clouds can occur i n L i TiS cathodes (Whittingham 1976b, Dahn 1980). This e f f e c t , X 6~ when present, leads to a degradation of a p o r t i o n of the cathode m a t e r i a l . Thompson (1978) r e p o r t s that by the use of l e s s p o l a r s o l v e n t s l i k e d i oxolane, e l e c t r o l y t e c o - i n t e r c a l a t i o n can be avoided. However, dioxolane i s more v o l a t i l e and more d i f f i c u l t to handle than PC and can lead to explosions i n e l e c t r o c h e m i c a l c e l l s . Although the e l e c t r o l y t e c o - i n t e r -c a l a t i o n process i s not w e l l understood, i t appears that once the PC i n t e r c a l a t e d m a t e r i a l i s formed i t subsequently remains e l e c t r o c h e m i c a l l y i n e r t f o r v o l t a g e s above 1.6 v o l t s . Therefore u s e f u l information can be gained from L i / L i TiS„ c e l l s u s i n g PC based e l e c t r o l y t e s . This p o i n t w i l l now be addressed i n d e t a i l . Figure 13a shows the f i r s t discharge of c e l l JD-63, which contained 5.8 mg of 2p63 T i S ^ , at a constant current of 275 yA. x i s determined from the weight of the cathode and the current u s i n g equation 3.1. A discharge of c e l l JD-235, t y p i f y i n g V(x) of L i / L i TiS c e l l s (see s e c t i o n X £. 3.5), i s p l o t t e d i n Figure 13a f o r comparison. The JJJi-63 discharge extends only to x ~ .44 at 1.7 v o l t s and has a shape that i s d i f f e r e n t from the JD-235 discharge. The JD-63 behaviour i s due t o the c o - i n t e r c a l a t i o n of e l e c t r o l y t e i n t o a p o r t i o n of the cathode m a t e r i a l on the f i r s t discharge which proceeds i n p a r a l l e l w i t h normal l i t h i u m i n t e r c a l a t i o n . The co-i n t e r c a l a t i o n process gives r i s e to the f l a t p o r t i o n of the JD-63 discharge curve near 2.32 v o l t s . The small c a p a c i t y (Ax ~ .44) i s due to the f a c t that c o - i n t e r c a l a t e d PC, molecules occupy more than one o c t a h e d r a l s i t e i n the host. The f i r s t and subsequent c y c l e s of JD-63 are shown i n J i_L 2 .4 .6 X in L i x T i S 2 .8 1.0 3 4 TIME (hrs) 13. a) F i r s t discharge of c e l l JD-63 (dashed curve) which contained 5.8 mg of 2p63 TiS2, at a constant current of 275 uA. The s o l i d curve i s a discharge of c e l l JD-235 from Figure 17, plotted here for comparison. b) F i r s t and subsequent cycles of c e l l JD-63 at a constant current of 275 uA. The time corresponding to Ax=l i s depicted i n the fig u r e (see t e x t ) . 55 Figure 13b. The subsequent c y c l e s e x h i b i t V(x) behaviour s i m i l a r to the JD-235 data i n Figure 13a i f normalized to x=l . C e l l JD-63 behaves, a f t e r the f i r s t d ischarge, as i f i t had a f u l l y u t i l i z e d 1.2 mg TiS^ cathode. This i s taken as evidence that the PCt. c o - i n t e r c a l a t e d m a t e r i a l , once formed, remains e l e c t r o c h e m i c a l l y i n e r t above 1.6 v o l t s . The c o - i n t e r c a l a -t i o n of the e l e c t r o l y t e causes a l a r g e expansion p a r a l l e l to the c - a x i s ; o the r e s u l t i n g c - a x i s measured to be 17.8 A (Dahn 1980). C e l l s constructed w i t h t h i s same cathode m a t e r i a l (2p63) have been given what we c a l l the "quick d i s c h a r g e " by holding them f i x e d a t 1.6 v o l t s w i t h a PAR p o t e n t i o s t a t immediately a f t e r assembly. Currents of about 2mA/mg flow i n i t i a l l y and a f t e r a few hours decay to a few yA/mg as the c e l l e q u i l i b r a t e s . The charge t h a t has passed during discharge i s obtained by i n t e g r a t i n g the curr e n t and one obtains Ax ~ .95 by weight ( t y p i c a l l y ) f o r c e l l s discharged i n t h i s manner. X-rays (Chapter 5) taken of c e l l s given the "quick d i s c h a r g e " show almost no e l e c t r o l y t e c o - i n t e r c a l a t i o n . When the c e l l s are subsequently c y c l e d between 1.7 and 2.7 v o l t s , very l i t t l e e l e c t r o l y t e c o - i n t e r c a l a t i o n occurs and e f f e c t i v e cathode u t i l i z a t i o n s of - 90% are obtained. I t was found t h a t d i f f e r e n t batches of TiS2 behaved d i f f e r e n t l y w i t h respect to the e l e c t r o l y t e c o - i n t e r c a l a t i o n phenomenon. Batches of TIS^ that were s i m i l a r to batch 2p63 where a quick discharge could e f f e c t i v e l y defeat the e l e c t r o l y t e c o - i n t e r c a l a t i o n are described as having a moderate c o - i n t e r c a l a t i o n problem i n Table 2. Other batches, w i t h smaller p a r t i c l e s i z e , were found to c o - i n t e r c a l a t e l a r g e f r a c t i o n s of e l e c t r o l y t e even on quick discharge and i n these cases the problem i s described as severe. Other batches, t y p i c a l l y made at higher temperatures, showed l i t t l e evidence f o r PC c o - i n t e r c a l a t i o n r e g a r d l e s s of how the c e l l s were c y c l e d . 56 C e l l JD-49 (Figure A2.1) i s an example of a case where the e l e c t r o l y t e c o - i n t e r c a l a t i o n problem i s minor. Thompson et a l . (1975) have shown that the presence of i n t e r l a y e r titanium dramatically reduces the rate of i n t e r c a l a t i o n of large molecules in TiS2« The, i n t e r l a y e r titanium increases the strength of the bonds between layers making i t hard to i n s e r t the large molecules due to t h e i r s i z e . These researchers claim that the presence of .1% excess titanium was s u f f i c i e n t to prevent the i n t e r c a l a t i o n of 2, 4, 6±r;i- methylpyr idine which i n t e r c a l a t e s r e a d i l y in the stoichiometric compound. Because the stoichiometry of T i S 2 i s hard to c o n t r o l , i t i s reasonable to expect the PC c o - i n t e r c a l a t i o n process to be sample dependent. In t h i s context, the "quick discharge" method can be understood. The PC c o - i n t e r c a l a t i o n process, involving large molecules and large l a t t i c e expansions,is probably k i n e t i c a l l y slower.than l i t h i u m i n t e r c a l a t i o n . By discharging quickly, the i n t e r l a y e r s i t e s are f i l l e d with l i t h i u m before the PC c o - i n t e r c a l a t i o n process can get started. Subsequent recharges can never f u l l y remove the l i t h i u m from the cathode, (equation 2.19b) so i f the l i t h i u m increases the bonding between layers somewhat, e l e c t r o l y t e c o - i n t e r c a l a t i o n i s i n h i b i t e d . The nature of the e l e c t r o l y t e co-intercalated material i s unknown. Either a layer of PC co-intercalated material i s present on the surface of each grain, or perhaps some grains are i n t a c t and others are t o t a l l y degraded. 57 3.5 R e s u l t s and D i s c u s s i o n In F i g ure 14, V(x) and -9x/9V f o r c e l l JD-211 are shown. This c e l l contained 2.06 mg of 3p38 TiS2 (Table 2). The c e l l was c y c l e d s e v e r a l times at 150 yA and then a slow c y c l e at 5 yA between 1.75 and 2.55 v o l t s was measured. The discharge time was 80.1 hrs which corresponds to 400.5 yA-hrs or Ax = .82 by weight f o r t h i s cathode. The charge time was 76.03 hrs corresponding to 380.5 yA-hrs or Ax = .80 by weight. This small asymmetry i s most probably due to side r e a c t i o n s . The v o l t a g e versus time data were d i g i t i z e d by:hand and normalized to x=l at 1.800 v o l t s to compensate f o r cathode u t i l i z a t i o n . The numeric voltammogram was c a l c u l a t e d and i s p l o t t e d as -9x/9V vs x i n Figure 14. The e r r o r bars i n the f i g u r e represent the d i f f e r e n c e s between charge and discharge. The data p o i n t s have been placed at the average of charge and discharge. This data i s i n good agreement w i t h the data reported by Thompson (1978) (Figure 6). I t can be concluded t h a t e l e c t r o l y t e c o - i n t e r c a l a t i o n , which i s present i n t h i s m a t e r i a l but not i n the dioxolane based c e l l s of Thompson, does not a f f e c t the e l e c t r o c h e m i c a l behaviour of the L i / L i TiS„ x 2 system. The p o s i t i o n of the l a r g e peak i n -9x/9V i s at x = .24 ± .02 and V = 2.327 ± .005 v o l t s . In Figure 15, the constant c u r r e n t voltammogram, measured w i t h the U.B.C. 9t/9V u n i t , f o r c e l l JD-235 i s shown. The data was taken at a current of 40 yA which lead t o an 82 hr discharge time f o r t h i s cathode. The p o i n t s i n the f i g u r e f o r charge and discharge are those obtained a f t e r t h e quasi-continuous output from the 9t/3V u n i t was d i g i t i z e d , stored i n the U.B.C. computer, normalized to x=l at 1.800 v o l t s and then 58 .2 .4 .6 X in L i x T i S 2 8 1.0 Figure 14. V(x) and -9x/9V f o r c e l l JD-211 measured at a current of 5.0 uA. The c e l l contained 2.06 mg of 3p38 TiS . 5 9 4 . 0 | r 1 . 8 2 . 0 2 . 2 2 . 4 2 . 6 Vol ts Figure 15. -3x/3V for c e l l JD-235 measured at a current of 40 uA. The c e l l contained 17.6 mg of 5pl51 T i S 2 . The diamonds are the charge and the t r i a n g l e s are the discharge. 60 p l o t t e d . This data was then, i n t e g r a t e d n u m e r i c a l l y u s i n g the t r a p e -z o i d a l r u l e and -3x/3V vs x and V.(x) obtained. These r e s u l t s are p l o t t e d i n Figures 16 and 17 r e s p e c t i v e l y . The agreement between charge and discharge i s evident i n the f i g u r e s . There i s no evidence of a sharp fea t u r e at x=6/7 i n con t r a s t w i t h the r e s u l t s of Thompson (1978). The l a r g e peak i n -3x/3V f o r t h i s m a t e r i a l (5pl51) i s lo c a t e d at 2.328 ± .005 v o l t s and x = .215 ± .01. The presence of a w e l l defined minimum near x = .11 and V = 2.362 ± .005 and a second peak near x = .07 i s e s t a b l i s h e d by the data. -3x/3V vs x and V'(x) f o r cell.JD-234 ( 5 p l l l T i S 2 ) are shown i n Figures 18 and 1 9 . r e s p e c t i v e l y . This data was obtained i n the same way as the JD-235 data, except t h a t the current was 12.5 uA which lead to a discharge time of 128 hrs f o r t h i s cathode. This m a t e r i a l e x h i b i t e d very l i t t l e e l e c t r o l y t e c o - i n t e r c a l a t i o n and the l a r g e peak in.-3x/3V occurs a t x = .21 ± .01 and V = 2.318 ± .002 v o l t s . The small peak near x = .07 i s not r e s o l v e d and appears as a shoulder on the side of the l a r g e peak. This could be due t o the "smearing" of the f e a t u r e s by the d i f f u s i o n overvoltage n ^ . Because the time taken f o r a discharge (128 hrs) i s longer than the times taken to c o l l e c t the JD-235 data and the c r y s t a l l i t e s i z e of the two batches are s i m i l a r , t h i s explanation r e q u i r e s a smaller l i t h i u m d i f f u s i o n constant f o r the JD-234 m a t e r i a l . Measurements taken at twice the current showed l i t t l e change i n the f e a t u r e s e x h i b i t e d i n Figures 18 and 19. This suggests that the fea t u r e near x = ..07 i s weaker i n m a t e r i a l where e l e c t r o l y t e c o - i n t e r c a l a t i o n i s minor. This point w i l l be addressed i n Chapter 7, Section 7.5.4, where i t w i l l be :shown t h a t i n t e r -l a y e r t i t a n i u m would cause a r e d u c t i o n i n t h i s f e a t u r e and a s h i f t of the 61 Figure 16. -9x/9V vs x for c e l l JD-235 obtained from the data i n Figure 15 as described i n the text. The t r i a n g l e s are the discharge and the diamonds are the charge. Figure 17. V(x) for c e l l JD-235 obtained from the data i n Figure 15 as described i n the text. The t r i a n g l e s are the discharge and the diamonds are the charge. 63 4 . 0 1 — r 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1/0 X in L i x T i S 2 Figure 18. -9x/9V vs x for c e l l JD-234 measured at 12.5 uA. This c e l l contained 6.3 mg of 5 p l l l T1S2. The t r i a n g l e s are the discharge and the diamonds are the charge. Figure 19. V(x) f o r c e l l JD-234 obtained from the data i n Figure 18. The t r i a n g l e s are the discharge and the diamonds are the charge. 65 peak i n -3x/8V to lower v o l t a g e s , as i s observed. A l i n e a r sweep voltammogram of c e l l JD-212 i s included i n Figure 20. The measured cur r e n t at a sweep r a t e of 3.83 uV/sec. i s p l o t t e d as a f u n c t i o n of c e l l v o l t a g e . This data looks very s i m i l a r to the data p l o t t e d i n Figure 15, except f o r the general r e d u c t i o n i n charge current w i t h respect to discharge current due to a side r e a c t i o n i n t h i s p a r t i c u l a r c e l l . The double peaked character of -3x/9V at high v o l t a g e ( s m a l l x) i s again noted. The general f e a t u r e s observed i n -&x/8V are independent of the measurement technique used. These measurements i n d i c a t e that the p o s i t i o n of the l a r g e peak to -8x/9V may not be f i x e d at x = 1/4, as suggested by Thompson (1978), but may i n f a c t be sample dependent. Measurements of the peak p o s i t i o n gave x = .24 ± .02, .215 ± .01 and .21 ± .01 f o r three d i f f e r e n t samples. A l l samples show evidence of double peaked chara c t e r although i n some samples the peak near x = .07 i s not r e s o l v e d . In those samples where the peaks are r e s o l v e d , a w e l l defined minimum between the peaks i s observed. These peaks are not due to f i r s t order phase t r a n s i t i o n s because the "notches" (Dahn and Haering 1981a, b) expected i n l i n e a r sweep voltammetry at f i r s t order t r a n s i t i o n s are absent. I f the l a r g e peak i s a c t u a l l y a divergence i n -8x/9V produced by a higher order phase t r a n s i t i o n , the peak should become sharper i f the c u r r e n t s or sweep r a t e s are reduced and the measurement i s performed c l o s e r t o e q u i l i b r i u m . No changes i n peak shapes were observed at slow sweep r a t e s or small c u r r e n t s i n agreement w i t h McKinnon (1980). The sample dependence of the r e s u l t s may be due to the presence of i n t e r l a y e r t i t a n i u m as w i l l be discussed i n Chapter 7. 1.8 2 .0 2 . 2 2 . 4 2 . Volts Figure 20. Linear sweep voltammogram of c e l l JD—212 taken at a sweep rate of 3.83 yV/'/sec. This c e l l contained 5.39 of 4pl00 TiS„. The V(x) data for room temperature in Figures 14, 17 and 19 can be used as a " c a l i b r a t i o n " of l i t h i u m concentration as a function of voltage for the respective materials. C e l l s fixed at constant voltage and allowed to e q u i l i b r a t e , w i l l have a value of x "bracketed" by the values of x determined f or charge and discharge from the f i g u r e s . Due to subtle differences between samples, a un i v e r s a l V(x) curve cannot be obtained. In subsequent experiments, the c e l l voltage i s monitored c a r e f u l l y and to obtain x, the V(x) r e l a t i o n s measured in t h i s section are used. 68 C H A P T E R FOUR THE MEASUREMENT OF 3V/3T) x 4.1 Experimental D e t a i l s Experiments involving the measurement, of 3V/3T to obtain information about the entropy i n electrochemical systems have been reported i n the l i t e r a t u r e . For example, Wen and Huggins (1981) performed measurements on Li / L i - S n c e l l s , Dudley et a l . (1980) investigated the Cu Mo,S0 system and x b o Thompson (1981) studied L i / L i TiS„ c e l l s . The magnitude of 3 V / 3 T ) x for i n t e r c a l a t i o n systems i s expected to be of order k/e ('= 86.2 U.V/K) from equations 2.10 and 2.19 d. Experiments measuring 3V/3T) as a function of x should have s e n s i t i v i t i e s of a few x yV/K. An experimental electrochemical c e l l t y p i c a l l y contains many junctions between d i s s i m i l a r materials. The presence of temperature gradients i n a c e l l w i l l lead to thermoelectric voltages which w i l l be the order of 30 yV/R for each junc t i o n . Because the expected magnitudes of 3 V / 3 T ) x are of the order of thermoelectric voltages, we must be very c a r e f u l that temperature gradients i n the c e l l are absent when measurements are made. For t h i s reason,, the optimum experiment would be to measure the open c i r c u i t voltage, V^, of a c e l l e q u i l i b r a t e d at temperature T^, change the temperature to T„, wait f o r thermal equilibrium and measure 69 the new voltage, V^- The temperature could then be recycled to T^ and upon thermal equilibrium, the voltage attained again. Then, provided T 1~T 2 i s small, V - V„ 3V\ = J J . [2 4.1 The presence of thermoelectric e f f e c t s makes i t u n l i k e l y that measurements involving an o s c i l l a t i o n of the c e l l temperature while monitoring the corresponding voltage o s c i l l a t i o n s w i l l be meaningful. S p e c i a l l y designed copper and aluminum flange c e l l s with thermistors imbedded i n each flange were used for the measurement of 9V/3T) x. The inner surface of each flange was n i c k e l plated. An exploded view of one of these c e l l s i s shown in Figure 21. Cathode preparation, e l e c t r o l y t e p u r i f i c a t i o n and c e l l construction were c a r r i e d out as in Chapter 3. These c e l l s were immersed in the f l u i d r e s e r v o i r of a Haake F3-K temperature c o n t r o l l e r . The f l u i d used was e l e c t r i c a l l y i n s u l a t i n g S h e l l Diala O i l Ax transformer o i l . The o i l (which was c i r c u l a t e d by a pump) ensured that the e l e c t r i c a l l y i s o l a t e d c e l l flanges would have the same temperature. At equilibrium, the temperature in the r e s e r v o i r was stable to ± .02 K. The c e l l voltage was measured using a Fluke 335D secondary voltage s t a n d a r d / d i f f e r e n t i a l voltmeter. The experiments were performed by allowing the c e l l to reach equilibrium at a given temperature by f i x i n g i t s voltage for about 24 hours and allowing the current to decay to near zero. The c e l l voltage i s then monitored under quasi-open c i r c u i t conditions with the high impedance Fluke 335D. One u s u a l l y observes a slow d r i f t i n the open c i r c u i t c e l l voltage which i s due to the 70 INSULATED ASSEMBLY SCREW TERMINAL BLOCK THERMISTOR COPPER FLANGE VITON O-RING LITHIUM ANODE SEPARATOR O-RING GROOVE  POWDER CATHODE NICKEL PLATED INNER SURFACE Figure 21. "Exploded" view of an electrochemical c e l l used for 3V/3T) measurements, x 71 presence of side reactions and s o l i d state d i f f u s i o n of the l i t h i u m in the i n t e r c a l a t i o n cathode i f equilibrium has not yet been reached. The d r i f t rates t y p i c a l l y do not exceed a few hundred microvolts per hour. The d r i f t rate i s monitored for some time by p l o t t i n g V against time. Then the temperature of the c e l l i s changed, while we continue to monitor the c e l l voltage. After thermal equilibrium has been attained (measured by the thermistors), the temperature i s returned to the s t a r t i n g temperature. A run taken on c e l l JD-218 which contained 3p38 TiS2 at an open c i r c u i t voltage of 2.09040 v o l t s i s shown in Figure 22a. The background d r i f t rate was established at 17 . 6 0°C, the temperature was changed to 20 . 6 0°C, we waited for equilibrium, returned to 17 . 6 0°C and then cycled back to 20 . 6 0°C. A voltage change of -420 ±10.yVis induced by the +3°C temperature change. This y i e l d s 9 V / 9 T ) x = -140 ± 3 yV/°C. A s i m i l a r run on the same c e l l at an open c i r c u i t voltage of 2 . 3 5 2 6 0 v o l t s i s shown i n Figure 22b. In t h i s case, the large peaks which occur immediately a f t e r the temperature i s changed, are due to the thermoelectric e f f e c t s discussed e a r l i e r . Once thermal equilibrium has been reached, the i n i t i a l d r i f t rate i s re-established. Thermoelectric e f f e c t s are more pronounced i n t h i s run because the c e l l was in a le s s symmetric l o c a t i o n i n the bath; the heating rates for each flange were s l i g h t l y d i f f e r e n t . 9 V / 9 T ) x = 17 ± 3 j i V/°C for t h i s voltage. An experiment to measure 9 V / 9 T ) x at roughly 50 values of x takes on the order of 50 days using t h i s method. 72 i r 20.60 WC 20.60 °C 17.60 °C CO O > 2.35240 2.35280 .25 .50 .75 1.0 TIME (hours) 1.25 Figure 22. 8V/3T) X measurements at two vol t a g e s on c e l l JD-218. a) 2.09040 v o l t s , b) 2.35360 v o l t s . 73 4.2 Results and Discussion In Figure 23 3V/3T) x versus V for c e l l JD-218 i s plotted. This c e l l was given the "quick discharge", discussed in Section 3.4, then recharged to near 2.420 v o l t s . After a given measurement, the voltage was f i x e d at a lower value. Data was taken u n t i l 2.156040 v o l t s i n t h i s manner.; These data points are referred to as discharge points although the c e l l i s in equilibrium during the measurement. After t h i s point the c e l l was charged to 2.337740 v o l t s and a measurement taken. The next 5 points were taken i n the charge d i r e c t i o n . The f i n a l four points were taken i n the discharge d i r e c t i o n . The small discrepancy in the JD-218 data for charge and discharge i s not understood, although i t may be due to aging of the c e l l . The nature of the passivating f i l m on the l i t h i u m anode may have changed with time and caused the anode contribution to the data to change. This data can be replotted as a function of x using the V(x) data for t h i s material in Figure 14 and t h i s i s done i n Figure 24. In Figure 25, 8V/3T versus x for c e l l JD-156, which contained RM9 TiS^, i s p l o t t e d . The data for the two c e l l s are i n good agreement. The s o l i d curve in Figures 24 and 25 i s a p l o t of - — l n ( x / ( l - x ) ) - const (eqn. 2.10 and 2.19d). The constant needed to make the c a l c u l a t i o n pass roughly through the data i s 120 yV/K. The entropy of l i t h i u m metal at 298K i s 6.7 cal/moleK which y i e l d s f(T) = 291 pV/K at 298K in equation 2.10. We expect the anode contribution to be 291 yV/K, not 120 yV/K. However, there are contributions to the entropy of the L i TiS cathode that have not been taken into account X c. in the., c a l c u l a t i o n s . For instance, the v i b r a t i o n of i n t e r c a l a t e d atoms in t h e i r s i t e s w i l l add a term in the entropy proportional to the number of 200 lOOh It 0 > -100 -2001 2.5 2.4 2.3 2.2 2.1 2.0 VOLTS 1.9 Figure 23. 8V/8T) X versus V measured f o r c e l l JD-218. 17.60°C and 20.60 C f o r these measurements, The temperature change was 3 C, between • - discharge, x - charge. (see text) 75 200 > > -I00h -200--300 0 .2 .4 .6 X in L i x T i S 2 . 8 Figure 24. 8V/8T versus x for c e l l JD-218 obtained from the data i n Figures 14 and 23. •- discharge, x - charge. The s o l i d curve i s the pre d i c t i o n of simple mean f i e l d theory (see t e x t ) . Figure 25. 9V/3T) versus x for c e l l JD-156. The s o l i d curve i s the p r e d i c t i o n of simple mean f i e l d theory (see t e x t ) . 77 i n t e r c a l a t e d atoms. When d i f f e r e n t i a t e d w i t h respect to x, t h i s w i l l l ead to another constant s h i f t i n SV/ST)^. When one c a l c u l a t e s the c e l l v o l t a g e , a l l terms l i n e a r i n x can be included w i t h the s i t e energy. T h i s i s what i s u s u a l l y done. One immediately n o t i c e s a fe a t u r e at x = .16 ± .01 i n d i c a t i v e of an entropy minimum (see Figure 3). T h i s v a l u e of x agrees w i t h the p o s i t i o n of the minimum i n -3x/3V i n Figure 14, which suggests an ordered composition. No obvious f e a t u r e s i n 3V/3T) x are observed at x = 1/3, 1/4, 1/7 .... where they would be expected i f the ordered s t r u c t u r e s proposed by Thompson (1978) were to form. Because 3V/3T) x i s s e n s i t i v e to order of very short range (Appendix 1 ) , there i s no evidence f o r " i n c i p i e n t " o r d e r i n g as suggested by Thompson (1981) at x = 1/3, 1/4, 1/7 ..... . The general agreement of the data w i t h the simple mean f i e l d theory p r e d i c t i o n f o r 3V/3T) x i s good except near x = .16. The 3V/3T) data reported by Thompson (1981) f o r L i / L i TiS c e l l s X X z does not agree w i t h t h i s data. Thompson c o l l e c t e d h i s data during a constant current discharge by o s c i l l a t i n g the c e l l temperature while monitoring the r e s u l t i n g v o l t a g e o s c i l l a t i o n s . Thermoelectric e f f e c t s (Figure 22b) w i l l be l a r g e i n Thompson's measurements and f o r t h i s reason h i s data are suspect. Because the measurement of 3V/3T) i s q u i t e t e d i o u s , a l l batches of x TiS^ were not measured i n t h i s way. The experiments to be described i n the r e s t of the t h e s i s were s p e c i a l i z e d to 3p38. and m a t e r i a l s i m i l a r to i t (such as RM9 T i S 2 ^ where e l e c t r o l y t e c o - i n t e r c a l a t i o n i s moderate and a w e l l res o l v e d double peak s t r u c t u r e i n -3x/3V) i s observed. 78 The feature i n 3V/8T) at x = .16, incommensurate w i t h the ordered x s t r u c t u r e s expected on the two dimensional t r i a n g u l a r l a t t i c e , when combined w i t h the -8x/9V r e s u l t s presents a pu z z l e . What i s the nature of the increased order (entropy minimum) at t h i s composition? With t h i s i n mind, s t r u c t u r a l i n v e s t i g a t i o n s u s i n g i n s i t u x-ray d i f f r a c t i o n techniques were undertaken and w i l l now be reported. 79 C H A P T E R F I V E IN SITU X-RAY DIFFRACTION 5.1 Advantages of the In S i t u Technique I n t e r c a l a t i o n induced changes i n the host structure may be studied by X-ray d i f f r a c t i o n even though t h i s technique does not allow one to locate the positions of the l i t h i u m guest atoms because of t h e i r low X-ray scattering power. Moreover, such experiments are complicated by the fact that most of the compounds of i n t e r e s t (eg. L i x T i S 2 ) are unstable i n a i r . A number of studies employing X-ray d i f f r a c t i o n have been reported i n the l i t e r a t u r e (eg. Wainwright 1978, Whittingham and Thompson 1975 and Bichon et a l . 1973). These inv e s t i g a t i o n s involved the preparation of separate samples by chemical or electrochemical means for each l i t h i u m concentration studied. This type of experiment i s very time consuming because the de t a i l e d c h a r a c t e r i z a t i o n of a sing l e system involves the preparation of many samples. In order to compare the X-ray information obtained on separate samples, great care must be taken i n the preparation and alignment of the X-ray d i f f r a c t i o n samples. In e f f e c t , an absolute determination of the l a t t i c e constants must be made on each sample. C h i a n e l l i et a l . (1978) described an in s i t u X-ray d i f f r a c t i o n technique used to study L i x T i S 2 i n an electrochemical c e l l . This involved 80 a s p e c i a l l y designed c e l l which could be charged or discharged while an X-ray d i f f r a c t i o n p r o f i l e was taken. One advantage of t h i s technique i s that i t provides one with a means of continuously varying the i n t e r c a l a n t content without moving the c e l l from the diffractometer, making the deter-mination of r e l a t i v e changes in the l a t t i c e parameters very accurate. I f , however, the X-ray d i f f r a c t i o n information i s to be correlated with the c e l l voltage i n a meaningful way, great care must be taken in the design of the c e l l . This i s because X-ray d i f f r a c t i o n probes the bulk of the i n t e r c a l a t e d p a r t i c l e s whereas the voltage i s determined by the concentration of l i t h i u m at the p a r t i c l e surfaces. A well designed c e l l should achieve rapid e q u i l i b r a t i o n of the e n t i r e i n t e r c a l a t i o n electrode. Otherwise the presence of l i t h i u m concentration gradients w i l l y i e l d broad X-ray peaks or, i n some cases, e n t i r e l y spurious peaks associated with p a r t i c l e s of the electrode which are i n a c t i v e . The design u t i l i z e d 2 by C h i a n e l l i et a l . (1978), which incorporated thick (- 100 mg/cm ) cathodes did not s a t i s f y these c r i t e r i a . Thick porous cathodes require long e q u i l i b r a t i o n times due to the f i n i t e d i f f u s i o n rate of l i t h i u m ions i n the e l e c t r o l y t e f i l l e d pores. The currents at which data were c o l l e c t e d by C h i a n e l l i et a l . resulted i n measurements made f a r from equilibrium and an incorrect c o r r e l a t i o n of the s t r u c t u r a l information with V(x). In t h i s chapter we describe the design and operation of electrochemical c e l l s that allow one to study the i n t e r c a l a t i o n cathode by means of X-ray d i f f r a c t i o n . These c e l l s , f o r a l l intents and purposes, function exactly l i k e the test c e l l s described in Chapter 3, except that a beryllium X-ray window i s incorporated into the design. Because V(x,T) for an i n t e r c a l a -t i o n compound i s analogous to an equation of state (Chapter 2 ) , c e l l s allowed to e q u i l i b r a t e at fixed voltage and temperature w i l l have 81 homogeneously i n t e r c a l a t e d cathodes. This fact allows equilibrium measurements on i n t e r c a l a t i o n compounds to be made e a s i l y and reproducibly. If e q u i l b r a t i o n times are long, (for instance due to a k i n e t i c a l l y slow phase t r a n s i t i o n i n the cathode) the reaction can be followed by taking X-ray d i f f r a c t i o n p r o f i l e s as a function of time. Dynamic e f f e c t s l i k e s o l i d state d i f f u s i o n can also be studied when f i n i t e currents are flowing. The advantages of the i n s i t u X-ray d i f f r a c t i o n technique make i t an extremely powerful technique, as we demonstrate with s t r u c t u r a l measure-ments on L i TiS~. 5.2 Experimental X-ray measurements were taken using a P h i l i p s powder diffT a c t o m e t e r . This system consists of a P h i l i p s 1730/10 X-ray generator equipped with a copper tube (PW 2253/20) and a v e r t i c a l powder goniometer (PW 1050/70). X-rays of undesirable wavelength are eliminated by a graphite monochromator c r y s t a l mounted between the sample and the detectors. A P h i l i p s 1386/50 automatic divergence s l i t i s used to ill u m i n a t e a constant area of the sample as the s c a t t e r i n g angle.is changed. The geometry of t h i s system and the e f f e c t s of the automatic divergence s l i t on i n t e n s i t y c a l c u l a t i o n s are described i n Appendix 4. Receiving s l i t widths of 0.1 mm and 0.2 mm are t y p i c a l l y used. This system i s capable of locating the scattering angles of Bragg peaks to the nearest 0.01°. In Figure 26 we show an exploded view of the electrochemical c e l l used i n our i n s i t u X-ray d i f f r a c t i o n experiments. Minor modifications to the goniometer and r a d i a t i o n s h i e l d (1381/30) were made so that the c e l l could be accommodated. 0.700 cm of material was removed from the end of the goniometer 0-shaft (by m i l l i n g ) to allow f or proper alignment of the c e l l cathode in the X-ray beam. A cover for the r a d i a t i o n shield having feedthroughs for e l e c t r i c a l and thermal co n t r o l of the c e l l was constructed. When the c e l l i s mounted i n the goniometer, the top of the glass spacer (Figure 26) i s i n contact with the accurately m i l l e d portion of the 6-shaft. E l e c t r i c a l and thermal i s o l a t i o n from the goniometer are provided by the glass spacer. The c e l l i s 4.7 cm x 3.6 cm x .95 cm i n size and f i t s e a s i l y i n s i d e the r a d i a t i o n s h i e l d . The c e l l top, made of brass or s t a i n l e s s s t e e l , holds the .25 mm t h i c k beryllium window. The window mates with a recess on the bottom 83 POLYPROPYLENE GASKET  SEPARATOR  LITHIUM ANODE Figure 26. Exploded view of the e l e c t r o c h e m i c a l c e l l used f o r i n s i t u X-ray d i f f r a c t i o n s t u d i e s of l i t h i u m i n t e r c a l a t i o n compounds. 84 of the c e l l top and i s "tacked" there with s i l i c o n high vacuum grease. Removal of the beryllium f or cleaning and cathode preparation i s therefore simple. The c e l l base i s made of s t a i n l e s s s t e e l , or i n some cases, n i c k e l plated brass. If thermal c o n t r o l of the c e l l i s desired the n i c k e l plated brass bases are used. A channel through these bases allows for the passage of f l u i d , (ethylene g l y c o l / ^ O ) whose temperature i s c o n t r o l l e d by a Haake F-3 temperature c o n t r o l l e d bath. The connection to the bath i s made by th i n walled rubber hoses. The temperature of the c e l l , measured by a thermistor, can be varied between -10°C and 60°C. This corresponds to the useful thermal range for operation of these c e l l s . The c e l l has a c a p a b i l i t y of reaching angles as low as 20 = 8° before the s c a t t e r i n g from the cathode material i s t o t a l l y l o s t . For 29 > 14° the area of the cathode illuminated i s constant, due to the automatic d i v e r -gence s l i t and the 7° taper i n the c e l l top ( F i g . 26). The electrochemical c e l l i s constructed i n the region between the beryllium top and the c e l l base and i s sealed by a .40 mm polypropylene gasket. C e l l s are constructed i n either one of two ways. In- the f i r s t , depicted i n Figure 26, the cathode material i s a f f i x e d to the bottom of the beryllium window. In t h i s geometry, the X-rays enter the c e l l through the beryllium window, are d i f f r a c t e d by the cathode and then e x i t . In the second case, the c e l l i s made upside down with the l i t h i u m anode in contact with the beryllium. The X-rays pass through the beryllium, the l i t h i u m and the separator before s t r i k i n g the cathode material. In t h i s case, the cathode i s prepared on a t h i n n i c k e l f o i l . In both cases, Bragg peaks due to the beryllium and l i t h i u m metal are observed. However, i n the former case the l i t h i u m metal peaks are very weak due to the high scattering power of the i n t e r c a l a t i o n compound. 85 Powder cathodes of density 10-20 mg/cm^ are prepared on the beryllium or on a n i c k e l substrate as described i n Chapter 3. Because cathodes of t h i s type require some pressure to keep the p a r t i c l e s e l e c t r i c a l l y contacted to the substrate, i t was found necessary to include a center beam in the c e l l top to prevent the beryllium from f l e x i n g excessively under the needed load. C e l l s were constructed as described i n Chapter 3. The li t h i u m metal f o i l used was .38 mm thick and i t s surface was cleaned by scraping p r i o r to c e l l assembly. This l i t h i u m thickness, when combined with the separator and cathode thicknesses ensured the proper c e l l pressure when the c e l l top was tightened to seal against the polypropylene gasket. C e l l s constructed i n t h i s way were found to be leak t i g h t and experiments on several c e l l s have lasted many months. There are three problems associated with a c e l l design of t h i s type: (1) contamination of the d i f f r a c t i o n p r o f i l e with unwanted peaks (Be, L i , e t c ) , (2) preferred o r i e n t a t i o n of the cathode p a r t i c l e s and (3) c e l l alignment. These problems are discussed below. Peaks due to beryllium, beryllium oxide, l i t h i u m metal, n i c k e l and the separator can show up i n our X-ray d i f f r a c t i o n p r o f i l e s . As i l l u s -trated i n Figure 27, these peaks are e a s i l y i d e n t i f i e d and have fixed angular positions; they do not s h i f t as i n t e r c a l a t i o n proceeds. Because there are many peaks from the i n t e r c a l a t i o n compound i n the range 10° < 26 < 150°, the contaminant peaks usually i n t e r f e r e with only a few peaks and are an annoyance only. Many of the compounds of i n t e r e s t have layered structures and exhibit a large degree of preferred o r i e n t a t i o n i n these c e l l s due to the pressure f i x i n g the cathode p a r t i c l e s i n place. P l a t e l e t s tend 86 60° 50° 40° SCATTERING ANGLE (DEGREES) Figure 27. X-ray d i f f r a c t i o n p r o f i l e obtained from a L i / L i x T i S 2 c e l l . The peaks a r i s i n g from the L i x T i S 2 cathode are l a b e l l e d by M i l l e r indices only, while the contaminant peaks are denoted by the contaminant and the M i l l e r i n d i c e s . In t h i s c e l l the l i t h i u m metal anode was in contact with the beryllium window and the cathode was i n contact with the c e l l base. This leads to the large l i t h i u m metal peaks i n the p r o f i l e s , (a) T i S 2 (b) L i 1 T i S 2 . 87 to l i e f l a t and hence the (00&) peaks are enhanced with respect to the (hkO) peaks. This i s evident i n Figure 27 where i t can be seen that the (004) peak i s roughly f i v e times as intense as the (101) peak. Intensity c a l c u l a t i o n s , accounting for the e f f e c t of the automatic divergence s l i t , (see Appendix 4) show that the (101) peak should be about 10 times as intense as the (004) peak for a randomly oriented powder. The amount of preferred o r i e n t a t i o n i s a function of c r y s t a l l i t e s i z e . Smaller grains show a smaller, but s t i l l present, e f f e c t . This d i f f i c u l t y must be recognized i n an i n t e r p r e t a t i o n of X-ray i n t e n s i t i e s from t h i s type of c e l l . The c e l l cathode i s usually displaced normal to the goniometer axis. This i s because the cathode and l i t h i u m thicknesses can vary from c e l l to c e l l . The posi t i o n s of the measured Bragg peaks depend on the magnitude of t h i s displacement. In our c e l l s , t y p i c a l o f f - a x i s displacements are i n the range of .05 mm to .35 mm. These displacements are small enough so that the focussing condition of the goniometer i s not s i g n i f i c a n t l y affected i . e . i n h i g h l y c r y s t a l l i n e materials the Bragg peaks are s t i l l sharp. The s h i f t i n angle i s given by A(29) = cos6 . 5.1 In t h i s expression R i s the goniometer radius (173.0 mm), 6 i s the displacement of the cathode from the goniometer axis and 0 i s the measured Bragg angle. The corrected angle i s 20 = 20 - A(20) corr 5.2 88 This can be converted into a correction for the plane spacing, d. To f i r s t order « , ( . , 0 C O S 0 "\ _ „ d = d , 1 + — . n , 5.3 corr obs 1 R sin0 J ' where d , i s the d spacing calculated from the measured (uncorrected) obs Bragg angle 0. We have found that correcting the data f or t h i s e f f e c t i s much easier than t r y i n g to a l i g n each cathode on the goniometer axis. 89 5.3 S t r u c t u r a l Measurements on L i x T i S 2 The data shown i n Figure 27 indicates the q u a l i t y of the d i f f r a c t i o n p r o f i l e s which can be obtained with these c e l l s . This data was taken on c e l l JX-15 which contained 12.2 mg of .'3p38 T i S 2 . The c e l l was given the "quick discharge" to 1.700 v o l t s as described i n Section 3.4 to avoid e l e c t r o l y t e c o - i n t e r c a l a t i o n . The Ka doublet i s e a s i l y resolved for s c a t t e r i n g angles greater than 55°. The Bragg peaks from L i x T i S 2 are narrow at the compositions shown. The magnitudes of the s h i f t s i n peak p o s i t i o n that occur upon i n t e r c a l a t i o n are c l e a r l y shown i n Figure 27. For instance, the L i x T i S 2 (004) Ko^peak. s h i f t s 5.61° from 65.51° at x=0 to 59.90° at x=l. The a b i l i t y to gather data of t h i s q u a l i t y indicates that the i n s i t u X-ray d i f f r a c t i o n technique should be a very powerful t o o l . The r e v e r s i b i l i t y of the l a t t i c e changes that occur i n L i x T i S 2 are i l l u s t r a t e d by what we c a l l the "breathing battery" experiment. In Figure 28 we show the r e s u l t of f i x i n g the X-ray detector at 28 = 15.53° and monitoring the X-ray i n t e n s i t y as L i / L i TiS„ c e l l JX-14 was cycled over a small range of x. This angular p o s i t i o n places the detector on the side of the (001) peak. As the c e l l i s charged and discharged, the peak s h i f t s and o s c i l l a t i o n s i n the measured X-ray i n t e n s i t y are observed. JX-14 contained 19.3 mg of 3p38 T i S 2 and was cycled at ±150 UA. An eight minute discharge at t h i s rate corresponds to an i n s e r t i o n of Ax & .004 i n L i x T i S 2 . The voltage range corresponds to an average value of x = .04 ± .02. The L i T i S 9 c r y s t a l l i t e s expand and contract ("breathe") as we charge and X z discharge the c e l l . The phase s h i f t between the observed voltage and X-ray responses to the current i s caused by the f i n i t e d i f f u s i o n rate of the l i t h i u m atoms in the H S 2 host. This i s because the c e l l voltage i s determined by the 90 2.475 co ^ 2.425h o > o o Id CO tr Q_ CO H z o u 2.375h 500h 400 300h 200 32 48 TIME (MINUTES) Figure 28. Result of monitoring the c e l l voltage and scattered X-ray i n t e n s i t y of c e l l JX-14 as a function of time while the c e l l was cycled at ±150 uA between fixed voltage l i m i t s . The X-ray detector was fixed at 26 = 15.53° which i s on the side of the (001) peak. See text. 91 li t h i u m concentration at the surface of the p a r t i c l e s and the Bragg peak p o s i t i o n by a weighted average (due to the absorption of X-rays by the cathode material) of the lithium concentration in the bulk of the cathode p a r t i c l e s . By varying the amplitude and frequency of the d r i v i n g current and monitoring the X-ray and voltage responses, i t should be possible to learn about s o l i d state d i f f u s i o n of the i n t e r c a l a n t . In an attempt to understand the V(x), -3x/3V and 3V/3T) x behaviour of L i / L i x T i S 2 c e l l s described i n Chapters 3 and 4, a d e t a i l e d study of the changes in the c r y s t a l l a t t i c e which occur as i n t e r c a l a t i o n proceeds was performed. The v a r i a t i o n of the c and a l a t t i c e parameters i s shown in Figure 29. This data was obtained from c e l l s JX-12, 13, 14 and 17, a l l of which contained 3p38 TiS2• This data i s i n good agreement with recent r e s u l t s by Thompson and Symon (1981). Each data point was obtained a f t e r the c e l l was f i x e d at a given voltage for 10-20 hours u n t i l the 2 current density had decayed to a value les s than 1 uA/cm . A subsequent X-ray d i f f r a c t i o n p r o f i l e taken about 10 hours l a t e r was i d e n t i c a l to the f i r s t , which indicated that the cathode had i n fact reached equilibrium. The dependence of the l a t t i c e parameters on the c e l l voltage was measured and c(x) and a(x) were obtained using the data i n Figure 14. The l a t t i c e parameters were calculated from a least squares f i t to at l e a s t eight Bragg peak positions as described i n Appendix 3. In most cases values of X, 1 less than 0.01 were obtained. An example of the agreement between calculated and observed plane spacings i s given in Table 3. I t i s evident Figure 29. V a r i a t i o n of the l a t t i c e parameters c and a with x i n Li„TiS 0. Table 3 Observed and calculated layer spacings for L i x T i S 2 i n ce JX-13. The c e l l voltage was 2.296 V which corresponds to x = .342 ± .008. Results of the l e a s t squares f i t were: a = 3.4205 A; c = 6.0931 A and x = .0093. The o f f - a x i s displacement was .056 mm. h k d observed o A d c a l c u l a l o A 0 0 2 3.0455 3.0465 1 0 1 2.6629 2.6641 1 0 2 2.1238 2.1238 1 0 3 1.6755 1.6751 0 0 4 1.5230 1.5232 1 0 4 1.3552 1.3547 0 0 5 1.2183 1.2186 2 0 3 1.1966 1.1967 1 1 4 1.1380 1.1375 1 0 5 1.1272 1.1270 94 that the c-axis increases by about 10% between x=0 and x=l with most of the increase coming before x=0.4. In contrast, the a-axis increases most r a p i d l y near x=l and increases by about 1.5% over the range 0<x<l. We note no anomalous v a r i a t i o n of the (101) plane spacing as reported by C h i a n e l l i et a l . (1978). It should be pointed out that the l a t t i c e parameters measured were independent of the type of f i r s t discharge given the c e l l s . C e l l s JX-12, 13 and 14 were given the "quick discharge", discussed i n section 3.4, to try to defeat e l e c t r o l y t e c o - i n t e r c a l a t i o n , while JX-17 was discharged at constant current and cycled several times i n i t i a l l y . In a l l cases, we observed the growth of a set of Bragg peaks associated with the e l e c t r o l y t e co-intercalated material. The speed of the process i s dependent on the s p e c i f i c batch of TiS2 used; for 3p38 T1S2 the time scale i s on the order of several days. In Table 4 we report the plane spacings of the peaks associated with the e l e c t r o l y t e co-intercalated material. These peaks do not s h i f t i n p o s i t i o n provided the c e l l voltage i s maintained above 1.6 v o l t s . I t i s evident from the table that most o probably, the layer spacing of t h i s material i s 17.8 A, even though we have been unable to solve i t s c r y s t a l structure. We f i n d that the e l e c t r o l y t e c o - i n t e r c a l a t i o n process proceeds only for voltages above 2.275 v o l t s . C e l l s given the "quick discharge" i n i t i a l l y showed l i t t l e evidence for c o - i n t e r c a l a t i o n (eg. JX-15 i n Figure 27b), however when charged above 2.275 v o l t s , e l e c t r o l y t e c o - i n t e r c a l a t i o n begins. The Bragg peaks corresponding to L i T i S 9 gradually decrease in i n t e n s i t y as X Z. e l e c t r o l y t e c o - i n t e r c a l a t i o n proceeds but do not s h i f t or broaden with time at f i x e d voltage. The fact that the l a t t i c e parameters of L i x T i S 2 do not vary or depend i n any way on the degree of c o - i n t e r c a l a t i o n suggests Table 4 The observed plane spacings of the e l e c t r o l y t e co-intercalated material Plane spacing Tentative Product of plane (A) M i l l e r spacing a n d Q M i l l e r indices index (A) 8.85 002 17.7 5.918 003 17.75 4.456 004 17.82 3.558 005 17.79 2.969 006 17.81 2.849 2.706 2.529 2.344 2.321 1.979 009 17.81 1.879 1.677 1.487 0012 17.84 96 that the co-intercalated material does not a f f e c t the behaviour of the L i x T i S 2 system i n agreement with our findings i n Chapter 3. We w i l l return to t h i s point shortly. Electrochemical evidence for the sample dependence of e l e c t r o l y t e c o - i n t e r c a l a t i o n was confirmed by X-ray measurements performed on the d i f f e r e n t samples. It was observed that the widths of the (00&) peaks changed as i n t e r c a l a t i o n proceeded i n L i TiSo. The (hkO) peaks remained sharp and peaks of mixed character broadened to a degree determined by the r a t i o of SL to h and k. The (004) peak measured at 2.373, 2.314 and 1.900 v o l t s for c e l l JX-17 i s shown i n Figure 30. The values of x corres-ponding to these voltages are also given i n the f i g u r e . The peak widths measured are independent of c e l l cycle number and are reproducible. This i s demonstrated by Figures 31 and 32. In Figure 31a we show the (004) peak measured on March 15/1981 for c e l l JX-17 at 2.332 v o l t s . In Figure 31b, the (004) peak measured, again at 2.332 v o l t s on March 27 i s p l o t t e d . During the twelve days between these measurements the c e l l was used to c o l l e c t data at other voltages. To check the independence of peak width with c e l l cycle number, JX-17 was also given several high rate cycles shown i n Figure 32. The peak width i s i d e n t i c a l and i s equal to .35° i n Figures 31a and b. Tests at other voltages and values of x also showed the peak width to be independent of cycle number. The peak broadening i s a r e v e r s i b l e phenomenon and therefore cannot be associated with a breaking up of the c r y s t a l l i t e s into smaller p a r t i c l e s . The loss i n i n t e n s i t y i n Figure 31 i s due to the gradual c o - i n t e r c a l a t i o n of e l e c t r o l y t e during the twelve days between these measurements, which reduces the amount of L i x T i S 2 phase present. The fa c t that the breadth of the peak i s independent of the amount of e l e c t r o l y t e c o - i n t e r c a l a t i o n SCATTERING ANGLE (DEGREES) Figure 30. Typical (004) Bragg Peaks for 3p38 L i TiS . a) x = .12 b) x = .29 c) x = 1.0. a) CO "D C o o CD CO CN CD CL CO - i — * c D O o 3000 2400 h 1800 h 1200 h 600 h 0 b) CO "D C O O CU CO CD CL CO c D O o 3400 2800 2200 h 1600 1000 400 61.2 61.6 62.0 62.4 62.8 Scattering Angle (Degrees) 61.2 61.6 62.0 62 .4 62.8 Scattering Angle (Degrees) figure 31. The (.004) Bragg peak of I.i T i S 2 measured at 2.332 v o l t s i n c e l l JX-17, a) March 15/81 b) March 27/81. The c e l l was,used to measure data at other voltages and given several high rate cycles, shown in Figure 32, during the twelve days between a) and b) . 0 0 99 Figure 32, C e l l JX-17 was- given the high rate cycles- shown above bet^ ween March 15 and March 27, 1981, The data i n Figure 31b was taken a f t e r these cycles. 100 that has occurred indicates that the breadth i s a function of the L i T i S 0 alone, x 2 In Figure 33 the (004) peak width as a function of x i n L i x T i S 2 from 3p38 TIS^ i s plotted. This data was c o l l e c t e d with c e l l s JX-13 and JX-17. To obtain the peak width plotted i n Figure 33, we have treated the measured peak as comprised of two gaussian peaks of equal width a r i s i n g from the Ka^, and Ka^ peaks. The i n t e n s i t y of the Ka^ peak has been taken to be twice that of the Ko^ peak i n the standard way (Klug and Alexander 1974). The plotted width i s the f u l l width at hal f maximum,A(28), of one of the Ka peaks and i s determined by the best f i t of t h i s model to the data. The instrumental r e s o l u t i o n at t h i s angle i s equal to A(20) = 0.13° and has not been subtracted from the data. As i n Chapter 3 we have observed differences i n the behaviour of the d i f f e r e n t batches of TiS 2- 3p38 T i S 2 and 5pl51 T i S 2 ( c e l l JX-32) showed (004) peak broadening behaviour characterized by Figure 33. However, i n cases where the electrolyte c o - i n t e r c a l a t i o n was minor, the peak braodening e f f e c t was not detected. We have seen i n Chapter 3 that the feature i n -3X/3V, which i s present for 3p38 and 5pl51 TiS2 i s weaker i n materials where e l e c t r o l y t e c o - i n t e r c a l a t i o n i s minor (eg. Figure 18). We believe that the differences are due to excess titanium i n the van der Waals gaps of the material as suggested i n Chapter 3. We w i l l return to t h i s point i n Chapter 7. One immediately notices the c o r r e l a t i o n of the features observed i n Figure 33 with those i n Figure 14 (-3x'/3V vs x ) . Broadening of the (00£) peaks i s i n d i c a t i v e of imperfect order i n the c - d i r e c t i o n . The minimum i n the peak breadth near x ~ .16 indicates that the system 101 Figure 33. V a r i a t i o n of the width of the (004) Bragg peak for L i x T i S 2 . This data was c o l l e c t e d on c e l l s JX-13 and JX-17. 102 has become more ordered at t h i s composition. This i s consistent with the 3V/3T) data (Figure 24) which shows that the entropy of L i T i S 9 i s X X z. reduced near x. = .16. The observed broadening of the (00£) peaks cannot be explained by equation 2.13, which r e l a t e s peaks i n -3x/9V with increased f l u c t u a t i o n s i n average l i t h i u m concentration. A f l u c t u a t i o n of the l i t h i u m concentration of a given layer implies, according to Figure 29, a corresponding f l u c t u a t i o n of the l o c a l value of the c l a t t i c e parameter. This produces a broadening of the (00&) l i n e s which c o r r e l a t e s with peaks i n -3x/3V v i a equation 2.13. However, these e f f e c t s w i l l be n e g l i g i b l e because of the appearance of the factor N i n 2.13. In f a c t , f l u c t u a t i o n e f f e c t s i n macroscopic systems w i l l always be n e g l i g i b l e unless -3x/3V -*• °° as near a phase t r a n s i t i o n . As we have seen i n Chapter 3, experimental evidence indicates that -3x'/3V i s f i n i t e for a l l values of x i n L i TiS„. Thus f l u c t u a t i o n e f f e c t s can be ruled x 2 out as a cause of peak broadening. The phenomenon of staging i n i n t e r c a l a t i o n systems i s well known. In f a c t Na xTiS2 (eg. Zanini et a l . 1981, Rouxel et a l . 1971) has a stage two structure for 0.17 < x. < .33. With t h i s i n mind i t was hypo-thesized that the increased order at x = .16 i n L i TIS could be due to x; 2 the formation of a stage 2 phase. The broadening of the (00£) peaks away from x = .16 would occur i n t h i s model as the system changed from stage 2 to stage 1. A c a r e f u l examination of the L i ^gTiS2 structure to search for s u p e r l a t t i c e peaks a r i s i n g from a possible stage 2 structure was deemed to be i n order. A X-ray i n t e n s i t y c a l c u l a t i o n for L i ^ T i S 2 w a s m a < ^ e u s i n S the program described i n Appendix 4. This c a l c u l a t i o n was performed to get an idea of which s u p e r l a t t i c e peaks should be the most intense. This c a l c u l a t i o n assumed that a stage two structure was formed with every 2nd van der Waals gap empty 103 and the others with .32 of t h e i r octahedral s i t e s f i l l e d randomly with lith i u m . The layer spacing of the empty layers was taken to be that o of unintercalated TiS~, c, = 5.700 ± .001 A, and of the occupied laye r s , i- b o _ o c = 6.100 A. This gives an average l a t t i c e constant of c = 5.900 A a & & which i s roughly the value observed for x = .16 i n Figure 29. The r e s u l t s of t h i s c a l c u l a t i o n (taking account of the automatic divergence s l i t ) .for a randomly oriented powder i n an X-ray c e l l are given i n Table 5. The s u p e r l a t t i c e peaks have f r a c t i o n a l I i n t h i s table. It should be pointed out that even though the scattering factor of the l i t h i u m has been included, i t s e f f e c t on the i n t e n s i t i e s i s minimal. It turns out that the i n t e n s i t i e s of the s u p e r l a t t i c e peaks are proportional to 2 (c - c, ) i f c - c, << c. Because of the preferred o r i e n t a t i o n i n a b a b ^ these c e l l s (see Section 5.2), the (00£) peaks are the most intense and a search for s u p e r l a t t i c e peaks should therefore begin with (00£) s a t e l l i t e s . Of these, the (009/2) peak i s quite intense (13% of (004)) and unobstructed by nearby peaks. In Figure 34 the r e s u l t of a search for the (009/2) peak i s shown. The t r i a n g u l a r data points were taken at 2.366 v o l t s on c e l l JX-19 which contained 3p38 TiS2« The (004) peak i s narrow at t h i s voltage, which corresponds to x = .136 ± .01. The peaks that are present i n the Figure are the Kq^ and Kc^ peaks of the (202) planes of some i n a c t i v e material (TiS_) on the cathode. [The cathode u t i l i z a t i o n , see Chapter 3, i s never 100% i n these c e l l s . ] The dashed arrow at 73.38° indicates the expected p o s i t i o n of the (009/2) peak. The (004) peak has a peak height of 500,000 counts on t h i s scale and therefore, according to the c a l c u l a t i o n given i n Table 5, we expect a (009/2) peak of 65,000 counts i n height i f the widths are the same. C l e a r l y , there i s no evidence Table 5 Relative i n t e n s i t y c a l c u l a t i o n s f o r L i ^ 6TiS2 (see text) c a = 5.700 A, c b = 6.100 A, c D= 5.900 A, a = 3.411 A and \ = 1.5417 A. h k % Intensity ( r e l a t i v e ) 29 (degrees) 0 0 1/2 .07 7.492 0 0 - 1 12.72 15.015 0 0 3/2 <.01 22.605 1 0 0 2.40 30.255 0 0 2 1.19 30.297 1 0 1/2 .10 31.212 l ' 0 -1/2 1 0 1 100.00 33.937 1 0 -1 1 0 3/2 2.96 38.097 1 0 -3/2 0 0 5/2 .01 38.131 1 0 2 79.86 43.346 0 0 3 3.74 46.155 1 0 5/2 4.27 49.428 1 0 -5/2 1 1 0 54.74 53.744 1 1 1/2 .14 54.344 ....cont'd Table 5/cont'd Intensity ( r e l a t i v e ) 20 (degrees) 0 1 1 1 2 1 0 1 1 2 2 2 2 2 1 0 1 2 2 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 7/2 1 3 -3 0 2 4 7/2 -7/2 1 -1 3/2 -3/2 2 4 9/2 3 5/2 -5/2 1.70 16.76 37.50 .72 1.84 11.05 1.87 30.80 1.00 30.18 0.83 1.43 8.60 1.85 54.427 56.135 56.186 62.923 62.944 63.019 63.542 65.100 67.768 71.421 71.488 72.026 73.495 76.00 106 c o o CD W O O co 00 c Z5 o o 132000 128000 124000 CD 120000 Q_ 116000 112000 TiS 2 (202) K a 1 A A <><j>A A 0A .ft <!> A •A A* > TiS 2 (202) K a 2 AOA AAAA I <!><!> Ay A «AA AOA . • *AAAA - A 71.9 72.1 72.3 72.5 72.7 Scattering Angle (Degrees) Figure 34. The (009/2) region of L i T i S 2 measured using JX-19. The t r i a n g l e s are the l a t a measured at 2.366 v o l t s and the diamonds are the data measured at 2.372 v o l t s . The dashed arrow marks the expected p o s i t i o n of (009/2) at 2.366 v o l t s and the s o l i d arrow at 2.372 v o l t s . 107 for such a peak. The c e l l voltage was then changed to 2.372 v o l t s (x = .120 ± .01) and the scan repeated a f t e r equilibrium was attained. The 2.372 v o l t data are the diamond shaped data points i n Figure 34. The expected (009/2) p o s i t i o n at t h i s voltage i s 72.50° and i s marked i n the Figure with the s o l i d arrow. Again, no evidence f o r the (009/2) peak i s seen. We notice that the (202) peak from the i n a c t i v e TiS2 does not s h i f t as the voltage i s changed as expected. Other positions where strong s u p e r l a t t i c e peaks were expected were also checked and no evidence f o r a long range ordered stage 2 structure was found at room temperature. If a long range stage 2 structure had formed, we would have expected to see evidence of the phase t r a n s i t i o n from stage 1 to stage 2 i n the electrochemical data. The lack of a long range stage 2 structure for compositions near x = .16 i n L i T i S 9 at room temperature x ^ i s consistent with the r e s u l t s reported so f a r . If there i s a tendency for L i TiS„ to have a non-periodic l i t h i u m x z d i s t r i b u t i o n i n the c - d i r e c t i o n broad (00£) peaks would r e s u l t due to the dependence of c on x. A simple model which produces broad (00£) peaks without producing s u p e r l a t t i c e peaks considers the stacking of equal numbers of layers of two c h a r a c t e r i s t i c l i t h i u m concentrations x^ and x^ at random. These layers w i l l have c h a r a c t e r i s t i c c l a t t i c e parameters c & and c^. Intensity c a l c u l a t i o n s for disordered layered structures have been performed by Hendricks and T e l l e r (1942) and are discussed i n Appendix 5. Generalizing t h e i r r e s u l t to the problem of random stacking of layers of two d i f f e r e n t c-axes, we f i n d (see equation A5.27 i n Appendix 5). 108 A- . I X 2c = c + c, = a b sin9 5.4 and 4ir sine tan6 , . N2 5.5 A(2ej = — ( c - c, ) A c + c , a b In these expressions, 6 i s the Bragg angle, A (26) i s the f u l l width at half maximum and X i s the X-ray wavelength. It has been assumed that c -c. << c +cT . A (004) peak at 26 = 61.9° , of width a b a b A(29) = .40 implies c = (c +c,)/2 = 6.00 A and c -c, = .18 A. This 3 - D 3 b corresponds to the measured (004) peak near x = .25 i n L i T i S 9 and x z implies a difference of roughly Ax = .21 for the two types of layers (see Figure 29). We show i n Appendix 5, that as the c o r r e l a t i o n length, 5c, for staging increases (stacking sequence c , c, , c , c K ... becomes a b a D more preferred), the Bragg peaks corresponding to the average, c, l a t t i c e become narrow much fas t e r than the s u p e r l a t t i c e peaks become sharp enough to "see" i n an experiment of t h i s kind. We f i n d i n Appendix 5 that the f u l l width at half maximum, A:(29 ) , of the (00£) peaks • av corresponding to the average l a t t i c e i s (using equations A 5.27a and A 5.27b) 4ir sine tane , \ 2 ^ ^ / 1 N C , A ( 2 ° a J = ~T ~r.—Z~7— ( c a - c v ) tanh( - ) 5.6 av A c + c , a b It, where, £c, i s the c o r r e l a t i o n length for staging and 5 i s defined by 109 equation A5.5 i n Appendix 5. The f u l l width at hal f maximum of the (00£) s u p e r l a t t i c e peaks, A(29 g) i s found to be (equation A5.28b). A(29'£) = —. —\ o s i n h ( ^ ) . 5.7 s ir(c + c_) cos9 2? a b For a c o r r e l a t i o n length, £c, between zero (random stacking) and 2 c, the width of the peaks due to the average l a t t i c e can vary by a factor of about 4. The width of the (009/2) peak at 70.64° f or % = 2 and o c = 6.00 A i s 2.90 . The other s u p e r l a t t i c e peaks w i l l be s i m i l a r l y broad due to the r e l a t i v e i n s e n s i t i v i t y of equation 5.7 to 9. Weak peaks of t h i s breadth are d i f f i c u l t to observe i n a powder d i f f r a c t i o n experiment. It i s clear from these arguments that e f f e c t s a r i s i n g due to the short range c o r r e l a t i o n s can explain the fac t that we observe (00£) peaks with v a r i a b l e width without observing s u p e r l a t t i c e peaks. Equation 5.6 makes s p e c i f i c predictions for the fu n c t i o n a l dependence of the (00£) peaks of the average l a t t i c e as a function of I. Detailed measurements of t h i s dependence were not made, although preliminary measurements, show a v a r i a t i o n i n peak width as a function of I i n q u a l i t a t i v e agreement with Equation 5.6. We have seen i n Appendix one that 3V/3T) x i s s e n s i t i v e to order of a very short range. The feature i n 3V/3T) x at x = .16 i s most l i k e l y caused by the formation of a short range stage 2 structure. The minimum i n - 3x/3V at th i s composition i s also consistent with t h i s p i c t u r e . 110 5.4 Discussion The data that we have presented i n t h i s Chapter show that the l a t t i c e changes i n L i x T i S 2 are r e v e r s i b l e . The c o r r e l a t i o n between the (004) peak width data and the features i n 3V/3T) x and -3x/3V indic a t e that the three dimensional aspects of the L l x T i S 2 system are important. These data are i n q u a l i t a t i v e agreement with a model that proposes short range staging near x~.16. This should not be s u r p r i s i n g because of the tendency of layered i n t e r c a l a t i o n compounds to form staged structures, eg. Na xTiS2 (Zanini et a l . 1981). This implies that two dimensional l a t t i c e gas models must be extended to three dimensions i f they are to give an adequate account of the behaviour of the L i T i S 9 system.;. This point i s addressed i n Chapter 7. In many cases phase t r a n s i t i o n s to long range ordered states occur as the temperature of systems that exhibit short range order i s lowered. For t h i s reason we decided to perform a neutron d i f f r a c t i o n study of L i ^^TiS2 to see i f the short range order becomes long range as the temperature i s lowered. This experiment w i l l be discussed i n Chapter 6. The power of the i n s i t u X-ray technique has been demonstrated. High q u a l i t y s t r u c t u r a l data can be gathered quickly and e a s i l y . Measurements of the v a r i a t i o n of Bragg peak width with i n t e r c a l a n t content can be made r e l i a b l y and reproducibly. Because of the dependence of the l a t t i c e parameters of most i n t e r c a l a t i o n systems on the i n t e r -calant content, measurements of Bragg peak width on separately prepared samples are l i k e l y to be meaningless due to the d i f f i c u l t y i n ; preparing homogeneously in t e r c a l a t e d samples. The i n s i t u d i f f r a c t i o n technique i s by no means l i m i t e d to l i t h i u m i n t e r c a l a t i o n compounds and should be applicable to other systems with m e t a l l i c anodes. 111". C H A P T E R S I X THE NEUTRON DIFFRACTION EXPERIMENT 6.1 Sample Preparation and Experimental D e t a i l s Low temperature experiments on i n t e r c a l a t i o n compounds can be complicated by the fa c t that i n t e r c a l a n t d i f f u s i o n becomes incr e a s i n g l y slow as the temperature i s lowered. At a low enough temperature, i n t e r -calant motion w i l l e f f e c t i v e l y cease. Phase t r a n s i t i o n s l i k e staging, which involve a large scale rearrangement of the i n t e r c a l a n t within the host are affected by the "freezing out" of i n t e r c a l a n t motion. Phase t r a n s i t i o n s may be k i n e t i c a l l y hindered because even i f i t i s thermo-dynamically favourable for a phase t r a n s i t i o n to occur at low temperature, i t may be k i n e t i c a l l y impossible for the i n t e r c a l a n t to rearrange to "meet the s p e c i f i c a t i o n s " of the low temperature phase. Staging t r a n s i t i o n s i n L i x T i S 2 may be k i n e t i c a l l y hindered because the residence time, T, of a lithium atom i n an octahedral s i t e becomes large as the temperature _11 3370K/T i s lowered. Kleinberg et a l . (1982) found T = 1.9 x 10 e "s,:. —11 -3370rf /T ' 7 f o r L i 3 3 T i S 2 and x = 4.9 x 10 e ' s- for L i 7 Q T i S 2 using L i NMR. If T = 2 x 1 0 - 1 1 e 3 3 7 0 K / - T s- for L i 1 £ T i S 0 , T = 4 x 10~ 4 s at 200K,.. .1 s .16 I at 150K and 9000 s at 100K.. The formation of a long range stage two 112 phase i n L i ^^TiS2 may be complicated by poor l i t h i u m d i f f u s i o n i f the t r a n s i t i o n temperature i s low enough. Nevertheless, a low temperature experiment searching for s u p e r l a t t i c e peaks a r i s i n g from a stage two structure was made and w i l l be described i n t h i s Chapter. The X-ray i n t e n s i t y c a l c u l a t i o n for L i .,-TiS„ i n Table 5 shows J .16 2 that i f a long range stage two structure forms the i n t e n s i t i e s of the strongest s u p e r l a t t i c e peaks w i l l be about 3% of the i n t e n s i t y of the most intense peak from the average l a t t i c e . S u p e rlattice peaks of t h i s i n t e n s i t y should be e a s i l y detectable i n a low temperature X-ray d i f f r a c t i o n experiment. Unfortunately, our laboratory i s not equipped with low temperature X-ray f a c i l i t i e s , so i t was decided to use neutron d i f f r a c t i o n because the f a c i l i t i e s f o r low temperature measurements were already i n existence. However, the use of neutron d i f f r a c t i o n i s complicated by the large homogeneously i n t e r c a l a t e d samples needed as i s discussed below. We have seen i n Chapters 3, 4 and 5, that the hypothesized short range stage 2 structure i s only present f o r x ~ .16 i n L i x T i S 2 based on 3p38 TiS2« Samples for the neutron d i f f r a c t i o n experiment must therefore be homogeneously i n t e r c a l a t e d to a composition near x ~ .16. This c r i t e r i o n excludes the use of the n-butyl l i t h i u m sample preparation technique (Dines 1975) due to the d i f f i c u l t y i n preparing homogeneous samples (Dahn et a l . 1980). Homogeneous samples can be prepared i n electrochemical c e l l s at f i x e d voltage. Because the sc a t t e r i n g cross--24 2 section of most n u c l e i i s only a few barns (10 cm ), neutron d i f f r a c t i o n samples should be large and t y p i c a l l y have a mass of ten grams. A large s t a i n l e s s s t e e l flange c e l l which can accomodate a 10 gram cathode was b u i l t to prepare these samples. An exploded view of the c e l l i s given i n Figure 35. 113 &3 l 2 7 / 8 -I S " ANODE F L A N G E  H A N D L E  1 0 - 3 2 HOLE LITHIUM 4"XI05/R" S E P A R A T O R 5 " x l l 5 / f t P O L Y P R O P Y L E N E G A S K E T CATHODE F L A N G E MOAT CATHODE POWDER B A K E D DIRECTLY ON F L A N G E 4"XI0?B 1 0 - 3 2 BOLT INSULATED F R O M C A T H O D E F L A N G E Figure 35. "Exploded" view of the s t a i n l e s s s t e e l flange c e l l used to prepare neutron d i f f r a c t i o n samples. The "moat" i s a V wide by 1/8" deep groove around the perimeter of the cathode. 114 L i ^ T i S ^ samples prepared electrochemically with PC (C^H^O^) based e l e c t r o l y t e s s u f f e r from some degree of e l e c t r o l y t e c o - i n t e r c a l a t i o n (see Chapters 3 and 5). Because hydrogen has a very large incoherent neutron s c a t t e r i n g cross-section of 81 barns, e l e c t r o l y t e c o - i n t e r c a l a t i o n should be minimized i f possible i n electrochemically prepared samples (N.B. Incoherent s c a t t e r i n g leads to high background count rates i n neutron powder experiments). By using the "quick discharge" method described i n Section 3.4, e l e c t r o l y t e c o - i n t e r c a l a t i o n was reduced i n our sample preparation. As described i n Chapter 5, preferred o r i e n t a t i o n i s easy to obtain by applying pressure to p l a t e l i k e c r y s t a l l i t e s . We w i l l show l a t e r that several (00&) s u p e r l a t t i c e peaks are among the strongest s u p e r l a t t i c e peaks expected f o r L i n,TiS„. Our neutron d i f f r a c t i o n samples were . ±b z designed to have a large degree of preferred o r i e n t a t i o n to a i d i n a search for (005,) s u p e r l a t t i c e peaks. The TiS2 used i n preparing these samples was 5pl51 and 6pl3 (see Table 2) which behaved i n a manner s i m i l a r to 3p38 TiS2- Cathodes were prepared by spreading a s l u r r y of the TiS2 powder and propylene g l y c o l i n a uniform layer on the lower flange i n Figure 35. The usable 2 area of the flange i s 275 cm which implies a coverage of approximately 2 35 mg/cm for a 10 gram cathode. "The cathode was dried i n a vacuum furnace at 120°C. The large flange c e l l (Figure 35) was assembled i n a way completely analogous to the small c e l l s described i n Chapter 3. The polypropylene gasket used to separate the flanges was not designed to produce a s e a l , so the c e l l was l e f t i n the glove box for the duration of the sample preparation. The c e l l s were given the "quick discharge" to 1.7 v o l t s as described, i n Section 3.4. Currents of 2 to 3 amps 115 flowed i n i t i a l l y f o r 10 gram cathodes and decayed to a few milliamps a f t e r 10 hours. The c e l l s were then charged to the voltage corresponding to the desired value of x and allowed to reach equilibrium. Then the c e l l s were opened, the cathode powder recovered and rinsed with clean PC to remove the LiAsF^ s a l t i n the e l e c t r o l y t e . The powder was then drie d under vacuum at 55°C. An X-ray d i f f r a c t i o n p r o f i l e was taken at t h i s point to check that the samples had the desired composition and homogeneity. The samples were then loaded into aluminum sample holders. The sample holders were aluminum "envelopes" consisting of a f l a t back of .032" aluminum and a front "pouch" made of .020" aluminum (Figure 36). These pieces were welded together around three edges. The L i TiS„ powder was inserted through the unsealed edge and the sample was packed u n t i l the pouch was f i l l e d . The unsealed edge of the sample holder was crimped closed and sealed with "Torr Seal" low vapour pressure epoxy. F i l l i n g and sealing of the sample holders was performed i n the argon f i l l e d glove box. The samples were then pressed between 2 f l a t metal plates at a pressure of 1000 l b s / i n to orient the p l a t e l i k e c r y s t a l l i t e s . A cadmium mask (cadmium has a very high neutron absorption cross-section) was placed around the edge of the sample to prevent sc a t t e r i n g from the hydrogen atoms i n the "Torr Seal". The samples were then ready for the neutron d i f f r a c t i o n experiment. Sample JN-3, prepared from 5pl51 T i S 2 was held fixed at 2.355 v o l t s which corresponds to the average (charge and discharge) voltage of the minimum i n -9x/8V (see Figure 15) for t h i s m aterial. This corresponds to x = .14 ± .02 from Figure 17. JN-7 was prepared from 6pl3 T i S 2 and was held at 2.320 v o l t s , corresponding to x = .25 ± .02. The l a t t i c e o o parameters of the samples are a = 3.412 ± .002 A and c = 5.915 ± .002 A 116 CADMIUM MASK 211 -7 II x 3 SAMPLE AREA CRIMPED AND SEALED END b) WELD .010" Cd MASK .032" AL .020" AL SAMPLE .040" CRIMP AFTER FILLING 'TORR SEAL" / INCIDENT BEAM DIFFRACTED BEAM Figure 36. a) Front and b) top view of the aluminum sample holders used i n the neutron d i f f r a c t i o n experiment. The geometry of the sample i n the neutron beam i s shown i n b. 117 for JN-3 and a = 3.416 ± .002 A and c = 6.041 + .002 A for JN-7. Other samples were prepared as backups i n case problems were encountered, but were not needed. E l e c t r o l y t e c o - i n t e r c a l a t i o n i n these samples was moderate and indicated about a 20% loss of the L i TiS„ sample to the x 2 r e l e c t r o l y t e co-intercalated material. This f r a c t i o n i s determined by the r e l a t i v e i n t e n s i t i e s of the e l e c t r o l y t e co-intercalated Bragg peaks to the L i TiS Bragg peaks from our experience with X-ray c e l l s . A chemical analysis performed on a portion of sample JN-6 (based on 6pl3 TiS2), a backup to JN-3, by Canadian M i c r o a n a l y t i c a l Services gave .4 ± .2% hydrogen and 5.3 ± .2% carbon by weight i n t h i s sample. PC i s C^HgO^, so we expect a r a t i o of 8:1 i n the percentages by weight of carbon and hydrogen i n the sample. The error of .2% i n the hydrogen determination allows for agreement between the analysis and the p r e d i c t i o n . We w i l l assume the carbon determination i s accurate and set the weight percent of hydrogen to .6%. JN-3 and JN-7 had a smaller co-intercalated f r a c t i o n than JN-6 by roughly a factor of two. The hydrogen content i n these samples i s estimated to be .3 ± .1% by weight. An estimate of the e f f e c t i v e l i n e a r absorption constant, y'', of the samples can be made, now that the r e l a t i v e f r a c t i o n of the atoms present are known, y' includes contributions from a l l processes which remove neutrons from the incident beam, i . e . scattering and true absorption. The absorption, a and s c a t t e r i n g , a , cross-sections of the relevant 9. s atoms are given i n Table 6. The e f f e c t i v e l i n e a r absorption constant, y', i s (a ( i ) + a ( i ) ) a s M(i) w(i) 6.1 Table 6 Absorption, a , and s c a t t e r i n g , a g , cross-sections f o r the n u c l e i present i n samples JN-3 and JN-7 at a wavelength of 1.572 A. A f t e r Bacon:.: (1975) Nucleus a a a s a a + o s barns barns barns H .28 81.5 81.8 C .004 5.5 5.5 0 .0001 4.2 4.2 T i 5.1 4.4 9.5 S .41 1.2 1.6 L i 58. 1.2 59.2 119 where p i s the density of the f i n a l compound, M(i) i s the molecular t i l weight and w(i) the f r a c t i o n by weight of the i element i n the compound. We estimate .003 ± .001, .025 ± .001 and .025 ± .001 for w(i) of hydrogen, oxygen and carbon r e s p e c t i v e l y i n samples JN-3 and JN-7. This y i e l d s u' - (.7 + x) ± .2 cm - 1 where x i s the lit h i u m concentration of the sample. Hydrogen contributes about .5 cm ^ to t h i s estimate of u'. For samples JN-3 and JN-7 we fi n d u' = .8 ± . 2 cm ^ and u'' = .9 ± . 2 cm re s p e c t i v e l y . A neutron powder d i f f r a c t i o n i n t e n s i t y c a l c u l a t i o n was made for L i , T i S 0 to get an idea of which s u p e r l a t t i c e peaks would be the . l o Z strongest. This c a l c u l a t i o n p a r a l l e l s the X-ray c a l c u l a t i o n discussed i n Section 5.3. Equation A4.8 i n Appendix 4, which i s appropriate for the geometry used i n t h i s experiment, was used, with y' = 1 cm ^ and a sample thickness, d = .1 cm, for t h i s c a l c u l a t i o n . The sca t t e r i n g lengths -12 -12 fo r li t h i u m , titanium and s u l f u r are -.214 x 10 cm, -.34 x 10 cm -12 and .28 x 10 cm re s p e c t i v e l y (Bacon 1975). The r e s u l t s of t h i s c a l c u l a t i o n , f o r a randomly oriented powder i n the geometry of Figure A4.4b, are given i n Table 7. Superlattice peaks have f r a c t i o n a l £. One observes that the (003/2) peak i s the most intense s u p e r l a t t i c e peak. Neutron d i f f r a c t i o n experiments were performed using the C5 t r i p l e axis spectrometer at the Atomic Energy of Canada Ltd. NRU reactor. The components of a t r i p l e axis spectrometer are described i n many books (eg. Bacon 1975 and Cocking and Webb 1965) and w i l l not be discussed here. o A neutron wavelength, £, of 1.572 A was used f o r these experiments which Table 7 Relative neutron d i f f r a c t i o n i n t e n s i t i e s f o r L i # i 6 T i S 2 (see t e x t ) . c a = 5.700 A, c b = 6.100 A, c = 5.900 A, a = 3.412 A and X = 1.572 I h k I Intensity 20 (r e l a t i v e ) (degrees) 0 0 1/2 .59 7.64 0 0 1 16.18 15.31 0 0 3/2 2.21 23.05 1 0 0 78.80 30.85 0 0 2 51.10 30.90 1 0 1/2 .13 31.83 1 0 -1/2 1 0 1 100. 34.61 1 0 -1 1 0 3/2 1.60 38.86 1 0 -3/2 0 0 5/2 1.09 38.90 1 0 2 1.82 44.23 0 0 3 2.66 47.11 1 0 5/2 1.42 50.46 1 0 -5/2 1 1 0 2.65 54.86 1 1 1/2 .21 55.49 0 0 7/2 .01 55.58 121 was obtained using the (111) planes of a germanium monochromator c r y s t a l . In order to increase counting rates, a r e l a t i v e l y poor r e s o l u t i o n c o n f i -guration was used. The incident beam co l l i m a t i o n was 0.6° and there was no co l l i m a t i o n of the scattered beam p r i o r to the germanium (111) analyser c r y s t a l . The C5 spectrometer i s computer controlled and can be operated i n a v a r i e t y of ways. In these experiments, the scattering angle, 29, was changed by the computer i n a way to make the quantity, 2 c' sine , „ Vobs " A 6 ' 2 vary i n user defined steps. In equation 6.2, c' i s a length input to the computer by the user. If c' = c, where c i s the c l a t t i c e parameter of the specimen, (00A) powder peaks w i l l be found at v , ='Z and (hk£) obs peaks at V'-^g = c'/d(hk£). d(hk£) i s the (hk£) plane spacing. The computer requires c'' and A before i t can solve equation 6.2 for 9 i f given v D k s « If o n e sets c = c at a given temperature and does not modify c for scans taken at d i f f e r e n t temperatures, (00£) peaks w i l l appear at non-integral v 0^s due to changes i n c r e s u l t i n g from thermal expansion. In these experiments, c' was fixed and the data corrected by measuring the (001) and (002) peak positions at each temperature to obtain c and using v = —, v , , 6.3 c' obs where v ^ g i s given by equation 6.2. v i s the so l u t i o n to equation 6.2 when c = c'. Data i s obtained by pos i t i o n i n g the neutron detector at 122 a user defined value of v , for a f i x e d number of neutrons incident on obs the specimen. The number of incident neutrons i s monitored by a device c a l l e d the monitor,which counts a f r a c t i o n of the neutrons i n the incident beam. Aft e r a user defined number of monitor counts, the number of neutrons counted by the detector i s stored. The detector i s then moved to a new value of v ^ and the process repeated. At the end of the data c o l l e c t i o n process, the values of V Q ^ S a r e corrected using equation 6.3. Temperature control between 17K and 300K was achieved using an A i r Products "Displex" based cryostat. This cryostat uses a two stage Joule-Thompson gaseous expansion r e f r i g e r a t i o n system. The samples were mounted on the ce n t r a l post of the cryostat which was c r i t i c a l l y aligned with respect to the spectrometer. By using a s i l i c o n temperature sensor and a computer c o n t r o l l e d heater i n the cryostat, temperature s t a b i l i t y of ±1K was maintained. In Figure 37 we show the r e s u l t of a scan through the A l ( l l l ) peak which a r i s e s from the aluminum i n the sample holder. This data was obtained on sample JN-3 at 17K and shows the r e s o l u t i o n obtained with the spectrometer configuration described e a r l i e r . The f u l l width at h a l f maximum i s , Av = .037. This peak i s located at 20 = 39.34° and has an angular f u l l width at hal f maximum, A20 = .60°. 123 Al (111) 17K 1800 i L O LU CN 1400 o 1000 CD Q_ GO -+-> C 3 o o 600 A A 200 I I I I 1 I 1 I I 2.48 2.50 2.52 2.54 2.56 2.58 V Figure 37. The A l ( l l l ) peak from the JN-3 sample holder at 17 K. This demonstrates the spectrometer r e s o l u t i o n i n the c o n f i g u r a t i o n described i n the t e x t . Monitor 2E5 means 2 x 10^. The background i s roughly 400 counts. The parameter c' = 5.9090 A f o r t h i s scan and X = 1.572 A. 124 6.2 Results f o r Sample JN-3, L i 1 4 T i S 2 This sample underwent a complex thermal cycle, given i n Table 8, due to r e c u r r i n g problems with the heater i n the cryostat. Each time the heater f a i l e d , the sample had to be warmed to room temperature and removed from the cryostat to repair the heater. The precise alignment of the f l a t sample i n the geometry of Figure A4.4b i s d i f f i c u l t because the sample mounting arrangement i s designed to accommodate c y l i n d r i c a l samples. Differences i n the absolute i n t e n s i t i e s of Bragg peaks, measured at the same temperature, of about 10% were observed a f t e r remounting the sample. The sample was maintained at each temperature fo r the duration of the desired scans, t y p i c a l l y one to two days. In Figure 38, the (001) and (002) peaks from runs 142 and 143 at 17K are p l o t t e d . The large background i s due p r i m a r i l y to incoherent sc a t t e r i n g from the hydrogen i n the samples. The assymetry observed i n the (002) peak i s due to the (100) peak which appears at v = 2.003 for L i ^ T i S 2 at room temperature. The i n t e n s i t y c a l c u l a t i o n for L i ^ T i S 2 , Table-7, predicts that the (100) peak should be more intense than the (002) peak for a randomly oriented powder. The (005,) peaks are enhanced by a factor of 3 with respect to the (hko) peaks i n t h i s sample due to the preferred o r i e n t a t i o n created by pressing the samples. This f a c t o r i s determined by comparing the measured (100):(002) r a t i o , measured for sample JN-7 where the peaks are resolved (Figure 45), to the calculated r a t i o f o r a randomly oriented powder. The shape of the (100),(002) composite peak changes as a function of temperature due to the ahisqtropy i n the thermal expansion p a r a l l e l and perpendicular to the layers which i s c h a r a c t e r i s t i c of many layer compounds (Wiegers 1980). Table 8. The thermal h i s t o r y of sample JN-3. The temperature and scan numbers discussed i n the thesis are l i s t e d chronologically i n the table. Heater problems are also noted. A l l scans are through v = 3/2 except scans 142 and 143. Temperature Scan number(s) Figure(s 300 K J 100 K 122 39 17 K Heater F a i l u r e — 300 K 139, 140, 141 Sample Removed, Heater Repaired 39 17 K 142, 143, 144 Heater F a i l u r e 38,40a 300 K - -- Sample Removed, Heater Repaired -150 K 148 40b 100 K 156 40c 60 K - -200 K 159 40d 250 K 162, 163 40e 300 K 170 40f 175 K 174 40g a) CD LxJ o 1800 1600 h 1400 \— CD CL CO *E 1200 13 O o (001) 17K b) 1000 CD L U O O 6000 5000 C 4000 o j £ 3000 CO -«-' c 3 2000 (002) 17K 0.950 0.975 1.000 1.025 1.050 1000 1.950 1.975 2.000 2.025 2.050 V V Figure 38. The (001) (a) and (002) (b) peaks of sample JN-3, L i # 1 4 T i S 2 , at 17K..... The (002) peak shape i s complicated by the" presence of the unresolved (100) peak (see tex t ) . 127 The width of the (001) peak i n Figure 38a i s r e s o l u t i o n l i m i t e d . One observes that the background i n Figure 38 at v =1, 1110 counts, i s l e s s than the background at v = 2, 1600 counts. This i s due p r i m a r i l y to the cadmium mask which blocks an increasing portion of the sample as v becomes smaller. This e f f e c t causes the e f f e c t i v e sample siz e to change as v v a r i e s . The i n t e n s i t y of the (001) peak r e l a t i v e to the (002) peak should be scaled by the r a t i o of background i n t e n s i t i e s to correct f or t h i s e f f e c t . We estimate that the peak height of the (002) peak (Figure 38b) i s 2700 counts, 2/3 of the height of the composite (100),(002) peak. The (001) peak height i s 600 counts which, when scaled by the r a t i o of background counts, becomes 870 counts. Assuming the peak widths are the same, the (001):(002) i n t e n s i t y r a t i o i s therefore .3. This i s i n good agreement with the c a l c u l a t i o n (Table 7) which predicts .317. On the basis of t h i s c a l c u l a t i o n we expect a \ (003/2) peak of - 80 counts i n height, provided the peak has r e s o l u t i o n l i m i t e d width, at monitor 1E6 (1E6 means 1 x 10 ) i f long range staging develops. Each data point measured at monitor 1E6 required 3 minutes of counting time. In Figure 39a the r e s u l t of scan 122 (see Table 8) through v = 3/2 at 100K i s shown. Each data point represents 30 minutes of counting time at monitor 1E7. The presence of a peak v = 3/2 of roughly 400 to 500 counts i n height and a f u l l width of Av ~ .06 i s observed. The sample was then cooled to 17K . and another scan through v = 3/2 was taken which showed a s i m i l a r peak. The standard deviation a, of a data point of s counts xs a = /s" . For s = 14000, a ~ 120. The sample was then warmed to 300K . and another scan taken which i s shown i n Figure 39b. Counts per Monitor 1E7 OQ l-i CO C D G O ( U O a rt I O O J C - 0 9 • o II H CO • ho Cn CO cn o o cu oo o o o o ft o o o o o o a. CO o o 3 CD cn ro Counts per Monitor 1E7 a1 o o 7\ 129 At t h i s temperature clear evidence for a peak at v = 1.50 i s not observed and the data i s consistent with the absence of a peak. The r i s e i n count rate near v = 1.60 i s due to a weak peak at v = 1.61 believed to be due to the e l e c t r o l y t e c o-intercalated f r a c t i o n of the sample. The temperature dependence of the background l e v e l i n Figure 39 i s due to the temperature dependence of the Debye-Waller factor f o r the inco-herent scattering from the hydrogen n u c l e i . This data i s consistent with the ideas developed e a r l i e r about short range staging i n L i ^^TiS^. As the temperature i s lowered, the c o r r e l a t i o n length for staging increases and a peak i s observed at (003/2).. The d e t a i l e d temperature dependence of the (003/2)^ peak was measured and these r e s u l t s w i l l now be reported. In Figure 40 the r e s u l t of a seri e s of scans through v = 3/2 at d i f f e r e n t temperature are shown. The scans are given i n chronological sequence (see Table 8 ) , i . e . the data i n Figure 40a was measured f i r s t , 40b second, etc. The s o l i d curve through the data points i s the r e s u l t of a l e a s t squares f i t of the function W V ) = P 1 + ( v - p 4 ) p 2 + ^ e - ( ^ 4 ) 'PS + p 5 5 6.4 + P , e - ( V - P 7 ) 2 / P 8 2 6 where p^ are the parameters i n the f i t . . . In these f i t s , p 2 and p^ were' held fixed at 660 counts and 1.61 re s p e c t i v e l y , p^, p 2 , p^ a n& have units of counts and the other parameters are u n i t l e s s . This function represents two Gaussian peaks on a sloping background, one free to move and 130 a) 17K b) 14000 LU c o CX. CO 12500 12300 c 11900 H A O o 11700 150K A A A 12100 \ - * J I I I I I L 1.40 1.45 1.50 1.55 1.60 V 100K 200K 12700 V-3 12500 \ -c o t- 12300 CD Q_ W § 12100 f -O o 11900 1.55 1.60 Figure 40. Scans through v = 1.50 taken at various temperatures f o r sample JN-3, L i # i 4 T i S 2 (see t e x t ) . The points are the data and the s o l i d curve i s the le a s t squares f i t to the data described i n the text. The v e r t i c a l scale i n c i s d i f f e r e n t from a,b,d-g and should be noted. Counts per Monitor 1E7 Counts per Monitor 1E7 132 one f i x e d at v = 1.61. The purpose of these f i t s was to e x t r a c t the "best" peak p o s i t i o n ( p ^ ) , i n t e g r a t e d i n t e n s i t y ( p r o p o r t i o n a l to p^) and f u l l width at h a l f maximum (1.66 p^) of the peak near v = 1.5 as a f u n c t i o n of temperature, p^ and Pg describe the peak at v = p^ = 1.61 and are of l i t t l e concern to us. The f i t s minimized the q u a n t i t y 1=1 where I , (v.) and I ., (v.) are the observed and calculated number of obs l c a l c x t i l counts at the i value o f v. N i s the number of data points. The computer program MLSQQ i n the A.E.C.L. f o r t r a n l i b r a r y was used to perform these f i t s . The program returned the "best" values of the parameters, the expected standard deviation i n these values and R f 1 N -1 ^ (N-M) 1 N ^ obs" i 6.6 In equation 6.6, M i s the number of f r e e parameters i n equation 6.4, which i s s i x as p 2 and Py have been f i x e d , x' I s a measure of the q u a l i t y of the f i t , x' z 1 i n d i c a t i n g a good f i t . [ O r d i n a r i l y the q u a l i t y of f i t i s tested w i t h X = N 1 h-r T v2 (I , (v.) - I 1 (v.)) -M) .t, o(v.) obs x c a l c x 6.7 where a(v^) i s the standard deviation of the i * " * 1 data point (Bevington 1969). In our case 133 i- 1 because the v a r i a t i o n i n •'- o b s( v ;_) ^ s o n l y a few percent. We expect X"'" ~ X - l • In Table 9 the r e s u l t s of the f i t t i n g are tabulated. In Figure 41 the i n t e g r a t e d i n t e n s i t y of the peak near v = 1.50 as a f u n c t i o n of temperature i s shown. Figure 42 shows the peak width and Figure 43 shows the peak p o s i t i o n both as a f u n c t i o n of temperature. The e r r o r bars i n the f i g u r e are plus and minus the standard d e v i a t i o n of the r e l e v a n t parameter c a l c u l a t e d by the f i t t i n g program. I t i s obvious from the s i z e of the e r r o r bars that the data i s f a r from optimum. One n o t i c e s i n F igure 40f (300K)' that a peak i s s t i l l present near v = 3/2. This i s i n con t r a s t to the e a r l i e r data shown i n Figure 39b. The f i t t e d peak i n Figure 40f i s centered at v = 1.488, not v = 1.500. We can only speculate as to the cause of t h i s e f f e c t . I t i s p o s s i b l e that the staging process does not c y c l e w e l l thermally. I f t h i s i s the case, the (003/2) peak should s t i l l appear at v = 1.500. A f i t of the two peak model used e a r l i e r to the 300K . data shown i n Figure 40f, c o n s t r a i n i n g one peak to remain centered at v = 1.500 and the other at v = 1.61 produced a f i t to the data w i t h x' = 1.26. This i s s l i g h t l y l a r g e r than x' = 1.13 obtained when the peak near v = 1.5 was f r e e to move. A second p o s s i b i l i t y i s that the peak at v = 1.488 i n Figure 40f i s due to a process other than s t a g i n g , f o r instance from an i r r e v e r s i b l e phase t r a n s i t i o n i n the e l e c t r o l y t e c o - i n t e r c a l a t e d f r a c t i o n of the sample. However, t h i s p o s s i b i l i t y does not e x p l a i n the absence of a peak i n Figure 39b. A t h i r d p o s s i b i l i t y i s that the sample holder developed a leak and the sample d e t e r i o r a t e d . I t i s do u b t f u l that a leak developed Table 9. Least squares parameters and t h e i r standard deviations T(K) P l °1 P_ a 2 P3 a3 P4 °4 P5 a5 P 6 °6 P 8 a8 X' 17 13365 107 660 - 18.0 8.1 1.500 .003 .040 .010 229 121 1.61 - .032 .014 1.01 100 12210 54 660 - 18.4 4.5 1.500 .003 .037 .006 358 86 1.61 - .027 .007 1.11 150 11935 112 660 - 20.4 10.5 1.490 .004 .048 .015 267 115 1.61 - .023 .010 1.04 175 11791 34 660 - 9.24 3.0 1.497 .005 .030 .008 42 183 1.61 - .015 - 1.18 200 11659 22 660 - 9.8 1.7 1.500 .003 .030 .005 150 51 1.61 - .021 .009 0.69 250 11347 42 660 - 7.4 3.1 1.491 .005 .030 .010 229 76 1.61 - .0298 .013 1.14 300 10884 32 660 - 7.4 2.3 1.488 .004 .027 .008 364 75 1.61 - .018 .005 1.13 O J 135 to* » 100 200 T (K) 300 Figure 41. v = 3/2 peak i n t e n s i t y vs temperature f o r the data and f i t s of Figure 40. The t r i a n g u l a r data point a t 300K . i s the r e s u l t of scans 139, 140, 141 shown i n Figure 39b (see t e x t ) . 136 0.02 100 200 300 T (K) Figure 42. The f u l l width at hal f maximum (FWHM), A v , of the v = 3/2 peak determined from the f i t s to the data of Figure 40. 137 o (/> O CL < UJ u l.505h l.500h + 1.495h l.490h 1.485 100 200 300 T (K) Figure 43. The v =3/2 peak p o s i t i o n determined from the f i t s to the data of Figure 40. 138 because the (001) and (002),(100) peaks from L i ^ TIS^ remained r e s o l u t i o n l i m i t e d with constant i n t e n s i t y during thermal c y c l i n g . The integrated i n t e n s i t y of the peak at v = 1.500 i n Figures 40a, b and c i s consistent with the r e s u l t s of the c a l c u l a t i o n given i n Table 7. Assuming the (001) and (003/2) peaks are Gaussian and s c a l i n g the (001) peak i n t e n s i t y with respect to the (003/2) peak i n t e n s i t y by the r a t i o of background counts as explained e a r l i e r we f i n d that the r a t i o of (001) to (003/2)/ i n t e n s i t y i s 9.1 ± 3. The predicted r a t i o , computed i n Table 7 i s 7.36. Using the simple model of imperfect staging developed i n Chapter 5 and Appendix 5, we can extract the c o r r e l a t i o n length f o r staging, £c, from the width of the (003/2).' peak. We obtain £c ~ 45A using equation A5.24 and the data i n Figures 40a and 40c. D i f f i c u l t i e s are encountered when one t r i e s to deconvolute the instrumental r e s o l u t i o n , assumed to-be Gaussian (Rietveld 1969), from the observed peak shape. The q u a l i t y of the data i s too low for an assignment of peak type (Lorentzian, Gaussian etc.) to be made. Lorentzian peak shapes are predicted by the model discussed i n Appendix 5. We have obtained the c o r r e l a t i o n length given above using tables given by van de Hulst and Reesinck (1947) which describe the deconvolution of composite peaks made up of Lorentzian and Gaussian components. The f u l l width of the (003/2) peak i n Figure 40a i s Av = .064 and the Gaussian r e s o l u t i o n function of the spectrometer has Av = .037. If the natural peak shape of the (003/2) peak i s Lorentzian, i t must have a f u l l width Av = .042 (van de Hulst and Reesinck 1947) to produce the measured width when convoluted with the spectrometer r e s o l u t i o n function. This y i e l d s £ =7.6 using equation A5.24c. 139 The v a r i a t i o n of the c-axis with temperature was measured using the (001) p o s i t i o n and i s shown i n Figure 44. The (002) peak was not used f o r t h i s measurement due to the d i f f i c u l t y i n obtaining i t s exact p o s i t i o n r e s u l t i n g from the presence of the unresolved (100) peak. The points i n Figure 44 were measured a f t e r the f i n a l heater repair with the sample i n a fixed l o c a t i o n for a l l the measurements. The s o l i d l i n e i n the f i g u r e represents the behaviour of a s o l i d with a thermal expansion c o e f f i c i e n t , = ^  ^ = 47 x 10 ^/K. Whittingham and Thompson (1975) measured the thermal expansion of TiS2 and L i x T i S 2 samples. They —ft obtained a = 19.4 x 10 /K f o r TiS„ between 20 K and 300 K and c I —ft ac = 16 x 10 /K for L i T i S 2 between 150 K and 300 K. For temperatures below 100 K, a ~ 0 for the Li..TiS 0 sample. The c-axis of L i . , T i S 0 c 1 2 .14 I appears to become i n s e n s i t i v e to temperature changes below about 200 K. The temperature dependence of the c-axis above 200 K for L i ^ T i S 2 i s —6 large, ~ 50 x 10 /K, compared to the values f or TiS2 and LiTiS2-We did attempt to locate other s u p e r l a t t i c e peaks at low temperature, without success. The (103/2), (10-3/2) peak and the (005/2) peak, which would not be resolved i n t h i s experiment, are expected to have i n t e n s i t i e s comparable to the (003/2) peak (see Table 7). However, these peaks o f a l l near the 2.344 A peak of the e l e c t r o l y t e c o-intercalated f r a c t i o n (Table 4) which would make t h e i r detection d i f f i c u l t . The (105/2), (10-5/2) peak i s the next strongest s u p e r l a t t i c e peak predicted i n Table 7. Since there are no other peaks which l i e at the expected (105/2), (10-5/2) p o s i t i o n , a search at monitor 1E7 was made for th i s peak at 100K;.:. No evidence f or a peak was observed. This i s most l i k e l y due to the preferred o r i e n t a t i o n present i n the sample which could reduce 140 0 100 200 300 T (K) Figure 44. c vs T for sample JN-3, L i ^TIS^- The s o l i d l i n e i n the f i g u r e represents the behaviour of a s o l i d 0 w i t h a = 47 x 10~6/K and a c-axis of 5.9175 A at 300 K. 141 the expected peak Intensity below the background noise. Several scans were taken at values of v where s u p e r l a t t i c e peaks, which would a r i s e from a /3a ~ s u p e r l a t t i c e , would be expected. No evidence f o r peaks of t h i s type were observed. Because of time l i m i t a t i o n s , no further experiments were performed on t h i s sample. 142 6.3 R e s u l t s f o r Sample JN-7, L i 2ST±S2 F i g u r e 45 shows the r e s u l t of a scan through v = 2.0 at 300 K. The (002) and (100) peaks are p a r t i a l l y r e s o l v e d at x = .25 and are i n d i c a t e d i n the f i g u r e . The (002) peak i s twice as intense as the (100) peak. This i n d i c a t e s a t h r e e f o l d enhancement of (00£) peak i n t e n s i t y to (hkO) peak i n t e n s i t y r e s u l t i n g from p r e f e r r e d o r i e n t a t i o n . The r e s u l t s of scans at 100K . and 300K .. through v = 1.5 are shown i n F i gure 46a and b r e s p e c t i v e l y . There appears to be a weak broad peak near v = 1.5. Apart from the d i f f e r e n c e i n background, no temperature dependent e f f e c t s near v = 1.5 were observed. The v = 1.500 peak from L i ^ T i S 2 would appear at a p o s i t i o n , v = 1.530 i n Figure 46. This i s f O o because c has been changed from 5.919 A ( L i ^ T i S 2 ) to 6.041 A ( L i 2 ^ T i S 2 ) i n the spectrometer computer. The e l e c t r o l y t e c o - i n t e r c a l a t e d f r a c t i o n s of samples JN-3 and JN-7 were comparable. The absence of a peak at v = 1.530 i n F i g u r e 46 suggests that the peak at v = 1.500 i n Figures 39 and 40 i s due to the L i ^ T i S 2 and not the e l e c t r o l y t e c o - i n t e r c a l a t e d f r a c t i o n . The c-a x i s of L i 2 ^ T i S 2 at 100K was measured o u s i n g the (001) and (002) peak p o s i t i o n s and i s 6.006 ± .003 A. 3 0 0 K 3600 3000 t-C 2400 r -O g_ 1800 1200 \— 600 1.92 1.96 2.00 2.04 2.08 V gure 45. Scan through v = 2.00 at 300K . taken on sample JN-7, L i 9 C . T i S 9 . a) 100K 8900 LU 8800 \ -o CO 8500 h-c O 8400 O 8300 8700 h-8600 h A A A AA A A A J I I L 1.400 1.450 1.500 1.550 1.600 V b) 300K o o CD QL CO 8600 1\ LU 8500 8400 8300 8200 c O 8100 |— O 8000 A A A A A A A A AA A AA A A A A * A A A A A J I I L 1.400 1.450 1.500 1.550 1.600 V Figure 46. Scans through v = 1.50 at 100K (a) and 300K (b) taken on sample JN-7, L i t 2 5 T i S 2 . The measurement depicted i n (b) was performed f i r s t . 145 6.4 Discussion The data presented i n section 6.2 show strong evidence for an (00 3/2) s u p e r l a t t i c e peak at low temperatures i n L i ^TiS2« There i s no evidence for a temperature dependent peak at v = 1.50 for the L i 25^^2 s a m P x e # These r e s u l t s are consistent with the - 3.x/3V, 3V/3I) and 004 peak width measurements reported i n Chapters 3, 4 and 5 r e s p e c t i v e l y . A disordered stage 2 structure which forms near x = .16 i s responsible f or the structure present i n the data of Chapters 3, 4 and 5. As the temperature i s lowered, the degree of order improves and a (003/2) s u p e r l a t t i c e peak i s observed i n L i ^^TiS^. Even though the long range ordered stage two state may be thermo-dynamically favorable f o r L i ^^TiS^ at temperatures below 150K, i t i s most l i k e l y that the phase t r a n s i t i o n i s k i n e t i c a l l y hindered by poor l i t h i u m d i f f u s i o n as discussed i n section 6.1. This most l i k e l y explains why the (003/2) peak (Figure 40) does not sharpen further as the temperature i s lowered from 100K . to 17K... The differences observed i n the r e s u l t s at 300K.. shown i n Figures 39b and 40f are not understood. Further scans to help c l a r i f y the thermal c y c l i n g behaviour of sample JN-3, L i ^ T i S 2 were not c a r r i e d out due to l i m i t a t i o n s i n spectrometer time. The r e s u l t s of t h i s neutron d i f f r a c t i o n experiment suffered from the high background and contaminant peaks caused by e l e c t r o l y t e c o - i n t e r -c a l a t i o n which occurs i n electrochemical sample preparation with PC based e l e c t r o l y t e s . The samples were prepared electrochemically because of the high degree of i n t e r c a l a n t homogeneity desired. It i s clear that much better data could be obtained with samples where e l e c t r o l y t e c o - i n t e r c a l a t i o n i s not involved. The use of an e l e c t r o l y t e based on dioxolane during sample preparation would most l i k e l y y i e l d "clean" homogeneous samples with an absence of co-intercalated e l e c t r o l y t e . It should be pointed out that low temperature X-ray d i f f r a c t i o n on the present samples should prove u s e f u l . This i s because many of the expected s u p e r l a t t i c e peaks have i n t e n s i t i e s r e l a t i v e to the main l a t t i c e peaks comparable to the i n t e n s i t y of the (003/2) peak r e l a t i v e to the (002) i n t e n s i t y i n the neutron d i f f r a c t i o n experiment (see Tables 5 and 7). The co-intercalated e l e c t r o l y t e produces only a minor contribution to the background i n an X-ray d i f f r a c t i o n measurement and thus would not cause a major problem. The smaller sample sizes which can be used for X-ray d i f f r a c t i o n are also a t t r a c t i v e . I t i s clear that further s t r u c t u r a l work remains to be done. We now turn to the problem of understanding the data reported i n Chapters 3, 4, 5 and 6 i n terms of a l a t t i c e gas model for L i TiS„. 147 PART III LATTICE GAS MODELS OF INTERCALATION SYSTEMS C H A P T E R S E V E N THE ROLE OF LATTICE EXPANSION AND ELASTIC ENERGY IN INTERCALATION SYSTEMS 7.1 Introduction In Chapters 5 and 6 of the thesis we learned from d i f f r a c t i o n experiments that L i TiS„ forms an imperfect stage two structure for compositions near x ~ .16. The features near x ~ .16 in -3x/8V and 3V/3T) x reported in Chapters 3 and 4 i n d i c a t e an increase in order or an entropy minimum at t h i s composition. At a given i n t e r c a l a n t con-centration the entropy of a disordered stage one compound, with the int e r c a l a n t d i s t r i b u t e d randomly throughout the host, i s greater than the entropy of a stage 2 compound, where the i n t e r c a l a n t i s no longer d i s t r i b u t e d randomly throughout the host. In f a c t , because the random d i s t r i b u t i o n has the highest c o n f i g u r a t i o n a l entropy, any constraint on the d i s t r i b u t i o n of in t e r c a l a n t w i l l r e s u l t i n a decrease i n entropy. Therefore the data reported i n Part II of the t h e s i s i s consistent and indicates that the features in the electrochemical data a r i s e from the three dimensional aspects of L i TiS,,. x 2 Early attempts to explain V(x) and -3x/8V of L i / L i TiS c e l l s , X discussed i n Chapter 2 , were based on two dimensional models. These models assumed that the i n t e r a c t i o n between l i t h i u m atoms intercalated 149 i n d i f f e r e n t l a y e r s was weak enough to be neglected. This assumption i s c l e a r l y i n c o r r e c t because a staged s t r u c t u r e cannot form without some i n t e r a c t i o n between i n t e r c a l a n t i n d i f f e r e n t l a y e r s . Three dimensional l a t t i c e gas models f o r L i TiS„ must be considered i n order t o e x p l a i n the data reported i n Part I I of the t h e s i s . The i n c l u s i o n of two body r e p u l s i v e i n t e r a c t i o n s between i n t e r c a l a n t atoms i n d i f f e r e n t l a y e r s can lead to the formation of staged s t r u c t u r e s (e.g. see Safran 1980). However, phase diagrams and -8x/9V c a l c u l a t e d f o r models w i t h two body i n t e r a c t i o n s always d i s p l a y symmetry about x = 1/2. The symmetry can be broken by the i n c l u s i o n of three body i n t e r a c t i o n s (Osorio and F a l i c o v 1981) and presumably one could improve the f i t s t o the V(x) and -8x/9V data f o r L i / L i TiS„ c e l l s by t a k i n g t h i s approach. We have X £. decided to take a more basic approach which i n v o l v e s the i n c l u s i o n of the e l a s t i c energy, c a l c u l a t e d u s i n g a simple model, i n the thermodynamics. 7.2 Importance of the E l a s t i c Energy The l a t t i c e changes in L i TiS„ are r e v e r s i b l e f o r 0 < x < 1 as ° x 2 — — reported in Chapter 5. The c-axis expands by roughly 10% and the a-axis by 1.5% as x increases from 0 to 1. Ty p i c a l expansions for a and c are 1% and 10% r e s p e c t i v e l y f o r most Li^MX^ systems as x increases from 0 to 1 (Whittingham and Gamble 1975, Murphy et a l . 1976). An estimate of the e l a s t i c energy stored i n a t y p i c a l L i MX. system can be X c. . made in the case of L i NbSe„ where the e l a s t i c constants are known. x 2 For x=l, the l a t t i c e expands such that Aa/a = .013 = e^=e^and Ac/c=.080 = £j (Dahn and Haering 1982). The measured e l a s t i c constants are A.1 11 11 2 c^^+c^2 = 28.5 x 1CT , c^^ = 6.7 x 10 and c ^ = -1.0 x 10 dynes/cm (Sezerman et a l . 1980). The e l a s t i c energy computed per unit volume i s 9 3 2 E = 1/2 c.. £.£. = 2.4 x 10 ergs/cm , of which 90% a r i s e s from the c„_e„ I J 1 j *> > 33 3 term. (Neglecting the a-axis expansion causes only a 10% error i n the e l a s t i c energy.) The e l a s t i c energy per unit c e l l i s .102 eV z 4 kT at T=300K. This energy i s of the same order of magnitude as the nearest neighbour i n t e r a c t i o n energies, - 2.5 kT, needed to q u a l i t a t i v e l y f i t the V(x) data for L i TiS c e l l s (see Figure 8). E l a s t i c e f f e c t s are therefore important and should be taken into account i n a t h e o r e t i c a l treatment of i n t e r c a l a t i o n compounds l i k e L i TiS.. x 2 .Intercalated atoms d i s t o r t the host atoms and set up a s t r a i n f i e l d in the host. Since t h i s s t r a i n f i e l d a f f e c t s the energy of a second int e r c a l a t e d atom, t h i s leads to a s t r a i n mediated or e l a s t i c i n t e r a c t i o n between in t e r c a l a n t atoms. The e l a s t i c i n t e r a c t i o n has been treated f o r hydrogen in metals (Wagner 1978) , GICs (Safran and Hammann 1980) and l i t h i u m i n t e r c a l a t i o n compounds (McKinnon and Haering 198 0, McKinnon 151 1980). In these treatments the s t r a i n produced by the intercalant atoms i s characterized by an e l a s t i c dipole tensor, which i s the f i r s t moment of the forces that the atom exerts on the host around i t . These authors show that i n anisotropic media, the s t r a i n mediated i n t e r a c t i o n can be a t t r a c t i v e i n some d i r e c t i o n s and repulsive i n others. For , instance, in layered hosts one expects an a t t r a c t i v e i n t e r a c t i o n between inte r c a l a n t atoms in the same layer and a repulsive i n t e r a c t i o n between atoms placed i n d i f f e r e n t layers on a l i n e perpendicular to the layers (McKinnon and Haering 1980). Estimates of the magnitude of the i n t e r a c t i o n energies show them to be the order of kT for nearest neighbours. Safran (198 0) included two body inte r a c t i o n s which were a t t r a c t i v e i n plane and repulsive f o r atoms in d i f f e r e n t planes i n an attempt to treat e l a s t i c e f f e c t s i n a d e s c r i p t i o n of staging i n GICs. McKinnon (1980) has also replaced the d e t a i l e d e l a s t i c i n t e r a c t i o n with two body i n t e r a c t i o n s i n a discussion of L i MX„ systems. x 2 The l a t t i c e gas models discussed by Safran (1980) and McKinnon and Haering (1980) assumed that the e l a s t i c i n t e r a c t i o n energies remained constant as a function of i n t e r c a l a n t concentration. The assumption of constant i n t e r a c t i o n energies implies that the strength of the e l a s t i c dipole representing the in t e r c a l a n t atoms i s independent of inte r c a l a n t concentration. This leads to l a t t i c e expansion which i s l i n e a r i n int e r c a l a n t concentration provided there i s no staging or c l u s t e r i n g of the i n t e r c a l a n t . L a t t i c e expansion i n L i TiS i s non-linear (Figure 29). The c-axis increases r a p i d l y f o r x £ .3 and i s nearly independent of l i t h i u m concentration f o r x > .5. We expect e l a s t i c e f f e c t s in L i ^ T i S ^ to be large when x ~ .3, where dc/dx i s large, and small when x ~ .5. Rather than attempt to deal with the v a r i a t i o n of the strength of the e l a s t i c d i p o l e tensor within the formalism of the s t r a i n mediated i n t e r a c t i o n , a very simple model, which describes the e l a s t i c energy in L i TiS arid other layered i n t e r c a l a t i o n compounds, i s introduced. Interactions between atoms which a r i s e due to e l e c t r o n i c e f f e c t s w i l l be treated i n the t r a d i t i o n a l manner by introducing two body in t e r a c t i o n s between inte r c a l a n t atoms. 153 7.3 The Spring and Plate Model 7.3.1 Zero Applied Pressure The estimate of the e l a s t i c energy of L i NbSe. made in the previous X /, section showed that i f the a-axis expansion was neglected, a 10% error i n the e l a s t i c energy was introduced. Because the a-axis expansion i n most L i MX compounds i s s i m i l a r l y small, (Whittingham and Gamble 1975, Murphy X z et a l . 197 6) only the co n t r i b u t i o n of the c-axis expansion to the e l a s t i c energy i s considered i n the following model. When a li t h i u m atom i s inter c a l a t e d into a MX2 structure i t produces l o c a l d i s t o r t i o n s i n the host l a t t i c e . The competition between the l o c a l d i s t o r t i o n s and the van der Waals forces binding the layers together determines the r e s u l t i n g c-axis behaviour. In cases where the inte r c a l a t e d l i t h i u m tends to expand the l a t t i c e , the MX^  compound can be modeled " by r i g i d plates connected together by springs of spring constant K and equilibrium length c , corresponding to the c-axis of the MX^  compound (Figure 47.) These springs correspond to the van der Waals bonds between the MX^  "sandwiches". The e l a s t i c e f f e c t s ( l o c a l d i s t o r t i o n s ) r e s u l t i n g from the int e r c a l a t e d l i t h i u m atoms are modeled by springs with spring constant k and equilibrium length C ^ > C q . To ca l c u l a t e the c-axis v a r i a t i o n , we normalize the number of l a t t i c e springs to one per metal atom; i . e . N l a t t i c e springs. We now ins e r t n l i t h i u m atoms randomly throughout the host ( r e c a l l x = n/N) and apply the condition for equilibrium at zero pressure: 154 Figure 47. The springs which represent the van der Waals bonds between the host layers (7C,c0) and those which represent the l o c a l d i s t o r t i o n s near the int e r c a l a n t (fe,c^) . The host layers are modeled as r i g i d plates. 1 NK(c(x)-c ) = nfe(c T-c(x)) 7.1 o L which y i e l d s o = x K c.-c a + x ' a Tz ' 1 - 2 "L o The r e s u l t i n g e l a s t i c energy per metal atom i s | = % K ( c ( x ) - c o ) 2 + J5xfe(c L-c(x)) 2 = J X 7.3 a + x where J = % K ( c - c ) 2 . 7.4 L o In the case of L i NbSe„, J - 0.1 ev. x 2 A f i t of equation 7.2 to the measured c-axis v a r i a t i o n of L i ^ T i S ^ i s shown i n Figure 48. The parameters used were c =5..7A*, c —6..305A and a= O L i " The s i m i l a r i t y between equations 7.2 and 7.3 i n d i c a t e s that the e l a s t i c energy given by equation 7.3 increases most r a p i d l y f o r x 1 a . Because the agreement between the model and the data (Figure 48) i s reasonably good, the e f f e c t s of the e l a s t i c energy on the thermodynamics of L i MX„ X z systems w i l l be considered. This i s done i n Section 7.4 of the t h e s i s . The bonding between atoms w i t h i n a MX2 sandwich i s much stronger than the bonding between atoms i n d i f f e r e n t l a y e r s . A t a b l e of bonding a n i -s o t r o p i c s f o r layered compounds can be found i n the review by Wieting and Verble (1979) which shows that the r a t i o of f o r c e constants w i t h i n and 1 5 6 I 1 1 1 1 I I . 1 . 1 0 0.2 0.4 0.6 0.8 1.0 X in L i x T i S 2 Figure 48. The measured l a t t i c e parameters for L ^ T i S ^ from Figure 29. The s o l i d curve i s a f i t to the e-axis expansion using the spring and plate model described i n the text. 157 between the layers i s t y p i c a l l y 5 to 50 f o r MX^  compounds. This has two implications for the spring and plate model. F i r s t l y , strong i n t r a -layer bonding implies that the assumption of r i g i d MX2 layers should be good. Secondly one would expect a = K/k < 1 because the d i s t o r t i o n s near the i n t e r c a l a n t , represented by k, involve the strong i n t r a l a y e r bonds, while K represents the weak 'inter-layer bonds.. The value of -ot "= .2 needed to describe the c-axis v a r i a t i o n of L i TiS„ i s i n agreement with the bond x 2 strength arguments given above. In the spring and plate model, the spacing between two layers i s determined s o l e l y by the intercalant content in the gap between those p a r t i c u l a r layers because the plates are r i g i d . One finds that the spacing of the i t h i n t e r l a y e r gap, c^ i s c.(x.)-c x. 1 1 ° = L _ 7.5 c T - c a + x. L o l where x^ i s the f r a c t i o n of occupied s i t e s in the i t h gap. The e l a s t i c energy of the e n t i r e c r y s t a l can be found by summing the e l a s t i c energies, E_^ , for each la y e r , where E. x. — = j i _ 7.6 N T a + x. L i J i s given by equation 7.4 and N i s the number of metal atoms in a IJ single layer. The fact that the spring and plate model i s a single layer model allows for the easy treatment of staging, where d i f f e r e n t layers can have d i f f e r e n t i n t e r c a l a n t content. We note in passing that the e l a s t i c energy, predicted by the spring and plate model, of a set of two layers with x^ = 0 and x 2 = 2x i s l e s s than the e l a s t i c energy when 158 = = x. This implies that the elastic energy stored in a stage two structure is less than that stored in a stage one structure at the same intercalant composition. The elastic energy is important in determining the s t a b i l i t y of staged intercalated phases. The minimization of the elastic energy alone tends to lead to f u l l and empty layers, whose arrangement is immaterial. A repulsive interaction between intercalant in different layers is needed to produce staged structures. We w i l l return to this point in Section 7 . 4 . 159 7.3.2 Applied Pressure The changes i n l a t t i c e constant and e l a s t i c energy as a function of applied pressure can be calculated simply within the framework of the spring and plate model. Because the a-axis i s incompressible i n the model, no d i s t i n c t i o n between hydrostatic and u n i a x i a l (along c) pressure w i l l be made. For a pressure, P, the force per l a t t i c e spring, fj, due to the applied pressure i s A/N^ i s the area per l a t t i c e spring or per metal atom, as we have taken one l a t t i c e spring per metal atom. The condition for equilibrium i s I = PA/N L 7.7 I + K(c.,(x.,)-c ) = x.fe(c-c..(x.)) g X X O X L 1 1 7.8 The layer spacing and e l a s t i c energy are given by c. (x.)-c x x o x. x i 7.9a c L - c o a + x . ~ fe(a+x.)(c -c ) x x L o x 7.9b a + x. x and 7.10 where 1 6 0 L o 7 . 1 1 {, i s u n i t l e s s . The mechanical p o t e n t i a l energy, Pv, where v i s the volume of the system, i s needed i n the Gibbs free energy. The volume of the i t h layer. v., xs x v. = N — c . x L N x J _ i 7.12 Using equations 7.4, 7.9b and 7.11 Pv. ] = 6c. = 23 x a + x. a a - a x *> J c T -c L o 7 . 1 3 The sum of the e l a s t i c and mechanical energies i s E. + Pv. x x x L o 7 . 1 4 We notice that constant applied pressure causes an increase i n the v a r i a t i o n of (E + Pv^)/N^ due to changes i n x_^  by renormalizing the 2 magnitude of J to J ( l + ^'/a) • A s h'~* 0» equation 7.14 reduces to equation 7.6. For $--'/a ~ 1 the applied pressure w i l l have a large e f f e c t on the thermodynamics. Using equations 7.4, 7.7, the d e f i n i t i o n of ot and K = c 3 3 A / c o N L we fi n d /a = 1 when P = c 0 0 °L . Co i f c Q O i s the 3 3 ' 3 3 e l a s t i c constant of the unintercalated host material. For (c: -c )/c =.1 L o o 1 6 1 ( s i m i l a r to L i ^ T i S ^ and c = 5 x 10 dynes/cm , ft'/a = 1 when P = 5 x 1 0 1 0 dynes/cm 2 = 5 x 10^ bar. For P « c (c -c ) / c the change i n o ' o mechanical energy per unit c e l l upon i n t e r c a l a t i o n of x. atoms i s 7.15 For Ax = 1 in L i TiS. PAy, • x = 3.7 ueV/bar Interaction energies of several meV are needed to f i t V(x) of L i / L i x T i S 2 c e l l s (Figure 8). On the scale of meV, energies of a few ueV can be neglected. Our estimate given above implies that pressure of a few atmospheres w i l l have very l i t t l e e f f e c t on the thermodynamics of i n t e r -c a l a t i o n systems l i k e L i ^ T i S ^ . The Helmholtz and Gibbs free energies of the i n t e r c a l a t i o n compound need not be distinguished at pressures of a few atmospheres as stated i n Section 2.1 of the t h e s i s . However at pressures 3 of order 10 bar, the d i s t i n c t i o n must be made. 162 7.4 I n c l u s i o n of the E l a s t i c Energy i n L a t t i c e Gas Models of I n t e r c a l a t i o n 7.4.1 Mean F i e l d Theory The s t a t i s t i c a l mechanics of l a t t i c e gas models has been discussed b r i e f l y i n Section 2.4. The n o t a t i o n used there w i l l a l s o be used here. We begin by t r e a t i n g the l a t t i c e gas i n the absence of two body i n t e r -a c t i o n s and show the e f f e c t s of the i n c l u s i o n of the e l a s t i c energy, c a l c u l a t e d u s i n g the sprin g and p l a t e model, on the thermodynamics. I n t e r a c t i o n s w i l l be included i n the mean f i e l d approximation l a t e r i n t h i s s e c t i o n and i n more complex approximations i n Sections 7.4.2 and 7.4.3. As i n the previous s e c t i o n a system of N equivalent s i t e s , w i t h L l a y e r s and N s i t e s per l a y e r i s considered. The s i t e energy i s denoted J_» as E^. The i n t e r n a l energy, u\, of the i t h l a y e r , having a f r a c t i o n , x_^ , of i t s s i t e s occupied i s U. = N E x. + E.(x.) 7 - 1 6 i L o i i i where E^ Cx )^ i s the e l a s t i c energy per l a y e r given by equation 7.10. The entropy of the i t h l a y e r , S_^  i s given by the logarithm of the number of ways of arranging x.N atoms on N s i t e s , 1 L i I J S. = k In l V ( x . N L ) ! (N L-x.N L)! 7.17a = —kN ( x . l n ( x . ) + ( l - x . ) l n ( l - x . ) ) 7.17b where k i s - Boltzmann's constant. S t i r l i n g ' s approximation has been used to obtain 7.17b from 7.17a. The Gibbs f r e e energy, G., of the i t h l a y e r 163 G. = U. - TS. + Pv. 1 1 1 1 i s G . = N l L E x. + kT(x.ln(x.) + (1-x.) ln( 1-x.) ) + o 1 1 1 1 1 + J / a ) 2 — ^ (i'/a)2 + ^--J^- 1 1 0 a + x. 0 a c T - c J ^ -• L o 1 J 7.18a In obtaining 7.18a, we have used equation 7,. 14 for the e l a s t i c and mechanical energy of the i t h la y e r . The Gibbs free energy of the ent i r e system, G = T G. i-1 1 ist be a minimum i n equilibrium at each value of x l L L .L, l i = l 9G. In equilibrium, each of the layer chemical p o t e n t i a l s , = — - — } 1 3G must be equal to u = — — . The layer chemical p o t e n t i a l i s . N dX T dX . L 1 y . = E + Jd+K'/a) 2 ^ — ^ 7 + kT l n ( x . / ( l - x . ) ) T " U , • • / . Z 1 1 (a + x.) 1 o7.18b and 3xj 9P ± kT x.(l-x.) I l - 2J(l+j(*/a)' (a + x ±) -1 7.18c 164 At t h i s point we note that the e f f e c t s of pressure appear as a factor multiplying J i n equations 7.18a, b and c i f the zero of free energy i s 2 redefined properly. For t h i s reason (l+^'/ a) w i l l be dropped throughout the following discussion. If pressure e f f e c t s are desired, J should be 2 replaced by J(l+jJ'/ a) i n what follows. For 9x./9u. > 0 at a l l values of x., there i s a one to one i i x correspondence between and x^; i . e . the function x^(y_^) i s single valued. 9x./9u. > Q at a l l values of x. i f x x x , m 2x.(l-x.)a .- . -(a + x J at a l l values of x.. In t h i s case, each of the layers has x. = x. The x x maximum of the function g(a,x^) occurs at x c given by x c = 1 + a - / l + a + a i . 7.20 For kT/J < ( k T / j ) c = g(a,x c) , 7.21 the function x(vu) i s t r i p l e valued over some range of vu. This means that d i f f e r e n t layers can have d i f f e r e n t compositions when kT/J < (kT/J) c-In Figure 49, G ± versus x for kT/J > ( k T / j ) c and kT/J < CkT/J) c i s shown. When kT/J < (kT/J) , we observe that the t o t a l free energy, G, can be c minimized for x± <_x <x„ (see Figure 49) i f a f r a c t i o n -—-— of the x x layers have composition x^ and a f r a c t i o n 2•• have composition x ^ x 2 - x 1 165 Figure 49. The Gibbs free energy of layer i , G i 5 versus x. at kT/J =1.0 and k T / j =1.5 with a"= 0.2. 1 For a = 0.2, (kT/J) =1.344. The dashed : l i n e c shows the Gibbs free, energy per layer i n the two phase region (see t e x t ) . 166 For x^ <_ x < _ X 2 > the free energy per layer, G/L, i s given by the tangent l i n e which i s dashed i n Figure 49. For kT/J < (kT/J) , the system phase separates into layers of composition x^ and layers of composition when x 1 < x < x„. The model makes no predictions as to the arrangement of the calculated for a - 0.2. The maximum temperature for two phase formation the dependence of T^ and x c on a i s given. The a p p l i c a t i o n of pressure 2 (replace J with J(l+$'/a) i n equation 7.21) causes an increase i n ( k T / j ) ^ . In Figure 52, V(x) and -9x/8V are plotted at d i f f e r e n t values of J for a = 0.2. The e f f e c t of the ^elastic energy i s to decrease the voltage and to increase-8x/8V at small x. For j/kT > (J/kT) the system i s driven c to a f i r s t order phase t r a n s i t i o n which i s evidenced by 8V/9x = 0 in the two phase region. The peak in -9x/9V i s found at x - x^ when the system remains single phase. The symmetry about x = 1/2 in -9x/3V when J=0 i s removed when J^O. We now consider the i n c l u s i o n of an inte r a c t i o n , u , between int e r c a l a n t atoms occupying nearest neighbor s i t e s i n the same layer which w i l l be treated i n mean f i e l d theory (see Section 2.4). The s i t e s within a si n g l e layer are taken to l i e on a two dimensional tri a n g u l a r l a t t i c e . At zero layers of d i f f e r e n t composition. Figure 50 shows the phase diagram i s T c = J ( k T / j ) c / k which corresponds to the composition x c- In Figure 51 pressure x. x 7.22a G. = N. a + x. x X u = E + 6ux. + *x o x Ja - + kT l n ( x . / ( l - x . ) ) 7.22b 167 1.5 I.2H 0.9H I -0.6 0.3H 1 1 1 1 Cf = 0.2 Single Phase Co - Existing — Phases i i .2 .4 .6 .8 1.0 Figure 50. The phase diagram of the l a t t i c e gas with Gibbs free energy per layer given by equation 7.18a when j$'=0 and a=0.2 (see t e x t ) . The e l a s t i c energy, calculated using the spring and plate model, has been included i n the Gibbs free energy. 168 Figure 51. The v a r i a t i o n s of ( k T / j ) c ( s o l i d curve) and x (dashed curve) with a (see t e x t ) . Figure 52. V(x) and -9x/8V for the l a t t i c e gas described by equations 7.18a, b and c with ^'=0, a=0.2 and J-0, J=.67kT and J=kT (see t e x t ) . 170 and Ja + 6u - 7.22c The e f f e c t of pressure on the system can be calculated by replacing J 2 with' j£l+|f/-/a). i n equations 7.22. In Figure 53, the phase diagram for J=u and a=0.2 i s given. We note that when u^O the two phase region does not extend to x=l at zero temperature. An attempt to f i t the V(x) and -3x/3V data of Li/-Li TiS c e l l s i s shown in Figure 54. The points i n the fi g u r e are the data presented i n Figure 14. The s o l i d curves are V(x) and -3x/3V calculated using equations and a=0.2. a i s determined from the f i t to the c-axis expansion (Figure 48). The dashed curves i n Figure 54 have been given previously i n Figure 8 and correspond to y =0. E =-^ 2.30 eV, u=.064 eV, J=0 and T=300K. The in c l u s i o n a o of the elastic.energy improves the f i t to V(x) and breaks the symmetry about x=l/2 i n -3x/3V. No provision has been made for staging i n the c a l c u l a t i o n , which i s an important process i n L i TiS„. The c a p a b i l i t y to X Z-deal with staging w i l l now be included i n the t h e o r e t i c a l treatment. 2.2 and 7.22 with T=300K, y. =0.0, E = -2.464 eV, u=.087 eV, J=.032 eV a o 171 1.2 " 5 0.6 1 i i l u = J — or = 0.2 Single — »l \ Phase " C o - > — Existing Phases \ i i i .2 .4 .6 .8 1.0 Figure 53. The phase diagram for the l a t t i c e gas with v..'. r i n t e r a c t i o n s described by equations 7.22 a, b and c for u=J and a=0.2. The two body in t e r a c t i o n s have been treated i n mean f i e l d theory. The e l a s t i c energy, calculated using the spring and plate model, has been included i n the Gibbs free energy (see text) 172 0 .2 .4 .6 .8 1.0 X in Li xTiS2 Figure 54. F i t s to V(x) and - d x / d V of L i / L i TiS„ c e l l s using the mean f i e l d theory described i n the text. The points are the data of Figure 14. The s o l i d curves are the f i t s obtained when the e l a s t i c energy i s included i n the free energy. The dashed curves are obtained when the e l a s t i c energy i s not included i n the free energy (see t e x t ) . 173 7.4.2 The Bragg-Williams Approximation In order to s t a b i l i z e staged structures, i n t e r a c t i o n s between int e r c a l a n t atoms i n d i f f e r e n t layers must be included i n the l a t t i c e gas model. For the time being a repulsive i n t e r a c t i o n , u', between atoms in nearest neighbor layers i s included. We w i l l take t h i s i n t e r a c t i o n to depend only on the average occupancy of the involved l a y e r s . At low temperatures, stage two structures are expected to form to avoid the in t e r a c t i o n u'. We div i d e the system into two s u b l a t t i c e s commensurate with the expected ground state structure (Lee et a l . 1980, Dahn et a l . 1982) . The two sub l a t t i c e s are assumed to have f r a c t i o n s x^ and x 2 of t h e i r s i t e s f i l l e d . The mean f i e l d approximation i s made within each l a y e r , that i s , the atoms.within a si n g l e layer are assumed to be randomly d i s t r i b u t e d i n spite of the i n t e r a c t i o n , u. The entropy, S^, of layer i i s given by equation 7.17b. This procedure of decomposing the l a t t i c e of s i t e s into s u b l a t t i c e s and t r e a t i n g each s u b l a t t i c e i n mean f i e l d theory i s known as the Bragg-Williams Approximation (Bragg and Williams 1934) . The Gibbs free energy i s N L 2 r Jx. ^ G = -=- ) 1 E.x. + u'x.x... + 3ux. + — ^ - - TS. 2 . L. v o x x x+1 x a + x. x x=l x J 7.23 Modulo two arithmetic has been used i n the subscripts. In equilibrium, the chemical p o t e n t i a l , y\ for each s u b l a t t i c e must be equal to ]l, i . e . 9G V = 9n~ P;T I 31 N 9x 9G ] 9n., P,T t' = U i 3n, = 2 9G m N 9x . i „ „ 7.24 where x ^(x^+x^)/ 2, n=xN and n_^=x^N/2. Using 7.23 and 7.24, s e l f consistent 174 equations for each:!df the x^ are obtained as described by Lee et al.(1980) We f i n d x. = i where 1 + e e { X i } / k T J " 1 ^ . = l j 2 7 2 5 e(x.} = E - y + 2u' x.,. + 6ux. + - — — — K . 7.26 i o l+l I . • ,-,,2 (a + x-£) These equations can be solved i t e r a t i v e l y (Harris and Berlinsky 1979) . When the procedure converges ( s e e Appendix 6), one obtains x^ as a function of \x and T. When equations 7.26 have more than one sol u t i o n , only the solut i o n which gives the lowest free energy i s taken. The values of x^ , and x 2 which minimize G for each value of x are found i n t h i s way. The stage 1 structure (x^=x 2) gives the lowest free energy for a l l values of x at high temperature. At low temperatures, the stage two structure ( x ^ x 2 ) may have the minimum free energy at c e r t a i n values of x. The Landau expansion (Landau and L i f s h i t z 1969), a standard method of determining phase diagrams, i s unsuccessful when applied to equation 7.23 except i n c e r t a i n cases due to the n o n - l i n e a r i t y of the e l a s t i c energy (see Appendix 6). Phase diagrams have been determined by the i t e r a t i v e technique described above which was implemented on a computer. In Figure 55 phase diagrams for u=0 and a=0.2 with a) J=0, b) u'=2j and c) u'=0 are shown. When J=0, the phase t r a n s i t i o n from stage 1 to stage 2 i s second order and the phase diagram i s symmetric about x=l/2. When u'=2j, the t r a n s i t i o n from stage 1 to stage 2 i s f i r s t order, except 175 a) 1.0 > 0.5 b) i 1 a) J = 0 / / V- h Stage 1 Stage 2 \ b) u'«=2J O 1.0 c) u '=0 Stage 1 Co-Existing Stage 1 Phases .2 .4 . 6 X .8 1.0 Figure 55. Phase diagrams for a=0.2, u=0 and a) J=0, b) u'=2J and c) u'=0 (see t e x t ) . Second order t r a n s i t i o n s are denoted by dashed curves. The co-existing phase regions associated with f i r s t order t r a n s i t i o n s are indicated i n b) by a plus (+) sign. For example 1+2 means co-existing stage 1 and stage 2 phases. 176 f o r the dashed p o r t i o n of the diagram at h i g h temperature which i s second order. The symmetry about x=l/2, evident when J-0, i s removed when J^Q. The e f f e c t of decreasing u' w i t h respect to J i s to decrease the width of the s i n g l e phase stage 2 r e g i o n . When u'=0, the stage two phase i s absent and the system e x i s t s as a mixture of c o e x i s t i n g stage 1 phases over a wide range of x at low temperature. In Figure 56 we show V(x) and -9x/3V c a l c u l a t e d f o r kT/j=1.6, u'=2J and u=0 (phase diagram i n Figure 55b). We note that -3x/3V d i s p l a y s a step at the continuous t r a n s i t i o n , and an i n f i n i t y at the f i r s t order t r a n s i t i o n . -3x/3V i s not-symmetric about x=l/2 and d i s p l a y s the two peaked s t r u c t u r e c h a r a c t e r i s t i c of -3x/3V f o r L i / L i TiS„ c e l l s . x 2 In Figure 57 the phase diagram f o r J=.4u'-= .7u and a=0.2 i s shown. The mixed phase regions do not extend to x=l at zero temperature as i n Figure 55 when u=0.' Of s p e c i a l i n t e r e s t i s the f a c t that the stage two s t r u c t u r e i s not formed at any temperature at x=l/2. The problem of i n plane i n t e r c a l a n t o rdering can be t r e a t e d simply i n the Bragg-Williams approximation (Osorio and F a l i c o v 1981, McKinnon 1980). Each of the two s u b l a t t i c e s considered e a r l i e r i n .this s e c t i o n i s f u r t h e r subdivided i n t o three s u b l a t t i c e s based on the decomposition of the t r i a n g u l a r l a t t i c e i n t o s u b l a t t i c e s shown i n Figure 9. The Gibbs f r e e energy i s w r i t t e n i n terms of the f r a c t i o n a l occupations, x^, f o r each of the 6 s u b l a t t i c e s . The c o n d i t i o n of equal chemical p o t e n t i a l s (see equation 7.24) i s a p p l i e d as before to o b t a i n s e l f c o n s i s t e n t equations for the x^ as a f u n c t i o n of ] i and T. In Figure 58, the phase diagram f o r u=u'=.5J and a =0.1 i s shown. The i n t e r a c t i o n between atoms i n d i f f e r e n t l a y e r s , u*, has been taken to depend only on the average co n c e n t r a t i o n of the involved l a y e r s as before. The phases 1' Figure 56. V(x) and -3x/9V calculated for kT/J=1.6 in Figure 55b. 178 l 1 1 r J =.4u' = .7u , or =0.2 Figure 57. The phase diagram for J=.4u'=.7u and a=0.2. The stage 1 to 2 t r a n s i t i o n i s continuous where indicated by the dashed curve. The two phase regions associated with the f i r s t order t r a n s i t i o n s are indicated by the regions 1+2 and 2+1. The horizontal arrow i n the fi g u r e corresponds to the value of kT/J used for the ca l c u l a t i o n s of V(x) , -9x/9V and 9 s / 9 x ) T given i n Figures 59, 6Q and 61. 179 Figure 58. The phase diagram f o r u=u'=.5J and a=0.1 (see t e x t ) . The dotted portions of the curves have not been determined exactly. 1 8 0 and 2' have the i n t e r c a l a n t ordered on the /3a s u p e r l a t t i c e (see Figure 9) and are stage 1 and stage 2 r e s p e c t i v e l y . The t r a n s i t i o n s between the phases are f i r s t order in a l l cases. Where the mixed phase regions are not indicated (e.g. 1' to 1 t r a n s i t i o n ) the region of coexisting phases i s about the width of the l i n e separating the phases. The model i s capable of describing staging and ordering. I t i s c l e a r that in the absence of experimental data to a i d in the choice of parameters, a wide range of behaviour can be predicted with t h i s model. Higher stage structures can be considered a l s o , by including longer ranged i n t e r a c t i o n s between int e r c a l a n t atoms i n d i f f e r e n t l a y e r s . To include stage 3 structures an i n t e r a c t i o n , u", between atoms separated by two layers must be included. When we are not interested in in-plane ordering, the system can be divided into 6 s u b l a t t i c e s , commensurate with the expected stage 2 and stage 3 structures. The c a l c u l a t i o n of phase diagrams, V(x) and -9x/9V i s e a s i l y c a r r i e d out using the Bragg-Williams approximation as described e a r l i e r . In Figure 71a the phase diagram for u'=J=10u", u=0 and a=0.2 i s shown. The i n t e r a c t i o n between atoms in d i f f e r e n t layers has been taken to depend on the average concentration of the involved layers as before. This phase diagram w i l l be discussed further i n Section 7.5. In Figures 59, 60, 61 and 62, V(x) , -3x/9V, 9 s / 9 x ) T and c"(x) r e s p e c t i v e l y have been calculated by minimizing equation 7.23 with respect to x± and x 2 for a=0.2, J=1.4 kT, u=2.0 kT, u«=3.5 kT, T=300K, Eq=-2.45 eV, o o. c =5.700 A and c T = 6.305 A. These values of parameters correspond to a o L cut across the phase diagram in Figure 57 at kT/J=.71, marked by the h o r i z o n t a l arrow. Equations 7.17 and 7.5 are used to c a l c u l a t e S and c = (c 1+c_)/2 r e s p e c t i v e l y . The V(x) and -9x/9V c a l c u l a t i o n s (Figures 59 Figure 59. V(x) c a l c u l a t e d f o r a=Q.2, J=1.4 kT, u=2.0 kT, u'=3.5 kT, E 0 = -2.45 eV and T=30QK as described i n the t e x t . Figure 60. -3x/3V corresponding to V(x) shown i n Figure 59. Figure 61. 3S/9x) vs x calculated f o r a=0.2, J='l.4 kT, u=2.Q kT, U'=3.5 kT, EQ=-2.45 eV and T=300K as described i n the text. 184 185 and 60) are in q u a l i t a t i v e agreement with the L i ^ T i S ^ data (e.g. Figures 17 and 16). The c a l c u l a t i o n predicts a single phase stage two state for .55 <_x < .155. The f i r s t order t r a n s i t i o n s between stage 1 and stage 2 are evidenced by i n f i n i t i e s i n - 9 x / 9 V (Figure 60). The reader i s reminded that 9S/9x) i s related to 9V/9T) by equation 2.10, so the c a l c u l a t i o n i n Figure 61 should be compared to the data in Figures 24 and 25. The double peaked structure i n the -3x/9V c a l c u l a t i o n and the prediction of a stage two state for .55 <^  x < .155 are encouraging. The fact that the stage two structure i s s t a b i l i z e d at small x i s due s o l e l y to the presence of the e l a s t i c energy in the Gibbs free energy. The c a l c u l a t i o n s (Figures 59-62) are i n q u a l i t a t i v e agreement with the data except for the p r e d i c t i o n of a perfect stage two structure which i s not formed i n L i x T i S 2 at room temperature. The f i r s t order t r a n s i t i o n s to and from the stage two structure are also not observed experimentally. The Bragg-Williams approximation i s not capable of dealing with short range order and imperfect staging so we do not expect the r e s u l t s of the c a l c u l a t i o n described above to be q u a n t i t a t i v e l y applicable to Li^TiS,,. To deal with short range e f f e c t s , Monte Carlo simulations were made and w i l l now be discussed 186 7.A.3 Monte Carlo Simulations Monte Carlo simulations on the Hamiltonian, H = NE x + o f I ( u' x k x k + i + ^ T x 7 } + l I. u x i k x i j ' 7 ' 2 7 k ^ k •* k <x,j> J using the method developed by Metropolis et a l . (1953) have been made in an attempt to understand L i TiS„. The Monte Carlo method i s described i n X Z. Appendix 7. In equation 7.27 x i s the average intercalant concentration of the l a t t i c e , x^ i s the average concentration of layer k and x ^ i s the occupancy of s i t e i in layer k ( x ^ = 0 or 1) . The other quantities i n equation 7.27 have the same meaning as in the previous section. £ denotes « i j > the sum over nearest neighbor p a i r s i n layer k. The simulations were made on ZxZxm l a t t i c e s , t r i a n g u l a r in two dimensions, with periodic boundary 2 conditions. Z = N/L i s the number of s i t e s per layer and m=L i s the number of layers. Values of Z between 3 and 6 and m between 4 and 8 were examined to determine the e f f e c t of the boundary conditions on the r e s u l t s . Stage two structures are compatible with periodic boundary conditions only when m i s even. The /Ja in plane ordered structure (Figure 9) i s commensurate with the boundary conditions when Z i s an i n t e g r a l multiple of 3. We expect staging to be p r e f e r e n t i a l l y enhanced and suppressed for m even and odd respectively because of the small sample sizes considered. Simulations with free boundary conditions were not performed. 187 The i n t e r p r e t a t i o n of Monte Carlo r e s u l t s i s often d i f f i c u l t due to f i n i t e sample siz e (Binder 1979). This i s because of the boundary e f f e c t s discussed above and the fac t that f i n i t e systems do not exhibit "sharp" phase t r a n s i t i o n s . By t h i s we mean that the i n f i n i t i e s i n thermodynamic response functions which occur at phase t r a n s i t i o n s for i n f i n i t e systems are not observed f o r f i n i t e systems. This makes the detection of phase t r a n s i t i o n s d i f f i c u l t i n some cases. One also expects differences between thermodynamic quantities calculated for the i n f i n i t e system and those obtained for an &c£x£ system of order 1/Z (Binder 1979). However, the amount of computer time required to obtain Monte Carlo r e s u l t s of the same accuracy on d i f f e r e n t s i z e systems increases r a p i d l y as L increases. This makes i t economically u n a t t r a c t i v e to study systems approaching i n f i n i t e s i z e . The Monte Carlo simulations made i n an attempt to describe L i TiS„ v x 2 were motivated in the following way. The experimental data indicates that the staged structure which forms near x ~ .16 i s imperfect at room temperature, i . e . the c o r r e l a t i o n s are short ranged (see Chapter 5 and Appendix 5). The Monte Carlo technique i s capable of dealing with short range c o r r e l a t i o n s . Even i f a sharp phase t r a n s i t i o n to a long range ordered stage 2 state would occur for an i n f i n i t e system described by equation 7.27, the study of f i n i t e systems using Monte Carlo allows one to "suppress" the long range order so that the e f f e c t s of short range order can be studied. Therefore, we expect Monte Carlo simulations on f i n i t e l a t t i c e s using the Hamiltonian i n equation 7.27 to be h e l p f u l i n understanding L i TiS„. The r e s u l t s of these simulations should not be interpreted as corresponding to those of an i n f i n i t e system described by equation 7.27 because of the f i n i t e s i z e . e f f e c t s discussed e a r l i e r . The Monte Carlo method i s well suited to the c a l c u l a t i o n of ensembl averages and c o r r e l a t i o n functions. As i s described in Appendix 7, the ensemble averages <x>, <x >, i m c = - Y <:c.(x.)> 7.28 m : L x x x=l and = - J ( < x > - <x> ) 7.29 m . L , x x=l are calculated as functions of y and T. c ^ ( x ^ ) I s g l v e n by equation 7.5 9x/9y can be obtained from <x2>-<x>2 using equation 2.13. The quantity Y measures the tendency of the system to have d i f f e r e n t amounts of i n t e r calant in d i f f e r e n t l a y e r s . The d i f f e r e n c e between f i r s t and second neighbor layer c o r r e l a t i o n functions, W 1 m f V W - m I < ( 2 x . - l ) ( 2 x . + 1 - l ) > - < ( 2 x . - l ) ( 2 x . + 2 - l ) > 7.30 x= 1 •> vanishes for a l l staged structures except stage 2. Contact with the simple 2 layer model of imperfect staging developed in Appendix 5 can be made so that the (00£) peak width can be calculated from the Monte Carlo r e s u l t s . For a system with equal numbers of two c h a r a c t e r i s t i c layer spacings, c and c, (corresponding to x and x, ), arranged with staging 3 . D 3. D c o r r e l a t i o n s described by the quantity A in equation A5.3, Y = f 7.31a 189 and W = 16A(1-4A)Y . 7.31b The Monte Carlo r e s u l t s for Y and W can be used to obtain A and x -x. , a b' hence and c^ using equation 7.5, for the system of layers characterized by c a » ' c k a n c^ ^ with the same values of Y and W . The (00£) peak width i s then obtained using equation A5.27a. The r e s u l t s of Monte Carlo simulations on 4x4x4 and 6x6x6 l a t t i c e s are shown i n Figures 63 to 70. The parameters used in these c a l c u l a t i o n s o o are u=2.0 kT, u'=3.5 kT, a=0.2, c =5.700 A, c=6.305 A, E =-2.455 eV, ' ' ' o ' L o T=300K, J=1.4 kT for the 6x6x6.lattice and J=1.5kT for the 4x4x4 l a t t i c e . In Figure 63 the Monte Carlo r e s u l t s are compared with the V(x) data f o r L i / L i x T i S 2 c e l l s given previously i n Figure 17.. In Figures 64 and 65 the -9x/9V r e s u l t s for the 4x4x4 and 6x6x6.lattices r e s p e c t i v e l y are plotted along with the data of Figure 16. The double peaked structure evident in the -9x/9V data i s reproduced quite well by the c a l c u l a t i o n , although the c a l c u l a t i o n gives peaks of equal strength. The minimum in -9x/9V near x ~ .16 i s due to the formation of a short ranged stage 2 structure. A general increase in the strength and sharpness of the peaks in -9x/9V was observed as the si z e of the basal planes of the samples was increased. The -9x/3V r e s u l t s do not depend strongly on whether the boundary conditions are commensurate or incommensurate with" the v^ 5a ordered structure. This i s because u=2kT i s much l e s s than u ~ 2.9kT which marks the onset of long range order in the t r i a n g u l a r l a t t i c e gas (Berlinsky et a l . 1979). -9x/3V for a 4x4x5 l a t t i c e also showed the c h a r a c t e r i s t i c double peaked structure even though the boundary conditions are not commensurate with 190 Figure 63. V(x) for L i / L i x T i S 2 c e l l s . The s o l i d curves are the data for charge and discharge previously shown in Figure 17. The Monte Carlo r e s u l t s for the 6x6x6 l a t t i c e (diamonds) have been sh i f t e d by +.050 v o l t s for c l a r i t y and the 4x4x4 r e s u l t s (crosses) have been shifted by +.100 v o l t s . 1 9 1 o > > X 4 . 0 -w 3 . 0 -2 . 0 1 . 0 0 . 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 X in L i x T i S 2 1 . 0 Figure 64. -dx/9V for L i / L i x T i S 2 c e l l s . The data previously shown in Figure 16 i s reproduced here for comparison with the Monte Carlo r e s u l t s from the 4x4x4 l a t t i c e . The t r i a n g l e s are the charge data, the diamonds are the discharge data and the x's are the Monte Carlo r e s u l t s . The s o l i d curve through the Monte Carlo points has been drawn to guide the eye. X in L i x T i S 2 Figure 65. -9x/3V for L i / L i x T i S 2 c e l l s . This f i g u r e i s i d e n t i c a l to Figure 64 except that the Monte Carlo r e s u l t s are for the 6x6x6 l a t t i c e . 193 a stage 2 s t r u c t u r e . Figure 66 and 67 show the Monte Carlo r e s u l t s f o r 9V/3T) x c a l c u l a t e d f o r the 4x4x4 and 6x6x6 l a t t i c e s r e s p e c t i v e l y . The c a l c u l a t i o n s were performed by t a k i n g f i n i t e d i f f e r e n c e s between Monte Carlo runs at 300K and 330K. The data, of Figure 24 i s reproduced i n Figures 66 and 67 f o r comparison. As discussed i n Chapter 4, a constant s h i f t of -12 0 yV/K has been added to the c a l c u l a t i o n . The 4x4x4 r e s u l t s agree e x c e p t i o n a l l y w e l l w i t h the data. The 6x6x6 c a l c u l a t i o n overestimates the s t r e n g t h of the f e a t u r e near x ~ .16. The regions near x = 1/3 and x = 2/3 i n Figure 67 where 9V/9T) . i s s l o w l y v a r y i n g are i n d i c a t i v e of short range order on the /3~a s u p e r l a t t i c e w i t h i n each l a y e r . The f a c t that the data shows no evidence f o r short range order at x = 1/3 or x = 2/3 i s most l i k e l y due to the presence of longer range i n t e r a c t i o n s (e.g. 2nd neighbor) between i n t e r c a l a n t atoms i n each l a y e r which would suppress the formation of the v^ 3a ordered s t a t e . Figure 68 shows c(x) c a l c u l a t e d f o r the 4x4x4 l a t t i c e . The r e s u l t s f o r the 6x6x6 l a t t i c e are s i m i l a r . The L i TiS„ d a t a . i s reproduced here x 2 f o r comparison. The formation of the short range stage two s t r u c t u r e near x .16 causes a r e d u c t i o n i n c compared to the value of c p r e d i c t e d by equation 7.2 (Figure 48) f o r 0 ~ x ~ .4. The f i t to the data i s q u i t e good. 1 9c The thermal expansion c o e f f i c i e n t , a = — -) has a l s o been c a l c u l a t e d r c c aT x f o r 300K. The r e s u l t s f o r the 6x6x6 l a t t i c e are shown i n Figure 69. In t h i s c a l c u l a t i o n the anharmonic e f f e c t s that lead to thermal expansion i n simple s o l i d s have been neglected. The expansion shown i n Figure 69 comes about due to the temperature dependence of the staging process. 194 200 100 > 0 I -> -lOOh -200h -300 i r i r • 0 • • T J I I I I I L .2 .4 .6 . 8 1.0 X in L i x T i S 2 F i g u r e 66. dV/dT) f o r L i / L i x T i S 2 c e l l s . T h e M o n t e C a r l o " r e s u l t s f o r t h e 4x4x4 l a t t i c e ( ) a n d t h e d a t a o f F i g u r e 24 C O ) , d i s c h a r g e ; (x) , c h a r g e ) a r e shown. 195 200 I00h > > 0 -- l O O h -200 -300 0 .2 .4 .6 X in L i x T i S 2 Figure 67. 3V/3T) for L i / L i x T i S 2 c e l l s . This f i g u r e i s i d e n t i c a l to Figure 66 except that the Monte Carlo r e s u l t s (A) are for the 6x6x6 l a t t i c e . 196 .4 .6 X in Li x TiS 2 Figure 68. c"(x) for L i x T i S 2 . The data from Figure 29 (') and the Monte Carlo r e s u l t s for the 4x4x4 l a t t i c e (a) are shown. 1 dC Figure 69. The thermal expansion c o e f f i c i e n t , = ~ " " g r j r ) . calculated using Monte Carlo simulations on the 6x6x6 l a t t i c e as described i n the text. 198 This i s most e a s i l y understood by considering the phase diagram shown i n Figure 57. At high temperatures the int e r c a l a n t i s randomly d i s t r i b u t e d (stage 1) throughout the host and c(x) predicted by the spring and plate model shows no v a r i a t i o n with temperature. As the temperature i s lowered and a stage two structure forms, the average l a t t i c e parameter, predicted by the spring and plate model, contracts, only at l i t h i u m concentrations where staged structures appear. In Chapter 6 we noted that the thermal expansion c o e f f i c i e n t of L i ^^TiS^ at 300K i s much larger than a for TiS„ and LiTiS„ which i s consistent with Figure 69. c 2 2 F i n a l l y , i n Figure 70 the predicted (004) peak width for the 6x6x6 l a t t i c e i s p l o t t e d . This c a l c u l a t i o n i s i n q u a l i t a t i v e agreement with the data in Figure 33. The peak width plotted in Figure 7 0 was obtained as described e a r l i e r i n t h i s section. The Monte Carlo r e s u l t s for the other s i z e l a t t i c e s are s i m i l a r . One notes that u'>u i n these f i t s to the data. These i n t e r a c t i o n s represent combinations of e l e c t r o n i c and e l a s t i c e f f e c t s not taken into e x p l i c i t account in our model. The s t r a i n mediated i n t e r a c t i o n (McKinnon 1980) i s expected to be a t t r a c t i v e between atoms i n the same plane and repulsive between atoms i n d i f f e r e n t l a y e r s . The screened Coulomb i n t e r a c t i o n i s expected to be repulsive i n both cases. The sum of these e f f e c t s could lead to the values needed to describe the data. The q u a l i t a t i v e and in some cases quantitative f i t s , shown i n Figures 63-70, to the L i TiS„ data have been made with a sin g l e model X £, using a single set of parameters. It i s the i n c l u s i o n of the e l a s t i c energy in the l a t t i c e gas model for L i TiS which i s responsible for X £. t h i s success. The spring and plate model, although admittedly 199 i 1 1 r Figure 7 0. The (004) peak width calculated using Monte Carlo simulations on the 6x6x6 l a t t i c e as described in the text. 200 ov e r s i m p l i f i e d , allows one to gain an understanding of the r o l e of l a t t i c e expansion and e l a s t i c energy in Ll^TlS^' The success of the Monte Carlo simulations on f i n i t e s i z e samples suggests that the Daumas-Herold (1969) domain model of staging may be important in describing the equilibrium thermodynamics of L i ^ T i S ^ . This model i s described i n Appendix 5, Section A5.2. Figure A5.5a depicts a stage two compound given by the Daumas-Herold model. It i s c l e a r from Figure A5.5a that as the domain s i z e decreases, the energy associated with the bending of the host layers becomes s i g n i f i c a n t . This energy i s not included i n the r i g i d layer spring and plate model and i t s i n c l u s i o n , i n a more exact theory, may improve the d e s c r i p t i o n of L i T i S 0 . x 2 201 7.5 A p p l i c a t i o n of the Theory to other I n t e r c a l a t i o n Compounds. 7.5.1. Graphite I n t e r c a l a t i o n Compounds The phase diagram for the case u=0, u'=J=10u" and a=0 .2, calculated using the Bragg-Williams approximation for solving the l a t t i c e gas model discussed in Section 7 . 4 . 2 , i s shown in Figure 71a. The width of the stage 2 and stage 3 phases in Figure 71a are c o n t r o l l e d by the magnitudes of u' and u'' r e s p e c t i v e l y . The phases 6' and 6" have 6 layer u n i t c e l l s and are described at T=0 as having 4 and 5 of every 6 layers f i l l e d and the others empty. This phase diagram should be compared to the one produced by Safran (1980), i n an attempt to describe staging in GICs, shown in Figure 71b. The o v e r a l l features i n Figure 71a and b are quite s i m i l a r ; the presence of stages 1, 2 and 3, the f i r s t order phase t r a n s i t i o n s between these phases at low temperature and a region at high temperature where the stage 1-2 t r a n s i t i o n becomes continuous. In our case i t i s the i n c l u s i o n of the e l a s t i c energy which makes the t r a n s i t i o n s f i r s t order at low temperature. Safran introduces an a t t r a c t i v e i n t e r -action between in t e r c a l a n t atoms in the same plane which dr i v e s the f i r s t order t r a n s i t i o n s and a repulsive i n t e r a c t i o n between int e r c a l a n t atoms in d i f f e r e n t layers which decays as a power law as the layer separation increases. The major d i f f e r e n c e between the theories i s the breaking of the symmetry about x = 1 /2 exhibited by Safran's diagram. It i s the i n c l u s i o n of the e l a s t i c energy, which v a r i e s most r a p i d l y at small x, that breaks the symmetry. Graphite i n t e r c a l a t i o n compounds produce f u l l y staged structures at room temperature with well defined i n p l a . n e concentrations ( i n most 202 Figure 71 a) Phase diagram for u'=10u"=J, u=0 and a=0.2 calculated as described in the text. The shaded regions denote two phase regions. The t r a n s i t i o n from stage 1 to stage 2 becomes continuous over the dashed portion of the diagram. The dotted portions of the single phase stage 3, 6' and 6" phases have not been determined p r e c i s e l y . b) Phase diagram a f t e r Safran (1980). Although the phase diagram exhibits symmetry about x=l/2, Safran has omitted the phases related by symmetry to 3, 4 and 5 from the diagram. cases) and l a t t i c e parameters. This i s i n accord with the low temperature portion of the phase diagram in Figure 71b. It would of i n t e r e s t to examine the high temperature behaviour of GIC's to determine how well our model can explain the l a t t i c e parameter v a r i a t i o n and departures from stoichiometry of GIC's. 204 7.5.2 The Effects of Pressure on Staged Intercalation Compounds Wada et a l . (1980) have shown that under the application of hydrostatic pressure stage two K^24 undergoes a reversible f i r s t order t r a n s i t i o n to a stage 3 phase. The mixed phase region exists from about 2 kbar to 6 kbar. Pressure induced staging transformations of t h i s type can be understood with our model. We do not attempt to describe KC^^ quantitatively, but show instead that the model predicts pressure induced phase tr a n s i t i o n s . In Figure 72 the phase diagram for J=.4u' = .7u, a=0.2 and £'=0.1 11 (corresponding to a pressure of - 25 i kbar for a host with, c^g = 5x10 2 dynes/cm and (c -c )/c =.1) i s shown. This diagram was obtained by L i • O O replacing J in equation 7.23 with J(l+fy'/a) 2 as described i n Section 7.4.1 and performing the Bragg-Williams solution described in Section 7.4.2. This phase diagram should be compared to the one shown in Figure 57 computed for the same parameters except £'=0, corresponding to zero applied pressure. When x=.25 and kT/J=1.0, the hypothetical compound described by these phase diagrams i s stage one at zero pressure and stage two when $' = 0.1-. The phase t r a n s i t i o n between these states i s f i r s t order as a function of pressure. Although higher stages have not been included i n t h i s model cal c u l a t i o n , i t i s clear that the application of pressure w i l l induce phase tran s i t i o n s to higher stage structures, consistent with the data of Wada et a l . (1980). The portions of the phase diagrams in Figures 57 and 72 where the stage 1 to 2 t r a n s i t i o n i s continuous (dashed curves in the Figures) can be obtained using Landau theory (see Appendix S i x ) . The vari a t i o n 205 Figure 72. The phase diagram for J=.4u'=.7u, a=0.2 and fi'=0.1 calculated as described in the text. The phase t r a n s i t i o n from stage 1 to stage 2 i s continuous where indicated by the dashed curve. 206 of the t r a n s i t i o n temperature, = J(kT/J) ^ /k, with applied pressure can be obtained simply for the regions where the t r a n s i t i o n i s continuous. We find 8T 4j(l+£'/a) x ( i _ x ) c c u c c . a o 8 P r M \ 3 k c „ ( c T - c ) (a + x ) 33 L o • c 7.32 for the model c a l c u l a t i o n described above. In equation 7.32, x £ i s the 11 2 value of x corresponding to T c. For c ^ = 5x10 dynes/cm , x=.07, a=0.2, ( c T - c )/c = .1, J = 1.4 kT , T = 300K and £' = 0 we f i n d dT /dP ~ 22K/kbar. J - i O O K K C This shows that the phase diagrams of i n t e r c a l a t i o n compounds should be se n s i t i v e to pressures of the order of kbar. Str u c t u r a l and electrochemical measurements as a function of pressure on L i TiS„ should provide more information on the d e t a i l s of the structure at x=.16. The comparison of t h e o r e t i c a l predictions with pressure dependent experimental r e s u l t s should f a c i l i t a t e the refinement of the spring and plate model and the l a t t i c e gas model which have been used to describe L i TiS„. x 2 207 7.5.3 L i NbSe„ x 2 It has been discovered (Dahn and Haering 1982) that L i NbSe X 6. exhibits a phase diagram much l i k e the one shown in Figure 71a. As li t h i u m i s in t e r c a l a t e d into NbSe2, several regions of co-existing phases are observed between x=0 and x=.3. L i NbSe„ with x ~ .15 i s a r x 2 stage 2 compound, and with x ~ .3 i t i s a stage 1 compound. For con-pos i t i o n s 215 $ x $ .3, a mixture of stage 2 and stage 1 compounds i s observed. An imperfect stage 3 structure i s observed near L i ^NbSe^ and co- e x i s t i n g stage 2 and stage 3 regions are observed f o r .07 ~ x ~ .15. The existence of higher stages at smaller x cannot be ruled out by these experiments. The quantitative d e s c r i p t i o n of L i NbSe with our model has not been performed, although we f e e l i t can be done. \ 208 7.5.4 L i TL. .S, x 1.1 2 V(x) and -dx/dV f o r L i / L i T i S c e l l s , measured by Thompson (1979) X X • x Z are shown in Figure 73. The voltage of L i / L i ^ T i ^ ^S^ c e l l s i s roughly .2 v o l t s lower than L i / L i TiS„ c e l l s of the same l i t h i u m composition. x 2 -8x/3V for L i Ti.. 1S i s a smooth featureless curve, when the noise X X • X z present in the data i s smoothed, un l i k e -3x/3V for L i x T i S 2 . The change in electrochemical behaviour r e s u l t i n g from the excess titanium can be understood with the spring and plate model of the e l a s t i c energy. Excess titanium occupies octahedral s i t e s within each van der Waals gap and behaves quite d i f f e r e n t l y from int e r c a l a t e d l i t h i u m . The c-axis l a t t i c e expansions r e s u l t i n g from excess T i are about a factor of 10 l e s s than those r e s u l t i n g from an equal amount of inte r c a l a t e d l i t h i u m in TiS2 (Thompson et a l , 1975). The excess T i increases the binding between neighbouring l a y e r s . This can be modelled as an increase in the l a t t i c e spring constant K. Increases i n K cause corresponding increases in J and 2 a ( r e c a l l J=% K ( c -c ) and a=K/k) which appear in equations 7.22a, b XJ O and c for the Gibbs free energy, chemical p o t e n t i a l and -3x/3y, calculated in mean f i e l d theory, for our l a t t i c e gas model. In Figure 74 the e f f e c t of increases in K on V(x) i s shown. In Figure 75, the -3x/3V curves corresponding to the V(x) curves of Figure 74 are plotted. As K increases, the voltage at fixed x i s reduced. The asymmetry exhibited by -3x/3V i s also reduced as K increases. Both of these predictions are in agreement with the L i Ti.. 1S and L i TiS„ data. X X • X Z X z The spring and plate model predicts that a for L i ^ T i ^ ^S^ should be much greater than a for L i TiS-. Unfortunately, l a t t i c e expansion data for 209 a) i—i—i—i—i—i—i—i—r | 61—1 I " I I ' " " L-0.0 0.2 0.4 0.6 0.8 1.0 X •b) 2.00 ^ 1.50 h o > > \ X 1.00 0.50 0.00 0.0 0.2 0.4 0.6 0.8 1.0 X Figure 73. a) V(x) f o r L i / L i x T i S and L i / L ^ T ^ 1 S 2 c e l l s , b) -8x/3V f o r L i / L x T i ^ .jS c e l l s . A f t e r Thompson (197 9).* 2.5 2 .3 CO — 2 1 o ^ •1 > 1.9 h 1.7 0.0 0 .2 0 .4 0.6 0 .8 1 X Figure 74. V(x) for u=3.4 kT, T=300K, E =-2.45 eV and a) a=0.2, J=1.2kT; b) a=0.6° J=3.6 kT; c) a=1.0, J=6.0 kT. The r a t i o of K for curve a to K for curve b to K for curve c i s 1:3:5. These curves were calculated as described i n the text. 4.0 Figure 75. -9x/8V corresponding i n Figure 74. to the V(x) curves 212 L i T i S are hot available,, so:experimentally determined values of a and c-^  cannot be estimated. Electrochemical and s t r u c t u r a l measure-ments on L i Ti., , S. as a function of x and y should prove useful i n x 1+y 2 furthering the understanding of l a t t i c e expansion and e l a s t i c energy in i n t e r c a l a t i o n compounds. The reduction of the magnitude of the small peak near x - .07 in -8x/8V for c e l l JD-234 (Figure 18) i s most l i k e l y due to the presence of excess titanium. V(x) for c e l l JD-234 i s also s h i f t e d to lower voltages, consistent with an increase i n K. for t h i s p a r t i c u l a r batch of T i S 2 . R e c a l l that 5 p l l l T i S 2 (used i n JD-234) showed very l i t t l e e l e c t r o l y t e c o - i n t e r c a l a t i o n which was interpreted as an i n d i c a t i o n of excess titanium. PART IV C H A P T E R E I G H T SUMMARY AND SUGGESTIONS FOR FUTURE WORK 8.1 Summary of the Thesis The introduction of the thesis showed that even though su b s t a n t i a l amounts of experimental and t h e o r e t i c a l work on L i x T i S 2 had been performed, L i x T i S 2 was s t i l l poorly understood. In part II of the t h e s i s we described experimental methods f o r studying l i t h i u m i n t e r c a l a t i o n compounds and applied them to L i TiS„. Chapter 3 described how measurements of V(x) and -3x/3V can be made. Measurements of -3x/3V f o r L i / L i TiS„ c e l l s confirmed the r e s u l t s x 2 reported by Thompson (1978), although we noticed sample dependent r e s u l t s believed to be due to small amounts of i n t e r l a y e r titanium i n some samples. In chapter 4 we discussed the measurement and i n t e r -p retation of 3V/3T) x f o r i n t e r c a l a t i o n based c e l l s . The 3V/3T) x r e s u l t s f o r L i / L i x T i S 2 c e l l s established the presence of ah entropy minimum, i n d i c a t i v e of increased order, at a l i t h i u m composition near X-.16 i n L i x T i S 2 . No evidence of entropy minima near x =/l/3, j1/4, 1/7. or any other compositions commensurate with the ordered structures of the two dimensional t r i a n g u l a r l a t t i c e gas were observed. The experi-215 mental details: of an i n s i t u X-ray d i f f r a c t i o n technique f o r i n v e s t i g a -t i n g the c r y s t a l structures of l i t h i u m i n t e r c a l a t i o n systems were pre-sented i n Chapter 5. In s i t u X-ray d i f f r a c t i o n studies of L i x T i S 2 were made to help understand the electrochemical behavior of L i / L i TiS„ c e l l s • r X 2 and were also discussed i n Chapter 5. The v a r i a t i o n of the l a t t i c e para-meters of L i x T i S 2 was measured as a function of x (Figure 29).. V a r i a -tions i n the widths of (00£) peaks were observed (Figure 33).and were taken as evidence f o r the formation of a short range ordered stage 2 structure near L i ^ T i S 2 . The i n t e r p r e t a t i o n of the peak width data was described i h Chapter 5 and Appendix 5. In Chapter 6 a neutron d i f f r a c -t i o n experiment to search f o r a stage two structure i n L i ^ T i S 2 was described. The r e s u l t s of t h i s experiment established that as the temp-erature was lowered the length over which the staging c o r r e l a t i o n s i n L i ^ T i S 2 extended increased. Measurements oh L i 2^TiS2 showed no evidence f o r a staged structure. The experimental r e s u l t s presented i n part II of the t h e s i s i d e n t i f i e d staging (although short range) as the dominant phys i c a l mechanism i n L i T i S 0 . x 1 The r o l e of l a t t i c e expansion and e l a s t i c energy i n layered i n t e r -c a l a t i o n compounds was considered i n part I I I of the t h e s i s . The spring and plate model, discussed i n section 7,3, allowed us to c a l c u l a t e the changes i n the e l a s t i c energy and the l a t t i c e parameters of i n t e r c a l a t i o n compounds as a function of i n t e r c a l a n t concentration and applied external pressure. We showed how to include the e l a s t i c energy i n three dimensional l a t t i c e gas models of i n t e r c a l a t i o n compounds and discussed the e f f e c t s of i t s i n c l u s i o n on the electrochemical behaviour of model systems. The symmetry about x = % exhibited by phase diagrams ." and -3x/3V, calculated for l a t t i c e gas models with two body in t e r a c t i o n s 216 was removed when the e l a s t i c energy was included i n the free energy. Many phase diagrams were calculated to explore the wide range of possible behavior predicted by the three dimensional l a t t i c e gas model considered in section 7.4.2. The e l a s t i c energy was shown to favor staged structures at low intercalant content, consistent with data on systems l i k e L i NbSe_. x 2 The three dimensional l a t t i c e gas model described above was applied to Li xTiS2> Q u a l i t a t i v e and, i n some cases, quantitative agreement with the data reported in part II. was obtained. The i n c l u s i o n of the e l a s t i c energy i n the Hamiltonian was shown to be responsible f o r t h i s success. The L i TiS„ data and the predictions of the model were com-x 2 pared i n Figures 63-70 and the accompanying text i n Section 7.4.3. We showed that at room temperature, L i x T i S 2 has a disordered, or short range, stage two structure f o r X-.16. For x-0 and x^.3, L i x T i S 2 i s a stage 1 compound, the peaks i n -9x/3V near x=.07 and x-.21 (Figures 64 and 65) a r i s e as L i x T i S 2 changes from stage 1 to short range stage 2. The spring and plate model was applied to other i n t e r c a l a t i o n systems i n Section 7.5. Q u a l i t a t i v e descriptions of pressure induced staging t r a n s i t i o n s in GICs and of the e f f e c t of i n t e r l a y e r titanium on the electrochemical behaviour of L i T i , , S„ were obtained. A phase diagram, x 1+y 2 which agrees well with the phases observed i n Li^NbSe 2, was c a l c u l a t e d . 217 8.2 Suggestions f o r Future Work The experimental methods described i n part II of the thesis can be applied to any i n t e r c a l a t i o n compound which can be prepared e l e c t r o -chemically i n a c e l l with a m e t a l l i c anode. Careful a p p l i c a t i o n of these experimental techniques to other i n t e r c a l a t i o n systems should help to i d e n t i f y the dominant phys i c a l mechanisms present. Further s t r u c t u r a l work on L i TiS„. as a function of temperature x 2 i s needed to obtain the d e t a i l s of the short range ordered, or i n c i p i e n t , stage 2 structure which develops near x-.16. Electrochemical and s t r u c -t u r a l studies on L i x T i S 2 as a funtion of applied pressure should help to c l a r i f y i t s behaviour as well as to r e f i n e the spring and plate model. The c a l c u l a t i o n s given i n section 7.4.3 predict that the thermal expan-sion c o e f f i c i e n t , q^, of L i x T i S 2 should be highly dependent on x. Measurements of as a function of x and T should be performed. Measure-ments of the changes i n e l e c t r o n i c properties of L i x T i S 2 as a function of x would be u s e f u l . The quantitative addition of excess titanium i n Ti^ +^S2 should allow the experimentalist to change the e l a s t i c properties of the host i n a c o n t r o l l e d fashion. Electrochemical and s t r u c t u r a l measurements on L i x T i j S 2 as a function of x and y should help to further elucidate the r o l e of l a t t i c e expansion:and e l a s t i c energy i n layered i n t e r c a l a t i o n systems. These experiments should also help to c l a r i f y the e l e c t r o l y t e c o - i n t e r c a l a t i o n phenomenon. On the t h e o r e t i c a l side, further work i s needed to improve the ., spring and plate model f o r the e l a s t i c changes i n layered i n t e r c a l a t i o n compounds. The e f f e c t s of the energies associated with the bending of 218 the host layers at domain boundaries should be investigated. The extension of the simple ideas, which motivated the spring and plate model, to non-layered i n t e r c a l a t i o n compounds should prove u s e f u l . Detailed c a l -culations of the e l e c t r o n i c i n t e r a c t i o n and i t s v a r i a t i o n f o r L i x T i S 2 are needed, to see i f the i n t e r a c t i o n energies we have taken i n our l a t t i c e gas models of L i TiS„ are r e a l i s t i c . ° x 2 A P P E N D I X ONE 9V/3T) FOR THE ONE DIMENSIONAL LATTICE GAS The one dimensional l a t t i c e gas (equivalent to the Ising model for magnetic systems) with nearest neighbor in t e r a c t i o n s can be solved exactly and provides a good te s t i n g ground f o r our ideas about BV/dT)^. We consider the c y c l i c chain of N l a t t i c e s i t e s . The Hamiltonian i s where E q i s the s i t e energy (the energy diff e r e n c e between an empty and f i l l e d s i t e ) , p i s the chemical p o t e n t i a l and u i s the nearest neighbor i n t e r a c t i o n energy, x^ i s the s i t e occupation l a b e l and x^=l corresponds to a f i l l e d s i t e and x^=0 to an empty s i t e . The c y c l i c nature of the chain implies s i t e N+1 i s s i t e 1. The p a r t i t i o n function can be found by the transfer matrix method (Thompson 1972) and we f i n d N 1=1 A l . l A1.2 where -u/4kT r , ,y-E -u N , r . ,2,u-E -u, , u/kT \ A 1 Q A + = e ( cosh( o ) ± ( smh ( o ) + e J J • A1.3 2kT~ 2kT In t h i s expression, k i s Boltzmann's constant, and T i s the Kelvin temperature. Since A > A , we f i n d as N becomes large. The grand p o t e n t i a l (since we have included u i n the Hamiltonian) i s given by Q = -kT ln(Z) , and from Q. we can obtain the thermodynamic quantities of i n t e r e s t . The entropy, S,and the average occupation of each s i t e on the l a t t i c e , x, are o - f^R) , A1.4 b ^TJp and 2 2 1 We can solve A1.5 for u as a function of x and T. Then we can d i f f e r e n t i a t e to obtain 3y/9T) x. The chemical p o t e n t i a l , y, i s related to the voltage measured i n a battery by V = — ( -14 + const. ) ez where e i s the ele c t r o n i c charge and z i s the number of electrons trans-ferred from anode to cathode per i n t e r c a l a n t . In Figure A l . l we plot S/Nk, -(y-E Q)/kT and e/k 9V/9T) x as a function of x for u = -4kT, 0 and 4 kT. When u > 0, the inte r a c t i o n s between intercalant atoms are repulsive, and the system t r i e s to arrange i t s e l f so that f i l l e d s i t e s are adjacent to empty s i t e s . At x: = 1/2 (half the s i t e s f i l l e d ) the number of ways of doing t h i s i s small, hence the dip in S at x = 1/2 for u = 4 kT and the corresponding feature i n 9V/9T) x > The drop i n -y at x = 1/2 i s due to the cost i n energy (^2u) for f i l l i n g a s i t e located between two already f i l l e d s i t e s . When u < 0, the entropy i s a maximum at x = 1/2 and we observe the c h a r a c t e r i s t i c behaviour i n 9V/9T) x for an entropy maximum. The one dimensional l a t t i c e gas model does not exhibit any long range ordered states at non-zero temperature, so the features exhibited i n Figure A l . l are due so l e l y to short range order. To get an idea of the degree of order needed to produce a feature i n 3V/9T) x as i s observed f o r u = 4 kT, we consider the c o r r e l a t i o n function < X J X . >. At x = 1/2 we f i n d (Thompson 1972) 222 Figure A l . l . S/Nk, - ( p - E Q)/kT and ~ — as a function of x for the one dimensional l a t t i c e gas with u = -4 kT, 0 and 4 kT. <x.x. x x+r > = h ( tanh(-u/4kT))r + % A1.6 The number of s i t e s over which the order p e r s i s t s can be s p e c i f i e d i n terms of the quantity £, defined i m p l i c i t l y (at x = 1/2) by The c o r r e l a t i o n length i s ? m u l t i p l i e d by the l a t t i c e spacing. By comparing A1.6 and A1.7 we f i n d When u = 4 kT, £ = 3.68. The feature i n 3V/3T) at x = 1/2 in Figure A l . l for u = 4 kT i s due to cor r e l a t i o n s extending over about 4 l a t t i c e spacings. Thus, the measurement of 3V/3T) x i s a technique that i s t r u l y s e n s i t i v e to short range order and can be used to supplement X-ray d i f f r a c t i o n , which i s s e n s i t i v e to long range order. A1.7 5 = 1 A1.8 ln([tanh(-u/kT) |) 224 A P P E N D I X TWO x > 1 IN L i TiS„ x 2 A2.1 Introduction The structure of HS2 suggests that up to three l i t h i u m atoms per titanium atom could be i n t e r c a l a t e d . I t i s known that f o r 0 <^  x <_ 1 the l i t h i u m atoms occupy octahedral s i t e s within each van der Waals gap (Dahn et a l . 1980). C l e a r l y , when x > 1, the tetrahedral s i t e s must begin to f i l l or a major s t r u c t u r a l phase t r a n s i t i o n of the host must occur. Electrochemical measurements of the V(x) behaviour of L i / L i TiS x ^ c e l l s have been made previously (Murphy and Carides 1979, Dahn and Haering 1979) and were in good agreement. These authors suggested that f o r 1 < x < 2 i n L i T i S 0 the material i s made up of L i . T i S 0 and L i ~ T i S 0 — — x 2 r 1 2 2 z phases. This has been confirmed by in s i t u X-ray d i f f r a c t i o n experiments (Dahn et a l . 1982). Dahn and Haering (1979) found that f o r x > 2, some i r r e v e r s i b l e process occurs and that the subsequent cycles are very d i f f e r e n t from.the i n i t i a l discharge behaviour. We have performed i n s i t u X-ray experiments for x > 1 and report the r e s u l t here. A de s c r i p t i o n of L i TiS„ f or 0 < x < 2 i n terms of a l a t t i c e gas model with the i n c l u s i o n of the tetrahedral s i t e s i s given at the end of the appendix. 225 A2.2 Experimental Figure A2.1 shows the f i r s t discharge at constant current of a L i / L i C 1 0 4 , PC/TiS 2 c e l l ( c e l l JD-49) to a voltage of 0.20 v o l t s and i t s subsequent recharge to 2.80 v o l t s . The T i S 2 used i n t h i s c e l l was prepared by grinding large si n g l e c r y s t a l s made by iodine vapour transport. E l e c t r o l y t e c o - i n t e r c a l a t i o n i n t h i s material was very minimal (see Chapter 3). If the c e l l i s cycled further between these voltage l i m i t s , the behaviour i s s t i l l t y p i f i e d by Figure A2.1. The " f l a t n e s s " of the V(x) curve for 1 <_ x <_ 2 suggests that a mixture of co-existing L i T i S 2 and L i 2 T i S 2 phases i s present. In Figure A2.2 we show the v a r i a t i o n of the voltage of a L i / L i TiS„ i n s i t u X-ray d i f f r a c t i o n c e l l (JX-15), X z, containing 12.2 mg of T i S 2 , when discharged and charged at 290 yA. X-ray d i f f r a c t i o n p r o f i l e s were taken at various times during the i n t e r c a l a t i o n process. The region of the d i f f r a c t i o n p r o f i l e near 60° i s shown at times indicated by the numbered arrows i n the f i g u r e . At time 1, the (004) peak c h a r a c t e r i s t i c of L i ^ T i S 2 i s observed. As the discharge proceeds, the i n t e n s i t y of the L i ^ T i S 2 peaks decreases and a new set of peaks, i d e n t i f i e d with L i 2 T i S 2 , appear and begin to grow. At time 2, the L i ^ T i S 2 peak has diminished and a new peak from L i 2 T i S 2 i s appearing at lower angle. The discharge continued as shown in the f i g u r e and then the c e l l was held at 0.400 v o l t s f o r 10 hours. A c a r e f u l scan of the material was taken (time 3) and only a small remnant of the L i ^ T i S 2 peak remains. The peaks a r i s i n g from L i 2 T i S 2 reached t h e i r maximum i n t e n s i t y at t h i s time. As the c e l l i s charged (time 4) the L i ^ T i S 2 peaks grow and the L i 2 T i S 2 peaks shrink. When the c e l l i s charged to 1.80 v o l t s (time 5) no L i ~ T i S 9 peaks remain and the LiTiS„ structure i s retained. This i s a 226 Figure A2.1. Voltage of a L i / L i C 1 0 4 , PC/TiS 2 c e l l (JD-49) containing 1.8 mg of TiS2 cycled at 75 yA. 5.8 hours corresponds to x = 1 in Li xTiS2» Note that the x scale applies only to the discharge curve. 227 CO I-_l O > 2 I 0 i i 2 i i 3 \ , i 4 \ i i A-1 i i i i _ r 1 A 1 i i i i 1.0 1.2 1.4 1.6 X 1.8 " 1.8 in L i x T i S 2 1.6 1.4 1.2 1.0 o LU CO \ CO r-Z O CJ >-CO z LJ r-600h 400h 200h 57 60 57 60 57 ' 60 57 SCATTERING ANGLE (DEGREES) Figure A2.2. V a r i a t i o n of the voltage of a L i / L i x T i S 2 X-ray c e l l (JX-15) cycled at constant current between x=l and x=2. X-ray d i f f r a c t i o n p r o f i l e s of the cathode were taken at the times indicated by the numbers 1 to 5 i n the f i g u r e . The (004) region of these p r o f i l e s are shown. For d e t a i l s see the text. 228 c l e a r example of a f i r s t order phase t r a n s i t i o n . It should be pointed out that the peaks of the charged Li^TiS2 material (time 5) are broader than those measured before a discharge to x=2 (time 1). This indicates some i r r e v e r s i b l e s t r a i n i n g of the c r y s t a l l a t t i c e , u nlike the r e v e r s i b l e peak broadening observed for 0 _< x <_ 1 i n Chapter 5 of the t h e s i s . In Figure A2.3 we show a portion of the X-ray p r o f i l e taken at time 3 of Figure A2.2. The peaks corresponding to Li2TiS2 are numbered i n increasing order as a function of t h e i r s cattering angle. The contaminant peaks i n the spectrum are l a b e l l e d also. In Table A2.1 we l i s t the scattering angles and computed plane spacings of the Li2TiS2 peaks. No off axis correction was included when c a l c u l a t i n g the plane spacings. The peaks l a b e l l e d 1, 4, 9 and 14 are seen to be related by integer multiples of t h e i r d spacings (Table A2.1). This fact and the proximity of these peaks to the (00£) peaks of L i j T i S 2 suggest that peaks 1, 4, 9 and 14 are the (00£) peaks f o r the new structure. However, a l l attempts to index the remaining peaks on a hexagonal l a t t i c e with a 1, 2 or 3 layer unit c e l l f a i l e d . The unit c e l l of Li2TiS2 remains unsolved, o although the dimension, 6.33 A, i s probably c h a r a c t e r i s t i c of the layer spacing. [The structure may now be t r i c l i n i c with the angle between a and c no longer equal to 90°.] The fa c t that upon recharge the c r y s t a l /reverts back to the o r i g i n a l 1-T structure suggests that the s t r u c t u r a l change between x=l and x=2 may not be too great. Figure A2.4 shows the constant current c y c l i n g behaviour of c e l l JD-49 to a lower voltage l i m i t of 0.05 v o l t s . This data was taken immediately a f t e r the data i n Figure A2.1. The c e l l discharges to x > 3 i n i t i a l l y . One notices that the shape of the i n i t i a l recharge curve and subsequent discharge curve are very d i f f e r e n t from that of Ni 2 0 0 t Z ZD >-< m rr < in z LU I -BelOl Be 0 0 2 B e O 0 0 2 Ni III I 8 Li 110 Separator Li,TIS2 0 0 4 Li,TiS2 103 0 0 2 60 1 50 40 SCATTERING ANGLE (DEGREES) 30 Figure A2.3, The X-ray d i f f r a c t i o n p r o f i l e of L i ^ T i S numbered from 2-14 on t h i s portion of the p r o f i l e Figure A2.2. The contaminant peaks are l a b e l l e d . 2 — 2' T ^ e P e a^- S corresponding to Li2TiS2 are This data was taken -at"time 3 of 230 Table A2.1 Bragg Peaks of L i TiS Peak Scattering Plane Number Angle Spacing (degrees) (A) 1 14.04 6.308 x l = 6.308 A 2 23.38 3.805 3 26.79 3.328 4 28.23 3.161 x2 = 6.322 A 5 32.26 2.775 6 38.88 2.316 7 39.62 2.275 8 41.21 2.191 9 42.80 2.113 x3 = 6.339 A 10 46.28 1.962 11 47.64 1.909 12 49.20 1.852 13 57.35 1.606 14 58.27 1.583 x4 = 6.334 A 231 Figure A2.4. Constant current cycles of c e l l JD-49 at 75 yA. This data was taken immediately a f t e r the data i n Figure A2..1. Cycle numbers are indicated. (The data i n Figure A2.1 i s cycle zero.) 232 the i n i t i a l discharge. Also apparent i s the "washing out" of the features with increasing cycle number. C e l l JX-15, a f t e r producing the data i n Figures A2.2 and A2.3 was discharged to a lower voltage l i m i t of 0.024 v o l t s at constant current. X-rays were taken on the plateau between x=2 and x=3 and showed the Li2TiS2 peaks shrinking, and only a s i n g l e broad peak, near 29 = 27°, growing. The c e l l was allowed to e q u i l i b r a t e at 0.025 v o l t s and a d i f f r a c t i o n p r o f i l e was taken. The region of the spectrum near 27° i s shown i n Figure A2.5. The c e l l was then charged to 2.150 v o l t s and an X-ray p r o f i l e was taken. Again, the only discernable feature character-i s t i c of the material was the s i n g l e broad . peak, appearing t h i s time near 20 = 27.40° . Dahn and Haering (1979) noticed that c e l l s cycled i n t h i s new phase at high voltages (1.7-<->-2.7 volts)showed increasing capacity with cycle number. This was interpreted as a slow conversion of the material back to the o r i g i n a l ! T - L i x T i S 2 phase characterized by Figure A2.1. We cycled c e l l JX-15 at high voltages and did observe an increase i n capacity with cycle number s i m i l a r to that reported by Dahn and Haering (1979), but X-rays taken a f t e r many cycles showed no evidence of the L i x T i S 2 c r y s t a l l i n e phase. It remains unclear exactly what happens when L i TiS„ i s discharged through the i r r e v e r s i b l e phase t r a n s i t i o n f or x > 2. I t i s possible that decomposition to Li2S and other products may occur v i a a reaction l i k e 2Li + Li„TiS„ -»• 2Li„S + T i . This i s a t t r a c t i v e because c a l c u l a t i o n s show that the most intense Bragg X 233 Figure A2.5. Portion of the X-ray p r o f i l e of c e l l JX-15 at 0.025 v o l t s . This data was taken a f t e r the data shown i n Figure A2.2. 234 peak from L i 2 S (cubic, a = 5.720 A) should f a l l at 26 = 27.00°. The next most intense peak should be located at 44.82°. We observed no evidence of a peak near 44.82° or peaks associated with free titanium. However, we expect the peak at 44.82° to have a f u l l width at half maximum of about 2° which would make i t d i f f i c u l t to observe. Thompson et a l . (1980) have reported the electrochemical behaviour of L i / L i ^ S c e l l s and have shown that the reaction proceeds by a two step process; 1 ) . L i + S £ LiS 2) L i + LiS t L i 2 S . The reactions are r e v e r s i b l e according to Thompson et a l . This i s of in t e r e s t because i f L12S formation was present i n our case, we would expect to be able to recharge the c e l l v i a the reactions hypothesized by Thompson et a l . However, the d e t a i l s of the V(x) r e l a t i o n s that we have measured (Figure A2.4) and those reported by Thompson et a l . f o r the L i / L i ^ S system are very d i f f e r e n t . The d e t a i l s of the L ^ T i S ^ system for x > 2 remain a mystery and further work i s needed i n t h i s area. In summary we have confirmed the existence of a two phase region for 1 < x < 2 i n L i TiS„. The structure of Li„TiS„ remains unsolved — — x 2 2 2 o although i t i s l i k e l y that the layer spacing i s 6.33 A. For x > 2, one observes an i r r e v e r s i b l e t r a n s i t i o n to a new phase. The s t r u c t u r a l d e t a i l s of t h i s new phase remain unclear although the electrochemical behaviour of t h i s phase has been measured. 235 A2.3 A L a t t i c e Gas Model f or L i TiS„, 0 < x < 2 x 2' — — In t h i s section we w i l l draw heavily on the notation and formalism developed i n Chapters 2 and 7 of the t h e s i s . Consider the s i t e s a v a i l a b l e i n the TiS2 host. The tr i a n g u l a r l a t t i c e of s i t e s consists of one s u b l a t t i c e of octahedral s i t e s , which we know are occupied for 0 <_x1 and two s u b l a t t i c e s of geometrically smaller tetrahedral s i t e s . Because of the ph y s i c a l s i z e of the s i t e s i t i s l i k e l y that the octahedral s i t e energy, E q , w i l l be l e s s than the tetrahedral s i t e energy, E ^ . . One notices that a large drop i n voltage at x=l could be produced by a large s i t e energy d i f f e r e n c e between octahedral and tetrahedral s i t e s . However, the f a c t that l i t h i u m atoms must move through tetrahedral s i t e s even when x < 1 to d i f f u s e through the c r y s t a l suggests that the energy di f f e r e n c e cannot be as large as the -1 eV needed to produce the drop in voltage observed at x=l. The s i t e energy diffe r e n c e model would not predict a f i r s t order phase t r a n s i t i o n f o r 1 <_ x 2. A d i f f e r e n t approach must be taken. The tetrahedral s i t e s l i e above and below the plane of the octa-hedral s i t e s (Chapter 2), so the distance between adjacent tetrahedral s i t e s i s larger than the nearest neighbour octahedral-tetrahedral s i t e distance. We expect the i n t e r a c t i o n energy between atoms on nearest neighbour octahedral and tetrahedral s i t e s , u -t> to D e larger than that fo r atoms on adjacent tetrahedral s i t e s , u t t ' Thus i t i s possible that even though octahedral s i t e s f i l l f i r s t , i t may be favourable to have only tetrahedral s i t e s f i l l e d for x > 1 to avoid the larger i n t e r a c t i o n u ot We include an i n t e r a c t i o n between atoms on nearest neighbor octa-hedral s i t e s , u ^ . The in t e r a c t i o n s and the s u b l a t t i c e s are depicted 236 schematically i n Figure A2.6. We expect u << u^ ., < u ^ due to the J ° v oo t t ot distances between the s i t e s . We have seen i n Chapter 7 how the in c l u s i o n of the e l a s t i c energy and staging improve the f i t to V(x) for 0 <_ x <_ 1. However, at t h i s point we w i l l treat the problem on the two dimensional tr i a n g u l a r l a t t i c e since we expect u ^ t and u o t > which w i l l determine the behaviour for x > 1, to be much larger than the other i n t e r a c t i o n s d i s -cussed i n Chapter 7. The ideas developed i n Chapter 7 can, of course, be included to improve the f i t for 0 <_ x <_ 1. The fundamental aspects of t h i s problem can be obtained from the zero temperature case. Let x , x^ and be the f r a c t i o n a l occupations of the octahedral and the two tetrahedral s i t e s u b l a t t i c e s r e s p e c t i v e l y . The Helmholtz free energy at T=0 i s given by ? F = N ( E O X O + 3 u O O X Q + E t ( x r + ^ :+ S u ^ x ^ + x ^ ) + ^ 3 u t t X l X 2 } ' where we have included u i n mean f i e l d theory. For a given set of OO J O parameters such that E > E + 3u A2.2 t o oo the system w i l l f i l l octahedral s i t e s f o r 0 <_ x <_ 1 to minimize i t s free energy. The system must then begin to f i l l tetrahedral s i t e s . However, i f E + E + 3u + 3u > 2E + 3u A2.3 t o oo ot J t t t 237 t o t ° - U o t - t t t o" t t t - U t t - t t Figure A2.6. View p a r a l l e l to the c-axis of TiS2 showing the octahedral (o) and tetrahedral (t) s i t e p ositions within a s i n g l e Van der Waals gap. The i n t e r a c t i o n energies, u O Q , u t t and u o t., between atoms occupying neighbouring s i t e s are also shown. then at x=2 the configuration with lowest free energy w i l l be found with Xq=0 and x^=X2=l (both tetrahedral s i t e s f i l l e d ) . Assuming that both A2.2 and A2.3 are true, we plot schematically the v a r i a t i o n of F/N with x for the case of octahedral s i t e s f i l l i n g f i r s t (curve a) and tetrahedral s i t e s f i l l i n g f i r s t (curve b) i n Figure A2.7a. We observe that the system can minimize i t s free energy for 1 <_ x <_ 2 by e x i s t i n g as a mixture of a phase with a l l octahedral s i t e s f i l l e d (x=l) and a phase with a l l tetrahedral s i t e s f i l l e d (x=2). The dependence of F on x for 1 <^  x <^  2 i s given by the dashed l i n e i n Figure A2.7a. The voltage curve corresponding to t h i s model i s shown schematically i n Figure A2.7b. When the temperature i s not zero, we can solve the problem simply within the framework of the Bragg-Williams approximation as described i n Chapter 7. Values of the s i t e and i n t e r a c t i o n energies which produce a reasonable f i t to the experimental V(x) behaviour are given i n Table A2. (It should be noted that equally good f i t s to the data can be obtained with other values of E^, u Q t and u t t ' ) The voltage curve for these parameters and an experimental. L i / L i x T i S 2 c e l l discharge curve are plotted f or comparison i n Figure A2.8. The theory does not f i t f o r x > 2 since the system undergoes an i r r e v e r s i b l e phase t r a n s i t i o n to an unknown structure as discussed e a r l i e r . When these ideas are combined with those developed i n Chapter 7 one obtains a s e l f consistent theory capable of describing L i x T i S 2 over i t s e n t i r e r e v e r s i b l e range, 0 _< x _< 2. The a p p l i c a b i l i t y of the model for x > 1 hinges on the i n t e g r i t y of the tri a n g u l a r l a t t i c e of s i t e s in the compound Li2TiS2. Unfortunately, since we have not been able to solve the structure of Li2TiS2, we cannot claim that the i n t e g r i t y of the t r i a n g u l a r l a t t i c e i s preserved. 239 Figure A2.7. Form of the a) free energy and b) voltage versus x for three interpenetrating s u b l a t t i c e s of octahedral and tetrahedral s i t e s (see text) at T=0. The s i t e energies and interactions s a t i s f y equations A2.2 and A2.3. Table A2.2 Values of the s i t e energies and i n t e r a c t i o n parameters used i n the model c a l c u l a t i o n to describe L i TiS„, 0 < x < 2. E = -2.35. eV o E = -2.10 eV u = 0.062 eV oo u = 0.62 eV ot u = 0.51 eV T = 300°K 241 Figure A2.8. The experimental (dashed curve) V(x) behaviour for a L i / L i x T i S 2 c e l l (L038) discharged at constant current. The s o l i d curve i s a c a l c u l a t i o n using the model described in the text with the parameters i n Table A2.2. A P P E N D I X T H R E E LEAST SQUARES REFINEMENT OF LATTICE PARAMETERS OF HEXAGONAL CRYSTALS The problem of determining the best l a t t i c e parameters, a and c, from the indexed powder d i f f r a c t i o n pattern of a hexagonal c r y s t a l was encountered i n the analysis of the data i n Part I I of t h i s t h e s i s . I t turns out that one can solve exactly for the l a t t i c e parameters a and c which minimize the quantity where d Q(hk£) i s the observed value of the plane spacing corresponding to the M i l l e r indices h, k and I. d(hk£) i s the calculated plane spacing and i s given by A3.1 o 1 4 h 2 + k 2 + hk A3.2 (d(hk£)) 2 3 2 a One can e a s i l y show that minimizing A3.1 with respect to a and c i s equivalent to minimizing „ n (hk£) - n ( h k £ ) „ R = I (-5 ^ ) 2 A3.3 2 2 w i t h respect to 1/a and 1/c , where 1 * < h U ) = ( d ( h k O ) 2 ' A 3 ' 4 We introduce the f o l l o w i n g n o t a t i o n a l conveniences, n(hk£) = A(hk) g o + B(£)g 3. C where A(hk) = -| ( h 2 + k 2 + hk) B(£) = lZ , A3.5 g a = l / a 2 . and g c = 1/c 2 2 The c o n d i t i o n f o r minimizing R w i t h respect to g and g i s that a. C 2 2 8*c = 9 * a This c o n d i t i o n y i e l d s the two equations and n (hkZ) - n(hk£) I i— o ) A(hk) hk£ n (hk£r 0 = I {-2- ) B G e ) hk£ nQ (Tuc£) A3,6 Since n(hk£) i s given by A3.5, these equations are two simultaneous l i n e a r equations for the two unknowns g^ and g^. The solution to the equations i s t r i v i a l and we f i n d , writing i n matrix notation I hk£ no(hk£):J - I hk£ A(hk)B(.£) n (hk£)' o y fA(hk)B(-d)] y f A(hk) I 2 h ^ n (hk£)2 J h k £ K ( h k £ ) J v 0 I A(hk) ) hk£ B(£) hk£ V h k € ) I x A3.7 I hk£ A(hk) V 1 * ^ hk£ f B(£) 1 2 r [nQ(hk£)J A(hk)B(£) hkl n (hk£)' o -1 The values of g and g computed using A3.7 are those that minimize a c 2 2 R , hence x • To use A3.7 one need only input the observed nQ(hkX,) ["obtained from the experimental d (hkJl)] and the functions A(hk) and o B(£). This procedure involves no i t e r a t i o n s and i s e a s i l y implemented on the computer. The "best" l a t t i c e constants, a and c are obtained from g and g using A3.5. The o f f - a x i s c o r r e c t i o n (Chapter 5) to the observed d Q(hk£) values can be implemented p r i o r to computing the r e s u l t of equation A3.7. 246 A P P E N D I X FOUR INTENSITY CALCULATIONS FOR X-RAY AND NEUTRON DIFFRACTION A4.1 X-ray D i f f r a c t i o n The purpose of t h i s appendix i s not to derive the standard equations pertaining to powder d i f f r a c t i o n but to show what modifications to these equations must be made for the s p e c i f i c geometries, samples and instruments used i n these experiments. A computer program written by Richard Marsolais which calculates the r e l a t i v e i n t e n s i t i e s of Bragg peaks w i l l also be discussed. The P h i l i p s PW 1050/70 v e r t i c a l goniometer uses the Bragg-Brentano pseudofocusing geometry shown i n Figure A4.1. In t h i s geometry the angle between the incident beam and the sample, 0, i s 1/2 the angle between the incident beam and the receiving s l i t , 26. For a goniometer with a divergence s l i t of angular divergence, <5, and a d i f f r a c t e d beam monochromator, the integrated i n t e n s i t y of a Bragg peak centered at angle 26 i s (for a thick sample) I = I 'm F 2 <5 1 + cos 29 cos 2<j) A4.1 o sin0 sin26 w h e r e 6 a n d <J> a r e s h o w n i n F i g u r e A4.1. I n t h i s e x p r e s s i o n F i s t h e 247 Figure A4.1. The Bragg-Brentano focusing geometry. In most gonio-. meters, the sample i s f l a t , not curved. Because the sample dimensions are usually much les s than R, the focusing condition i s approximately s a t i s f i e d , hence pseudofocusing. , r r 2 " r r M s a m p l e Figure A4.2 The incident beam portion of the goniometer, which defines the quantities i n equation A4.2 s t r u c t u r e f a c t o r , m i s the m u l t i p l i c i t y of the r e f l e c t i o n and I Q i s a constant (Warren 1969). I f <S i s f i x e d t h i s i s the standard form f o r Bragg-Brentano geometry. Our goniometer i s equipped w i t h the PW1386/50 automatic divergence s l i t , which attempts to keep the i l l u m i n a t e d area of the sample constant The divergence s l i t opens and c l o s e s as 6 changes i n order to do t h i s . The number of i n c i d e n t X-ray photons s t r i k i n g the sample i s p r o p o r t i o n a l to the angular divergence, 6 , of the divergence s l i t . With reference t Figure A4.2, one f i n d s r l + r2 = R sine sinS 2 2 ( s i n 0 - s i n cS) A4.2 To i l l u m i n a t e a constant area, one must vary S i n such a way so that rl + r2 '*"S kept constant as 6 v a r i e s . U s u a l l y 6 << 6 so A4.2 reduces to r, + r 0 = ? f . A4.3 1 2 sm6 I f 6 a s i n 6 then r ^ + i s constant. Mechanically, i t i s d i f f i c u l t to s a t i s f y t h i s c o n s t r a i n t . P h i l i p s has approximated t h i s f u n c t i o n a l form w i t h a divergence s l i t mechanism based on r o t a t i n g rods. The t h e o r e t i c a l v a r i a t i o n of 6(see P h i l i p s 1386/50 manual) p r e d i c t e d f o r t h i s assembly i s 6 d sin(11.16 + 796/90) - .19355 . A4.4 Our divergence s l i t has a small defect and t h e r e f o r e does not obey 2 4 9 A4.4 exactly. We have measured the r a t i o of automatic s l i t i n t e n s i t y to fixed s l i t i n t e n s i t y as a function of angle to c a l i b r a t e the v a r i a t i o n of <S. The data was f i t with a function of the form 4 8; a C + I B sin(nO) . A 4 - 5 n=l n The values of C and B obtained from the f i t are given i n Table A4.1. n & The v a r i a t i o n of s l i t width versus angle for 6 a s i n 9, 6 given by A4.4 and 6 given by A4.5 i s plotted in Figure A4.3. One can see that for angles between 10° and 45° the experimentally determined s l i t width (curve c) i s a good approximation to 6 a s i n 6(curve a). The absorption of the incident and d i f f r a c t e d X-ray beam by the beryllium window must be taken into account i n i n t e n s i t y c a l c u l a t i o n s for X-ray c e l l s . An absorption f a c t o r , A(9) , A ( 0 ) = e-(2yd/sin9) M > 6 must be included i n A4.1 i n t h i s case, d i s the thickness of the Be window and y i s the l i n e a r absorption c o e f f i c i e n t f o r Be. A computer program based on A4.1 to c a l c u l a t e X-ray i n t e n s i t i e s has been written by Richard Marsolais. The user must input the l a t t i c e parameters and the positions and scattering f actors of the atoms within the unit c e l l . The atomic scattering f a c t o r s are obtained from the International Tables f o r X-ray Crystallography (Ibers and Hamilton 1974). The program outputs the r e l a t i v e i n t e n s i t i e s of the Bragg peaks which w i l l be observed i n the powder pattern. Three i n t e n s i t i e s are c a l c u l a t e d . The f i r s t assumes standard Bragg-Brentano geometry with a fixed divergence s l i t (<S = const) and no d i f f r a c t e d beam monochromator. This allows easy 250 Table A4.1 Values of the parameters C and B n needed to f i t the experimental v a r i a t i o n of <5 C -.34160 B 1 11.55511 B 2 -2.03072 B 3 2.40173 B, -0.78281 251 Figure A4,3, V a r i a t i o n of the s l i t width versus 0 f o r a) 6 <=? sinG, b)<5 given by A4.4 and c.) 8 given by A.4,5 252 comparison with standard published X-ray data. The second i n t e n s i t y includes the e f f e c t s of the automatic s l i t through A4.5 and the d i f f r a c t e d beam monochromator. The t h i r d c a l c u l a t i o n includes the e f f e c t of the Be window using A4.6. The program examines a l l possible h, k and £ and groups r e f l e c t i o n s , where h, k and I lead to the same scattering angle and structure f a c t o r , into forms. The number of sets of h, k and & in a form i s the m u l t i p l i c i t y , m. This number i s needed i n A4.1 and does not have to be input by the user. The program sorts the Bragg peaks, l a b e l l e d by the M i l l e r indices h, k and £, according to t h e i r scattering angle and p r i n t s a nice table giving the r e s u l t s of the c a l c u l a t i o n . The program i s c a l l e d SPECTRUM.S and an i n s t r u c t i o n set for the use of t h i s program can be found i n the f i l e SPECTRUM.W under U.B.C. computer number XBAT. 253 A4.2 Neutron D i f f r a c t i o n The r e l a t i v e i n t e n s i t y of Bragg peaks measured in the geometry of Figure A4.4a i s m F 1 0 1 sine sin29 A f ( 6 ) A4.7 where A'(8) i s the angle dependent absorption factor (Bacon 1975). The experiments described in Chapter 6 involved a f l a t plate specimen, t o t a l l y immersed in the neutron beam at a l l angles as shown i n Figure A4.4b, The specimen i s half angled with the detector as in the Bragg-Brentano geometry. The number of neutrons incident on the sample i s proportional to the cross-sectional area of the beam cut by the specimen. I t i s easy to show that i n t h i s case ! a m 1F12 i f , ! . e-(2U'd/sine) > 1 0 1 sin2e 2y' e J > A 4 - 8 where d i s the specimen thickness and y' i s the e f f e c t i v e l i n e a r absorption c o e f f i c i e n t which includes a l l processes (scattering and absorption) which remove neutrons from the incident beam. There are two l i m i t i n g cases: 1) y'd/sin 6 << 1 which y i e l d s m |F| d n l a sine'si^e and 2) y'/d sin 6 >:> 1 which yields _ m IF1 2 d . / m I a -rr—\—1 . — o a . A4.10 2y' sm20 Figure A4.4 Two geometries used f o r neutron d i f f r a c t i o n experiments. 255 In the f i r s t case, where the absorption i s s m a l l , the s c a t t e r e d i n t e n s i t y s c a l e s w i t h the t h i c k n e s s , or amount of m a t e r i a l . The second case corresponds to the i n f i n i t e t hickness l i m i t , which can be reached at small thicknesses i f y' i s s u f f i c i e n t l y l a r g e . 256 A P P E N D I X F I V E IMPERFECT STAGING A5.1 The Hendricks-Teller Solution to D i f f r a c t i o n from Disordered Layer Structures The c a l c u l a t i o n s presented f o r disordered layer structures i n t h i s appendix w i l l help us to understand the v a r i a b l e (004) Bragg peak width of L i x T i S 2 discussed i n Chapter 5. The c l a s s i c paper by Hendricks and T e l l e r (1942) on X-ray interference from p a r t i a l l y ordered layer l a t t i c e s i s . t h e basis f o r the ideas presented here. However, another method of solving the problem, based on the t r a n s f e r matrix method, i s presented. The s p e c i f i c problem that we want to treat i s the case of an imperfect stage 2 structure. This i s i l l u s t r a t e d in Figure A5.1. A series of planes, with equal numbers of two c h a r a c t e r i s t i c spacings, c and c,, cl D are stacked such that perfect staging (c , c, , c , c, •••) i s not a b a b n e c e s s a r i l y obtained. The degree of perfection can be described i n terms ab of the p r o b a b i l i t y , P , of f i n d i n g c a f t e r c . Because the number of D cL a and b layers are equal P = P . I f P = 1 we obtain perfect staging. If P = h, the layers are randomly stacked and i f P =0, regions of c and c, coalesce and a phase mixture i s obtained. We would l i k e to a b v P E R F E C T IMPERFECT Figure A5.1. Perfect (stage two) and imperfect staging 258 ab examine the e f f e c t s of changes i n P on the predicted d i f f r a c t i o n pattern. We assume that the atomic content of each layer i s i d e n t i c a l and that there i s no layer dependent s h i f t i n the atom positions p a r a l l e l to the l a y e r s . The scattering from the i n t e r c a l a t e d l i t h i u m w i l l be neglected because i t s X-ray scattering cross-section i s small. The only disorder in the problem i s the two d i f f e r e n t layer spacings, and c^. For the incident and scattered X-rays as shown in Figure A5.2, we can represent the amplitude and the phase of the scattered r a d i a t i o n from layer k as a vector in the complex plane V k(q) = |V k(q)| e i < f )k . In t h i s expression, q = k.-k , i s the scattering vector (| k. [ = | k | =~) , — —1 — r i r A |Vk(c[)| i s the amplitude of the scattered r a d i a t i o n , which i n general w i l l be a function of q, and <j>^  i s the phase. ^ ( 3 ) c a n be thought of as the layer structure f a c t o r . The amplitude of the r a d i a t i o n scattered by a c r y s t a l containing 2n+l planes labeled -n, -n+l,»'-*, 0,,1, •••,n i s I | V k ( 2 ) | e 1 ^ k=-n and the i n t e n s i t y per layer, I, i s k=-n m=-n Because the atomic content of each of the layers i s i d e n t i c a l , |VfcCq>! = |V m(q)| = | V ( 3 ) | 259 layer k+1 layer k layer k-1 Figure A5.2. The incident (k.) and scattered (_k^ ) X-rays from a layered c r y s t a l . q =k-~iSf * s t^ie s c a t t e r i n g vector. The phase shif ' | ^ + ^ - < { ) , =<j)^  between the scattered r a d i a t i o n from layers k+1 and k i s cjj^q-c^ 2 6 0 Making the s u b s t i t u t i o n k' = k-m we obtain 1 -'2ST lv(a>l2 T I e ± ( w " *m> ] k'=-n-m n "I I rti - - _ rt, \ -I A5.1a m=-n The term i n brackets i n equation A5.1a i s the sum over the l a t t i c e of the phase s h i f t s between l a y e r s separated by k' i n t e r l a y e r spacings. We de f i n e <e 1 ( <f ,k-*o)> = — — i — £ ei(W~ *m> , A5.1b 2n+l ^ ' i t * 0 0 m=-n the average val u e of the phase s h i f t between l a y e r s separated by k i n t e r l a y e r spacings. Thus when n-*°° equation A5.1a becomes ( r e p l a c i n g k' wi t h k) I = | V ( q ) | 2 I <e1((|>k-<U> . A 5 . 2 a k=-°° Equation A5.2a can be r e w r i t t e n i n terms of the phase s h i f t s between i n d i v i d u a l l a y e r s , OO I = | v ( q ) | 2 I <e l { ( < ! ,k-*k-I> + ( ^ k - l ^ k - 2 ) + + (*l-*o ) }>. A5.2b k = _ o o We know that ^ " ^ k ^  only take on two va l u e s : 1 } V * k - 1 = *a = S V o r 2) • k - + k _ 1 = 4>b = qc b z . We set up the problem as shown i n f i g u r e A5.3. ^VJ-I"^i ~ $ ^ la y e r k i s what we c a l l a type a l a y e r and ^ k + l ^ k = ^ D ^ l a y e r Is a type b l a y e r . The p r o b a b i l i t y of f i n d i n g <j> a f t e r <j> (the same as f i n d i n g D 3. 261 Type a Type b Type b Type a Type b Cb F i g u r e A5.3. A l a y e r spacing of c i s taken to come above a type a l a y e r and a spacing of c^ above a type b l a y e r . cj> =q-c a z, c()^=q>c^ z. a type b layer a f t e r a type a layer) i s ,ab (1 - 4A) where -\ < A 1 \. The subscript 1 r e f e r s to the f a c t that we are dealing ab with f i r s t neighbor layers. For A<0, P^ >Jg and staging i s favored. S i m i l a r l y , P a a = h (1 + 4A) i s the p r o b a b i l i t y of f i n d i n g an a layer a f t e r an a layer. We also have and „ba , ,, ,., „ab P 1 = h (1 - 4A) = P x P L = h (1 + 4A) = P 1 because the number of a and b layers are equal. If layer zero i s type a, the p r o b a b i l i t y that layer 2 i s also type a i s _aa aa aa ab ba P 2 = P ] _ P ] _ + P L P L = % (1 + (4A) 2) = P 2 b S i m i l a r l y P J A = p a b = Jj (1 - (4A) 2) The subscript 2 indicates we are dealing with 2nd neighbor l a y e r s . One can e a s i l y v e r i f y that 263 and „aa „bb , ,, . ,.k, P, = P, = h (1 + ( A A ) ) k k „ab „ba , ,.. ,\k, k k = ^ ( " ( ^ ) A5.3a A5.3b When A<0, a c o r r e l a t i o n parameter, £, f o r staging can be defined by and aa bb P k = P k k k h (1 + ( - l ) k - e - k / ? ) h (1 - ( - l ) k e " k / ? ) A5.4a A5.4b Using equations A5.3 and A5.4 we obtain £ = -(log|4A|) -1 A5.5 £ can be thought of as the number of layers over which the co r r e l a t i o n s extend. Returning to the c a l c u l a t i o n of the i n t e n s i t y , we p a r t i t i o n equation A5.26 into three parts as;:follows I = | V ( q ) | 2 I 1+ I <e i { (*k-*k-l> + + (*k-+o>>> + I k=l + £ < e i ^ ( * _ k - * - ( k - i ) ) + ••• + (*_i-* 05 }> ] : A5.6 k-1 It i s us e f u l to consider z = < e i { ( (* ,k- <' >k-l ) + + ( * l - * 0 ) } 5 A5.7 Half of the members of an ensemble of layer systems w i l l have (<j>1-(}> 1 O c and the other ha l f w i l l have (<f>..-<j> ) = (K. Thus i o D Z = U e 1 ^ + U e^b <e i { ^ k - * k - P + + ( * 2 - + l ) > *• . A5. If d>,-<|> = <j) , t h i s implies that layer zero was type a. The p r o b a b i l i t y 1 o a 3.3. that layer 1 i s also type a i s P^ . The p r o b a b i l i t y that layer 1 i s ab type b i s . Sim i l a r statements can be made f o r those members of the ensemble where = Now we perform the average over ^ - ^ j " which depends only on the type of layer 1 (a or b), as we have set up the problem. We f i n d Z = l i ^ a \ e 3 + h e J h (1 + 4A) e ± ( f )a + h (1 - 4A) e 1 < j >b j + % (1 - 4A) e i < f > a + 5s (1 + 4A) e 1 < l > b x A5.9 x <e i{(<j>kH >k-l) + • • • + (<(>3-(|>2^> Let us examine the second term of t h i s expression i . e . h(l- 4A) e i < f )b Here, layer zero was type a, which leads to the fac t o r e"^ a. The leading f a c t o r of \ r e s u l t s because one half of the systems i n the ensemble have layer zero type a. Layer 1 was type b, hence e ^ b , and the fac t o r *2 (.1 - 4A) i s the p r o b a b i l i t y of f i n d i n g a type b layer a f t e r a type a layer. The other terms a r i s e i n a s i m i l a r manner. We notice that jj e 1 * 3 ( h (1 +'4A) e 1 < f >a + (1 - 4A) e ^ a ) A5.10a and \ e 1 ^ = jg ( !j ( i + 4 A) e^b + % (1 - 4A) e ^ b ) . A5.10b 265 This enables us to rewrite equation A5.9 i n matrix form as < ei{(<^ k-(f> k_ 1) + • • • + C*3-*2> }> A 5 . l l z = ( h h) r 2 l where Q i s the 2x2 matrix f h (1 + 4A) e 1** % (1 - 4A) e ± ( i >a ^ h(l- 4A) e i < J )b % (1 + 4A) e i < f )b A5.12 Continuing i n t h i s manner i t i s easy to show that Z = ( h h) k f 1 1 '< Q 1 A5..13 At t h i s point one notices that what has been done i s very s i m i l a r to the t r a n s f e r matrix method f o r evaluating the p a r t i t i o n function f o r the one dimensional Ising model (e.g. see Huang 1963). From equation A5.13 i t i s straightforward to show that Trace r i 1 1 f i i The trace of a prOductiof three matrices i s invariant under c y c l i c permutations. Therefore Z = h Trace B = h Trace B 1 1 1 1 1 1 1 1 k B " 1 k B - X A5.14 where B i s the 2x2 matrix that diagonalizes Q. The matrix B also 266 diagonalizes Q since B Q k B" 1 = ( B Q B" 1 ) k -1 We denote the diagonal elements of B Q B by and Q 2 2 > Because k -1 B Q B i s diagonal, only the diagonal elements of the matrix R, R = B 1 1 1 1 ^ B" 1 A5.15 enter into the trace. These elements we define as R^ and R 2 2- Thus z = % I R - . ( Q ± i ) x = l A5.16 The elements and Q 2 2 are the eigenvalues of the matrix Q. Inserting the form of Z given by equation A5.16 into equation A5.6, we obtain I = |v(q) |v(q) I % < R i i ( ^ i i ) k + R i i ( Q i i ) k } J k= 1 i= 1 1 + % I 1=1 r R , , Q XI 1 1 1 - Q + 1 1 R. . Q.. i i i i 1 - Q". J i i J A5.17 The * indicates complex conjugation and a r i s e s due to the phase s h i f t s , ^(j) or - d ) i n the second summation i n equation A5.6. To solve the problem, we need only f i n d the matrix B that diagonalizes Q. The r e s u l t (equation A5.17) i s i d e n t i c a l to the s o l u t i o n one obtains by applying the general r e s u l t obtained by Hendricks and T e l l e r (1942) ( t h e i r equation 30) to our s p e c i f i c problem. In f a c t Hendricks and T e l l e r use a matrix approach 267 very s i m i l a r to the one described here to derive t h e i r general equation which can treat any number of d i f f e r e n t types of layers and phase s h i f t s between them. Our s o l u t i o n technique can also be generalized to more d i f f i c u l t problems. Omitting the algebra, the r e s u l t that we obtain f o r the case of two d i f f e r e n t layer spacings when A<0 i s I = h |V(q)| + r (a + 3 ) . . e : •' * 1 - Y + a 2 2 i i • , • - 2 i - (y + a) coscj) + ((y + OL) /2) (a - 3) 3 f 1 - Y - a 2 2 'i + 1 - (y - a) costj) + ((y - a)/2)' A5.18 where a = / (1 + 4A) 2cos 2 c J ) j - 16A A5.19a = (1 - 4A) cos<}> A5.19b Y = (1 + 4A) cost)) A5.19c • = C* + <k)/2 a b A5.19d *d = ( * a " V / 2 = e (j) and a b c + c t a b A5.19e A5.19f The f i r s t term i n equation A5.18 describes the Bragg peaks that occur at <j) = 2im where n i s an integer. These peaks correspond to the average l a t t i c e . The peaks a r i s i n g due to s u p e r l a t t i c e e f f e c t s at = Tr(2n - 1) are described by the second term. ."For instance, when A=-.25 (perfect 268 staging) i t i s easy to show that equation A5.18 predicts i n f i n i t e l y sharp Bragg peaks at <}> = 2-rrn with i n t e n s i t y proportional to (1 + cos<j>^ ) and i n f i n i t e l y sharp s u p e r l a t t i c e peaks at <j> = (2n — 1)TT with i n t e n s i t y proportional to (1 - coscj)^) . This i s exactly the r e s u l t one obtains by considering the geometrical structure f a c t o r f o r the two layer unit c e l l which i s formed when staging i s perfect. When A=0, corresponding to random stacking of the layers , or ^ ^O* n ° s u p e r l a t t i c e peaks are expected. Indeed the second term i n equation A5.18 vanishes when A=0 or 'I'^O* When A=0, the f i r s t term i n equation A5.18 becomes i 2 ^ „ 1 - C O S (j), i = | v ( q ) | r~ — A 5- 2 0 1 - 2 COS<j)^ COS(f> + cos <J>^  which agrees with Hendricks and T e l l e r ' s r e s u l t f o r random stacking ( t h e i r equation 6). <f> i s the average phase s h i f t between layers (equation A5.19d). I t i s e a s i l y , seen that <f> = 2ir£, where £ i s defined by q = h b + k b 2 + £ b_3 . A5.21 In t h i s expression b^, b^, and b^ are the r e c i p r o c a l l a t t i c e vectors of the average l a t t i c e and h, k and £ are the M i l l e r i n d i c e s . The layer structure f a c t o r V(q) has i n f i n i t e l y sharp peaks at h, k i n t e g r a l . When h and k are i n t e g r a l , we w i l l observe peaks corresponding to the average l a t t i c e at £=0,±1 ,±2, • • • and, peaks from the s u p e r l a t t i c e at £=±1/2 ,±*3 /2 , • • •. The scattered i n t e n s i t y per layer as a function of £, obtained using equation A5.18, near £ = 4 and £ = 4.5 i s shown i n f i g u r e A5.4 f o r various values of A. |v(q)| has been set equal to one i n t h i s c a l c u l a t i o n . The peak at £ = 4 sharpens r a p i d l y as |A| increases. The peak at £ = 4.5, 269 "co c CD CD CD D O CO 5 -4 -c CD TJ CD i _ CD O CJ CO 3 2 1 h 0 4.40 4 . 4 4 4 .48 4 .52 4 .56 4.60 I in (0 0 ft 240 200 160 120 80 40 0 3 — c /jc —\-^===^p 1 V — — i ^ S r r ^ H — 1 90 3.94 3.98 4.02 4.06 h n (0 0 I) 4.10 Figure A5.4, Calculations of the scattered i n t e n s i t y per layer, I, from equation A5.18, with |v(q)| = 1 and e (defined by equation A5.19f) equal to .022, as a function of I. a) A=0, 5=0.0, b) A=.l, 5=1-09, c) A=.2, 5=4.48 and d) A=.23, 5=12.0. 270 due to the s u p e r l a t t i c e , requires a c o r r e l a t i o n parameter, £, (see equation A5.5) of several layers before i t becomes sharp enough to "see" i n a d i f f r a c t i o n experiment. The peaks due to the average l a t t i c e have a half width at half maximum, A£ , av A£ = T T £ 2  2 \  + 4 A A5.22a av 1 - 4A = T T I2 e 2 tanhO^-) A5.22b where c - c, <j>, e = - 2 ^ = - ^ . A5.23 C a + Cb r In deriving t h i s r e s u l t from equation A5.18 we have assumed A £ & v < < £ . The half width of the s u p e r l a t t i c e peaks, A £ g , i s found to be A £ = J - 1 ~ 4 l A i A5.24a S 2 7 7 /4[AT = - sinh(-J-) A5.24b I T Zb, 1 f o r 5 » 1 , A5.24c 2 T T £ where again A £ g < < £ has been assumed i n obtaining A5.24 from A5.18. Cle a r l y , as £->-0, A£g->-«> (equation A5.24b) and we therefore do not expect equation A5.24 to be correct when A£ - £ . s In a powder d i f f r a c t i o n experiment one can only co n t r o l |q|. The half widths, A[q[, of the peaks due to the average l a t t i c e w i l l depend not only on £, but also on the r a t i o of h and k to Using equation A5.21, fo r a hexagonal c r y s t a l , we obtain 271 A | q | = ( ( h 2 + hk + k 2 ) b 2 + l2b*)~* lb2 A£ . A5.25 To convert A q into a ha l f width i n the Bragg angle, 0, we note that h i = y - sine which y i e l d s A6 = -. ~ ~ A I q[ A5.26a 4ir cos0 1 -( ( h 2 + hk + k 2 ) b 2 + l2bh~^2 lb\ M . A5.26b 4 T T cos6 £ Using equations A5.22 and A5.26b, the half width, A(6 ), of the (00£) av peaks from the average l a t t i c e i s . , „ £ > . i r sine tan0 , N2 1 + 4A » c m A(6 ) = T-T — r- (c - c,) -rr A5.27a av A (c + c ) a b 1 - 4A a b 2 »2 (c - c, ) , = T T A — ^ r - — - ^ tanh(-^-) A5.27b cos0 / , \J 2^ (c + c ) a b where a b sine For the s u p e r l a t t i c e peaks, £ = ±1/2, ±3/2, ACe*) = T — 7 ~ — \ s- 1 41A1 A5.28a 272 The half width of powder peaks from perfect c r y s t a l s of f i n i t e s i z e i s (Warren 1969) A6 = X . A5.29 • 4L cos9 where L i s the s i z e of the p a r t i c l e . This allows us to define T T £ ( C + c,)/2 a b as an e f f e c t i v e p a r t i c l e s i z e f o r the staged material. The e s s e n t i a l r e s u l t s of t h i s c a l c u l a t i o n w i l l now be summarized. We have found that the width of the Bragg peaks corresponding to the average l a t t i c e v a r i e s r a p i d l y due to changes i n E, when £ i s small (see Figure A5.4 and equations A5.22 and A5.27). The s u p e r l a t t i c e peaks remain too broad to detect e a s i l y for ££2. Therefore, changes i n width of the Bragg peaks corresponding to the average structure are expected as the c o r r e l a t i o n length, £c, f o r staging changes. S u p e r l a t t i c e peaks may not be observed i f £ i s small. We expect t h i s to be true i n more complicated models involving higher stages and many d i f f e r e n t c-spacings. 273 A5.2 The E f f e c t of the Daumas-Herold Domain Model of Staging The simple p i c t u r e of staging presented i n the l a s t section, consists of v a r i a t i o n s i n the i n t e r c a l a n t content perpendicular to the l a y e r s , but no v a r i a t i o n p a r a l l e l to the l a y e r s . I t i s d i f f i c u l t to understand how phase t r a n s i t i o n s between d i f f e r e n t stages (eg. stage"1 -> stage 2) can occur: i n t h i s model. Because the i n t e r c a l a n t cannot d i f f u s e perpen-d i c u l a r to the layers ( i n most cases), a staging transformation within t h i s model must involve.some of the i n t e r c a l a n t coming out of the l a t t i c e and re-entering i n d i f f e r e n t i n t e r l a y e r gaps. Experiments on graphite i n t e r c a l a t i o n compounds show that t h i s does not happen i n a staging transformation. This dilemma i s resolved by the Daumas-Herold (1969) domain model of staging. In the Daumas-Herold model each i n t e r l a y e r gap contains an equal amount of i n t e r c a l a n t , but regions (domains) of high and low i n t e r c a l a n t concentration are formed within each layer. Transitions between stages simply involve a rearrangement of the atoms within each layer. These domains, for a stage two compound might be arranged as shown schematically i n f i g u r e A5.5. We would l i k e to consider the e f f e c t s of domain structures on the observed d i f f r a c t i o n pattern. The r e s o l u t i o n of a diffractometer i s r e l a t e d to the coherence length which determines the s i z e of a specimen sampled coherently i n a d i f f r a c t i o n experiment. One can show using arguments given i n Born and Wolf (1975) that the coherence length, L , i s given approximately by ,. 2 T T . . A~TqT where A'|q| i s the uncertainty in the magnitude of the scattering vector 3. "A;! qJ. i s .one measure of the resolution' of .the d i f f ractometer.'" "For' 274 a) — r—»-J / / -\ • • C b / \ r • / \ / \ _^ u a T \ • • / \ v \ • y \ / \ r Figure A5.5. The Daumas-Herold domain model of staging f o r a stage 2 compound, a) equal s i z e domains, perfect staging; b) domains of random l a t e r a l extent and perfect staging; c) domains of random l a t e r a l extent and imperfect staging. example, a diffractometer with an angular r e s o l u t i o n A(29)=.10 u, using X = l.54178 A, has A| qj - . 006A - 1 or L-900 A at 26=30°. Domains of s i z e greater than L w i l l scatter independently, and thus appear i n f i n i t e to the diffractometer. Domains much smaller than L w i l l scatter coherently. (00£) s u p e r l a t t i c e peaks, corresponding to a two layer, c +c =2c, unit ce l l . , w i l l be observed for the p e r i o d i c stage two compound depicted i n f i g u r e A5.5a i f the domain s i z e , ,r, i s greater than L. If the domain s i z e , r, i s much smaller than L, (00£) s u p e r l a t t i c e peaks w i l l not be observed due to the f a c t that adjacent columns of domains are i d e n t i c a l except for a s h i f t of c p a r a l l e l to the c-axis. These columns w i l l i n t e r f e r e d e s t r u c t i v e l y when q i s given by a (00£) s u p e r l a t t i c e p o s i t i o n . The regular periodic array depicted i n Figure A5.5a w i l l produce (hk£) s u p e r l a t t i c e peaks, indexed on the unit c e l l given by 2r and c +c, , f o r some values of h and k even when r<<L. A more r e a l i s t i c model of a stage two compound i s shown i n Figure A5.5b, where the domains have been taken to have random l a t e r a l extent, although the staging c o r r e l a t i o n s p a r a l l e l to the c-axis have been taken to be i n f i n i t e i n length. For t h i s model, (00£) s u p e r l a t t i c e peaks w i l l be observed when the average domain s i z e , <r>, i s greater than L. I t i s c l e a r that (00£) s u p e r l a t t i c e peaks w i l l not be observed when <r>«L. (00£) Bragg peaks corresponding to the average l a t t i c e w i l l have .resolution l i m i t e d width i n both cases. S u p e r l a t t i c e peaks corres-ponding to q not p a r a l l e l to c w i l l not be observed due to the l a t e r a l randomness of the domains. In Figure A5.5c, domains with random l a t e r a l extent and imperfect staging c o r r e l a t i o n s are shown. In t h i s case we expect the arguments given i n the preceeding section of the appendix to hold when the average 276 domain siz e i s large compared to L. When the domain s i z e i s small com-pared to L, we expect to observe no s u p e r l a t t i c e peaks and only broad (00£) peaks from the average l a t t i c e due to the large degree of disorder p a r a l l e l to c. An exact mathematical treatment of poorly correlated domains of random l a t e r a l extent i s d i f f i c u l t and i s beyond the scope of t h i s t h e s i s . If the domains i n a staged compound have a l i m i t e d and v a r i a b l e l a t e r a l extent (eg.-10 l a t t i c e spacings), disorder i n the staging sequence might be expected. The q u a l i t a t i v e arguments given above predict that (00£) peaks of the average structure w i l l be broad and no s u p e r l a t t i c e peaks w i l l be observed. If the domain s i z e were to grow i n l a t e r a l extent (perhaps as the temperature was lowered), one expects the degree of order i n the staging sequence to improve. This i s due i n part to the reduction i n the importance of f l u c t u a t i o n s f o r large systems. Also i f one assumes that the staging mechanism i s driven by repulsive i n t e r a c -t i o n s , u', between int e r c a l a n t atoms i n adjacent planes i t w i l l become increasingly d i f f i c u l t to place two domains of high in t e r c a l a n t content i n contact as the domain s i z e grows. (00£) Bragg peaks from the average l a t t i c e w i l l sharpen and (00-O s u p e r l a t t i c e peaks w i l l s t a r t to appear as the domain s i z e increases. The v a r i a b l e (004) peak width of L i TiS„ (Chapter 5) and the appear-ance of the (00^)' s u p e r l a t t i c e peak i n L i 1 4 T i s 2 a t l o w temperatures (Chapter 6) are most l i k e l y due to the coupled e f f e c t s of v a r i a b l e domain s i z e and changes i n the c o r r e l a t i o n length f o r staging. 277 APPENDIX 6 DETERMINING PHASE DIAGRAMS'OF LATTICE GAS MODELS SOLVED WITH THE BRAGG-WILLIAMS APPROXIMATION The Landau expansion (Landau and L i f s h i t z 1969) i s of l i t t l e use when applied to the free energy given i n equation 7.23. The reasons for t h i s w i l l be discussed i n t h i s appendix. The i t e r a t i v e technique, presented i n Section 7.4.2, which was used to determine most of the phase diagrams presented i n the thesis w i l l also be discussed. The notation used here i s the same that was used i n Section 7.4.2. A Landau expansion for the Gibbs free energy (equation 7.23) can be made i f we write x^ = x+ri and x^ = x-r). T] i s c a l l e d the order para-meter. When Tl=0 the l a t t i c e gas described by equation 7.23 w i l l be stage 1 and when n^O i t w i l l be stage 2. Expanding equation 7.23 i n powers of n we f i n d where G(x,0) = N_L E x + (3u + u')x + ' L o Jx + kT(xln(x) + a + x A6.1 + ( l - x ) l n ( l - x ) ) 1-x 1 (a' + x) Ja •3 + 3u - u' A6.2a (1 - x ) 3 J (a + x ) 5 1 Ja A6.2b 278 and r - M L 30 K + 1 x ( 1 - x ) -^2 . A6.2c (a + x) Because a l l cubic terms (n 3) vanish i n the expansion, one i s tempted to state that the phase t r a n s i t i o n s should be continuous (Landau and L i f s h i t z 1969). However, the c o e f f i c i e n t s B and C can be negative which makes the system unstable to f i r s t , order t r a n s i t i o n s over an extended range of x and T. This i s because i f B and C are negative, G can be made a r b i t r a r i l y small by making r\ a r b i t r a r i l y large. The Landau expansion, which involves expanding i n powers of the small quantity n'» f a i l s for t h i s problem. When J=0, the higher order c o e f f i c i e n t s (B,C,—) are always p o s i t i v e and then the phase diagram i s found by solving the equation A=0 (Landau and L i f s h i t z 1969). For J=0, kT = 2x(l - x) c For kT/u' <(kT/u') c a stage two structure w i l l be stable and for kT/u ,>(kT/u') c a stage 1 structure w i l l be stable. The phase t r a n s i t i o n i s continuous for J=0. Phase diagrams for non-zero J can be found by solving the s e l f consistent equations 7.25 and 7.26, x. = i : i + ^ ( x ^ x i + i l / k T j " 1 ? > 2 5 where Ja e{x.,x...} = E - y + 2u' x + 6ux + j 1 1 + 1 ° 1 + 1 1 (a + x . ) 2 7.26 279 This i s done by an i t e r a t i v e procedure. For given values of y, T, E Q , u*, u, a and J i n i t i a l guesses for x^ and x 2 are made. The i n i t i a l guesses w i l l be termed X^Q and * 2Q« f i r s t i t e r a t i o n y i e l d s x ^ and x 2^ as follows; x 11 ' ! + e e { x 1 0 f x 2 0 } / k T yl + (1_g)XiQ and A6.3 V 2 1 f 1 + eE { x 2 0 ' x l l } / k T 1 -1 "g + ( l - g ) x . 20 where . 1~ g £ 1, The procedure continues i n t h i s manner with x K n + l ) "2 (n+1) f i + e e { x l n ' x 2 n > / k T 1 -1 g + (l-g)x In i + e e { x 2 n » x l ( n + l ) } / k T g + (l-g)x 2n A6.4 When x./ - A ^ x. < x. , , 1 N + A for i= l and i=2, the procedure i s i(n+l) - — i n — i(n+l) ' v deemed to have converged and x^ n and x 2 n are approximate solutions to equations 7.25 and 7.26. A i s an a r b i t r a r i l y small quantity taken i n the calc u l a t i o n s described i n the thesis to be A = 10.^x. . We have found m that at high temperatures t h i s procedure converges r a p i d l y for g=l, while at low temperatures the i t e r a t i o n can o s c i l l a t e between two values i f g=l. For g<l the range of convergence of the i t e r a t i o n i s extended to lower temperatures. However, the rate of convergence of the i t e r a t i o n slows as g i s made smaller. Usually one t r i e s to pick the largest value 280 of g, at a given temperature, which avoids the o s c i l l a t i o n problem. Several sets of i n i t i a l guesses for X^Q and X^Q were examined to determine the dependence of the ehdpoint of the i t e r a t i o n on the s t a r t i n g point. It was found that near f i r s t order phase t r a n s i t i o n s more than one sol u t i o n to equations 7.25 and 7.26 could be found, due to l o c a l minima in G associated with metastable phases. In these cases the solution which gave the lowest free energy was taken. In cases where the i t e r a t i v e procedure established that the phase t r a n s i t i o n was continuous i t was found that the r e s u l t s of the Landau theory were correct. For example, Landau theory i s in agreement with the i t e r a t i v e s o l u t i o n on the dashed portions (continuous t r a n s i t i o n s ) of figures 55, 57, 71a and 72. Because of t h i s , the v a r i a t i o n of the maximum temperature for the formation of the stage two phase as a function of applied pressure can be determined using Landau theory. This was done i n Section 7.5.2. However, we have found no a p r i o r i procedure (other than a complete i t e r a t i v e solution) f or deciding which portions of a phase diagram may be r e l i a b l y analysed using the Landau theory. 281 Appendix 7 MONTE CARLO .SIMULATIONS The purpose of t h i s appendix i s to show how the Monte Carlo method was applied to the l a t t i c e gas model described by the Hamiltonian given in equation 7.27. The notation used i n t h i s appendix i s the same as in Section 7.4.3. The method we have used i s based on the Monte Carlo technique introduced by Metropolis et a l . (1953) and discussed in d e t a i l by Binder (1979). In the grand canonical ensemble, the ensemble average of any observable Q i s given by I Q(i) e - ( H ( ± ) - n n ( « ) / k T _ _ i . A7.1 " v "(H(i) - un(i))/kT I e i i l a b e l s a p a r t i c u l a r state of the system. H(i) i s the i n t e r n a l energy of the system, n ( i ) i s the number of p a r t i c l e s and Q(i) i s the value of the observable Q, a l l in state i . For systems with many possible states, equation A7.1 i s of l i m i t e d usefulness unless a convenient method of performing the sum over states i s found. When equation A7.1 cannot be evaluated simply, one often resorts to Monte Carlo'methods, which e f f e c t i v e l y perform a sum over a f r a c t i o n of the possible states of the system, to approximate <Q>. The states included in the Monte Carlo sum are generated by a computer and selected as determined by t h e i r p r o b a b i l i t y of occurring in the manner described by Metropolis et a l . ( 1 9 5 3 ) . 282 The l a t t i c e of s i t e s i s f i l l e d i n an a r b i t r a r y manner to i n i t i a l i z e the Monte Carlo process. Then each s i t e i s examined i n turn as follows. If the s i t e i s i n i t i a l l y empty ( f i l l e d ) the change i n H-yn, A(H-yn) , r e s u l t i n g from the f i l l i n g (emptying) of the s i t e i s computed. I f A(H-yn)< 0, the s i t e i s f i l l e d (emptied). If A(H-yn)>. 0, the s i t e i s f i l l e d (emptied) i f e~ A^ H _ y n )/ k T>,p and l e f t empty ( f i l l e d ) for < e A(H yn)/kT ^ ^ where p i s a random number between 0 and 1. The examination of a single s i t e i n t h i s manner w i l l be c a l l e d a Monte Carlo step. We re f e r to the examination of every s i t e of the l a t t i c e i n the above manner as one Monte Carlo pass over the l a t t i c e . For a l a t t i c e of N s i t e s , N Monte Carlo steps are needed to make up a Monte Carlo pass over the l a t t i c e . The Hamiltonian, equation 7.27, depends on the occupancy of each s i t e , x ^ , and the average concentration i n each l a y e r , x^. After each Monte Carlo step the average concentration i n each l a y e r , must be evaluated because t h i s quantity i s needed for.the next Monte Carlo step. For a given set of y, u, u', T, J , a and E q in H-yn, the r e s u l t s of the f i r s t several hundred Monte Carlo passes over the l a t t i c e were discarded due to possible dependences on the s t a r t i n g configuration of the l a t t i c e . After each subsequent Monte Carlo pass, N. L A7.2 A7.3a A7.3b 283 x ^ , x Z and x ^ were evaluated and stored i n the computer, c^Cx^) i s given by equation 7.5. The r e s u l t s of several thousand Monte Carlo passes were averaged to obtain <x>, < x 2 > , and c = <c>. The dependences of y and c on <x> were obtained by p l o t t i n g the Monte Carlo data. 9x/9y i s obtained by d i f f e r e n t i a t i n g y(<x>) using f i n i t e differences or by using equation 2.13 which i s 2 2 <x > - <x> kT N 3x 9y 2.13 9y/9T) i s obtained from 9x/9y) and 9T/9x) using 3y 9T 3Tl 1 9x7 3x A7.4 (Stanley 1971). 9T/9x) i s calculated by taking f i n i t e differences between Monte Carlo runs taken at d i f f e r e n t temperature. The c o r r e l a t i o n function between j 1 " * 1 neighbour layer s , < I I ( 2 x k - l ) ( 2 x -1) > k = l requires the computation of I X ( 2 x k - l ) ( 2 x k + . - l ) k=l af t e r each Monte Carlo pass. F i r s t and second neighbour layer c o r r e l a t i o n s functions were evaluated to compute W/, given by equation 7.30. The assistance of Leo MacDonald and Geoff Johnson with the ptimization of the computer program.which performed the Monte imulations i s greatly appreciated. 285 BIBLIOGRAPHY A l e f e l d , G. ahdVblkl, J . 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"In Situ X-ray Diffraction Studies on Lithium Intercalation Compounds Can. J . Phys. 60, 307 (1982). PUBLICATIONS CONT'D. Dahn, J.R., Dahn, D.C. and Haering, R.R " E l a < ; t i r 

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