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Studies of phase transitions in lithium intercalation batteries by dQ/dV measurements Johnson, Geoffrey William 1982

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STUDIES OF PHASE TRANSITIONS IN LITHIUM INTERCALATION BATTERIES BY dQ/dV MEASUREMENTS by GEOFFREY WILLIAM JOHNSON B.S c , The University of B r i t i s h Columbia, 1980 A THESIS SUBMITTED, IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE 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) Geoffrey William Johnson, 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 Date DE-6 (3/81) ABSTRACT F i r s t order phase t r a n s i t i o n s have been detected electrochemically in i n t e r c a l a t i o n b a t t e r i e s . The quantity dQ/dV peaks when the phase front associated with a f i r s t order phase t r a n s i t i o n moves through an i n t e r -c a l a t i o n compound, where V i s the c e l l ' s voltage and Q i s the charge that flows. The divergence of dQ/dV i s analogous to the divergence of the compressibility of a gas when a f i r s t order phase t r a n s i t i o n occurs. dQ/dV can be measured by the standard electrochemical technique of l i n e a r sweep voltammetry, but re s o l u t i o n and c l a r i t y are v a s t l y improved by measuring dQ/dV at constant current, which minimizes the d i s t o r t i o n s i n dQ/dV due to k i n e t i c e f f e c t s . The phase diagram of the l i t h i u m i n t e r -calated layer compound has been constructed. Applications of dQ/dV measurements to phase t r a n s i t i o n s i n other i n t e r c a l a t i o n systems and to sin g l e phase regions are discussed. i i i TABLE OF CONTENTS Page ABSTRACT TABLE OF CONTENTS LIST OF FIGURES LIST OF SYMBOLS ACKNOWLEDGEMENTS x i i V l l CHAPTER 1 INTRODUCTION 1.1 1.2 1.3 1.4 In t e r c a l a t i o n Systems Phase Transitions and dQ/dV Ki n e t i c E f f e c t s i n an Experimental System Preparation of L i ^ V ^ Electrochemical C e l l s 1 1 8 11 14 CHAPTER 2 LINEAR SWEEP VOLTAMMETRY 2.1 dQ/dV Measured by Linear Sweep Voltammetry 2.2 Theory of K i n e t i c E f f e c t s on Phase Transitions 2.3 Experimental Apparatus and Technique 2.4 Results f o r L i VS„ X Z 2.5 Effectiveness of the Technique 21 21 23 37 39 49 CHAPTER 3 CONSTANT CURRENT dQ/dV 3.1 dQ/dV Measured at Constant Current 3.2 Theory of K i n e t i c E f f e c t s on Phase Transitions 3.3 Experimental Apparatus and Technique 3.4 Results f o r L i VS„ X z 3.5 Non-ideal Peak Shapes 3.6 Effectiveness of the Technique 51 51 53 62 65 84 90 CHAPTER 4 4.1 4.2 4.3 4.4 OTHER APPLICATIONS AND EXAMPLES OF CONSTANT CURRENT dQ/dV S e n s i t i v i t y of the Technique Applications to Single Phase Regions "Supercooling" Staging 92 92 97 101 106 i v Page CHAPTER 5 CONCLUSIONS 108 5.1 dQ/dV Measurements • 108 5.2 Linear Sweep Voltammetry 110 5.3 Constant Current dQ/dV 112 BIBLIOGRAPHY 116 V LIST OF FIGURES Figure Page 1. The structure of MX2 layer compounds. 3 2. Schematic diagram of an i n t e r c a l a t i o n c e l l . 5 3. A t y p i c a l flange c e l l . 17 4. Charge and discharge voltage curves for the L i VS„ c e l l GJ-9. X 19 5. Surface voltage and current due to a phase t r a n s i t i o n , i n l i n e a r sweep voltammetry theory. Fast s o l i d state d i f f u s i o n . 25 6. Voltammogram and current against charge f or a phase t r a n s i t i o n , i n theory with f a s t s o l i d state d i f f u s i o n . 27 7. Voltammograms with s o l i d state d i f f u s i o n added to theory. 32 8. Current against charge with s o l i d state d i f f u s i o n added to theory. 33 9. Voltammogram for the L_ XVS 2 c e l l GJ-9. 40 10. Current against charge f o r the Li^VS,, c e l l GJ-9. 41 11. Voltammogram for a Li^VS. c e l l of Dahn, J.R., and Haering. 42 12. Current against charge f or a L i VS„ c e l l of Dahn, J.R., and Haering. 43 13. Phase diagram for L i VS„. 46 & x 2 14. P a r t i a l phase diagram f or L i x V S 2 derived from x-ray data. 48 15. Theoretical peak i n constant current dQ/dV against c e l l voltage due to a phase t r a n s i t i o n . 58 16. Theoretical peak i n constant current dQ/dV against charge due to a phase t r a n s i t i o n . 59 17. Constant current dQ/dV against c e l l voltage for the L i VS 2 c e l l GJ-9, at 35 yamps. 66 18. Constant current dQ/dV against x and Q f o r the L_ xVS 2 c e l l GJ-9, at 35 yamps. 67 19. Comparison of l i n e a r sweep voltammetry and constant current dQ/dV f o r c e l l GJ-9. 68 Constant current dQ/dV against c e l l voltage for the L i VS^ c e l l GJ-9, at 140 vamps. Constant current dQ/dV against x and Q for the L i c e l l GJ-9, at 140 yamps. Positions of the leading edges of charge and discharge peaks for the 2.470 vo l t t r a n s i t i o n , extrapolated to remove the I R s h i f t and obtain the tr a n s i t i o n voltage. Positions of the leading edges of peaks for the 2.384 vo l t t r a n s i t i o n , similar to figure 22. Positions of the leading edges of peaks for the 2.201 vo l t t r a n s i t i o n , similar to figure 22. Hysteresis i n voltage and x from expected equilibrium values due to energy dissipation i n a two phase region. Peak height against Q Q / | I | for the 2.470 v o l t t r a n s i t i o n . Non-ideal peak of dQ/dV against time i f phase fronts are nucleated i n a Gaussian d i s t r i b u t i o n i n time. Non-ideal peak of dQ/dV against voltage i f phase fronts are nucleated in a Gaussian d i s t r i b u t i o n i n time. Constant current dQ/dV against c e l l voltage for the LixMoS2 c e l l PM-30, made with naturally occurring M0S2. Expanded view of the high voltage data from figure 29, showing a peak due to an impurity. Constant current dQ/dV against c e l l voltage f o r Li^Mo^S^. Material i n i t i a l l y Cu, ,Mo0S,. J 1.4 3 4 Constant current dx/dV against c e l l voltage for the L i TiS„ c e l l JD-235. x 2 Constant current dx/dV against x for the L i TiS„ c e l l JD-235. X Voltage curve of f i r s t discharge of the lithium against pyrite c e l l M-34, showing "supercooling" effect. Constant current dQ/dV against c e l l voltage for the lithium against p y r i t e c e l l M-47, showing "supercooling". Staging in L i NbSe v i i LIST OF SYMBOLS A Surface area through which i n t e r c a l a t i o n occurs. a Crystallographic a x i s , or dp/dV a (a constant) i n l i n e a r sweep voltammetry theory. a^ Value of c r y s t a l l o g r a p h i c axis a for a p a r t i c u l a r hexagonal unit c e l l . b Crystallographic a x i s , or -dp/dV AeR (a constant) i n l i n e a r sweep voltammetry theory. C "Capacitance" of a c e l l , c Crystallographic axis. C Q Value of c f o r a p a r t i c u l a r hexagonal unit c e l l . D D i f f u s i o n c o e f f i c i e n t or d i f f u s i o n constant. e Magnitude of the e l e c t r o n i c charge. F Helmholtz free energy of the cathode F a Helmholtz f r e e energy of the anode F Helmholtz free energy of a gas. G Gibbs f r e e energy. I dQ/dt. Current. I Peak current. P J Surface number current density, s L Halfwidth or radius of a p a r t i c l e . M A t r a n s i t i o n metal atom. n Number of int e r c a l a n t atoms , or Stage number, as i n a stage n layer compound. P Pressure. Q Net p o s i t i v e charge flowing from the cathode to the anode in the external c i r c u i t ; net charge of l i t h i u m ions that have been in t e r c a l a t e d . v i i i QQ Width i n Q of a f i r s t order phase t r a n s i t i o n . R C e l l resistance. r P o s i t i o n i n a p a r t i c l e . f P o s i t i o n of a phase front i n a p a r t i c l e . S Entropy. T Temperature. t Time. t Time at which Q n i s reached i f t = 0 when Q = 0. t ^ I n i t i a l time. t^ Time at which i s reached. U Int e r n a l energy. u f / L . Normalized p o s i t i o n of a phase front i n a p a r t i c l e . V Voltage at the surface of a p a r t i c l e , or C e l l voltage i n derivatives such as dx/dV, dQ/dV, or dt/dV unless otherwise s p e c i f i e d (as i n dp/dV f o r example). V C e l l voltage, c V IT: /C (a constant). Roughly, the voltage change i n a time T ^ . V Q Equilibrium voltage of a f i r s t order phase t r a n s i t i o n . V^ Voltage at which a t r a n s i t i o n peak ends on charge, i n l i n e a r sweep voltammetry. V2 Voltage at which a t r a n s i t i o n peak ends on discharge, i n l i n e a r sweep voltammetry. v Volume. w Gaussian d i s t r i b u t i o n . X A chalcogen atom. x Intercalant concentration, as i n L i MX„ •, or ' x 2 ' Dummy v a r i a b l e i n the d e f i n i t i o n of $. y A v a r i a b l e of integration, a dV/dt. Sweep ra t e , or Phase of the Li^VS 9 system, or of the Li vMoS 9 system. i x 3 Slope of a l i n e a r r e l a t i o n s h i p between voltage and charge, or Phase of the L i VS„ system, or of the L i MoS„ system, or X Z. X z. C r y s t a l l o g r a p i c angle. Y Dimensionality of a phase fr o n t . AF Change in the t o t a l Helmholtz free energy. An Change i n n. Ap Change i n i n t e r c a l a n t concentration across a f i r s t order phase t r a n s i t i o n . K Isothermal compressibility of a gas. u Chemical p o t e n t i a l of the cathode, p Chemical -potential of the anode. cl p Intercalant concentration. p^ Intercalant concentration at a p a r t i c l e ' s surface. p^ Intercalant concentration i n phase 1 near a f i r s t order phase t r a n s i t i o n . p^ Intercalant concentration i n phase 2 near a f i r s t order phase t r a n s i t i o n . a . Halfwidth of a Gaussian d i s t r i b u t i o n . T Variable of i n t e g r a t i o n . 2 L /D. D i f f u s i o n time constant. $ P r o b a b i l i t y i n t e g r a l . X ACKNOWLEDGEMENTS It i s a great pleasure to thank my supervisor, Rudi Haering, f o r the help and encouragement he has given me. I have greatly benefitted from his i n t e r e s t and physical i n s i g h t . My co-workers in Rudi Haering's group have provided many h e l p f u l discussions and ideas. In p a r t i c u l a r , I would l i k e to thank J e f f Dahn, Doug Dahn, Peter Mulhern, and Richard Marsolais, each of whom has provided a d d i t i o n a l examples of dQ/dV measurements for t h i s t h e s i s . J e f f Dahn's previous work on l i n e a r sweep voltammetry was invaluable. I would also l i k e to thank fellow lab members Marcel Py, Rod McMillan, Andre van Schyndel, Rick Clayton, and Alec Rivers-Bowerman. The construction of dQ/dV Analyser Model F06 by the Physics Department E l e c t r o n i c s Shop, and i t s programming by M. A. Potts, are g r a t e f u l l y acknowledged. F i n a l l y , I would l i k e to thank the Natural Sciences and Engineering Research Council f o r the f i n a n c i a l support they have given me over the past two years. 1 CHAPTER ONE INTRODUCTION 1.1 I n t e r c a l a t i o n Systems In an i n t e r c a l a t i o n system, guest atoms or molecules are inserted r e v e r s i b l y into a host structure, which does not undergo any s i g n i f i c a n t s t r u c t u r a l change during either the i n s e r t i o n or removal of the i n t e r -calant. I n t e r c a l a t i o n has been observed i n a wide v a r i e t y of materials, a l l characterized by open passages i n t h e i r structures through which an int e r c a l a n t can d i f f u s e . Among other systems, hydrogen has been i n t e r -calated into metals, a v a r i e t y of atoms and molecules have been i n t e r -calated into graphite, and various a l k a l i metals, p a r t i c u l a r l y l i t h i u m , have been i n t e r c a l a t e d into a v a r i e t y of t r a n s i t i o n metal dichalcogenides. Although other types of structures, such as r u t i l e s , i n t e r c a l a t e l i t h i u m , t h i s thesis concentrates on the detection of phase t r a n s i t i o n s i n the t r a n s i t i o n metal dichalcogenides. These materials are denoted by where M i s a t r a n s i t i o n metal and X i s sulphur, selenium, or t e l l u r i u m . Many MX2 materials are layer compounds. Layers of metal atoms are sandwiched between layers of chalcogens. Adjacent planes of chalcogens are bound together only by weak van der Waals forces, so the MX2 sandwiches are e a s i l y pushed apart from each other. Intercalant atoms can r e a d i l y be inserted into the van der Waals gaps between MX2 sandwiches, par-t i c u l a r l y such small atoms as lithi u m . Note that i f li t h i u m i s i n t e r -calated to a concentration of x l i t h i u m atoms f or each M atom, the 2 r e s u l t i n g i n t e r c a l a t i o n compound w i l l be denoted L i MX . Figure 1(a) shows the general form of an MX2 layer structure, while f i g u r e 1(b) shows the two possible ways in which an M atom can be coordinated by chalcogen atoms. The MX2 c r y s t a l s have hexagonal symmetry with t h e i r c r y s t a l l o -graphic c-axes perpendicular to the plane of the MX2 sandwiches. Figure 1(c) shows the structure of several common layer compounds. A l l the MX^  layer compounds tend to shear along t h e i r van der Waals gaps, forming p l a t e l e t s when the material i s crushed. Intercalant atoms d i f f u s e through a layer compound by hopping, or t u n n e l l i n g , between s i t e s i n the van der Waals gap. There i s one s i t e octahedrally coordinated by chalcogen atoms f o r each M atom, arid two s i t e s t e t r a h e d r a l l y coordinated by chalcogen atoms f o r each M atom, with the octahedral s i t e s no t i c eati'ly larger than the tetrahedral s i t e s . Neu-tron d i f f r a c t i o n has shown i n the case of L i TiS» (Dahn 1980) that the x 2 octahedral s i t e s are occupied f o r 0 _< x _< 1. I t i s to be expected that they should have a lower chemical p o t e n t i a l than the smaller tetrahedral s i t e s . I t has been found that the d i f f e r e n c e i n chemical p o t e n t i a l between an i n t e r c a l a t e d a l k a l i metal atom i n an i n t e r c a l a t i o n compound, such as an MX2 layer compound, and an atom i n a piece of a l k a l i metal i s often s u f f i c i e n t l y large that many i n t e r c a l a t i o n systems are being considered as high energy density rechargeable b a t t e r i e s . I n t e r c a l a t i o n can be performed by chemical means, such as i n t e r c a l a t i n g l i t h i u m by mixing a host material with n-butyl l i t h i u m , but an electrochemical c e l l can be constructed instead. A high energy rechargeable battery based on the n a t u r a l l y occurring layer compound MoS2 i n t e r c a l a t e d with l i t h i u m (Haering et a l 1980) i s being developed commercially by Moli Energy Ltd. of Burnaby, B.C., Canada. 3 (a) General form van der Waals gap (b) Coordination units for MX2 layer structures AbA trigonal prism AbC octahedron (c) 2H-MoS2 1120 2H-NbS2 V 1120 l T - T i S 2 1120 AbA BaB AbA CbC AbC AbC Figure 1 - The structure of MX^  layer compounds. (a) The general form of an MX2 layer compound sandwich. (b) Two possible ways an M atom can be coordinated by X atoms. (c) The three most common types of MX2 layer compounds. A schematic electrochemical c e l l i s shown i n f i g u r e 2 f o r a l i t h i u m i n t e r c a l a t e d MX2 layer compound. In general f o r such compounds i t i s en e r g e t i c a l l y favorable f o r a l i t h i u m atom to occupy a s i t e i n the com-pound's van der Waals gaps instead of a s i t e i n a sheet of l i t h i u m metal. When an anode of l i t h i u m metal and a cathode of MX^  are immersed i n an e l e c t r o l y t e containing L i + ions, and a path f o r electrons to t r a v e l between the anode and the cathode i s provided, l i t h i u m atoms i n the anode d i s -sociate into L i + ions and electrons. The electrons t r a v e l through the external path and are capable of doing work i n an external load. The electrons combine with L i + ions at the surface of the MX2 to form l i t h i u m atoms, which d i f f u s e into the host structure. Since L i + ions are gen-erated at the anode and disappear at the cathode, charge n e u t r a l i t y i s maintained i n the sol u t i o n by the migration of L i + ions from anode to cathode. The c e l l i s recharged by d r i v i n g a current through the external path so that electrons flow from cathode to anode. This reverses the i n t e r c a l a t i o n process, and L i + ions migrate from cathode to anode, where they combine with electrons and plate onto the l i t h i u m metal. The cathode normally consists of a powder. Large si n g l e c r y s t a l s break up during i n t e r c a l a t i o n , since although no s i g n i f i c a n t s t r u c t u r a l change occurs i n the host } the van der Waals gaps change in s i z e , causing the c-axis to vary by as much as ten percent during i n t e r c a l a t i o n . Intercalant content, and hence the c-axis, varies from point to point, s t r a i n i n g the host s u f f i c i e n t l y to break i t apart. I f single c r y s t a l s must be used i n an experiment i n t e r c a l a t i o n must be allowed to proceed very slowly. The e l e c t r o l y t e i s a polar organic solvent which reacts minimally with l i t h i u m and the other c e l l components, with a l i t h i u m s a l t dissolved i n i t such as LiC10 A , Li B r , or LiAsF^. The solvent must be polar to 5 anion f r o m -Li salt Li metal anode External load o o o N Li + o o o o o Li i d -eation L i x M X 2 cathode 1M Li salt in PC J Figure 2 - Schematic diagram of an i n t e r c a l a t i o n c e l l . 6 d i s s o l v e and solvate an i o n i c l i t h i u m s a l t . A l l electrochemical c e l l s discussed i n t h i s thesis used the solvent 1 ,2-propanediol c y c l i c carbonate, also c a l l e d propylene carbonate or PC, with either LiClO^ or LiAsF^ d i s -solved to a one molar concentration i n the PC. I n t e r c a l a t i o n c e l l s i n general have a complicated dependence of voltage V on i n t e r c a l a n t content x. The f i n e structure of t h i s dependence c . should be v i s i b l e in dx/dV, where V i n dx/dV w i l l be the c e l l voltage, since t h i s i s the inverse of the slope of V"c plotted against x. Let us define a charge Q equal to the charge on the L i + ions that have been i n t e r -calated into the cathode. Q i s d i r e c t l y proportional to x. The quantity dQ/dV, where V again w i l l be the c e l l voltage, may be measured experimen-t a l l y by the standard chemical technique of l i n e a r sweep voltammetry (see Chapter 2) or more c l e a r l y by constant current dQ/dV (see Chapter 3). We s h a l l be p r i m a r i l y concerned with the e f f e c t of f i r s t order phase tra n -s i t i o n s on dQ/dV, using l i t h i u m i n t e r c a l a t e d VS 2 f o r i l l u s t r a t i o n of dQ/dV measurements. dQ/dV i s also of i n t e r e s t i n s i n g l e phase regions (see Section 4.2). The voltage 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 chemical p o t e n t i a l y of the anode and the chemical p o t e n t i a l y of the cathode at cl i t s surface, where i n t e r c a l a t i o n i s occurring. If the anode has a Helmholtz free energy F and the cathode has a Helmholtz free energy F, then i f the 3. number n of i n t e r c a l a n t atoms changes by An the t o t a l Helmholtz f r e e energy changes by: AF = 8F 9 F a ^ 3n 3n An = (y - y ) An (1.1) a The electrons do work t o t a l l i n g eAnV^, where e i s the magnitude of the e l e c t r o n i c charge and V i s the c e l l voltage. The change i n free energy 7 i s the negative of the work done, so ( 1 . 1 ) becomes: V c = l ( y a - u ) ( 1 . 2 ) We s h a l l assume that y of the l i t h i u m metal remains constant, and hence a V"c i s proportional to - u . We s h a l l show how t h i s r e l a t i o n s h i p i s d i s t o r t e d by k i n e t i c e f f e c t s . Note that the Gibbs' free energy G must be used instead of the Helmholtz f r e e energy F i f pressures above a few atmospheres are present. Since G = F + Pv, where P i s the pressure and v i s the volume, G i s close to F for t y p i c a l experimental pressures, but can be very d i f f e r e n t i f high pressures are used. 8 1.2 Phase Transitions and dQ/dV S t r i c t l y speaking, in an i n t e r c a l a t i o n system i n s e r t i o n occurs by d i f f u s i o n of i n t e r c a l a n t into a c r y s t a l without any s i g n i f i c a n t s t r u c t u r a l changes occurring in the host. In general, however, the term " i n t e r -c a l a t i o n system" i s applied to any system that i n t e r c a l a t e s over some range of ranges of composition and temperature. In addition to i n t e r -c a l a t i o n , f i r s t order phase t r a n s i t i o n s w i l l occur i f d i f f e r e n t host c r y s t a l structures are e n e r g e t i c a l l y favored at d i f f e r e n t i n t e r c a l a n t concentrations. I t i s also possible that the free energy may be minimized by a two phase region where both phases have the same host c r y s t a l sym-metry but d i f f e r e n t i n t e r c a l a n t concentrations. In e i t h e r case, a new phase w i l l nucleate when the i n t e r c a l a n t concentration at the surface of the c r y s t a l enters the e n e r g e t i c a l l y unfavorable range between the i n t e r -calant concentrations of the two phases i n the two phase region. A phase front w i l l move from the surface of the c r y s t a l to the c r y s t a l ' s center, converting the e n t i r e c r y s t a l to the new phase. The motion of the phase front i s driven by a concentration gradient between the phase front and the c r y s t a l ' s surface that a r i s e s from the f i n i t e speed of s o l i d state d i f f u s i o n . Four d i f f e r e n t c r y s t a l structures have been observed i n the l i t h i u m i n t e r c a l a t e d layer compound i n f i v e phases (Murphy et a l 1977). Both two phase regions and single phase regions were observed using a v a r i e t y of i n t e r c a l a n t concentrations. L i VS„ w i l l ' b e discussed i n d e t a i l i n x 2 Section 1.4 and in Chapters 2 and 3. The motion of a sharp front through a c r y s t a l has been observed when s i l v e r atoms are i n t e r c a l a t e d into the layer compound NbSe2 (Folinsbee et a l 1981). S i l v e r concentrations were measured with an electron beam probe. The sharp front has been a t t r i b u t e d by the authors to a f i r s t 9 order phase t r a n s i t i o n , although a concentration dependent d i f f u s i o n c o e f f i c i e n t could conceivably cause such a front. Since there i s evidence of f i r s t order phase t r a n s i t i o n s occurring i n i n t e r c a l a t i o n systems, l e t us consider what e f f e c t such a phase tra n -s i t i o n has on the voltage of an i n t e r c a l a t i o n c e l l . I f we ignore k i n e t i c e f f e c t s f o r the moment, the surface concentration of in t e r c a l a n t should remain f i x e d while the phase front moves through the c r y s t a l . As a r e s u l t the chemical p o t e n t i a l of the cathode w i l l remain constant i n the two phase region, and thus the c e l l voltage w i l l also remain constant while the phase front moves. This means that a plateau i n the c e l l voltage measured against x or Q w i l l frequently signal the presence of a f i r s t order phase t r a n s i t i o n . The inverse of the slope of a voltage curve plotted against x or Q w i l l c l e a r l y peak when a f i r s t order phase t r a n -s i t i o n occurs. Thus peaks i n dx/dV or dQ/dV can si g n a l the presence of f i r s t order phase t r a n s i t i o n s . dx/dV or dQ/dV can be measured e l e c t r o -chemically. I t i s the objective of t h i s thesis to understand and i n t e r -pret such dQ/dV measurements and t h e i r peaks. To understand why dx/dV or dQ/dV should be us e f u l i n detecting f i r s t order phase t r a n s i t i o n s , i t i s h e l p f u l to draw an analogy between the thermodynamics of an i n t e r c a l a t i o n c e l l and the thermodynamics of a gas. Rec a l l that the pressure i s s u f f i c i e n t l y low that the Helmholtz free energy and i t s d i f f e r e n t i a l f o r an i n t e r c a l a t i o n system are: where U i s the i n t e r n a l energy, T i s the temperature, S i s the entropy, P i s the pressure, v i s the volume, and n i s the number of int e r c a l a t e d atoms or molecules. For a f i x e d volume F = F(n,T) and (1.4) becomes: F(n,v,T) = U - TS (1.3) dF = -SdT + ydn + Pdv (1.4) dF = -SdT + ydn (1.5) 10 A gas with a f i x e d number of p a r t i c l e s n has a free energy F = F (v,T) with a d i f f e r e n t i a l : dF = -SdT - Pdv (1.6) g An analogy can be drawn between the gas and the i n t e r c a l a t i o n system by noting that the free energies of the two are formally i d e n t i c a l i f the v of the gas i s substituted by the n of the i n t e r c a l a t i o n system and i f the P of the gas i s substituted by -y of the i n t e r c a l a t i o n system, leaving the other thermodynamic quantities unchanged. The thermodynamics of a gas can be transformed by t h i s analogy into the thermodynamics of an i n t e r c a l a t i o n system. The equation of state f o r a gas i s P = P(v,T). By analogy, the equation of state of an i n t e r c a l a t i o n system i s u = y(n,T). Since when k i n e t i c e f f e c t s are ignored the c e l l voltage i s proportional to - u , the c e l l voltage as a function of n and T describes the equation of state of the i n t e r c a l a t i o n system. dQ/dV i s c l e a r l y s e n s i t i v e to structure i n an i n t e r c a l a t i o n system's equation of state. In a gas the isothermal compressibility i s given by: 1 dvT KT v P (1.7) T This i s known to diverge when a f i r s t order t r a n s i t i o n occurs. By analogy and (1.2) the quantity i n (1.7) becomes: n du ex dV (1.8) T At a fi x e d temperature dx/dV or dQ/dV i s thus analogous to the compressibil-i t y of a gas and should s i m i l a r l y diverge at f i r s t order phase t r a n s i t i o n s . See Dahn (1982) f or other applications of t h i s analogy. 11 1.3 K i n e t i c E f f e c t s i n an Experimental System When charge flows i n an i n t e r c a l a t i o n c e l l k i n e t i c e f f e c t s d i s t o r t measured quantities from t h e i r equilibrium thermodynamic values. Once the sources and nature of the k i n e t i c e f f e c t s i n an experimental c e l l are understood, the behavior of dQ/dV i n an experimental s i t u a t i o n can be predicted. We s h a l l assume that the powder cathodes are t h i n , so that each cathode p a r t i c l e i s i n good e l e c t r i c a l contact with the cathode sub-str a t e and i n good contact with the c e l l ' s e l e c t r o l y t e . K i n e t i c e f f e c t s can then be described by s o l i d state d i f f u s i o n e f f e c t s i n the c r y s t a l s and by the e l e c t r i c a l resistance of the c e l l . S o l i d state d i f f u s i o n e f f e c t s a r i s e since the cathode p a r t i c l e s are of f i n i t e s i z e . Intercalant atoms or molecules are introduced into the ' c r y s t a l at the c r y s t a l ' s surface, and a concentration gradient a r i s e s as the inte r c a l a n t d i f f u s e s towards the center of the p a r t i c l e . As mentioned, layer-compounds tend to form p l a t e l e t s when powdered, with i n t e r c a l a t i o n occurring through':the edges of the p l a t e l e t into the van der Waals gaps. Here d i f f u s i o n has two dimensional symmetry. Once nucleated, a phase front can be expected to become c y l i n d r i c a l to minimize the surface tension at the two phase i n t e r f a c e . D i f f u s i o n i n layer com-pounds w i l l therefore be assumed to have c y l i n d r i c a l symmetry when doing t h e o r e t i c a l c a l c u l a t i o n s . A material could instead have a structure of, say, channels into which in t e r c a l a n t d i f f u s e s from a l l c r y s t a l faces. Phase fronts and d i f f u s i o n i n such cases w i l l be assumed to have sphe r i c a l symmetry. In t e r c a l a t i o n can conceivably occur only from one edge of a c r y s t a l . In such a s i t u a t i o n phase fronts and d i f f u s i o n would have a one dimen-sio n a l symmetry. The phase front would be a plane. While t h i s configura-t i o n can occur i n experiments using s i n g l e c r y s t a l s , i t does not a r i s e 12 when a powder cathode i s used, and w i l l be considered i n less d e t a i l than the two and three dimensional cases. The d i f f u s i o n c o e f f i c i e n t f o r inte r c a l a n t i n a host c r y s t a l has proved very d i f f i c u l t to measure experimentally. When there i s a d i s t r i b u t i o n of p a r t i c l e sizes and perhaps, as for layer compounds, i n t e r c a l a t i o n only occurs over part of the surface area of each p a r t i c l e , i t can be very d i f f i c u l t to determine the area through which d i f f u s i o n i s occurring.. It should also be noted that i f a layer compound cathode i s pressed, the p a r t i c l e s tend to a l i g n t h e i r c-axes normal to the cathode's surface, mostly exposing faces through which no i n t e r c a l a t i o n occurs. I n t e r c a l a t i o n then occurs mainly i n the pores of the cathode, and the area through which d i f f u s i o n occurs i s almost impossible to measure. An accurate measurement of the d i f f u s i o n c o e f f i c i e n t requires accurate knowledge of the s i z e and shape of the cathode p a r t i c l e s . Single c r y s t a l measurements can be made, but a sing l e c r y s t a l w i l l break up i f i n t e r c a l a t i o n proceeds too r a p i d l y . I t i s possible (see Chapter 3) to deduce some information electrochemically about the r e l a t i v e values of the d i f f u s i o n c o e f f i c i e n t s i n the various phases of an i n t e r c a l a t i o n system. An i n t e r c a l a t i o n c e l l has an e l e c t r i c a l resistance to which there are several contributions. These are thoroughly discussed by McKinnon (1980). There are ohmic contributions to the resistance from the e l e c t r i c a l contact to the anode and the e l e c t r i c a l contact to the powder cathode. S u f f i c i e n t pressure must be maintained on the c e l l to ensure good e l e c t r i c a l contact with each p a r t i c l e i n the cathode. The p a r t i c l e s themselves may contribute some ohmic resistance. Transport across the interfaces of the electrodes with the e l e c t r o l y t e w i l l contribute to the resistance, since there w i l l be a c t i v a t i o n energies associated with each electrode reaction that must be overcome. There may also be a contribution to the resistance from 13 l i m i t s to the e l e c t r o l y t e ' s a b i l i t y to transport i n t e r c a l a n t ions from the anode to the cathode. Also, e l e c t r o l y t e i n the pores of the cathode can be depleted of int e r c a l a n t ions and contribute to the resistance. The resistance of an .intercalation c e l l mostly a r i s e s from contribu-tions that are u n l i k e l y to vary r a p i d l y with the in t e r c a l a n t concentration in the cathode. If the current i s not changed r a p i d l y or d r a s t i c a l l y , the resistance should remain roughly constant. The resistance w i l l be assumed a constant i n t h e o r e t i c a l c a l c u l a t i o n s . 14 1.4 Preparation of L i VS„ Electrochemical C e l l s x 2 As mentioned in Section 1.2, Murphy et a l (1977) have shown that L i VS„ has several phases. As a r e s u l t , t h i s material was used as a x 2 test of the a b i l i t y of dQ/dV measurements to detect f i r s t order phase t r a n s i t i o n s . I t i s d i f f i c u l t to prepare the layer compound VS 2 d i r e c t l y from i t s constituent elements, since the vanadium - sulphur phase diagram i s quite complicated. Li^VS^ with x-1 can be prepared -.'.relatively e a s i l y . Murphy et a l f i r s t reported the existence, of L i VS„. L i VS. with x-1 was pre-x 2 x 2 pared by heating Li 2C02 and V^O^ i n a carbon c r u c i b l e with an H^S atmosphere at 500°C for 10 hours. The product was powdered and then heated i n an Yi^S atmosphere for 24 hours, a f t e r which i t was slowly cooled. Murphy et a l prepared Li^VS.^ f o r a dozen d i f f e r e n t values of x by chemically removing i n t e r c a l a t e d l i t h i u m . L i V S 2 was placed i n a c e t o n i t r i l e and treated with iodine, obtaining L i VS„ from the reaction: ° x 2 LiVS„ + %(1 - x) I_ -»- L i VS„ + (1 - x) L i l • (1.9) I 2 x 2 The Li^VS,^ obtained by t h i s proceedure was studied by x-ray and neutron d i f f r a c t i o n to determine i t s structure and phase diagram. Results of t h i s study w i l l be presented as needed. I t has been found that L i VS„ with x ^ l can be prepared from a x 2 stoichiometric mixture of the elements at high temperature i n an evacuated quartz tube (Dahn, J.R., and Haering 1981). An x-ray powder d i f f r a c t i o n pattern was made of t h e i r product. This revealed a layer compound with a IT polytype and a hexagonal unit c e l l with l a t t i c e parameters a = 3. 35A> and c = 6.16A*. The c r y s t a l symmetry i s i n agreement, and the l a t t i c e parameters are i n rough agreement, with the x-ray data of Murphy et a l f o r L i _ o r V S „ which gives a = 3.382& and c = 6.1 50&. U. o_> 2 Li*VS. with x-1 was made using the l a t t e r of the two techniques, x 2 15 A s t o i c h i o m e t r i c m i x t u r e o f Ll^S, s u l p h u r , and v a n a d i u m w e i g h i n g 3 . 1 1 grams was p r e p a r e d i n an a r g o n f i l l e d g l o v e b a g , s i n c e I ^ S r e a c t s w i t h a t m o s p h e r i c m o i s t u r e , and p l a c e d i n a q u a r t z t u b e w i t h a v o l u m e o f r o u g h l y 3 20 cm . The t u b e was e v a c u a t e d o f a r g o n w i t h a d i f f u s i o n pump and i t s end s e a l e d w i t h a b l o w t o r c h . The t u b e was p l a c e d i n a L i n d b e r g t h r e e zone f u r n a c e and h e a t e d t o 7 0 0 ° C o v e r a t h r e e h o u r p e r i o d . The t u b e was l e f t a t t h i s t e m p e r a t u r e f o r n i n e h o u r s , and t h e n a l l o w e d t o c o o l i n a i r t o room t e m p e r a t u r e . The t u b e was opened i n an a r g o n f i l l e d g l o v e b a g . The i n n e r s u r f a c e o f t h e q u a r t z t u b e had p a r t i c i p a t e d i n a r e a c t i o n t h a t r e n -d e r e d much o f t h e t u b e opaque . The m a t e r i a l i n t h e t u b e t h a t was f u r t h e s t away f r o m t h e w a l l s c o n s i s t e d o f b l a c k c r y s t a l l i t e s t h a t r e a d i l y c r u s h e d i n t o a powder . T h i s m a t e r i a l was s e p a r a t e d f r o m t h e g r a y e r and h a r d e r m a t e r i a l n e a r t h e t u b e w a l l s and t r a n s f e r r e d i n t o an a r g o n f i l l e d Vacuum A t m o s p h e r e s g l o v e b o x . The g l o v e box c o n t a i n e d p r e - p u r i f i e d g r a d e a r g o n gas c l e a n s e d by a H y d r o x p u r i f i e r t o m a i n t a i n oxygen and w a t e r c o n c e n t r a - ' t i o n s on t h e o r d e r o f 1 ppm, w i t h s l i g h t l y h i g h e r "^n i t rogen c o n c e n t r a t i o n s . The L i x V S 2 m a t e r i a l o b t a i n e d was p l a c e d i n a P h i l i p s X - r a y Powder D i f f r a c t o m e t e r t o o b t a i n an x - r a y powder d i f f r a c t i o n p a t t e r n . The m a t e r i a l i s s e n s i t i v e t o a t m o s p h e r i c m o i s t u r e , so i t was x - r a y e d i n s i d e o f a c a s e n o r m a l l y u s e d t o h o l d a c o m p l e t e e l e c t r o c h e m i c a l c e l l f o r i n s i t u x - r a y d i f f r a c t i o n ( D a h n , P y , and H a e r i n g 1 9 8 2 ) . The c a s e i n c o r p o r a t e s a b e r y l -l i u m window t o a l l o w x - r a y s t o e n t e r t h e c a s e . A l l s t r o n g B r a g g peaks n o t due t o t h e b e r y l l i u m window c o u l d be i n d e x e d t o a IT s t r u c t u r e w i t h h e x a g o n a l u n i t c e l l p a r a m e t e r s a = ( 3 . 3 4 5 + 0 .001) A and c = ( 6 . 1 6 3 + 0 .001 ) A*. The c r y s t a l symmetry i s i n a g r e e m e n t , and t h e l a t t i c e p a r a -m e t e r s i n r o u g h a g r e e m e n t , w i t h t h e x - r a y d a t a o f Murphy e t a l f o r L i ^ -j^2 which - g i v e s a = 3 . 3 4 5 A and c = 6 . 1 5 2 A . I n a d d i t i o n t o t h e peaks i n d e x e d t o t h e IT s t r u c t u r e t h e r e were a few weak and b r o a d peaks t h a t w o u l d a p p e a r 16 to be due to the presence of impurities formed i n the reaction with the quartz tube. The i n t e n s i t i e s of the Bragg peaks were d i s t o r t e d by pre-ferred o r i e n t a t i o n . The p l a t e l e t s making up the powder tended to have t h e i r c-axes perpendicular to the plane of the sample. The i n t e n s i t i e s are not known accurately enough to d i s t i n g u i s h between octahedral and t r i g o n a l prismatic coordination of the vanadium atoms by the sulphur atoms. Electrochemical c e l l s were prepared from the L i VS„ material. A x 2 t y p i c a l c e l l design i s shown in f i g u r e 3. Two s t e e l flanges act as the contacts to the cathode and anode of the c e l l . The flanges are e l e c t r i c a l l y i s o l a t e d from each other by a vi t o n rubber O-ring gasket that also seals the i n t e r i o r of the c e l l from the atmosphere. Assembly i s performed i n an argon f i l l e d glove box. Flange c e l l s with polypropylene flanges and neoprene O-rings were also used. E l e c t r i c a l contact with the anode and cathode i s made by ce n t r a l brass posts electroplated with n i c k e l on t h e i r i n t e r i o r surface. The cathode of a flange c e l l i n general consists of an i n t e r c a l a t i o n compound, i n the form of a powder, mixed with PC or propylene g l y c o l and spread onto a n i c k e l or aluminum substrate. The cathode i s baked to drive off the organic material, leaving the cathode s t i c k i n g to the substrate s u f f i c i e n t l y to allow handling and weighing. Since L i ^ V ^ i s a i r s e n s i t i v e , a mixture of L i ^ V ^ and PC was spread on n i c k e l inside a glove box. The cathode was then sealed i n a s t e e l box, which was removed from the glove box. The s t e e l box was evacuated with a mechanical pump and heated i n a Hotpoint oven to a nominal 120°C u n t i l the pressure dropped to a steady value around 20 m i l l i t o r r . The s t e e l box was taken back into the glove box and the cathode used to construct a c e l l . The anode of a flange c e l l i s a piece of l i t h i u m metal, cut to s i z e and scraped with a s c a l p e l to remove any surface layer of, for example, Figure 3 - A t y p i c a l flange c e l l . 18 l i t h i u m oxide. The separators used were pieces of Celgard #2500 micro-porous polypropylene f i l m wetted with e l e c t r o l y t e under pressure. The PC used i n the e l e c t r o l y t e was vacuum d i s t i l l e d to reduce water and other impurities, p a r t i c u l a r l y propylene g l y c o l , below concentrations on the order of 40 ppm and 10 ppm r e s p e c t i v e l y . The l i t h i u m s a l t used was U. S. Steel Agri-Chemicals LiAsF, i n 1M concentration. Vacuum dried LiClO. has 6 4 also been used. Electrochemical c e l l s made with L l ^ f t ^ charged and discharged repeat-edly at various constant currents. C e l l s were cycled between 1.8 and 2.8 v o l t s . The 18th cycle of the c e l l GJ-9 i s shown i n f i g u r e 4. This was at a low constant current of 70 .yamps, requiring 57 hours for a charge or discharge. Notice the clear plateaus near 2.38 and 2.46 v o l t s . R e call that such plateaus are to be expected i f f i r s t order phase t r a n s i t i o n s occur. The length of the charge or discharge corresponds to a change in x of about 0.7 by weight. I t i s believed that although a portion'of the cathode may not be i n good e l e c t r i c a l contact, t h i s i s p r i m a r i l y due to the presence of impurities i n the cathode material, since s i m i l a r changes i n x were obtained f o r other c e l l s . This i s i n agreement with Dahn, J.R., and Haering (1981) who achieved a change i n x during c y c l i n g of 0.95 with e s s e n t i a l l y i d e n t i c a l voltage behavior to that shown i n f i g u r e 4. The change i n x between 1.8 v o l t s and->2.8 v o l t s has been normalized to 1 with t h i s i n mind. C e l l GJ-9 l o s t about ten per cent of i t s capacity by the 24th cycle and a further ten per cent by the 38th c y c l e . Sinceclow current c y c l i n g was being done to obtain the r e s u l t s presented i n Chapter 3, the f i r s t 40 cycles took place over a period of three and a half months. The f i r s t charge of c e l l GJ-9 took 0.70 + 0.03 of the time taken by the f i r s t discharge, c y c l i n g at constant current. This indicates that the cathode material was i n i t i a l l y L i n 7VS 9, i n agreement with the x-ray 19 Q (coulombs) Figure 4 - Voltage curves for the 18th charge ( s o l i d l i n e ) and discharge (dashed l i n e ) at 70 yamps for L i VS„ c e l l GJ-9. 20 data. S i m i l a r l y , Dahn, J.R., and Haering found that the f i r s t charge took 0.85 of the time taken by the f i r s t discharge for t h e i r material, i n d i c a t i n g that t h e i r cathode material was i n i t i a l l y L i ^ 85^2' '*"n agreement with the x-ray data. L i ^ V ^ cycles r e v e r s i b l y in the range 1 <_ x <^  2 in addition to the range 0 _< x _< 1. The c e l l voltage stays on a plateau at about 1.0 v o l t s on discharge, and a plateau at about 1.3 v o l t s on charge, probably i n -dicating that x changes between 1 and 2 by a r e v e r s i b l e f i r s t order phase t r a n s i t i o n . Further discussions of L i VS„ w i l l be r e s t r i c t e d to the x 2 range 0 <_ x _< 1. 21 CHAPTER TWO LINEAR SWEEP VOLTAMMETRY 2.1 dQ/dV Measured by Linear Sweep Voltammetry The f i r s t measurements of dQ/dV for a L i MX„ electrochemical c e l l x 2 were performed by Thompson (1979). Thompson stepped the voltage of e l e c t r o -chemical c e l l s , which had cathodes of the layer compound T1S2, and measured the charge which flowed u n t i l the c e l l reached equilibrium. This technique requires c a r e f u l achievement of equilibrium a f t e r each step, which can become d i f f i c u l t and time consuming when high r e s o l u t i o n i s desired. The techniques presented i n Chapters 2 and 3 achieve considerable experimental s i m p l i c i t y at the expense of having d i s t o r t i o n s due to k i n e t i c e f f e c t s . Once such d i s t o r t i o n s are understood, dQ/dV can be c l e a r l y interpreted. Linear sweep voltammetry i s a technique i n which the voltage of a c e l l i s swept at a constant rate a = dV/dt, where t i s the time and V in dV/dt w i l l be the c e l l voltage. The current I = dQ/dt which flows as a r e s u l t of the changing voltage i s measured. Note that because of the d e f i n i t i o n of Q the current i s defined as the flow of p o s i t i v e charge from the cathode to the anode. If Q decreases monotonically with voltage, as i s normally the case, I has the opposite sign to a. This technique i s of i n t e r e s t since i f we ignore k i n e t i c e f f e c t s : T = dfi = dV dO, _ dQ , . 1 - dt dt dv " a dv u ' i ; Since a i s constant, the current i s i d e a l l y proportional to dQ/dV. The usefulness of l i n e a r sweep voltammetry has been noted previously (Jacobsen et a l 1979, McKinnon 1980). This chapter p a r a l l e l s and confirms the theory and experimental r e s u l t s presented by Dahn, J.R., and Haering (1981). Linear sweep voltammetry w i l l be compared and contrasted i n Chapter 3 with the new technique of constant current dQ/dV. 23 2.2 Theory of K i n e t i c E f f e c t s on Phase Transitions We s h a l l now model how the k i n e t i c e f f e c t s discussed i n Section 1.3 influence the appearance of a f i r s t order phase t r a n s i t i o n when doing l i n e a r sweep voltammetry. It i s assumed that the cathode i s thi n and consists of uniformly sized p a r t i c l e s , and that a l l p a r t i c l e s are i n good e l e c t r i c a l contact with t h e i r substrate and with the e l e c t r o l y t e . A l l p a r t i c l e s are then i n the same state of i n t e r c a l a t i o n and at the same chemical p o t e n t i a l . It i s further assumed, as mentioned i n Section 1.3, that k i n e t i c e f f e c t s can be modelled by s o l i d state d i f f u s i o n i n the p a r t i c l e s together with a series resistance R that i s assumed constant. The equilibrium c e l l voltage i s assumed to decrease monotonically with Q. If the voltage at the surface of the cathode p a r t i c l e s i s V , the c e l l voltage V i s : V = V - IR (2.2) c On discharge the c e l l voltage i s depressed by the IR l o s s , while on charge the voltage applied to the c e l l i s above the equilibrium voltage by the IR l o s s . I f the c e l l voltage i s swept l i n e a r l y at a rate a, st a r t i n g at a time t ^ at a c e l l voltage ^ (tg) : V (t) = V (t_) + a ( t - t - ) (2.3) c c U 0 We s h a l l assume a f i r s t order phase t r a n s i t i o n occurs at an equilibrium voltage of V Q , and on discharge Q increases from 0 to as the phase front moves through the system. On recharge Q changes from Q Q back to 0. Assuming that no capacity e x i s t s except f o r that associated with the phase t r a n s i t i o n , when there are no k i n e t i c e f f e c t s then no.current flows i f the voltage i s above or below V Q . The current i s a delta function i n voltage, and i n f i n i t e while Q changes between 0 and Q Q . Let us i n i t i a l l y consider the case of very rapid s o l i d state d i f f u s i o n , so that the series resistance R i s the only s i g n i f i c a n t k i n e t i c e f f e c t . 24 D i f f u s i o n w i l l be properly incorporated l a t e r . Consider a recharging c e l l , a i s p o s i t i v e and V (t ) 1 S below V ^ . Since the current flows only when V i s equal to V Q , equals V u n t i l V Q i s reached at t ^ . V then stays at u n t i l Q has dropped from Q Q to 0, while V C continues to increase by (2.3). From (2.2) : I = - ( V - V J / R (2.4) c (J | l | r i s e s l i n e a r l y with slope 1/R. At a voltage V ^ = V ^ and a time t^ the charge Q has dropped to zero. A f t e r t h i s , no more current flows and V equals V again, while V continues to r i s e . The v a r i a t i o n of V with V M c ° ' c c i s shown in f i g u r e 5(a). The charge as a function of time, from (2.3) and (2.4), i s : Q(t) = Q Q + ft , ft a ( t - t ) 2 a ( t - t 0 ) dt = Q Q - — ( 2 . 5 ) '0 0 Using (2.5) and noting at t^ that Q = 0: v i ~ v o = a ( t r t o ) = ( 2 a R Q 0 ) l s ( 2 , 6 ) Also at , the current peaks at: I = -(2aQ n/R) J s (2.7) P u using (2.4) and (2.6). The current i s shown i n f i g u r e 5(b). On discharge r e s u l t s s i m i l a r to f i g u r e 5 are obtained. During the phase t r a n s i t i o n Q r i s e s from 0 to Q„ while V drops from V _ to V _ and V i s constant at V _ . 0 c 0 2 0 The current has the opposite sign to the current iri f i g u r e 5, but the same magnitude, and V g ~ V2 = V l ~ V0" 0 1 1 c n a r 8 e > f r o m (2.3), (2.4), and (2.5): I = -(2a {Q Q - Q(t)} / R)^ (2.8) | l | increases q u a d r a t i c a l l y with the change i n Q. On discharge | l | i s given by (2.8) with {Q Q - Q(t)} replaced by Q(t). | l | p lotted against Q i s a parabolic curve. 25 V(t)t v 0 V, Vc(t) Figure 5 - (a) Relationship between p a r t i c l e surface voltage V and and c e l l voltage V , when V i s swept l i n e a r l y through a f i r s t order phase t r a n s i t i o n on charge. Fast s o l i d state d i f f u s i o n , (b) Current flowing due to the r e l a t i o n s h i p i n (a). Figure 6(a) shows a l i n e a r sweep voltammogram f o r a f i r s t order phase t r a n s i t i o n when there i s a serie s resistance R and very f a s t s o l i d state d i f f u s i o n . This i s constructed from f i g u r e 5(b) and i t s analogue f o r a discharging c e l l . The two peaks form a "notch" at the t r a n s i t i o n voltage V Q . This d i s t i n c t i v e feature w i l l be used experimentally to i d e n t i f y f i r s t order phase t r a n s i t i o n s and t h e i r equilibrium voltages. Figure 6(b) shows the current as a function of Q. The width i n Q of the two phase region can be obtained experimentally from the l i m i t s i n Q of the charge and discharge peaks. Notice that the addition of a serie s resistance has d i s t o r t e d the i d e a l delta function at V Q of current against voltage into a sawtooth, but the t r a n s i t i o n voltage can s t i l l be found. Both I and - V Q are proportional to a 2 . The peaks in f i g u r e 6(a) become narrower and shorter as a decreases. To determine how d i s t i n c t i v e the peaks are when compared to s i n g l e phase capacity, consider charging a si n g l e phase region of width Q Q with an equilibrium voltage l i n e a r i n Q: V = 3 ( Q Q - Q ) + V(0) (2.9) I d e a l l y , there i s a constant dQ/dV = -1/3. Since a i s also constant, i t i s reasonable to expect a solution f o r constant I. If there i s such a sol u t i o n , Q - Q Q i s simply I ( t - t 0 ) . Then from (2.3) and (2.9), (2.2) becomes: (a+BI) ( t - t j - V(0) - V (0) - IR (2.10) 0 c which i s s a t i s f i e d i f V (0) = V(0) + ctR/3 and I = -a/g. In t h i s simple case, c % s i n g l e phase current scales with a instead of a . As a i s decreased, peaks due to f i r s t order phase t r a n s i t i o n s should become increasingly prominent. We can estimate the order of magnitude of a necessary so that peaks due to a f i r s t order phase t r a n s i t i o n are twice the height of sing l e phase capacity. I f we equate I i n (2.7) with 21 = -2a/B and square both sides: 27 Figure 6 - (a) "Notch" formed by charge and discharge voltammograms of a f i r s t order phase t r a n s i t i o n . Fast s o l i d state d i f f u s i o n , (b) Current against charge f or the case shown i n (a). a - - ^ (2.11) If in a s i n g l e phase a one milliamp-hour c e l l charges and discharges over a 0.5 v o l t range, dQ/dV i s ^-7.2 coulombs/volt. Since dQ/dV = -1/3, 3 ^ 0.14 volts/coulomb. If R ^ 250 ohms and Q Q ^ 0.2 coulombs, from (2.11) a ^ 8 uvolts/sec. C l e a r l y a very slow sweep rate i s necessary f o r phase t r a n s i t i o n s to be v i s i b l e . A sweep rate of 8 yvolts/sec over a 0.5 v o l t range corresponds to about a 17 hour charge or discharge. Now we s h a l l r e c a l c u l a t e the peak shapes incorporating the e f f e c t of s o l i d state d i f f u s i o n , as well as that of a s e r i e s resistance. Again we s h a l l consider the c e l l to only have capacity a r i s i n g from the f i r s t order phase t r a n s i t i o n . At the star t of the phase t r a n s i t i o n the c e l l i s i n equilibrium in phase 1 with a uniform i n t e r c a l a n t concentration p ^ . Phase 2 nucleates at the p a r t i c l e s ' surfaces with an i n t e r c a l a n t concentration p £ , and phase fronts move into the p a r t i c l e s . The surface concentration p g r i s e s as l i t h i u m gradients, governed by the d i f f u s i o n equation, form to push the phase f r o n t s . Say the d i f f u s i o n c o e f f i c i e n t i n phase 2 i s a constant D. I t w i l l be assumed that the current i s small enough that steady state solutions to the d i f f u s i o n equation can be used. This means that the phase f r o n t s ' motion i s s u f f i c i e n t l y slow that the concentration gradients f o r a given phase front p o s i t i o n are time independent. The phase front i s assumed to be sharp, and as i t moves the concentration jumps by Ap = p^ - p^ . The i n t e r i o r of each p a r t i c l e i s assumed to remain at a constant p ^ . The phase front i s nucleated at t = 0. Three possible geometric shapes f o r the phase front w i l l be considered distinguished by a l a b e l y. If y = 1, each p a r t i c l e i s an i n f i n i t e slab with thickness 2L, with i n t e r c a l a t i o n occurring in a f l a t plane through each side. If y = 2, each p a r t i c l e i s an i n f i n i t e c y l i n d e r of radius L, with i n t e r c a l a t i o n occurring through the side w a l l . The case y = 2 applies to the Li xMX2 layer compounds. If y = 3, each p a r t i c l e i s a sphere of radius L, with i n t e r c a l a t i o n occurring through the sphere's en t i r e surface. A p o s i t i o n i n a p a r t i c l e w i l l be denoted by r , which i s the distance from the midpoint of the slab, from the axis of the c y l i n d e r , or the center of the sphere. The phase front i s at f . Define u = r/L. If J g i s the surface current number density, the steady state concen-t r a t i o n p r o f i l e gives at r = L (Carslaw and Jaeger 1959): J L P - P 9 = - f p (1 - u), Y = 1, (2.12a) s 2 D J L S l n u, Y = 2, (2.12b) D J L s ^ - 1| , y = 3. (2.12c) D If we assume that the amount of i n t e r c a l a n t required f o r the concentration gradient i s small compared to the amount moving the phase boundary, then to change the concentration by Ap between f and f + df, where df = L du < 0, we f ind: Y - l du J s / o i O N U d t - = - L A ^ ( 2 ' 1 3 ) which governs the motion of the phase fr o n t . This a r i s e s since the number of i n t e r c a l a n t atoms flowing through a p a r t i c l e ' s surface i n a time dt i s J g dt m u l t i p l i e d by the p a r t i c l e ' s surface area. This must equal Ap(-df) m u l t i p l i e d by the surface area of the phase fr o n t . The r a t i o of the two Y - l surface areas i s u We assume in equilibrium the voltage decreases l i n e a r l y with p i n the small range of p close to that occurs i n the d i f f u s i o n gradient, so dp/dV i s a negative constant. I f the i n t e r c a l a n t i s singly ionized i n the e l e c t r o l y t e , and the surface area through which i n t e r c a l a t i o n i s occurring 30 i s A, then: I = eAJ (2.14) s Recall we s t i l l have: V = V - IR (2.2) c where, since the phase front nucleates at t = 0: V = V n + at (2.15) c 0 Notice from (2.14) we see: ft 0 rt rt I dt = eA 0 J g dt ( c e l l discharging) (2.16a) Q = Q 0 + I dt = Q + eA 0 J dt ( c e l l charging) (2.16b) 0 s When discharging, Ap > 0, p g > p^, J g > 0, I > 0, V c < V, and a < 0. When charging, a l l these i n e q u a l i t i e s are reversed. Since dp/dV i s a negative constant, from (2.2), (2.14), and (2.15): p s " P2 " f ( V - V - ~dV ( a t + A e R J s } = at - bJ (2.17) s where a and b are defined by: a = a (2.18) b = - ^  AeR (2.19) and are not to be confused with the cry s t a l l o g r a p h i c axes a and b. The equations (2.12), (2.13), and (2.17) can be solved f o r t , J , and t J dt, -which; by (2.14), (2.15), and (2.16) define V , I, and Q. - 0 S C S p e c i f i c a l l y , (2.12) and (2.17) combine to give J , and hence I, as functions of u and t f o r each y J g can then be substituted into (2.13) and the equation integrated to obtain t as a function of u. Equation (2.13) can also be integrated d i r e c t l y : 31 J dt = ^ (1 - u Y) s y which by (2.16) gives Q. The solutions are, f o r the case y phase f r o n t : t = ApL Da Dat 2 u 2 2bD Ji { (1 - U ) + — r — (1 - u) ) J = T s L 1 - u + bD -1 J dt = ApL (1 - u) For the case y = 2 of a c y l i n d r i c a l phase fr o n t : t = ApL Da Dat J s = L {u In u + -1 2 + L (1 -uV - l n u + bD 0 J dt = ^-ApL (1 - u 2) s 2 For the case y = 3 of a sphe r i c a l phase front U p L t = Da Dat j-2 3 . 1 2 2 bD ,. 3. -M J s = L I - 1 +M u L -1 0 J g dt = yApL (1 - u 3) (2.20) 1 of a planar (2.21a) (2.21b) (2.21c) (2.22a) (2.22b) (2.22c) (2.23a) (2.23b) (2.23c) Voltammograms of I against are shown i n f i g u r e 7 for each value of Y and for three values of the parameter bD/L. Figure 8 shows I as a function of Q when discharging f o r the same nine cases. The parameter bD/L i s the only constant that a f f e c t s the curves' shapes as opposed to sc a l i n g the curves. These figures should be compared to figures 6(a) and 6(b) which show only the e f f e c t of a series resistance. To compare, note from (2.21), 32 Figure 7 - Linear sweep voltammograms computed from (2.21), (2".22) , and (2.23), incorporating s o l i d state d i f f u s i i o n gradients. 33 LAp Ae/y Figure 8 - Current against charge f o r the cases shown in f i g u r e 7, c e l l discharging. (2.22) , and (2.23) that: Q Q = LApAe / Y (2.24) so that from (2.7): I I rv»* r b ^ Ae (DaAp)""5 ^ and from (2.6) : V, - V 1 0 aL 1 h f \ bD y. L < J (2.26) [DaJ Notice that i n a l l cases the expressions (2.21b), (2.22b), and (2.23b) f o r J^ , at small values of t (so that u ^ 1) reduce using (2.14) to: at ( V V 1 = " R = R ~ ( 2 ' 2 7 ) There i s s t i l l , i n a l l cases, an i n i t i a l l i n e a r r i s e i n current-with slope 1/R. The value of R can s t i l l be found from the slope of the i n i t i a l r i s e , and the d i s t i n c t i v e "notch" due to a f i r s t order phase t r a n s i t i o n i s s t i l l present. S o l i d state d i f f u s i o n a l t e r s the way in which the current f a l l s , but the i n i t i a l r i s e of the current i s c o n t r o l l e d by the series resistance. Due to the s c a l i n g r e f l e c t e d in (2.25) and (2.26), the f a c t that a l l of the peaks i n f i g u r e 7 have the same i n i t i a l l i n e a r ramp must be emphasized. Figure 7 makes cl e a r , however, that the f a l l of the current becomes much sharper as bD/L increases. The larger' D i s , and hence the f a s t e r d i f f u s i o n occurs, the more l i k e a sawtooth a voltammogram peak becomes. We can look at the parameter bD/L i n another way. Define the time constant associated with d i f f u s i o n : x D = L 2 / D (2.28) Define also a capacitance associated with the c e l l : C = - LAe (2.29) dv Then from (2.19) we see: ^ = ^ (2. -30) The parameter bD/L can be thought of as the r a t i o of two time constants. One (RC) i s associated with the speed with which the c e l l charges or d i s -charges, while the other ( T^) i s associated with d i f f u s i o n . When i s r e l a t i v e l y small, the peak shapes approach i d e a l , and when i s r e l a t i v e l y large, d i f f u s i o n rounds off the peak shapes. We can estimate bD/L. From (2.19), noting a p a r t i c l e ' s volume i s LA/ Y: > = < 2 - 3 » which gives: bD _ dQ RDY n - 7 v L ~ dV 2 Consider again a one milliamp-hour c e l l charging and discharging over a 0.5 v o l t range i n a s i n g l e phase, so -dQ/dV ^7.2 coulombs/volt. If R ^ -9 2 250 ohms again, y = 2, L ^ 10'iim, and D ^ 10 cm /sec, from (2.32) we f i n d bD/L ^ 3.6. This indicates that the range of bD/L shown in figures 7 and 8 i s l i k e l y to occur experimentally. Since the motion of the phase front i s governed by the concentration gradient between the phase front and the surface, the shape of a peak depends on the d i f f u s i o n constant only of the phase nucleated on the surface. Notice that the rounding of the peaks occurs as the current becomes larger and hence the concentration gradients become more severe. Despite the rounding of the peaks, the peak height i s s t i l l well within an order of magnitude of I , and the peak width in voltage i s s t i l l well within an order of magnitude of - V Q . Thus, the peak height and voltage width h s t i l l scale roughly as a , and the previous estimate of how slow a sweep rate i s necessary for a peak to be twice the height of sin g l e phase capacity i s s t i l l reasonable. The curves i n f i g u r e 8 of I against Q on discharge have been rounded from the parabolas of f i g u r e 6(b) into less d i s t i n c t i v e shapes, however they may s t i l l prove s u f f i c i e n t l y c l e a r to determine the range i n Q of a f i r s t order phase t r a n s i t i o n . In both figures 7 and 8, the cases y = 2 and y = 3 appear very s i m i l a r . The curves i n these two cases are not s u f f i c i e n t l y d i f f e r e n t to expect i t to be possible to experimentally deter-mine the dimensionality of the phase front from a voltammogram. The case Y = 1 i s d i s t i n c t i v e , however peak shapes due to a planar phase front were not observed in the materials discussed i n t h i s t h e s i s . 2.3 Experimental Apparatus and Technique Linear sweep voltammetry requires apparatus capable of providing an accurately swept voltage to a c e l l , and capable of providing simultaneously whatever current i s demanded by the c e l l , with a low output impedance com-pared to the impedance of the c e l l . The sweep rate must be s u f f i c i e n t l y slow that features i n a voltammogram due to f i r s t order phase t r a n s i t i o n s can be resolved. A Princeton Applied Research (PAR) Model 175 Universal Programmer with the 0.1 m i l l i v o l t / s e c Scan Rate Modification r e l i a b l y sweeps out voltages between -11 and 11 v o l t s at-a slowest rate of 0.1 m i l l i v o l t / s e c . The output from a PAR Model 175 was sent to a PAR Model 173 Potentiostat/ Galvanostat with a 10 kilo-ohm input impedance. A r e s i s t o r i n serie s with the input was selected to cut the sweep range from ^22 v o l t s to, t y p i c a l l y , 0.5 v o l t s . The sweep range was p r e c i s e l y adjusted using the voltage sweep l i m i t s of the PAR Model 175. A fi x e d p o t e n t i a l from the PAR Model 173 was summed i n t e r n a l l y to the input to s h i f t the voltage range to the desired l o c a t i o n . A PAR Model 173 may either c o n t r o l the voltage, as i n t h i s case, or control the current that flows. The current supplied may have a magnitude of one ampere or l e s s , and the output impedance i s n e g l i g i b l e . A PAR Model 17 9 D i g i t a l Coulometer was used to measure the charge Q that flowed while charging and discharging a c e l l . The slowest sweep rate possible with the PAR Model 173 and PAR Model 175 covers the i n i t i a l l y ^22 v o l t range at 0.1 m i l l i v o l t / s e c , which takes about 61 hours. The system was normally used at close to t h i s sweep l i m i t to ensure maximum re s o l u t i o n . Flange c e l l s were connected to the c e l l output of the PAR Model 173, and an electrometer attachment was used by the PAR Model 173 to monitor the c e l l voltage. The output of the electrometer, together with the output 38 of the PAR Model 173's current meter, were recorded as a voltammogram. The current meter's output was also fed into the PAR Model 179, and the output of the coulometer was used to record the current against Q. The temperatures of the flange c e l l s were held f i x e d , when possible, i n temperature baths. 39 2.4 Results f o r L i VS„ x 2 Figure 9 shows a voltammogram and f i g u r e 10 shows current against Q measured f o r the c e l l GJ-9 on i t s eighth charge and discharge. The voltage curve of t h i s c e l l has been shown previously in f i g u r e 4. The measurements were made at room temperature at a sweep rate of 2.35 uvolts/sec, which required 59.22 hours to sweep 0.5 v o l t s . On t h i s c y c l e , c e l l GJ-9 had a capacity of 14.6 coulombs, or about x = 0.71 by weight. These r e s u l t s can be compared with those of Dahn, J.R., and Haering (1981), shown in f i g u r e s 11 and 12. Their measurements used the same equipment as described in Section 2.3, and were made at a sweep rate of 2.82 yvolts/sec, which required 49.25 hours to sweep 0.5 v o l t s , at a temperature of (23 ± 0.5) °C. Their c e l l had a capacity of about 1.5 coulombs. In both cases, the voltammograms were done at sweep rates very close to the slowest rate possible with the equipment. Since the measurements shown are for two c e l l s which d i f f e r by a f a c t o r of more than nine i n capacity, notice from equations (2.6) and (2.7) that peak height and width both vary l i k e Q Q as well as a . One expects that peaks due to f i r s t order phase t r a n s i t i o n s i n c e l l GJ-9 should have roughly t r i p l e : the height of peaks found for the smaller capacity c e l l . However, i n s i n g l e phase regions I v a r i e s with Q Q , so s i n g l e phase capacity in c e l l GJ-9 should have roughly nine times the current flowing as i s the case f o r the smaller capacity c e l l . This scaling appears roughly correct both f o r the peaks and what appears to be si n g l e phase capacity between about 2.2 and 2.3 v o l t s . The peak widths for c e l l GJ-9 should also be three times as large, however t h i s i s not the case. Two very c l e a r "notches" apparently due to f i r s t order phase t r a n -s i t i o n s occur at (2.380 ± 0.005) v o l t s and (2.460 ± 0.005) v o l t s . The i ~i i i i i — i — r 2.1 2.2 2.3 2.4 2.5 2.6 Cell Voltage (volts) Figure 9 - Voltammogram for the 8th charge ( s o l i d l i n e ) and discharg (dashed li n e ) of the L i VS,, c e l l GJ-9, at a sweep rate of 2.35 uvolts/sec. Q (coulombs) 14 12 10 8 6 4 2 0-2 0 0 II I I I I I I I I I I I I I I I 1.0 0.8 0.6 0.4 0.2 0.0 X in LI X V S 2 Figure 10 - Current against charge for the 8th charge ( s o l i d l i n e ) and discharge (dashed l i n e ) of the L i c e l l GJ-9, at a sweep rate of 2.35 yvolts/sec. 2.1 2.2 2.3 2.4 2.5 2.6 VOLTS Figure 11 - Voltammogram for a L i VS^ c e l l , showing a charge ( s o l i d l i n e ) and a d i scharge (dasheS l i n e ) at a sweep r a t e of 2.82 >uvolts/sec. A f t e r Dahn, J.R., and Haering (1981). Figure 12 - Current against charge f or a L i c e l l , showing a charge ( s o l i d l i n e ) and discharge (dasned l i n e ) at a sweep rate of 2182 yvolts/sec. After Dahn, J.R., and Haering (1981). 44 charge and discharge peaks forming the "notches" are large and c l e a r , although c l a r i t y i s somewhat l i m i t e d by the overlap of t r a n s i t i o n s with d i f f u s i o n t a i l s and with sin g l e phase capacity. The peaks f o r these two possible t r a n s i t i o n s a l l show an i n i t i a l l i n e a r ramp, which should have a slope of 1/R. This gives R = (140 ± 20)' ohms for c e l l GJ-9 and R = (650 ± 50) ohms f o r the c e l l of Dahn, J.R., and Haering. This diffe r e n c e i s roughly accounted f o r by c e l l GJ-9 having about four times the cathode area. This diffe r e n c e also accounts f o r the unexpectedly narrow peaks fo r c e l l GJ-9, since b f o r c e l l GJ-9 i s then one quarter the s i z e of b for the other c e l l . A l l of the peaks show noticable rounding due to d i f f u s i o n , and are sim i l a r to those peaks expected f o r y = 2 or y = 3 with bD/L less than one. These two "notches" occur at voltages correspond-ing to the cl e a r plateaus mentioned i n Section 1.4 and shown in f i g u r e 4. Another "notch", although le s s c l e a r , appears superimposed on low voltage si n g l e phase capacity at (2.190 ± 0.005) v o l t s . There i s also what appears to be a "notch" at (2.235 ± 0.010) v o l t s . This l a s t "notch" has the most poorly defined of the peaks, but they s t i l l appear s u f f i c i e n t l y c l e a r to be considered a "notch". They have been taken as such i n the past (Dahn, J.R., and Haering) and w i l l be treated as such f o r the moment, however the existence of a f i r s t order phase t r a n s i t i o n at t h i s voltage s h a l l be reconsidered i n Section 3.4. The graphs of I against Q i n f i g u r e 10 and fi g u r e 12 show the range i n x, normalized to the c e l l capacity, f o r the four suspected phase tra n -s i t i o n s . The peaks i n these graphs r i s e quickly to t h e i r maximum values before f a l l i n g off almost l i n e a r l y . The peaks, l i k e the peaks i n the voltammograms, appear s i m i l a r to those expected f o r y = 2 or y = 3 with bD/L les s than one. Consider f i r s t the p a i r of peaks at low x, ending at x = 0. On charge, the current has a minimum at x = 0.34 ± 0.01. The minimum on discharge i s displaced to lower x, since a second peak begins before the d i f f u s i o n t a i l of the f i r s t i s completed. In the case of GJ-9, on discharge the current i s quite high at the s t a r t of the second peak, and i t i s d i f f i c u l t to extrapolate the value of x at which the f i r s t peak ends, although on charge the l i m i t i n g value of x i s quite c l e a r . In a s i m i l a r manner the p a i r of peaks that next appear with increasing x occur between x = 0.34 ± 0.01 and x = 0.52 ± 0.01. Also, the p a i r of peaks appearing at high x c l e a r l y occur in the range of x from x = 0.90 ± 0.02 to x = 1.00 ± 0.01. Since there i s l i t t l e capacity on recharge below 2.19 v o l t s or on discharge above 2.46 v o l t s , the s i n g l e phase regions at x = 0 and x = 1 that must exist i f the peaks are indeed due to f i r s t order phase tr a n -s i t i o n s are very narrow, with a width below 0.02 i n x. Any si n g l e phase region at x = 0.34 ± 0.01 must be s i m i l a r l y narrow. The peaks near (2.325 ± 0.010) v o l t s do not have very c l e a r l y defined l i m i t s i n Q, so larger error estimates are necessary. The peaks are i n the range of x from x = 0.57 ± 0.05 to x = 0.63 ± 0 . 0 3 . A si n g l e phase region appears to extend from x = 0.63 ± 0.03 to x = 0.90 ± 0.02. Dahn, J.R., and Haering have done voltammograms at temperatures from 23 C to 45 C. The only change i n the voltammogram with increasing tem-perature was that the peaks between x = 0.90 ± 0.02 and x = 1.00 ± 0.01 narrowed the range of x they covered, a range always l i m i t e d by x = 1.00 ± 0.01, u n t i l f i n a l l y disappearing by 40°C. This information, together with the analysis j u s t presented, leads to the phase diagram shown in f i g u r e 13, and was o r i g i n a l l y presented by Dahn, J.R., and Haering. The large si n g l e phase region i s l a b e l l e d IT i n accordance with the x-ray powder d i f f r a c t i o n r e s u l t s mentioned in Section 1.4. R e c a l l from that section that Murphy et a l (1977) found that several phases exist 46 43 Figure 13 - Phase diagram f o r L i derived from l i n e a r sweep voltammetry r e s u l t s and the x-ray data of Murphy et a l (1977). After Dahn, J.R., and Haering (1981). 47 when they analysed x-ray powder d i f f r a c t i o n patterns at a dozen values of x. The p a r t i a l phase diagram derived from those x-ray patterns i s shown in f i g u r e 14. Noting that the boundaries i n x of the phases were not well defined by the lim i t e d number of x-ray patterns, the p a r t i a l phase diagram i s i n agreement with f i g u r e 13, i f the phases and two phase regions i n fig u r e 13 are l a b e l l e d as shown, using Murphy et a l ' s notation f o r the phases. There are two a d d i t i o n a l two phase regions added i n fi g u r e 13 over and above those present i n the p a r t i a l phase diagram. The 3 + IT region corresponds to the poorly defined peaks near (2.325 ± 0.010) v o l t s , and w i l l be discussed further i n Section 3.4. Murphy et a l found that the phase l a b e l l e d 3S has a hexagonal unit c e l l with c r y s t a l l o g r a p h i c axes a_ = 3.380/2 and c„ = 6. 138ft, and that weak x-ray l i n e s are present consistent with a 3c s u p e r l a t t i c e . The IT phase was discussed i n Section 1.4. The 3 phase appears to be a d i s -t o r t i o n of the IT phase. I t was indexed f o r L i ^ ^ VS 2 as monoclinic with a = 5.756A*, b = 3.280&, c = 6.164/2, and 3 = 91.28°. Notice that a ^  3 ^ , b ^ a^, and c ^  c^. The a phase also appears to be a d i s t o r t i o n of IT, indexed as monoclinic with a = 5.65 9/2, b = 3.240/2, C = 6.050/2, and 3 = 91.0°, with the same approximate r e l a t i o n s h i p to the IT hexagonal unit c e l l as exi s t s f o r the 3 phase. The VS^ phase i s once again IT with a hexagonal unit c e l l f o r which a = 3.21 7& and c = 5.745°.. Figure 14 - P a r t i a l phase diagram f o r L i derived from x-ray powder d i f f r a c t i o n r e s u l t s . A f t e r Murphy et a l (1977). 49 2.5 Effectiveness of the Technique The a b i l i t y both to resolve and unambiguously i d e n t i f y peaks due to f i r s t order phase t r a n s i t i o n s i s l i m i t e d both by the a v a i l a b l e apparatus and the nature of the technique i t s e l f . The apparatus used sweeps through a 0.5 v o l t range i n at most 61 hours. Even though the peak r e s o l u t i o n i n the previous section, l e f t something to be desired, i t was not possible to slow the sweep rate further. I t i s possible to improve the re s o l u t i o n somewhat by using a l i g h t cathode. Unless the sweep rate i s s u f f i c i e n t l y slow, sin g l e phase features may mask, or be mistaken f o r , peaks due to f i r s t order phase t r a n s i t i o n s . I t should be noted that i n many chemical applications l i n e a r sweep voltammetry i s often performed in minutes, f a r too f a s t to detect phase t r a n s i t i o n s . In l i n e a r sweep voltammetry the peak shapes are not very d i s t i n c t i v e unless they are very large compared to surrounding si n g l e phase regions. Since d i f f u s i o n gradients become larger as the current becomes lar g e r , the peaks are rounded by d i f f u s i o n e f f e c t s . The peaks also have an i n i t i a l l i n e a r r i s e due to a f i n i t e c e l l resistance. As a r e s u l t , the peaks' shapes may not be s u f f i c i e n t l y d i s t i n c t i v e to unambiguously s i g n a l two phase :regions. I t i s thus pos s i b l e that a bump in sing l e phase capacity can be mistaken f o r a two phase region, p a r t i c u l a r l y since i t may not be possible to use a slow enough sweep rate to make peaks due to phase t r a n s i t i o n s r e l a t i v e l y large. If the sweep rate i s s u f f i c i e n t l y slow f o r peaks due to f i r s t order phase t r a n s i t i o n s to be c l e a r l y resolved, then, as shown ,in the previous section, i t i s possible to q u a n t i t a t i v e l y i d e n t i f y t r a n s i t i o n voltages, t y p i c a l l y to within 0.005 v o l t s , and to determine the range i n x of a two phase region, t y p i c a l l y to within 0.01. In the case of the L i VS„ system, the phase diagram was obtained q u a n t i t a t i v e l y using l i n e a r sweep voltam-metry and r e s u l t s from x-ray powder d i f f r a c t i o n , although the 3 + IT two phase region w i l l be discussed further in Section 3 .4 . Better r e s o l u t i o n of the peaks in the voltammograms would s t i l l be desirable, however, to ensure that the form of the phase diagram could have been found unambig-uously even i f no x-ray data had been a v a i l a b l e . Peak positions and widths are not as'easy to p r e c i s e l y determine i n some cases as they are in others, due to the l i m i t s on r e s o l u t i o n . The c e l l resistance can be e a s i l y found from the i n i t i a l ramps of well resolved peaks. S t i l l , l i n e a r sweep voltammetry does provide an electrochemical technique with which f i r s t order phase t r a n s i t i o n s can be found, or at l e a s t t h e i r p o s s i b i l i t y indicated. C l e a r l y resolved peaks appear to correspond, as expected, with plateaus i n the curve of voltage against x. 51 CHAPTER THREE CONSTANT CURRENT dQ/dV 3.1 dQ/dV Measured at Constant Current We have seen that measurements of dQ/dV can be used to q u a n t i t a t i v e l y determine a phase diagram f o r an i n t e r c a l a t i o n system. Linear sweep v o l -tammetry, however, does not have s u f f i c i e n t l y good r e s o l u t i o n of the peaks caused by f i r s t order phase t r a n s i t i o n s to be consistently unam-biguous. The apparatus described i n Section 2.3 must be used near the l i m i t s of i t s c a p a b i l i t i e s to get any meaningful r e s u l t s . Also, since the current i s varying with time, the peak shapes are rounded by i n -creasing d i f f u s i o n gradients. As w e l l , the IR s h i f t i n the voltage i s cont i n u a l l y changing as the current changes. C l e a r l y the shapes of the peaks could be improved i f the current was held constant. D i f f u s i o n gradients would not worsen during a phase t r a n s i t i o n , arid, assuming that the c e l l resistance was constant, the IR s h i f t i n the voltage would remain constant. If d i f f u s i o n gradients could be minimized, dQ/dV could be measured with the c e l l c l o s e r to equilibrium. Notice that i f we ignore k i n e t i c e f f e c t s f o r the moment: = -da dt = dt o n dV ~ dt dV dV K ' J where V i n dt/dV w i l l be the c e l l voltage. I f the current i s held constant, dt/dV i s i d e a l l y proportional to dQ/dV. dt/dV can be measured e a s i l y with either a voltmeter or an analogue to d i g i t a l converter attached to a micro-computer. This chapter w i l l develop the theory and technique of constant current dQ/dV measurements, and then compare the r e s u l t s of such measure-ments f o r the L i ^ V ^ system with those described i n Section 2.4 made using l i n e a r sweep voltammetry. 3.2 Theory of K i n e t i c E f f e c t s on Phase Transitions We s h a l l now model, i n a s i m i l a r manner to Section 2.2, how the k i n e t i c e f f e c t s discussed i n Section 1.3 influence the appearance of a f i r s t order phase t r a n s i t i o n when doing constant current dQ/dV measure-ments. The same physical assumptions made i n Section 2.2 regarding the behavior of an i n t e r c a l a t i o n c e l l w i l l be made here. The c e l l ' s cathode i s t h i n and has uniformly sized p a r t i c l e s . A l l p a r t i c l e s are at the same chemical p o t e n t i a l and in the same state of i n t e r c a l a t i o n . K i n e t i c e f f e c t s can be modelled by s o l i d state d i f f u s i o n and a serie s resistance R. The only c e l l capacity i s that associated with a f i r s t order phase t r a n s i t i o n at V Q which changes Q between 0 and Q Q . The c e l l i s i n i t i a l l y i n equilibrium i n phase 1 at p^. Phase 2 nucleates on the surfaces at p^, and phase fronts move into the p a r t i c l e s . The d i f f u s i o n c o e f f i c i e n t i s again a constant D i n phase 2. The concentration changes at a phase front by Ap =~&2 ~ p l ' a n ( ^ t ' i e P n a s e fronts nucleate at t = 0. The surface concentration i s p g . The same three cases, l a b e l l e d by y, f o r the symmetry of the p a r t i c l e s and t h e i r phase fronts as discussed i n Section 2.2 w i l l be considered here. In the case y = 1 each p a r t i c l e i s an i n f i n i t e slab of thickness 2L, while i n the case y = 2 each p a r t i c l e i s an i n f i n i t e c y l i n d e r of radius L, and in the case y = 3 each p a r t i c l e i s a sphere of radius L. Also, once again i t i s assumed that i n e q u i l i b -rium dp/dV i s a negative constant f o r the values of p, a l l close to p^, which occur i n the d i f f u s i o n gradient. Recall that i f there are no d i f f u s i o n e f f e c t s , then dQ/dV i s a delta function i n voltage at V Q and i s i n f i n i t e while Q changes between 0 and Q Q . We s t i l l have: V = V - IR (2.2) c Since the current i s constant, and R i s assumed constant, V i s merely V 54 s h i f t e d by a constant to lower voltages on discharge, and to higher voltages on charge. The seri e s resistance does not a f f e c t the appearance of dQ/dV, whether during a phase t r a n s i t i o n or i n a sin g l e phase, except to s h i f t dQ/dV plotted against voltage by a constant. dQ/dV i s s t i l l a d e l t a func-t i o n against voltage and i s s t i l l i n f i n i t e against Q between 0 and Q Q . Constant current dQ/dV separates the two k i n e t i c e f f e c t s we are considering. Only s o l i d state d i f f u s i o n can a f f e c t peak shapes. With only a series resistance considered, a dQ/dV measurement against voltage consists of two delta functions, one at V Q - J11R and the other at V Q + jIj R . The t r a n -s i t i o n voltage V Q i s centered between the two delta functions. Now consider how the delta functions and dQ/dV against Q are affected by s o l i d state d i f f u s i o n . Define again f as the p o s i t i o n of the phase front and u = f/L. Once again, assume that the amount of int e r c a l a n t . required f o r the concentration gradient i s small compared to the amount moving the phase boundary. This leads again to: u Y - l d H . _ i (2.13) dt LAp Taking the i n t e r c a l a n t to again be si n g l y ionized i n the e l e c t r o l y t e , and taking the surface area through which i n t e r c a l a t i o n occurs to be A, we have: I = eAJ (2.14) s Using (2.14), (2.13) becomes: dt AeLAp dt Integrating (3.2): Q = A e ^ A p (1 - u Y) ( c e l l discharging) (3.3a) Q = - — e ^ P UY ( c e l l charging) (3.3b) Y Notice that as i n Section (2.2) we must have from (3.3): Q 0 = AeL|Ap| / y ( 3- 4) For s i m p l i c i t y write (3.3) as: Q = Q Q (1 - u Y) ( c e l l discharging) (3.5a) Q = Q Q u Y ( c e l l charging) (3.5b) Since I i s constant, we can immediately write: Q = I t ( c e l l discharging) (3.6a) Q = Q Q + I t ( c e l l charging)' (3.6b) and hence from (3.5): t = Q Q (1 - u Y) / | l | (3.7) u = (1 - | l | t / Q Q ) 1 / Y (3.8) Now define f o r convenience a constant i n terms of two constants we have used before, namely the d i f f u s i o n time constant x ^ and the capacitance C associated with the c e l l : T D = L 2 / D (2.28) C = - J LAe (2.29) dv Define a change i n voltage associated with the time x ^ : I T - J L F , ^-1 V = ' D s dp dV (3.9) D C D Once again, make the steady state approximation, which assumes the motion of the phase front i s s u f f i c i e n t l y slow that the concentration gradients f o r a given phase front p o s i t i o n are time independent. This allows the use of (2.12) again f o r the surface concentration p g . Using (3.9), (2.12) may be written as: 56 p s " P2 = "Iv VD ( 1 " U ) ' i p v dV D To f i n d dQ/dV, notice: - 1 , Y = 1, 1 = 2 , Y = 3. dp dt dQ_ = dQ_ dp dt dV dt dV dpj = dV dpj P P s ' ' s dQ/dV i s a constant divided by the time d e r i v a t i v e of (3.10)', P=P, t i a t i n g (3.10) and using (2.13) , 2 dp dt P=P. s -2( Y-1) T5A7u (3.10a) (3.10b) (3.10c) (3.11) D i f f e r e n -(3.12) Using (2.14), (3.4), (3.9), and (3.12), (3.11) becomes: Y'Q-r dV V. 0 2( Y-1) — u D 1 which by (3.8) becomes: dc: = Y Q o dV (1 - | l | t / Q 0 )2 ( Y - 1 ) / Y D 1 (3.13) (3.14) Note the exponent i s 0 for y = 1, 1 f o r y = 2, and 4/3 f o r y = 3, Now we can write down V . Since: c p s " p2 " Iv" ( V - V (3.15) equation (2.2)' becomes: V + ^ 0 dV \~1 ( P s - P 2 ) - IR (3.16) From (3.10) t h i s becomes: 57 V Y = 1, (3.17a) = V Q + V D l n u - IR, Y (3.17b) Y = 3. (3.17c) Using (3.8) we f i n d : (3.18a) (3.18b) -1/3 - 1} - IR, y = 3. (3.18c) Thus, f o r constant current dQ/dV we have dQ/dV given by (3.14), V"c given by (3.18), and Q given by (3.6). Figure 15 shows dQ/dV against and f i g u r e 16 shows dQ/dV against Q fo r a f i r s t order phase t r a n s i t i o n . dQ/dV against V"c i s no longer a delta function at V Q ± 11 ]R, but instead jumps to — / Q Q / 1 V D I A T V 0 ± K d i f f u s i o n t a i l , roughly of width V^, points away from V^. As in l i n e a r sweep voltammetry, the case y = 1 has a unique appearance, but the cases y = 2 and y = 3 are u n l i k e l y to be experimentally d i s t i n g u i s h a b l e . For the case y = 2, which corresponds to the i n t e r c a l a t i o n of an KX^ layer compound, the t a i l f a l l s to one tenth of the maximum peak height when the voltage has changed by V . The t a i l f o r both y = 2 arid-y = 3 does not f a l l to zero properly, diverging to i n f i n i t e voltage as the phase front reaches the center of the p a r t i c l e . This i s due to the steady state approximation. Time dependent terms that do not appear i n (3.10) prevent divergence without otherwise s i g n i f i c a n t l y a l t e r i n g the peak shape. Notice that the- peak shapes are no longer dependent on a parameter i n the way l i n e a r sweep voltammetry peak shapes are dependent on bD/L. Con-stants only scale the constant current dQ/dV peak shapes. The constant v a r i e s d i r e c t l y with I and inversely with D dp/dV. I f the current i s halved, the peak height doubles while the peak width i s cut i n two. As Figure 15 - Constant current dQ/dV against V for a f i r s t order phase t r a n s i t i o n at V n. 59 0.0 0.2 0.4 0.6 0.8 1.0 Q / Q 0 Figure 16 - Constant current dQ/dV against Q for a f i r s t order phase t r a n s i t i o n , c e l l discharging. 60 the current i s decreased, peak heights increase r a p i d l y . One would expect sing l e phase capacity, i n general, to have a constant dQ/dV, only s l i g h t l y modified by d i f f u s i o n e f f e c t s and experiencing a constant IR s h i f t . To get an idea how the features observed with constant current dQ/dV compare with those observed in l i n e a r sweep voltammetry, consider a s p e c i a l case. R e c a l l that in Section 2 .2 i t was found that a l i n e a r sweep voltam-metry peak would have twice the height of s i n g l e phase capacity i f the voltage of a 1 milliamp-hour c e l l was swept over a 0 .5 v o l t range at ^8 yvolts/sec, taking about 17 hours. At a constant current of ^58 yamps such a c e l l would also charge or discharge i n about 17 hours. Since a p a r t i c l e ' s v:61ume. i s AL/y, we can rewrite ( 3 . 9 ) using ( 2 . 1 4 ) : -1 v - I L D yD dV ( 3 . 1 9 ) where i n t h i s case dQ/dV i s the i d e a l value in the d i f f u s i o n gradient, and -9 2 i s proportional to dp/dV. Once again, take L 'v 10 ym and D ^ 10 cm /sec, and consider y = 2 . Again, i n a s i n g l e phase f o r t h i s case we would have dQ/dV ^ - 7 . 2 coulombs/volt. We s h a l l use t h i s value both f o r single phase capacity i n general, and f o r the i d e a l value i n the d i f f u s i o n gradient. Then ( 3 . 1 9 ) gives ^ -4 m i l l i v o l t s , negative since the current i s p o s i t i v e and so the c e l l i s discharging. If we again take QQ ^ 0.2 coulombs, the peak height i s -YQo/| v nl % -100 coulomb/volt. The peak height i s almost 14 times the height of s i n g l e phase capacity. The peak i s only 4 m i l l i - ' v o l t s wide. The peak i n constant current dQ/dV i s not only very narrow, but has seven times the height of a comparable peak i n l i n e a r sweep voltam-metry. F i r s t order phase t r a n s i t i o n s now r e s u l t i n sharp peaks that should be f a r easier to resolve i n a given material than the peaks from l i n e a r sweep voltammetry. Figure- 16 again has a unique shape for y = 1. For y = 2 , the peak i n dQ/dV against Q i s a perfect sawtooth. For y = 3, the peak has a s l i g h t l y concave t a i l , but i s quite s i m i l a r to the y = 2 case. I t would again be experimentally d i f f i c u l t to d i s t i n g u i s h between the two cases, since the peak t a i l could conceivably be curved i n the y = 2 case by sing l e phase capacity^ and inte r c a l a n t gradients between the phase fronts and the par-t i c l e s ' centers. The peak shapes, l i k e those of the peaks plotted against V , are very d i s t i n c t i v e when compared with s i n g l e phase capacity, par-t i c u l a r l y since the peaks have sudden i n i t i a l jumps to t h e i r maxima. Charging and discharging over a f i r s t order phase t r a n s i t i o n at constant current r e s u l t s i n a pair of sharp peaks against V , with t h e i r i n i t i a l jumps separated by 2IR and equidistant from V Q . The peaks have t a i l s with a width of roughly V~D pointing away from V Q . F i r s t order phase t r a n s i t i o n s should be c l e a r l y d i s t i n g u i s h a b l e , and V Q should be easy to determine q u a n t i t a t i v e l y . The peaks against Q are for y = 2 a p a i r of s a w t o o t h s o r f o r y = 3 close to sawtooths i n shape, overlapping between 0 and Q Q. This should allow a clear determination of an i n t e r c a l a t i o n system's phase diagram. 3.3 Experimental Apparatus and Technique To measure dQ/dV at constant current, r e c a l l from (3.1) that dQ/dV i s j u s t dt/dV m u l t i p l i e d by the known constant I. I f a c e l l i s charged or discharged by a constant current source, then a voltmeter or analogue to d i g i t a l converter can be used with a microcomputer to monitor the c e l l ' s voltage and compute dt/dV. Both .the apparatus and the micro-computer program required are straightforward. One member of our research group (Mulhern, 1982) was able to assemble the necessary apparatus and program the microcomputer i n a day. A Tektronix 4052 microcomputer was attached through a GPIB bus to a Hewlett-Packard HP3455A D i g i t a l Voltmeter and a Hewlett-Packard HP59309A D i g i t a l Clock. A c e l l was charged and discharged at constant current with i t s voltage monitored by the highly s e n s i t i v e voltmeter. When the voltage changed by at l e a s t one m i l l i v o l t , dt/dV was recorded as the time taken f o r the voltage change to occur, divided by the voltage change. Also recorded were the average time and average voltage during the measurement. This gave dQ/dV as a function of both c e l l voltage and, since Q i s l i n e a r i n I t , charge. If such a system i s used, the data may be displayed using whatever graphics or p r i n t i n g system one i s w i l l i n g to i n s t a l l . An example of measurements made with t h i s system w i l l be given i n Section 4.1. A more sophisticated system was developed to be dedicated to the measurement of constant current dQ/dV f o r four c e l l s simultaneously. This was constructed by the UBC Physics Department E l e c t r o n i c s Shop and was programmed by M. A. Potts. The voltage i s measured by a c i r c u i t based on the I n t e r s i l ICL8068/ICL7104 16 b i t analogue to d i g i t a l converter, which has a conversion speed of approximately one second. Four analogue voltage inputs are multiplexed to t h i s converter.'- The converter i s attached in r o t a t i o n to whichever input channels are in use. The multiplexing and conversion i s c o n t r o l l e d by a Cromemco Single Card Computer with i t s program residing i n EPROMs (Erasable Programmable Read Only Memories). The d i g i t a l output of the converter i s used by the computer together with an i n t e r n a l clock to compute dt/dV f o r each a c t i v e input channel. The information i s sent by the computer to eight Teledyne-Philbrick Model 4025 12 b i t d i g i t a l to analogue converters. The converter outputs are displayed using X-Y recorders to record dt/dV against V , and time base recorders to record dt/dV against time, and hence against Q. The input voltages are assumed to f a l l between 0 and 3 v o l t s . This gives the 16 b i t converter a r e s o l u t i o n of ^ 46 uv o l t s . The computer only outputs dt/dV i f the voltage has changed by at least 0.5 m i l l i v o l t s . The 12 b i t dt/dV output converter outputs between 5 v o l t s and -5 v o l t s . The computer scales dt/dV so that t h i s output range corresponds to a r e s u l t 8 8 between 10 sec/volt and -10 sec/volt. A m u l t i p l i e r may be set to any power of two from 1 to 256. The computer scales up i t s dt/dV output by g t h i s amount, and the ±5 v o l t output range then corresponds to ±10 sec/volt divided- by the m u l t i p l i e r . This p a r t i c u l a r choice of ranges has success-f u l l y handled v i r t u a l l y every experimental s i t u a t i o n encountered thus f a r . The c e l l voltage output i s between 0 and 3 v o l t s , so those 12 b i t con-verters have a re s o l u t i o n l i m i t of ^ 0.7 m i l l i v o l t s . The minimum required change i n voltage of 0.5 m i l l i v o l t s ensures that noise i n dt/dV due to the r e s o l u t i o n l i m i t of the input converter i s mini-mized, and i t ensures that the computer cannot t r y to divide by zero. To reduce noise, some averaging of the "input voltage i s done. Since the input converterrtakes about one second to make a measurement, each c e l l voltage i s measured approximately every four seconds i f a l l four input channels are i n use. A group of such voltage samples f or a p a r t i c u l a r c e l l i s c o l l e c t e d and averaged to obtain a voltage with l e s s noise, to be used in c a l c u l a t i n g dt/dV. The number of samples may be any power of two between 1 and 132. T y p i c a l l y , around 16 samples produces a very clear r e s u l t . The time taken for an average i s c l e a r l y small compared to the t o t a l time i n which an experiment i s performed. Noise i s also reduced by holding a c e l l at a constant temperature i n a Haake F3 temperature bath. Experiments were performed using the flange c e l l s that were described i n Section 1.4. 65 3.4 Results f o r L i VS„ x 2 A seri e s of dQ/dV measurements were made on the c e l l GJ-9 using the equipment described i n the l a t t e r part of Section 3.3. The voltage curve fo r t h i s L i ^ V ^ c e l l was previously shown i n f i g u r e 4, a l i n e a r sweep voltammogram was shown i n f i g u r e 9, and a l i n e a r sweep I against Q curve was shown in f i g u r e 10. The t r a n s i t i o n voltages and range i n x of the two phase regions were discussed i n Section 2.4 based on these f i g u r e s . Constant current dQ/dV measurements were made on c e l l GJ-9 at 35 yamps, 52.5 vamps, 70 yamps, 87.5 yamps, 105 vamps, 122.5 yamps, and 140 yamps. The c e l l ' s temperature was held constant at (20.0 ± 0.5) °C. The c e l l ' s capacity declined during the measurements by about 20 percent. Figure 17 shows dQ/dV against c e l l voltage measured at 35 yamps. Figure 18 shows dQ/dV against Q and x f o r t h i s case. Three p a i r s of remarkably clear peaks are v i s i b l e i n f i g u r e 17, and three corresponding p a i r s of peaks are v i s i b l e i n f i g u r e 18. The charge took 104.3 hours while the discharge took 99.8 hours. Figure 19 d i r e c t l y compares the l i n e a r sweep voltammogram previously shown in f i g u r e 9, which took roughly 59 hours, for a charge or discharge, with constant current dQ/dV measured at 52.5 yamps, f o r which the charge took 62.3 hours and the discharge 60.8 hours. The gain i n re s o l u t i o n i s c l e a r . Figure 20 shows dQ/dV against c e l l voltage measured at 140 yamps. Figure 21 shows dQ/dV against Q--and,-x" ': for t h i s case. The charge took 21.6 hours, while the discharge took 20.7 hours. The peaks are s t i l l c l e a r and d i s t i n c t i v e at 140 yamps. When examining the data i n figures 17 through 21, i t should be noted that while the l i n e a r sweep voltammogram i n f i g u r e 9 was made at t h e l l i m i t of the equipment's c a p a b i l i t i e s , constant current dQ/dV measurements could be made e a s i l y with the current an order of magnitude below 35 yamps. The peaks against voltage i n these figures are q u a l i t a t i v e l y very '66 800 o > 600 Z5 o o > 200 400 h Cell 2.3 2.4 2.5 Voltage (volts) 2.6 Figure 17 - Constant current dQ/dV against c e l l voltage f o r the 31st charge ( s o l i d l i n e ) and discharge (dashed li n e ) of the Li xVS2 c e l l GJ-9, at a current of 35 yamps. Q (coulombs) 12 10 8 6 4 2 0 1.0 0.8 0.6 0.4 0.2 0.0 X in L i x V S 2 Figure 18 - Constant current dQ/dV against x and Q for the 31st charge ( s o l i d l i n e ) and discharge (dashed li n e ) of the L i VS^ c e l l GJ-9, at a current of 35 yamps. Individual data points are shown when x changes by more than ^0.005 between points. 68 2.3 2.4 Cell Voltage (volts) 2.5 2.6 2.3 2.4 Cell Voltage (volts) 2.5 2.6 igure 19 - Comparison of l i n e a r sweep voltammetry and constant current dQ/dV. (a) The same voltammogram as i n fi g u r e 9, showing the 8th charge ( s o l i d l i n e ) and discharge (dashed l i n e ) f o r c e l l GJ-9, each taking 59 hours. (b) Constant current dQ/dV against c e l l voltage f o r the 32nd charge ( s o l i d l i n e ) and discharge (dashed l i n e ) f o r c e l l GJ-9, each taking 62 hours at a current of 52.5 yamps. Figure 20 - Constant current dQ/dV against c e l l voltage f o r the 29th charge ( s o l i d l i n e ) and discharge (dashed line) of the L i VS„ c e l l GJ-9, at a current of 140 yamps. Q (coulombs) 10 8 6 4 2 0 2 5 0 n — i — i — i — i — i — i — i — i — i — i — r 1.0 0.8 0.6 0.4 0.2 0.0 X in L i x V S 2 Figure 21 - Constant current dQ/dV against x and Q for the 29th charge ( s o l i d l i n e ) and discharge (dashed l i n e ) of the L i c e l l GJ-9, at a current of 140 yamps. 71 s i m i l a r i n shape to the i d e a l peaks discussed i n Section 3.2. The peaks have a rr-api'd* i n i t i a l r i s e followed by a t a i l . The peak width decreases with the current while the peak height increases, as expected. The peaks against Q are no t i c a b l y more rounded than the i d e a l sawtooths predicted i n Section 3.2. Non-ideal peak shapes w i l l be discussed further i n Section 3.5. The peaks are s t i l l s u f f i c i e n t l y d i s t i n c t i v e to be unambiguous signals of f i r s t order phase t r a n s i t i o n s . Notice the small precursors to the t r a n s i t i o n peaks. I t i s p a r t i c u l a r l y c l e a r i n f i g u r e 17 that there are three f i r s t order phase t r a n s i t i o n s i n L i VS„, not four as was believed on the basis of l i n e a r x 2 sweep voltammetry data. There i s no f i r s t order phase t r a n s i t i o n near 2.325 v o l t s . There i s a not i c a b l e bump in the sing l e phase capacity near t h i s voltage, however t h e i r are no peaks. The bump appears to have the same shape on charge and discharge, and remains the same height r e l a t i v e to s i n g l e phase capacity independent of current. R e c a l l i n Section 2.4 that the bump was a t t r i b u t e d to a B + IT two phase region. This two phase region does not appear to e x i s t . Since the B phase i s a monoclinic d i s t o r t i o n of the IT hexagonal unit c e l l , i t would appear very l i k e l y that the t r a n s i t i o n between B and IT i s second order, the change from one unit c e l l to the other occurring i n a continuous manner. Further i n v e s t i g a t i o n of the change between the B and IT phases, including confirmation of the x-ray data of Murphy et a l (1977) discussed i n Section 2.4, i s necessary i n order to confirm or deny t h i s hypothesis. I t i s possible that such a second order t r a n s i t i o n would become f i r s t order at lower temperatures. If the t r a n s i t i o n i s second order, then the B + IT two phase region i n the phase diagram shown i n fi g u r e 13 should be replaced by a second order l i n e near x = 0.6. The three f i r s t order phase t r a n s i t i o n s that are v i s i b l e with constant current dQ/dV appear i n roughly the same places noted e a r l i e r in Section 2.4, so there are no other major modifications to be made to the phase diagram. We s h a l l now consider q u a n t i t a t i v e l y the constant current dQ/dV r e s u l t s . In order to determine the t r a n s i t i o n voltages, the p o s i t i o n of each peak's leading edge was plotted as a function of current. The base of the i n i t i a l r i s e was used, ignoring the small precursors. Extra-polating the peak p o s i t i o n to zero current eliminates the IR s h i f t . Data fo r the three phase t r a n s i t i o n s i s shown in figures 22, 23, and 24. Linear least squares f i t s are also shown. For each of the two peaks at higher voltages, a l l of the data was used i n the f i t except f o r the points at 140 yamps. At 140 yamps the peak shapes were notic a b l y l e s s i d e a l than at lower currents, moving the leading edges closer to the t r a n s i t i o n voltage and somewhat f l a t t e n i n g the i n i t i a l ramp. The least squares f i t s shown i n f i g u r e 24 to the data f o r the low voltage peak only make use of the data f o r the three lowest currents. At higher currents, the peak shape i s poorly defined due to the presence of IT sing l e phase capacity, so the leading edges become more and more d i f f i c u l t to locate with cer-t a i n t y . Notice that slopes of the le a s t squares f i t s w i l l equal 1/R, so we can determine the c e l l r esistance during each phase t r a n s i t i o n . On discharge, there i s e s s e n t i a l l y no capacity at voltages above the highest voltage phase t r a n s i t i o n . Shown i n f i g u r e 22, the best f i t l i n e to the peak's leading edge gives a c e l l resistance R = (195 ± 15) ohms, and the zero current intercept i s (2.466 ± 0.001) v o l t s . The same tra n -s i t i o n oh charge gives a c e l l resistance of R = (95 ± 10) ohms and a zero current intercept of ("2.473 ± 0.001) v o l t s . Notice the intercepts d i f f e r by 7 m i l l i v o l t s . We s h a l l take (2.470 ± 0.003) v o l t s to be the t r a n s i t i o n voltage, and comment on the di f f e r e n c e between the intercepts s h o r t l y . Notice that the c e l l resistance when the c e l l i s s t a r t i n g i t s discharge 73 2.43 2.45 2.47 2.49 2.51 V o l t a g e o f L e a d i n g E d g e ( v o l t s ) Figure 22 - The p o s i t i o n of the leading edges of the charge (triangles) and discharge (squares) peaks f o r the 2.470 v o l t t r a n s i t i o n . The s o l i d l i n e s are least squares f i t s used to extrapolate to zero current, eliminating the IR s h i f t s . Data was taken on c e l l GJ-9. A l l voltages are ±1 m i l l i v o l t . 74 2.35 2.37 2.39 2.41 2.43 Voltage of Leading Edge (volts) Figure 23 - The p o s i t i o n of the leading edges of the charge (triangles) and discharge (squares) peaks for the 2.384 v o l t t r a n s i t i o n . The s o l i d l i n e s are least squares f i t s as in f i g u r e 22. Data was taken on c e l l GJ-9. A l l voltages are ±1 m i l l i v o l t . 75 2.16 2.18 2.20 2.22 2.24 Voltage of Leading Edge (volts) Figure 24 - The p o s i t i o n of the leading edges of the charge (triangles) and discharge (squares) peaks for the 2.201 v o l t t r a n s i t i o n . The s o l i d l i n e s are l e a s t squares f i t s as i n f i g u r e 22. Data was taken on c e l l GJ-9. A l l voltages are ±1 m i l l i v o l t . 76 i s double the resistance when charging. Figure 23 shows the best f i t l i n e s f o r the t r a n s i t i o n near 2.38 v o l t s . On discharge, the c e l l resistance i s R = (95 ± 25) ohms and the zero current intercept i s (2.382 ± 0.002) v o l t s . On charge, the c e l l resistance i s R = (100 ± 15) ohms and the zero current intercept i s (2.387 ± 0.001) v o l t s . The intercepts d i f f e r by 5 m i l l i v o l t s . We s h a l l take (2.384 ± 0.003) v o l t s to be the t r a n s i t i o n voltage. The c e l l resistance i s roughly the same on charge and discharge, and roughly the same as the c e l l resistance found when charging through the 2.470 v o l t t r a n s i t i o n . Figure 24 shows the best f i t l i n e s f o r the t r a n s i t i o n near 2.20 v o l t s . On discharge, the c e l l resistance i s R''= (115 ± 10) ohms and the zero current intercept i s (2.200 ± 0.001) v o l t s . On charge, there i s e s s e n t i a l l y no capacity at voltages below t h i s t r a n s i t i o n . The c e l l resistance i s R = (200 ± 15) ohms and the zero current intercept i s (2.203 ± 0.001) v o l t s . The intercepts d i f f e r by 3 m i l l i v o l t s . We s h a l l take the t r a n s i t i o n voltage to be (2.201 ± 0.003) v o l t s . The c e l l r e s i s -tance when s t a r t i n g to charge the c e l l i s s i m i l a r to the c e l l resistance when s t a r t i n g to discharge the c e l l . This resistance i s roughly double the resistance of a l l other cases. With the exception of the i n i t i a l r esistance of about 200 ohms on both charge and discharge,- the c e l l r e s i s -tance appears to be roughly a constant 100 ohms. The difference between the charge and discharge intercepts changes from 3 m i l l i v o l t s f o r the 2.201 v o l t t r a n s i t i o n , to 5 m i l l i v o l t s f o r the 2.384 v o l t t r a n s i t i o n , and to 7 m i l l i v o l t s f o r the 2.470 v o l t t r a n s i t i o n . The bases of the peaks' i n i t i a l r i s e were located i n the same manner for each t r a n s i t i o n , so while a 3 m i l l i v o l t d i f f e r e n c e could be explained away as inaccurately i d e n t i f y i n g the points at which each i n i t i a l r i s e began, the 5 and 7 m i l l i v o l t differences cannot be e a s i l y eliminated by such an argument. To understand what i s occurring, we need to consider the ranges i n x of the two phase regions. The data shown i n f i g u r e 18 and the corresponding data measured at other currents was examined to determine the ranges i n x of the sing l e phase and two phase regions. The peaks due to the phase t r a n s i t i o n s are, as mentioned, quite rounded compared to the i d e a l sawtooth, making i t d i f f i c u l t to determine x values to better than one percent. A l l values of x quoted .below are ±0.01. I t was found on both charge and discharge that both the VS 2 and 3S phases have widths i n x of no more than 0.01. Recall that the VS 2 phase appears at x-= 0 and that the 3S phase appears at x = 1 . On charge, the 2.470 v o l t t r a n s i t i o n occurs i n the range 0.01 <_ x <_ 0.33. This i s the VS 2 + a two phase region. The 2.381 v o l t t r a n s i t i o n occurs i n the range 0.34 j< x <^  0.52. This i s the a + g two phase region. The a sing l e phase region has a width of no more than 0.01 at x = 0.33. Single phase capacity due to the g and IT phases occurs in the range 0.52 < x < 0.90. The 2.201 v o l t t r a n s i t i o n occurs i n the range 0.90 <_ x <_ 0.99. This i s the IT + 3S two phase region. On discharge, the r e s u l t s are s l i g h t l y d i f f e r e n t . The VS 2 + a two phase region occurs i n the range 0.01 <_ x _< 0.28. Also,- the a + 3 two phase region occurs i n the range 0.29 _< x _< 0.43. The a sing l e phase region i s s t i l l very narrow. The range i n x of the VS 2 + a two phase region on discharge i s 85 percent of the range on charge. S i m i l a r l y , the range i n x of the a + g two phase region on discharge i s 80 percent of the range on charge. The IT + 3S region s t i l l l i e s i n the range . 0.90 <_ x <^  1.00, although a di f f e r e n c e of a few percent i n width compared to the range on charge may be possible given the error assigned to the endpoints. To compensate f o r the reduced ranges i n x of the two phase regions, the s i n g l e phase capacity due to the B and IT phases ranges on discharge over 0.43 _< x _< 0.90, a 25 percent increase i n width over the charge case. I t i s clear that there i s s i g n i f i c a n t hysteresis i n x f o r both the + ct and a -f $ two phase regions. This hysteresis appears to be independent of the current. Also r e c a l l that there were small differences between the zero current extrapolations f o r the leading edges of the charge and discharge peaks that cannot be e a s i l y explained away f o r these two phase regions. Thus, there i s evidence of hysteresis i n voltage as well as i n x. If hysteresis i s occurring i n the IT + 3S two phase region, i t i s not dramatic enough to be unambiguously detected. McKinnon (1982) has shown for i n t e r c a l a t i o n systems that i f energy i s d i s s i p a t e d when one phase i s converted into another, then the two phases can coexist over a range of voltages and of l i m i t s i n x of the two phase region. I t i s very reasonable to expect that heat w i l l be generated by p l a s t i c deformation during such s t r u c t u r a l phase t r a n s i t i o n s as occur i n L i ^ V ^ . The energy loss depresses the voltage below the expected e q u i l i b -rium voltage on discharge, as shown i n f i g u r e 25. The area between the equilibrium voltage and the s h i f t e d voltage i s proportional to the heat generated, so the s h i f t must be constant since the phase conversion of any given amount of material w i l l generate the same heat, regardless of the p o s i t i o n of the phase f r o n t . Such a s h i f t i s independent of current. S i m i l a r l y , on charge the voltage w i l l be s h i f t e d by a constant above the equilibrium voltage one would normally expect. The s h i f t on charge may w e l l d i f f e r from the s h i f t on discharge. D i f f e r e n t widths i n x for the two phase regions on charge and discharge w i l l occur i f the equilibrium dV/dx i s d i f f e r e n t i n the si n g l e phase regions on either side of the two phase regions, as shown in f i g u r e 25. 79 X Figure 25 - Hysteresis i n voltage and x due to energy d i s s i p a t i o n i n a two phase region. Charge and discharge voltages are displaced from the expected equilibrium value (dashed l i n e ) . Since dV/dx i s d i f f e r e n t i n the sin g l e phase regions, the two phase regions are of d i f f e r e n t lengths. 80 The behavior of the f i r s t order phase t r a n s i t i o n s i n L i VS„ i s x I consistent with the hysteresis discussed by McKinnon. The hysteresis i n voltage i s only a few m i l l i v o l t s , but t h i s i s s u f f i c i e n t to noticeably a f f e c t the ranges i n x of the two phase regions. Some two phase capacity that appears on charge reappears as single phase capacity on discharge. The t r a n s i t i o n voltages given e a r l i e r are ac t u a l l y centered i n small ranges, within the quoted error, i n which the two phases can coexist. The hys-t e r e s i s i n x causes phase boundaries i n the phase diagram to s h i f t depending on whether charge or discharge i s being considered. It i s d i f f i c u l t to derive further quantitative information from the peaks. The widths of the peaks against voltage are expected to be roughly V^, so the widths should decrease l i n e a r l y with current. While the peak widths do, i n general, decrease monotonically with current, at higher currents s i n g l e phase capacity d i s t o r t s the peak shapes. At lower currents the peaks' i n i t i a l increases, which are ramps rather than jumps, d i s t o r t the widths of the narrow peaks s u f f i c i e n t l y to give quantitative measure-ments a high uncertainty. The peak height i d e a l l y has a maximum value of ~ Y Q Q / | V ^ | , so when graphed the maximum peak height should be proportional to Q Q / | I | . An example set o f data i s presented i n fi g u r e 26 for the peaks of the + ct two phase region. Again, at high currents the data i s poor since the peak shapes are not as close to i d e a l as they are at low currents. At low currents the peak height v a r i e s by as much as ten percent from measurement to measurement, since the reso l u t i o n l i m i t of the apparatus i s being reached and the peak shapes are s t i l l not i d e a l . The peak heights shown i n f i g u r e 26 are roughly l i n e a r i n Q / | l | , as i s corresponding data f o r the other t r a n s i t i o n peaks, but the data i s not accurate enough to f i t with a straight l i n e and obtain a r e l i a b l e slope. Since the peak 81 8 0 0 o > oo 6 0 0 _Q 13 O O 4 0 0 > TD •o 2 0 0 0 0 20 40 60 8 0 Q 0 / |I| (hours) 100 Figure 26 - Peak height against Q_/JI[ f o r the charge (triangles) and discharge (squares) peaks or the 2.470 v o l t t r a n s i t i o n . Data was taken on c e l l GJ-9. height i s i d e a l l y ~ Y Q Q / | V |, the r a t i o of such slopes f o r d i f f e r e n t peaks would be the r a t i o of D dp/dV for the phases nucleated on the surfaces of the p a r t i c l e s i n each case. One would expect that the discharge peak for the + CL region would have roughly the same height as the charge peak for the a + 3 region, since i n both cases the a phase i s nucleated on the p a r t i c l e s ' surfaces. Also, i f dp/dV doesn't change much i n the large sin g l e phase region, the discharge peak f o r the a + 3 region and the charge peak f o r the IT + 3S region would have roughly the same height i f D has not changed s i g n i f i c a n t l y . This i s p l a u s i b l e , since no f i r s t order s t r u c t u r a l change occurs between 3 and IT. Noting that the peak heights also scale with each t r a n s i t i o n ' s Q Q , the data in f i g u r e 17 i s i n q u a l i t a t i v e agree-ment with these comments. I t would also appear that D dp/dV increases markedly with voltage. This may be due to improved d i f f u s i o n f o r phases at low x, but i t i s not known how dp/dV i s changing. It i s very clear that constant current dQ/dV outperforms l i n e a r sweep voltammetry i n c l a r i t y and r e s o l u t i o n . The presence of hysteresis, f or example, was not detected by l i n e a r sweep voltammetry. The t r a n s i t i o n voltages obtained with constant current dQ/dV are s l i g h t l y higher than those obtained with l i n e a r sweep voltammetry. In t h i s section, the t r a n s i t i o n voltages obtained with constant current dQ/dV were found to be (2.470 ± 0.003) v o l t s , (2.384 ± 0.003) v o l t s , and (2.201 ± 0.003) v o l t s . In section 2.4, the t r a n s i t i o n voltages obtained with l i n e a r sweep voltammetry were re s p e c t i v e l y (2.460 ± 0.005) v o l t s , (2.380 ± 0.005) v o l t s and (2.190 ± 0.005) v o l t s . The r e s u l t s are within the quoted error of each other for the t r a n s i t i o n near 2.38 v o l t s , and the r e s u l t s are close to the quoted error l i m i t s f o r the other two t r a n s i t i o n s . The s l i g h t discrepancies could a r i s e because the c e l l s used for the l i n e a r sweep voltammetry were at room temperature, rather than i n a temperature bath 83 at 20°C. The output voltage of the dQ/dV apparatus was c a l i b r a t e d follow-ing the measurements described in t h i s t h e s i s , and an error of s l i g h t l y l e s s than 3 m i l l i v o l t s was found. The data in t h i s thesis was corrected for t h i s error. The PAR equipment described i n section 2.3 may have had a c a l i b r a t i o n error in i t s output voltage of a few m i l l i v o l t s . The ranges i n x f o r the three phase t r a n s i t i o n s as obtained by l i n e a r sweep voltammetry agree with the ranges obtained on charge using constant current dQ/dV. Recall that the l i n e a r sweep I against Q had the c l e a r e s t peaks on charge, so they were used to determine the ranges i n x. The discharge peaks are only c l e a r enough i n constant current dQ/dV for the hysteresis to be detected. F i n a l l y , note that peaks due to f i r s t order phase t r a n s i t i o n s are unambiguous i n constant current dQ/dV, which i s not the case with l i n e a r sweep voltammetry. Constant current dQ/dV demonstrated that at 20°C there i s no f i r s t order phase t r a n s i t i o n between the (3 and IT phases as had been believed on the basis of l i n e a r sweep voltammetry. 84 3.5 Non-ideal Peak Shapes In Section 3.4, i t was noted that the peaks i n dQ/dV due to f i r s t order phase t r a n s i t i o n s do not have exactly the shapes predicted i n Section 3.2. The peaks against voltage do not simply jump to t h e i r maximum value, although the peaks may be very sharp. There i s a slope to the leading edge of the peaks which, as noted i n Section 3.4, makes i t d i f f i c u l t to determine the widths of narrow peaks accurately. Peaks may appear s l i g h t l y rounded at t h e i r maxima, but the i n i t i a l slope i s the most obvious deviation from the i d e a l peak shape. In addition, there i s a small precursor to each t r a n s i t i o n peak. This occurs to some extent i n a l l materials tested. This low upturn i n -dQ/dV i s no t i c a b l e i n the figures 17, 19, and 20, for example, and begins well before the t r a n s i t i o n voltage i s reached by the cathode material. No explanation has been found f o r these precursors. I t i s not clear why such an e f f e c t would occur i n the s i n g l e phase before the t r a n s i t i o n begins. The peaks against Q are noticably rounded with a prominent leading edge. They appear f a r less i d e a l than the voltage peaks. One can specu-l a t e about the source of the non-ideal behavior. The rounding of the peaks indicates that the cathode p a r t i c l e s are not a l l nucleating the new phase on t h e i r surfaces at the same time. Two p o s s i b i l i t i e s suggest themselves. One p o s s i b i l i t y i s that some p a r t i c l e s i n the cathode f i n d i t easier to nucleate the new phase than other p a r t i c l e s . This could conceivably occur i f the phase nucleation occurred more r e a d i l y f o r , say, small p a r t i c l e s than large p a r t i c l e s due to differences i n s t r a i n energies. Experiments in which only a l i m i t e d range of p a r t i c l e sizes i s used i n the cathode have been performed by P.J. Mulhern with L i M0S2, but they do not demon-strate convincingly that the peak shape i s sharper f o r a narrow p a r t i c l e d i s t r i b u t i o n . In addition, the three phase t r a n s i t i o n s i n L i VS„ have 85 very s i m i l a r peak shapes against Q, and i t i s not clear why d i f f e r e n t s t r u c t u r a l phase t r a n s i t i o n s with d i f f e r e n t hysteresis should experience such s i m i l a r nucleation problems, p a r t i c u l a r l y since the d i f f e r e n t amounts of hysteresis i n d i c a t e that the p l a s t i c deformation involved i n the phase conversion i s s i g n i f i c a n t l y d i f f e r e n t f o r each t r a n s i t i o n . A second p o s s i b i l i t y i s that the cathode p a r t i c l e s are not a l l at the same voltage because of r e s i s t i v e e f f e c t s . I f the cathode i s thi c k , then p a r t i c l e s at d i f f e r e n t depths i n the cathode w i l l experience d i f f e r e n t resistances, both because of the e l e c t r i c a l resistance added by the cathode material between a p a r t i c l e and the cathode's substrate, and because of resistance caused by any d i f f i c u l t i e s i n getting good contact with the e l e c t r o l y t e . Such a case could occur simply by having large c r y s t a l l i t e s that are d i f f i c u l t to spread evenly on the substrate. In e f f e c t , each p a r t i c l e would experience a d i f f e r e n t IR s h i f t , and the i d e a l peak shapes would be convoluted with the d i s t r i b u t i o n of the time at which the par-t i c l e s reached the t r a n s i t i o n voltage. It would be useful to accurately model a th i c k cathode, however that w i l l not be attempted here. Instead, we s h a l l pick a simple d i s t r i b u t i o n for the time and see i f the r e s u l t i n g peak shapes are q u a l i t a t i v e l y s i m i l a r to the experimentally observed peaks. We s h a l l use a Gaussian d i s t r i b u t i o n f o r mathematical s i m p l i c i t y . I f there i s a q u a l i t a t i v e s i m i l a r i t y to the •experimental peaks, t h i s merely indicates that a more c a r e f u l l y chosen d i s t r i b u t i o n might indeed be used to explain the peak shapes. Consider the case y = 2 . The i d e a l peak against Q i s a sawtooth between 0 and QQ with a maximum height of -2QQ/[V"d| . Take the c e l l to be discharging. From (3.14): dV 2Q 0 V. ( 3 . 20 ) D 86 Take the time t to be 0 when Q = 0 and t when Q = Q . The Gaussian r U d i s t r i b u t i o n i s : 1 2 2 w(t) = — exp(-t /2a ) (2TJ)"2O (3.21) Since the contribution from any p a r t i c l e has a width i n time of t , the r convolution of (3.20) with (3.21) gives: da 2% r dt W(T) 1 - t-T (3.22) Define the p r o b a b i l i t y i n t e g r a l : $(x) = (2ir)' dy exp(-y ) (3.23) After i n t e g r a t i o n (3.22) becomes: dV V D1 2h (t -t) {$(t/2%) - *({t-t_}/2 J sa)} F r + W% {exp(-{t-t } 2/2a 2) - exp(-t 2/2c 2)} r (3.24) Figure 27 shows the r e s u l t i n g peak shape compared to the i d e a l peak for the case a = t /10. F If we ignore a l l capacity except f o r that due to the phase t r a n s i t i o n , we can convolute the c e l l voltage given by (3.18b) with the Gaussian. Noting that the portion of the cathode which has not yet begun to phase convert i s s t i l l nominally at the c e l l voltage: V - V_ + IR = c 0 1 (2TT) 2a 2 2 ^  dx exp(-x /2a ) (V - V n + IR) c 0 + V, rt t - t . dT W(T) h l n 1 - t-x rt (2ir)li $(t/2 J f ia) j t - t 2 2 dT exp(-x /2a ) % l n t-T (3.25) 87 Figure 27 - The s o l i d l i n e i s the i d e a l peak shape of f i g u r e 16 f o r Y = 2, with the c e l l discharging. The squares show sample points calculated f o r a non-ideal peak, i f the time at which p a r t i c l e s nucleate t h e i r phase fronts i s Gaussian d i s t r i b u t e d . The case shown i s f o r a Gaussian halfwidth equal to one tenth of the time taken to discharge from 0 to Q n. 88 The i n t e g r a l i s divergent, due to the steady state approximation, at T = t - t . The i n t e g r a l was computed numerically excluding the region of F T near t - t . Figure 28 shows the r e s u l t i n g peak shape, again compared F to the i d e a l peak for the case a = t^/lO. The peak shapes shown i n f i g u r e s 27 and 28 are q u a l i t a t i v e l y s i m i l a r to the experimentally observed peaks. The peak against Q i s noticably rounded with a s i g n i f i c a n t leading edge, s i m i l a r to what i s seen exper-imentally although more exaggerated. The peak has a l i n e a r f a l l back to low dQ/dV, s i m i l a r to the o r i g i n a l sawtooth. The peak against c e l l voltage i s not as d r a s t i c a l l y a f f e c t e d as the peak against Q, but i t now has an i n i t i a l slope and i s somewhat rounded o f f . This i s also in qual-i t a t i v e agreement with the experimental peaks. I t appears that use of a d i s t r i b u t i o n of times at which the p a r t i c l e s reach the t r a n s i t i o n voltage i s a f r u i t f u l approach to understanding non-ideal peak shapes. Figure 28 - The s o l i d l i n e i s the i d e a l peak shape of f i g u r e 15 for Y = 2. The squares show sample points calculated f o r a non-ideal peak, i f the time at which p a r t i c l e s nucleate t h e i r phase fronts i s Gaussian d i s t r i b u t e d . The case shown i s f o r a Gaussian h a l f -width equal to one tenth of the time taken to discharge from 0 to V 90 3.6 Effectiveness of the Technique It has been made clear in t h i s chapter that constant current dQ/dV measurements give c l e a r and unambiguous evidence of the presence of f i r s t order phase t r a n s i t i o n s . The r e s o l u t i o n of the peaks caused by two phase regions i s f a r superior to that obtained with l i n e a r sweep voltammetry. The peak shapes against voltage are d i s t i n c t i v e and sharpen markedly with decreasing current. This leads to a c l e a r d e r i v a t i o n of the phase diagram. The apparatus i s capable of measuring dQ/dV over much longer times than i s possible with the l i n e a r sweep voltammetry apparatus described i n Section 2.3. This allows the c e l l to be c l o s e r to equilibrium during dQ/dV measurements than i s possible with l i n e a r sweep voltammetry. If measurements are made at a sequence of currents, a graph of peak p o s i t i o n as a function of current allows a quantitative determination of the c e l l resistance during the phase t r a n s i t i o n , as w e l l as of the equilibrium t r a n s i t i o n voltage. Graphs of dQ/dV against Q allow an accurate determination of the range i n x of a two phase region. The ranges i n x and the extrapolated values for the t r a n s i t i o n voltages are s u f f i c i e n t l y accurate to detect hysteresis i n the phase t r a n s i t i o n s . The peaks' non-ideal shapes, discussed in the l a s t section, make i t d i f f i c u l t to extract quantitative r e s u l t s f o r from the peak widths or for D dp/dV from the peak heights, although the peak widths and heights change q u a l i t a t i v e l y with the current i n the expected manner. The r e s o l u t i o n of constant current dQ/dV i s such that very f i n e d e t a i l s , such as hysteresis i n the t r a n s i t i o n voltages, can be examined. Constant current dQ/dV peaks increase in dQ/dV and become thinner when the current decreases, while s i n g l e phase dQ/dV changes l i t t l e . This i s in contrast with l i n e a r sweep voltammetry, whose peaks i n I decrease with increasing sweep rate, although they decrease le s s than the s i n g l e phase capacity decreases. The c l a r i t y of dQ/dV peaks due to phase t r a n s i t i o n s , as exemplified by fi g u r e 17, prevents such mistakes as the i d e n t i f i c a t i o n of a 3 + IT two phase region i n L i VS 9, as was mistakenly done using l i n e a r sweep voltammetry r e s u l t s . I f an ambiguous peak shape appears i n constant current dQ/dV, the current can be decreased u n t i l i t i s clear whether or not a f i r s t order phase t r a n s i t i o n a c t u a l l y occurs. 92 CHAPTER FOUR OTHER APPLICATIONS AND EXAMPLES OF CONSTANT CURRENT dQ/dV 4.1 S e n s i t i v i t y of the Technique We have seen i n Chapter 3 that constant current dQ/dV can be used to c l e a r l y and unambiguously i d e n t i f y f i r s t order phase t r a n s i t i o n s . The peaks due to phase t r a n s i t i o n s remain sharp and c l e a r l y v i s i b l e at r e l -a t i v e l y high currents, as i n the example of L i ^ V ^ at 140 yamps, previously shown i n f i g u r e 20. There, charge or discharge took about a day. One would expect that i f a material, such as L i VS„, with one or more peaks x 2 i n dQ/dV due to f i r s t order phase t r a n s i t i o n s , was present as an impurity i n another material, then i t s presence might be detected by dQ/dV measure-ments. A small impurity would be driven quickly through a phase t r a n s i t i o n , since a large f r a c t i o n of the c e l l ' s current would be used for the phase conversion u n t i l the cathode's voltage was able to s h i f t away from the t r a n s i t i o n voltage. This reduces the dQ/dV peak heights, making an im-pur i t y more d i f f i c u l t to detect. An impurity would not be detected i f i t had phase t r a n s i t i o n s at voltages close to those of phase t r a n s i t i o n s i n the main material, but a dQ/dV peak from an impurity may well-be v i s i b l e i f superimposed on single phase capacity. A s t r i k i n g example of the s e n s i t i v i t y of constant current dQ/dV was recently discovered by Mulhern (1982). A flange c e l l , PM-30, with a cathode made from the n a t u r a l l y occurring layer compound MoS9, was discharged f o r 93 the f i r s t time. The M0S2 was obtained from the Endako mine. The discharge was at a constant 25 yamps. Constant current dQ/dV was measured using the apparatus described i n the f i r s t part of Section 3.3, and the r e s u l t s are shown in f i g u r e 29. The capacity below 1.4 v o l t s i s due to the M0S2 con-verting from i t s i n i t i a l a phase into 3 phase. There i s a small bump near 2.08 v o l t s that does not appear to be associated with the phase conversion. Figure 30 shows the same data as f i g u r e 29 but only above 1.4 v o l t s , with the v e r t i c a l scale increased by a fac t o r of 30. A peak i s c l e a r l y v i s i b l e . The peak at 2.08 v o l t s was not found i n s y n t h e t i c a l l y prepared M0S2, so i t was cl e a r that t h i s peak could be simply due to an impurity. The impurity has now been i d e n t i f i e d by Mulhern as Cu^ ^ Mo^S^. Only 0.3 percent of the capacity of c e l l PM-30 was caused by the impurity, yet i t s presence was s t i l l detected. Constant current dQ/dV i s capable of detecting e f f e c t s involving only f r a c t i o n s of a percent of a c e l l ' s capacity. This s e n s i t i v i t y allows straightforward i n v e s t i g a t i o n of a material f o r the presence of trace amounts of other materials that also i n t e r c a l a t e . Such s e n s i t i v i t y also permits c a r e f u l examination of very small e f f e c t s i n a material's voltage, and hence i n i t s equation of state. A constant current dQ/dV measurement made on Cu^ ^ Mo^S^ by Mulhern i s shown i n f i g u r e 31. The copper i s removed on the f i r s t discharge, leaving Mo^S^ i n t e r c a l a t e d with l i t h i u m . The f i r s t order phase t r a n s i t i o n at 2.08 v o l t s on subsequent cycles, shown i n fi g u r e 31, appears to involve a change in i n t e r c a l a n t content without any change i n the symmetry of the host l a t t i c e (Mulhern 1982). A v i r t u a l l y i d e n t i c a l dQ/dV curve appears follow-ing the removal on f i r s t discharge of iron from Fe, ,Mo„S.. 0 0 1 . 4 3 4 94 1.0 1.2 1.4 1.6 1.8 2 . 0 2 . 2 2 . 4 Cell Voltage (volts) Figure 29 - Constant current dQ/dV against c e l l voltage for the f i r s t discharge of the L i M 0 S 2 c e l l PM-30, made with n a t u r a l l y occurring MoS^. The current was 25 pamps. Data provided by P. J . Mulhern. 1.4 1.6 1.8 2 . 0 2 . 2 2 . 4 Cell Voltage (volts) Figure 30 - Expanded view of the high voltage data shown i n f i g u r e 29. This was the f i r s t evidence of an impurity involving 0.3 percent of the capacity of c e l l s made with natural Mx^. Data provided by P. J . Mulhern. 30 2 0 > TD 0 - 1 0 •20 - 3 0 l l l l l J Discharge — 1— Y~ \ ^- Charge — I I I l l l l l 1.8 2.0 2.2 2.4 Cell Voltage (volts) Figure 31 - Constant current dQ/dV against c e l l voltage f o r l i t h i u m i n t e r c a l a t e d Mo0S,. Material was Cu, ,Mo„S. p r i o r to i t s f i r s t 3 4 1.4 3 4 r discharge. Data provided by P. J . Mulhern. k.2 Applications to Single Phase Regions When dQ/dV i s measured against V , by d e f i n i t i o n the curve shows the c e l l capacity as a function of c e l l voltage. Also, since dQ/dV i s s e n s i -t i v e to f i n e structure i n the equation of state of an i n t e r c a l a t i o n c e l l , dQ/dV i s an extremely s e n s i t i v e probe of the thermodynamics of a material. As a r e s u l t , understanding dQ/dV in s i n g l e phase regions implies under-standing of the i n t e r c a l a t i o n system's equation of state. I t i s to be expected that since d i f f e r e n t materials w i l l normally have quite d i f f e r e n t equations of state, that dQ/dV w i l l normally be quite d i s t i n c t i v e f o r a given material. This allows the detection of impurities, as discussed in the previous section. Also, i f two materials have s i m i l a r dQ/dV curves, then t h e i r thermodynamics and p h y s i c a l properties may be expected to be s i m i l a r . I d e n t i c a l dQ/dV curves strongly indicate that two c e l l s have i d e n t i c a l i n t e r c a l a t i n g materials. The d i f f i c u l t y of deriving dQ/dV from a model of an i n t e r c a l a t i o n compound has been recently i l l u s t r a t e d by the successful model of Dahn (1982) f o r the l i t h i u m i n t e r c a l a t e d layer compound L i x T i S 2 (see also Dahn, Dahn, and Haering 1982). Figures 32 and 33 i l l u s t r a t e dx/dV measured by Dahn for L i TiS„ i n the range 0 < x < 1. The i n c l u s i o n in x 2 ° — — a l a t t i c e gas Hamiltonian of the e l a s t i c energy due to the expansion of the van der Waals gaps as i n t e r c a l a t i o n proceeds predicts f l a t t e n i n g of the voltage curve at low x, causing the higher dx/dV values observed at low x. The dip i n dx/dV near x = 0.16 has been a t t r i b u t e d to the formation of a short range stage 2 structure. A stage n structure i s one i n which every nth van der Waals gap i s f i l l e d at one i n t e r c a l a n t concentration, with the intervening gaps f i l l e d at a lower concentration. Although t h i s model i s very successful in p r e d i c t i n g the s t r u c t u r a l and thermodynamic behavior of L i TiS„ and some other MX„ layer compounds, dx/dV obtained 4.0 | r 1.8 2.0 2.2 2.4 2. Volts Figure 32 - Constant current dx/dV against c e l l voltage f o r the L i T1S2 c e l l JD-235 at 40 yamps. Diamonds in d i c a t e charge, X t r i a n g l e s i n d i c a t e discharge. From Dahn (1982). 99 Figure 33 - Constant current dx/dV against x for the L i TiS^ c e l l JD-235 at 40 yamps. Diamonds indic a t e charge, t r i a n g l e s i n d i c a t e discharge. From Dahn (1982). 100 from Monte Carlo c a l c u l a t i o n s s t i l l has peaks of equal height on either side of x = 0.16, unlike the data. I t i s possible that t h i s discrepancy may be removed by the i n c l u s i o n of a more accurate expression f o r the e l a s t i c energy, however t h i s example serves to i l l u s t r a t e the s e n s i t i v i t y of dQ/dV to the thermodynamics of the system, and i t s usefulness i n testing models of i n t e r c a l a t i o n systems. Other predictions of thermodynamic quan-t i t i e s are i n noticably better agreement with data. It should also be noted that information about a c e l l ' s k i n e t i c s can be obtained from sin g l e phase dQ/dV. dQ/dV s t i l l experiences an IR s h i f t i n s i n g l e phase regions, so measurements of dQ/dV as a function of current can be used to determine the c e l l ' s resistance at any point. D i f f u s i o n e f f e c t s , however, are not l i k e l y to be c l e a r l y v i s i b l e , and i t i s u n l i k e l y that s i n g l e phase measurements can extract information about s o l i d state d i f f u s i o n . 101 4 . 3 "Supercooling" An analogy was presented in Section 1.2 between the thermodynamics of a gas and the thermodynamics of an intercalation system. Effects analogous to supercooling and supersaturation have been observed in intercalation systems. Supercooling occurs in a gas when i t s volume i s r i s i n g and i t s pressure, as a r e s u l t , i s dropping. The pressure under-shoots the c r i t i c a l pressure at which a f i r s t order phase t r a n s i t i o n occurs. The pressure then returns to the c r i t i c a l pressure as the f i r s t order phase t r a n s i t i o n proceeds. Supersaturation i s the analogous effect with the volume dropping and the pressure r i s i n g . These effects occur because of the nucleation process for a new phase. Once the c r i t i c a l pressure i s reached, i t i s possible to form a two phase system. U n t i l a new phase i s nucleated, however, the system continues in i t s o r i g i n a l phase. This i s a metastable region. A new phase w i l l form once regions are nucleated above a c r i t i c a l size at which they become stable and grow. Consider supercooling. Regions of different sizes have different surface areas through which gas can enter and leave. For a given pressure, there w i l l be a c r i t i c a l size at which the surface area i s just right for no net flow of gas to occur across the surface. Clearly, as the pressure decreases the size becomes smaller for which no outflow of gas occurs from the nucleation region. Regions of varying sizes are formed by fluctuations in the gas. I f a region's size i s below the c r i t i c a l s i z e , a net outflow of gas occurs from the nucleation region, and the region dissipates. I f a region's size i s above the c r i t i c a l s i z e , however, a net inflow of gas occurs and the region grows. The pressure increases back to the t r a n s i t i o n pressure as the system attempts to reach equilibrium with the newly nucleated phase as i t continues to grow. Supersaturation proceeds s i m i l a r l y . 102 The overshoot i n pressure i s needed to overcome whatever energy b a r r i e r i s associated with the s t a r t of phase conversion. Analogously, i n an i n t e r c a l a t i o n system an overshoot i n voltage i s required to nucleate a new phase. By analogy, the voltage enters a metastable region and moves away from the t r a n s i t i o n voltage, u n t i l the c r i t i c a l s i z e f or a nucleated region decreases s u f f i c i e n t l y f o r the nucleated regions to grow and phase conversion to begin. Recall that the volume i s analogous to Q or x. The c r i t i c a l s i z e decreases with voltage since x i n the o r i g i n a l phase r i s e s as the voltage drops. A nucleation region must have a smaller surface area f o r a larger x i n the o r i g i n a l phase i f there i s to be no net inflow of i n t e r c a l a n t into a nucleation region. Once again, when the c r i t i c a l s i z e becomes small enough, the new phase nucleates and the voltage relaxes back to the t r a n s i t i o n voltage. "Supercooling" in an i n t e r c a l a t i o n system w i l l occur i f i t i s d i f f i c u l t to nucleate a new phase. For most phase t r a n s i t i o n s "super-cooling" i s not observed, however i t i s sometimes seen during the con-version from a phase to $ phase in M0S2 that was mentioned i n Section 4.1. "Supercooling" was frequently seen i n c e l l s with a l i t h i u m anode and a p y r i t e (FeS2) cathode prepared by R. Marsolais. The majority of the capacity of such c e l l s i s i n a f i r s t order phase t r a n s i t i o n . I t i s not yet clear what the end product of the phase t r a n s i t i o n i s , a l -though the f i r s t discharge of a p y r i t e c e l l usually changes x from 0 to ^3.5 in Li xFeS2. Such c e l l s can only be p a r t i a l l y recharged and have poor cycle l i f e , however they make excellent primary b a t t e r i e s , par-t i c u l a r l y since the t r a n s i t i o n voltage i s very conveniently close to 1.5 v o l t s . Figure 34 shows the "supercooling" e f f e c t i n the f i r s t d i s -charge voltage curve of c e l l M-34. The c e l l took about 60 hours to discharge at 400 yamps. 103 1.50 1.45 \-O > 8. 1.40 o H—' O > <u ^ 1.35 1.30 0 1 0 - 2 0 30 40 Discharge Time (hours) 50 Figure 34 - Voltage curve f o r the f i r s t discharge of the l i t h i u m against p y r i t e c e l l M-34. A very strong "supercooling" e f f e c t occurs at the s t a r t of a f i r s t order phase t r a n s i t i o n involving most of the c e l l ' s capacity. C e l l discharged to 1.0 v o l t s i n roughly 60 hours, at a current of 400 yamps. Data provided by R. Marsolais. 104 Even a small "supercooling" e f f e c t should appear prominently i n dQ/dV. As the voltage reaches a minimum and st a r t s increasing again, dQ/dV diverges to - O T, suddenly switches sign to +00, and then decreases i n magnitude. Then when the voltage reaches a maximum and starts decreasing again, dQ/dV diverges to +°°, changes sign to -°°, and then s e t t l e s down to record the f i r s t order phase t r a n s i t i o n . One would expect that once the new phase was nucleated, the c e l l voltage would return to the value normally expected during a phase t r a n s i t i o n , and the usual dQ/dV peak should appear with i t s i n i t i a l portion missing, and some d i s t o r t i o n i n the remainder. Figure 35 shows how the "supercooling" e f f e c t shown i n f i g u r e 34 ap-pears i n a dQ/dV curve. The data shown i s for the f i r s t discharge of c e l l M-47. This was done at 100 yamps and took roughly a week. The data i n f i g u r e 34 thus has a considerably higher IR s h i f t than the data i n f i g u r e 35, causing a roughly 20 m i l l i v o l t d i f f e r e n c e i n voltage between the two sets of data. The e f f e c t appears i n dQ/dV as expected, followed by what appears to be the l a s t portion of a dQ/dV peak from the f i r s t order phase t r a n s i t i o n . 105 Figure 35 - Constant current dQ/dV against c e l l voltage f o r the lith i u m against p y r i t e c e l l M-47. The "supercooling" e f f e c t shown in f i g u r e 34 causes the changes i n sign i n dQ/dV. Data provided by R. Marsolais. 106 4.4 Staging As mentioned i n Section 4.2, a stage n structure in a layer compound has every nth van der Waals gap f i l l e d at one in t e r c a l a n t concentration, with the intervening gaps a l l f i l l e d at a lower concentration. Frequently, staged structures have every nth gap f i l l e d , with the intervening gaps empty. F i r s t order phase t r a n s i t i o n s may occur i n a material to change from one staged structure to another. This has recently been observed i n l i t h i u m i n t e r c a l a t e d NbSe 2 (Dahn, D.C., and Haering 1982). The voltage curve f o r L i NbSe„ i s shown in f i g u r e 36(a) and dx/dV i s shown in f i g u r e x z 36(b). X-ray studies have shown that a stage 3 structure exists at x - 0.08, a stage 2 structure e x i s t s at x - 0.14, and a stage 1 structure e x i s t s above x - 0.27. An i r r e v e r s i b l e change occurs on the f i r s t d i s -charge from the unintercalated material to the stage 3 structure. Rever-s i b l e t r a n s i t i o n s occur between stage 3 and stage 2 and between stage 2 and stage 1. The two r e v e r s i b l e t r a n s i t i o n s appear i n the curves for the f i r s t charge and second discharge. However, the f i r s t discharge only shows a peak at the voltage corresponding to the t r a n s i t i o n from stage 2 to stage 1. I t i s not known why no peaks due to f i r s t order phase tra n -s i t i o n s appear on f i r s t discharge due to the changes from unintercalated NbSe^ to stage 3, or from stage 3 to stage 2. The nature of the change from NbSe2 to stage 2, by way of stage 3, on f i r s t discharge requires further i n v e s t i g a t i o n . I t i s of i n t e r e s t to note that the staging in L i NbSe can be qual-i t a t i v e l y described by the same model, mentioned in Section 4.2, that was developed to describe L i TiS„ (Dahn, Dahn, and Haering 1982). 107 gure 36 - Staging i n L i NbSe 2 > A f t e r Dahn, D.C., and Haering (1982). (36a) Voltage'curve of L i NbSe 2 c e l l DD-36. Shown are (a) f i r s t discharge, (b) f i r s t charge, and (c) second discharge. (36b) Constant current dx/dV against c e l l voltage f o r the same cases as in f i g u r e 36(a). 108 CHAPTER FIVE CONCLUSIONS 5.1 dQ/dV Measurements This thesis has demonstrated the usefulness of dQ/dV measurements in detecting f i r s t order phase t r a n s i t i o n s . As discussed i n Section 1.2, a c e l l ' s voltage as a function of Q or x i s analogous to the equation of state of a gas. dQ/dV i s analogous to the isothermal compressibility of a gas, which diverges during a f i r s t order phase t r a n s i t i o n . dQ/dV i s thus not only s e n s i t i v e to f i n e structure i n a c e l l ' s equation of state, but w i l l peak when a f i r s t order phase t r a n s i t i o n occurs. dQ/dV measurements are s e n s i t i v e probes both of an i n t e r c a l a t i o n system's thermodynamics and of i t s phase diagram. There i s a wide v a r i e t y of i n t e r c a l a t i o n systems f o r which dQ/dV measurements can be u s e f u l . Examples were given i n t h i s thesis f o r the i n t e r c a l a t i o n systems L i VS„, L i MoS9, L i Mo~S,, L i T i S 0 , and L i NbSe„, X Z X Z. X - 5 H X Z X /-as w e ll as the system L i FeS„ ( l i t h i u m against p y r i t e ) . There are many X z. other systems to which dQ/dV measurements can be applied. The measurements for the l i t h i u m against p y r i t e system, which may or may not involve i n t e r -c a l a t i o n , i n d i c a t e that dQ/dV measurements may prove u s e f u l f o r e l e c t r o -chemical systems other than the i n t e r c a l a t i o n systems we have been d i s -cussing, p a r t i c u l a r l y i f more than a sin g l e , straightforward chemical reaction i s involved i n such a system. The experimental techniques d i s -cussed i n t h i s thesis can be applied 'without change to any electrochemical c e l l , and the i n t e r p r e t a t i o n of dQ/dV peaks as signals of f i r s t order phase t r a n s i t i o n s i s independent of the nature of the electrochemical system. 110 5.2 Linear Sweep Voltammetry Since dQ/dV can be measured by the standard chemical technique of l i n e a r sweep voltammetry, the detection of f i r s t order phase t r a n s i t i o n s i n i n t e r c a l a t i o n systems i s immediately possible f o r many researchers, and the necessary equipment i s commercially a v a i l a b l e . As discussed in Chapter 2, once k i n e t i c e f f e c t s due to the c e l l ' s resistance and to s o l i d state d i f f u s i o n are taken into account, the shapes of peaks i n dQ/dV due to f i r s t order phase t r a n s i t i o n s can be predicted. The phase diagram f o r Li^VS2 was obtained by Dahn, J.R., and Haering (1981) from the i n t e r p r e t a -t i o n of l i n e a r sweep voltammograms, and was presented in f i g u r e 13. The phases were i d e n t i f i e d from the x-ray data of Murphy et a l (1977). The voltages of the phase t r a n s i t i o n s were q u a n t i t a t i v e l y determined, as were the ranges i n x of both s i n g l e and two phase regions. Linear sweep voltammetry i s not i d e a l f o r dQ/dV measurements. The apparatus that was used had to perform near the l i m i t s of i t s c a p a b i l -i t i e s before any useful r e s u l t s could be obtained. The r e s o l u t i o n and c l a r i t y of peaks i n dQ/dV were not as good as one would desire. This i s p a r t i c u l a r l y the case since the peaks are not very d i s t i n c t i v e i n shape, having the rounded appearances predicted by theory. Also, although peak heights do. . increase with a compared to single phase regions,. they only do so l i k e a , where a i s the voltage sweep rate. I t was revealed by constant current dQ/dV measurements, described i n Chapter 3, that the poor r e s o l u t i o n i n the l i n e a r sweep voltammetry measurements on Li^VS2 led to the misinterpretation of some single phase capacity as a 3 + IT two phase region. This error i l l u s t r a t e s the u t i l i t y of the higher r e s o l u t i o n constant current dQ/dV technique. I f phase t r a n s i t i o n s are close together i n voltage, the breadth of dQ/dV peaks i n l i n e a r sweep voltammetry may mask some of the t r a n s i t i o n s I l l and make i n t e r p r e t a t i o n d i f f i c u l t . A l i m i t e d amount of information can be obtained from the peak shapes. The slope of the i n i t i a l r i s e of a peak i s 1/R. The general peak shape gives some i n d i c a t i o n of how quickly s o l i d state d i f f u s i o n i s occurring, but only a q u a l i t a t i v e i n d i c a t i o n . In general, since the current, and hence the severity of the d i f f u s i o n gradients, v a r i e s s i g n i f i c a n t l y during a l i n e a r sweep voltammogram, the peak shapes are s u f f i c i e n t l y rounded and broad to mask small e f f e c t s , and an unambiguous i n t e r p r e t a t i o n of peaks i s not always possible. 112 5.3 Constant Current dQ/dV" Constant current dQ/dV i s both simple to measure and clear i n i n t e r -pretation. Measurements require only that a c e l l be charged or discharged at constant current, and that the c e l l voltage be monitored by a voltmeter or analogue to d i g i t a l converter attached to a microcomputer. The apparatus and measurement technique were described i n Chapter 3. Peaks i n constant current dQ/dV due to f i r s t order phase t r a n s i t i o n s are sharp and narrow, p a r t i c u l a r l y when compared with the peaks generated by l i n e a r sweep voltam-metry (see f i g u r e 19). Since i t i s experimentally simple to measure dQ/dV at low currents, so that a c e l l ' s charge or discharge takes considerably longer than the 61 hours that l i n e a r sweep voltammetry i s l i m i t e d to with PAR equipment, r e s o l u t i o n and c l a r i t y can be improved u n t i l there i s no possible ambiguity i n the i n t e r p r e t a t i o n of peaks due to f i r s t order phase t r a n s i t i o n s . The peaks i n constant current dQ/dV are not rounded by d r a s t i c a l l y changing d i f f u s i o n gradients or d i s t o r t e d by a continuously changing IR s h i f t . Unlike the peaks i n l i n e a r sweep voltammetry, the peaks i n constant current dQ/dV against voltage merely have a t a i l added to the i d e a l d e l t a function by d i f f u s i o n , and experience a constant IR s h i f t . The peak height v a r i e s inversely with current, while the peak width v a r i e s l i n e a r l y with current. In an experiment, the current can simply be decreased u n t i l any peaks are s u f f i c i e n t l y t a l l and narrow to be c l e a r l y resolved. I f the c e l l r e s i s t a n c e i s constant, charge and discharge peaks are separated by 2IR. If the resistance d i f f e r s on charge and discharge, the leading edges of the peaks can be plo t t e d as a function of current, and t h e i r p o s i t i o n s extrapolated to zero current to obtain the t r a n s i t i o n voltage. The slope of such an extrapolation i s 1/R. Both R and the t r a n s i t i o n voltage can thus be obtained q u a n t i t a t i v e l y with good accuracy. 113 The peak height i s roughly l i n e a r i n Q / | l | , but the accuracy of the peak height i s l i m i t e d by the voltage r e s o l u t i o n of the apparatus. It i s d i f f i c u l t to obtain more than q u a l i t a t i v e information regarding the r e l a t i v e values of D dp/dV during each phase t r a n s i t i o n from the slope of peak height against Q Q / | I | . Despite t h i s , i t i s s t i l l c l e a r that more and better q u a l i t y information can be obtained from constant current dQ/dV peaks than from l i n e a r sweep voltammetry peaks. When coupled with x-ray d i f f r a c t i o n studies, constant current dQ/dV measurements can be used to q u a n t i t a t i v e l y determine phase diagrams. Peaks i n dQ/dV against Q are c l e a r and quite d i s t i n c t i v e , more so than in l i n e a r sweep voltammetry. These can be used to determine the ranges in x of f i r s t order phase t r a n s i t i o n s to within 0.01. The L i VS 0 i n t e r c a l a t i o n system was extensively examined with constant current dQ/dV. The phase diagram that was previously determined by l i n e a r sweep voltammetry and by the x-ray studies of Murphy et a l (1977), was mostly confirmed by constant current dQ/dV. As mentioned, the 3 + IT two phase region was not found. I t appears l i k e l y that the t r a n s i t i o n between the 3 and IT phases i s second order rather than f i r s t order i n nature. This i s p l a u s i b l e since the 6 phase i s a monoclinic d i s t o r t i o n of the IT structure. Further i n v e s t i g a t i o n of the nature of t h i s t r a n s i t i o n , including confirmation of the x-ray r e s u l t s , i s needed. The t r a n s i t i o n s between the VS^ and a phases and between the a and 3 phases appear to undergo s i g n i f i c a n t hysteresis i n both voltage and x. Some hysteresis i s p o s s i b l e i n the t r a n s i t i o n from IT to 3S, but i t must be le s s than ten percent i n x i f i t occurs. The two t r a n s i t i o n s f o r which hysteresis occurs have a voltage hysteresis of several m i l l i v o l t s and a hysteresis i n x of 15 to 20 percent. This can be q u a l i t a t i v e l y explained by the model of McKinnon (1982), which predicts such hysteresis 114 w i l l occur i f heat i s generated i n the phase conversion process. Further i n v e s t i g a t i o n of t h i s hysteresis i s needed, perhaps using microcalorimetry to study the heat evolution during the t r a n s i t i o n s . The r e s u l t s f o r L i VS„ i l l u s t r a t e the f i n e d e t a i l which can be x 2 studied with constant current dQ/dV. The s e n s i t i v i t y of the technique has been well i l l u s t r a t e d by the discovery of Cu^ ^Mo^S^ i n cathodes prepared from n a t u r a l l y occurring M0S2 by Mulhern (1982). Even though only 0.3 percent of the material was Cu^ ^Mo^S^, the f i r s t order phase t r a n s i t i o n that occurs during i n t e r c a l a t i o n of t h i s material was v i s i b l e . I t would appear that impurities which i n t e r c a l a t e and undergo f i r s t order phase t r a n s i t i o n s can be detected even when present i n very small concentrations. Since dQ/dV i s s e n s i t i v e to f i n e structure i n an i n t e r c a l a t i o n system's equation of state, d i f f e r e n t materials should i n general each have t h e i r own d i s t i n c t i v e dQ/dV pattern. dQ/dV in sing l e phase regions i s a good test f o r any model of the thermodynamics of an i n t e r c a l a t i o n system. dQ/dV's s e n s i t i v i t y to the equation of state makes i t very d i f f i c u l t to duplicate with a model. This has been i l l u s t r a t e d by the model f o r L i x T i S 2 of Dahn (1982). dQ/dV in s i n g l e phase regions also provides some information regarding the k i n e t i c s of a c e l l , since the c e l l resistance at any point can be deter-mined by measuring dQ/dV at a serie s of currents. Other examples have been given i n t h i s thesis of the uses of constant current dQ/dV. Since the voltage of a c e l l i s con t r o l l e d i n l i n e a r sweep voltammetry by the apparatus, "supercooling" e f f e c t s cannot appear. "Super-cooling" i s a very prominent e f f e c t i n constant current dQ/dV. Its "presence indicates that i t i s d i f f i c u l t to nucleate the new phase at the st a r t of a f i r s t order phase t r a n s i t i o n . Also, t r a n s i t i o n s between d i f f e r e n t staged structures have been observed i n L i NbSe„ by Dahn, D.C, and Haering (1982). 115 A t r a n s i t i o n involving a change i n inte r c a l a n t content, but no change i n the host l a t t i c e ' s symmetry, has been observed by Mulhern (1982) i n Li^Mo^S^. There i s no doubt that many more i n t e r e s t i n g examples of constant current dQ/dV applications w i l l a r i s e as the technique i s applied to an ever wider v a r i e t y of materials. Constant current dQ/dV i s a very s e n s i t i v e probe into the thermo-dynamics of i n t e r c a l a t i o n systems. I t i s to be hoped that other types of electrochemical systems can also benefit from constant current dQ/dV measurements. Constant current dQ/dV, p a r t i c u l a r l y when combined with x-ray d i f f r a c t i o n a n a l y s i s , can be used to q u a n t i t a t i v e l y determine phase diagrams. In addition, constant current dQ/dV measurements in singl e phase regions are sure to pose many more challanges to those attempting to model the behavior of electrochemical systems. 1 1 6 BIBLIOGRAPHY Carslaw, H.S., and Jaeger, J.C. ( 1 9 5 9 ) Conduction of Heat i n Solids ( 2 n d ed.), Oxford University Press, London. Dahn, D.C, and Haering, R.R. ( 1 9 8 2 ) S o l i d State Comm., in press. Dahn, J.R. ( 1 9 8 0 ) M.Sc. Thesis, The University of B r i t i s h Columbia, Vancouver, Canada. Dahn, J.R. ( 1 9 8 2 ) Ph.D. Thesis, The University of B r i t i s h Columbia, Vancouver, Canada. Dahn, J.R. , Dahn, D.C, and Haering, R.R. ( 1 9 8 2 ) S o l i d State Comm. _ 4 2 , 1 7 9 . Dahn, J.R., and Haering, R.R. ( 1 9 8 1 ) S o l i d State Ionics _ 2 , 1 9 . Dahn, J.R., Py, M.A., and Haering, R.R. ( 1 9 8 2 ) Can. J . Phys. 60_, 3 0 7 . Folinsbee, J.T. , Jericho, M.H., March, R.H., and T i n d a l l , D.A. ( 1 9 8 1 ) Can. J . Phys. J 5 9 , 1 2 6 7 . Haering, R.R., S t i l e s , J.A.R., and Brandt, K. ( 1 9 8 0 ) United States Patent 4 , 2 2 4 , 3 9 0 . Jacobsen, T., West, K., and Atlung, S. ( 1 9 7 9 ) J . Electrochem. Soc. 1 2 6 , 2 1 6 9 . McKinnon, W.R. ( 1 9 8 0 ) Ph.D. Thesis, The University of B r i t i s h Columbia, Vancouver, Canada. McKinnon, W.R. ( 1 9 8 2 ) J . Less Common Met., in press. Mulhern, P.J. ( 1 9 8 2 ) M.Sc. Thesis, The University of B r i t i s h Columbia, Vancouver, Canada. Murphy, D.W. , Cros, C , DiSalvo, F.J., and Waszczak, J.V. ( 1 9 7 7 ) Inorg. Chem. 1_6, 3 0 2 7 . Thompson, A.H. ( 1 9 7 9 ) J . Electrochem. Soc. 1 2 6 , 6 0 8 and e a r l i e r references contained therein. 

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