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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.Sc,  The U n i v e r s i t y 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  We accept to  this  of P h y s i c s )  t h e s i s as conforming  t h e r e q u i r e d standard  THE UNIVERSITY OF BRITISH COLUMBIA September 1982  (c) G e o f f r e y W i l l i a m Johnson, 1982  In p r e s e n t i n g  t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of  requirements f o r an advanced degree at the  the  University  o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make it  f r e e l y a v a i l a b l e f o r reference  and  study.  I  further  agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may department or by h i s or her  be granted by  the head o f  representatives.  my  It is  understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain  s h a l l not be allowed without my  permission.  Department of The U n i v e r s i t y of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date  DE-6  (3/81)  written  ABSTRACT  First in  order phase t r a n s i t i o n s have been d e t e c t e d  intercalation batteries.  associated  with a f i r s t  electrochemically  The q u a n t i t y dQ/dV peaks when t h e phase f r o n 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 v o l t a g e and Q i s t h e charge flows.  that  The d i v e r g e n c e of dQ/dV i s analogous t o t h e d i v e r g e n c e of the  compressibility  o f a gas when a f i r s t  order phase t r a n s i t i o n o c c u r s .  dQ/dV can be measured by t h e s t a n d a r d e l e c t r o c h e m i c a l t e c h n i q u e of l i n e a r sweep voltammetry, but r e s o l u t i o n  and c l a r i t y a r e v a s t l y  improved by  measuring dQ/dV a t constant c u r r e n t , which m i n i m i z e s t h e d i s t o r t i o n s i n dQ/dV due t o k i n e t i c e f f e c t s . calated  l a y e r compound  The phase diagram of t h e l i t h i u m  has been c o n s t r u c t e d .  Applications  interof dQ/dV  measurements t o phase t r a n s i t i o n s i n o t h e r i n t e r c a l a t i o n systems and t o s i n g l e phase r e g i o n s a r e d i s c u s s e d .  iii  TABLE OF CONTENTS  Page ABSTRACT TABLE OF CONTENTS  xii  LIST OF FIGURES LIST OF SYMBOLS  Vll  ACKNOWLEDGEMENTS  CHAPTER 1 1.1 1.2 1.3 1.4  CHAPTER 2  1  INTRODUCTION I n t e r c a l a t i o n Systems Phase T r a n s i t i o n s and dQ/dV K i n e t i c E f f e c t s i n an E x p e r i m e n t a l System P r e p a r a t i o n of L i ^ V ^ E l e c t r o c h e m i c a l C e l l s  LINEAR SWEEP VOLTAMMETRY  2.1 2.2 2.3 2.4  dQ/dV Measured by L i n e a r Sweep Voltammetry Theory of K i n e t i c E f f e c t s on Phase T r a n s i t i o n s E x p e r i m e n t a l Apparatus and Technique R e s u l t s f o r L i VS„  2.5  E f f e c t i v e n e s s of t h e Technique  CHAPTER 3  X  Z  CONSTANT CURRENT dQ/dV  3.1 3.2 3.3 3.4  dQ/dV Measured a t Constant C u r r e n t Theory o f K i n e t i c E f f e c t s on Phase T r a n s i t i o n s E x p e r i m e n t a l Apparatus and Technique R e s u l t s f o r L i VS„  3.5 3.6  N o n - i d e a l Peak Shapes E f f e c t i v e n e s s of t h e Technique  CHAPTER 4  4.1 4.2 4.3 4.4  X  z  OTHER APPLICATIONS AND EXAMPLES OF CONSTANT CURRENT dQ/dV S e n s i t i v i t y o f t h e Technique A p p l i c a t i o n s t o S i n g l e Phase Regions "Supercooling" Staging  1 8 11 14  21 21 23 37 39 49  51 51 53 62 65 84 90  92 92 97 101 106  iv  Page CHAPTER 5 5.1 5.2 5.3  CONCLUSIONS dQ/dV Measurements L i n e a r Sweep Voltammetry Constant Current dQ/dV  BIBLIOGRAPHY  108 •  108 110 112  116  V  LIST OF FIGURES  Figure  Page  1.  The s t r u c t u r e of MX  2.  Schematic diagram of an i n t e r c a l a t i o n  3.  A t y p i c a l flange c e l l .  4.  Charge and d i s c h a r g e v o l t a g e curves GJ-9.  5.  2  l a y e r compounds.  3 cell.  5 17  f o r t h e L i VS„  cell 19  X  S u r f a c e v o l t a g e and c u r r e n t due t o a phase t r a n s i t i o n , i n l i n e a r sweep voltammetry t h e o r y . Fast s o l i d s t a t e diffusion.  25  Voltammogram and c u r r e n t a g a i n s t charge f o r a phase t r a n s i t i o n , i n theory w i t h f a s t s o l i d s t a t e d i f f u s i o n .  27  7.  Voltammograms w i t h  32  8.  Current  6.  s o l i d s t a t e d i f f u s i o n added t o t h e o r y .  a g a i n s t charge w i t h  solid  state diffusion  added  to t h e o r y . 9.  33  Voltammogram f o r the L _ V S X  2  c e l l GJ-9.  40  10.  Current  11.  Voltammogram f o r a L i ^ V S . c e l l of Dahn, J.R., and H a e r i n g .  42  12.  Current a g a i n s t charge f o r a L i VS„ c e l l of Dahn, J.R., and H a e r i n g .  43  13.  a g a i n s t charge f o r t h e Li^VS,, c e l l GJ-9.  Phase diagram f o r L i VS„. x 2 P a r t i a l phase diagram f o r L i V S  41  46  &  14. 15.  16.  17.  18.  19.  x  2  d e r i v e d from x-ray  data.  T h e o r e t i c a l peak i n constant c u r r e n t dQ/dV a g a i n s t v o l t a g e due t o a phase t r a n s i t i o n .  cell  T h e o r e t i c a l peak i n constant due t o a phase t r a n s i t i o n .  charge  c u r r e n t dQ/dV a g a i n s t  Constant c u r r e n t dQ/dV a g a i n s t c e l l GJ-9, a t 35 yamps.  48  58  59 c e l l v o l t a g e f o r the L i V S  66  Constant c u r r e n t dQ/dV a g a i n s t x and Q f o r t h e L _ V S GJ-9, a t 35 yamps. x  Comparison of l i n e a r sweep voltammetry and c u r r e n t dQ/dV f o r c e l l GJ-9.  2  2  cell 67  constant 68  Constant current dQ/dV against c e l l voltage f o r the L i VS^ c e l l GJ-9, at 140 vamps. Constant current dQ/dV against x and Q f o r the L i GJ-9, at 140 yamps.  cell  P o s i t i o n s of the leading edges of charge and discharge peaks f o r the 2.470 v o 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 t r a n s i t i o n voltage. P o s i t i o n s of the leading edges of peaks f o r the 2.384 v o l t t r a n s i t i o n , s i m i l a r to f i g u r e 22. P o s i t i o n s of the leading edges of peaks f o r the 2.201 v o l t t r a n s i t i o n , s i m i l a r t o f i g u r e 22. Hysteresis i n voltage and x from expected e q u i l i b r i u m values due to energy d i s s i p a t i o n i n a two phase region. Peak height against  QQ/|I|  f o r 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 f r o n t s 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 f r o n t s are nucleated i n 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 f o r the Li MoS2 c e l l PM-30, made with n a t u r a l l y occurring M0S2. x  Expanded view of the high voltage data from f i g u r e 29, showing a peak due to an impurity. Constant current dQ/dV against c e l l voltage f o r Li^Mo^S^. M a t e r i a l i n i t i a l l y Cu, ,Mo S,. 1.4 3 4 Constant current dx/dV against c e l l voltage f o r the L i TiS„ c e l l JD-235. x 2 0  J  Constant current dx/dV against x f o r the L i TiS„ c e l l JD-235. X  Voltage curve of 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, showing "supercooling" e f f e c t . Constant current dQ/dV against c e l l voltage f o r the l i t h i u m against p y r i t e c e l l M-47, showing "supercooling". Staging i n L i NbSe  vii  LIST OF SYMBOLS  A  S u r f a c e a r e a through which i n t e r c a l a t i o n o c c u r s .  a  Crystallographic dp/dV a  axis,  or  (a c o n s t a n t ) i n l i n e a r sweep voltammetry t h e o r y .  a^  V a l u e of c r y s t a l l o g r a p h i c cell.  b  Crystallographic -dp/dV AeR  a x i s a f o r a p a r t i c u l a r hexagonal u n i t  a x i s , or  (a c o n s t a n t ) i n l i n e a r sweep voltammetry t h e o r y .  C  " C a p a c i t a n c e " of a  c  Crystallographic  CQ  V a l u e of c f o r a p a r t i c u l a r hexagonal u n i t  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 c o n s t a n t .  e  Magnitude of the e l e c t r o n i c charge.  F  Helmholtz f r e e energy of t h e cathode  F  a  cell,  axis.  Helmholtz f r e e energy of the anode  F  Helmholtz f r e e energy of a gas.  G  Gibbs f r e e  I  dQ/dt.  I J  P s  cell.  energy.  Current.  Peak c u r r e n t . S u r f a c e number c u r r e n t  density,  L  H a l f w i d t h or r a d i u s of a p a r t i c l e .  M  A t r a n s i t i o n m e t a l atom.  n  Number of i n t e r c a l a n t atoms , or Stage number, as i n a s t a g e n l a y e r  compound.  P  Pressure.  Q  Net p o s i t i v e charge f l o w i n g from the cathode to the anode i n the external c i r c u i t ; net charge of l i t h i u m ions that have been intercalated.  viii  QQ  Width i n Q of a f i r s t  R  Cell  r  Position i n a particle.  f  P o s i t i o n of a phase f r o n t i n a p a r t i c l e .  S  Entropy.  T  Temperature.  t  Time.  t  Time a t which Q  t^  I n i t i a l time.  t^  Time a t which  U  I n t e r n a l energy.  u  f / L . N o r m a l i z e d p o s i t i o n of a phase f r o n t  V  V o l t a g e a t t h e s u r f a c e of a p a r t i c l e , or  o r d e r phase t r a n s i t i o n .  resistance.  n  i s reached i f t = 0 when Q = 0.  i s reached.  in a particle.  C e l l v o l t a g e i n d e r i v a t i v e s such as dx/dV, dQ/dV, o r dt/dV u n l e s s o t h e r w i s e s p e c i f i e d (as i n dp/dV f o r example). V  c  Cell  voltage, Roughly, t h e v o l t a g e change i n a time T ^ .  V  IT: /C (a c o n s t a n t ) .  VQ  E q u i l i b r i u m v o l t a g e of a f i r s t  V^  V o l t a g e a t 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  V o l t a g e a t which a t r a n s i t i o n peak ends on d i s c h a r g e , 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  I n t e r c a l a n t c o n c e n t r a t i o n , as i n L i MX„ •, ' x 2'  order phase t r a n s i t i o n .  or  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 i n t e g r a t i o n ,  a  dV/dt.  Sweep r a t e , or  Phase of t h e L i ^ V S  9  system, o r of the L i M o S v  9  system.  ix  3  S l o p e of a l i n e a r r e l a t i o n s h i p between v o l t a g e and Phase of the L i VS„ X  Crystallograpic  Z.  system, or of the L i MoS„ system, X  z.  or  angle.  Y  D i m e n s i o n a l i t y of a phase f r o n t .  AF  Change i n the  An  Change i n n.  Ap  Change i n i n t e r c a l a n t c o n c e n t r a t i o n a c r o s s a f i r s t transition.  K  Isothermal c o m p r e s s i b i l i t y  u  Chemical p o t e n t i a l of the  cathode,  Chemical - p o t e n t i a l of the  anode.  p  charge, or  t o t a l Helmholtz f r e e energy.  of a  o r d e r phase  gas.  cl  p  Intercalant  concentration.  p^  Intercalant  c o n c e n t r a t i o n at a p a r t i c l e ' s  p^  I n t e r c a l a n t c o n c e n t r a t i o n i n phase 1 near a f i r s t transition.  order phase  p^  I n t e r c a l a n t c o n c e n t r a t i o n i n phase 2 near a f i r s t transition.  order phase  a T  .  H a l f w i d t h of a Gaussian d i s t r i b u t i o n . Variable  of  integration.  2 L $  /D.  D i f f u s i o n time c o n s t a n t .  Probability integral.  surface.  X  ACKNOWLEDGEMENTS  It  i s a g r e a t p l e a s u r e to thank my  s u p e r v i s o r , Rudi H a e r i n g ,  h e l p and encouragement he has g i v e n me. his  f o r the  I have g r e a t l y b e n e f i t t e d from  i n t e r e s t and p h y s i c a l i n s i g h t . My co-workers i n Rudi Haering's  d i s c u s s i o n s and i d e a s .  group have p r o v i d e d many h e l p f u l  In p a r t i c u l a r , I would l i k e t o thank J e f f Dahn,  Doug Dahn, P e t e r Mulhern, and R i c h a r d M a r s o l a i s , each of whom has p r o v i d e d a d d i t i o n a l examples of dQ/dV measurements  for this thesis.  p r e v i o u s work on l i n e a r sweep voltammetry was  invaluable.  J e f f Dahn's I would a l s o  l i k e t o thank f e l l o w l a b members M a r c e l Py, Rod M c M i l l a n , Andre van Schyndel,  R i c k C l a y t o n , and A l e c  Rivers-Bowerman.  The c o n s t r u c t i o n of dQ/dV A n a l y s e r Model F06 by t h e P h y s i c s Department E l e c t r o n i c s Shop, and i t s programming acknowledged.  by M. A. P o t t s , a r e g r a t e f u l l y  F i n a l l y , I would l i k e to thank the N a t u r a l S c i e n c e s and  E n g i n e e r i n g R e s e a r c h C o u n c i l f o r the f i n a n c i a l support over  the past two y e a r s .  they have g i v e n  me  1  CHAPTER ONE  INTRODUCTION  1.1  Intercalation  Systems  In an i n t e r c a l a t i o n system, guest atoms or m o l e c u l e s a r e i n s e r t e d reversibly  i n t o a host s t r u c t u r e ,  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 d u r i n g e i t h e r t h e i n s e r t i o n o r removal of t h e i n t e r calant.  Intercalation  a l l characterized intercalant  has been observed i n a wide v a r i e t y  by open passages i n t h e i r s t r u c t u r e s  can d i f f u s e .  Among  of  through which an  other systems, hydrogen has been i n t e r -  calated  into metals, a v a r i e t y  calated  i n t o g r a p h i t e , and v a r i o u s a l k a l i m e t a l s , p a r t i c u l a r l y  have been i n t e r c a l a t e d  materials,  of atoms and m o l e c u l e s have been i n t e r -  into a variety  A l t h o u g h other types of s t r u c t u r e s ,  of t r a n s i t i o n m e t a l d i c h a l c o g e n i d e s .  such as r u t i l e s ,  t h i s t h e s i s c o n c e n t r a t e s on the d e t e c t i o n t r a n s i t i o n metal dichalcogenides.  lithium,  intercalate  lithium,  of phase t r a n s i t i o n s i n t h e  These m a t e r i a l s a r e denoted by  where M i s a t r a n s i t i o n m e t a l and X i s s u l p h u r , selenium, or t e l l u r i u m . Many MX2 m a t e r i a l s a r e l a y e r compounds. sandwiched between l a y e r s  of c h a l c o g e n s .  L a y e r s o f m e t a l atoms a r e  Adjacent p l a n e s of chalcogens  a r e bound t o g e t h e r only by weak van der Waals f o r c e s ,  so t h e MX2 sandwiches  are  atoms can r e a d i l y  be  e a s i l y pushed a p a r t from each o t h e r . inserted  ticularly calated  Intercalant  i n t o t h e van der Waals gaps between MX2 sandwiches, p a r -  such s m a l l atoms as l i t h i u m .  Note that  i f lithium i s inter-  to a c o n c e n t r a t i o n of x l i t h i u m atoms f o r each M atom, t h e  2  resulting  i n t e r c a l a t i o n compound w i l l be denoted  shows the g e n e r a l form  of an MX  2  L i MX  .  Figure  l a y e r s t r u c t u r e , w h i l e f i g u r e 1(b) shows  the two  p o s s i b l e ways i n which an M atom can be c o o r d i n a t e d by  atoms.  The MX  2  1(a)  c r y s t a l s have hexagonal symmetry w i t h t h e i r  g r a p h i c c-axes p e r p e n d i c u l a r t o the p l a n e of the MX  2  chalcogen  crystallo-  sandwiches.  1(c) shows the s t r u c t u r e of s e v e r a l common l a y e r compounds.  Figure  A l l the  MX^  l a y e r compounds tend t o shear a l o n g t h e i r van der Waals gaps, forming p l a t e l e t s when the m a t e r i a l i s crushed. I n t e r c a l a n t atoms d i f f u s e through a l a y e r compound by hopping, 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  o c t a h e d r a l l y c o o r d i n a t e d by chalcogen atoms f o r each M atom, arid  or  site two  s i t e s t e t r a h e d r a l l y c o o r d i n a t e d by chalcogen atoms f o r each M atom, w i t h t h e o c t a h e d r a l s i t e s no t i c eati'ly l a r g e r than the t e t r a h e d r a l s i t e s . t r o n d i f f r a c t i o n has  shown i n the case of L i TiS» (Dahn 1980) x 2  o c t a h e d r a l s i t e s a r e o c c u p i e d f o r 0 _< x _< 1.  Neu-  t h a t the  I t i s t o be expected  that  they should have a lower c h e m i c a l p o t e n t i a l than the s m a l l e r t e t r a h e d r a l sites. I t has been found t h a t 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 an MX  2  l a y e r compound, and an atom i n a p i e c e of a l k a l i m e t a l  s u f f i c i e n t l y l a r g e t h a t many i n t e r c a l a t i o n systems a r e being as h i g h energy performed  density rechargeable b a t t e r i e s .  as  i s often considered  I n t e r c a l a t i o n can  be  by c h e m i c a l 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  host m a t e r i a l w i t h n - b u t y l l i t h i u m , but an e l e c t r o c h e m i c a l c e l l can constructed  instead.  A h i g h energy  r e c h a r g e a b l e b a t t e r y based  n a t u r a l l y o c c u r r i n g l a y e r compound MoS (Haering et a l 1980) of Burnaby, B.C.,  2  i s being developed  Canada.  i n t e r c a l a t e d with commercially  on  a be  the  lithium  by M o l i Energy L t d .  3  (a) General form  van der Waals gap  (b) Coordination units for MX  2  layer structures  AbC octahedron  AbA trigonal prism  (c)  2H-MoS  2H-NbS  2  lT-TiS  2  2  V 1120  1120  AbA BaB  Figure 1 (a) (b) (c)  AbA CbC  1120  AbC  AbC  - The s t r u c t u r e of MX^ l a y e r compounds. The g e n e r a l form of an MX2 l a y e r compound sandwich. Two p o s s i b l e ways an M atom can be c o o r d i n a t e d by X atoms. The t h r e e most common types of MX2 l a y e r compounds.  A schematic e l e c t r o c h e m i c a l c e l l i n t e r c a l a t e d MX  l a y e r compound.  2  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 g e n e r a l f o r such compounds i t i s  e n e r g e t i c a l l y f a v o r a b l e f o r a l i t h i u m atom to occupy a s i t e i n the compound's van der Waals gaps i n s t e a d of a s i t e i n a sheet When an anode of l i t h i u m m e t a l and a cathode of MX^ e l e c t r o l y t e containing L i  +  +  a r e immersed i n an  i o n s , and a path f o r e l e c t r o n s t o t r a v e l between  the anode and t h e cathode i s p r o v i d e d , sociate into L i  of l i t h i u m m e t a l .  ions and e l e c t r o n s .  l i t h i u m atoms i n t h e anode  dis-  The e l e c t r o n s t r a v e l through the  e x t e r n a l path and a r e c a p a b l e of doing work i n an e x t e r n a l l o a d . e l e c t r o n s combine w i t h L i  +  i o n s at the s u r f a c e of the MX  atoms, which d i f f u s e i n t o the host e r a t e d a t the anode and disappear  structure.  The c e l l  t o form l i t h i u m  2  +  ions a r e gen-  at t h e cathode, charge n e u t r a l i t y i s  m a i n t a i n e d i n t h e s o l u t i o n by the m i g r a t i o n cathode.  Since L i  The  of L i  +  ions from anode to  i s recharged by d r i v i n g a c u r r e n t through the e x t e r n a l  p a t h so t h a t e l e c t r o n s f l o w from cathode t o anode. i n t e r c a l a t i o n process,  and L i  +  T h i s r e v e r s e s the  ions m i g r a t e from cathode to anode, where  they combine w i t h e l e c t r o n s and p l a t e onto the l i t h i u m m e t a l . The cathode n o r m a l l y break up d u r i n g change o c c u r s  c o n s i s t s of a powder.  i n t e r c a l a t i o n , s i n c e a l t h o u g h no s i g n i f i c a n t s t r u c t u r a l  i n the host  }  the van der Waals gaps change i n s i z e ,  the c - a x i s t o v a r y by as much as t e n percent I n t e r c a l a n t content, straining  Large s i n g l e c r y s t a l s  the host  during  intercalation.  and hence t h e c - a x i s , v a r i e s from p o i n t t o p o i n t ,  s u f f i c i e n t l y t o break i t a p a r t .  I f single crystals  must be used i n an experiment i n t e r c a l a t i o n must be allowed very  causing  t o proceed  slowly. The e l e c t r o l y t e i s a p o l a r o r g a n i c  s o l v e n t which r e a c t s  minimally  w i t h l i t h i u m and the o t h e r c e l l components, w i t h a l i t h i u m s a l t d i s s o l v e d i n i t such as L i C 1 0  A  , L i B r , or L i A s F ^ .  The s o l v e n t must be p o l a r t o  5  External  anion from Li salt Li  o  o  o  Li + o  anode  o  Li  id-  eation  N  metal  load  o  Li MX x  o  o  1M Li salt  in P C J  F i g u r e 2 - Schematic diagram of an intercalation c e l l .  2  cathode  6  d i s s o l v e and  solvate  an  ionic lithium salt.  d i s c u s s e d i n t h i s t h e s i s used the  solvent  a l s o c a l l e d p r o p y l e n e carbonate or PC, solved  to a one  on  c  with either LiClO^  c e l l s i n g e n e r a l have a c o m p l i c a t e d dependence of  intercalant  content x. .  The  fine structure  of t h i s dependence  s i n c e t h i s i s the  against  inverse  of the  a charge Q e q u a l to the i n t o the  cathode.  the  Chapter 2) s h a l l be  s l o p e of V"  charge on  the L i  the  c e l l voltage,  s t a n d a r d c h e m i c a l t e c h n i q u e of  or more c l e a r l y by  primarily  plotted  c  may  constant current  concerned w i t h the  Section  cell  that  voltage, x.  Let  us  have been i n t e r -  to x.  The  quantity  be measured experimen-  l i n e a r sweep voltammetry dQ/dV (see  e f f e c t of f i r s t  s i t i o n s on dQ/dV, u s i n g l i t h i u m i n t e r c a l a t e d VS measurements.  ions  +  Q i s d i r e c t l y proportional  dQ/dV, where V a g a i n w i l l be t a l l y by  dis-  PC.  the  calated  carbonate,  or L i A s F ^  should be v i s i b l e i n dx/dV, where V i n dx/dV w i l l be  define  cells  1 ,2-propanediol c y c l i c  molar c o n c e n t r a t i o n i n the  Intercalation voltage V  A l l electrochemical  2  (see  Chapter 3 ) .  order phase  We  tran-  f o r i l l u s t r a t i o n of dQ/dV  dQ/dV i s a l s o of i n t e r e s t i n s i n g l e phase r e g i o n s  (see  4.2).  The  v o l t a g e of  potential y  an i n t e r c a l a t i o n c e l l can  of the anode and  the  be r e l a t e d to the  c h e m i c a l p o t e n t i a l y of the  chemical  cathode at  cl  its  surface,  where i n t e r c a l a t i o n i s o c c u r r i n g .  f r e e energy F  and  the  cathode has  I f the anode has  a Helmholtz f r e e energy F,  a Helmholtz  then i f  the  3.  number n of  i n t e r c a l a n t atoms changes by An  energy changes AF  = =  The  8F  3n  the t o t a l Helmholtz  free  by: 9 F  a^  3n  An  (y - y ) An a  electrons  (1.1)  do work t o t a l l i n g  e l e c t r o n i c charge and  V  i s the  eAnV^, where e i s the magnitude of c e l l voltage.  The  change i n f r e e  the energy  7  i s the n e g a t i v e of the work done, so ( 1 . 1 )  V  c  = l ( y  a  - u )  (1.2)  s h a l l assume t h a t y  We  becomes:  of the l i t h i u m metal  remains c o n s t a n t , and  hence  a V"  c  i s p r o p o r t i o n a l to - u .  We  s h a l l show how  this relationship is distorted  by k i n e t i c e f f e c t s . Note t h a t the Gibbs' f r e e energy G must be used i n s t e a d of Helmholtz f r e e energy F i f p r e s s u r e s above a few  the  atmospheres are p r e s e n t .  S i n c e G = F + Pv, where P i s the p r e s s u r e and v i s the volume, G i s c l o s e to F f o r t y p i c a l experimental p r e s s u r e s a r e used.  p r e s s u r e s , but can be v e r y d i f f e r e n t i f h i g h  8  1.2  Phase T r a n s i t i o n s Strictly  and  dQ/dV  s p e a k i n g , i n an  i n t e r c a l a t i o n system i n s e r t i o n o c c u r s  d i f f u s i o n of i n t e r c a l a n t  i n t o a c r y s t a l without any  changes o c c u r r i n g  host.  i n the  c a l a t i o n system" i s a p p l i e d  range of ranges of c o m p o s i t i o n and calation, f i r s t  concentrations.  are  It i s also possible  different intercalant  c r y s t a l e n t e r s the  that  intercalant  c o n c e n t r a t i o n s of the  c o n v e r t i n g the  the  In a d d i t i o n  f a v o r e d at d i f f e r e n t the  to  inter-  two  by  be  minimized  same host c r y s t a l symIn e i t h e r c a s e , a  c o n c e n t r a t i o n at the  new  s u r f a c e of  u n f a v o r a b l e range between the  phases i n the  s u r f a c e of the  two  phase.  inter-  phase r e g i o n .  c r y s t a l to the  e n t i r e c r y s t a l to the new  i s driven  intercalant  f r e e energy may  concentrations.  energetically  f r o n t w i l l move from the  front  temperature.  energetically  phase w i l l n u c l e a t e when the  calant  i n t e r c a l a t e s over some  phase r e g i o n where both phases have the  metry but  the  system that  term " i n t e r -  o r d e r 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  by a two  significant structural  In g e n e r a l , however, the  to any  by  The  crystal's  A phase center,  motion of the  phase  a c o n c e n t r a t i o n g r a d i e n t between the phase f r o n t  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  solid  and  state  diffusion. Four d i f f e r e n t c r y s t a l s t r u c t u r e s intercalated two  l a y e r compound  phase r e g i o n s and  of i n t e r c a l a n t  S e c t i o n 1.4 The silver  et a l 1981). probe.  s i n g l e phase r e g i o n s were observed u s i n g a  i n Chapters 2 and  The  intercalated  in  3. through a c r y s t a l has  i n t o the  been observed when  l a y e r compound NbSe2 ( F o l i n s b e e  S i l v e r c o n c e n t r a t i o n s were measured w i t h an sharp f r o n t has  Both  variety  L i VS„ w i l l ' b e d i s c u s s e d i n d e t a i l x 2  motion of a sharp f r o n t  atoms a r e  lithium  i n f i v e phases (Murphy et a l 1977).  concentrations.  and  have been observed i n the  been a t t r i b u t e d  by  the  electron  authors to a  beam  first  9  order phase t r a n s i t i o n , a l t h o u g h coefficient  could conceivably  S i n c e t h e r e i s evidence  a c o n c e n t r a t i o n dependent  diffusion  cause such a f r o n t . 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, l e t us c o n s i d e r what e f f e c t s i t i o n has on t h e v o l t a g e o f an i n t e r c a l a t i o n c e l l .  occurring  such a phase I f we i g n o r e  e f f e c t s f o r t h e moment, t h e s u r f a c e c o n c e n t r a t i o n o f i n t e r c a l a n t remain f i x e d w h i l e t h e phase f r o n t moves through t h e c r y s t a l . the chemical  potential  o f the cathode w i l l remain constant  phase r e g i o n , and thus the c e l l v o l t a g e w i l l a l s o the phase f r o n t moves.  trankinetic  should  As a r e s u l t  i n t h e two  remain constant  T h i s means t h a t a p l a t e a u i n the c e l l  while  voltage  measured a g a i n s t x o r Q w i l l f r e q u e n t l y s i g n a l t h e p r e s e n c e of a f i r s t o r d e r phase t r a n s i t i o n . plotted  a g a i n s t x or Q w i l l c l e a r l y peak when a f i r s t  s i t i o n occurs. first  The i n v e r s e of t h e s l o p e of a v o l t a g e  o r d e r phase  tran-  Thus peaks i n dx/dV or dQ/dV can s i g n a l t h e p r e s e n c e of  o r d e r phase t r a n s i t i o n s .  chemically.  curve  dx/dV o r dQ/dV can be measured  I t i s t h e o b j e c t i v e of t h i s t h e s i s  electro-  t o understand and i n t e r -  p r e t such dQ/dV measurements and t h e i r peaks. To understand why dx/dV or dQ/dV should be u s e f u l order phase t r a n s i t i o n s ,  i t i s helpful  i n detecting  first  t o draw an analogy between the  thermodynamics o f an i n t e r c a l a t i o n c e l l and t h e thermodynamics of a gas. Recall  that the pressure  i s s u f f i c i e n t l y low t h a t t h e Helmholtz  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  free  system a r e :  F(n,v,T) = U - TS  (1.3)  dF = -SdT + ydn + Pdv  (1.4)  where U i s t h e i n t e r n a l energy, T i s t h e temperature, S i s t h e entropy, P i s the p r e s s u r e , v i s the volume, and n i s the number of i n t e r c a l a t e d atoms o r m o l e c u l e s . dF = -SdT + ydn  F o r a f i x e d volume F = F(n,T) and (1.4) becomes: (1.5)  10  A gas w i t h a f i x e d number of p a r t i c l e s n has a f r e e 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  noting that  the f r e e e n e r g i e s o f t h e two a r e f o r m a l l y i d e n t i c a l i f t h e v  of the gas i s s u b s t i t u t e d  by t h e n of t h e i n t e r c a l a t i o n  P o f the gas i s s u b s t i t u t e d  system and i f t h e  by -y o f t h e i n t e r c a l a t i o n system, l e a v i n g the  o t h e r thermodynamic q u a n t i t i e s be  system by  unchanged.  The thermodynamics of a gas can  transformed by t h i s analogy i n t o t h e thermodynamics o f an i n t e r c a l a t i o n  system. The  e q u a t i o n of s t a t e  equation of state  f o r a gas i s P = P ( v , T ) .  of an i n t e r c a l a t i o n  By analogy, t h e  system i s u = y ( n , T ) .  S i n c e when  k i n e t i c e f f e c t s a r e ignored the c e l l voltage i s proportional  to - u , the  c e l l v o l t a g e as a f u n c t i o n  of n and T d e s c r i b e s t h e e q u a t i o n o f s t a t e of  the  dQ/dV i s c l e a r l y s e n s i t i v e  i n t e r c a l a t i o n system.  intercalation  T  1 dvT v dP  i s g i v e n by: (1.7)  T  T h i s i s known t o d i v e r g e when a f i r s t and  i n an  system's e q u a t i o n of s t a t e .  In a gas t h e i s o t h e r m a l c o m p r e s s i b i l i t y K  to s t r u c t u r e  order t r a n s i t i o n o c c u r s .  By analogy  (1.2) t h e q u a n t i t y i n (1.7) becomes:  n du  ex dV  (1.8) T  At a f i x e d temperature dx/dV o r dQ/dV i s thus analogous t o the c o m p r e s s i b i l ity  of a gas and should s i m i l a r l y d i v e r g e a t f i r s t  See  Dahn (1982) f o r other a p p l i c a t i o n s  o r d e r phase  of t h i s analogy.  transitions.  11  1.3  Kinetic  Effects  i n an E x p e r i m e n t a l System  When charge f l o w s 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 measured q u a n t i t i e s the  from t h e i r e q u i l i b r i u m  distort  thermodynamic v a l u e s .  Once  s o u r c e s and n a t u r e of t h e k i n e t i c e f f e c t s i n an e x p e r i m e n t a l c e l l a r e  understood, t h e b e h a v i o r of dQ/dV i n an e x p e r i m e n t a l s i t u a t i o n predicted.  We s h a l l assume that  can be  t h e powder cathodes a r e 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 c o n t a c t w i t h t h e cathode subs t r a t e and i n good c o n t a c t w i t h t h e c e l l ' s e l e c t r o l y t e . can  then be d e s c r i b e d by s o l i d  and  by t h e e l e c t r i c a l r e s i s t a n c e Solid  state  diffusion effects  Kinetic  i n the c r y s t a l s  of the c e l l .  s t a t e d i f f u s i o n e f f e c t s a r i s e s i n c e t h e cathode p a r t i c l e s a r e  of f i n i t e s i z e .  Intercalant  atoms o r m o l e c u l e s a r e i n t r o d u c e d i n t o t h e '  c r y s t a l at t h e c r y s t a l ' s s u r f a c e , and a c o n c e n t r a t i o n g r a d i e n t as  effects  the i n t e r c a l a n t  diffuses  arises  towards t h e c e n t e r of t h e p a r t i c l e .  As mentioned, layer-compounds tend t o form p l a t e l e t s when powdered, w i t h i n t e r c a l a t i o n o c c u r r i n g through':the edges of t h e p l a t e l e t i n t o t h e van  der Waals gaps.  Here d i f f u s i o n has two d i m e n s i o n a l symmetry.  n u c l e a t e d , a phase f r o n t the  Once  can be expected t o become c y l i n d r i c a l t o minimize  s u r f a c e t e n s i o n at t h e two phase i n t e r f a c e .  Diffusion  i n layer  com-  pounds w i l l t h e r e f o r e be assumed t o have c y l i n d r i c a l symmetry when doing theoretical say,  calculations.  A m a t e r i a l could instead  channels i n t o which i n t e r c a l a n t  Phase f r o n t s  diffuses  have a s t r u c t u r e o f ,  from a l l c r y s t a l f a c e s .  and d i f f u s i o n i n such cases w i l l be assumed t o have  spherical  symmetry. Intercalation In  can c o n c e i v a b l y o c c u r only from one edge of a c r y s t a l .  such a s i t u a t i o n phase f r o n t s  sional  symmetry.  and d i f f u s i o n would have a one dimen-  The phase f r o n t would be a p l a n e .  While t h i s  configura-  t i o n can occur i n experiments u s i n g s i n g l e c r y s t a l s , i t does n o t a r i s e  12  when a powder cathode i s used, and w i l l be c o n s i d e r e d i n l e s s d e t a i l the two and t h r e e d i m e n s i o n a l The  cases.  diffusion coefficient  very d i f f i c u l t  than  f o r intercalant  t o measure e x p e r i m e n t a l l y .  i n a host c r y s t a l has proved  When t h e r e i s a d i s t r i b u t i o n of  p a r t i c l e s i z e s and perhaps, as f o r l a y e r compounds, i n t e r c a l a t i o n o n l y o c c u r s over p a r t of the s u r f a c e a r e a of each p a r t i c l e , i t can be v e r y difficult  t o determine t h e area through which d i f f u s i o n  I t should a l s o be noted  i s occurring..  t h a t i f a l a y e r compound cathode i s p r e s s e d , the  p a r t i c l e s tend t o a l i g n t h e i r c-axes normal t o t h e cathode's s u r f a c e , mostly  exposing  f a c e s through which no i n t e r c a l a t i o n o c c u r s .  then o c c u r s mainly  i n t h e pores  which d i f f u s i o n o c c u r s i s almost  of t h e cathode,  and t h e a r e a  i m p o s s i b l e t o measure.  measurement of t h e d i f f u s i o n c o e f f i c i e n t  Intercalation through  An a c c u r a t e  r e q u i r e s a c c u r a t e knowledge of  the s i z e and shape of t h e cathode p a r t i c l e s .  S i n g l e c r y s t a l measurements  can be made, but a s i n g 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 p o s s i b l e (see Chapter  3) t o deduce some i n f o r m a t i o n  e l e c t r o c h e m i c a l l y about t h e r e l a t i v e v a l u e s of t h e d i f f u s i o n i n t h e v a r i o u s phases of an i n t e r c a l a t i o n An  coefficients  system.  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 r e s i s t a n c e t o which t h e r e a r e  several contributions.  These a r e t h o r o u g h l y d i s c u s s e d by McKinnon  (1980).  There a r e ohmic c o n t r i b u t i o n s t o t h e r e s i s t a n c e from t h e e l e c t r i c a l c o n t a c t to t h e anode and the e l e c t r i c a l c o n t a c t t o t h e powder cathode. p r e s s u r e must be maintained  on t h e c e l l t o ensure good e l e c t r i c a l c o n t a c t  w i t h each p a r t i c l e i n t h e cathode. some ohmic r e s i s t a n c e .  Sufficient  The p a r t i c l e s themselves may c o n t r i b u t e  T r a n s p o r t a c r o s s the i n t e r f a c e s of t h e e l e c t r o d e s  w i t h the e l e c t r o l y t e w i l l c o n t r i b u t e t o the r e s i s t a n c e , s i n c e t h e r e w i l l be a c t i v a t i o n e n e r g i e s a s s o c i a t e d w i t h each e l e c t r o d e r e a c t i o n t h a t must be overcome.  There may a l s o be a c o n t r i b u t i o n t o t h e r e s i s t a n c e 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 the  anode to the cathode.  can  be d e p l e t e d of i n t e r c a l a n t The r e s i s t a n c e  t i o n s that  intercalant  i o n s from  e l e c t r o l y t e i n the pores of the cathode  i o n s and c o n t r i b u t e  to the r e s i s t a n c e .  of an . i n t e r c a l a t i o n c e l l m o s t l y a r i s e s from  a r e u n l i k e l y t o v a r y r a p i d l y w i t h the i n t e r c a l a n t  i n t h e cathode. resistance  Also,  to t r a n s p o r t  I f the c u r r e n t  concentration  i s not changed r a p i d l y or d r a s t i c a l l y , the  should remain r o u g h l y c o n s t a n t .  a constant i n t h e o r e t i c a l  contribu-  calculations.  The r e s i s t a n c e  w i l l be assumed  14  1.4  P r e p a r a t i o n of L i VS„ E l e c t r o c h e m i c a l C e l l s x 2 As mentioned i n S e c t i o n 1.2,  L i VS„ has s e v e r a l phases. x 2  Murphy et a l (1977) have shown t h a t  As a r e s u l t , t h i s m a t e r i a l was  t e s t of the a b i l i t y of dQ/dV measurements to d e t e c t f i r s t  used as a o r d e r phase  transitions. It  is difficult  t o prepare t h e l a y e r compound V S  c o n s t i t u e n t elements, complicated. et a l f i r s t  2  d i r e c t l y from i t s  s i n c e the vanadium - s u l p h u r phase diagram  L i ^ V S ^ w i t h x-1  can be prepared -.'.relatively e a s i l y .  r e p o r t e d the e x i s t e n c e , of L i VS„. x 2  pared by h e a t i n g L i C 0 2  and V^O^  at 500°C f o r 10 hours.  The product was  2  i s quite Murphy  L i VS. w i t h x-1 was x 2  pre-  i n a carbon c r u c i b l e w i t h an H^S powdered and then heated  atmosphere f o r 24 hours, a f t e r which i t was  slowly cooled.  atmosphere i n an Yi^S  Murphy et a l  prepared Li^VS.^ f o r a dozen d i f f e r e n t v a l u e s of x by c h e m i c a l l y removing intercalated lithium. L i V S was p l a c e d i n a c e t o n i t r i l e and t r e a t e d w i t h i o d i n e , o b t a i n i n g L i VS„ from the r e a c t i o n : ° x 2 2  LiVS„ + % ( 1 - x) I_ -»- L i VS„ + (1 - x) L i l I 2 x 2 The Li^VS,^ o b t a i n e d by t h i s proceedure was d i f f r a c t i o n t o determine  • (1.9)  s t u d i e d by x-ray and  i t s s t r u c t u r e and phase diagram.  neutron  R e s u l t s of t h i s  study w i l l be p r e s e n t e d as needed. I t has been found t h a t L i VS„ w i t h x ^ l can be prepared from a x 2 s t o i c h i o m e t r i c m i x t u r e of the elements q u a r t z tube  (Dahn, J.R.,  and H a e r i n g  p a t t e r n was  made of t h e i r p r o d u c t .  a IT p o l y t y p e and a hexagonal and  c = 6.16A*.  a t h i g h temperature  1981).  i n an  evacuated  An x-ray powder d i f f r a c t i o n  T h i s r e v e a l e d a l a y e r compound w i t h  u n i t c e l l w i t h l a t t i c e parameters  The c r y s t a l symmetry i s i n agreement, and the  a = 3. 35A> lattice  parameters a r e i n rough agreement, w i t h the x-ray data of Murphy et a l f o r L i _ V S „ which g i v e s a = 3.382& and c = 6.1 50&. U. o_> 2 o r  L i * V S . w i t h x-1 was x 2  made u s i n g the l a t t e r of the two  techniques,  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 , g r a m s was p r e p a r e d  i n an a r g o n f i l l e d  atmospheric moisture,  and vanadium w e i g h i n g  glove bag,  and p l a c e d i n a q u a r t z  since I ^ S  3.11  reacts  with  tube w i t h a volume of  roughly  3 20 cm .  T h e t u b e was e v a c u a t e d  end s e a l e d w i t h a b l o w t o r c h .  T h e t u b e was p l a c e d  furnace  and heated  at  temperature for nine hours,  this  to  of argon w i t h a d i f f u s i o n  room t e m p e r a t u r e . inner  surface  of the  d e r e d much o f t h e  quartz  tube opaque.  away f r o m t h e w a l l s c o n s i s t e d i n t o a powder.  The m a t e r i a l  i n the  and  on t h e  order  of  crushed harder  i n t o an a r g o n f i l l e d  Vacuum  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  1 ppm, w i t h s l i g h t l y h i g h e r " ^ n i t r o g e n  ren-  furthest  readily  grayer  The  that  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 tions  to  glove bag.  t u b e t h a t was  from the  zone  left  in air  in a reaction  of b l a c k c r y s t a l l i t e s that  t u b e w a l l s and t r a n s f e r r e d  Atmospheres g l o v e box.  cool  i n an a r g o n f i l l e d  T h i s m a t e r i a l was s e p a r a t e d  m a t e r i a l near the  T h e t u b e was  and t h e n a l l o w e d t o  t u b e had p a r t i c i p a t e d  its  i n a Lindberg three  700°C over a t h r e e hour p e r i o d .  T h e t u b e was o p e n e d  pump a n d  argon  concentra-'  concentrations.  T h e L i 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 P o w d e r x  Diffractometer is  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 .  s e n s i t i v e to atmospheric  moisture,  s o i t was x - r a y e d i n s i d e o f a  n o r m a l l y used to hold a complete e l e c t r o c h e m i c a l c e l l diffraction  (Dahn, P y , and H a e r i n g 1 9 8 2 ) .  l i u m window t o a l l o w x - r a y s not  due t o t h e  hexagonal u n i t + 0.001)  A*.  to enter the  for  in situ  The c a s e i n c o r p o r a t e s case.  cell  a  A l l strong Bragg  parameters a = (3.345 + 0.001)  The c r y s t a l s y m m e t r y i s  i n agreement, and t h e x - r a y data  which- g i v e s a = 3 . 3 4 5 A and c = 6 . 1 5 2 A . IT s t r u c t u r e  A and c =  In a d d i t i o n to  with  (6.163 para-  a l f o r L i ^ -j^2  the peaks  t h e r e w e r e a f e w weak a n d b r o a d p e a k s  beryl-  peaks  lattice  o f Murphy et  case  x-ray  b e r y l l i u m w i n d o w 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  meters i n rough agreement, w i t h the  to the  The m a t e r i a l  indexed  that would  appear  16  to  be due t o t h e presence  quartz tube.  of i m p u r i t i e s formed i n t h e r e a c t i o n w i t h the  The i n t e n s i t i e s of t h e Bragg peaks were d i s t o r t e d by p r e -  ferred orientation.  The p l a t e l e t s making up t h e powder tended  t h e i r c-axes p e r p e n d i c u l a r t o the p l a n e of t h e sample.  t o have  The i n t e n s i t i e s  a r e not known a c c u r a t e l y enough t o d i s t i n g u i s h between o c t a h e d r a l and t r i g o n a l p r i s m a t i c c o o r d i n a t i o n of t h e vanadium atoms by t h e s u l p h u r atoms. E l e c t r o c h e m i c a l c e l l s were prepared  typical cell  from t h e L i VS„ m a t e r i a l . x 2  d e s i g n i s shown i n f i g u r e 3.  c o n t a c t s t o t h e cathode  Two s t e e l f l a n g e s a c t as t h e  and anode of t h e c e l l . The f l a n g e s a r e 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 v i t o n rubber O - r i n g gasket the i n t e r i o r o f t h e c e l l an argon  from t h e atmosphere.  f i l l e d g l o v e box.  that also seals  Assembly i s performed i n  F l a n g e c e l l s w i t h p o l y p r o p y l e n e f l a n g e s and  neoprene O-rings were a l s o used. cathode  A  E l e c t r i c a l c o n t a c t w i t h t h e anode and  i s made by c e n t r a l b r a s s p o s t s e l e c t r o p l a t e d w i t h 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 f l a n g e c e l l  compound, i n t h e form  i n g e n e r a l c o n s i s t s of an i n t e r c a l a t i o n  of a powder, mixed w i t h PC o r p r o p y l e n e g l y c o l and  spread onto a n i c k e l or aluminum s u b s t r a t e . off  t h e o r g a n i c m a t e r i a l , l e a v i n g the cathode  s u f f i c i e n t l y t o a l l o w h a n d l i n g and weighing. a mixture cathode box.  The cathode  i s baked t o d r i v e  s t i c k i n g to the substrate Since L i ^ V ^ i s a i r s e n s i t i v e ,  of L i ^ V ^ and PC was spread on n i c k e l i n s i d e a g l o v e box. The  was then s e a l e d i n a s t e e l box, which was removed from t h e g l o v e  The s t e e l box was evacuated  w i t h a mechanical  pump and heated  ina  H o t p o i n t oven t o a nominal  120°C u n t i l t h e p r e s s u r e dropped t o a steady  v a l u e around 20 m i l l i t o r r .  The s t e e l box was taken back i n t o t h e g l o v e  box  and t h e cathode  used t o c o n s t r u c t a c e l l .  The anode of a f l a n g e c e l l and  i s a piece of l i t h i u m metal, cut t o s i z e  scraped w i t h a s c a l p e l t o remove any s u r f a c e l a y e r o f , f o r example,  Figure 3 - A t y p i c a l flange  cell.  18  l i t h i u m oxide.  The  s e p a r a t o r s used were p i e c e s of C e l g a r d #2500 m i c r o -  porous p o l y p r o p y l e n e f i l m wetted w i t h e l e c t r o l y t e under p r e s s u r e . 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  i m p u r i t i e s , p a r t i c u l a r l y propylene order of 40 ppm  and  S t e e l Agri-Chemicals  10 ppm  to reduce water and  The other  g l y c o l , below c o n c e n t r a t i o n s on  respectively.  The  the  l i t h i u m s a l t used was  L i A s F , i n 1M c o n c e n t r a t i o n . 6  PC  U.  S.  Vacuum d r i e d L i C l O . has 4  a l s o been used. E l e c t r o c h e m i c a l c e l l s made w i t h L l ^ f t ^ e d l y at v a r i o u s c o n s t a n t c u r r e n t s . volts. at  The  18th c y c l e of the c e l l  charged  and d i s c h a r g e d  C e l l s were c y c l e d between 1.8 GJ-9  i s shown i n f i g u r e 4.  repeatand  This  2.8 was  a low c o n s t a n t c u r r e n t of 70 .yamps, r e q u i r i n g 57 hours f o r a charge  or discharge. that  N o t i c e the c l e a r p l a t e a u s near 2.38  such p l a t e a u s a r e to be expected  occur.  The  2.46  by weight.  to a change i n  I t i s b e l i e v e d t h a t a l t h o u g h a p o r t i o n ' o f the  of i m p u r i t i e s i n the cathode m a t e r i a l , s i n c e s i m i l a r  i n x were o b t a i n e d f o r o t h e r c e l l s .  to  changes  T h i s i s i n agreement w i t h Dahn,  and Haering  (1981) who  essentially  i d e n t i c a l v o l t a g e behavior  J.R.,  a c h i e v e d a change i n x d u r i n g c y c l i n g of 0.95  change i n x between 1.8  to t h a t shown i n f i g u r e 4.  v o l t s and->2.8 v o l t s has been n o r m a l i z e d  t h i s i n mind.  C e l l GJ-9  24th c y c l e and  a f u r t h e r ten per cent by the 38th c y c l e .  first  Recall  not be i n good e l e c t r i c a l c o n t a c t , t h i s i s p r i m a r i l y due  the presence  c y c l i n g was  volts.  o r d e r phase t r a n s i t i o n s  l e n g t h of the charge or d i s c h a r g e corresponds  x of about 0.7 cathode may  if first  and  with The  to 1 w i t h  l o s t about ten per cent of i t s c a p a c i t y by  being done to o b t a i n the r e s u l t s presented  the  Sinceclow  i n Chapter  current  3,  the  40 c y c l e s took p l a c e over a p e r i o d of t h r e e and a h a l f months. The  first  charge of c e l l GJ-9  took 0.70  the f i r s t d i s c h a r g e , c y c l i n g a t constant the cathode m a t e r i a l was  initially  L i  n 7  + 0.03  current.  of the time taken This i n d i c a t e s that  V S , i n agreement w i t h the 9  by  x-ray  19  Q  (coulombs)  F i g u r e 4 - V o l t a g e curves f o r the 18th charge ( s o l i d l i n e ) and d i s c h a r g e (dashed l i n e ) a t 70 yamps f o r L i VS„ c e l l GJ-9.  20  data.  S i m i l a r l y , Dahn, J.R.,  took 0.85  and Haering  of the time taken by the f i r s t  indicating  t h a t the f i r s t  initially  L i ^ 85^2'  '*"  n  data.  c y c l e s r e v e r s i b l y i n the range 1 <_ x <^ 2 i n a d d i t i o n to  range 0 _< x _< 1.  The  c e l l v o l t a g e s t a y s on a p l a t e a u at about 1.0  on d i s c h a r g e , and a p l a t e a u at about 1.3  v o l t s on charge,  the volts  probably i n -  d i c a t i n g t h a t x changes between 1 and 2 by a r e v e r s i b l e f i r s t transition.  charge  discharge f o r t h e i r m a t e r i a l ,  t h a t t h e i r cathode m a t e r i a l was  agreement w i t h the x-ray Li^V^  found  o r d e r phase  F u r t h e r d i s c u s s i o n s of L i VS„ w i l l be r e s t r i c t e d t o the x 2 range 0 <_ x _< 1.  21  CHAPTER  TWO  LINEAR SWEEP VOLTAMMETRY  2.1  dQ/dV Measured by L i n e a r Sweep Voltammetry The f i r s t measurements of dQ/dV f o r a L i MX„ e l e c t r o c h e m i c a l c e l l x 2  were performed  by Thompson (1979).  chemical c e l l s ,  which had  cathodes  Thompson stepped the v o l t a g e of of the l a y e r compound T1S2,  the charge which flowed u n t i l the c e l l reached  equilibrium.  electro-  and measured This  technique  r e q u i r e s c a r e f u l achievement of e q u i l i b r i u m a f t e r each s t e p , which can become d i f f i c u l t  and time consuming when h i g h r e s o l u t i o n i s d e s i r e d .  techniques presented  i n Chapters 2 and  3 achieve considerable experimental  s i m p l i c i t y a t the expense of having d i s t o r t i o n s due Once such d i s t o r t i o n s a r e understood,  The  to k i n e t i c  dQ/dV can be c l e a r l y  effects.  interpreted.  L i n e a r sweep voltammetry i s a t e c h n i q u e i n which the v o l t a g e of a cell  i s swept a t a c o n s t a n t r a t e a = dV/dt, where t i s the time and V i n  dV/dt w i l l  be the c e l l v o l t a g e .  r e s u l t of the changing  The c u r r e n t I = dQ/dt which f l o w s as a  v o l t a g e i s measured.  Note t h a t because of the  d e f i n i t i o n of Q the c u r r e n t i s d e f i n e d as the f l o w of p o s i t i v e charge the cathode  t o the anode.  from  I f Q decreases m o n o t o n i c a l l y w i t h v o l t a g e , as i s  n o r m a l l y the case, I has the o p p o s i t e s i g n to a. T h i s t e c h n i q u e i s of i n t e r e s t T 1  = dfi = dV dO, _ - dt  dt  dv "  a  s i n c e i f we  ignore k i n e t i c  effects:  dQ  ,  dv  u  S i n c e a i s c o n s t a n t , the c u r r e n t i s i d e a l l y p r o p o r t i o n a l t o dQ/dV.  .  '  i ;  The  u s e f u l n e s s of l i n e a r sweep voltammetry has been noted p r e v i o u s l y (Jacobsen et a l 1979, McKinnon 1980). and  T h i s c h a p t e r p a r a l l e l s and confirms the t h e o r y  e x p e r i m e n t a l r e s u l t s presented by Dahn, J.R., and Haering  (1981).  L i n e a r sweep voltammetry w i l l be compared and c o n t r a s t e d i n Chapter 3 w i t h the new t e c h n i q u e of constant c u r r e n t dQ/dV.  23  2.2  Theory of K i n e t i c E f f e c t s on Phase T r a n s i t i o n s We s h a l l now model how the k i n e t i c e f f e c t s d i s c u s s e d i n S e c t i o n 1.3  i n f l u e n c e t h e appearance o f a f i r s t o r d e r phase t r a n s i t i o n when doing sweep voltammetry.  I t i s assumed t h a t t h e cathode  linear  i s t h i n and c o n s i s t s o f  u n i f o r m l y s i z e d p a r t i c l e s , and t h a t a l l p a r t i c l e s a r e i n good c o n t a c t w i t h t h e i r s u b s t r a t e and w i t h t h e e l e c t r o l y t e .  electrical  A l l particles are  then i n t h e same s t a t e o f i n t e r c a l a t i o n and a t the same c h e m i c a l  potential.  It  effects  i s f u r t h e r assumed, as mentioned i n S e c t i o n 1.3, t h a t k i n e t i c  can be modelled  by s o l i d  state diffusion  i n the p a r t i c l e s together with a  s e r i e s r e s i s t a n c e R t h a t i s assumed c o n s t a n t .  The e q u i l i b r i u m c e l l v o l t a g e  i s assumed t o decrease m o n o t o n i c a l l y w i t h Q. If  t h e v o l t a g e a t t h e s u r f a c e of t h e 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 d i s c h a r g e t h e c e l l v o l t a g e i s depressed the v o l t a g e a p p l i e d t o t h e c e l l loss.  by t h e IR l o s s , w h i l e on charge  i s above t h e e q u i l i b r i u m v o l t a g e by t h e IR  I f t h e c e l l v o l t a g e i s swept l i n e a r l y a t a r a t e a, s t a r t i n g a t a  time t ^ a t a c e l l v o l t a g e ^ (tg) : V (t) = V (t_) + a ( t - t - ) c c U 0  (2.3)  We s h a l l assume a f i r s t o r d e r phase t r a n s i t i o n o c c u r s a t an e q u i l i b r i u m v o l t a g e of V Q , and on d i s c h a r g e Q i n c r e a s e s from 0 t o phase f r o n t moves through t h e system. to  0.  On recharge Q changes from Q Q back  Assuming t h a t no c a p a c i t y e x i s t s except  f o r that a s s o c i a t e d with the  phase t r a n s i t i o n , when t h e r e a r e no k i n e t i c e f f e c t s then n o . c u r r e n t if  t h e v o l t a g e i s above o r below V Q .  as the  flows  The c u r r e n t i s a d e l t a f u n c t i o n i n  v o l t a g e , and i n f i n i t e w h i l e Q changes between 0 and Q Q . Let  us i n i t i a l l y c o n s i d e r t h e case of v e r y r a p i d s o l i d  state  so t h a t t h e s e r i e s r e s i s t a n c e R i s t h e o n l y s i g n i f i c a n t k i n e t i c  diffusion,  effect.  24  D i f f u s i o n w i l l be p r o p e r l y i n c o r p o r a t e d a i s p o s i t i v e and V ( t ) i s equal t o V Q ,  1  below V ^ .  S  later.  Consider  a recharging  cell,  S i n c e t h e c u r r e n t f l o w s o n l y when V  equals V u n t i l V Q i s reached a t t ^ .  u n t i l Q has dropped from Q Q t o 0, w h i l e V  C  continues  V then s t a y s a t  t o i n c r e a s e by ( 2 . 3 ) .  From (2.2) : I = |l|  - ( V  -  V  / R  J  c (J  (2.4) At a v o l t a g e V ^ = V ^ and a time t ^ the  r i s e s l i n e a r l y w i t h s l o p e 1/R.  charge Q has dropped to z e r o . equals V M  A f t e r t h i s , no more c u r r e n t f l o w s and V  again, while V continues ° ' c  c  i s shown i n f i g u r e 5 ( a ) .  The v a r i a t i o n of V w i t h V  to r i s e .  c  The charge as a f u n c t i o n of time, from (2.3) and  (2.4), i s : ,  ft  Q(t) = Q  a ( t - t ) dt = Q  +  Q  0  '0 Using  Q  a(t-t ) — ( 2  i  A l s o at I  P  ~  v  o  a ( t  =  r o t  )  =  ( 2 a R  Q  )  ) l s  (  = -(2aQ /R) u  ,  6  )  (2.7)  Js  n  (2.4) and ( 2 . 6 ) .  The c u r r e n t i s shown i n f i g u r e 5 ( b ) .  c u r r e n t has t h e o p p o s i t e V  V  I = -(2a {Q  During  On  discharge  t h e phase t r a n s i t i o n Q  Q  =  V  l ~ 0" V  (2.8) w i t h  {Q  curve.  Q  at V _ . 0  s i g n t o t h e c u r r e n t i r i f i g u r e 5, but t h e same 0 1 1  c n a r  8 > e  f  r  o  m  ( 2 . 3 ) , ( 2 . 4 ) , and ( 2 . 5 ) :  - Q(t)} / R)^  (2.8)  i n c r e a s e s q u a d r a t i c a l l y w i t h t h e change i n Q.  parabolic  2  , t h e c u r r e n t peaks a t :  magnitude, and g ~ 2  by  5  0  r i s e s from 0 t o Q„ w h i l e V drops from V _ t o V _ and V i s constant 0 c 0 2  |l|  .  0  r e s u l t s s i m i l a r to f i g u r e 5 are obtained.  The  2  (2.5) and n o t i n g a t t ^ that Q = 0:  v  using  ft  - Q ( t ) } r e p l a c e d by Q ( t ) .  |l|  On d i s c h a r g e  | l | i s given  p l o t t e d against Q i s a  25  V(t)t  v  0  V,  V (t) c  F i g u r e 5 - (a) R e l a t i o n s h i p between p a r t i c l e s u r f a c e v o l t a g e V and and c e l l v o l t a g e 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. F a s t s o l i d s t a t e d i f f u s i o n , (b) Current f l o w i n g due t o the r e l a t i o n s h i p i n ( a ) .  F i g u r e 6(a) shows a l i n e a r sweep voltammogram f o r a f i r s t  o r d e r phase  t r a n s i t i o n when t h e r e i s a s e r i e s r e s i s t a n c e R and v e r y f a s t s o l i d s t a t e diffusion.  T h i s i s c o n s t r u c t e d from f i g u r e 5(b) and i t s analogue f o r a  discharging c e l l . VQ.  The two peaks form a " n o t c h " a t t h e t r a n s i t i o n  voltage  T h i s d i s t i n c t i v e f e a t u r e w i l l be used e x p e r i m e n t a l l y t o i d e n t i f y  order phase t r a n s i t i o n s and t h e i r e q u i l i b r i u m v o l t a g e s . the c u r r e n t as a f u n c t i o n of Q.  The width  first  F i g u r e 6(b) shows  i n Q of t h e two phase r e g i o n can  be o b t a i n e d e x p e r i m e n t a l l y from t h e l i m i t s i n Q of t h e charge and d i s c h a r g e peaks.  N o t i c e t h a t t h e a d d i t i o n o f a s e r i e s r e s i s t a n c e has d i s t o r t e d t h e  i d e a l d e l t a f u n c t i o n a t V Q of c u r r e n t a g a i n s t v o l t a g e i n t o a sawtooth, but the t r a n s i t i o n v o l t a g e can s t i l l Both I  be found.  - V Q are proportional to a .  and  2  become narrower and s h o r t e r as a d e c r e a s e s .  The peaks i n f i g u r e 6(a)  To determine how  distinctive  the peaks a r e when compared t o s i n g l e phase c a p a c i t y , c o n s i d e r c h a r g i n g a s i n g l e phase r e g i o n of width Q Q w i t h an e q u i l i b r i u m v o l t a g e l i n e a r i n Q: V  =  3(QQ-Q)  +  V(0)  (2.9)  I d e a l l y , t h e r e i s a constant dQ/dV = -1/3. i s r e a s o n a b l e t o expect  Since a i s a l s o constant, i t  a s o l u t i o n f o r constant I .  s o l u t i o n , Q - Q Q i s simply I ( t - t ) . 0  I f t h e r e i s such a  Then from (2.3) and ( 2 . 9 ) , (2.2)  becomes: (a+BI) ( t - t j - V(0) - V (0) - IR 0 c which i s s a t i s f i e d  (2.10)  i f V (0) = V(0) + ctR/3 and I = -a/g. c  I n t h i s simple  case,  %  s i n g l e phase c u r r e n t s c a l e s w i t h a i n s t e a d of a . due  to f i r s t  As a i s decreased,  o r d e r phase t r a n s i t i o n s should become i n c r e a s i n g l y prominent.  We can e s t i m a t e t h e o r d e r of magnitude of a n e c e s s a r y due  to a f i r s t  capacity.  peaks  so t h a t peaks  o r d e r phase t r a n s i t i o n a r e twice t h e h e i g h t of s i n g l e phase  I f we equate I  i n (2.7) w i t h 21 = -2a/B and square both s i d e s :  27  F i g u r e 6 - (a) "Notch" formed by charge and d i s c h a r g e voltammograms of a f i r s t o r d e r phase t r a n s i t i o n . Fast s o l i d s t a t e d i f f u s i o n , (b) Current a g a i n s t charge f o r the c a s e shown i n ( a ) .  a - - ^  (2.11)  I f i n a s i n g l e phase a one m i l l i a m p - h o u r a 0.5 0.14  v o l t range, dQ/dV i s ^-7.2 volts/coulomb.  a ^ 8 uvolts/sec.  I f R ^ 250  coulombs/volt. ohms and Q  C l e a r l y a very  t r a n s i t i o n s to be v i s i b l e .  c e l l charges and  Q  solid  we  over  S i n c e dQ/dV = -1/3,  ^ 0.2  3 ^  coulombs, from (2.11)  slow sweep r a t e i s n e c e s s a r y  f o r phase  A sweep r a t e of 8 y v o l t s / s e c over a 0.5  range corresponds to about a 17 hour charge or Now  discharges  volt  discharge.  s h a l l r e c a l c u l a t e the peak shapes i n c o r p o r a t i n g the e f f e c t  s t a t e d i f f u s i o n , as w e l l as t h a t of a s e r i e s r e s i s t a n c e .  Again  s h a l l c o n s i d e r the c e l l to o n l y have c a p a c i t y a r i s i n g from the f i r s t phase t r a n s i t i o n .  p£, p  g  and  push the phase f r o n t s . D.  The  Say  surface  the d i f f u s i o n c o e f f i c i e n t  concentration form to  i n phase 2 i s a  I t w i l l be assumed t h a t the c u r r e n t i s s m a l l enough t h a t  s t a t e s o l u t i o n s to the d i f f u s i o n equation  t h a t the phase f r o n t s ' motion i s s u f f i c i e n t l y  can be used.  slow t h a t the  T h i s means  concentration  g r a d i e n t s f o r a g i v e n phase f r o n t p o s i t i o n a r e time independent. phase f r o n t i s assumed to be sharp, jumps by Ap = p^ at a constant  p ^.  -  p^.  The  and  as i t moves the  The  concentration  i n t e r i o r of each p a r t i c l e i s assumed to remain  The phase f r o n t i s n u c l e a t e d  at t =  0.  Three p o s s i b l e geometric shapes f o r the phase f r o n t w i l l be d i s t i n g u i s h e d by a l a b e l y.  I f y = 2,  considered  I f y = 1, each p a r t i c l e i s an i n f i n i t e  w i t h t h i c k n e s s 2L, w i t h i n t e r c a l a t i o n o c c u r r i n g i n a f l a t p l a n e each s i d e .  Phase  at the p a r t i c l e s ' s u r f a c e s w i t h an i n t e r c a l a n t c o n c e n t r a t i o n  r i s e s as l i t h i u m g r a d i e n t s , governed by the d i f f u s i o n e q u a t i o n ,  steady  order  i n t e r c a l a n t concentration p^.  phase f r o n t s move i n t o the p a r t i c l e s .  constant  we  At the s t a r t of the phase t r a n s i t i o n the c e l l i s i n  e q u i l i b r i u m i n phase 1 w i t h a u n i f o r m 2 nucleates  of  slab  through  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 r a d i u s  L,  with  i n t e r c a l a t i o n o c c u r r i n g through the s i d e w a l l .  to the Li MX2 l a y e r compounds. x  r a d i u s L, w i t h  case y = 2 a p p l i e s  The  I f y = 3, each p a r t i c l e i s a sphere of  i n t e r c a l a t i o n o c c u r r i n g through the sphere's e n t i r e s u r f a c e .  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 d i s t a n c e from the midpoint of the s l a b , from the a x i s of the c y l i n d e r , or the c e n t e r the sphere. If J  The  phase f r o n t i s at f .  i s the  g  Define u = r/L.  s u r f a c e c u r r e n t number d e n s i t y , the steady s t a t e concen-  t r a t i o n p r o f i l e g i v e s at r = L (Carslaw and  P  s  - P  9  J  l n u,  D s D  Y = 2,  (2.12b)  ^  - 1| ,  y = 3.  (2.12c) concentration  i s s m a l l compared to the amount moving the phase boundary, then by Ap  between f and  f + d f , where df = L du <  f ind: Y-l U  du s dt- -LA^ J  i O N  /o  =  ( 2  which governs the motion of the phase f r o n t . of i n t e r c a l a n t atoms f l o w i n g g  (2.12a)  L  t o change the c o n c e n t r a t i o n  J  Y = 1,  assume t h a t the amount of i n t e r c a l a n t r e q u i r e d f o r the  gradient  we  1959):  L S  I f we  Jaeger  J L = - f p (1 - u ) , 2 D  J  of  '  1 3 )  T h i s a r i s e s s i n c e the number  through a p a r t i c l e ' s s u r f a c e i n a time dt i s  dt m u l t i p l i e d by the p a r t i c l e ' s s u r f a c e a r e a .  m u l t i p l i e d by the s u r f a c e a r e a of the phase f r o n t .  T h i s must equal The  Ap(-df)  r a t i o of the  two  Y-l s u r f a c e areas i s u We  assume i n e q u i l i b r i u m the v o l t a g e decreases l i n e a r l y w i t h p i n the  s m a l l range of p c l o s e to dp/dV i s a n e g a t i v e e l e c t r o l y t e , and  constant.  t h a t occurs  i n the d i f f u s i o n g r a d i e n t ,  so  I f the i n t e r c a l a n t i s s i n g l y i o n i z e d i n the  the s u r f a c e area through which i n t e r c a l a t i o n i s o c c u r r i n g  0,  30  i s A, then: I = eAJ s  (2.14)  R e c a l l we s t i l l V  c  have:  = V - IR  (2.2)  where, s i n c e t h e phase f r o n t n u c l e a t e s a t t = 0: V  c  = V  0 n  + at  (2.15)  N o t i c e from (2.14) we see:  I dt = eA  ft  0  J  dt  g  rt Q = Q  0  ( c e l l discharging)  (2.16a)  ( c e l l charging)  (2.16b)  0 rt I dt = Q  +  J  + eA  0  0  When d i s c h a r g i n g , Ap > 0, p  g  dt s  > p^, J  g  > 0, I > 0, V  c  < V, and a < 0.  When  c h a r g i n g , a l l these i n e q u a l i t i e s a r e r e v e r s e d . S i n c e dp/dV i s a n e g a t i v e c o n s t a n t , from (2.2), (2.14), and (2.15):  p  s  " 2 " f P  (  V  -V  - ~dV  (  a  t  +  A  e  R  J  s  }  = at - bJ s  (2.17)  where a and b a r e d e f i n e d by: a =  b = - ^  a  (2.18)  AeR  (2.19)  and a r e not t o be confused w i t h t h e c r y s t a l l o g r a p h i c axes a and b. The e q u a t i o n s (2.12), (2.13), and (2.17) can be s o l v e d f o r t , J , and t J - 0  d t , -which; by (2.14), (2.15), and (2.16) d e f i n e V , I , and Q. S  Specifically,  C  (2.12) and (2.17) combine t o g i v e J , and hence I , as  f u n c t i o n s of u and t f o r each y  J  g  can then be s u b s t i t u t e d i n t o  and t h e e q u a t i o n i n t e g r a t e d to o b t a i n t as a f u n c t i o n of u . can a l s o be i n t e g r a t e d  directly:  (2.13)  Equation  (2.13)  31  J  dt = ^  s  (1 - u )  (2.20)  Y  y  which by (2.16) g i v e s Q.  The s o l u t i o n s a r e , f o r the case y  1 of a p l a n a r  phase f r o n t :  ApL  t =  J  u  Da Dat T  =  2  2  { (1  -  1 - u +  U  bD  2bD ) + —r—  Ji  (2.21a)  (1 - u) )  -1 (2.21b)  L  s  dt = ApL (1 - u)  J  (2.21c)  For t h e case y = 2 of a c y l i n d r i c a l phase f r o n t :  ApL  t =  J  {u  Da Dat L  s=  J 0  In u +  -ln u +  bD  2  +  (1  L  -uV  (2.22a)  -1 (2.22b)  dt = ^-ApL (1 - u )  (2.22c)  2  s  2  For the case y = 3 of a s p h e r i c a l phase f r o n t  UpL  t =  J  j-2 3 . 1  2  2 bD  ,.  3. -M  Da Dat L  s=  -1  I - 1 +M u  J  (2.23b)  L  dt = yApL (1 - u )  (2.23c)  3  g  (2.23a)  0 Voltammograms of I a g a i n s t  a r e shown i n f i g u r e 7 f o r each v a l u e of  Y and f o r t h r e e v a l u e s of the parameter bD/L. of Q when d i s c h a r g i n g f o r the same n i n e c a s e s . o n l y constant curves.  F i g u r e 8 shows I as a f u n c t i o n The parameter bD/L  i s the  t h a t a f f e c t s t h e c u r v e s ' shapes as opposed to s c a l i n g t h e  These f i g u r e s should be compared t o f i g u r e s 6(a) and 6(b) which  show o n l y the e f f e c t of a s e r i e s r e s i s t a n c e .  To compare, n o t e from (2.21),  32  F i g u r e 7 - L i n e a r sweep voltammograms computed from (2.21), (2".22) , and (2.23), i n c o r p o r a t i n g s o l i d s t a t e d i f f u s i i o n g r a d i e n t s .  33  LAp  F i g u r e 8 - Current  Ae/y  a g a i n s t charge f o r the cases shown i n f i g u r e c e l l discharging.  7,  (2.22) , and Q  (2.23) t h a t :  = LApAe /  Q  so t h a t from  (2.7):  II Ae and  (2.24)  Y  rv»* r ^ b  (DaAp)""  ^  5  from (2.6) : V, - V 1 0 aL  1  f \ bD L < J  h  y. [DaJ  N o t i c e t h a t i n a l l cases J^,  1  =  "  R  (  (2.14) t o :  Solid  '  ( 2  i n a l l c a s e s , an i n i t i a l still  the d i s t i n c t i v e "notch"  but the i n i t i a l Due  1) reduce u s i n g  (2.23b) f o r  V V  The v a l u e of R can  present.  (2.21b), (2.22b), and  R ~  =  There i s s t i l l ,  and  the e x p r e s s i o n s  at s m a l l v a l u e s of t (so t h a t u ^ at  1/R.  (2.26)  due  2 7 )  l i n e a r r i s e i n current-with  be found from the s l o p e of the i n i t i a l to a f i r s t  o r d e r phase t r a n s i t i o n i s  s t a t e d i f f u s i o n a l t e r s the way  i n which the c u r r e n t  slope rise, still falls,  r i s e of the c u r r e n t i s c o n t r o l l e d by the s e r i e s r e s i s t a n c e .  t o the s c a l i n g r e f l e c t e d  i n (2.25) and  the peaks i n f i g u r e 7 have the same i n i t i a l  (2.26),  the f a c t t h a t a l l of  l i n e a r ramp must be  emphasized.  F i g u r e 7 makes c l e a r , however, t h a t the f a l l of the c u r r e n t becomes much sharper  as bD/L  increases.  The  l a r g e r ' D i s , and  hence the f a s t e r  diffusion  o c c u r s , the more l i k e a sawtooth a voltammogram peak becomes. We constant x  D  can l o o k at the parameter bD/L associated with = L  2  dv  LAe  D e f i n e the time  diffusion:  / D  Define also a capacitance C = -  i n another way.  (2.28) a s s o c i a t e d w i t h the  cell: (2.29)  Then from (2.19) we ^  (2. -30)  = ^  The parameter bD/L One  (RC)  charges,  see:  can be thought of as the r a t i o of two  time  i s a s s o c i a t e d w i t h the speed w i t h which the c e l l w h i l e the o t h e r  relatively  ( ^)  charges or  i s associated with d i f f u s i o n .  T  constants. dis-  When  s m a l l , the peak shapes approach i d e a l , and when  is  is relatively  l a r g e , d i f f u s i o n rounds o f f the peak shapes. We  can e s t i m a t e bD/L.  From (2.19), n o t i n g a p a r t i c l e ' s volume i s  LA/ : Y  > =  < - » 2  3  which g i v e s : bD _ dQ RDY L ~ dV 2 Consider 0.5  a g a i n a one m i l l i a m p - h o u r  c e l l c h a r g i n g and  v o l t range i n a s i n g l e phase, so -dQ/dV ^ 7 . 2 -9  250  ohms a g a i n , y = 2, L ^ 10'iim, and D ^  f i n d bD/L 7 and  ^ 3.6.  10  coulombs/volt.  cm  /sec, from (2.32)  depends on the d i f f u s i o n constant  shown i n f i g u r e s  o n l y of the phase n u c l e a t e d on the s u r f a c e .  of the peaks o c c u r s as the c u r r e n t becomes l a r g e r  of the peaks, the peak h e i g h t  magnitude of I , and  - V Q .  h  s c a l e r o u g h l y as a , and  Despite  the  i s s t i l l w e l l w i t h i n an o r d e r of  the peak w i d t h i n v o l t a g e i s s t i l l  o r d e r of magnitude of still  concentration  the s u r f a c e , the shape of a peak  hence the c o n c e n t r a t i o n g r a d i e n t s become more severe.  rounding  we  experimentally.  g r a d i e n t between the phase f r o n t and  N o t i c e t h a t the rounding  If R ^  2  T h i s i n d i c a t e s t h a t the range of bD/L  8 i s l i k e l y to occur  7  d i s c h a r g i n g over a  S i n c e the motion of the phase f r o n t i s governed by the  and  - v  n  w e l l w i t h i n an  Thus, the peak h e i g h t and v o l t a g e w i d t h the p r e v i o u s e s t i m a t e of how  slow a sweep  rate is  i s n e c e s s a r y f o r a peak t o be t w i c e the h e i g h t of s i n g l e phase c a p a c i t y  still  reasonable.  The curves i n f i g u r e 8 of I a g a i n s t Q on d i s c h a r g e have been rounded from the p a r a b o l a s of f i g u r e they may first and y  still  6(b)  prove s u f f i c i e n t l y c l e a r t o determine  order phase t r a n s i t i o n . =  i n t o l e s s d i s t i n c t i v e shapes,  3 appear v e r y s i m i l a r .  In both f i g u r e s  the range  i t t o be p o s s i b l e  mine the d i m e n s i o n a l i t y of the phase f r o n t  i n Q of a  7 and 8, the cases y = 2  The c u r v e s i n these two  s u f f i c i e n t l y d i f f e r e n t t o expect  however  cases a r e not  to experimentally d e t e r -  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 p l a n a r phase f r o n t were not observed  i n the m a t e r i a l s d i s c u s s e d i n t h i s  thesis.  2.3  E x p e r i m e n t a l Apparatus and Technique L i n e a r sweep voltammetry r e q u i r e s apparatus c a p a b l e of p r o v i d i n g an  a c c u r a t e l y swept v o l t a g e t o a c e l l , whatever c u r r e n t  and c a p a b l e of p r o v i d i n g  simultaneously  i s demanded by t h e c e l l , w i t h a low output impedance com-  pared t o t h e impedance of t h e c e l l .  The sweep r a t e must be s u f f i c i e n t l y  slow t h a t f e a t u r e s i n a voltammogram due t o f i r s t order phase t r a n s i t i o n s can be r e s o l v e d . A P r i n c e t o n A p p l i e d Research (PAR) Model 175 U n i v e r s a l Programmer w i t h t h e 0.1 m i l l i v o l t / s e c Scan Rate M o d i f i c a t i o n r e l i a b l y v o l t a g e s between -11 and 11 v o l t s a t - a slowest The  output from a PAR Model 175 was sent  Galvanostat  sweeps out  r a t e o f 0.1 m i l l i v o l t / s e c .  t o a PAR Model 173 P o t e n t i o s t a t /  w i t h a 10 kilo-ohm input impedance.  A r e s i s t o r i n s e r i e s with  the i n p u t was s e l e c t e d t o c u t t h e sweep range from ^22 v o l t s t o , 0.5 v o l t s .  The sweep range was p r e c i s e l y a d j u s t e d  l i m i t s of t h e PAR Model 175.  using  typically,  the voltage  sweep  A f i x e d p o t e n t i a l from t h e PAR Model 173 was  summed i n t e r n a l l y t o t h e input t o s h i f t  the v o l t a g e range t o t h e d e s i r e d  location. A PAR Model 173 may e i t h e r c o n t r o l t h e v o l t a g e , as i n t h i s case, o r c o n t r o l the current  that flows.  The c u r r e n t  s u p p l i e d may have a magnitude  of one ampere or l e s s , and t h e 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 t o measure t h e charge Q t h a t while charging  and d i s c h a r g i n g a c e l l .  The slowest  w i t h t h e PAR Model 173 and PAR Model 175 covers  sweep r a t e p o s s i b l e  t h e i n i t i a l l y ^22 v o l t  range a t 0.1 m i l l i v o l t / s e c , which takes about 61 hours. normally  used a t c l o s e t o t h i s sweep l i m i t  an e l e c t r o m e t e r  the c e l l v o l t a g e .  The system was  t o ensure maximum r e s o l u t i o n .  F l a n g e c e l l s were connected t o t h e c e l l and  flowed  output of t h e PAR Model 173,  attachment was used by the PAR Model 173 t o monitor The output of t h e e l e c t r o m e t e r ,  together  w i t h t h e output  38  of the PAR Model 173's  c u r r e n t meter, were recorded  c u r r e n t meter's output was  as a voltammogram.  The  a l s o f e d i n t o the PAR Model 179, and the output  of the coulometer was used to r e c o r d the c u r r e n t a g a i n s t Q.  The  temperatures  of the f l a n g e c e l l s were h e l d f i x e d , when p o s s i b l e , i n temperature b a t h s .  39  2.4  Results Figure  f o r L i VS„ x 2 9 shows a voltammogram and  measured f o r the c e l l GJ-9 c u r v e of t h i s c e l l has  on  f i g u r e 10 shows c u r r e n t  i t s e i g h t h charge and  been shown p r e v i o u s l y  discharge.  i n f i g u r e 4.  were made at room temperature at a sweep r a t e of 2.35 required  59.22 hours t o sweep 0.5  c a p a c i t y of  14.6  volts.  On  f i g u r e s 11 and  12.  i n S e c t i o n 2.3,  and  The  uvolts/sec,  by weight.  and  Haering  were made at a sweep r a t e of 2.82  a capacity  coulombs.  voltammograms were done at sweep r a t e s v e r y  voltage  which had  a  These r e s u l t s  (1981), shown i n described  y v o l t s / s e c , which  v o l t s , at a temperature of  of about 1.5  Q  measurements  T h e i r measurements used the same equipment as  r e q u i r e d 49.25 hours to sweep 0.5 T h e i r c e l l had  The  t h i s c y c l e , c e l l GJ-9  coulombs, or about x = 0.71  can be compared w i t h those of Dahn, J.R.,  against  (23 ± 0.5)  In both c a s e s ,  °C.  the  c l o s e to the slowest  rate  p o s s i b l e w i t h the equipment. Since  the measurements shown are f o r two  c e l l s which d i f f e r by  f a c t o r of more than n i n e i n c a p a c i t y , n o t i c e from e q u a t i o n s (2.6) (2.7)  t h a t peak h e i g h t  and  expects t h a t peaks due should  have r o u g h l y  capacity c e l l .  to f i r s t order  t r i p l e : the h e i g h t  i n c e l l GJ-9  of peaks found f o r the  should  s c a l i n g appears r o u g h l y  for  c e l l GJ-9  should  and  2.3  One  GJ-9  smaller  n i n e times  capacity c e l l .  c o r r e c t both f o r the peaks and  s i n g l e phase c a p a c i t y between about 2.2  .  I v a r i e s w i t h Q Q , so  have roughly  f l o w i n g as i s the case f o r the s m a l l e r  and  as w e l l as a  phase t r a n s i t i o n s i n c e l l  However, i n s i n g l e phase r e g i o n s  s i n g l e phase c a p a c i t y current  like Q Q  w i d t h both v a r y  a  volts.  the  This  what appears to The  peak widths  a l s o be t h r e e times as l a r g e , however t h i s i s not  case. Two  very  c l e a r "notches" a p p a r e n t l y  s i t i o n s occur at  due  (2.380 ± 0.005) v o l t s and  to f i r s t order  be  phase t r a n -  (2.460 ± 0.005) v o l t s .  The  the  i  2.1  ~i  2.2  i  i  i  2.3 Cell Voltage  i — i — r  2.4  2.5  2.6  (volts)  F i g u r e 9 - Voltammogram f o r t h e 8 t h charge ( s o l i d l i n e ) and d i s c h a r g (dashed l i n e ) of the L i VS,, c e l l GJ-9, a t a sweep r a t e of 2.35 u v o l t s / s e c .  Q 2  0  0  (coulombs)  14 12 10 8 II I I I I I I I  1.0  0.8  6 I I  0.6  4 I I  0.4  X in LI V S X  2 I  0.2  I  0I I  0.0  2  F i g u r e 10 - C u r r e n t a g a i n s t charge f o r the 8 t h charge ( s o l i d and d i s c h a r g e (dashed l i n e ) of the L i c e l l GJ-9, at a sweep r a t e of 2.35 y v o l t s / s e c .  line)  2.1  2.2  2.4  2.3  2.5  2.6  VOLTS  F i g u r e 11 - Voltammogram f o r a L i VS^ c e l l , showing a charge l i n e ) and a d i s c h a r g e (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).  (solid  F i g u r e 12 - Current a g a i n s t charge f o r a L i c e l l , showing a charge ( s o l i d l i n e ) and d i s c h a r g e (dasned l i n e ) a t a sweep r a t e of 2182 y v o l t s / s e c . A f t e r Dahn, J.R., and Haering (1981).  44  charge and d i s c h a r g e peaks forming t h e "notches" a r e l a r g e and c l e a r , a l t h o u g h c l a r i t y i s somewhat l i m i t e d by t h e o v e r l a p o f t r a n s i t i o n s w i t h diffusion tails  and w i t h s i n g l e phase c a p a c i t y .  The peaks f o r these two  p o s s i b l e 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 s l o p e of 1/R.  T h i s g i v e s R = (140 ± 20)' ohms f o r c e l l GJ-9 and R =  (650 ± 50) ohms f o r t h e c e l l of Dahn, J.R., and H a e r i n g . i s r o u g h l y accounted area.  f o r by c e l l GJ-9 having about  This difference  f o u r times t h e cathode  T h i s d i f f e r e n c e a l s o accounts f o r t h e unexpectedly narrow peaks  for  c e l l GJ-9, s i n c e b f o r c e l l GJ-9 i s then one q u a r t e r t h e s i z e of b  for  the other c e l l .  A l l o f t h e peaks show n o t i c a b l e rounding due t o  d i f f u s i o n , and a r e s i m i l a r t o those peaks expected f o r y = 2 o r y = 3 w i t h bD/L l e s s than one. ing  These two "notches" occur a t v o l t a g e s c o r r e s p o n d -  t o t h e c l e a r p l a t e a u s mentioned Another  i n S e c t i o n 1.4 and shown i n f i g u r e 4.  "notch", a l t h o u g h l e s s c l e a r , appears  superimposed  v o l t a g e s i n g l e phase c a p a c i t y a t (2.190 ± 0.005) v o l t s . what appears t o be a " n o t c h " a t (2.235 ± 0.010) v o l t s . has t h e most p o o r l y d e f i n e d of t h e peaks, but they s t i l l c l e a r t o be c o n s i d e r e d a "notch". past  on low  There i s a l s o This last appear  "notch"  sufficiently  They have been taken as such i n t h e  (Dahn, J.R., and Haering) and w i l l  be t r e a t e d as such f o r t h e moment,  however t h e e x i s t e n c e of a f i r s t o r d e r phase t r a n s i t i o n at t h i s v o l t a g e s h a l l be r e c o n s i d e r e d i n S e c t i o n 3.4. The graphs of I a g a i n s t Q i n f i g u r e 10 and f i g u r e 12 show t h e range i n x, n o r m a l i z e d t o t h e c e l l c a p a c i t y , f o r t h e f o u r suspected phase t r a n sitions.  The peaks i n t h e s e graphs r i s e q u i c k l y t o t h e i r maximum v a l u e s  before f a l l i n g  o f f almost l i n e a r l y .  voltammograms, appear bD/L l e s s than one. x = 0.  The peaks,  l i k e t h e peaks i n t h e  s i m i l a r t o those expected f o r y = 2 o r y = 3 w i t h C o n s i d e r f i r s t t h e p a i r o f peaks a t low x, ending a t  On charge, t h e c u r r e n t has a minimum a t x = 0.34 ± 0.01. The  minimum on d i s c h a r g e i s d i s p l a c e d to lower x, b e f o r e the d i f f u s i o n t a i l of the f i r s t  s i n c e a second peak begins  i s completed.  on d i s c h a r g e the c u r r e n t i s q u i t e h i g h at the s t a r t and  i t is difficult  In the case of  of the second peak,  to e x t r a p o l a t e the v a l u e of x at which the f i r s t  ends, a l t h o u g h on charge the l i m i t i n g v a l u e of x i s q u i t e c l e a r . s i m i l a r manner the p a i r of peaks t h a t next between x = 0.34 appearing to  ± 0.01  and x = 0.52  at h i g h x c l e a r l y occur  x = 1.00  ±  In a  ± 0.01.  A l s o , the p a i r of peaks  i n the range of x from x = 0.90  ±  t h a t must e x i s t  i f the peaks a r e indeed due  r e g i o n a t x = 0.34  v o l t s or on  v o l t s , the s i n g l e phase r e g i o n s at x = 0 and x = 1 to f i r s t  s i t i o n s a r e v e r y narrow, w i t h a width below 0.02 ± 0.01  o r d e r phase t r a n -  i n x.  must be s i m i l a r l y narrow.  The  Any  s i n g l e phase  peaks near  (2.325 ± 0.010) v o l t s do not have v e r y c l e a r l y d e f i n e d l i m i t s i n Q, l a r g e r e r r o r estimates are necessary.  extend  ± 0.05  to x = 0.63  from x = 0.63  Dahn, J.R., 23 C to 45 C. p e r a t u r e was  0.02  0.01.  d i s c h a r g e above 2.46  to  peak  appear w i t h i n c r e a s i n g x occur  S i n c e t h e r e i s l i t t l e c a p a c i t y on r e c h a r g e below 2.19  from x = 0.57  GJ-9,  ± 0.03  and Haering  The  so  The peaks a r e i n the range of x  ±0.03.  to x = 0.90  A s i n g l e phase r e g i o n appears ±  0.02.  have done voltammograms a t temperatures  from  o n l y change i n the voltammogram w i t h i n c r e a s i n g tem-  t h a t the peaks between x = 0.90  ± 0.02  and x = 1.00  ±  0.01  narrowed the range of x they c o v e r e d , 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 d i s a p p e a r i n g by 40°C.  t o g e t h e r w i t h the a n a l y s i s j u s t p r e s e n t e d , shown i n f i g u r e 13, and was Haering.  This information,  l e a d s to the phase diagram  o r i g i n a l l y p r e s e n t e d by Dahn, J.R.,  The l a r g e s i n g l e phase r e g i o n i s l a b e l l e d  and  IT i n accordance  the x-ray powder d i f f r a c t i o n r e s u l t s mentioned i n S e c t i o n 1.4.  with  Recall  from t h a t s e c t i o n t h a t Murphy et a l (1977) found t h a t s e v e r a l phases e x i s t  46  43  F i g u r e 13 - Phase diagram f o r L i d e r i v e d from l i n e a r sweep voltammetry r e s u l t s and t h e x-ray d a t a of Murphy et a l (1977). A f t e r Dahn, J.R., and Haering (1981).  47  when they a n a l y s e d x-ray powder d i f f r a c t i o n p a t t e r n s a t a dozen v a l u e s of x.  The p a r t i a l phase diagram d e r i v e d from those x-ray p a t t e r n s i s shown  i n f i g u r e 14.  Noting t h a t t h e boundaries  i n x of t h e phases were not w e l l  d e f i n e d by t h e l i m i t e d number of x-ray p a t t e r n s , t h e p a r t i a l phase diagram i s i n agreement w i t h f i g u r e  13, i f t h e phases and two phase r e g i o n s i n  f i g u r e 13 a r e l a b e l l e d as shown, u s i n g Murphy et a l ' s n o t a t i o n f o r t h e phases.  There a r e two a d d i t i o n a l two phase r e g i o n s added i n f i g u r e 13  over and above those p r e s e n t r e g i o n corresponds and w i l l  i n t h e p a r t i a l phase diagram.  t o t h e p o o r l y d e f i n e d peaks near  The 3 + IT  (2.325 ± 0.010) v o l t s ,  be d i s c u s s e d f u r t h e r i n S e c t i o n 3.4.  Murphy e t a l found  t h a t t h e phase l a b e l l e d 3S has a hexagonal u n i t  c e l l w i t h 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 t h a t weak x-ray l i n e s a r e p r e s e n t c o n s i s t e n t w i t h a 3c s u p e r l a t t i c e . IT phase was d i s c u s s e d i n S e c t i o n 1.4. t o r t i o n of t h e IT phase.  The 3 phase appears  t o be a d i s -  I t was indexed f o r L i ^ ^VS as m o n o c l i n i c w i t h 2  a = 5.756A*, b = 3.280&, c = 6.164/2, and 3 = 91.28°. b ^ a^, and c ^ c^.  The  The a phase a l s o appears  Notice that a ^ 3 ^ ,  t o be a d i s t o r t i o n of IT,  indexed as m o n o c l i n i c w i t h a = 5.65 9/2, b = 3.240/2, C = 6.050/2, and 3 = 91.0°, w i t h t h e same approximate r e l a t i o n s h i p t o t h e IT hexagonal u n i t c e l l as e x i s t s f o r t h e 3 phase.  The VS^ phase i s once a g a i n IT w i t h  a hexagonal u n i t c e l l f o r which a = 3.21 7& and c = 5.745°..  F i g u r e 14 - P a r t i a l phase diagram f o r L i 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).  d e r i v e d from x-ray  49  2.5  E f f e c t i v e n e s s of the Technique The a b i l i t y both t o r e s o l v e and unambiguously i d e n t i f y peaks due t o  first  o r d e r phase t r a n s i t i o n s i s l i m i t e d both by t h e a v a i l a b l e apparatus  and t h e n a t u r e o f t h e t e c h n i q u e i t s e l f . a 0.5 v o l t range i n a t most 61 h o u r s . the p r e v i o u s s e c t i o n , l e f t  The apparatus used sweeps through Even though t h e peak r e s o l u t i o n i n  something t o be d e s i r e d , i t was not p o s s i b l e t o  slow t h e sweep r a t e f u r t h e r .  I t i s p o s s i b l e t o improve t h e r e s o l u t i o n  somewhat by u s i n g a l i g h t cathode.  U n l e s s t h e sweep r a t e i s s u f f i c i e n t l y  slow, s i n g l e phase f e a t u r e s may mask, o r be mistaken f o r , peaks due t o first  order phase t r a n s i t i o n s .  I t should be n o t e d t h a t  i n many c h e m i c a l  a p p l i c a t i o n s l i n e a r sweep voltammetry i s o f t e n performed i n m i n u t e s , f a r too f a s t  t o d e t e c t phase t r a n s i t i o n s .  In l i n e a r  sweep voltammetry t h e peak shapes a r e not v e r y  u n l e s s they a r e v e r y l a r g e compared  distinctive  t o surrounding s i n g l e phase r e g i o n s .  S i n c e d i f f u s i o n g r a d i e n t s become l a r g e r as t h e c u r r e n t becomes l a r g e r , the peaks a r e rounded by d i f f u s i o n e f f e c t s . l i n e a r r i s e due t o a f i n i t e c e l l r e s i s t a n c e .  The peaks a l s o have an i n i t i a l As a r e s u l t , t h e 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 t o unambiguously s i g n a l two phase :regions.  I t i s thus p o s s i b l e t h a t a bump i n s i n g l e phase c a p a c i t y  can be mistaken f o r a two phase r e g i o n , p a r t i c u l a r l y s i n c e i t may not be p o s s i b l e t o use a slow enough sweep r a t e t o make peaks due t o phase transitions relatively  large.  I f t h e sweep r a t e i s s u f f i c i e n t l y slow f o r peaks due t o f i r s t  order  phase t r a n s i t i o n s t o be c l e a r l y r e s o l v e d , then, as shown ,in t h e p r e v i o u s s e c t i o n , i t i s p o s s i b l e 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 v o l t a g e s , t y p i c a l l y t o w i t h i n 0.005 v o l t s , and t o determine t h e range i n x o f a two phase r e g i o n , t y p i c a l l y  to w i t h i n 0.01.  In t h e case of the L i VS„ system,  the phase diagram was o b t a i n e d q u a n t i t a t i v e l y u s i n g l i n e a r sweep v o l t a m -  metry and r e s u l t s from x-ray powder d i f f r a c t i o n , a l t h o u g h the 3 + IT phase r e g i o n w i l l be d i s c u s s e d f u r t h e r i n S e c t i o n 3 . 4 . of  the peaks i n the voltammograms would s t i l l  ensure t h a t the form of the phase diagram  Better resolution  be d e s i r a b l e , however, t o  c o u l d have been found unambig-  u o u s l y even i f no x - r a y d a t a had been a v a i l a b l e . widths a r e not as'easy to p r e c i s e l y determine in  o t h e r s , due to the l i m i t s on r e s o l u t i o n .  Peak p o s i t i o n s  first  The c e l l r e s i s t a n c e can Still,  be linear  does p r o v i d e an e l e c t r o c h e m i c a l t e c h n i q u e w i t h which  o r d e r phase t r a n s i t i o n s can be found, or a t l e a s t t h e i r  indicated.  and  i n some cases as they a r e  e a s i l y found from the i n i t i a l ramps of w e l l r e s o l v e d peaks. sweep voltammetry  two  C l e a r l y r e s o l v e d peaks appear  possibility  to c o r r e s p o n d , as expected, w i t h  p l a t e a u s i n the c u r v e of v o l t a g e a g a i n s t x.  51  CHAPTER THREE  CONSTANT CURRENT dQ/dV  3.1  dQ/dV Measured a t Constant  Current  We have seen t h a t measurements of dQ/dV can be used t o 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.  L i n e a r sweep v o l -  tammetry, however, does n o t have s u f f i c i e n t l y good r e s o l u t i o n o f t h e peaks caused by f i r s t biguous.  o r d e r phase t r a n s i t i o n s t o be c o n s i s t e n t l y unam-  The apparatus d e s c r i b e d  i n S e c t i o n 2.3 must be used near t h e  l i m i t s o f i t s c a p a b i l i t i e s t o get any m e a n i n g f u l r e s u l t s .  Also, since  the c u r r e n t i s v a r y i n g w i t h time, t h e peak shapes a r e rounded by i n creasing d i f f u s i o n gradients.  As w e l l , t h e IR s h i f t  c o n t i n u a l l y changing as t h e c u r r e n t changes.  i n the voltage i s  C l e a r l y t h e shapes o f t h e  peaks c o u l d be improved i f the c u r r e n t was h e l d c o n s t a n t .  Diffusion  g r a d i e n t s would not worsen d u r i n g a phase t r a n s i t i o n , arid, assuming t h a t the c e l l r e s i s t a n c e was c o n s t a n t , constant.  t h e IR s h i f t  i n t h e v o l t a g e would remain  I f d i f f u s i o n g r a d i e n t s c o u l d be minimized, dQ/dV c o u l d be  measured w i t h t h e c e l l c l o s e r t o e q u i l i b r i u m . N o t i c e t h a t i f we i g n o r e k i n e t i c e f f e c t s f o r t h e moment: =  dV  -da dt  =  ~ dt dV  on  dt dV  K  where V i n dt/dV w i l l be t h e c e l l v o l t a g e . dt/dV i s i d e a l l y p r o p o r t i o n a l t o dQ/dV. either a voltmeter  I f the current i s held  '  J  constant,  dt/dV can be measured e a s i l y  o r an analogue t o d i g i t a l c o n v e r t e r a t t a c h e d  with  to a micro-  computer.  T h i s chapter w i l l develop  t h e t h e o r y and t e c h n i q u e of c o n s t a n t  c u r r e n t dQ/dV measurements, and then compare the r e s u l t s of such measurements f o r t h e L i ^ V ^ system w i t h those d e s c r i b e d i n S e c t i o n 2.4 made u s i n g linear  sweep voltammetry.  3.2  Theory of K i n e t i c E f f e c t s on Phase T r a n s i t i o n s We s h a l l now model, i n a s i m i l a r manner t o S e c t i o n 2.2, how t h e  k i n e t i c e f f e c t s discussed first  i n S e c t i o n 1.3 i n f l u e n c e t h e appearance of a  o r d e r phase t r a n s i t i o n when doing  ments. behavior  c u r r e n t dQ/dV measure-  The same p h y s i c a l assumptions made i n S e c t i o n 2.2 r e g a r d i n g t h e 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.  i s t h i n and has u n i f o r m l y same c h e m i c a l  sized p a r t i c l e s .  The c e l l ' s cathode  A l l p a r t i c l e s are at the  p o t e n t i a l and i n t h e same s t a t e o f i n t e r c a l a t i o n .  e f f e c t s can be modelled by s o l i d R.  constant  Kinetic  s t a t e d i f f u s i o n and a s e r i e s r e s i s t a n c e  The o n l y c e l l c a p a c i t y i s t h a t a s s o c i a t e d w i t h a f i r s t  order phase  t r a n s i t i o n a t V Q which changes Q between 0 and Q Q .  The c e l l  i n e q u i l i b r i u m i n phase 1 a t p ^ .  on t h e s u r f a c e s a t  Phase 2 n u c l e a t e s  p ^ , and phase f r o n t s move i n t o t h e p a r t i c l e s . i s a g a i n a constant f r o n t by Ap ~&2 =  ~  D i n phase 2. p  l '  surface concentration  a n (  ^ ' t  i e  P  is p .  n a s e  The d i f f u s i o n  The c o n c e n t r a t i o n  coefficient  changes a t a phase  f r o n t s n u c l e a t e a t t = 0.  The same t h r e e c a s e s ,  g  is initially  The  l a b e l l e d by y , f o r  t h e symmetry of the p a r t i c l e s and t h e i r phase f r o n t s as d i s c u s s e d i n S e c t i o n 2.2 w i l l be c o n s i d e r e d  here.  I n t h e case y = 1 each p a r t i c l e i s  an i n f i n i t e s l a b of t h i c k n e s s 2L, w h i l e  i n t h e 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 o f r a d i u s L, and i n t h e case y = 3 each p a r t i c l e i s a sphere of r a d i u s L. rium dp/dV i s a n e g a t i v e which occur  A l s o , once a g a i n constant  i n the d i f f u s i o n  i t i s assumed t h a t i n e q u i l i b -  f o r the v a l u e s  of p , a l l c l o s e t o  p^,  gradient.  R e c a l l t h a t i f t h e r e a r e no d i f f u s i o n e f f e c t s , then dQ/dV i s a d e l t a f u n c t i o n i n v o l t a g e a t V Q and i s i n f i n i t e w h i l e Q changes between 0 and QQ.  We s t i l l  have:  V  = V - IR c Since the current i s constant,  (2.2) and R i s assumed c o n s t a n t , V  i s merely V  54 s h i f t e d by a constant on charge.  to lower v o l t a g e s on d i s c h a r g e , and to h i g h e r  The s e r i e s r e s i s t a n c e does not a f f e c t  voltages  the appearance of dQ/dV,  whether d u r i n g a phase t r a n s i t i o n or i n a s i n g l e phase, except t o s h i f t dQ/dV p l o t t e d a g a i n s t v o l t a g e by a c o n s t a n t . t i o n a g a i n s t v o l t a g e and i s s t i l l Constant c u r r e n t dQ/dV s e p a r a t e s Only s o l i d  dQ/dV i s s t i l l  i n f i n i t e a g a i n s t Q between  a delta  func-  0 and Q Q .  the two k i n e t i c e f f e c t s we a r e c o n s i d e r i n g .  s t a t e d i f f u s i o n can a f f e c t peak shapes.  With only a s e r i e s  r e s i s t a n c e c o n s i d e r e d , a dQ/dV measurement a g a i n s t v o l t a g e c o n s i s t s of two d e l t a f u n c t i o n s , one a t V Q -  J11R and t h e o t h e r a t V Q +  s i t i o n v o l t a g e V Q i s centered Now  c o n s i d e r how  by s o l i d  between the two d e l t a f u n c t i o n s .  D e f i n e a g a i n f as the p o s i t i o n of the phase  Once a g a i n , assume t h a t the amount of i n t e r c a l a n t  r e q u i r e d f o r the c o n c e n t r a t i o n g r a d i e n t i s s m a l l compared moving the phase boundary.  u  Taking  .  t o the amount  This leads again t o :  Y - l d H . _ i dt  The t r a n -  the d e l t a f u n c t i o n s and dQ/dV a g a i n s t Q a r e a f f e c t e d  state diffusion.  f r o n t and u = f / L .  jIj R .  (2.13)  LAp  the i n t e r c a l a n t  to a g a i n be s i n g l y i o n i z e d i n t h e e l e c t r o l y t e , and  t a k i n g the s u r f a c e a r e a through which i n t e r c a l a t i o n o c c u r s  to be A, we  I = eAJ  (2.14)  s (2.14), (2.13) becomes:  Using  dt  AeLAp dt  Integrating Q =  A  e  ^  (3.2): A  p  (1 - u )  Q = - — ^P Y  Y  e  U  Y  N o t i c e t h a t as i n S e c t i o n Q  0  = AeL|Ap|  / y  ( c e l l discharging)  (3.3a)  ( c e l l charging)  (3.3b)  (2.2) we must have from ( 3 . 3 ) : ( - ) 3  4  have:  For s i m p l i c i t y w r i t e (3.3) a s : Q = Q  Q  Q = Q  Q  (1 - u )  ( c e l l discharging)  (3.5a)  u  (cell  (3.5b)  Y  Y  S i n c e I i s c o n s t a n t , we can immediately  write:  Q = It Q = Q  + It  Q  and hence from t = Q  ( c e l l discharging)  (3.6a)  ( c e l l charging)'  (3.6b)  (3.5):  (1 - u )  / |l|  Y  Q  charging)  u = (1 - | l | t / Q )  (3.7)  (3.8)  1 / Y  Q  Now  d e f i n e f o r convenience  a constant  i n terms of two c o n s t a n t s we have  used b e f o r e , namely the d i f f u s i o n time constant x ^ and the c a p a c i t a n c e C a s s o c i a t e d w i t h the c e l l : T  D  = L  2  / D  (2.28)  C = - J LAe dv  (2.29)  D e f i n e a change i n v o l t a g e a s s o c i a t e d w i t h the time x ^ : IT-  V  D  =  ' C D  J L s  D  F  ^-1 dp dV  ,  (3.9)  Once a g a i n , make the steady s t a t e approximation, motion of the phase f r o n t  is sufficiently  which assumes the  slow t h a t t h e c o n c e n t r a t i o n  g r a d i e n t s f o r a g i v e n phase f r o n t p o s i t i o n a r e time independent.  This  a l l o w s the use of (2.12) a g a i n f o r the s u r f a c e c o n c e n t r a t i o n p .  Using  g  ( 3 . 9 ) , (2.12) may  be w r i t t e n a s :  56  s  p  "Iv  =  " 2 P  V  D  (  i p  "  1  U  '  - 1 ,  v  dV  )  D  Y = 1,  (3.10a)  1 = 2 ,  (3.10b)  Y = 3.  (3.10c)  To f i n d dQ/dV, n o t i c e : dQ_ dV  dQ_ dp d t d t dV dpj  =  dp dt dV dpj  =  P  P  dQ/dV i s a constant tiating  dp dt  Using  s  , 2 s  — V. D  dV  (2.13)  (3.12)  Y  u  ( 3 . 4 ) , ( 3 . 9 ) , and (3.12),  Y'Q-r0  Differen-  -2( -1)  T5A7  (2.14),  'P=P, ' s  d i v i d e d by t h e time d e r i v a t i v e of (3.10)',  (3.10) and u s i n g  P=P.  (3.11)  (3.11)  becomes:  2( -1)  u  (3.13)  Y  1  which by (3.8) becomes:  dc:  Y Q =  dV  o  (1 -  |l|t/Q ) 0  2 ( Y  -  D  Note t h e exponent i s 0 f o r y = 1, 1 f o r y = 2, Now we can w r i t e down V . c p  s  equation  " Iv"  " 2 p  ( V  -  and 4/3 f o r y = 3,  Since: (3.15)  V  (2.2)' becomes:  V From  (3.14)  1 ) / Y  1  0  +  (3.10) t h i s  \~1  ^ dV  (P  s  becomes:  -  P ) 2  - IR  (3.16)  57  Y =  V = V  + V  Q  D  l n u - IR,  (3.8)  we  (3.17a) (3.17b)  Y Y =  Using  1,  (3.17c)  3.  find: (3.18a) (3.18b) -1/3  Thus, f o r constant by  c u r r e n t dQ/dV we  (3.18), and Q g i v e n by  a first  Q  diffusion t a i l ,  ± 11]R,  have dQ/dV g i v e n by  (3.18c) (3.14), V"  c  given  and f i g u r e 16 shows dQ/dV a g a i n s t Q  order phase t r a n s i t i o n .  f u n c t i o n at V  y = 3.  (3.6).  F i g u r e 15 shows dQ/dV a g a i n s t for  - 1} - IR,  dQ/dV a g a i n s t V"  c  but i n s t e a d jumps to — / Q Q / 1 V D I  r o u g h l y of w i d t h V^,  i s no l o n g e r a d e l t a A T  V  p o i n t s away from V^.  0  ±  As  K  in linear  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 e x p e r i m e n t a l l y d i s t i n g u i s h a b l e .  For  the case y = 2, which corresponds compound, the t a i l  f a l l s to one  v o l t a g e has changed by V .  The  to the i n t e r c a l a t i o n of an KX^  layer  t e n t h of the maximum peak h e i g h t when the tail  f o r both y = 2 arid-y = 3 does not  f a l l to zero p r o p e r l y , d i v e r g i n g to i n f i n i t e v o l t a g e as the phase f r o n t reaches  the c e n t e r of the p a r t i c l e .  approximation. divergence  T h i s i s due  state  Time dependent terms t h a t do not appear i n (3.10) prevent  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.  N o t i c e t h a t the- peak shapes are no the way  to the steady  longer dependent on a parameter i n  l i n e a r sweep voltammetry peak shapes a r e dependent on bD/L.  s t a n t s only s c a l e the constant v a r i e s d i r e c t l y w i t h I and  c u r r e n t dQ/dV peak shapes. i n v e r s e l y w i t h D dp/dV.  h a l v e d , the peak h e i g h t doubles w h i l e the peak w i d t h  The  Con-  constant  I f the c u r r e n t i s i s cut i n two.  As  Figure  15 - Constant c u r r e n t dQ/dV a g a i n s t V phase t r a n s i t i o n at V . n  for a f i r s t  order  59  0.0  0.2  0.4  Q /  0.6 Q  0.8  1.0  0  F i g u r e 16 - Constant c u r r e n t dQ/dV a g a i n s t Q f o r a f i r s t phase t r a n s i t i o n , c e l l d i s c h a r g i n g .  order  60  the c u r r e n t i s d e c r e a s e d , peak h e i g h t s  increase rapidly.  s i n g l e phase c a p a c i t y , i n g e n e r a l , to have a constant modified  by d i f f u s i o n e f f e c t s and  To get an i d e a how compare w i t h those  would expect  dQ/dV, o n l y  a constant  IR  the f e a t u r e s observed w i t h c o n s t a n t  slightly  shift. c u r r e n t dQ/dV  observed i n l i n e a r sweep voltammetry, c o n s i d e r a  R e c a l l t h a t i n S e c t i o n 2.2  case.  experiencing  One  i t was  special  found t h a t a l i n e a r sweep v o l t a m -  metry peak would have twice the h e i g h t of s i n g l e phase c a p a c i t y i f the v o l t a g e of a 1 milliamp-hour ^8  c e l l was  y v o l t s / s e c , t a k i n g about 17  hours.  swept over a 0 . 5  particle's  v:61ume. i s AL/y,  we  c u r r e n t of ^ 5 8  At a c o n s t a n t  i n about 17  such a c e l l would a l s o charge or d i s c h a r g e  can r e w r i t e ( 3 . 9 )  v o l t range at  using  hours.  yamps  Since a  (2.14):  -1  vD -  (3.19)  I L  yD  dV  where i n t h i s case dQ/dV i s the i d e a l v a l u e  i n the d i f f u s i o n g r a d i e n t , -9  Once a g a i n , take L 'v 10 ym  i s p r o p o r t i o n a l to dp/dV. and  c o n s i d e r y = 2.  dQ/dV ^ - 7 . 2  Again,  coulombs/volt.  c a p a c i t y i n g e n e r a l , and Then ( 3 . 1 9 ) and  so the c e l l  peak h e i g h t 14  -  v  n  times the h e i g h t  v o l t s wide.  We  s h a l l use  The  %  cm  /sec,  t h i s v a l u e both f o r s i n g l e phase i n the d i f f u s i o n  m i l l i v o l t s , negative  i s discharging.  i s YQo/| l  D ^ 10  2  i n a s i n g l e phase f o r t h i s case we would have  f o r the i d e a l v a l u e  ^ -4  gives  and  and  -100  I f we  of s i n g l e phase c a p a c i t y .  peak i n constant  s i n c e the c u r r e n t i s p o s i t i v e  a g a i n take QQ ^  coulomb/volt.  gradient.  The  0.2  coulombs,  peak h e i g h t  The  the  i s almost  peak i s o n l y 4  milli-'  c u r r e n t dQ/dV i s not o n l y v e r y narrow,  but has  seven times the h e i g h t of a comparable peak i n l i n e a r sweep v o l t a m -  metry.  First  order phase t r a n s i t i o n s now  result  i n sharp peaks t h a t  be f a r e a s i e r to r e s o l v e i n a g i v e n m a t e r i a l than the peaks from  should  linear  sweep voltammetry. Figure- 16  a g a i n has  a unique shape f o r y = 1.  For y = 2 ,  the peak i n  dQ/dV a g a i n s t Q i s a p e r f e c t concave t a i l ,  sawtooth.  F o r y = 3, t h e peak has a s l i g h t l y  but i s q u i t e s i m i l a r t o t h e y = 2 case.  experimentally d i f f i c u l t  I t would a g a i n be  t o d i s t i n g u i s h between t h e two c a s e s , s i n c e t h e  peak t a i l c o u l d c o n c e i v a b l y be curved i n t h e y = 2 case by s i n g l e phase capacity^ and i n t e r c a l a n t g r a d i e n t s between the phase f r o n t s and t h e p a r ticles'  centers.  The peak shapes, l i k e those of t h e peaks p l o t t e d a g a i n s t  V , a r e v e r y d i s t i n c t i v e when compared w i t h s i n g l e phase c a p a c i t y , p a r ticularly  s i n c e t h e peaks have sudden i n i t i a l jumps t o t h e i r maxima.  Charging  and d i s c h a r g i n g over a f i r s t  order phase t r a n s i t i o n a t  c o n s t a n t c u r r e n t r e s u l t s i n a p a i r of sharp peaks a g a i n s t V , w i t h  their  i n i t i a l jumps separated by 2IR and e q u i d i s t a n t from V Q .  The peaks have  t a i l s w i t h a width of r o u g h l y V~ p o i n t i n g away from V Q .  F i r s t order phase  D  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 t o determine q u a n t i t a t i v e l y . sawtoothsor 0 and Q Q .  The peaks a g a i n s t Q a r e f o r y = 2 a p a i r o f  f o r y = 3 c l o s e t o sawtooths i n shape, o v e r l a p p i n g between  T h i s should a l l o w a c l e a r d e t e r m i n a t i o n 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 c o n s t a n t  c u r r e n t , r e c a l l from (3.1)  i s j u s t dt/dV m u l t i p l i e d by the known constant or d i s c h a r g e d by a constant to  I.  If a cell  c u r r e n t source, then a v o l t m e t e r  t h a t dQ/dV i s charged or analogue  d i g i t a l c o n v e r t e r can be used w i t h a microcomputer to monitor  c e l l ' s v o l t a g e and  compute dt/dV.  Both .the apparatus and  the  the  micro-  computer program r e q u i r e d a r e s t r a i g h t f o r w a r d . One member of our r e s e a r c h group (Mulhern, 1982) the n e c e s s a r y  apparatus and  attached  Packard HP3455A D i g i t a l Voltmeter A c e l l was  charged and  through a GPIB bus  d i s c h a r g e d at constant  changed by a t l e a s t one m i l l i v o l t ,  to a H e w l e t t -  current with i t s  dt/dV was  When the v o l t a g e  r e c o r d e d as the time  the v o l t a g e change to occur, d i v i d e d by the v o l t a g e change.  r e c o r d e d were the average time and  I f such a system i s used, the data may  whatever g r a p h i c s or p r i n t i n g system one  A more s o p h i s t i c a t e d system was  was  is willing  be d i s p l a y e d u s i n g  to i n s t a l l .  the I n t e r s i l  developed  An  example  4.1.  to be d e d i c a t e d to the  c u r r e n t dQ/dV f o r f o u r c e l l s s i m u l t a n e o u s l y .  c o n s t r u c t e d by the UBC  programmed by M.  Also  since Q i s linear  of measurements made w i t h t h i s system w i l l be g i v e n i n S e c t i o n  measurement of constant  taken  average v o l t a g e d u r i n g the measurement.  T h i s gave dQ/dV as a f u n c t i o n of both c e l l v o l t a g e and, i n I t , charge.  A  and a Hewlett-Packard HP59309A D i g i t a l  v o l t a g e monitored by the h i g h l y s e n s i t i v e v o l t m e t e r .  for  a b l e to assemble  program the microcomputer i n a day.  T e k t r o n i x 4052 microcomputer was  Clock.  was  A. P o t t s .  P h y s i c s Department E l e c t r o n i c s Shop and  This was  The v o l t a g e i s measured by a c i r c u i t based  on  ICL8068/ICL7104 16 b i t analogue to d i g i t a l c o n v e r t e r , which  has a c o n v e r s i o n  speed of approximately  one  i n p u t s a r e m u l t i p l e x e d to t h i s converter.'r o t a t i o n to whichever i n p u t channels  second. The  Four analogue v o l t a g e  converter  a r e i n use.  i s attached i n  The m u l t i p l e x i n g and  conversion  i s c o n t r o l l e d by a Cromemco S i n g l e Card Computer w i t h i t s  program r e s i d i n g i n EPROMs ( E r a s a b l e Programmable Read Only Memories). The  d i g i t a l output  of the converter  i s used by t h e computer t o g e t h e r  an i n t e r n a l c l o c k t o compute dt/dV f o r each a c t i v e input channel.  with  The  i n f o r m a t i o n i s sent by t h e computer t o e i g h t T e l e d y n e - P h i l b r i c k Model 4025 12 b i t d i g i t a l t o analogue c o n v e r t e r s .  The c o n v e r t e r outputs a r e  d i s p l a y e d u s i n g X-Y r e c o r d e r s t o r e c o r d dt/dV a g a i n s t V , and time base r e c o r d e r s t o r e c o r d dt/dV a g a i n s t time, and hence a g a i n s t Q. The  input v o l t a g e s a r e assumed t o f a l l  between 0 and 3 v o l t s .  g i v e s t h e 16 b i t c o n v e r t e r a r e s o l u t i o n of ^46 u v o l t s . outputs  The computer only  dt/dV i f t h e v o l t a g e has changed by a t l e a s t 0.5 m i l l i v o l t s .  12 b i t dt/dV output  converter  outputs  between 5 v o l t s and -5 v o l t s .  computer s c a l e s dt/dV so t h a t t h i s output 8 8 between 10 s e c / v o l t and -10 s e c / v o l t . power of two from 1 t o 256.  divided- by t h e m u l t i p l i e r .  The  range corresponds t o a r e s u l t A m u l t i p l i e r may be s e t t o any  range then corresponds t o ±10  sec/volt  T h i s p a r t i c u l a r c h o i c e of ranges has s u c c e s s -  f u l l y handled v i r t u a l l y every c e l l v o l t a g e output  The  The computer s c a l e s up i t s dt/dV output by g  t h i s amount, and the ±5 v o l t output  The  This  experimental  s i t u a t i o n encountered thus f a r .  i s between 0 and 3 v o l t s ,  so those  12 b i t con-  v e r t e r s have a r e 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 r e q u i r e d change i n v o l t a g e o f 0.5 m i l l i v o l t s ensures t h a t n o i s e i n dt/dV due t o t h e r e s o l u t i o n l i m i t of t h e i n p u t c o n v e r t e r  i s mini-  mized, and i t ensures t h a t t h e computer cannot t r y t o d i v i d e by z e r o . reduce n o i s e , some a v e r a g i n g input converterrtakes  o f the "input v o l t a g e i s done.  channels a r e i n u s e . cell  Since the  about one second t o make a measurement, each  v o l t a g e i s measured approximately  every  To  f o u r seconds i f a l l f o u r  cell  input  A group of such v o l t a g e samples f o r a p a r t i c u l a r  i s c o l l e c t e d and averaged t o o b t a i n a v o l t a g e w i t h l e s s n o i s e , to be  used i n c a l c u l a t i n g dt/dV. between 1 and 132. result.  The number of samples may  T y p i c a l l y , around  be any power of  16 samples produces a v e r y  two  clear  The time taken f o r an average i s c l e a r l y s m a l l compared t o the  t o t a l time i n which an experiment  i s performed.  N o i s e i s a l s o reduced by  h o l d i n g a c e l l a t a constant temperature i n a Haake F3 temperature b a t h . Experiments were performed u s i n g the f l a n g e c e l l s t h a t were d e s c r i b e d i n Section  1.4.  65  3.4  R e s u l t s f o r L i VS„ x 2 A s e r i e s of dQ/dV measurements were made on t h e c e l l GJ-9 u s i n g t h e  equipment d e s c r i b e d i n the l a t t e r p a r t of S e c t i o n 3.3. for  The v o l t a g e  curve  t h i s L i ^ V ^ c e l l was p r e v i o u s l y 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 a g a i n s t Q curve was shown i n f i g u r e 10.  The t r a n s i t i o n v o l t a g e s and range i n x of t h e  two phase r e g i o n s were d i s c u s s e d  i n S e c t i o n 2.4 based on these f i g u r e s .  Constant c u r r e n t dQ/dV measurements were made on c e l l GJ-9 a t 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 h e l d c o n s t a n t  capacity d e c l i n e d during  a t (20.0 ± 0.5) °C.  the measurements by about 20 p e r c e n t .  17 shows dQ/dV a g a i n s t c e l l v o l t a g e measured a t 35 yamps. shows dQ/dV a g a i n s t Q and x f o r t h i s case.  peaks a r e v i s i b l e i n f i g u r e 18. took 99.8 hours.  Three p a i r s of remarkably  F i g u r e 19 d i r e c t l y compares t h e l i n e a r sweep  a charge or d i s c h a r g e , w i t h constant  gain i n r e s o l u t i o n i s c l e a r .  v o l t a g e measured a t 140 yamps. for  t h i s case.  hours.  59 hours,  c u r r e n t dQ/dV measured a t 52.5  yamps, f o r which t h e charge took 62.3 hours and t h e d i s c h a r g e The  p a i r s of  The charge took 104.3 hours w h i l e t h e  voltammogram p r e v i o u s l y shown i n f i g u r e 9, which took r o u g h l y for  Figure  F i g u r e 18  c l e a r peaks a r e v i s i b l e i n f i g u r e 17, and t h r e e c o r r e s p o n d i n g  discharge  The c e l l ' s  60.8 hours.  F i g u r e 20 shows dQ/dV a g a i n s t  cell  F i g u r e 21 shows dQ/dV a g a i n s t Q--and x" ': ,-  The charge took 21.6 hours, w h i l e t h e d i s c h a r g e  took 20.7  The peaks a r e s t i l l c l e a r and d i s t i n c t i v e a t 140 yamps.  When  examining t h e data i n f i g u r e s 17 through 21, i t should be noted t h a t w h i l e t h e l i n e a r sweep voltammogram i n f i g u r e 9 was made a t 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  c u r r e n t dQ/dV measurements c o u l d  be made e a s i l y w i t h t h e c u r r e n t an order The  of magnitude below 35 yamps.  peaks a g a i n s t v o l t a g e i n these f i g u r e s a r e q u a l i t a t i v e l y  very  '66  800 o > 600 Z5  o o  400  h  > 200  2.3 Cell Voltage  2.4  2.5  (volts)  F i g u r e 17 - Constant c u r r e n t dQ/dV a g a i n s t c e l l v o l t a g e f o r t h e 31st charge ( s o l i d l i n e ) and d i s c h a r g e (dashed l i n e ) of the L i V S 2 c e l l GJ-9, a t a c u r r e n t of 35 yamps. x  2.6  Q 12  1.0  10  0.8  (coulombs)  8  6  4  0.6  0.4  X in L i V S x  2  0.2  0  0.0  2  F i g u r e 18 - Constant c u r r e n t dQ/dV a g a i n s t x and Q f o r t h e 31st charge ( s o l i d l i n e ) and d i s c h a r g e (dashed l i n e ) o f t h e L i VS^ c e l l GJ-9, a t a c u r r e n t of 35 yamps. I n d i v i d u a l data p o i n t s a r e shown when x changes by more than ^0.005 between p o i n t s .  68  2.3 Cell Voltage  2.3 Cell Voltage i g u r e 19 - Comparison  2.4  2.5  2.6  2.5  2.6  (volts)  2.4 (volts)  of l i n e a r sweep voltammetry and c o n s t a n t c u r r e n t dQ/dV. (a) The same voltammogram as i n f i g u r e 9, showing the 8 t h charge ( s o l i d l i n e ) and d i s c h a r g e (dashed l i n e ) f o r c e l l GJ-9, each t a k i n g 59 hours. (b) Constant c u r r e n t dQ/dV a g a i n s t c e l l v o l t a g e f o r t h e 32nd charge ( s o l i d l i n e ) and d i s c h a r g e (dashed l i n e ) f o r c e l l GJ-9, each t a k i n g 62 hours a t a c u r r e n t of 52.5 yamps.  F i g u r e 20 - Constant c u r r e n t dQ/dV a g a i n s t c e l l v o l t a g e f o r t h e 29th charge ( s o l i d l i n e ) and d i s c h a r g e (dashed l i n e ) of t h e L i VS„ c e l l GJ-9, a t a c u r r e n t of 140 yamps.  Q 10 2 5 0  8  (coulombs) 6  4  2  0  n—i—i—i—i—i—i—i—i—i—i—r  1.0  0.8  0.6  0.4  X in L i V S x  0.2  0.0  2  F i g u r e 21 - Constant c u r r e n t dQ/dV a g a i n s t x and Q f o r t h e 29th charge ( s o l i d l i n e ) and d i s c h a r g e (dashed l i n e ) of the L i c e l l GJ-9, a t a c u r r e n t of 140 yamps.  71  s i m i l a r i n shape t o t h e i d e a l peaks d i s c u s s e d  i n S e c t i o n 3.2.  have a rr-api'd* i n i t i a l r i s e f o l l o w e d by a t a i l . w i t h t h e c u r r e n t w h i l e t h e peak h e i g h t  The peaks  The peak w i d t h d e c r e a s e s  i n c r e a s e s , as expected.  The peaks  a g a i n s t Q a r e n o t i c a b l y more rounded than t h e i d e a l sawtooths p r e d i c t e d i n S e c t i o n 3.2. 3.5.  Non-ideal  peak shapes w i l l be d i s c u s s e d f u r t h e r i n S e c t i o n  The peaks a r e s t i l l  of f i r s t  s u f f i c i e n t l y d i s t i n c t i v e t o be unambiguous s i g n a l s  o r d e r phase t r a n s i t i o n s .  Notice the small precursors  t o the  t r a n s i t i o n peaks. It  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 t h a t t h e r e a r e t h r e e f i r s t  order  phase t r a n s i t i o n s i n L i VS„, not f o u r as was b e l i e v e d on t h e b a s i s of l i n e a r x 2 sweep voltammetry d a t a . 2.325 v o l t s .  There i s no f i r s t  o r d e r phase t r a n s i t i o n near  There i s a n o t i c a b l e bump i n t h e s i n g l e phase c a p a c i t y near  t h i s v o l t a g e , however t h e i r a r e no peaks.  The bump appears t o have t h e  same shape on charge and d i s c h a r g e , and remains t h e same h e i g h t to  s i n g l e phase c a p a c i t y independent of c u r r e n t .  R e c a l l i n S e c t i o n 2.4  t h a t t h e bump was a t t r i b u t e d t o a B + IT two phase r e g i o n . phase r e g i o n does n o t appear t o e x i s t .  relative  T h i s two  S i n c e t h e B phase i s a m o n o c l i n i c  d i s t o r t i o n o f t h e IT hexagonal u n i t c e l l ,  i t would appear v e r y  likely  that  the t r a n s i t i o n between B and IT i s second o r d e r , t h e change from one u n i t c e l l t o t h e other o c c u r r i n g i n a continuous  manner.  Further i n v e s t i g a t i o n  of t h e change between t h e B and IT phases, i n c l u d i n g c o n f i r m a t i o n of t h e x-ray  data  of Murphy et a l (1977) d i s c u s s e d  i n S e c t i o n 2.4, i s n e c e s s a r y  i n o r d e r t o c o n f i r m o r deny t h i s h y p o t h e s i s . second order t r a n s i t i o n would become f i r s t If  I t i s p o s s i b l e t h a t such a  order a t lower temperatures.  t h e t r a n s i t i o n i s second o r d e r , then t h e B + IT two phase r e g i o n i n t h e  phase diagram shown i n f i g u r e 13 should be r e p l a c e d by a second order near x =  0.6.  w i t h constant  The t h r e e f i r s t  line  o r d e r phase t r a n s i t i o n s t h a t a r e v i s i b l e  c u r r e n t dQ/dV appear i n roughly  t h e same p l a c e s noted  earlier  i n S e c t i o n 2.4, so t h e r e a r e no o t h e r major m o d i f i c a t i o n s t o be made t o the phase diagram. We s h a l l now c o n s i d e r results.  I n order  q u a n t i t a t i v e l y t h e constant  current  t o determine t h e t r a n s i t i o n v o l t a g e s ,  dQ/dV  the p o s i t i o n  of each peak's l e a d i n g edge was p l o t t e d as a f u n c t i o n of c u r r e n t .  The  base of t h e i n i t i a l r i s e was used, i g n o r i n g t h e s m a l l p r e c u r s o r s .  Extra-  p o l a t i n g t h e peak p o s i t i o n t o zero c u r r e n t e l i m i n a t e s the IR s h i f t . for  Data  the t h r e e phase t r a n s i t i o n s i s shown i n f i g u r e s 22, 23, and 24.  Linear least  squares f i t s a r e a l s o shown.  F o r each o f t h e two peaks a t  h i g h e r v o l t a g e s , a l l of t h e data was used i n t h e f i t except f o r t h e p o i n t s at  140 yamps.  At 140 yamps the peak shapes were n o t i c a b l y l e s s i d e a l  at  lower c u r r e n t s , moving t h e l e a d i n g edges c l o s e r t o t h e t r a n s i t i o n  v o l t a g e and somewhat f l a t t e n i n g  the i n i t i a l ramp.  The l e a s t squares  than  fits  shown i n f i g u r e 24 t o t h e data f o r t h e low v o l t a g e peak o n l y make use of the data  f o r t h e t h r e e lowest c u r r e n t s .  At h i g h e r  c u r r e n t s , t h e peak  shape i s p o o r l y d e f i n e d due t o t h e presence o f IT s i n g l e phase c a p a c i t y , so the l e a d i n g edges become more and more d i f f i c u l t tainty.  N o t i c e t h a t s l o p e s of t h e l e a s t  squares f i t s w i l l equal  we can determine t h e c e l l r e s i s t a n c e d u r i n g On d i s c h a r g e ,  to locate with  cer-  1/R, so  each phase t r a n s i t i o n .  t h e r e i s e s s e n t i a l l y no c a p a c i t y a t v o l t a g e s above t h e  h i g h e s t v o l t a g e phase t r a n s i t i o n .  Shown i n f i g u r e 22, t h e best  f i t line  to t h e peak's l e a d i n g edge g i v e s a c e l l r e s i s t a n c e R = (195 ± 15) ohms, and  t h e zero c u r r e n t  i n t e r c e p t i s (2.466 ± 0.001) v o l t s .  The same t r a n -  s i t i o n oh charge g i v e s a c e l l r e s i s t a n c e o f R = (95 ± 10) ohms and a zero current  i n t e r c e p t of ("2.473 ± 0.001) v o l t s .  by 7 m i l l i v o l t s .  Notice the intercepts d i f f e r  We s h a l l take (2.470 ± 0.003) v o l t s t o be t h e t r a n s i t i o n  v o l t a g e , and comment on t h e d i f f e r e n c e between t h e i n t e r c e p t s s h o r t l y . Notice  that the c e l l  r e s i s t a n c e when t h e c e l l  i s s t a r t i n g i t s discharge  73  2.43  2.45 Voltage  of  2.47 Leading  2.49 Edge  2.51  (volts)  F i g u r e 22 - The p o s i t i o n of the l e a d i n g edges of the charge ( t r i a n g l e s ) and d i s c h a r g e (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 a r e l e a s t squares f i t s used to e x t r a p o l a t e t o z e r o c u r r e n t , e l i m i n a t i n g the IR s h i f t s . Data was taken on c e l l GJ-9. A l l v o l t a g e s a r e ±1 m i l l i v o l t .  74  2.35  2.37  2.39  Voltage of Leading  2.41 Edge  2.43  (volts)  F i g u r e 23 - The p o s i t i o n of the l e a d i n g edges of the charge ( t r i a n g l e s ) and d i s c h a r g e (squares) peaks f o r 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 a r e 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 v o l t a g e s a r e ±1 m i l l i v o l t .  75  2.16  2.18  2.20  Voltage of Leading  2.22 Edge  2.24  (volts)  F i g u r e 24 - The p o s i t i o n of the l e a d i n g edges of the charge ( t r i a n g l e s ) and d i s c h a r g e (squares) peaks f o r 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 a r e 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 v o l t a g e s a r e ±1 m i l l i v o l t .  76  i s double  the r e s i s t a n c e when c h a r g i n g .  F i g u r e 23 shows t h e b e s t f i t l i n e s f o r t h e t r a n s i t i o n near 2.38 volts.  On d i s c h a r g e , t h e c e l l r e s i s t a n c e i s R = (95 ± 25) ohms and t h e  zero current i n t e r c e p t  i s (2.382 ± 0.002) v o l t s .  On c h a r g e , t h e c e l l  r e s i s t a n c e i s R = (100 ± 15) ohms and the zero c u r r e n t i n t e r c e p t i s (2.387 ± 0.001) v o l t s . s h a l l take  The i n t e r c e p t s d i f f e r by 5 m i l l i v o l t s .  (2.384 ± 0.003) v o l t s  We  t o be the t r a n s i t i o n v o l t a g e .  The  c e l l r e s i s t a n c e i s roughly t h e same on charge and d i s c h a r g e , and r o u g h l y the same as t h e c e l l r e s i s t a n c e found when c h a r g i n g through volt  the 2.470  transition. F i g u r e 24 shows t h e best f i t l i n e s f o r t h e t r a n s i t i o n near 2.20  volts.  On d i s c h a r g e , t h e c e l l r e s i s t a n c e i s R''= (115 ± 10) ohms and t h e  zero c u r r e n t i n t e r c e p t  i s (2.200 ± 0.001) v o l t s .  On charge,  e s s e n t i a l l y no c a p a c i t y a t v o l t a g e s below t h i s t r a n s i t i o n .  there i s The c e l l  r e s i s t a n c e i s R = (200 ± 15) ohms and t h e zero c u r r e n t i n t e r c e p t i s (2.203 ± 0.001) v o l t s .  The i n t e r c e p t s d i f f e r by 3 m i l l i v o l t s .  take t h e t r a n s i t i o n v o l t a g e t o be (2.201 ± 0.003) v o l t s . tance when s t a r t i n g when s t a r t i n g  t o charge t h e c e l l  t o discharge the c e l l .  the r e s i s t a n c e of a l l o t h e r cases.  i s similar  We  shall  The c e l l  resis-  t o the c e l l r e s i s t a n c e  This r e s i s t a n c e i s roughly  double  W i t h the e x c e p t i o n of the i n i t i a l  r e s i s t a n c e of about 200 ohms on both charge and discharge,- t h e c e l l tance appears t o be r o u g h l y a c o n s t a n t The  100 ohms.  d i f f e r e n c e between t h e charge and d i s c h a r g e i n t e r c e p t s changes  from 3 m i l l i v o l t s f o r t h e 2.201 v o l t 2.384 v o l t The  resis-  transition,  to 5 m i l l i v o l t s f o r the  t r a n s i t i o n , and t o 7 m i l l i v o l t s f o r the 2.470 v o l t  transition.  bases of the peaks' i n i t i a l r i s e were l o c a t e d i n t h e same manner f o r  each t r a n s i t i o n , so w h i l e a 3 m i l l i v o l t d i f f e r e n c e c o u l d be e x p l a i n e d away as i n a c c u r a t e l y i d e n t i f y i n g  t h e p o i n t s a t which each i n i t i a l  rise  began, the 5 and 7 m i l l i v o l t such an argument. the  d i f f e r e n c e s cannot be e a s i l y e l i m i n a t e d by  To understand what i s o c c u r r i n g , we need t o c o n s i d e r  ranges i n x of the two phase r e g i o n s . The data shown i n f i g u r e 18 and t h e c o r r e s p o n d i n g data measured a t  other c u r r e n t s was examined  t o determine t h e ranges i n x of t h e s i n g l e  phase and two phase r e g i o n s .  The peaks due t o t h e phase t r a n s i t i o n s a r e ,  as mentioned, q u i t e rounded compared t o the i d e a l sawtooth, making i t difficult of  t o determine x v a l u e s t o b e t t e r than one p e r c e n t .  x quoted .below a r e ±0.01.  t h a t both t h e V S  2  x =1.  I t was found on both charge and d i s c h a r g e  and 3S phases have widths i n x of no more than 0.01.  R e c a l l that the V S at  2  phase appears a t x-= 0 and t h a t t h e 3S phase  On charge, the 2.470 v o l t  0.01 <_ x <_ 0.33.  This i s the V S  2  appears  t r a n s i t i o n o c c u r s i n t h e range  + a two phase r e g i o n .  t r a n s i t i o n o c c u r s i n t h e range 0.34 j< x <^ 0.52. phase r e g i o n .  A l l values  The 2.381 v o l t  T h i s i s t h e a + g two  The a s i n g l e phase r e g i o n has a width of no more than 0.01  at  x = 0.33.  S i n g l e phase c a p a c i t y due t o t h e g and IT phases o c c u r s  in  t h e range 0.52 < x < 0.90.  range 0.90 <_ x <_ 0.99.  The 2.201 v o l t t r a n s i t i o n o c c u r s i n t h e  T h i s i s t h e IT + 3S two phase  region.  On d i s c h a r g e , t h e r e s u l t s a r e s l i g h t l y d i f f e r e n t .  The V S  2  + a two  phase r e g i o n o c c u r s i n t h e range 0.01 <_ x _< 0.28.  Also,- t h e a + 3 two  phase r e g i o n o c c u r s i n t h e range 0.29 _< x _< 0.43.  The a s i n g l e  r e g i o n i s s t i l l v e r y narrow. r e g i o n on d i s c h a r g e i s 85 the of  The range i n x of t h e V S  percent  2  phase  + a two phase  of t h e range on charge.  Similarly,  range i n x of t h e a + g two phase r e g i o n on d i s c h a r g e i s 80 p e r c e n t t h e range on charge.  The IT + 3S r e g i o n s t i l l  lies  i n t h e range .  0.90 <_ x <^ 1.00, a l t h o u g h a d i f f e r e n c e of a few p e r c e n t i n w i d t h compared to  t h e range on charge may be p o s s i b l e g i v e n t h e e r r o r a s s i g n e d t o t h e  endpoints.  To compensate f o r t h e reduced ranges i n x of the two phase  r e g i o n s , the s i n g l e phase c a p a c i t y due d i s c h a r g e over charge It  0.43  _< x _< 0.90,  to the B and  a 25 p e r c e n t  IT phases ranges on  i n c r e a s e i n w i d t h over  the  i s c l e a r t h a t t h e r e i s s i g n i f i c a n t h y s t e r e s i s i n x f o r both  the  case.  + ct and  a -f $ two  phase r e g i o n s .  independent of the c u r r e n t .  T h i s h y s t e r e s i s appears to  A l s o r e c a l l t h a t t h e r e were s m a l l d i f f e r e n c e s  between the zero c u r r e n t e x t r a p o l a t i o n s f o r the l e a d i n g edges of charge and two  the  d i s c h a r g e peaks t h a t cannot be e a s i l y e x p l a i n e d away f o r these  phase r e g i o n s .  w e l l as i n x. it  be  Thus, t h e r e i s evidence  of h y s t e r e s i s i n v o l t a g e  I f h y s t e r e s i s i s o c c u r r i n g i n the  i s not dramatic  enough to be unambiguously  McKinnon (1982) has d i s s i p a t e d when one  IT + 3S two  phase r e g i o n ,  detected.  shown f o r i n t e r c a l a t i o n systems t h a t i f energy i s  phase i s c o n v e r t e d  i n t o another,  then the two  phases  can c o e x i s t over a range of v o l t a g e s and  of l i m i t s i n x of the two  region.  t h a t heat w i l l be generated  I t i s very reasonable  p l a s t i c deformation Li^V^.  The  to expect  energy l o s s depresses  e q u i l i b r i u m v o l t a g e and  any  phase by  d u r i n g 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 the v o l t a g e below the expected  rium v o l t a g e on d i s c h a r g e , as shown i n f i g u r e 25.  generated,  equilib-  The a r e a between the  the s h i f t e d v o l t a g e i s p r o p o r t i o n a l to the  so the s h i f t must be constant  heat  s i n c e the phase c o n v e r s i o n  g i v e n amount of m a t e r i a l w i l l generate  the p o s i t i o n of the phase f r o n t .  the same heat,  Such a s h i f t  the e q u i l i b r i u m v o l t a g e one would n o r m a l l y  i s independent of c u r r e n t .  expect.  w e l l d i f f e r from the s h i f t on d i s c h a r g e .  the two  The  s h i f t on  charge  occur  i f the e q u i l i b r i u m  i n the s i n g l e phase r e g i o n s on e i t h e r s i d e of the  phase r e g i o n s , as shown i n f i g u r e  25.  above  D i f f e r e n t widths i n x f o r  phase r e g i o n s on charge and d i s c h a r g e w i l l  dV/dx i s d i f f e r e n t  of  r e g a r d l e s s of  S i m i l a r l y , on charge the v o l t a g e w i l l be s h i f t e d by a constant  may  as  two  79  X  F i g u r e 25 - H y s t e r e s i s i n v o l t a g e and x due to energy d i s s i p a t i o n i n a two phase r e g i o n . Charge and d i s c h a r g e v o l t a g e s a r e d i s p l a c e d from the expected e q u i l i b r i u m v a l u e (dashed l i n e ) . S i n c e dV/dx i s d i f f e r e n t i n the s i n g l e phase r e g i o n s , the two phase r e g i o n s a r e of d i f f e r e n t l e n g t h s .  80  The  behavior  of t h e 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  c o n s i s t e n t w i t h t h e h y s t e r e s i s d i s c u s s e d by McKinnon.  The h y s t e r e s i s  i n v o l t a g e 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 affect  the ranges i n x of the two phase r e g i o n s .  t h a t appears on charge reappears The  to noticeably  Some two phase c a p a c i t y  as s i n g l e phase c a p a c i t y on d i s c h a r g e .  t r a n s i t i o n voltages given e a r l i e r are a c t u a l l y centered  i n small  w i t h i n t h e quoted e r r o r , i n which t h e two phases can c o e x i s t . t e r e s i s i n x causes phase boundaries on whether charge or d i s c h a r g e It peaks. V^,  is difficult  depending  considered.  t o d e r i v e f u r t h e r q u a n t i t a t i v e i n f o r m a t i o n from the  The widths o f t h e peaks a g a i n s t v o l t a g e a r e expected  so t h e widths should decrease  widths do, i n g e n e r a l , decrease  l i n e a r l y with current.  monotonically  t o be r o u g h l y  W h i l e t h e peak  with current, at higher  c u r r e n t s s i n g l e phase c a p a c i t y d i s t o r t s t h e peak shapes. the peaks' i n i t i a l  The hys-  i n t h e phase diagram t o s h i f t  i s being  ranges,  At lower c u r r e n t s  i n c r e a s e s , which a r e ramps r a t h e r than jumps, d i s t o r t  the widths of t h e narrow peaks s u f f i c i e n t l y t o g i v e q u a n t i t a t i v e measurements a h i g h u n c e r t a i n t y . The peak h e i g h t  i d e a l l y has a maximum v a l u e of ~ Y Q Q / | V ^ | ,  graphed t h e maximum peak h e i g h t example s e t o f data i s p r e s e n t e d + ct two phase r e g i o n .  should be p r o p o r t i o n a l to  so when An  QQ/|I|.  i n f i g u r e 26 f o r t h e peaks of t h e  A g a i n , a t h i g h c u r r e n t s the data i s poor s i n c e  the peak shapes a r e not as c l o s e t o i d e a l as they a r e a t low c u r r e n t s . At low c u r r e n t s the peak h e i g h t v a r i e s by as much as t e n percent measurement t o measurement, s i n c e t h e r e s o l u t i o n l i m i t i s being reached  of t h e apparatus  and t h e peak shapes a r e s t i l l not i d e a l .  h e i g h t s shown i n f i g u r e 26 a r e r o u g h l y  linear i n Q / | l | ,  The peak as i s c o r r e s p o n d i n g  data f o r t h e o t h e r t r a n s i t i o n peaks, but t h e data i s not a c c u r a t e to  f i t with a straight  from  l i n e and o b t a i n a r e l i a b l e s l o p e .  enough  S i n c e t h e peak  81  800  o > oo 6 0 0 _Q  13 O O  400  >  TD  •o 2 0 0  0  0  20  40  Q  0  / |I|  60  80  100  (hours)  F i g u r e 26 - Peak h e i g h t a g a i n s t Q_/JI[ f o r the charge ( t r i a n g l e s ) and d i s c h a r g e (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 s l o p e s 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 f o r the phases n u c l e a t e d on the s u r f a c e s of the p a r t i c l e s i n each case.  One would expect  t h a t the d i s c h a r g e peak f o r  + CL r e g i o n would have r o u g h l y the same h e i g h t as the charge  the  peak  f o r the a + 3 r e g i o n , s i n c e i n b o t h cases the a phase i s n u c l e a t e d on particles'  surfaces.  A l s o , i f dp/dV doesn't  the  change much i n the l a r g e  s i n g l e phase r e g i o n , the d i s c h a r g e peak f o r the a + 3 r e g i o n and the  charge  peak f o r the IT + 3S r e g i o n would have r o u g h l y the same h e i g h t i f D has not changed s i g n i f i c a n t l y .  T h i s i s p l a u s i b l e , s i n c e no f i r s t  change occurs between 3 and  IT.  order  structural  N o t i n g t h a t the peak h e i g h t s a l s o  scale  w i t h each t r a n s i t i o n ' s Q Q ,  the d a t a i n f i g u r e 17 i s i n q u a l i t a t i v e  ment w i t h these comments.  I t would a l s o appear t h a t D dp/dV i n c r e a s e s  markedly w i t h v o l t a g e .  T h i s may  at low x, but i t i s not known how  be due  to improved d i f f u s i o n f o r phases  dp/dV i s changing.  I t i s v e r y c l e a r t h a t c o n s t a n t c u r r e n t dQ/dV outperforms voltammetry i n c l a r i t y and r e s o l u t i o n . example, was  agree-  The presence  l i n e a r sweep  of h y s t e r e s i s , f o r  not d e t e c t e d by l i n e a r sweep voltammetry.  The  transition  v o l t a g e s o b t a i n e d w i t h c o n s t a n t c u r r e n t dQ/dV a r e s l i g h t l y h i g h e r those o b t a i n e d w i t h l i n e a r sweep voltammetry.  In t h i s s e c t i o n ,  than  the  t r a n s i t i o n v o l t a g e s o b t a i n e d w i t h c o n s t a n t c u r r e n t dQ/dV were found be (2.470 ± 0.003) v o l t s , volts.  In s e c t i o n 2.4,  (2.384 ± 0.003) v o l t s , and  (2.190 ± 0.005) v o l t s .  (2.460 ± 0.005) v o l t s ,  (2.380 ± 0.005) v o l t s  The r e s u l t s a r e w i t h i n the quoted  each other f o r the t r a n s i t i o n near 2.38 to the quoted  (2.201 ± 0.003)  the t r a n s i t i o n v o l t a g e s o b t a i n e d w i t h l i n e a r sweep  voltammetry were r e s p e c t i v e l y and  to  v o l t s , and  e r r o r l i m i t s f o r the o t h e r two  voltammetry were a t room temperature,  the r e s u l t s are c l o s e  transitions.  d i s c r e p a n c i e s c o u l d a r i s e because the c e l l s used  e r r o r of  The  slight  f o r the l i n e a r sweep  r a t h e r than i n a temperature  bath  83  at ing  20°C.  The  output  v o l t a g e of the dQ/dV apparatus was  the measurements d e s c r i b e d  l e s s than 3 m i l l i v o l t s was for  this error.  The  PAR  found.  The  data  an e r r o r of  i n t h i s t h e s i s was  equipment d e s c r i b e d i n s e c t i o n 2.3  c a l i b r a t i o n e r r o r i n i t s output The  i n t h i s t h e s i s , and  calibrated follow-  v o l t a g e of a few  slightly corrected  may  have had  millivolts.  ranges i n x f o r the t h r e e phase t r a n s i t i o n s as o b t a i n e d  sweep voltammetry agree w i t h the ranges o b t a i n e d c u r r e n t dQ/dV.  on charge u s i n g  R e c a l l t h a t the l i n e a r sweep I a g a i n s t Q had  d i s c h a r g e peaks a r e o n l y c l e a r enough i n constant  the c a s e w i t h l i n e a r sweep voltammetry. t h a t at 20°C t h e r e i s no phases as had  first  linear  constant  The  c u r r e n t dQ/dV f o r the  F i n a l l y , note t h a t peaks due  phase t r a n s i t i o n s a r e unambiguous i n c o n s t a n t  by  the c l e a r e s t  peaks on charge, so they were used to determine the ranges i n x.  h y s t e r e s i s to be d e t e c t e d .  a  to f i r s t  order  c u r r e n t dQ/dV, which i s not  Constant c u r r e n t dQ/dV demonstrated  order phase t r a n s i t i o n between the (3 and  been b e l i e v e d on the b a s i s of l i n e a r  sweep voltammetry.  IT  84  3.5  Non-ideal  Peak Shapes  In S e c t i o n 3.4, i t was noted t h a t t h e peaks i n dQ/dV due t o f i r s t order phase t r a n s i t i o n s do not have e x a c t l y t h e shapes p r e d i c t e d i n S e c t i o n 3.2.  The peaks a g a i n s t v o l t a g e do not simply jump t o t h e i r maximum v a l u e ,  although  t h e peaks may be v e r y  sharp.  There i s a s l o p e t o t h e l e a d i n g  edge of t h e peaks which, as noted i n S e c t i o n 3.4, makes i t d i f f i c u l t t o determine t h e widths of narrow peaks a c c u r a t e l y . rounded a t t h e i r maxima, but t h e i n i t i a l from t h e i d e a l peak shape. each t r a n s i t i o n peak. T h i s low upturn  cathode m a t e r i a l .  s l o p e i s t h e most obvious d e v i a t i o n  In a d d i t i o n , there i s a small precursor t o  T h i s occurs  t o some extent  The edge.  i n a l l materials tested.  w e l l b e f o r e t h e t r a n s i t i o n v o l t a g e i s reached by the No e x p l a n a t i o n has been found f o r these  i s n o t c l e a r why such an e f f e c t would occur  the t r a n s i t i o n  slightly  i n -dQ/dV i s n o t i c a b l e i n t h e f i g u r e s 17, 19, and 20, f o r  example, and begins  It  Peaks may appear  precursors.  i n t h e s i n g l e phase  before  begins.  peaks a g a i n s t Q a r e n o t i c a b l y rounded w i t h a prominent l e a d i n g  They appear f a r l e s s i d e a l than the v o l t a g e peaks.  l a t e about t h e source  of t h e n o n - i d e a l b e h a v i o r .  One can specu-  The rounding  of t h e peaks  i n d i c a t e s t h a t t h e cathode p a r t i c l e s a r e not a l l n u c l e a t i n g t h e new phase on t h e i r s u r f a c e s a t t h e same time. One p o s s i b i l i t y  Two p o s s i b i l i t i e s  i s t h a t some p a r t i c l e s i n the cathode f i n d  n u c l e a t e t h e new phase than o t h e r p a r t i c l e s . if  suggest  themselves.  i t easier to  This could conceivably  the phase n u c l e a t i o n o c c u r r e d more r e a d i l y f o r , say, s m a l l  than l a r g e p a r t i c l e s due t o d i f f e r e n c e s i n s t r a i n e n e r g i e s .  occur  particles Experiments  i n which o n l y a l i m i t e d range of p a r t i c l e s i z e s i s used i n the cathode have been performed by P.J. Mulhern w i t h L i M0S2, but they do not demons t r a t e c o n v i n c i n g l y t h a t t h e peak shape i s sharper distribution.  f o r a narrow p a r t i c l e  I n a d d i t i o n , t h e t h r e e phase t r a n s i t i o n s i n L i VS„ have  85 very similar structural  peak shapes a g a i n s t Q , and i t i s not c l e a r why  phase t r a n s i t i o n s  with different  different  h y s t e r e s i s should experience  such s i m i l a r n u c l e a t i o n problems, p a r t i c u l a r l y s i n c e the d i f f e r e n t amounts of h y s t e r e s i s i n d i c a t e t h a t t h e p l a s t i c d e f o r m a t i o n i n v o l v e d i n the  phase c o n v e r s i o n 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  transition.  A second p o s s i b i l i t y i s t h a t t h e cathode p a r t i c l e s a r e not a l l at the  same v o l t a g e because of r e s i s t i v e e f f e c t s .  then p a r t i c l e s a t d i f f e r e n t  I f the cathode i s t h i c k ,  depths i n the cathode w i l l e x p e r i e n c e d i f f e r e n t  r e s i s t a n c e s , b o t h because of t h e e l e c t r i c a l r e s i s t a n c e added  by t h e cathode  m a t e r i a l between a p a r t i c l e and t h e cathode's s u b s t r a t e , and because of r e s i s t a n c e caused by any d i f f i c u l t i e s electrolyte.  Such a case c o u l d o c c u r simply by h a v i n g l a r g e c r y s t a l l i t e s  that are d i f f i c u l t p a r t i c l e would would  i n g e t t i n g good c o n t a c t w i t h t h e  t o spread e v e n l y on t h e s u b s t r a t e .  experience a d i f f e r e n t  I t would  of t h e time a t which t h e p a r -  voltage.  be u s e f u l t o a c c u r a t e l y model a t h i c k cathode, however t h a t  w i l l not be attempted here.  I n s t e a d , we s h a l l p i c k a s i m p l e d i s t r i b u t i o n  f o r t h e time and see i f the r e s u l t i n g to t h e e x p e r i m e n t a l l y observed peaks. f o r mathematical s i m p l i c i t y .  peak shapes a r e q u a l i t a t i v e l y We s h a l l use a Gaussian  I f there i s a q u a l i t a t i v e  d i s t r i b u t i o n might  d  0  V. D  chosen  The i d e a l peak a g a i n s t Q i s a sawtooth  between 0 and QQ w i t h a maximum h e i g h t of -2QQ/[V" | .  2Q  distribution  indeed be used t o e x p l a i n t h e peak shapes.  C o n s i d e r t h e case y = 2 .  be d i s c h a r g i n g .  similar  s i m i l a r i t y to the  • e x p e r i m e n t a l peaks, t h i s merely i n d i c a t e s t h a t a more c a r e f u l l y  dV  each  IR s h i f t , and t h e i d e a l peak shapes  be c o n v o l u t e d w i t h t h e d i s t r i b u t i o n  t i c l e s reached t h e t r a n s i t i o n  In e f f e c t ,  Take t h e c e l l t o  From (3.14):  (3.20)  86  Take the time t t o be 0 when Q = 0 and t  when Q = Q .  r  The Gaussian  U  distribution i s : 1  w(t) =  2 2 e x p ( - t /2a )  —  (3.21)  (2TJ)" O 2  S i n c e t h e c o n t r i b u t i o n from any p a r t i c l e has a w i d t h i n time of t , the r  c o n v o l u t i o n of (3.20) w i t h (3.21) g i v e s :  da  2  % r  dt  Define the p r o b a b i l i t y  $(x) =  After  V  2 D  (3.22)  integral:  (3.23)  (3.22) becomes:  h  dV  t-T  dy exp(-y )  (2ir)'  integration  1 -  W(T)  1  ( t - t ) {$(t/2%) F  + W%  - *({t-t_}/2 a)} Js  r  {exp(-{t-t } /2a ) - e x p ( - t / 2 c ) } 2  2  2  2  (3.24)  r  F i g u r e 27 shows the r e s u l t i n g peak shape compared the  to t h e i d e a l peak f o r  case a = t /10. F I f we i g n o r e a l l c a p a c i t y except f o r t h a t due t o t h e phase t r a n s i t i o n ,  we can c o n v o l u t e t h e c e l l v o l t a g e g i v e n by (3.18b) w i t h t h e G a u s s i a n . N o t i n g t h a t t h e p o r t i o n o f t h e cathode which has n o t y e t begun t o phase convert i s s t i l l n o m i n a l l y a t t h e c e l l V  c  - V_ + IR = 0  1  voltage:  dx exp(-x  2  2^ /2a )  (2TT) a 2  rt  + V,  dT W ( T )  hln  1 -  (V  c  - V  0 n  + IR)  t-x  t-t. rt (2ir)  li  $(t/2 a) Jfi  j  t-t  dT exp(-x  2  2 /2a ) % l n  t-T  (3.25)  87  F i g u r e 27 - The s o l i d l i n e i s t h e i d e a l peak shape of f i g u r e 16 f o r Y = 2, w i t h t h e c e l l d i s c h a r g i n g . The squares show sample p o i n t s c a l c u l a t e d f o r a n o n - i d e a l peak, i f the time a t which p a r t i c l e s n u c l e a t e t h e i r phase f r o n t s 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 w i d t h e q u a l t o one t e n t h of t h e time taken t o d i s c h a r g e from 0 t o Q . n  88  The  i n t e g r a l i s d i v e r g e n t , due  T = t - t . F  The  T near t - t . F to  s t a t e approximation,  computed n u m e r i c a l l y  excluding  at  the r e g i o n of  F i g u r e 28 shows the r e s u l t i n g peak shape, a g a i n  compared  the i d e a l peak f o r the case a = t ^ / l O . The  to  i n t e g r a l was  t o the steady  peak shapes shown i n f i g u r e s 27 and  the e x p e r i m e n t a l l y  observed peaks.  rounded w i t h a s i g n i f i c a n t  The  28 a r e q u a l i t a t i v e l y  peak a g a i n s t Q i s n o t i c a b l y  l e a d i n g edge, s i m i l a r to what i s seen exper-  i m e n t a l l y a l t h o u g h more exaggerated.  The  peak has  low dQ/dV, s i m i l a r to the o r i g i n a l sawtooth.  The  a l i n e a r f a l l back to peak a g a i n s t  v o l t a g e 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 a g a i n s t Q,  has  s l o p e and  an i n i t i a l  i s somewhat rounded o f f .  i t a t i v e agreement w i t h the e x p e r i m e n t a l  peaks.  cell but  approach to understanding  it  now  This i s also in qual-  I t appears t h a t use  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 is a f r u i t f u l  similar  n o n - i d e a l peak shapes.  of a  voltage  F i g u r e 28 - The s o l i d l i n e i s t h e i d e a l peak shape of f i g u r e 15 f o r Y = 2. The squares show sample p o i n t s c a l c u l a t e d f o r a n o n - i d e a l peak, i f t h e time a t which p a r t i c l e s n u c l e a t e t h e i r phase f r o n t s 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 w i d t h e q u a l t o one t e n t h o f the time taken t o d i s c h a r g e from 0 t o  V  90 3.6  E f f e c t i v e n e s s of the Technique It  has been made c l e a r i n t h i s chapter  t h a t constant  measurements g i v e c l e a r and unambiguous evidence order phase t r a n s i t i o n s . regions The  The  of the p r e s e n c e of  first  r e s o l u t i o n of the peaks caused by two  phase  i s f a r s u p e r i o r to t h a t o b t a i n e d w i t h l i n e a r sweep voltammetry.  peak shapes a g a i n s t v o l t a g e a r e d i s t i n c t i v e and  decreasing The  c u r r e n t dQ/dV  current.  sharpen markedly  with  T h i s l e a d s to a c l e a r d e r i v a t i o n of t h e phase diagram.  apparatus i s capable  of measuring dQ/dV over much l o n g e r times than i s  p o s s i b l e w i t h the l i n e a r sweep voltammetry apparatus d e s c r i b e d i n S e c t i o n 2.3.  T h i s a l l o w s the c e l l to be c l o s e r to e q u i l i b r i u m during  dQ/dV  measurements than i s p o s s i b l e w i t h l i n e a r sweep voltammetry. I f measurements a r e made at a sequence of c u r r e n t s , a graph of peak p o s i t i o n as a f u n c t i o n of c u r r e n t a l l o w s a q u a n t i t a t i v e d e t e r m i n a t i o n the c e l l r e s i s t a n c e d u r i n g  the phase t r a n s i t i o n , as w e l l as of  equilibrium transition voltage. accurate determination ranges i n x and  to  phase r e g i o n .  an The  the e x t r a p o l a t e d v a l u e s f o r the t r a n s i t i o n v o l t a g e s  s u f f i c i e n t l y accurate peaks' n o n - i d e a l  the  Graphs of dQ/dV a g a i n s t Q a l l o w  of the range i n x of a two  of  are  to d e t e c t h y s t e r e s i s i n the phase t r a n s i t i o n s .  shapes, d i s c u s s e d  i n the l a s t  extract quantitative results for  D dp/dV from the peak h e i g h t s , a l t h o u g h  The  s e c t i o n , make i t d i f f i c u l t  from the peak widths or f o r the peak widths and  heights  change  q u a l i t a t i v e l y w i t h the c u r r e n t i n the expected manner. The details,  r e s o l u t i o n of constant  c u r r e n t dQ/dV i s such t h a t v e r y  such as h y s t e r e s i s i n the t r a n s i t i o n v o l t a g e s , can be  Constant c u r r e n t dQ/dV peaks i n c r e a s e i n dQ/dV and the c u r r e n t decreases,  while  fine examined.  become t h i n n e r when  s i n g l e phase dQ/dV changes l i t t l e .  This i s  i n c o n t r a s t w i t h l i n e a r sweep voltammetry, whose peaks i n I decrease i n c r e a s i n g sweep r a t e , a l t h o u g h  with  they d e c r e a s e l e 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 t o phase t r a n s i t i o n s ,  as e x e m p l i f i e d by f i g u r e  17, prevents  such m i s t a k e s as t h e i d e n t i f i c a t i o n  of a 3 + IT two phase r e g i o n i n L i V S , as was m i s t a k e n l y 9  l i n e a r sweep voltammetry r e s u l t s . constant  I f an ambiguous peak shape appears i n  c u r r e n t dQ/dV, the c u r r e n t can be decreased  whether or n o t a f i r s t  done u s i n g  until  order phase t r a n s i t i o n a c t u a l l y  i t i s clear  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 o f t h e Technique We have seen i n Chapter 3 t h a t c o n s t a n t  to  c l e a r l y and unambiguously i d e n t i f y f i r s t  c u r r e n t dQ/dV can be used order phase t r a n s i t i o n s .  The  peaks due t o 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 a t r e l a t i v e l y h i g h c u r r e n t s , as i n t h e example of L i ^ V ^ a t 140 yamps, p r e v i o u s l y shown i n f i g u r e 20.  There, charge or d i s c h a r g e  took about a day.  One  would expect t h a t i f a m a t e r i a l , such as L i VS„, w i t h one o r more peaks x 2 in  dQ/dV due t o f i r s t  o r d e r phase t r a n s i t i o n s , was p r e s e n t  as an i m p u r i t y  i n another m a t e r i a l , then i t s presence might be d e t e c t e d by dQ/dV measurements.  A s m a l l i m p u r i t y would be d r i v e n q u i c k l y through a phase t r a n s i t i o n ,  s i n c e a l a r g e f r a c t i o n o f the c e l l ' s c u r r e n t would be used f o r the phase c o n v e r s i o n u n t i l t h e cathode's v o l t a g e was a b l e t o s h i f t away from the transition voltage. p u r i t y more d i f f i c u l t  T h i s reduces t h e dQ/dV peak h e i g h t s , making an imto detect.  An i m p u r i t y would not be d e t e c t e d  if it  had phase t r a n s i t i o n s a t v o l t a g e s c l o s e t o those o f phase t r a n s i t i o n s i n the main m a t e r i a l , but a dQ/dV peak from an i m p u r i t y may w e l l - b e if  visible  superimposed on s i n g l e phase c a p a c i t y . 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  r e c e n t l y d i s c o v e r e d by Mulhern (1982).  c u r r e n t dQ/dV was  A f l a n g e c e l l , PM-30, w i t h a cathode  made from the n a t u r a l l y o c c u r r i n g l a y e r compound MoS , was d i s c h a r g e d f o r 9  93  the f i r s t was  time.  The  a t a constant  25 yamps.  apparatus d e s c r i b e d  2.08  The d i s c h a r g e  Constant c u r r e n t dQ/dV was measured u s i n g t h e  i n the f i r s t  shown i n f i g u r e 29. verting  M0S2 was o b t a i n e d from t h e Endako mine.  p a r t of S e c t i o n 3.3, and t h e r e s u l t s a r e  The c a p a c i t y below 1.4 v o l t s i s due t o t h e  from i t s i n i t i a l a phase i n t o 3 phase.  M0S2 con-  There i s a s m a l l bump near  v o l t s t h a t does n o t appear t o be a s s o c i a t e d w i t h t h e phase  conversion.  F i g u r e 30 shows t h e same data as f i g u r e 29 but o n l y above 1.4 v o l t s , the v e r t i c a l The  s c a l e i n c r e a s e d by a f a c t o r of 30.  A peak i s c l e a r l y  peak a t 2.08 v o l t s was n o t found i n s y n t h e t i c a l l y prepared  so i t was c l e a r t h a t t h i s peak c o u l d be simply  due t o an i m p u r i t y .  i m p u r i t y has now been i d e n t i f i e d by Mulhern as Cu^ ^Mo^S^. percent  with  visible.  M0S2, The  Only 0.3  of t h e c a p a c i t y of c e l l PM-30 was caused by t h e i m p u r i t y , y e t i t s  presence was s t i l l  detected.  Constant c u r r e n t dQ/dV i s capable  e f f e c t s i n v o l v i n g o n l y f r a c t i o n s of a percent s e n s i t i v i t y allows  of d e t e c t i n g  of a c e l l ' s c a p a c i t y .  s t r a i g h t f o r w a r d i n v e s t i g a t i o n of a m a t e r i a l f o r t h e  presence of t r a c e amounts of other m a t e r i a l s t h a t a l s o i n t e r c a l a t e . s e n s i t i v i t y a l s o permits  of s t a t e .  c u r r e n t dQ/dV measurement made on Cu^ ^Mo^S^ by Mulhern i s  shown i n f i g u r e 31. Mo^S^  Such  c a r e f u l examination of v e r y s m a l l e f f e c t s i n a  m a t e r i a l ' s v o l t a g e , and hence i n i t s e q u a t i o n A constant  This  The copper i s removed on t h e f i r s t  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  discharge, leaving  o r d e r phase t r a n s i t i o n a t 2.08  v o l t s on subsequent c y c l e s , shown i n f i g u r e 31, appears t o i n v o l v e a change in  i n t e r c a l a n t content  lattice ing 0  without  (Mulhern 1982).  t h e removal on f i r s t  any change i n t h e symmetry of t h e host  A v i r t u a l l y i d e n t i c a l dQ/dV curve d i s c h a r g e of i r o n from Fe, 0  ,Mo„S..  1.434  appears f o l l o w -  94  1.0  1.2  1.4  1.6  Cell Voltage  1.8  2.0  2.2  2.4  (volts)  F i g u r e 29 - Constant c u r r e n t dQ/dV a g a i n s t c e l l v o l t a g e f o r t h e f i r s t d i s c h a r g e o f t h e L i M 0 S 2 c e l l PM-30, made w i t h n a t u r a l l y o c c u r r i n g MoS^. The c u r r e n t was 25 pamps. Data p r o v i d e d by P. J . Mulhern.  1.4  1.6  1.8 Cell Voltage  2.0  2.2  2.4  (volts)  F i g u r e 30 - Expanded view of t h e h i g h v o l t a g e data shown i n f i g u r e 29. T h i s was t h e f i r s t evidence o f an i m p u r i t y i n v o l v i n g 0.3 p e r c e n t of t h e c a p a c i t y of c e l l s made w i t h n a t u r a l Mx^. Data p r o v i d e d by P. J . Mulhern.  30  l  l  20  l  l  Discharge  J  > TD  l  0  1— ~ Y  —  -10  \  ^ - Charge  —  •20 -30  I 1.8  I  I 2.0  l  l l 2.2  Cell Voltage  l  l 2.4  (volts)  F i g u r e 31 - Constant c u r r e n t dQ/dV a g a i n s t c e l l v o l t a g e f o r l i t h i u m i n t e r c a l a t e d Mo S,. M a t e r i a l was Cu, ,Mo„S. p r i o r t o i t s f i r s t 3 4 1.4 3 4 discharge. Data p r o v i d e d by P. J . Mulhern. 0  r  k.2  A p p l i c a t i o n s to S i n g l e Phase Regions When dQ/dV i s measured a g a i n s t V  cell  , by  d e f i n i t i o n the c u r v e shows the  c a p a c i t y as a f u n c t i o n of c e l l v o l t a g e .  A l s o , s i n c e dQ/dV i s s e n s i -  t i v e to f i n e s t r u c t u r e i n the e q u a t i o n of s t a t e of an  intercalation cell,  dQ/dV i s an extremely s e n s i t i v e probe of the thermodynamics of a m a t e r i a l . As a r e s u l t , u n d e r s t a n d i n g dQ/dV i n s i n g l e phase r e g i o n s standing  implies  of the i n t e r c a l a t i o n system's e q u a t i o n of s t a t e .  under-  I t i s to  be  expected t h a t s i n c e d i f f e r e n t m a t e r i a l s w i l l n o r m a l l y have q u i t e d i f f e r e n t e q u a t i o n s of s t a t e , t h a t dQ/dV w i l l n o r m a l l y be q u i t e d i s t i n c t i v e f o r a given m a t e r i a l . the p r e v i o u s  This allows  section.  the d e t e c t i o n of i m p u r i t i e s , as d i s c u s s e d  Also,  i f two  then t h e i r thermodynamics and similar.  materials  have s i m i l a r dQ/dV  p h y s i c a l p r o p e r t i e s may  be  curves,  expected to  I d e n t i c a l dQ/dV curves s t r o n g l y i n d i c a t e t h a t two  in  be  c e l l s have  identical intercalating materials. The  d i f f i c u l t y of d e r i v i n g dQ/dV from a model of an i n t e r c a l a t i o n  compound has  been r e c e n t l y i l l u s t r a t e d by the s u c c e s s f u l 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 l a y e r compound L i T i S 2  Dahn, Dahn, and  Haering  1982).  Figures  32 and  33  a l a t t i c e gas H a m i l t o n i a n of the e l a s t i c energy due  The  inclusion in  to the expansion of  der Waals gaps as i n t e r c a l a t i o n proceeds p r e d i c t s f l a t t e n i n g of  the v o l t a g e  curve at low  low  d i p i n dx/dV near x = 0.16  x.  also  i l l u s t r a t e dx/dV  measured by Dahn f o r L i TiS„ i n the range 0 < x < 1. x 2 ° — —  the van  (see  x  The  x,  causing  of a s h o r t range stage 2 s t r u c t u r e . every n t h van  der Waals gap  the higher has  dx/dV v a l u e s  been a t t r i b u t e d to the  A stage n s t r u c t u r e i s one  i s f i l l e d at one  intercalant  w i t h the i n t e r v e n i n g gaps f i l l e d at a lower c o n c e n t r a t i o n . model i s v e r y  s u c c e s s f u l i n p r e d i c t i n g the s t r u c t u r a l and  b e h a v i o r of L i TiS„ and  observed at  some other MX„  formation  i n which  concentration, Although t h i s thermodynamic  l a y e r compounds, dx/dV  obtained  4.0 |  1.8  r  2.0  2.2  2.4  2.  Volts  F i g u r e 32 - Constant c u r r e n t dx/dV a g a i n s t c e l l v o l t a g e f o r the L i T1S2 c e l l JD-235 a t 40 yamps. Diamonds i n d i c a t e charge, t r i a n g l e s indicate discharge. From Dahn (1982). X  99  F i g u r e 33 - Constant c u r r e n t dx/dV a g a i n s t x f o r t h e L i T i S ^ c e l l JD-235 a t 40 yamps. Diamonds i n d i c 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 C a r l o c a l c u l a t i o n s s t i l l  has peaks of e q u a l h e i g h t  s i d e o f x = 0.16, u n l i k e t h e d a t a .  I t i s p o s s i b l e that t h i s discrepancy  may be removed by t h e i n c l u s i o n o f a more a c c u r a t e e x p r e s s i o n  on e i t h e r  f o r the  e l a s t i c energy, however t h i s example serves t o i l l u s t r a t e t h e s e n s i t i v i t y of dQ/dV t o t h e thermodynamics o f the system, and i t s u s e f u l n e s s models of i n t e r c a l a t i o n systems.  i n testing  Other p r e d i c t i o n s of thermodynamic quan-  t i t i e s a r e i n n o t i c a b l y b e t t e r agreement w i t h d a t a . It  should a l s o be noted t h a t  information  be o b t a i n e d from s i n g l e phase dQ/dV. in can  s i n g l e phase r e g i o n s ,  about a c e l l ' s k i n e t i c s can  dQ/dV s t i l l  e x p e r i e n c e s an IR s h i f t  so measurements of dQ/dV as a f u n c t i o n of c u r r e n t  be used t o determine t h e c e l l ' s r e s i s t a n c e a t any p o i n t .  Diffusion  e f f e c t s , however, a r e n o t l i k e l y t o 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 t h a t s i n g l e phase measurements can e x t r a c t diffusion.  information  about s o l i d  state  101  4.3  "Supercooling" An analogy was presented i n Section 1.2 between the thermodynamics  of a gas and the thermodynamics of an i n t e r c a l a t i o n system.  Effects  analogous to supercooling and supersaturation have been observed i n i n t e r c a l a t i o n systems.  Supercooling occurs i n 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 e f f e c t  with the volume dropping and the pressure r i s i n g . These e f f e c t s occur because of the n u c l e a t i o n process f o r a new phase.  Once the c r i t i c a l pressure i s reached, i t i s p o s s i b l e to form  a two phase system.  U n t i l a new phase i s nucleated, however, the system  continues i n 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 s i z e at which they become s t a b l e and grow.  Consider supercooling.  Regions of  d i f f e r e n t sizes have d i f f e r e n t 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 s i z e at which  the surface area i s j u s t r i g h t f o r no net flow of gas to occur across the surface.  C l e a r l y , as the pressure decreases the s i z e becomes smaller  f o r which no outflow of gas occurs from the nucleation region. of varying s i z e s are formed by f l u c t u a t i o n s i n the gas.  Regions  I f a region's  s i z e 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 d i s s i p a t e s .  I f a region's s i z e i s  above the c r i t i c a l s i z e , however, a net i n f l o w 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 e q u i l i b r i u m with the newly nucleated phase as i t continues t o 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 a s s o c i a t e d w i t h the s t a r t of phase c o n v e r s i o n . i n an i n t e r c a l a t i o n system an overshoot a new  phase.  By analogy,  i n v o l t a g e i s r e q u i r e d to n u c l e a t e  the v o l t a g e e n t e r s a m e t a s t a b l e  moves away from the t r a n s i t i o n v o l t a g e , u n t i l the c r i t i c a l n u c l e a t e d r e g i o n decreases  Q or x.  The  critical  region  and  size for a  s u f f i c i e n t l y f o r the n u c l e a t e d r e g i o n s to  grow and phase c o n v e r s i o n to begin. to  Analogously,  R e c a l l t h a t the volume i s analogous  s i z e decreases  w i t h v o l t a g e s i n c e x i n the  o r i g i n a l phase r i s e s as the v o l t a g e drops.  A n u c l e a t i o n r e g i o n must  have a s m a l l e r s u r f a c e a r e a f o r a l a r g e r x i n the o r i g i n a l phase i f t h e r e i s to be no net i n f l o w of i n t e r c a l a n t  into a nucleation region.  Once a g a i n , when the c r i t i c a l s i z e becomes s m a l l enough, the new n u c l e a t e s and  the v o l t a g e r e l a x e s back t o the t r a n s i t i o n v o l t a g e .  "Supercooling" difficult  i n an i n t e r c a l a t i o n system w i l l occur  to n u c l e a t e a new  c o o l i n g " i s not observed,  phase.  " S u p e r c o o l i n g " was  i f i t is  For most phase t r a n s i t i o n s  "super-  however i t i s sometimes seen d u r i n g the con-  v e r s i o n from a phase to $ phase i n M0S2 t h a t was 4.1.  phase  mentioned i n S e c t i o n  f r e q u e n t l y seen i n c e l l s w i t h a l i t h i u m anode  and a p y r i t e (FeS2) cathode prepared  by R. M a r s o l a i s .  The m a j o r i t y of  the c a p a c i t y 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 s not y e t c l e a r what the end product  It  of the phase t r a n s i t i o n i s , a l -  though the f i r s t d i s c h a r g e of a p y r i t e c e l l u s u a l l y changes x from 0 to ^3.5  i n Li FeS2. x  poor c y c l e l i f e ,  Such c e l l s can only be p a r t i a l l y recharged however they make e x c e l l e n t primary  and  batteries,  have par-  t i c u l a r l y s i n c e the t r a n s i t i o n v o l t a g e i s v e r y c o n v e n i e n t l y c l o s e to 1.5  volts.  F i g u r e 34 shows the " s u p e r c o o l i n g " e f f e c t  charge v o l t a g e curve of c e l l M-34. d i s c h a r g e at 400  yamps.  The  i n the f i r s t  c e l l took about 60 hours t o  dis-  103  1.50  1.45  \-  O >  8. 1.40 o  H—'  O >  <u  ^  1.35  1.30  0  1 0 - 2 0 Discharge Time  30  40  50  (hours)  F i g u r e 34 - V o l t a g e curve f o r t h e f i r s t d i s c h a r g e of t h e l i t h i u m a g a i n s t p y r i t e c e l l M-34. A very strong "supercooling" e f f e c t o c c u r s a t t h e s t a r t of a f i r s t o r d e r phase t r a n s i t i o n i n v o l v i n g most of t h e c e l l ' s c a p a c i t y . C e l l d i s c h a r g e d t o 1.0 v o l t s i n r o u g h l y 60 hours, a t a c u r r e n t o f 400 yamps. Data p r o v i d e d by R. M a r s o l a i s .  104  Even a s m a l l " s u p e r c o o l i n g " e f f e c t As the v o l t a g e reaches to  - , suddenly OT  should appear p r o m i n e n t l y  i n dQ/dV.  a minimum and s t a r t s i n c r e a s i n g a g a i n , dQ/dV  switches  s i g n t o + , and then decreases 00  Then when t h e v o l t a g e reaches  diverges  i n magnitude.  a maximum and s t a r t s d e c r e a s i n g  a g a i n , dQ/dV  d i v e r g e s t o +°°, changes s i g n t o -°°, and then s e t t l e s down t o r e c o r d t h e first  o r d e r phase t r a n s i t i o n .  was n u c l e a t e d , t h e c e l l  One would expect  t h a t once t h e new phase  v o l t a g e would r e t u r n t o t h e v a l u e n o r m a l l y  expected  d u r i n g a phase t r a n s i t i o n , and t h e u s u a l dQ/dV peak should appear w i t h i t s i n i t i a l p o r t i o n m i s s i n g , and some d i s t o r t i o n i n t h e remainder. F i g u r e 35 shows how t h e " s u p e r c o o l i n g " e f f e c t pears M-47.  i n a dQ/dV curve.  The data shown i s f o r the f i r s t  d i s c h a r g e of c e l l  T h i s was done a t 100 yamps and took r o u g h l y a week.  f i g u r e 34 thus has a c o n s i d e r a b l y h i g h e r IR s h i f t 35,  shown i n f i g u r e 34 ap-  causing  a r o u g h l y 20 m i l l i v o l t  s e t s of d a t a .  The data i n  than t h e data i n f i g u r e  d i f f e r e n c e i n v o l t a g e between the two  The e f f e c t appears i n dQ/dV as expected,  f o l l o w e d by what  appears t o be t h e l a s t p o r t i o n o f a dQ/dV peak from the f i r s t transition.  o r d e r phase  105  F i g u r e 35 - Constant c u r r e n t dQ/dV a g a i n s t c e l l v o l t a g e f o r t h e l i t h i u m a g a i n s t p y r i t e c e l l M-47. The " s u p e r c o o l i n g " e f f e c t shown i n f i g u r e 34 causes the changes i n s i g n i n dQ/dV. Data p r o v i d e d by R. M a r s o l a i s .  106  4.4  Staging As mentioned i n S e c t i o n 4.2,  has every n t h van der Waals gap  a stage n s t r u c t u r e i n a l a y e r compound  f i l l e d a t one  intercalant concentration,  w i t h the i n t e r v e n i n g gaps a l l f i l l e d  at a lower c o n c e n t r a t i o n .  staged s t r u c t u r e s have every n t h gap  f i l l e d , w i t h the i n t e r v e n i n g gaps  empty. from  First  o r d e r phase t r a n s i t i o n s may  one staged s t r u c t u r e to another.  l i t h i u m i n t e r c a l a t e d NbSe  2  (Dahn, D.C.,  occur i n a m a t e r i a l to change  T h i s has r e c e n t l y been observed i n and Haering  curve f o r L i NbSe„ i s shown i n f i g u r e 36(a) x z 36(b).  X-ray  x - 0.08,  Frequently,  1982).  The v o l t a g e  and dx/dV i s shown i n f i g u r e  s t u d i e s have shown t h a t a stage 3 s t r u c t u r e e x i s t s at  a stage 2 s t r u c t u r e e x i s t s a t x - 0.14,  e x i s t s above x - 0.27.  An  and a stage 1 s t r u c t u r e  i r r e v e r s i b l e change o c c u r s on the f i r s t  charge from the u n i n t e r c a l a t e d m a t e r i a l to the stage 3 s t r u c t u r e . s i b l e t r a n s i t i o n s occur between s t a g e 3 and and stage 1. first  The two  charge and  disRever-  stage 2 and between stage 2  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 f o r the  second  discharge.  However, the f i r s t  discharge only  shows a peak a t the v o l t a g e c o r r e s p o n d i n g 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  s i t i o n s appear on f i r s t  no peaks due t o f i r s t  d i s c h a r g e due t o the changes from  NbSe^ t o stage 3, or from stage 3 to stage 2. from NbSe2 t o stage 2, further It  order phase t r a n -  by way  unintercalated  The n a t u r e of the change  of stage 3, on f i r s t  discharge requires  investigation. i s of i n t e r e s t t o n o t e t h a t the s t a g i n g i n L i NbSe  can be q u a l -  i t a t i v e l y d e s c r i b e d by the same model, mentioned i n S e c t i o n 4.2, developed  to d e s c r i b e L i TiS„ (Dahn, Dahn, and H a e r i n g  1982).  that  was  107  gure 36 - S t a g i n g i n L i N b S e A f t e r Dahn, D.C., and Haering (1982). (36a) V o l t a g e ' c u r v e of L i NbSe c e l l DD-36. Shown a r e (a) f i r s t d i s c h a r g e , (b) f i r s t charge, and (c) second d i s c h a r g e . (36b) Constant c u r r e n t dx/dV a g a i n s t c e l l v o l t a g e f o r t h e same cases as i n f i g u r e 36(a). 2>  2  108  CHAPTER FIVE  CONCLUSIONS  5.1  dQ/dV  Measurements  T h i s t h e s i s has demonstrated t h e u s e f u l n e s s o f dQ/dV in detecting f i r s t  order phase t r a n s i t i o n s .  measurements  As d i s c u s s e d i n S e c t i o n 1.2,  a c e l l ' s v o l t a g e as a f u n c t i o n of Q or x i s analogous t o t h e e q u a t i o n o f s t a t e of a gas.  dQ/dV i s analogous t o t h e i s o t h e r m a l c o m p r e s s i b i l i t y o f  a gas, which d i v e r g e s d u r i n g a f i r s t not  of  dQ/dV i s thus  o n l y s e n s i t i v e t o f i n e s t r u c t u r e i n a c e l l ' s e q u a t i o n o f s t a t e , but  w i l l peak when a f i r s t are  order phase t r a n s i t i o n .  o r d e r phase t r a n s i t i o n o c c u r s .  dQ/dV  measurements  s e n s i t i v e probes b o t h of an i n t e r c a l a t i o n system's thermodynamics and 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 g i v e n i n t h i s t h e s i s f o r t h e  i n t e r c a l a t i o n systems L i VS„, L i MoS , L i Mo~S,, L i T i S , and L i NbSe„, 9  X  as  Z  X  0  Z.  X - 5 H  Z  w e l l as t h e system L i FeS„ ( l i t h i u m a g a i n s t p y r i t e ) . X  z.  other systems t o which dQ/dV measurements for  X  can be a p p l i e d .  X  The measurements  t h e l i t h i u m a g a i n s t p y r i t e system, which may o r may not i n v o l v e  c a l a t i o n , i n d i c a t e t h a t dQ/dV measurements  i f more than a s i n g l e , s t r a i g h t f o r w a r d  r e a c t i o n i s i n v o l v e d i n such a system.  inter-  may prove u s e f u l f o r e l e c t r o -  c h e m i c a l systems o t h e r than t h e i n t e r c a l a t i o n systems we have been cussing, p a r t i c u l a r l y  /-  There a r e many  dis-  chemical  The e x p e r i m e n t a l t e c h n i q u e s  dis-  cussed i n t h i s t h e s i s can be a p p l i e d 'without change t o any e l e c t r o c h e m i c a l  cell,  and the i n t e r p r e t a t i o n  of dQ/dV peaks as s i g n a l s  of f i r s t  order  phase t r a n s i t i o n s i s independent of the n a t u r e of the e l e c t r o c h e m i c a l system.  110  5.2  L i n e a r Sweep Voltammetry S i n c e dQ/dV can be measured by the s t a n d a r d  l i n e a r sweep voltammetry, the d e t e c t i o n of f i r s t in and  chemical  technique  of  o r d e r phase t r a n s i t i o n s  i n t e r c a l a t i o n systems i s immediately p o s s i b l e f o r many r e s e a r c h e r s , the n e c e s s a r y  equipment i s commercially  Chapter 2, once k i n e t i c e f f e c t s due  available.  As d i s c u s s e d  to the c e l l ' s r e s i s t a n c e and  to  in solid  s t a t e d i f f u s i o n a r e taken i n t o account, the shapes of peaks i n dQ/dV to  first  o r d e r phase t r a n s i t i o n s can be p r e d i c t e d .  Li^VS2 was  obtained  by Dahn, J.R.,  and  Haering  t i o n of l i n e a r sweep voltammograms, and was phases were i d e n t i f i e d from the x-ray  The  due  phase diagram f o r  (1981) from the i n t e r p r e t a -  presented  i n f i g u r e 13.  The  data of Murphy et a l (1977).  The  v o l t a g e s 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  L i n e a r sweep voltammetry i s not apparatus t h a t was i t i e s b e f o r e any  used had  phase r e g i o n s .  i d e a l f o r dQ/dV measurements.  The  to perform near the l i m i t s of i t s c a p a b i l -  u s e f u l r e s u l t s c o u l d be o b t a i n e d .  c l a r i t y of peaks i n dQ/dV were not  The  r e s o l u t i o n and  as good as one would d e s i r e .  This i s  p a r t i c u l a r l y the case s i n c e the peaks a r e not v e r y d i s t i n c t i v e i n shape, having  the rounded appearances p r e d i c t e d by t h e o r y .  h e i g h t s do. . i n c r e a s e do so l i k e a constant  A l s o , although  w i t h a compared to s i n g l e phase r e g i o n s , . they  , where a i s the v o l t a g e sweep r a t e .  I t was  poor r e s o l u t i o n i n the l i n e a r  only  revealed  c u r r e n t dQ/dV measurements, d e s c r i b e d i n Chapter 3, t h a t  by the  sweep voltammetry measurements on Li^VS2  led  to the m i s i n t e r p r e t a t i o n of some s i n g l e phase c a p a c i t y as a 3 +  two  phase r e g i o n .  r e s o l u t i o n constant If  peak  T h i s e r r o r i l l u s t r a t e s the u t i l i t y c u r r e n t dQ/dV  of the  IT  higher  technique.  phase t r a n s i t i o n s a r e c l o s e t o g e t h e r  dQ/dV peaks i n l i n e a r sweep voltammetry may  i n v o l t a g e , the breadth mask some of the  of  transitions  Ill  and  make i n t e r p r e t a t i o n  difficult.  be o b t a i n e d from the peak shapes. peak i s 1/R. solid  state  A l i m i t e d amount o f i n f o r m a t i o n can The s l o p e o f t h e i n i t i a l r i s e o f a  The g e n e r a l peak shape g i v e s some i n d i c a t i o n d i f f u s i o n i s o c c u r r i n g , but only a q u a l i t a t i v e  In g e n e r a l , s i n c e t h e c u r r e n t , and hence t h e s e v e r i t y gradients, varies  of how  quickly  indication.  of t h e d i f f u s i o n  s i g n i f i c a n t l y d u r i n g a l i n e a r sweep voltammogram, t h e  peak shapes a r e s u f f i c i e n t l y rounded and broad t o mask s m a l l e f f e c t s , and an unambiguous  interpretation  of peaks i s n o t always  possible.  112 5.3  Constant  Current dQ/dV"  Constant  c u r r e n t dQ/dV i s both simple t o measure and c l e a r i n i n t e r -  pretation.  Measurements r e q u i r e o n l y t h a t a c e l l  be charged  or discharged  at  c o n s t a n t c u r r e n t , and t h a t t h e c e l l v o l t a g e be monitored  or  analogue t o d i g i t a l c o n v e r t e r a t t a c h e d t o a microcomputer.  and measurement t e c h n i q u e were d e s c r i b e d i n Chapter  3.  by a v o l t m e t e r The apparatus  Peaks i n c o n s t a n t  c u r r e n t dQ/dV due t o f i r s t order phase t r a n s i t i o n s a r e sharp and narrow, p a r t i c u l a r l y when compared w i t h t h e peaks generated  by l i n e a r sweep v o l t a m -  metry (see f i g u r e 19). S i n c e i t i s e x p e r i m e n t a l l y simple t o measure dQ/dV at  low c u r r e n t s , so t h a t a c e l l ' s charge o r d i s c h a r g e takes c o n s i d e r a b l y  l o n g e r than t h e 61 hours t h a t l i n e a r  sweep voltammetry i s l i m i t e d t o w i t h  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 t h e r e i s no p o s s i b l e ambiguity  i n t h e i n t e r p r e t a t i o n of peaks due t o f i r s t o r d e r phase  transitions. The peaks i n constant c u r r e n t dQ/dV a r e not rounded by d r a s t i c a l l y changing shift.  d i f f u s i o n g r a d i e n t s o r d i s t o r t e d by a c o n t i n u o u s l y changing IR U n l i k e t h e peaks i n l i n e a r sweep voltammetry, t h e peaks i n c o n s t a n t  c u r r e n t dQ/dV a g a i n s t v o l t a g e merely  have a t a i l added t o the i d e a l  f u n c t i o n by d i f f u s i o n , and e x p e r i e n c e a c o n s t a n t IR s h i f t .  The peak h e i g h t  v a r i e s i n v e r s e l y w i t h c u r r e n t , w h i l e t h e peak width v a r i e s l i n e a r l y current.  I n an experiment,  delta  with  t h e c u r r e n t can simply be decreased u n t i l any  peaks a r e s u f f i c i e n t l y t a l l and narrow t o be c l e a r l y r e s o l v e d . If  t h e c e l l r e s i s t a n c e i s c o n s t a n t , charge and d i s c h a r g e peaks a r e  separated by 2IR.  I f t h e r e s i s t a n c e d i f f e r s on charge and d i s c h a r g e , t h e  l e a d i n g edges of the peaks can be p l o t t e d as a f u n c t i o n of c u r r e n t , 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 s l o p e of such an e x t r a p o l a t i o n i s 1/R.  Both R and t h e  t r a n s i t i o n v o l t a g e can thus be o b t a i n e d q u a n t i t a t i v e l y w i t h good a c c u r a c y .  113  The  peak h e i g h t  peak h e i g h t  i s roughly  linear in Q / | l | ,  the a c c u r a c y  of  the  i s l i m i t e d by the v o l t a g e r e s o l u t i o n of the apparatus.  It  to o b t a i n more than q u a l i t a t i v e i n f o r m a t i o n r e g a r d i n g  the  is d i f f i c u l t  r e l a t i v e v a l u e s of D dp/dV d u r i n g of peak h e i g h t a g a i n s t more and  but  each phase t r a n s i t i o n from the  Despite t h i s ,  QQ/|I|.  i t is s t i l l  b e t t e r q u a l i t y i n f o r m a t i o n can be o b t a i n e d  slope  c l e a r that  from constant  current  dQ/dV peaks than from l i n e a r sweep voltammetry peaks. When coupled  w i t h x-ray  d i f f r a c t i o n s t u d i e s , constant  c u r r e n t 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 a g a i n s t Q a r e c l e a r and in  linear  sweep voltammetry.  i n x of f i r s t The constant  L i VS  These can be used to determine the  order phase t r a n s i t i o n s to w i t h i n i n t e r c a l a t i o n system was  0  c u r r e n t dQ/dV.  The  m o s t l y confirmed  t h e 3 + IT two  by constant  phase r e g i o n was  t r a n s i t i o n between the 3 and order i n nature.  e x t e n s i v e l y examined  by the x-ray  not  s t u d i e s of Murphy et a l As mentioned,  I t appears l i k e l y  that  IT phases i s second order r a t h e r than  T h i s i s p l a u s i b l e s i n c e the 6 phase i s a  d i s t o r t i o n of the IT s t r u c t u r e .  Further  t r a n s i t i o n s between the VS^  and  the first  monoclinic  i n v e s t i g a t i o n of the n a t u r e  t h i s t r a n s i t i o n , i n c l u d i n g c o n f i r m a t i o n of the x-ray The  with  p r e v i o u s l y determined  c u r r e n t dQ/dV.  found.  ranges  0.01.  phase diagram t h a t was  by l i n e a r sweep voltammetry and (1977), was  q u i t e d i s t i n c t i v e , more so than  a phases and  of  r e s u l t s , i s needed. between the a  and  3 phases appear t o undergo s i g n i f i c a n t h y s t e r e s i s i n both v o l t a g e and Some h y s t e r e s i s 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, be l e s s than ten percent  i n x i f i t occurs.  The  two  but  x.  i t must  transitions for  which h y s t e r e s i s o c c u r s have a v o l t a g e h y s t e r e s i s of s e v e r a l m i l l i v o l t s and  a h y s t e r e s i s i n x of 15 to 20 p e r c e n t .  explained  T h i s can be  qualitatively  by the model of McKinnon (1982), which p r e d i c t s such h y s t e r e s i s  114 w i l l occur  i f heat  i s generated  i n t h e phase c o n v e r s i o n p r o c e s s .  i n v e s t i g a t i o n of t h i s h y s t e r e s i s i s needed, perhaps u s i n g to  study  the heat  Further  microcalorimetry  e v o l u t i o n d u r i n g 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 t h e f i n e d e t a i l which can be x 2 studied with constant  c u r r e n t dQ/dV.  The s e n s i t i v i t y  of t h e t e c h n i q u e has  been w e l l i l l u s t r a t e d by t h e d i s c o v e r y of Cu^ ^Mo^S^ i n cathodes from n a t u r a l l y o c c u r r i n g percent  M0S2 by Mulhern (1982).  Even though o n l y 0.3  of t h e m a t e r i a l was Cu^ ^Mo^S^, t h e f i r s t  order phase t r a n s i t i o n  t h a t o c c u r s d u r i n g i n t e r c a l a t i o n of t h i s m a t e r i a l was v i s i b l e . appear t h a t i m p u r i t i e s which i n t e r c a l a t e and undergo f i r s t t r a n s i t i o n s can be d e t e c t e d  even when present  i n very small  of s t a t e , d i f f e r e n t m a t e r i a l s should  I t would  order phase concentrations.  S i n c e dQ/dV i s s e n s i t i v e t o f i n e s t r u c t u r e i n an i n t e r c a l a t i o n equation  prepared  system's  i n g e n e r a l each have t h e i r  own d i s t i n c t i v e dQ/dV p a t t e r n . dQ/dV i n s i n g l e phase r e g i o n s i s a good t e s t f o r any model of t h e thermodynamics of an i n t e r c a l a t i o n system. equation  of s t a t e makes i t v e r y d i f f i c u l t  dQ/dV's s e n s i t i v i t y t o t h e t o d u p l i c a t e w i t h a model.  T h i s has been i l l u s t r a t e d by t h e model f o r L i T i S x  in  2  o f Dahn (1982).  dQ/dV  s i n g l e phase r e g i o n s a l s o p r o v i d e s some i n f o r m a t i o n r e g a r d i n g t h e  k i n e t i c s of a c e l l ,  since the c e l l  r e s i s t a n c e a t any p o i n t can be d e t e r -  mined by measuring dQ/dV a t a s e r i e s of c u r r e n t s . Other examples have been g i v e n i n t h i s t h e s i s of the uses of constant c u r r e n t dQ/dV.  S i n c e t h e v o l t a g e of a c e l l  voltammetry by t h e apparatus,  i n d i c a t e s that i t i s d i f f i c u l t  i n constant  c u r r e n t dQ/dV.  "Super-  I t s "presence  t o n u c l e a t e t h e new phase a t t h e s t a r t of  o r d e r phase t r a n s i t i o n .  s t r u c t u r e s have been observed  i n l i n e a r sweep  " s u p e r c o o l i n g " e f f e c t s cannot appear.  c o o l i n g " i s a v e r y prominent e f f e c t  a first  i s controlled  A l s o , t r a n s i t i o n s between d i f f e r e n t  i n L i NbSe„ by Dahn, D . C , and Haering  staged (1982).  115  A t r a n s i t i o n i n v o l v i n g a change i n i n t e r c a l a n t c o n t e n t , but no change i n the host l a t t i c e ' s symmetry, has been observed Li^Mo^S^.  by Mulhern (1982) i n  There i s no doubt t h a t many more i n t e r e s t i n g examples o f  c o n s t a n t c u r r e n t dQ/dV a p p l i c a t i o n s w i l l a r i s e as the t e c h n i q u e i s a p p l i e d to an ever wider v a r i e t y o f m a t e r i a l s . Constant  c u r r e n t dQ/dV i s a v e r y s e n s i t i v e probe i n t o t h e thermo-  dynamics of i n t e r c a l a t i o n systems.  I t i s t o be hoped t h a t o t h e r  types  of e l e c t r o c h e m i c a l systems can a l s o b e n e f i t from constant c u r r e n t dQ/dV measurements.  Constant  c u r r e n t dQ/dV, p a r t i c u l a r l y when combined  x-ray d i f f r a c t i o n a n a l y s i s , can be used t o q u a n t i t a t i v e l y phase diagrams.  with  determine  In a d d i t i o n , c o n s t a n t c u r r e n t dQ/dV measurements i n  s i n g l e phase r e g i o n s a r e sure t o pose many more c h a l l a n g e s t o those attempting  t o model t h e behavior of e l e c t r o c h e m i c a l systems.  116  BIBLIOGRAPHY  Carslaw, H.S., and Jaeger, J.C. ( 1 9 5 9 ) Conduction ( 2 n d e d . ) , Oxford U n i v e r s i t y P r e s s , London. Dahn, D . C , and H a e r i n g ,  R.R. ( 1 9 8 2 )  Solid  of Heat i n S o l i d s  S t a t e Comm., i n p r e s s .  Dahn, J.R. ( 1 9 8 0 ) M.Sc. T h e s i s , The U n i v e r s i t y o f B r i t i s h Vancouver, Canada.  Columbia,  Dahn, J.R. ( 1 9 8 2 ) Ph.D. T h e s i s , The U n i v e r s i t y of B r i t i s h Columbia, Vancouver, Canada. Dahn, J.R. , Dahn, D . C , and H a e r i n g , Dahn, J.R., and H a e r i n g ,  R.R. ( 1 9 8 1 )  Dahn, J.R., Py, M.A., and H a e r i n g ,  R.R. ( 1 9 8 2 )  S o l i d S t a t e Comm. _ 4 2 ,  Solid State Ionics _2,  R.R. ( 1 9 8 2 )  179.  19.  Can. J . Phys. 60_, 3 0 7 .  F o l i n s b e e , J.T. , J e r i c h o , 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  4,224,390.  Jacobsen, T., West, K., and A t l u n g , S. ( 1 9 7 9 ) 126,  J . E l e c t r o c h e m . Soc.  2169.  McKinnon, W.R. ( 1 9 8 0 ) Ph.D. T h e s i s , The U n i v e r s i t y o f B r i t i s h Columbia, Vancouver, Canada. McKinnon, W.R. ( 1 9 8 2 )  J . Less Common Met., i n p r e s s .  Mulhern, P . J . ( 1 9 8 2 ) M.Sc. T h e s i s , The U n i v e r s i t y o f B r i t i s h Columbia, Vancouver, Canada. Murphy, D.W. , Cros, C , D i S a l v o , F . J . , and Waszczak, J.V. ( 1 9 7 7 ) I n o r g . Chem. 1_6, 3 0 2 7 . Thompson, A.H. ( 1 9 7 9 ) J . Electrochem. contained t h e r e i n .  Soc. 1 2 6 , 6 0 8 and e a r l i e r  references  

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