UBC Theses and Dissertations

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UBC Theses and Dissertations

On the condensation of a Van der Waals gas Strickfaden, William Ben 1970

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ON THE CONDENSATION OP A VAN DER WAALS GAS BY WILLIAM BEN STRICKPADEN B.Sc, M.Sc,  U n i v e r s i t y o f New Mexico, 1 9 5 9 U n i v e r s i t y of A l b e r t a , 1 9 6 2  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE 'REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY .IN THE DEPARTMENT OF PHYSICS  We accept t h i s t h e s i s r e q u i r e d standard  THE  as conforming  to the  UNIVERSITY OF BRITISH COLUMBIA JANUARY, 1 9 7 0  In p r e s e n t i n g  this  thesis  an a d v a n c e d d e g r e e a t the L i b r a r y I  further  for  agree  scholarly  by h i s of  shall  this  written  the U n i v e r s i t y  make  tha  it  freely  permission  for  It  of  of  Columbia,  British for  for extensive  gain  permission.  Department Columbia  the  requirements  reference copying o f  I agree and this  shall  that  not  copying or  for  that  study. thesis  by t h e Head o f my D e p a r t m e n t  is understood  financial  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada  fulfilment  available  p u r p o s e s may be g r a n t e d  representatives. thesis  in p a r t i a l  or  publication  be a l l o w e d w i t h o u t my  ABSTRACT A c l a s s i c a l gas whose p a r t i c l e s i n t e r a c t through a weak long range a t t r a c t i o n and s h o r t range r e p u l s i o n i s studied.  We study the p r o p e r t i e s  t i o n , previously derived  of a non-linear  i n t e g r a l equa-  by N.G. Van Kampen, whose s o l u t i o n s  g i v e the p o s s i b l e e q u i l i b r i u m d e n s i t y d i s t r i b u t i o n s . shown that above a c r i t i c a l  temperature the s o l u t i o n of the  equation i s unique, while below the c r i t i c a l multiple solutions exist.  It is  The e x i s t e n c e  temperature  of a two phase s o l u -  t i o n which must s a t i s f y the Maxwell r u l e i s proven by s o l v i n g the  equation e x a c t l y f o r a p h y s i c a l l y reasonable p o t e n t i a l f o r  the long range a t t r a c t i o n and a very g e n e r a l s h o r t range r e p u l s i o n .  Stability  form f o r the  and n e c e s s i t y c o n d i t i o n s f o r  the s o l u t i o n s to be extrema o f the f r e e energy are g i v e n and a p p l i e d to the v a r i o u s It  s o l u t i o n s of the i n t e g r a l  equation.  i s shown that In the l i m i t o f l a r g e volume the c r i t i c a l  point manifests i t s e l f  as a s o - c a l l e d b i f u r c a t i o n p o i n t  (point  where the number of s o l u t i o n s changes) of t h e . i n t e g r a l equation. Surface  t e n s i o n o f simple l i q u i d s  i s c a l c u l a t e d from the  i n t e g r a l equation and compared to experiment. excellent considering and  the s m a l l amount o f input data needed  the approximations used. The  non-equilibrium  coexistence  properties  o f the system i n the  r e g i o n are s t u d i e d by s o l v i n g a s e t of hydrodynamic  equations n u m e r i c a l l y . at  Agreement i s  I t i s shown t h a t the metastable s t a t e s ,  l e a s t i n the supercooled -  p o r t i o n of the isotherm i i  -  are indeed  u n s t a b l e w i t h r e s p e c t to l a r g e s c a l e p e r t u b a t i o n s . and decay r a t e s f o r s m a l l d r o p l e t s of condensing and ating l i q u i d  are  given.  - iii  -  Growth evapor-  TABLE OP CONTENTS Abstract  i i  T a b l e o f Contents  . . . .  . . . .'. . • i v  Table o f Tables  vi  Table o f F i g u r e s  v i i  Acknowledgement Chapter  I --  Chapter I I  --  viii General Theory and Review 1.  Introduction. . . . . . . . . . . .  2.  S t a t i s t i c a l Mechanical  3.  K i n e t i c Theory D e r i v a t i o n  I4..  The Pressure and E q u a t i o n of S t a t e .  13  5.  The Moment E q u a t i o n s .  17  Derivation .  1 I|. 9  . . . . . . .  Equilibrium Properties 6.  Assumptions  19  7.  The T r a d i t i o n Van der Waals Theory and The Maxwell C o n s t r u c t i o n as an Example  23  8.  E x i s t e n c e and Unioueness Finite) "  (Volume  32  9.  E x i s t e n c e and Uniqueness Infinite)  (Volume  37  10.  B i f u r c a t i o n and M u l t i p l e S o l u t i o n s , ij.0  11.  Stability  kk  12.  An Exact S o l u t i o n  50 '  13. - Numerical S o l u t i o n and C a l c u l a t i o n of S u r f a c e Tension  - iv -  60  Chapter I I I --  Some N o n - e q u i l i b r i u m II4.. 15.  Chapter  IV --  Metastable S t a t e s and Method of Calculation. . . . . R e s u l t s of Numerical S o l u t i o n o f Moment Equations .  C o n c l u s i o n and 16.  Results  Conclusion  Bibliography .  75 the  Discussion and D i s c u s s i o n  .98 •  - v -  8l  •  1 0 1  TABLE OP TABLES Table I -- C r i t i c a l P o i n t .  .62  - vi -  TABLE OF FIGURES Figure  1.  Assumptions on U/(n)  Figure  2.  The I n v e r s e F u n c t i o n  Figure  3.  An I s o t h e r m , (b) , and I t s D e r i v a t i v e , (a) , Above t h e C r i t i c a l P o i n t 2l|  Figure  1;.  An I s o t h e r m , (b) , and I t s D e r i v a t i v e , (a) , Below t h e C r i t i c a l P o i n t . . . . . . . . . . 25  Figure  5»  Free Energy D e n s i t y f o r A r b i t r a r y c< , (a) , and f o r c< S a t i s f y i n g t h e M a x w e l l R u l e , ( b ) . 29  Figure  6.  S o l u t i o n Curves i n the Phase P l a n e  Figure  7.  Coexistence  Figure  8.  D e n s i t y D i s t r i b u t i o n Near I n t e r f a c e . . . . .  Figure  9.  Excess P r e s s u r e Near I n t e r f a c e f o r D e n s i t y  21  n r .£  (.LP)  .  .  .  .  .  .  22  .  . . . . .  53  Curve, Theory and Experiment . . 6l|_ 68  D i s t r i b u t i o n of Figure 8  70  F i g u r e 10.  Surface Tension,  71  F i g u r e 11.  S u r f a c e T e n s i o n on Log-Log S c a l e , Theory and  Theory and Experiment . . .  Experiment  .73  o f S t a t e f o r Van d e r Waals "V^ . . .  F i g u r e 12.  Equation  F i g u r e 13.  P e r t u b a t i o n Above t h e - C r i t i c a l P o i n t . . . . 8[j.  Figure  Decay o f the C e n t e r ,  F i g u r e 15*  P e r t u b a t i o n Below t h e C r i t i c a l Supercooled  82  from F i g u r e 13. . • • • 85 Point,  Vapor. . . . . . .  87  F i g u r e 16.  Growth o f a S m a l l P e r t u b a t i o n  89  F i g u r e 17.  Evaporation  90  F i g u r e 18.  Equilibrium Distribution .  92  F i g u r e 19.  Growth Rates  93  F i g u r e 20.  Pertubation i n the Unstable Isotherm  of a Pertubation . . . . . . . .  - vii -  Region o f t h e 95  ACKNOWLEDGEMENT I wish t o express my a p p r e c i a t i o n to Dr. L u i s de S o b r i n o f o r suggesting  t h i s problem.  The many d i s c u s s i o n s with him  were i n v a l u a b l e and h i s p a t i e n c e a pleasant  and u n d e r s t a n d i n g produced  atmosphere i n which t o work.  F i n a n c i a l a s s i s t a n c e from the U n i v e r s i t y o f B r i t i s h -Columbia ( K i l l a m Graduate F e l l o w s h i p )  and r e s e a r c h  grants  from the N a t i o n a l Research C o u n c i l of Canada i s h a p p i l y and gratefully  acknowledged.  - viii -  1 CHAPTER I. .1.  INTRODUCTION  GENERAL THEORY AND REVIEW '  \'.*'  :  -'A"'-  :  One o f the o u t s t a n d i n g problems o f e q u i l i b r i u m s t a t i s t i c a l 1 mechanics i s the problem o f phase t r a n s i t i o n s .  2 Van d e r Waals  a t t a c k e d t h i s problem by s e p a r a t i n g the i n t e r m o l e c u l a r p o t e n t i a l i n t o two p a r t s ; a s h o r t range o r hard v e r y l o n g range a t t r a c t i v e p a r t .  core p a r t and a weak but  A heuristic  c a l c u l a t i o n of  - the e f f e c t o f these two p a r t s gave r i s e to the Van d e r Waals 3 equation of s t a t e , which shows a phase t r a n s i t i o n . was the f i r s t  Ornstein  t o e v a l u a t e the p a r t i t i o n f u n c t i o n on the b a s i s  of .separating the p o t e n t i a l i n t o two p a r t s and t o show how Van  d e r Waals' equation f o l l o w e d .  However, i n both o f these  t h e o r i e s the Maxwell r u l e had t o be evoked "ad hoc" t o e x p l a i n the l i q u i d - v a p o r e q u i l i b r i u m curve and thermodynamics had to . be used to r u l e out the p o s i t i v e s l o p e r e g i o n o f the isotherm from being p h y s i c a l .  I t was n o t u n t i l r e c e n t l y t h a t the Maxwell  r u l e was obtained from the s t a t i s t i c a l mechanical Uhlenbeck and Hemmer, i n a one-dimensional  theory,  t h e o r y . Kac, separate  the p o t e n t i a l i n t o two p a r t s , and take the l i m i t o f an i n f i n i t e l y long-range limit,  a t t r a c t i v e p a r t , the s o - c a l l e d Van der Waals  and o b t a i n r i g o r o u s l y i n t h i s way the Maxwell r u l e and 5  the Van der Waals equation.  Van Kampen went back t o O r n s t e i n ' s  theory and was able t o show how the Maxwell r u l e f o l l o w e d from the t h e o r y by e n l a r g i n g the c l a s s o f f u n c t i o n s allowed tions.  as s o l u -  Lebowitz and Penrose were able to show r i g o r o u s l y the  e x i s t e n c e o f the phase t r a n s i t i o n i n the t h r e e dimensional i n the Van d e r Waals L i m i t .  case  The theory o f Van Kampen i s not as  2  " r i g o r o u s as ""that of Lebowitz  and Penrose,  however, i t i s l e s s  r e s t r i c t i v e w i t h r e s p e c t to the c h o i c e of p o t e n t i a l s .  de  7 Sobrino was a p p a r e n t l y the f i r s t Van  to g i v e a k i n e t i c theory bf a  der Waals gas which g i v e s the same equation f o r the e q u i -  l i b r i u m d e n s i t y d i s t r i b u t i o n as the Van Kampen t h e o r y . T h i s work i s d i v i d e d  i n t o two main p a r t s , the e q u i l i b r i u m  t h e o r y and the n o n - e q u i l i b r i u m theory. i s found  i n Chapter  The e q u i l i b r i u m  theory  I I , and i s a study of Van Kampen's equation.  In some cases Van Kampen d i d not study t h i s . e q u a t i o n d i r e c t l y but made v a r i o u s approximations  which we t r y to a v o i d .  In  S e c t i o n s 7» 8 , 9 and 1 1 we reproduce most of Van Kampen's r e s u l t s , u s i n g d i f f e r e n t methods. is  Our main i n t e r e s t , however,  to study Van Kampen's equation with a view to r e l a t e some  of i t s mathematical  structure,  the s o - c a l l e d b i f u r c a t i o n p o i n t s ,  to the q u e s t i o n of the e x i s t e n c e o f a phase t r a n s i t i o n . i s done i n S e c t i o n s 8 , 9> and 1 0 . Kampen's equation i s a l s o presented  This  An exact s o l u t i o n to Van i n . S e c t i o n 1 2 f o r the case  i n which the d e n s i t y d i s t r i b u t i o n i s assumed to have plane symmetry, and some o f i t s p r o p e r t i e s s t u d i e d i n r e l a t i o n to the general  results.  As shown by Lebowitz  and Penrose,  one expects  the Van der  Waals theory to h o l d r i g o r o u s l y i n the Van der Waals l i m i t , which i s probably one  expects  implicit  i n the Van Kampen t h e o r y .  the approximations  However,  o f the Van Kampen theory to be  good i f there are enough p a r t i c l e s w i t h i n the range of the interaction.  To t e s t the v a l i d i t y of the Van Kampen t h e o r y  and the s e p a r a t i o n of the p o t e n t i a l  i n t o two p a r t s , we c a l c u -  l a t e the s u r f a c e t e n s i o n i n S e c t i o n 1 3 and f i n d  i t to be In  e x c e l l e n t agreement with experiment.  .  The n o n - e q u i l i b r i u m theory appears where we have used  the k i n e t i c  vapor  or superheated  that the metastable  liquid states,  i n S e c t i o n s l l | and  theory of de Sobrino  gate the t r a n s i t i o n from a metastable  • 15,  to i n v e s t i -  s t a t e of s u p e r c o o l e d  to a s t a b l e s t a t e .  I t i s found  at l e a s t those on the s u p e r c o o l e d  p o r t i o n of the isotherm, are indeed u n s t a b l e with r e s p e c t to l a r g e s c a l e p e r t u b a t i o n s In the d e n s i t y and some growth and decay r a t e s f o r these p e r t u b a t i o n s are c a l c u l a t e d . knowledge t h i s  i s the f i r s t  made u s i n g a k i n e t i c In  our  time such a c a l c u l a t i o n has  been  theory.  a l l t h i s work, the t w o - p a r t i c l e c o r r e l a t i o n s due  a t t r a c t i v e p o t e n t i a l have been n e g l e c t e d and sults  To  are not expected  to the  therefore a l l re-  to be v a l i d near the c r i t i c a l p o i n t .  2.  STATISTICAL We  MECHANICAL DERIVATION  p r e s e n t the d e r i v a t i o n 5  Let f  m  and n r e s p e c t i v e l y . <  V(frr )fr,)  and f  be c o o r d i n a t e s of p a r t i c l e s  n  Separate the i n t e r m o l e c u l a r  > i n t o two p a r t s ,  X  (  %  potential,  a v e r y s h o r t range p a r t , f„)  and a weak l o n g range a t t r a c t i v e p a r t , - %  Y(£Jl)= Q i L . l ) ~  equation  -»  due to Van Kampen. Ha  o f the e q u i l i b r i u m  f~X)  '  S  u  V , o f the c o n t a i n e r i n t o c e l l s  b  d  i  v  i  d  e  t  of volume  h  e  l a r  • Se  Q(P f ), rn) n  Hence volume,  h. o f l i n e a r  dimensions much 1 arger than the i n t e r p a r t i c l e d i s t a n c e , but s m a l l compared to the range o f X and f  are v a r i e d  n  , f  with  within  the i t h . c e l l  o f p a r t i c l e s i n that  erage p o t e n t i a l cell  the v a r i a t i o n i n  the t o t a l p o t e n t i a l  , of p a r t i c l e s In the same c e l l  denote a p o i n t ber  Assume, f u r t h e r t h a t when  i n the same c e l l  i s n e g l i g i b l e and that  n  due to X  •  cell.  is finite.  energy,  Let r{  and l e t A/* be the numt  Let  denote the av-  energy o f i n t e r a c t i o n between a p a r t i c l e i n  i and one i n c e l l  j  due to the p o t e n t i a l K.  .  For a  7  p a r t i c u l a r d i s t r i b u t i o n o f p a r t i c l e s among the c e l l s t o t a l energy o f i n t e r a c t i o n can then be Let particles tential Q be N (2,1)  .  <lO(N;)  written  be the amount o f phase space occupied by  i n c e l l I under the a c t i o n .  the  Let £ » ' / ^ T  and the t o t a l number of p a r t i c l e s  r  F o r any s e t of i n t e g e r s ,  2.  * i  of the s h o r t range po-  =  N  ;  > s u b j e c t to  5 the  canonical  p a r t i t i o n function  g i v e n by ( o m i t t i n g  f o r the system w i l l  the k i n e t i c energy p a r t  be  and the s h o r t  range i n t e r a c t i o n between the c e l l s )  The summation  means summation over a l l N-  condition  The f a c t o r  (2.1).  Nj/"TTN;j  are t h a t many ways o f c h o o s i n g the N celll, are  N  to  there  p a r t i c l e s to go i n t o  to go i n t o c e l l 2 , e t c . ; and s i n c e the p a r t i c l e s  z  to the p a r t i t i o n f u n c t i o n .  t i o n between c e l l s this contribution in  (  occurs because  i d e n t i c a l , these d i f f e r e n t ways g i v e  bution  subject  L  the same c o n t r i -  The s h o r t  range i n t e r a c -  i s n e g l e c t e d because f o r a l a r g e w i l l be small  system  compared to the terms  given  (2.2).  Now l e t  where  (2.5)  $(N<)=  ^  ^  f  .  The f r e e energy Is P = — -^JU^Q and hence H  $^Nij  t o t a l f r e e energy f o r a p a r t i c u l a r c o n f i g u r a t i o n ^0^0  ^  s  the c o n t r i b u t i o n  range i n t e r a c t i o n o f N '  L  and •  to the f r e e energy from the s h o r t  p a r t i c l e s i n c e l l L.  e x p r e s s i o n s o f the thermodynamic contribution  i s the  equilibrium  ( I n these q u a n t i t i e s , the  o f the terms p r o c e e d i n g from the k i n e t i c energy  i s omitted s i n c e  they can be c o n s i d e r e d  constant f o r our  purposes.) To determine the most probable e q u i l i b r i u m t i o n we must f i n d to the c o n d i t i o n s e t o f J^M^ this,  the maximum term i n the sum (2.1).  This  i s equivalent  which maximizes (2 J+)  assume t h a t N-<^< M  uous space-dependent  subject  to f i n d i n g the To do  V and t h a t passage t o a c o n t i n -  density  V\(r )  can be e f f e c t e d .  The summations i n (2.I4.)  c  =  N{ l& go over i n t o i n t e -  and with ^ ( N ^ ~ Al|>(>^\) becomes  (2.7)  Vy(n(f))dr + ^ p ^ K C r ^ ' J ^ C n n t P ' J c l r c A f ^  <|{Mr)]  while condition  (2.1)  becomes  [ n ( J r ) d r := H .  (2.8) <B"£>i(.f)]  i s now a f u n c t i o n a l o f h(r) and to f i n d the dens-  i t y d i s t r i b u t i o n which maximizes (2.8)  (2.3)  to ( 2 . 1 ) .  subject  (2.6)  grals  configura-  (2.7)  subject  to c o n d i t i o n  we need only a p p l y the c a l c u l u s of v a r i a t i o n s .  Using  the method o f Lagrange m u l t i p l i e r s , the problem i s equival e n t to f i n d i n g the s t a t i o n a r y v a l u e s o f  iw*^  (2.9)  $w*>3  - ^ h t ? ) a ?  w i t h o( an undetermined Lagrangian m u l t i p l i e r . gives  Physically,  the chemical p o t e n t i a l o f the system.  To c a r r y out the v a r i a t i o n l e t ri(p) = Il (.f) + 0  where ft Is the e q u i l i b r i u m d e n s i t y , Q  6  €  ,  a s m a l l parameter and  *X(v) hold  a v a r i a t i o n In the d e n s i t y .  Since condition  (2.8)  must  f o r the e q u i l i b r i u m s o l u t i o n , the p o s s i b l e v a r i a t i o n s  are r e s t r i c t e d by the c o n d i t i o n  (2.10)  ( %(f)d? =  O.  Expanding c j > { r \ + ( t h e s u b s c r i p t on  U  0  w i l l now be dropped),  with  -  (2.11)  +  p^Cf.fOnCvO^'-cK^cnclr  V  (2.12) '4*  and  respect is  i {^V" a  are the f i r s t to V \ .  (2.13)  ^ .HAYOXVJAC" -V  p ^ K C f j r O  and second d e r i v a t i v e s of lp w i t h  A necessary c o n d i t i o n f o r $ to be s t a t i o n a r y  t h a t (j) v a n i s h  obtain  V  f o r a r b i t r a r y v a r i a t i o n s *X(f) .  the n e c e s s a r y  Hence we  condition  ^ ' ( n . C r ) ) + ft  \  KCr,On.lf  Odr'  =  OC .  v  To f i n d  the e q u i l i b r i u m d e n s i t y  study the s o l u t i o n s Equation (2.13) If  d i s t r i b u t i o n we need to  to t h i s n o n - l i n e a r  i s the e q u a t i o n d e r i v e d  one c o n s i d e r s  i n t e g r a l equation. by Van Kampen.  hard spheres i n t e r a c t i n g through a  weak a t t r a c t i v e p o t e n t i a l , the f u n c t i o n C O ( N I J ) o f equation  8  (2.£) i s j u s t the amount o f phase space occupied by spheres i n a volume A pendent  ,  of temperature.  hard  I n a hard sphere system i t i s i n d e An approximate  form f o r t h i s  f o r h a r d spheres o f volume 8 i s g i v e n by the  function  interpolation  formula, (2.lli)' For  CO ( M ^  r  (  & -  N  T  & )  H  I  .  a one d i m e n s i o n a l system of hard rods t h i s formula i s  exact; see Tonks.  L a t e r we  s h a l l see that t h i s form f o r  CO(M-) l e a d s to the t r a d i t i o n a l Van der Waals equation o f state.  9  3.  KINETIC THEORY DERIVATION It  of Van  i s c l e a r t h a t the s t a t i s t i c a l mechanical d e r i v a t i o n Kampen i s only  reasonable  an approximation, a l b e i t  a physically  one.  In t h i s s e c t i o n we  present a b r i e f d e s c r i p t i o n of  de  7 Sobrino's assumptions used i n o b t a i n i n g h i s k i n e t i c equation from which f o l l o w s  (2.13).  the same e q u i l i b r i u m equation  However, i t must be borne i n mind t h a t because i t i s a more d e t a i l e d d e s c r i p t i o n , there kinetic  t h e o r y than i n the  are more assumptions i n  2.  e q u i l i b r i u m t h e o r y of s e c t i o n  C o n s i d e r a c l a s s i c a l system o f volume V, v e r y l a r g e number, N,  the  containing  of i d e n t i c a l p a r t i c l e s .  a  The p a r t i c l e s  i n t e r a c t through a h a r d core p o t e n t i a l of diameter CT, and l o n g range p o t e n t i a l , external  fields  e q u a t i o n , the  —K(|r^~ rjj).  are n e g l e c t e d .  one  Boundary e f f e c t s  bution  p a r t i c l e truncated  (3.D  }  f u n c t i o n , "\ , by z  a £ +53.a£  In the f i r s t cDi l^-V^'K^" :  +  the  d i s t r i b u t i o n function  Ara*  distri-  4- & i c ^ ' U s - d ^ ' .  i n t e g r a l r ' i s i n t e g r a t e d over the  (J  A truncated  0.  domain,  w h i l e i n the second i n t e g r a l i t i s  over a l l v a l u e s . as  truncated  e q u a t i o n developed by Grad,  4 f  over the sphere, § ; \ f-f'\-  and  S t a r t i n g from L i o u v i l l e ' s  (_f v "t) i s r e l a t e d to the t w o - p a r t i c l e }  a  ; *\) i n both cases i s  integrated integrated  d i s t r i b u t i o n function i s  defined  where ^  i s the u s u a l N p a r t i c l e - d i s t r i b u t i o n  N  the  domain eD i s r e s t r i c t e d to  <&i  b e i n g d e f i n e d by the i n e q u a l i t y  The  truncated d i s t r i b u t i o n  f u n c t i o n , but  (fj,-?, )•> CT  and a l l "0,; .  f u n c t i o n approximates  the u s u a l  •z.  distribution  f u n c t i o n i f N(T , the occupied volume, i s s m a l l  compared t o the volume of the box c o n t a i n i n g the m o l e c u l e s . To  arrive  at h i s k i n e t i c  equation, de Sobrino now makes  the p o t e n t i a l  K ( f , - - r j ) i s v e r y weak and  the  assumption t h a t  has  a range much g r e a t e r than the average i n t e r p a r t i c l e  tance but much s m a l l e r than the s i z e o f the system. volume i n t e g r a l  o f the p o t e n t i a l  able w i t h the t o t a l k i n e t i c tions is  tion  3  The  K ( r . - r j ) i s assumed compar-  energy o f the system.  are now n e g l e c t e d i n the f i r s t  £0»r^ S / o ' t ) = i ( f , ^ , t )  dis-  Correla-  of ( 3 . 1 ) ,  integral  that  » and t h e i n t e g r a -  £,(Y\^>"t)  extended t o w i t h i n the -sphere • | r - r ' } < ; 0" s i n c e the con-  tribution  to t h i s r e g i o n w i l l be s m a l l due to the weakness  of the p o t e n t i a l . To  express the second i n t e g r a l  distribution that  f u n c t i o n s , de S o b r i n o now assumes with Enskog'  just before a c o l l i s i o n ,  particles  i s due e x c l u s i v e l y  potential,  hence  (3.2)  f ,  where  °7^( t) r  In terms o f one p a r t i c l e  =  • o  l  (  r  i  the c o r r e l a t i o n to the e f f e c t  ( £ i f ' j J f  i  is. the p a i r d i s t r i b u t i o n  (  F  ^  i  t  between two  o f the hard  ) f  i  0  C  P  ;  ^ )  core  j  f u n c t i o n e v a l u a t e d at  c o n t a c t o f the hard spheres and i s a f u n c t i o n o f the d e n s i t y  11  Equation of  (3.1)  (3 . 2 )  i s now  substituted  and a f t e r expanding  good approximation,  and  i n t o the second  to f i r s t order  J* K  +  >>.2iL  I  equation,  i i '- £ fkt^nlr;-!.)^'*  J(£) + K C O .  i s the u s u a l n o n - l i n e a r Boltzmann c o l l i s i o n i s another  collision  that e q u a t i o n  operator  and  o p e r a t o r which takes i n t o account  n o n - l o c a l c h a r a c t e r of the hard sphere apparent  , a very  in<3"^|i  some standard t r a n s f o r m a t i o n s of  terms, de Sobrino o b t a i n s the k i n e t i c  (3.3) ' 2£  integral  (3.3)  interaction.  the  It is  i s a Boltzmann equation where the  l o n g range p o t e n t i a l i s taken i n t o account by a s e l f - c o n s i s t e n t f i e l d method, with the e f f e c t of the hard core i n c l u d e d i n the last  term  and i n ^ .  de Sobrino-::- now tion (3.3)  goes on to show that the k i n e t i c  obeys an K-theorem and  l u t i o n s must be sought  t h a t the e q u i l i b r i u m  among the l o c a l Maxwellian  equaso-  solutions.  T h i s l e a d s , i n the u s u a l manner, to the f o l l o w i n g equation for  the d e n s i t y  h.(p).  * de Sobrino i n r e f e r e n c e f d e r i v e d the theory u s i n g a p a r t i c u l a r v a l u e of . The theory f o r a r b i t r a r y % f o l l o w s almost t r i v i a l l y , and has been worked out by L.D. Chinh (unpublished work).  12 Thus, the e q u i l i b r i u m d e n s i t y  obeys the same k i n d o f equation  as was d e r i v e d by Van Kampen, except that now we have a specific  r e l a t i o n between the f u n c t i o n ty(r\) o f equation ( 2 . 1 3 ) and  the f u n c t i o n ^ ( n ) •  As w i l l be seen i n the next s e c t i o n ,  same r e l a t i o n can be d e r i v e d  i n the e q u i l i b r i u m  theory.  this  13 ij..  THE  PRESSURE AND  We Use  EQUATION OF-STATE  wish to o b t a i n the l o c a l p r e s s u r e i n the system.  i s made of the thermodynamic NjJL =  (l+.l)  F +  PV  relation  ,  where jd. and F are the chemical p o t e n t i a l and f r e e respectively. particles tion  S i n c e each o f the c e l l s , A ,  containing N  i s i n l o c a l thermodynamic e q u i l i b r i u m , t h i s  rela-  can be a p p l i e d to g i v e the l o c a l p r e s s u r e ( P ( A l ) In v  the c e l l  i .  tained.  From ( 2 . 1 3 )  ^jx-  (lj..2)  and from  A  energy  (2.7)  (2.6),  In the l i m i t ,  _c*  the p r e s s u r e * P ( r ) i s ob-  the chemical p o t e n t i a l i s  ^  -  p  J KCv.f^'vNCF'jAr' j  the l o c a l f r e e energy  i n the c e l l  o f volume  is  Solving  (I4..I) f o r the p r e s s u r e , one  PCr)  =  ^j^-  -  gets  F(r)/A  nC?)|A-  and hence, U.lj-)  p , PC?) =  lf(«Cr))-  r i C f ) V ( * l > ) " " i p ^(- )^KC^ f')n(rUr' . F  f  )  T h i s can be w r i t t e n (1^.5)  p?H,C, ~ i  nCv)^<(r r')nCr ) Jlf  where "PH.C. i s the p r e s s u r e due  1  ,  c  to the hard core p a r t of the  potential. We terms  •  now p r e s e n t  (3.^).  of equation  be w r i t t e n i n terms and  another  the potential  derivation  of the f i r s t  The p r e s s u r e  of the pair  o f a gas o r l i q u i d  correlation  of interaction  three c  f u n c t i o n , Oj(rJ  between t h e  molecules  V(r) as"  For  a hard  core  potential,  f oo  I  VCv)  the  above  equation |3^.c.  (I1.6)  in  ?<0",  ^  becomes  = ^  4  ^WV^'CoO,  w h i c h ^(o") i s t h e p a i r  correlation  and  Is a function of the density.  the  argument  CT f r o m oq(rj) •  the  equation  f o r the pressure,  This  i s a differential  grated  to give  (J+.7)  Comparing  and hence i f ' ,  this  with  -  equations  Henceforth  we w i l l  Comparing equation equation  equation  vyc^}--v\JUxV^  function at contact  (i|..6) we  f o r if'Cn') w h i c h m a y  drop with  have  be  inte-  thus,  "v^tm^A  (2.13)  +•  and  n xconstant  3  (3 •!•!-) we s e e t h a t  15 the  i d e n t i f i c a t i o n o f ^ with the p a i r c o r r e l a t i o n f u n c t i o n  contact i s We  now  function If  justified. d i s c u s s the v a r i o u s forms f o r the p a i r  correlation  that have been used i n t h i s work.. the  interpolation  formula (2.1b,.) i s used, the  i n g form f o r the p a i r c o r r e l a t i o n f u n c t i o n  (lj..9)  L  ° 7 C A ) = v-  I -~  b  ;  results;  b-|-Trcr\  ->  VN  follow-  o  Then U>  r  9^r\ —  :  —  on  1  }  |— b  arid the  equation of s t a t e  for  constant density solutions,  the  as c a l c u l a t e d  Y\  by  n(r)  equation (i|.Il-)  = n,  a constant  independent of r i s (li.ll)  pp'  where  -  (l|.12)  2\AJ k=.  der Waals  pWobn^  ___D_  a  6  T h i s i s j u s t the  To  ^ K C ^ r ' ) ^ ' .  e q u a t i o n of s t a t e  f o r the  traditional  improve the  sphere gas.  equation of s t a t e we  must have a  T h i s may  be  obtained from the  better  f o r a hard  work o f Ree  12.  Hover, who  Van  gas.  approximation f o r the p a i r c o r r e l a t i o n f u n c t i o n  series  at  calculated  the  f i r s t s i x terms o f the  f o r such a system and  found,  virial  and  16  (14-. 13 )  I ^ |-bv»+ 0 . 2 ^ 5 C b n t + - 0.ll"3(b^-+ O.Q38C,Ci=^*.  Equation  (I4..I3) gives f o r  ',  Ree and Hover, i n the same r e f e r e n c e , a l s o obtained a Pade approximant to the above v i r i a l  series.  This  approximation  should be good f o r h i g h e r d e n s i t i e s and indeed i t f i t s the "experimental" p o i n t s obtained from molecular dynamics c a l c u l a t i o n s very w e l l i n the d e n s i t y r e g i o n below where the phase t r a n s i t i o n occured tion.  i n the m o l e c u l a r dynamics  ( N e i t h e r the v i r i a l  s e r i e s nor the Pade approximant  show a phase t r a n s i t i o n f o r a hard sphere have a t h i r d -Y| r  system.)  Hence  approximation, l + C . U +• C ^ ( b n f / X  C, - 0 . o t 3 5 o 7 0.^  calcula-  Q.oMll.^  C  5  3  & * l  +  = -  j> C if —  C  3 ^  t  CbO  ^  j 0 - 0 8 1 3 1 3 .  T h i s gives f o r 147' ,  ( 1 4 - . 1 6 )  ^  Ya  The equation of s t a t e f o r these other w i l l be d i s c u s s e d i n a l a t e r  section.  V  \Aa  approximations  we  5.  THE MOMENT EQUATIONS One  o f the n o n - e q u i l i b r i u m p r o p e r t i e s o f the Van der  Waals gas i s the t r a n s i t i o n o f t h e gas from a metastable s t a t e , where the f r e e energy stable equilibrium state. to  i s o n l y a l o c a l minimum, to a  To study t h i s problem  study the s o l u t i o n s o f the n o n - l i n e a r k i n e t i c  (3.3).  one needs equation  I n view o f the computational d i f f i c u l t i e s ,  be p o s s i b l e that the f i r s t  i t might  few moment equations p r o v i d e a  s u f f i c i e n t l y good approximation to the problem.  Since this  process i s probably a slow one, the moment equations may indeed be a v e r y good approximation.  7 de S o b r i n o o b t a i n e d the equations f o r the f i r s t  three  moments i n the r e l a x a t i o n time approximation i n the standard manner.  Prom the form o f these equations and from the f a c t  t h a t the p o t e n t i a l K ( r , r ' ) does n o t c o n t r i b u t e to the sheer s t r e s s o r thermal c o n d u c t i v i t y one sees how to c o r r e c t the u s u a l moment equations f o r the e f f e c t of the hard core and l o n g range p o t e n t i a l .  The equations one o b t a i n s f o r the  d e n s i t y , hX?-); mean v e l o c i t y , c C f ) ; and temperature, T ( r ) are  (5.D  (5.2)  (5.3)  as f o l l o w s :  + n V- c = O ,  :Dc ^ _ ? D t V  - J _ P  -nv\KCf f')n(rOdr + 7-G j l  H  C  R  )  T  C  V - C -  _ 1 _ V - ( A V T ) +  _ Z _ Q : V C .  18 •In—thes-e—equations,  A  i s the thermal c o n d u c t i v i t y and  the sheer s t r e s s t e n s o r both computed f o r a hard sphere PH.C.  i s the hydrodynamic d e r i v a t i v e , and h  a hard c o r e gas,  Boltzman's c o n s t a n t .  For the work o f  10  where the e x p r e s s i o n given by Enskog, which i s o b v i o u s l y applicable,  b 3  w i l l be  used.  5  CJ" i s the hard core  gas,  the p r e s s u r e o f  s e c t i o n s llj. and 15 we need, o n l y the thermal c o n d u c t i v i t y  (5.6)  Q  diameter..  19  CHAPTER I I - EQUILIBRIUM PROPERTIES 6. -ASSUMPTIONS The  purpose of t h i s chapter i s to i n v e s t i g a t e  e q u i l i b r i u m equation ( 2 . 1 3 ) under the most g e n e r a l and  to Van  tion the  assumptions concerning  i n s e c t i o n s 6 to 13  Kampen s work and 1  Penrose.  as thoroughly as p o s s i b l e  the k e r n e l KCf,?') .  discussions  the and  the f u n c t i o n  I t i s to be noted t h a t do not  the  add much p h y s i c a l l y  are l e s s r i g o r o u s  than Lebowitz  and  However the d e t a i l e d study of the i n t e g r a l equa-  (2.13)  i s , we  existence  t h i n k , of i n t e r e s t because i t shows  of the c r i t i c a l p o i n t i s connected w i t h  how the 14  bifurcation properties has  used t h i s  i d e a i n a more g e n e r a l  transitions. w i l l h e l p us  of a n o n - l i n e a r  Thus, we  operator.  Grmela  treatment of phase  hope t h a t the study of equation  (2.13)  to understand, from a c e r t a i n p o i n t of view,  the mathematical s t r u c t u r e o f phase t r a n s i t i o n s . In most o f the mathematical theorems to and  KC^ r') }  can be v e r y  considerations  and  general  indeed, but  f o r the examples, we  will  follow, from p h y s i c a l consider  only  f u n c t i o n s w i t h the f o l l o w i n g p r o p e r t i e s : a)  We  s h a l l suppose t h a t  if Co)  finite,  and  , ^'(0 —* - oo  +a>  only discuss p r o p e r t i e s  of  , and in  the range O<V)< 1. b)  as w e l l as I t s f i r s t three d e r i v a t i v e s are uous f u n c t i o n s  c ) , d)  in  'VCn) has  contin-  of A .  o n l y one  zero,  at n = h  c  , and  is  monotonically  "decreasing. e) '4^'  e v a l u a t e d at h= W  i s negative.  c  f)  Ktr^r'J  i s symmetric, i n f a c t , i t i s equal to  g)  KC^r')  is positive definite,  for  a l l if(r) not i d e n t i c a l l y  The  KO^-^'O  that i s  zero.  f u n c t i o n ^ ( ^ J i s p r o p o r t i o n a l to the f r e e  energy  d e n s i t y of a hard core gas, so assumptions a) to e) o n l y p r o p e r t i e s of the hard c o r e .  concern  Assumption a) i m p l i e s  t h a t the p h y s i c a l l y a l l o w a b l e range of v a l u e s f o r the density  i s l i m i t e d t o 0< *\ < 1  been chosen so t h a t n = l density. for  and t h a t the d e n s i t y u n i t s have  corresponds  As w i l l be seen l a t e r ,  to the c l o s e p a c k i n g  assumption  c) i s necessary  the e x i s t e n c e of a c r i t i c a l p o i n t w h i l e assumption  necessary  e) i s  i f the hard core system i s t o have no phase t r a n -  s i t i o n . -"- Assumptions c) and e) are chosen f o r s i m p l i c i t y . A s h o r t range i n t e r a c t i o n f o r which c) does not h o l d would produce a more complicated phase diagram than the two  phase, s i n g l e c r i t i c a l p o i n t one  Assumption f ) w i l l  c o n s i d e r e d i n t h i s work.  always be s a t i s f i e d  p a i r wise c e n t r a l p o t e n t i a l s . f o r most of the mathematical  simple  i ^ one deals with  Assumption g) i s necessary theorems; i t w i l l  be t r u e f o r p o t e n t i a l s which are square  certainly  i n t e g r a b l e and  a t t r a c t i v e f o r a l l v a l u e s of the v a r i a b l e s . While m o l e c u l a r dynamics c a l c u l a t i o n s do I n d i c a t e t h a t the hard core system does have a phase t r a n s i t i o n f o r values of the d e n s i t y near c l o s e packing, no a n a l y t i c a l l y d e r i v e d e x p r e s s i o n f o r the p a i r c o r r e l a t i o n f u n c t i o n shows t h i s e f f e c t . (See d i s c u s s i o n f o l l o w i n g equation (i|.lij.).)  Prom assumptions a) to e) one various of  functions involved.  , if)"  d e f i n e d by  and  the  can-make sketches of  For future reference  the  sketches  appear below.  s o l u t i o n of the  <a/(n) - o c =  equation  4?.  T h i s f u n c t i o n i s a l s o sketched on the f o l l o w i n g page f o r f u t u r e convenience.  .The value  Yl  0  i s d e f i n e d by Lv'Cho") = 0.  F i g u r e 2.  The  Inverse  Function  v\ = +T(«f)  23 7.  THE TRADITIONAL VAN DER WAALS THEORY AND THE MAXWELL CONSTRUCTION AS AN EXAMPLE I n t h i s s e c t i o n we c o n s i d e r  a simple example which  illus-  t r a t e s the f o l l o w i n g p o i n t s ; 1 ) the t r a d i t i o n a l Van d e r Waals theory,  and 2 ) how the Maxwell c o n s t r u c t i o n may be  by e n l a r g i n g defined  the c l a s s o f s o l u t i o n s .  i n this section w i l l  obtained  Some of the f u n c t i o n s  a l s o be o f use i n l a t e r  discuss-  ions.. By  adding and s u b t r a c t i n g a p p r o p r i a t e  f a c t o r s , the  f u n c t i o n a l to be maximized, equation ( 2 . 9 ) can be w r i t t e n , K(f r ){n(fJ- (F0}dr ! ?' ,  )  h  C  l  j )  where (7.2)  The  first  term r e p r e s e n t s  a volume e f f e c t , w h i l e the second  term i s a l i q u i d - g a s s u r f a c e term s i n c e i t i s not zero when the d e n s i t y is  i s not uniform.  l a r g e and t h a t the s u r f a c e  state  Assuming t h a t the system  separating  the l i q u i d - v a p o u r  ( i f i t o c c u r s ) i s s m a l l compared to the volume, we now  neglect  the second term o f ( 7 . 1 ) and c o n s i d e r  t i o n a l problem o f maximizing only the f i r s t assume t h a t the volume i s so l a r g e that (7.2)  is a  the new v a r i a -  term.  We a l s o  W(r) o f equation  constant.  - The E u l e r equation f o r the new v a r i a t i o n a l problem becomes, (7.3)  only  Assume that the f u n c t i o n tyC^) has p r o p e r t i e s s e c t i o n 6.  Now suppose t h a t the .temperature i s such that  (7.It-)  lb"(n ) c  -i-  £W(3 <  T h i s means that the f u n c t i o n n  a) t o e) o f  O  .  4/'(") + 2Wj5n  as a f u n c t i o n o f  i s monotonic and hence the equation (7'3)  r o o t g i v e n by,  ty'l^*) + z  n = o<. •  The s i t u a t i o n i s sketched below.  has o n l y one  However, c o n s i d e r the s i t u a t i o n when  4*"(n ) +  (7.5)  |3 > O  c  t^W [2> r\  The f u n c t i o n  now develops a maxima and minima  at the d e n s i t i e s in, and n  t  d e f i n e d by the two r o o t s of the  equation (7.6)  *V'(*) + 2 W p = O .  Hence f o r c e r t a i n values t h r e e r o o t s ^3,  Figure Derivative,  o f C<  and n . &  the equation  ( 7 . 3 ) may have  The s i t u a t i o n i s sketched  An Isotherm, (b) , and I t s (a) , Below the C r i t i c a l P o i n t  below.  26  We  thus see t h a t a c r i t i c a l  (7.7)  temperature,  +  such t h a t when ^ < ^> p o s s i b l e while  for  jo > ^  m u l t i p l e s o l u t i o n s are p o s s i b l e .  c  respectively. o f o( ( r v )  The  |3 with  the  C  critical  with the d e n s i t y of the l i q u i d  5  two  previous  and we  and  der Waals isotherms  with  are p l o t t i n g OC which i s propor-  t i o n a l to the chemical p o t e n t i a l i n s t e a d of the p r e s s u r e is  as  usual. The  of  gas  F i g u r e s 3 and l+,  sketches,  y i e l d the t r a d i t i o n a l Van  a r b i t r a r y tyty or  defined  s o l u t i o n to the problem i s  P h y s i c a l l y i t i s n a t u r a l to a s s o c i a t e temperature and H ,  > can be  C  -  o n l y one  c  |J  <|>  root  KW  i s e a s i l y shown to correspond to a maxima  (and hence a minima o f the f r e e energy) and  f o r e an a c c e p t a b l e  physical solution.  This i s shown by  l o o k i n g at the second v a r i a t i o n of., equation the second term;  |  (7.8)  From equation The  \{H'"<*) +  pw]x^ i^. ?  (7.1+) i t immediately f o l l o w s t h a t  root  , however, i s not  s o l u t i o n , f o r from F i g u r e fi. between H, , and H between fl, and  2  ,$  z  Y\ , hence z  the f r e e energy.  a physically  V"\^  $  2  < O .  acceptable  i t i s c l e a r that for a density > O , and  the r o o t  n.  H  always  lies  corresponds to a maximum of  T h i s r u l e s out the n e g a t i v e  of the isotherms from being p h y s i c a l s t a t e s . argument  ( 7 . 2 ) . neglecting....  thus,  =  2  is there-  slope  region  By a s i m i l a r  and fie, are always seen to correspond to minima  o f the f r e e energy and hence are a c c e p t a b l e  solutions.  These  s o l u t i o n s give the l i q u i d and gas d e n s i t i e s as w e l l as the superheated l i q u i d and supercooled  vapor p o r t i o n s o f the i s o -  therms of Van der Waals. A l l o f the s o l u t i o n s found so f a r are constant  density  s o l u t i o n s , t h a t i s they do n o t depend on the c o o r d i n a t e T h i s i s the o n l y p o s s i b i l i t y  the E u l e r equation  °  c  a c e r t a i n range o f o(.j ^^n^.  for  f  f o r jB < ^> .  m  and  for  R,  1^5  for  R  z  equation  >  s o l u t i o n of the  s o l u t i o n s we now  problem of maximizing  (j)jv\j ,  ( ? . l ) , s t i l l n e g l e c t i n g the second term and con-  s i d e r more g e n e r a l  v a r i a t i o n s than were c o n s i d e r e d  previous  In p a r t i c u l a r we want to maximize  problem.  not o n l y with  respect  the regions  "R| and ^  2  , f o r example.  conditions i t i s s u f f i c i e n t S> .  f o r the  to v a r i a t i o n s of the form n+ £ "p  a l s o v a r i a t i o n s which -move the boundary,  surface  L  of V , i s possible.  the p o s s i b i l i t y o f d i s c o n t i n u o u s  want t o r e t u r n to the o r i g i n a l  However when P ( c ».  5  where R^R-z and ^ 3 are volume regions Admitting  .  are a l l s o l u t i o n s o f  so t h a t a d i s c o n t i n u o u s  fn  r  S, , which  but  separates  To o b t a i n the r e l e v a n t  to c o n s i d e r o n l y one s e p a r a t i n g  We c o n s i d e r the problem of f i n d i n g necessary  c o n d i t i o n s such that the f u n c t i o n a l ( . ) 7  9  $ ^ h i e ^  =  \  FC*C*))<JF  i s maximized under the above c o n d i t i o n s .  I n our case we have  (7.10)  F U ) = ^ (VN) + p W n - o c n  C a r r y i n g out the v a r i a t i o n , thus,  -  \ F'tr0£-)(cl> ^  I FC«+€t^Ar  +  W ( . ^ ^ € - y V ^ + •• • •  f p'CnKTcAr -r F ( VI (.8, + d))SV - F ( n t S - oj) ^ +- • • • , where the n o t a t i o n evaluated  n ( S , + o)  means t h a t the d e n s i t y i s t o be  s l i g h t l y to the i n t e r i o r  means s l i g h t l y to e x t e r i o r . which 8V = o equation.  of s u r f a c e  F i r s t consider v a r i a t i o n s f o r  ., we then o b t a i n Next f o r v a r i a t i o n s SV^O  where F t h ^ = 0.  o r the usual Eule r we o b t a i n , F(3,+ o)sF(9,-o)  I n order t o see i f t h i s c o n d i t i o n can be  s a t i s f i e d we p l o t the f o l l o w i n g  and n ( § , - o )  F ( n ) f o r v a r i o u s values  sketches.  o f c< and f i n d  F M  29  CK, ( M a x w e l l )  F i g u r e $, Free Energy D e n s i t y f o r A r b i t r a r y c< , (a) , and f o r c< S a t i s f y i n g the Maxwell Rule, (b) When the h e i g h t  of t h e two maxima are equal we indeed can  have F(r> )= F (n ) 5  that since  5  and c o n d i t i o n  (7.11) i s s a t i s f i e d .  Note  tl^ corresponds to a minimum i t i s a u t o m a t i c a l l y  excluded from the c o n s i d e r a t i o n s . discontinuous  Hence the only  possible  s o l u t i o n s are those t h a t correspond to a l i q u i d  gas mixture ( p a r t II3 and p a r t n  5  I m p l i c i t l y a c o n d i t i o n on  as can be seen by r e w r i t i n g i t  <X  ).  Condition  (7 - H ) i s ,  as, (7.12) and  n  3  i n t h i s , form i s seen to be j u s t the Maxwell equal  c o n s t r u c t i o n a p p l i e d t o the curve Figure  ^'(r\)+-2^> W H  area  (See  )  From the second v a r i a t i o n i t i s e a s i l y seen that the discontinuous  s o l u t i o n (part  to a maximum o f  and p a r t n$ ) corresponds  and hence a minimum of the f r e e energy.  When c< does not have the value  s p e c i f i e d by the Maxwell  r u l e , o n l y the constant d e n s i t y s o l u t i o n s are allowed. i s seen from the p r e v i o u s s k e t c h of the maxima c o r r e s p o n d i n g to one  stable physical state.  F(n) where i n that case  of the s o l u t i o n s w i l l  a r e l a t i v e maxima and hence cannot  correspond  to an  o n l y be  absolutel  In t h i s case we w i l l have a meta-  s t a b l e s t a t e i n which the gas i s supercooled or the superheated.  This  F u r t h e r , as Van Kampen  liquid  shows, the f r e e  i s l e s s f o r the two phase s t a t e than t h a t f o r the  energy  single  phase (meta-stable) s t a t e w i t h the same temperature  and num-  b e r of p a r t i c l e s .  I t i s f o r t h i s reason t h a t the s i n g l e  phase states>which  i n t e r l a c e fl  3  and H  computed when the  5  Maxwell r u l e i s s a t i s f i e d j a r e c a l l e d meta-stable.  I t must  be remembered t h a t t h i s w i l l happen o n l y when (i > P> I n s t e a d of s t a r t i n g from the f r e e energy (7«l)  and n e g l e c t i n g the second  i n t e g r a l equation (2.13) Again assume t h a t  W  term, we  of equation  can s t a r t w i t h the  (which i n c l u d e s the second  i s independent., o f  form s o l u t i o n s of the e q u a t i o n .  We  f  the uniform  solutions.  and the t r a d i t i o n a l Van  That t h i s  term of equation  is appli-  i s the case i s  (7.1)  i s .zero f o r  Thus the p r e c e d i n g d i s c u s s i o n has  shown t h a t the i n t e g r a l equation density solutions yields  equation  Hence a l l of the d i s -  c u s s i o n c o n c e r n i n g the uniform s o l u t i o n s o f (7«3) c a b l e i n a more g e n e r a l s e t t i n g .  term).  and l o o k f o r u n i -  are l e a d to an  which i s i d e n t i c a l to equation (7«3)»  c l e a r because the second  .  (2.13)  f o r the  equilibrium  the e x i s t e n c e of a c r i t i c a l  point  der Waals theory, i f the volume of  the system i s l a r g e enough so t h a t W  i s independent  F u r t h e r , i n t h i s case, below the c r i t i c a l  p o i n t the  of ? . solution  i s not unique (at l e a s t t h r e e s o l u t i o n s can be f o u n d ) .  The  q u e s t i o n remains, however, as to whether space dependent s o l u t i o n s e x i s t and  to the uniqueness of the s o l u t i o n above  the c r i t i c a l p o i n t as w e l l as what happens when the volume is  finite. One  f i n a l p o i n t i s worth mentioning;  l u t i o n s and  discontinuous  the Maxwell r u l e were obtained when the second  term of equation  (7*1)  that t h i s r e s u l t  i s e q u i v a l e n t , mathematically,  was  the volume to be i n f i n i t e by a d e l t a f u n c t i o n . i n t e g r a l equation solutions. of  now  neglected.  (2.13)  does indeed possess  one uses a r e a l i s t i c  see  to r e q u i r i n g  potential the  discontinuous  assumes a l o n g r a n g e p o t e n t i a l  £(?.>?') = constant,  two  can e a s i l y  With a p o t e n t i a l o f t h i s type  can have o n l y one uniform  between these  One  and r e p l a c i n g the  However, I f one  the form  (2.13)  so-  then  the i n t e g r a l  s o l u t i o n f o r a l l G>  p o t e n t i a l one would expect  extremes to exist."  equation .  If  somethin  32  8.  EXISTENCE AND We  UNIQUENESS (VOLUME FINITE)  b e g i n here to d i s c u s s  the m a t h e m a t i c a l ' p r o p e r t i e s  i n t e g r a l equation ( 2 . 1 3 ) by making as few  assumptions  of  as  possible. U l t i m a t e l y we  N/v  such that the r a t i o handle because now _ i  ^{^j  we  is finite.  but with the  to  ifsSW}J  to d e a l w i t h t h i s problem i s to s o l v e the v a r i a t i o n a l as i n s e c t i o n 2 w i t h f i n i t e volume as a  problem f o r nif^V) parameter. using  is d i f f i c u l t  quantity  V-> co V I P way  This  have to d e a l not w i t h the f r e e energy  ^ One  N-*°o , V-> oo  would l i k e to take the l i m i t  the  One hCF^v)  work; when we things:  then has  to show that the above l i m i t e x i s t s ,  j u s t found.  This w i l l not be done i n t h i s  speak of i n f i n i t e volume we  a) t h a t the above l i m i t  w i l l assume  e x i s t s , and b)  extremum curves f o r the i n f i n i t e volume, l i m i t  that  two the  are g i v e n  by  equation ( 2 . 1 3 ) w i t h the volume I n t e g r a l taken over a l l space. In t h i s s e c t i o n we  t r e a t the case of f i n i t e  V  but  i t i s to be remembered t h a t f o r p h y s i c a l a p p l i c a t i o n s  and  V  are v e r y l a r g e and  s m a l l compared to  V'^  but  the range o f the p o t e n t i a l  our s e l e c t i o n of u n i t s i s taken to be one. t i o n s are not met, physical  b  l a r g e compared to  , N  K(r r' ) J  i  , which by  I f these  the r e s u l t i n g s o l u t i o n s may  N  and  be  of  condino  consequence.  Equation ( 2 . 1 3 )  is a non-linear  the type s t u d i e d by Hammerstein  and  i n t e g r a l equation of i s g e n e r a l l y known as a  Hammerstein e q u a t i o n i n the mathematical l i t e r a t u r e .  The  33 d i s t i n c t i o n between a homogeneous, and equation  i s of l e s s  importance i n n o n - l i n e a r  i n l i n e a r equations. (8.1)  +  can be  T h i s i s because the  j3 I R i f . f O n c ? ' ) AT'=  r e w r i t t e n i n the  (8.2)  a non-homogeneous  <x  equations  than  equation,  }  form,  ( q»(f'))<^' = o >  ift?) +• p v  i f the  equation,  (8.3)  = if  4>*U)  ?  can be s o l v e d f o r the f u n c t i o n ways be  V\ := £ ( *-f)  •  This can a l -  done under the assumptions of s e c t i o n 6,  since  Uy(n)  i s monotonic. Hammerstein i n f a c t s t u d i e d a s l i g h t l y more than ( 8 . 2 ) ;  equation  (8.[|.)  . 4Kr)  +  the  equation he  studied  K^.r'jfLf'', *f(r'/)A?'  \  =  general  was,  O  .  v  The  f o l l o w i n g b a s i c hypotheses are made on the k e r n e l  the f o l l o w i n g three a) K  ( ^  z  , and  K ( V : F ' ) Jtrdf ' 7  2  J  the i t e r a t e d k e r n e l finite  K  z  and  Kj, = ^ K ( f v " ) K ( f " f )dr" continuous) )  b)  the k e r n e l i s symmetric,  c) the k e r n e l i s p o s i t i v e , positive.  for  theorems:  belongs to c l a s s <£  continuous;  K  KCv^r') = K(r',r) ; i . e . a l l i t s eigenvalues  are  is  16  Existence  Theorem I .  I f the k e r n e l c) and  the  continuous f u n c t i o n  i s of c l a s s dC  a n d  2  (8.5)  s a t i s f i e s the b a s i c c o n d i t i o n s p^Cr^a)  satisfies 0 0  z  where C| i s a constant l e s s than the s m a l l e s t K  the k e r n e l  and  i s any  c o n s t a n t , then the  i n t e g r a l equation (8.I4.) has b e l o n g i n g to c l a s s X  existence  at l e a s t one  to equation ( 8 . 1 )  of a s o l u t i o n f o r any  zero.  non-linear  continuous s o l u t i o n  |o  the  I f the assumptions a)  to  This i s e a s i l y seen s i n c e  assumptions of s e c t i o n 6,  •fO-p) i s , under the  f a c t bounded above and be  eigenvalue of  t h i s theorem a s s e r t s  g) of s e c t i o n 6 are s a t i s f i e d . function  j r 6 V j  •  z  Applied  and  i s such t h a t ^"C^jO)  p \ i C r , u ) | < C, I.U.U Q • -t»< u,<  '  a ) , b)  below, and  C, i n (8.5)  Note that t h i s theorem w i l l not  can be  in taken to  apply when V  is  i n f i n i t e s i n c e i n that case, pf(o) ,. a c onstant depending 0< , does not  belong to  ^  the  on  .  Uniqueness Theorem I I . I f the k e r n e l c) and  f o r any  satisfies  fixed  \~ i n V  non-decreasing f u n c t i o n of e q u a t i o n (8.I4J has This the  the b a s i c  UL  at most one  theorem cannot be  assumptions a ) , b)  , the f u n c t i o n -fCv^u.) i s a , then the n o n - l i n e a r  <f  .  applied  However,. i f we  a t t r a c t i v e p o t e n t i a l by  integral  solution. to equation ( 8 . 1 )  assumptions of s e c t i o n 6 because i(--f) i s a  f u n c t i o n of  and  consider  with  decreasing  replacing  the  a r e p u l s i v e p o t e n t i a l , one must r e -  place  K-> - K  and t o get (8.1) back to standard form  With t h i s s u b s t i t u t i o n , - f of  U  so t h a t  becomes a non-decreasing  the theorem a p p l i e s .  function  Hence we can a s s e r t  that  w i t h a r e p u l s i v e p o t e n t i a l , no phase t r a n s i t i o n i s p o s s i b l e i n the model under d i s c u s s i o n ; the s o l u t i o n i s always unique. 16  Uniqueness Theorem I I I . I f the kernel s a t i s f i e s  the b a s i c assumptions a ) , b) and (8.J4.) has at most one s o l u t i o n  c) then the i n t e g r a l equation provided  p^Cr^u)  the f u n c t i o n  s a t i s f i e s uniformly  a Lipshitz  c o n d i t i o n of the type,  p l K v S i U - f(?,Ua)| < C,| UZ-UJ^  (8.6)  where the p o s i t i v e constant value,  C, i s l e s s than t h e f i r s t  eigen-  X, , of the equation  U s i n g the c o n t i n u i t y and d i f f e r e n t i a b i l i t y o f and  t h e mean v a l u e theorem of d i f f e r e n t i a l c a l c u l u s , the  L i p s h i t z condition  (8.6) can be converted i n t o the s t r o n g e r  requirement,  Pi  hence, when  (8.7) the  Si  5~ s o l u t i o n i s unique.  < C,  u  h  P  But, u s i n g  assumptions a) to e) o f  s e c t i o n 6, -I  (8.8)  .r*W  2u.  Hence,  (8.7)  condition  (8.9)  j2>  <  >M  can be w r i t t e n ,  I MA^o) •>  a c o n d i t i o n s t r o n g l y r e m i n i s c e n t o f the uniqueness c o n d i t i o n (7.I4.) o f s e c t i o n 7« point  , d e f i n e d by  (8.10) is  I n f a c t , l a t e r we w i l l show t h a t the  ^  c  -  > WW}\  V  ,  a so-called b i f u r c a t i o n point  (a p o i n t where the number  of s o l u t i o n s changes) f o r equation ( 8 . 1 ) large  volume.  i n the l i m i t o f  37 9  EXISTENCE AND We now  the  UNIQUENESS (VOLUME INFINITE)  p r e s e n t a theorem  a p p l i c a b l e when the volume of  r e g i o n becomes i n f i n i t e .  To do t h i s we must c o n s i d e r a  space o f f u n c t i o n s w i t h norms d i f f e r e n t than the ©7f^ norm used i n the f i r s t  T h i s i s because the oC  three theorems.  z  norms  -Xlf-r'l  of k e r n e l s such as  6  are i n f i n i t e  g r a t i o n Is i n f i n i t e . for  theorems  not,  Furthermore  i f the range o f i n t e -  $(?.6) must b e l o n g to cC^  I, I I , and I I I to h o l d , which i t g e n e r a l l y  i f the range of i n t e g r a t i o n i s i n f i n i t e  t i o n s a) t o e) of s e c t i o n We  under the assump-  6.  c o n s i d e r a space of f u n c t i o n s w i t h norm such t h a t  (9.1)  ± \ \  T h i s space w i l l  ftCnfdr  is finite.  be d e s i g n a t e d  The subspace o f f u n c t i o n s  w i t h the above norm equal to zero w i l l be d e s i g n a t e d *M. 0  is  will  a c o n v e n i e n t space from a p h y s i c a l p o i n t o f view  quantities like  -j- \ v->«> V J  rfcfidr  are expected to be  7h  z  since finite.  v  17  E x i s t e n c e and Uniqueness Let  K be a d i f f e r e n c e k e r n e l ; i . e . K (P,T') = K( r - r ' ) .  Let  K(<^0 ~>  (9.2)  (9.3)  K(?)  be the F o u r i e r t r a n s f o r m of v I Go- f , .  K(Cb) = Let  with  Theorem IV.  £(r,o)  \ e  belong t o  C,(u -lA ')< 1  l  IA >U_, , C > o V  i s an c£  2  z  M O ,  KO)o!r . and  satisfy  $(r,^-fCf^ )^ -  ; C, and C  2  C ^ U , - ^  are c o n s t a n t s .  f u n c t i o n such t h a t  Also  suppose  38  (9-14-)  ^r K (F)oVr Z  is finite.  a  e x i s t s a unique s o l u t i o n to equation (8.I4.)  Then there  >  (9.50  s o l u t i o n w i l l belong to h\  The  r  tion in ^ i .  z  (c -c.) 2  and be unique up  l  to any  (This theorem appears i n r e f e r e n c e  0  it  i  i s proved f o r one  dimension.  provided  We  func-  and  there  are u s i n g the three d i -  mensional g e n e r a l i z a t i o n , which appears to be v a l i d , of theorem of reference.'"^ although we have not been able  the  to  prove i t r i g o r o u s l y . ) The  theorem can r e a d i l y be a p p l i e d  w i t h the assumptions a) to g) of s e c t i o n 6 . suppose t h a t the maximum of  (9.6)  0 ^  K(GJ; ^  K(^)  (8.1)  to equation Further,  we  shall  i s attained forcD=o, i . e . ,  K t o ) = ^KCr)dv =  2'W. ~\  From the graph o f ty\'n) we that C = 0 . 2  f i n d that  Hence c o n d i t i o n  (9.7)  fi<  - y "  (9.5) C  n  ^  C, - [  (rV)] < O  and  becomes  ,  o r p r e c i s e l y the c o n d i t i o n found i n the example of s e c t i o n 7 (see  C T-i+)) •  Thus we  see  t h a t above the c r i t i c a l p o i n t  s o l u t i o n to the e q u i l i b r i u m equation i n the l i m i t of  the  infinite  volume i s unique. I t i s a l s o r e a d i l y seen that f o r a r e p u l s i v e p o t e n t i a l and  assuming  K(ui)< O  condition.(9.5)  asserts  ( i . e . a negative d e f i n i t e that a unique s o l u t i o n may  potential) be  found  39  f o r a l l p o s i t i v e values of I I of s e c t i o n  8.  ^  .  -This i s the analog of Theorem  10.  BIFURCATION AND 'The  previous  MULTIPLE SOLUTIONS  theorems assure  the  e x i s t e n c e of a unique  s o l u t i o n when a c e r t a i n c o n d i t i o n on  |3  is satisfied.  a l r e a d y know that when the volume i s I n f i n i t e and dition  i s not  satisfied,  e q u i l i b r i u m equation  at l e a s t three s o l u t i o n s of  (2.13)  exist.  a s i m i l a r s i t u a t i o n probably  z  values  p o i n t s , i . e . , p o i n t s of the ^  where two  want to show t h a t  kernels  to r i g h t ,  g e n e r a l l y do  eigen-  axis at each of which, as  the number of s o l u t i o n s of the  or more s o l u t i o n s meet and  equa-  again.  ( F i n i t e a n d - I n f i n i t e Volume) 0  to be a b i f u r -  c a t i o n p o i n t f o r a g i v e n s o l u t i o n S£(r) of a n o n - l i n e a r  equa-  type ^ (f) +  (10.1) with  we  be a p o i n t  then separate  A n e c e s s a r y c o n d i t i o n f o r a p o i n t p= ^>  t i o n o f the  not  o f having b i f u r c a t i o n  Note that a b i f u r c a t i o n p o i n t may  B i f u r c a t i o n Theorem V.  finite.  However, Hammerstein equa-  t i o n s u s u a l l y possess the p r o p e r t y  t i o n s changes.  now  of having a denumerable s e t of  as do l i n e a r equations.  pass from l e f t  We  con-  the  e x i s t s when the volume i s  Hammerstein equations with oC possess the p r o p e r t y  this  We  kernel  p JKC^r^Cr'^iT'-^  KCr.r'J ..'Is that  j3  O  b e an eigenvalue  0  of  the  kernel (10.2) Krasnoselskii~has  -  ^  )  ^  g i v e n c o n d i t i o n s under which  theorem i s a l s o s u f f i c i e n t f o r the e x i s t e n c e  the  of a b i f u r c a t i o n  point. and  He  showed under r a t h e r broad assumptions on the  the f u n c t i o n  of the k e r n e l We  ^(rju)  kernel  t h a t every non-degenerate eigenvalue  (10.2) i s , i n f a c t , a b i f u r c a t i o n p o i n t .  cannot apply" the theorem r e a d i l y to equation ( 8 . 1 )  the assumptions of s e c t i o n 6, e r a l the s o l u t i o n ^  0  .  because we  However, i f we  do not know i n gen-  consider  which the volume of the system i s l a r g e and  the case i n  take f o r %  the  unique s o l u t i o n c l o s e to the c r i t i c a l p o i n t , then because system i s l a r g e <f w i l l be p r a c t i c a l l y independent of r 0  S't^o]  the f u n c t i o n The  can be  and  taken out  the  and  of the i n t e g r a l s i g n ,  theorem then s t a t e s t h a t every non-degenerate eigenvalue  of the  equation  (10.3)  4- j ^ f ' [ 4 ' ] ^ ^ > v ' ) X ( f ' ) q v ^  O  ,  0  Hence every value of p  is a b i f u r c a t i o n point.  which s a t -  o  isfies  (10.1,) where  A,  i s the lowest eigenvalue of the k e r n e l  Theorem I I I ) i s a b i f u r c a t i o n p o i n t .  K(v,,f') (see  E q u a t i o n (10.1}.) can  be  rewritten  (io.5) and  - -  fo  \  [  the s m a l l e s t v a l u e of  ft"-)] J3  Q  occurs f o r  s m a l l e s t b i f u r c a t i o n p o i n t occurs at (10.6)  ^  =  X, I y \ r o | .  Yl - Y\ 0  0  , hence the  k2  This  i s j u s t the upper l i m i t  (8.9)  and  of the uniqueness c o n d i t i o n  (8.10)).  From the p r e v i o u s theorem we w i t h the  (see  thus see  assumptions of s e c t i o n 6,  lj|*> . at which the  solutions  c  that  equation  (10.1),  admits a b i f u r c a t i o n p o i n t  of the  equation "meet" or change  when the volume i s l a r g e . E x a c t l y how of the  many s o l u t i o n s  c r i t i c a l point  equations have only  appear i n the  i s not known.  neighborhood  Generally,  Hammerstein  a f i n i t e number of s o l u t i o n s which change  by a f i n i t e number on p a s s i n g through the b i f u r c a t i o n p o i n t , but  t h i s i s not  lutions  is infinite.  i n the l i m i t is  always the  we  will  s o l u t i o n s may  give  an  volume where the number o f  I f the number of s o l u t i o n s  r i g h t of the f i r s t  of e q u a t i o n  Sometimes the number of  In s e c t i o n 12  of i n f i n i t e  infinite.  case.  b i f u r c a t i o n point,  one  so-  example solutions  i s f i n i t e near  the  or more of these  b i f u r c a t e again at the next e i g e n v a l u e ,  A  2  ,  (10.3).  I t Is i n t e r e s t i n g to s p e c u l a t e what t h i s b i f u r c a t i o n at finite  i n t e r v a l s means p h y s i c a l l y .  r e l a t e d to the  temperature we may  the p o s s i b l e number of s o l u t i o n s i n t e r v a l s of the  temperature.  dimensional kernel  of the  eigenvalues  change  only  at w e l l  found to be,  spaced  d i f f e r e n c e between the A  2  , has  been c a l c u l a t e d  form _\\*-y\  and  are  have the s i t u a t i o n i n which  The  lowest b i f u r c a t i o n p o i n t s , A, and a one  S i n c e the  f o r a system of l i n e a r dimension Jl ,  two for  where i t has been assumed t h a t cal  \i  >> 1  .  p  c  i s the  criti  temperature,  Even i f the l o n g range p o t e n t i a l had a range of the. order lOOCT that  , f o r a system o f 1 o r i , the above r e l a t i o n  -^ /T T  c  indicates  would have to be determined to b e t t e r than  10  to d e t e c t the above bifurcation^phenomena i f they e x i s t e d .  11.  STABILITY AND For  INSTABILITY  thermodynamic s t a b i l i t y  the s o l u t i o n s  of equation  ( 2 . 1 3 ) must correspond to a minimum of the f r e e energy of the system. negative of following  S i n c e the f r e e energy ^?{" ^} n  o  equation (2.5)  stable solutions  ^{ ^  r  K  F J  i s proportional }  (see the  and equations ( 2 . 7 )  discussion  f o r a maximium of  (2.9))  and  w i l l correspond to maxima of  A necessary c o n d i t i o n  t o the  §{ '( '} n  ${h(f)}  .  pi  i s that  the second v a r i a t i o n , equation ( 2 . 1 2 ) , be n e g a t i v e f o r a l l admissable v a r i a t i o n s In s e c t i o n 7 we density  "X(r) such t h a t  investigated  the s t a b i l i t y  o f the constant  s o l u t i o n s , i . e . , some of the s o l u t i o n s  i n the l i m i t  of i n f i n i t e volume.  s e c t i o n to i n v e s t i g a t e infinite  equation ( 2 . 1 0 ) h o l d s .  one o b t a i n s  I t i s the purpose  of t h i s  the s i t u a t i o n when the volume i s not  and to g i v e some r e s u l t s f o r s o l u t i o n s which are  space dependent. The second v a r i a t i o n , equation ( 2 . 1 2 ) ,  i s rewritten  here f o r convenience,  where the admissable v a r i a t i o n s must s a t i s f y  (11.2)  W(-f)dr = J  y  From the assumptions first  O.  a) t o g) i n s e c t i o n 6 we  term i s n e g a t i v e w h i l e the second  see that  Is p o s i t i v e .  With no l o s s of g e n e r a l i t y f o r the moment we pose the volume to be i n f i n i t e ;  the  shall  sup-  the c o r r e s p o n d i n g formulas f o r  f i n i t e volume are e a s i l y obtained.  Let 7-(">) and k(cs) be  the Fourier i n t e g r a l transforms of ^ ( v ) and KCv^p') = KClF-r'l) We have then f o r the second term of ( 1 1 . 1 ) ,  respectively. (11.3)  ^  ^  U*-r)%{.Tn{r<)^c\r>  ^KC<S^(uj)yC-w)i^.  'X.(f) i s r e a l , hence ^(i^O = % C-<-°; . mum of  k(£>) occurs f o r Co = O .  Suppose that the maxi-  This i s not an unreasonable  assumption f o r most p h y s i c a l l y acceptable kernels. (11.k)  ^ K C C b ) ! ^ ^ / ! ^  <  Hence,  p K ( ^ | 7 l ( C b ) f ^ ^ j3 K ( o ^ X ( r ) d r . Z  Furthermore, from the assumptions a) to e) on lf(*0 of section 6 , we have, (11.5)  ^V|/'COTCC^)t\r « l  ^"(n ^% Cv)A? .  Putting these two i n e q u a l i t i e s into  (11.6)  .  $  z  2  t  ^) we get,  < ^ " ( M + pKCo)^7.\ )A> , ?  hence i f pK(o)<  (£  z  \+"(>c)l  w i l l be negative and hence a stable physical s o l u t i o n .  This condition i s just the uniqueness condition ( 8 . 9 or 9.7). The preceding results are summarized i n : S t a b i l i t y Property I . I f max K(Cb) occurs at 63 - o and (11.7) then  |2> K ( o ) <  I q/'c^|  i s negative f o r any yiCv) which makes ^)  stationary.  Below the c r i t i c a l p o i n t the l i n e the "graph o f  ^KCo) moves up on  ip"c*) so that i t i n t e r s e c t s the f u n c t i o n  i n two p o i n t s ,  MA*)  and P i ^ , d e f i n e d by the r o o t s of the equa-  tion,  I t i s c l e a r that i n e q u a l i t y (11.Ii) i s s t i l l w i l l be v a l i d  i f h(*0< ^, or n(?) > Y\  p l a c e d by ft, .  Thus P r o p e r t y  a  n  valid  and (11.5)  ^e. i n y"(n ) i s r e -  d  z  c  I can be m o d i f i e d  to read:  S t a b i l i t y Property I I . If  |J&(o) > |  as the d e n s i t y  » then  as long  2  satisfies,  (11.8)  rUr)<r>, or  where 10, and n  (| i s c e r t a i n l y n e g a t i v e  r  mCr)>m . 2  J  are the two p o s i t i v e r o o t s o f the equation,  (11.9)  fo K ( o ) + 1 ^ % ) = o .  The  above r e s u l t s concern s t a b i l i t y .  itrary variations i t y properties.  .  They h o l d f o r arb-  We s h a l l now d i s c u s s  some i n s t a b i l -  By i n s t a b i l i t y here, we mean a s o l u t i o n f o r  which the f r e e energy i s a maximum, o r not a minimum. metastable states discussed according  i n s e c t i o n 7 are s t a b l e  The  states  t o the d e f i n i t i o n o f t h i s s e c t i o n .  For i n s t a b i l i t y  i t i s n o t necessary f o r the f r e e energy  to be a maximum f o r a r b i t r a r y v a r i a t i o n s ; i t i s only n e c e s s a r y t h a t i t be a maximum f o r some p a r t i c u l a r c h o i c e X (r) .  of v a r i a t i o n  However, i t i s necessary f o r 'X(r) to s a t i s f y  Hence i f we can f i n d  some v a r i a t i o n s a t i s f y i n g  (11.2).  (11.2) which  makes  §  p o s i t i v e we have an i n s t a b i l i t y  2  Using  (11.3)  $  can be w r i t t e n ,  z  = j-O'W  (11.10)  \  Consider  the case when  ft(r) i s h  2  property.  P*W}^WA?  t  ^{RW)-?M] tt3>*|y •  p 1< (<0 > Ity\"O |  e i t h e r a constant between  .  Yl, and H  2  Suppose t h a t » where  H| and  are the roots o f (11.9), o r v a r y i n g between these  limits.  To be s p e c i f i c l e t h(r) be bounded by in' and n" w i t h n,< n'^  (11.11)  n(.r)$ n"< n  For sake o f argument suppose  Hence  2  .  ^ (W) <! ^(r>").  Then we have,  (11.10) becomes,  p k(.°)>0the f i r s t  Since I f  term i s p o s i t i v e .  The second z  i n t e g r a l can be made as s m a l l as one chooses by t a k i n g to be a s h a r p l y peaked f u n c t i o n near the o r i g i n Hence  O  % (£>J  co =• o .  f o r t h i s p a r t i c u l a r v a r i a t i o n and we have,  I n s t a b i l i t y Property I I I * If  ^K(o) > |vf"(n )|  bounded by H, and h $  2  c  z  » t h e n ' f o r any s o l u t i o n which i s  , the two p o s i t i v e r o o t s o f equation  can be made p o s i t i v e o r zero f o r some v a r i a t i o n  (11.9),  'XCO .  We now p r e s e n t a p r o p e r t y which i s u s e f u l i n a p p l i c a t i o n s . The second v a r i a t i o n ,  (11.1), can be w r i t t e n ,  (11.12)  $  2  =  ^W)£%t*)J.r  ;  where c^C i s the l i n e a r o p e r a t o r ,  (11.13) It  • r:  q/"(n)-x-(f) + p ^ K C r ^ ^ - y C f O o l r / ,  cT-x=  i s c l e a r t h a t i f a "X(*0 e x i s t s such t h a t ^7.(J)^r-0  dCy. ~ O  then ^ = O  and the s i g n o f the v a r i a t i o n i s d e t e r -  mined by the t h i r d v a r i a t i o n , vanish.  Since  $  and  contains  $  3  , which i n g e n e r a l w i l l not  "Ji i t can be made e i t h e r p o s i 5  t i v e o r n e g a t i v e and hence the t o t a l v a r i a t i o n cannot be positive  o r n e g a t i v e d e f i n i t e , which i s a necessary c o n d i t i o n  for 1  t o be maximized o r minimized.  Thus we have,  I n s t a b i l i t y P r o p e r t y IV. If a  'X(r) e x i s t s such t h a t a)  XH= O  ,  b) ^ % t ? ) J ? - o ,  c)  3?  3  4 0 ,  then the extremum tor  . Yv(f) (which appears i n the e x p r e s s i o n  X ) cannot maximize o r minimize  the f r e e energy.  F i n a l l y we g i v e a method f o r determining of  X X - o  (ii.lU)  .  Suppose the s o l u t i o n  y'U)+  ia(fj which s a t i s f i e s  f> ^ ( f ^ ' j n C f ' ) ^ ' -  depends on a parameter A ,  = n ( f ) V)  (ll.lLj.) with r e s p e c t t o A we have,  the s o l u t i o n s  oi  .  Differentiating  4~  ctF'=  O  hence,  satisfies a t o r the  the  equation  X. %. ~ O  .  S i n c e oC  i s a l i n e a r oper  s o l u t i o n so obtained i s the unique (up  to an unimpor-  t a n t m u l t i p l i c a t e constant) s o l u t i o n . We  may  give  the  above p r o p e r t i e s : of s t a b i l i t y  and  the uniqueness c o n d i t i o n The  the s o l u t i o n  g r a l equation  property asserts  is stable.  t h a t s o l u t i o n s below the the p o s i t i v e slope  solutions tive. lying  f o r the i n t e g r a l  that  above the  S t a b i l i t y Property I I  critical  region  on  diagram are  I n s t a b i l i t y Property I I I asserts i n s i d e the posi-tive s l o p e As  inteimplies  stable.  of The  d i s t i n c t i o n between m e t a s t a b l e and—stable..—  because f o r both these s o l u t i o n s  be u n s t a b l e .  critical  p o i n t which l i e o u t s i d e  the P~V  the  notion  (which i s known to be unique) of the  (2.13)  p r o p e r t y makes no  i n t e r p r e t a t i o n to  S t a b i l i t y P r o p e r t y I connects the  equation ( 2 . 1 3 ) . point  following physical  w i l l be  P r o p e r t y IV r u l e s out  region  $  2  w i l l be  that any o f the P-V  seen i n s e c t i o n 12,  solution diagram w i l l  Instability  certain periodic solutions  ( 2 . 1 3 ) from being s t a b l e p h y s i c a l  nega-  of  equation  ones.  5  Van Kampen and  I I and  new  result.  proved by  a d i f f e r e n t method, P r o p e r t i e s  a less general version  of I I I .  I  P r o p e r t y IV i s a  50 12.  AN EXACT SOLUTION Even though we have been a b l e t o show t h a t as one passes  through the c r i t i c a l p o i n t the number of s o l u t i o n s o f the integral  equation  (2.13)  changes, we have not been able to  prove the e x i s t e n c e of space, dependent s o l u t i o n s which may correspond  t o the presence of two phases.  In the p r e s e n t  s e c t i o n we prove -the e x i s t e n c e of such s o l u t i o n s by s o l v i n g .exactly the i n t e g r a l and  with  kernel.  equation  i n the l i m i t  the c h o i c e o f a p a r t i c u l a r We w i l l a l s o study  o f i n f i n i t e volume  (and p h y s i c a l l y  in detail  reasonable)  some of the p r o p e r t i e s  of the s o l u t i o n s , such as the n e c e s s i t y o f the Maxwell r u l e and  stability  questions.  s a r y to choose a s p e c i f i c potential  so the r e s u l t s  I t turns out t h a t i t i s not necesform f o r the hard are s t i l l  L e t us c o n s i d e r equation (12.1) i n the l i m i t  somewhat  core p a r t o f the general.  (2.13),  <f'U) + |3 jf< (?,?') V0(r';c/?'=-  ;  o f i n f i n i t e volume and choose a long  range  potential (12.2)  =  ^ 2-TT  7~~"r7T~ |r - r'|  '  A l s o l e t us l o o k f o r s o l u t i o n s which depend on only one c a r t e s i a n c o o r d i n a t e , say X , thus,  (12.3)  K(r)= H(x).  T h e - i n t e g r a t i o n over be performed w i t h  the ^ and  the r e s u l t  coordinates  i n ( 1 2 . 1 ) can  ^'U^'Kc^r')  (-i2.il.)'  wxe  -XU-x'l  =  With these subs-ti t u t ions, e q u a t i o n ( 1 2 . 1 ) becomes + 00  r  (12.6)  lf/Cn)-*+  -Mx-<'i  n ^ K '  2(&WXlj^_  Now i t i s e a s i l y shown t h a t ^\ e  O.  =  i s the Green's f u n c t i o n  w i t h the boundary c o n d i t i o n the s o l u t i o n be f i n i t e at X ^ -  0 0  .  that  Hence the s o l u t i o n of the  i n t e g r a l equation ( 1 2 . 6 )  i s equivalent  the f o l l o w i n g n o n - l i n e a r  differential  to the s o l u t i o n o f equation plus boundary  conditions,  J^' ^  (12.7)  C  ^"\V<-^ + 2 v J ( 3 n - < * _  =  boundary c o n d i t i o n ; Letting  (12.8)  s  ^  b  !  >  t  n  e  This d i f f e r e n t i a l (12.9)  If (n(>o) f i n i t e  differential  " ^ ^ + iv"'(^^  equation can be w r i t t e n  X" [v|/C^ v 2 v o p 2  - wTj".  -  aWpyi-o(=0.  Hence we see t h a t the f u n c t i o n r o l e i n the d i s c u s s i o n .  ^ + 2Y/^y\  the p r o p e r t i e s o f 3 and l ; .  plays  an important  We s h a l l suppose that  assumptions a) t o e) o f s e c t i o n 6 . + 2Wpn.  satisfies  Under these assumptions  were i n v e s t i g a t e d i n s e c t i o n 7«  From t h a t d i s c u s s i o n we see that when  2.W|3 < j ^"(n )| , the equation t  a t . X = * c o .  equation has s i n g u l a r p o i n t s when,  sjazOj  See F i g u r e s  5  if'+2^Wh — oL  w i l l have one  r o o t , hc< , and hence one s i n g u l a r p o i n t , while when  2W|*> > I ^"Cvot')] t h e r e i s the p o s s i b i l i t y V \ , Y\  o f three r o o t s  L e t us now expand the d i f f e r e n t i a l  3  < n < r\  and t h r e e s i n g u l a r p o i n t s , where  5  h ,  H  equation  s i n g u l a r s o l u t i o n s t o determine the c h a r a c t e r  5  near the  o f the s o l u t i o n  near these p o i n t s . L e t  §60  with  small.  To f i r s t  order,  equation  6,  1  Butty"tr\-L)i s always  s i n c etyCh;)t 2 p W n , ; - i s zero. so t h i s  (^V,)|<Q^+-  ^"(niWajiwjS  Hence when u/"(n,)-t2^w'>o  ttX  6  negative  can be w r i t t e n  (12.10)  e  (12.8) becomes,  we  = O.  have s o l u t i o n s o f the form  and when *)>%;)+2pvV < O s o l u t i o n s o f the form .  C-"W,3jS'  From F i g u r e s 3 and a  the p o i n t s  n  d  we see that  that y (n;)-v2wp>o for \ = *f .  0(^3^5  are saddle  p o i n t s while  e , e  tx  iv"(n;)+-2p>W < O  ,  for  T h i s shows-, t h a t . singular point  ^  corresponds to a c e n t r e . With t h i s  information  i t i s now p o s s i b l e t o sketch the  s o l u t i o n curves i n the phase p l a n e . on the next page.  These curves a r e shown  The s i n g u l a r p o i n t s  , H  3  ,  , n w h i c h are roots s  o f equation  (12.9) are seen to be s o l u t i o n s of the d i f f e r e n t i a l  equation  (12.7) and hence are s o l u t i o n s of the i n t e g r a l equation as can be seen by d i r e c t s u b s t i t u t i o n . differential phase p l a n e . ing  a first  The o t h e r s o l u t i o n s  to the  equation (12.7) f o l l o w the curves drawn i n the These curves can be obtained e x a c t l y by i n t e g r a l o f (12.7).  ed by m u l t i p l y i n g  The f i r s t  (12.7) by iy"<\n  obtain-  i n t e g r a l i s obtain-  and i n t e g r a t i n g .  This  yields, (12.11)  J  To o b t a i n the s o l u t i o n s of the i n t e g r a l e q u a t i o n i t i s necessary to impose the boundary c o n d i t i o n s have up to now been n e g l e c t e d . diagram that when  i n (12.7) which  I t i s seen from the phase  the  W C*)  only s o l u t i o n with f i n i t e  i s the  singular  , c o r r e s p o n d i n g to a c o n s t a n t d e n s i t y . c l u s i o n has  been reached e a r l i e r on  the b a s i s  Nowwhen 2j3W > 1 ^"(.roj t h e r e w i l l be of the  3  solutions  equation.  w i l l be  5  i n s i d e the  bounded, and  general,  , H^. , n  hence are  solutions  These s o l u t i o n s will  I t can be  Not  as one  moves out  these p e r i o d i c  solutions  stant  a constant f o r p a r t  In the  of the  remaining p a r t .  H^.  We  range and  two  X-^-o©  H5  i n g such a s o l u t i o n can  the  and  E  2  f o r X-»+oo  be  density.  return  of  to  through  a d i f f e r e n t con-  .  The  H  s  .  If  E,  obtained  and  condition for  that  E,  i s to c o i n c i d e  the same f o r the  E  of  2  i n t h i s case  approaches  obtained by n o t i n g  constant i n (12.11) must be  •Hence,  will  I t i s seen t h a t only  for  3  hence, i n  jjieparatrixs E,  have a s o l u t i o n which asymptotically  through n  and  i n a s o l u t i o n which i s  w i l l we  and  the  are  Such a s o l u t i o n can be  "Maxwell case" when the  the phase plane c o i n c i d e .  say  of  later.  are p a r t i c u l a r l y i n t e r e s t e d  f o r the  a l s o any  from fi^. the p e r i o d  s o l u t i o n s becomes l o n g e r .  nearly  three  original integral  about  these p e r i o d i c  We  the  r e p r e s e n t p e r i o d i c v a r i a t i o n s i n the  shown t h a t  IV.  solutions  of the phase plane  of the  circle  of Theorem  only  s o l u t i o n s , but  shaded r e g i o n  same con-  other f i n i t e  d i f f e r e n t i a l equation (12.7).  constants Y\  The  solution  two  H  3  obtain-  posses with curves.  E  2  55  n  This  5  I -[oc -  (12.13)  lf'(h)  -  IWpn]  c o n d i t i o n i s an i m p l i c i t  lf"(n)cln  =r  o .  c o n d i t i o n f o r C<  and i t c o r r e s -  ponds t o the Maxwell c o n s t r u c t i o n .  To show t h i s l e t  /'=. ^/'(n) 2|3Wn. and s u b s t i t u t e i n t o  (12.13)..  T  Integrating,  we  obtain  1«p  oi.  But  f  the two numbers H. and n 3  satisfy  5  c/n =  O  .  (12.9) or f ( n ) = f Cl j= o( 3  s  so t h a t the equation reduces t o  or u s i n g  the d e f i n i t i o n of  $  j ,  (12.lb,) *3  T h i s shows that the two phase s o l u t i o n i s obtained ing- the Maxwell-cons truc-tion- to the --^'(«•)••+•• 2-|3W n.  by a p p l y curve.  The c o n d i t i o n f u r t h e r s t a t e s that the p r e s s u r e of the two phases i s e q u a l .  To show t h i s n,  4>'(*)•+ p w n J  Integrate -  z  (12.II4.)  to o b t a i n  c< h  <3  »3  M u l t i p l y i n g equation (12.9) by Kl and u s i n g the values o f the two r o o t s  Subtracting  Y\-> , H  5  one  obtains  the l a s t two equations  gives  which j u s t s t a t e s that the p r e s s u r e o f the two phases Is  -  5 6  equal,  see  e q u a t i o n (1|.£).  p o t e n t i a l , the  where yU(«) These are  two  conditions  just usual  (12.9) and  plane .  We  discuss  now  conditions  are e q u i v a l e n t  the ?-v  the v a r i o u s  i s the  (12.11;) can  T(n)  to the Maxwell  We  use  construction  the p r o p e r t i e s  For,  from e i t h e r the  see  that i f h(x)  to the same equations.  of the  Hence on the b a s i s  following I n s t a b i l i t y Property  is a  a solution discussion  IV,  ax  cfT~  e q u a t i o n dCX  only need to check to see  =  0 with % -  i f condition 5 )  P r o p e r t y IV h o l d s f o r the p r o p e r t y focus our  can  i n t e g r a l equation. (12.6) o r  a l s o be  first  1^3  to equation  s o l u t i o n of the equations then h(x-rX) w i l l  a s o l u t i o n of the  be  i s an u n s t a b l e s o l u t i o n  I n s t a b i l i t y P r o p e r t y IV  the d i f f e r e n t i a l e q u a t i o n (12.7) we  We  of  i s seen to  be used to i n v e s t i g a t e the p e r i o d i c s o l u t i o n s  w i l l be  two  By S t a b i l i t y P r o p e r t y I I ,  by I n s t a b i l i t y P r o p e r t y I I I .  (12.7-).  pressure.  s o l u t i o n s obtained from a  are s t a b l e s o l u t i o n s w h i l e  5  stated  11.  a stable physical solution. H  be  i s the  By S t a b i l i t y P r o p e r t y I, the s o l u t i o n  and  chemical  to be s a t i s f i e d f o r a  p o i n t of view of t h e i r s t a b i l i t y . section  -o<  i s the chemical p o t e n t i a l and  phase system and on  Thus,, s i n c e  to be  of  Instability  applicable.  a t t e n t i o n on the p e r i o d i c s o l u t i o n s  the shaded r e g i o n of the phase plane (not  Let  us  inside  i n c l u d i n g the  lines  E, -and E-3  ).  Because the phase plane i s symmetric about the  ft. a x i s and because we can s h i f t . t h e o r i g i n anywherej  h(x) =  we can make  ft(x±\)y  dn clx  periodic solutions.  Hence  P r o p e r t y IV are s a t i s f i e d will $  ,  and the c o n d i t i o n s o f  \%(_K)dx-o  (only under e x c e p t i o n a l  , e.g., V " ( r O  = O  3  an odd f u n c t i o n f o r the + t >  linear in n  circumstances  and ft =  ) w i t h the  c  r e s u l t t h a t the p e r i o d i c s o l u t i o n s a r e u n s t a b l e s o l u t i o n s . E-i  Since applies  lies  above and below the f\ axis the same argument  to t h i s s o l u t i o n and hence i t i s an u n s t a b l e s o l u t i o n .  However, f o r E Is not va l i d  which remains above the h  3  and  U«dx £ o  a x i s , the argument  and the c o n d i t i o n s  of P r o p e r t y IV  are not met.  Thus we see t h a t of a l l the s o l u t i o n s i n the  plane only  , ft , h  E3  3  5  are s t a b l e p h y s i c a l s o l u t i o n s  may be a s t a b l e p h y s i c a l s o l u t i o n .  while  A l l the o t h e r s o l u t i o n s  are d e f i n i t e l y u n s t a b l e s o l u t i o n s .  Note that  E 3 corresponds  to the two phase s o l u t i o n d i s c u s s e d  earlier.  We have not been  able t o show t h a t one  apparently  E 3 i s d e f i n i t e l y s t a b l e because to do so,  needs an e x p l i c i t  a n a l y t i c form f o r the s o l u -  tion. Van the  Kampen a l s o o b t a i n e d space dependent s o l u t i o n s o f  i n t e g r a l e q u a t i o n ( 2 . 1 3 ) , but he made the assumption  £h(rj-h(f')^  that  c o u l d be expanded i n a T a y l o r s e r i e s to the t h i r d  term.  The expansion w i l l  slowly  varying  only be v a l i d i f the d e n s i t y  function of distance,  an assumption which i s  not  always t r u e .  E.3  and argued t h a t i t would minimize the f r e e energy.  d i d t h i s by f i r s t  is a  Van Kampen a l s o obtained a s o l u t i o n l i k e  omitting  i n g the d i s c o n t i n u o u s  the second term of ( 7 . 1 )  s o l u t i o n s as we have done.  He  and f i n d -  He then  argued that when the second term was t h a t maximized 3? interface, since bution  and  variations  would be  i > { M v ) ^  however the  w i t h the  the s o l u t i o n  smallest  possible  the second term i s n e g a t i v e f o r any  distri-  c l e a r l y would tend to make (f) s m a l l e r when the i n the d e n s i t y were more numerous.  i s very a p p e a l i n g that  the one  included,  This  from a p h y s i c a l p o i n t of view.  for various  h(r)can  be  argument  We  compared i n t h i s  agree way,  argument i s not very r i g o r o u s .  • -  S i n c e the p o t e n t i a l (12.2) i s the Green's f u n c t i o n f o r the o p e r a t o r ( V *— X 7  ) one  need hot make the assumption  t h a t the d e n s i t y depends on one then o b t a i n  a non-linear  space c o o r d i n a t e .  differential  (12.7) except that  replaced  s i n c e the new  one  c a r r i e d out  non-linear  by  V  .  point  an  can  now  analysis  i n t h i s s e c t i o n cannot be" made"  (The  coordinate  i n the d i f f e r e n t i a l equation.)  of view one  One  d i f f e r e n t i a l equation f o r s p h e r i c a l  symmetry i s non-autonomous. plicitly  would  equation s i m i l a r to  l o o k f o r s o l u t i o n s with s p h e r i c a l symmetry but s i m i l a r to the  One  (12.3)  would expect the two  the lowest f r e e energy to be  \~ appears ex- -  Prom a p h y s i c a l phase s o l u t i o n w i t h  a s p h e r i c a l drop of  surrounded by vapor (or v i c e - v e r s a )  but  t h i s has  liquid not  been  proven o r shown i n an example although the argument by  Van  Kampen does support i t . I t i s not known what w i l l happen i n the case of volume, but out  an a n a l y s i s s i m i l a r to the  i f a Green's f u n c t i o n c o u l d  finite  above could be c a r r i e d  be found t h a t was,  same time, a r-easonable p h y s i c a l p o t e n t i a l .  at  the  In t h i s work, i n d i f f e r e n c e  to Van Kampen, we have  tried  to study h i s i n t e g r a l equation w i t h a view to d i s c l o s i n g how i t s mathematical s t r u c t u r e c r i t i c a l point  i s r e l a t e d to the e x i s t e n c e  and the f i r s t order phase t r a n s i t i o n .  of the In  p a r t i c u l a r we have shown that the s o l u t i o n to h i s equation i s unique above the c r i t i c a l manifests i t s e l f  and  shovm how  obtaining  and t h a t  as a b i f u r c a t i o n p o i n t  have f u r t h e r s o l v e d symmetry u s i n g  point  the I n t e g r a l  the c r i t i c a l of t h a t  point  equation.  equation e x a c t l y  i n plane  a p h y s i c a l l y reasonable a t t r a c t i v e p o t e n t i a l the Maxwell r u l e i s a necessary c o n d i t i o n f o r  two phase s o l u t i o n s .  Periodic solutions  of the  i n t e g r a l equation were a l s o found and were shown to be unstable physical  solutions.  We  -1-3-.—NUMERICAL SOLUTION AND  CALCULATION OF SURFACE TENSION  In order to t e s t the h y p o t h e s i s s h o r t and  long range f o r c e s we  of the s e p a r a t i o n Into  chose to c a l c u l a t e i n t h i s  s e c t i o n the s u r f a c e t e n s i o n , a q u a n t i t y which should depend s t r o n g l y on the nature  o f the i n t e r f a c e , and,  the a t t r a c t i v e p a r t of the  t h e r e f o r e , on  interaction.  Even though a s y m t o t i c a l l y the pressure  o f the l i q u i d  vapor phases are the same, the p r e s s u r e w i l l vary as one verses  the i n t e r f a c e between l i q u i d and vapor.  (1|.5>)«  who  T h i s was  pressure  I t remains to o b t a i n an  e x p r e s s i o n f o r the s u r f a c e t e n s i o n i n terms of the pressure.  trans-  I f the d e n s i t y  v a r i a t i o n i s known through the i n t e r f a c e , the l o c a l can be c a l c u l a t e d by equation  and  local  done i n H i r s c h f e l d e r , C u r t i s and  found f o r the s u r f a c e t e n s i o n , Y - »  f o r a plane  where the l o c a l p r e s s u r e depends o n l y on one  Bird  surface  coordinate,  (.13.1) where P - P(<*>) = Q  P(-°0  .  To o b t a i n t h i s formula  one uses  the  p r i n c i p l e of v i r t u a l work to f i n d the work done per u n i t i n crease i n area o f the cross s e c t i o n of a t a l l r e c t a n g u l a r w i t h ends f a r i n the l i q u i d and gas r e g i o n s .  The  box  surface  t e n s i o n i s j u s t the net work done per u n i t area when the v o l ume  o f the system i s r e t u r n e d to i t s o r i g i n a l volume. We  now  put the previous  calculations. experiment we equations  equations  i n a form s u i t a b l e f o r  Because we want to compare our r e s u l t s are no l o n g e r t a k i n g b ^ l  (I4..5}, (I4..6),  . (Ij..2) • and  with  in this section.  (I4-.8)  the p r e s s u r e  From  and  chemical p o t e n t i a l are:  (13.2)  pP(x^= h(l + U ^ )  (13.3)  o< -  V C«) +  (13.If-)  (13.5)  |TTCT  ip'( ) h  R\KCr f'J^(F']Ar' J  )  ;  3 ;  - j^h  =  ^Y(.T^<)^i7')Ar'  -U Y |  - lU(^)^Jo .  The e q u i l i b r i u m s i n g l e phase d e n s i t i e s  (13.6)  ^ p -  (13.7)  .<*'•= V o ^ +  (13.8) Equation  nC »+DK>|)-  a b W o  (13.6)  satisfy:  Yj \>pv\\  ipW.bh  e  j  jkCv^r')^'.  can be w r i t t e n i n the form,  X  P =  U ( l f b ^ ) ^ - .  - b V " .  Hence, i f d i m e n s i o n l e s s , p r e s s u r e , temperature and d e n s i t y are d e f i n e d by:  (13.9)  (13.10)  P* =  T * -  -k- p  _L  £ l ,  & 0  (13.11) equations  . n * c (13.6)  b n  ;  .  and (13.7) can be r e w r i t t e n i n dimensionless  form:  (13.12)  P*^ T V ( |+ v\*t[ (**))  (13.13)  o(  Vcw*) +  -  2 . K * / T * -  A p l o t of the reduced p r e s s u r e v r s . d e n s i t y u s i n g the s e r i e s approximation,  equation  f o r ^(^) was  (I4..I3),  virial  made, and  from the f i g u r e the f o l l o w i n g values f o r the c r i t i c a l p o i n t were o b t a i n e d . Table 1 .  C r i t i c a l Point  T h e o r y , Eq. (13.12) 0.O7I  0.  o% \\  0.1S77  O. ^+7 4  c  1.51  |.«5  0.2&4  The  experimental  data was  reduced  to dimensionless form  u s i n g c o n s t a n t s from a Lennard-Jones argon  1.0  (6-12) p o t e n t i a l f o r  as these c o n s t a n t s can be r e l a t e d to W  0  be seen l a t e r .  by  and  1  b  as  will  V i r t u a l l y no change i n these v a l u e s o f the  c r i t i c a l p o i n t occurs i f one uses the Pad£ approximant, equation ( l | . l 5 ) , To determine and  I n s t e a d of the v i r i a l  series.  the c o e x i s t e n c e curve between the  the l i q u i d one may  perform  the Maxwell c o n s t r u c t i o n on  the isotherms of the f i g u r e , or a l t e r n a t e l y , one may two  equations,  vapor  use  the  (13.15)  P*CrO-P*ChO=  (13.16) to f i n d  c* ( V ) - CK ( n ) z  and vapor.  s o l v e the two  equations  >  = o  In, and  the q u a n t i t i e s  of the l i q u i d  0  j n  z  , which are the d e n s i t y  A computer program was  (13.15)  (13.16)  and  w r i t t e n to  by use of Newton's  method.  Convergence was  very r a p i d u s i n g i n i t i a l  termined  from the graph of the equation of s t a t e .  values  de-  Figure 7  shows the c o e x i s t e n c e curve f o r l i q u i d  and  u s i n g the Pade approximant f o r 0 | .  the same f i g u r e i s  On  gaseous phases  £1  plotted  the experimental data  f o r sr-gon.  I t i s seen t h a t  the gaseous phase i s r e p r e s e n t e d q u i t e w e l l w h i l e the phase i s i n e r r o r by as much as 20$.  The  liquid  f a c t t h a t the  liquid 22.  phase i s not r e p r e s e n t e d w e l l i n t h i s model i s w e l l known. To o b t a i n the d e n s i t y d i s t r i b u t i o n f o r the two s o l u t i o n s one must now for  h(^) .  J+zCb)--  mcOcJir '  assume plane symmetry w i t h the 3, a x i s p e r p e n d i c u l a r to  the I n t e r f a c i a l p l a n e , w i t h ^ i n c r e a s i n g toward the phase.  Because of the symmetry, some of the  can be c a r r i e d and  (13.3)  Let  (13.17) and  s o l v e the i n t e g r a l e q u a t i o n  phase  x', ^ ' j ^ - '  out i n the i n t e g r a l  (13.17).  liquid  integrations Let  9,<fjr  be s p h e r i c a l and c a r t e s i a n c o o r d i n a t e s , r e s p e c t -  i v e l y , with o r i g i n at ^  .  Then, s i n c e  jU, ($)  can be w r i t t e n ,  2  We  o o  ,_TT  (T  o  r  1 T T  J  o  i n t e g r a t e over <f , change v a r i a b l e s from 0 and f to ^  r'  and  , where ^  II  -  r  i  t w ©  /  ^ r  ii  -  r  I  becomes (13.18)  ' a i r ^ V ^ C f ' O n l ^ t ^ rAr"  JAzt<b)x  9nj^"^KCT")h( 3' )v- Ar ,  M  C  -~  v  , ,  n  1t  $  -5"  We now s u b s t i t u t e a cut o f f g e n e r a l i z e d Lennard-Jones tial, (13.19)  into  KCO  (13.18).  =  Changing S'=  to d i m e n s i o n l e s s v a r i a b l e s ,  s ' V c r , S^r 3/0-  and i n t e g r a t i n g over 00  r  4-e  j  V~" one o b t a i n s ,  =  b ^  poten-  -With an obvious change i n v a r i a b l e s , the i n t e g r a l  can be  written, CO  -(13.20) ^  a  U )=  lae^Cs)-  -  '^^.^s^^^C^sr-^Cs'.srj 6+\  s-i  S+ I  S-I  The f o r c e constant tion  (13.8).  (13.21)  6  i s related  to the q u a n t i t y W  0  The r e l a t i o n i s found to be  W = - 12. £ 0  \ 3-w\  1 3> —2.  U s i n g t h i s r e l a t i o n and the d e f i n i t i o n of % integral  (13.22) A  by equa-  e q u a t i o n (13.3)  t  h  e  can be w r i t t e n ,  vyWc*)) ~ >j_  \ Xl T * ^  =  * '  computer program was w r i t t e n t o s o l v e the n o n - l i n e a r i n t e -  gral  e q u a t i o n (13.22) by u s i n g an i t t e r a t i v e procedure.  Numerical range  i n t e g r a t i o n of  ^(  was performed  I n the  $'=-10 by the t r a p e z o i d a l r u l e , the remaining i n t e g r a l s  over the i n f i n i t e range b e i n g approximated by  -Ct>  where <\-lo and f\§ , Y\ a r e the constant e q u i l i b r i u m gas and  liquid was  densities, respectively.  The i n t e r v a l o f i n t e g r a t i o n  taken t o be 0 . 1 and a step f u n c t i o n was chosen as i n p u t  f u n c t i o n f o r the i n i t i a l 0.001  integration.  o r b e t t e r was o b t a i n e d a f t e r 8-11 i t e r a t i o n s and a  machine time procedure  (IBM 70I4J4.) o f approximately  S .  10 minutes.  ^  can be e v a l u a t e d f o r each  The n o n - l i n e a r a l g e b r a i c  equation ( 1 3 . 2 2 )  s o l v e d by Newton's method t o g i v e a new H Cs) . was  The  f o l l o w e d was the f o l l o w i n g ; f o r a g i v e n d e n s i t y  d i s t r i b u t i o n the i n t e g r a l point  Convergence o f H (.£•) t o  repeated u n t i l  was then  The procedure  convergence was o b t a i n e d .  A t y p i c a l d e n s i t y d i s t r i b u t i o n i s shown i n F i g u r e 8 , f o r a T*=  ,- /r -0.515 . T  c  A f t e r t h e d e n s i t y d i s t r i b u t i o n has been o b t a i n e d i t i s a simple matter tension.  to use equation  (13.1)  to o b t a i n the s u r f a c e  The d i m e n s i o n l e s s l o c a l p r e s s u r e i s e a s i l y d e f i n e d  from e q u a t i o n  (13.2)  and i s  U s i n g e q u a t i o n ( 1 3 . l ) , one o b t a i n s a d i m e n s i o n l e s s * tension, ,  surface  -*-°°  (13.21,)  X- ^ W ^ P ^ o V b J  - O o  From t h i s  V ^ K * . b  e q u a t i o n one sees t h a t experimental v a l u e s o f the  s u r f a c e t e n s i o n can be reduced  (13.25)  E  to d i m e n s i o n l e s s form by  ** =  Hence f o r a g i v e n p o t e n t i a l d e f i n e d i n terms o f W <r one can 0i  compute a--dl-mensionless -surf ace t e n s i o n , X ;  experimental  values may  a l s u r f a c e t e n s i o n by In F i g u r e 9 we  , to which the  be compared by r e d u c i n g the  equation  experiment-  (13.25).  p l o t the q u a n t i t y  r 0  PCs)  f  d e n s i t y d i s t r i b u t i o n of F i g u r e 8.  o  the  r  .-  In F i g u r e 10 we have compared the experimental the s u r f a c e t e n s i o n reduced by constants  values  determined from  for  virial  c o e f f i c i e n t s f o r the Lennard-Jones ( 6 - 1 2 ) p o t e n t i a l [Yr\-ta. JL -C } for  v a r i o u s gases to the computed values u s i n g the v i r i a l  ies  and  the Pade approximation to the hard  potential.  core p a r t of  Even though the c r i t i c a l p o i n t values  the  are q u i t e  i n s e n s i t i v e to e i t h e r the Pade approximant or the v i r i a l ies,  the values  ser-  ser-  o f the s u r f a c e t e n s i o n are q u i t e s e n s i t i v e to  which form i s chosen.  The  s u r f a c e t e n s i o n i s a l s o q u i t e sen-  s i t i v e to the form of the l o n g range p o t e n t i a l as.can.be seen from the two  values  Jones ( 6 - 1 0 )  and  c a l c u l a t e d at  (6-13) potential.  T / T = O . 55 C  We  f o r a Lennard-  would expect the  best  agreement to experiment f o r s p h e r i c a l atoms o r molecules i n which quantum e f f e c t s are s m a l l . requirements best  i s probably  The  gas  which meets  these  k r y p t o n w h i l e quantum e f f e c t s  are l a r g e r f o r argon and much l a r g e r f o r neon.  The  nitrogen  molecule i s a l s o known to be s l i g h t l y a s p h e r i c a l which account f o r the disagreement. ues  of experimental  o f 1%.  e r r o r was  The  I t i s seen t h a t the theory  may  o n l y data with quoted  that f o r nitrogen with fits  val-  values  the experimental  s u r p r i s i n g l y w e l l i n the temperature range "Vr - 0 - 5  to  data 0.7.  T h i s seems to I n d i c a t e t h a t the s e p a r a t i o n of p o t e n t i a l i n t o s h o r t and  long range p a r t s has  been j u s t i f i e d .  3  Note the  F i g u r e 10.  S u r f a c e Tension, Theory and Experiment  comparison to the p r e v i o u s c a l c u l a t i o n s by KIrkwood and B u f f , H i l l , The  of the s u r f a c e t e n s i o n  P l e s n e r and P l a t z , and  c a l c u l a t i o n s of H i l l , P l e s n e r and P l a t z , and  Jouanin.  Jouanin were  v e r y s i m i l a r to what has been done here, but t h e i r approximat i o n of the l o n g and s h o r t range p a r t s of the p o t e n t i a l were not as p r e c i s e .  Kirkwood and B u f f c a l c u l a t e d  the s u r f a c e t e n -  s i o n by d e r i v i n g an exact e x p r e s s i o n f o r the s u r f a c e t e n s i o n in  terms o f the p a i r c o r r e l a t i o n f u n c t i o n and  potential.  Numerical  r e s u l t s were then obtained by  approximate experimental and p o t e n t i a l .  The  the i n t e r - m o l e c u l a r  expressions f o r the p a i r  temperature  range used  inserting  correlation  i n F i g u r e 10  chosen because the bulk of the experimental data was  was  i n that  range. In  F i g u r e 11 we have p l o t t e d  a g a i n s t (l — /T ) t  c  the s u r f a c e t e n s i o n j ¥ j  on a l o g - l o g s c a l e .  Previous  experiments p  i n d i c a t e t h a t the s u r f a c e t e n s i o n v a r i e s p=s. I.l"3 . is  Prom the f i g u r e we  'tf (l-~ /'r ')  with  r  6  c  see t h a t the behaviour ^i—T/^.^....  i n d i c a t e d but w i t h a value of p'=- 1.6  i m e n t a l v a l u e of 1.23  as  i n s t e a d o f the  exper-  which i s indeed shown by the data of  34 Stansfield. It  i s a l s o seen from F i g u r e 11 t h a t s i n c e the s l o p e of  the experimental  curve and  the computed curve i s d i f f e r e n t ,  the agreement with experiment  where the two  be f o r t u i t o u s .  We  reasons;  near the c r i t i c a l  first,  curves c r o s s  may  do not b e l i e v e t h i s , to be the case f o r tx^o p o i n t c o r r e l a t i o n s due  the l o n g range p o t e n t i a l b e g i n to p l a y an important these have not been taken i n t o account c o n s i d e r a t i o n ; and second,  role  to and  i n the model under  the experimental data f o r s u r f a c e  73 F i g u r e 11.  S u r f a c e Tension on Log-Log Theory and Experiment  Scale,  o  t e n s i o n were reduced t o d i m e n s i o n l e s s form u s i n g f o r c e c o n s t a n t s determined from v i r i a l  s e r i e s f o r the gases i n v o l v e d ; i f the  data had been reduced to d i m e n s i o n l e s s form u s i n g f o r c e  con-  s t a n t s determined by c r i t i c a l p o i n t data, the agreement w i t h experiment would have been b e t t e r near the c r i t i c a l  point.  CHAPTER I I I .  SOME NON-EQUILIBRIUM RESULTS  • 111.. . METASTABLE STATES AND METHOD OF CALCULATION We have seen i n s e c t i o n 7 t h a t the metastable s t a t e s a r e only l o c a l minima o f the f r e e energy. is  Being minima, however,  enough to show t h a t these s t a t e s a r e s t a b l e a g a i n s t  pertubations  o f the thermodynamic v a r i a b l e s .  small  With the k i n e t i c  t h e o r y o f s e c t i o n 3» one can study these metastable s t a t e s i n more d e t a i l . are  de Sobrino has shown that the metastable  indeed s t a b l e with r e s p e c t  small pertubations,  e.g.,  d i s t r i b u t i o n function.  t o much more g e n e r a l  types o f  small dynamical p e r t u b a t i o n s  S i n c e the k i n e t i c  states  o f the  equation i s n o n - l i n e a r ,  t h i s r e s u l t does n o t mean t h a t the s t a t e s w i l l be s t a b l e f o r large pertubations.  Indeed, the problem we wish t o study i n  t h i s s e c t i o n Is the p o s s i b l e growth o r decay o f l a r g e bations  pertu-  In the u n i f o r m metastable s t a t e .  As remarked i n s e c t i o n 5> the f i r s t  few moment equations  o f the k i n e t i c e q u a t i o n should be a good approximation f o r the study o f t h i s problem.  This i s because the t r a n s i t i o n from a  m e t a s t a b l e s t a t e t o a s t a b l e s t a t e i s slow and the assumption of q u a s i - l o c a l e q u i l i b r i u m f o r the d i s t r i b u t i o n f u n c t i o n i s then a good one.  de Sobrino s t u d i e d  metastable solutions  the s t a b i l i t y o f the  f o r small pertubations  namic v a r i a b l e s o f the f i r s t  i n the hydrody-  t h r e e moment equations and found  a d i f f e r e n t d i s p e r s i o n r e l a t i o n from the one given k i n e t i c equation. is  The important f e a t u r e s  concerned are, however, s t i l l  confidence that  present.  by the  as f a r as s t a b i l i t y Thus we have some  the moment equations w i l l s t i l l  approximate  the problem when they are not l i n e a r i z e d . In  t h i s s e c t i o n we w i l l be concerned  development o f l a r g e space  o n l y w i t h the time  dependent p e r t u b a t i o n s o f the hydro-  dynamic v a r i a b l e s i n the moment equations o f s e c t i o n %. w i l l not be concerned  We  w i t h how these p e r t u b a t i o n s occur  initially. I n s t e a d o f attempting to s o l v e the complete s e t of moment equations we w i l l f i r s t s i m p l i f y t r a c t a b l e s e t of e q u a t i o n s . proceed except  them so as to o b t a i n a more  S i n c e the process i s expected to  s l o w l y we n e g l e c t v i s c o s i t y and terms i n v o l v i n g the term  V-C  i n equation (£.1) which cannot  ed i n o r d e r to have c o n s e r v a t i o n o f p a r t i c l e s  C  be n e g l e c t -  and the term  V-C  i n equation (5*3) i n o r d e r t o have t r a n s f e r o f energy by convection.  The s i m p l i f i e d  (11;.1)  ^IL + r>V-£ -  (11;. 2)  VP«. . C  (11;.3)  '  Equation  V*"£  s e t of equations now reads,  in  +  +  Y\  O,  v j k t ^ r ' ^ C ^ c & t '=  - Z ^ . PH.C.V-C  =  ( l l ^ . l ) i s now used  oj  A V ' D I V T ) .  t o e l i m i n a t e the term  with  (II4..3); the hard core thermal c o n d u c t i v i t y , equations  (5»U-) and (5» 5), i s s u b s t i t u t e d i n t o (II4..3), as w e l l as the e x p r e s s i o n f o r the hard  core p r e s s u r e , equation  equations may now be reduced  to dimensionless  (1|.6).  These  form by l e t t i n g :  where,  UU-5) 3 Here T  , n * , r** and "t* are the dimensionless T  d e n s i t y , d i s t a n c e , and time r e s p e c t i v e l y . the boundary values of the temperature erature r e s p e c t i v e l y .  0  temperature, and T  0  are  and dimensionless  temp-  The two dimensionless equations o b t a i n -  ed a r e :  •  .  (XI4..7)  £ 1 * - %0 3t*  bvJ x j  n*  1  3  0  + ^ ) Z * £ i i . - £-L v - C ^ O v i * ^ ) ,  3  ft*  St"*  3 Y\*  where  and a l l the g r a d i e n t - o p e r a t o r s are in,, terms of.,the. dimension-.. l e s s c o o r d i n a t e V* . In what f o l l o w s we w i l l c o n s i d e r o n l y s o l u t i o n s with s p h e r i c a l symmetry, so the independent  v a r i a b l e s are the  r a d i a l c o o r d i n a t e , V , and the time^T .  Hence we need to  s o l v e ( l l j . . 6 ) and (1I4..7)  f o r the temperature T ( t ^ j i , * . )  the d e n s i t y ft (*"*.> "t* )  .  Henceforth we w i l l drop the s t a r s  on a l l the v a r i a b l e s f o r convenience understood  and  i n w r i t i n g , i t being  t h a t dimensionless v a r i a b l e s are b e i n g used.  Before c o n s i d e r i n g the n u m e r i c a l s o l u t i o n s of these equations we i n d i c a t e the procedure used (II4.•  to s o l v e them.  Since  T) i s r e m i n d f u l of a d i f f u s i o n type e q u a t i o n we assume  t h a t the boundary c o n d i t i o n s to be used are those a p p r o p r i a t e for  t h a t type equation.  TC^-fc)  we  Y~  0  dary v a l u e s at V~= O  and n ( r £ + 4 t ) > T Cr 0 i -t+M)  , T( -t+M;) 0)  i s some l a r g e number.  s p e c i f y hCr^-t) ,  .. and " H r , t+Ai,)' g i v e n the  can compute nCvji-VAt)  -values of V\ ( 0, i + At) where  T h i s means that i f we  0 i  I n s t e a d of s p e c i f y i n g boun-  , we w i l l s p e c i f y  that  These are the n a t u r a l c o n d i t i o n s f o r s p h e r i c a l l y solutions.  symmetric  F o r the o t h e r boundary c o n d i t i o n we w i l l  that the d e n s i t y and temperature  suppose  are kept at some c o n s t a n t  values,  Mr  (1^.9)  0 i  t)  =  "o  jT(.r  0 J  i) = To  .  These boundary c o n d i t i o n s correspond to system maintained at a constant p r e s s u r e and disturbance.  temperature  f a r from the c e n t e r of the  Perhaps- more-natural boundary-conditions would — -  be that of f i x e d number o f p a r t i c l e s i n which case we should take V  JX  -  o  where jA. i s the chemical p o t e n t i a l , however t h i s case i s harder to work w i t h n u m e r i c a l l y . initial  I t was  observed,however,that  f o r the  growth of the d r o p l e t , the r a t e i s q u i t e i n s e n s i t i v e to  the exact form of the boundary c o n d i t i o n s (11^.9), i . e . , d e r i v a t i v e c o n d i t i o n s g i v e much the same r e s u l t  i f V ^ , is large.  To s o l v e the equations we use the d i f f e r e n c e approxima-. tions,  at  A-t  i n equation (H4..IO)  SH  3  (1J4..7) to reduce  TU/t+At) =  At  i t to the form,  AC^t) n C f j t + A-fc)^ B ( r j i ) j  where,  We  now  r e w r i t e equation  (i + n ^ ) v T + T  (1U..11)  ( I I 4 . . 6 ) i n the form,  vv"vy\4 V X - o j  where  Note t h a t i n t h i s meter.  equation the time  Hence the equation can be taken at e i t h e r the time  or the time "t + A t  .  We  approximation  that  VI  stead of  .  and  t + A t  Supposing  t  c o n s i d e r the l a t t e r case w i t h the i s computed a t the former t i m e ^ t ^ i n -  S u b s t i t u t i n g equation  s o l v i n g f o r Vn  point r  appears.,.only as a p a r a -  (Hj.,10)  into  (li|.ll)  gives  the r i g h t hand s i d e to depend on c o n d i t i o n s a t the  and u s i n g the d i f f e r e n c e approximation  s p h e r i c a l symmetry, we get the p o s i t i o n  V" , thus,  h(r+Arj  for  Vr  in  i n terms of. q u a n t i t i e s at  (11;. 13)  h (v-+Ar) =  Ar-  T h i s method was found to be s t a b l e f o r values o f A"b s m a l l enough, w h i l e s e v e r a l  other methods of s o l u t i o n were u n s t a b l e  f o r a l l v a l u e s of Ai> . For and  actual  computation we need to s p e c i f y  pair correlation function.  section,  the p o t e n t i a l  In the r e s u l t s of the next  the.potential,  (Ik-Ik)  and  the Van der Waals  plicity  ^ y equation (1;.9), were used f o r sim-  of c a l c u l a t i o n .  correlation function  These c h o i c e s f o r p o t e n t i a l and p a i r  r e p r e s e n t the e s s e n t i a l p h y s i c s of the  problem and should g i v e good s e m i - q u a n t i t a t i v e r e s u l t s .  •15.  RESULTS OP NUMERIC AITS GLUTTON OF THE MOMENT EQUATIONS The moment equations with the approximations  described  i n the p r e v i o u s s e c t i o n were s o l v e d n u m e r i c a l l y u s i n g an 360  and IBM  70hM- computer.  are p r e s e n t e d i n t h i s  The r e s u l t s of these  IBM  computations  section.  In o r d e r t o have some f e e l i n g f o r the numbers i n v o l v e d we -relate.the potential  (II4..II4.) to .a cut. o f f Lennard-Jones (6-12)  p o t e n t i a l by equating the z e r o t h and second potentials. CT  find  D  - -| £  U s i n g the v a l u e s f o r €  argon,  and -L - (.Cor  , where 6  and  5 . 4 x to Ux  d/w\.  8  io h* tw£  %  2Z  (IJ4..4) we  Prom equation  , z  . j  33 O T * °K j T*-  ^ T-  0.2St .  can a l s o d e f i n e a d i m e n s i o n l e s s v e l o c by,  Z E U ^ X - D * .  -0 =  F o r argon,  this gives:  •-\3=. So  =  I.  83* io**-0*  ^ X I D ^ / S ^ .  F o r convenience  i n Figure 12,  <W$4C.  =  MSo'O*!  70=41.5°*,"^  o.lS •  i n v i s u a l i z i n g the r e s u l t s , the  of s t a t e u s i n g the Van plotted  « t = 3,oxi 0 t *  , r e l a t e d to the v e l o c i t y U  (15.2)  (15.3)  as g i v e n i n r e f e r e n c e '  by:  r-  "0  and 0~  the dimensionless v a r i a b l e s are r e l a t e d t o r e a l  variables  ity>  W  that  two  are the parameters o f the (6-12) p o t e n t i a l as i n e q u a t i o n  (13.19). for  We  moments of the  der Waal's ^ , e q u a t i o n on a l o g - l o g s c a l e .  The  (4,.9)  equation has  been  c r o s s e s on the  -figure r e f e r to p o s i t i o n s at which r e s u l t s are p r e s e n t e d i n t h i s work.  The numbers i n c i r c l e s by the c r o s s e s r e f e r to  f i g u r e numbers and  the other numbers are r a d i a l growth r a t e s  of l a r g e p e r t u b a t i o n s , F i g u r e s 13  as w i l l be d e s c r i b e d  through 20  show the r e s u l t s of the c a l c u l a t i o n s  For a l l of the c a l c u l a t i o n s the i n i t i a l was  later.  density  distribution  chosen to be a t r a p e z o i d p l u s a c o n s t a n t , hence there  f o u r parameters to be  chosen f o r the d e n s i t y ; the h e i g h t  are and  width o f the t r a p e z o i d , the s l o p e of the s i d e of the t r a p e z o i d , and  the v a l u e  of the c o n s t a n t .  The  the boundary c o n d i t i o n ( l l | . l l ; ) and  constant  corresponds to  i s j u s t the value of the  d e n s i t y , n*(r?)= h* , o f the vapor a l a r g e d i s t a n c e from the c e n t e r of the d i s t u r b a n c e . ways f i x e d  i n these  The  s l o p e of the t r a p e z o i d was  c a l c u l a t i o n s to f a l l  t r a p e z o i d to the bottom i n  Af*= 1.5  t i o n s r e p o r t e d here the d i s t a n c e c o n d i t i o n s Were imposed, was  lr  • 0  from the top o f the  In a l l o f the  calcula-  at which the boundary n  taken to be  because i t was  ^=-12.  found t h a t l a r g e r d i s t a n c e s d i d not s i g n i f i c a n t l y modify results. values  al-  the  • C a l c u l a t i o n times were a l s o - s h o r t e r f o r the s m a l l e r .  of  r* .  T (r*J =-T  In a d d i t i o n t h e r e i s a f i f t h  parameter,  , the temperature of the vapor at  0  F i g u r e 13  shows what happens when T  the c r i t i c a l p o i n t .  0  Fj* .  and  Y)  a  are above  In a l l of the f i g u r e s o f t h i s type  to  f o l l o w ^ t h e d e n s i t y d i s t r i b u t i o n i s drawn f u l l w i t h s c a l e to the l e f t right.  and The  time, ti .  the temperature i s drawn dashed w i t h s c a l e to numbers beside  each curve  I t i s seen t h a t the i n i t i a l  q u i c k l y , presumably i n t o the constant i f one at  waits  l o n g enough.  are the  dimensionless  disturbance one  the  phase  decays  distribution  F i g u r e 11; shows t h a t the  density  the c e n t e r appears to decay .exponentially with a decay  8i| F i g u r e 13.  P e r t u b a t i o n Above the C r i t i c a l  Point  c  = -2.3.^  -The-temperature  of the d r o p l e t i s lower  than -Its s u r r o u n d i n g s , which i s what one would expect i f the For argon, u s i n g ( 1 5 . 1 ) , we  d r o p l e t were e v a p o r a t i n g . the r a d i u s of the i n i t i a l of  d i s t u r b a n c e , 2oxlo CTA. ; the number  p a r t i c l e s i n the i n i t i a l d i s t u r b a n c e , 350 — ~ 11  the decay c o n s t a n t ,  find  v o x 10  ASLC  .  particles;  and  .  F i g u r e 15 i n d i c a t e s the s i t u a t i o n when the vapor i s superc o o l e d . " At" this" temperature  of T  0  = 0.\5  the d e n s i t y o f l i q u i d  and vapor f o r a plane i n t e r f a c e are g i v e n by, fi = 0.817 u  ^=O-O078 is  p  » respectively. .  -0.0OII5  The  taken to be n = 0 . 0 ^ o  P*=O.00&7 to  .  and  The p r e s s u r e at these d e n s i t i e s  s u p e r c o o l e d vapor d e n s i t y at \~ - \% i s 0  and the p r e s s u r e a t t h i s d e n s i t y i s  I f we d e f i n e the degree of s u p e r c o o l i n g , D  ,  be JT_) a-- P r e s s u r e of Supercooled Vapor - ^ , 5 , ^ * ^ E q u i l i b r i u m Vapor P r e s s u r e ^ p*(Maxwell construction)  then f o r F i g u r e 15, D = 5.8 t h a t the d r o p l e t i s growing dr*/£|t* = 0.070 •  .  From the f i g u r e i t i s c a l c u l a t e d r a d i a l l y at the average  Because o f the smallness o f  olr*/cit*  rate , the aver-  age v e l o c i t y of the d i s t u r b a n c e i s indeed much s m a l l e r than the v e l o c i t y of sound, and hence the n e g l e c t of the terms i n volving  C  i n the moment equations seems j u s t i f i e d f o r the  p r e s e n t case.  I t i s seen t h a t the temperature  of the d r o p l e t  i s h i g h e r than i t s s u r r o u n d i n g s , i n d i c a t i n g t h a t the heat o f c o n d e n s a t i o n brought  i n by the incoming molecules does not have  time to d i f f u s e outwards.  Furthermore,  edge of the d r o p l e t i s lowered  toward  the d e n s i t y near the  the plane f a c e e q u i l i b r i u m  F i g u r e l£.  P e r t u b a t i o n Below the C r i t i c a l Supercooled Vapor • *  \—  O  CN)  00  -7  &  =-  Point,  -  "value of the vapor. let  The d e n s i t y near the c e n t e r of the  i s lower than the e q u i l i b r i u m v a l u e of the l i q u i d .  can be accounted  dropThis  f o r because s u r f a c e t e n s i o n w i l l tend to make  the p r e s s u r e d i f f e r e n t on the i n s i d e and o u t s i d e of the d r o p l e t . F u r t h e r , the temperature  i n s i d e i s h i g h e r , thus causing the  d e n s i t y to be lower i f we  are on the l i q u i d  s i d e of the  isotherm.  These d e t a i l s are i n q u a l i t a t i v e agreement with a phenomenolo g i c a l theory of the condensation  of water vapor  i n the a i r  36 to form r a i n d r o p l e t s .  However, that theory only deals with  drops much l a r g e r than the ones we  are c o n s i d e r i n g here.  As s t a t e d e a r l i e r , the mathematical  a n a l y s i s shows t h a t  i f the d i s t u r b a n c e i s s m a l l enough, i t w i l l not grow f o r the metastable  states.  F i g u r e 16 shows what happens i f the  i a l h e i g h t of the d e n s i t y i s taken to be 0 . 1 5 . there are 65 p a r t i c l e s  i n this disturbance.  q u i c k l y grows to a l a r g e r one.  s u p e r c o o l i n g n*s h e i g h t 0.25  0.05  The  i t was  dis-  but a lower  » the. d i s t u r b a n c e w i t h  grew w h i l e one with h e i g h t 0.15  F i g u r e 17 shows what happens i f ft  Q  below the e q u i l i b r i u m vapor v a l u e .  changed to  found t h a t the  At the same temperature, » "D = 4 . 7  argon  disturbance  I f the h e i g h t was  .0.12, with 52 p a r t i c l e s f o r argon, turbance would decay.  For  init-  decayed.  i s lowered  to a value  In t h i s case the vapor i s  no l o n g e r supercooled and we would expect the d r o p l e t to evapo r a t e i n s t e a d of grow. one  would expect  T h i s , indeed, i s what happens.  i f the d r o p l e t i s e v a p o r a t i n g , the tempera-  t u r e o f the d r o p l e t i s lower than that of the The wiggles round  As  i n the temperature  curve near  o f f . e r r o r s i n the program due  surroundings.  V" - 10  are due  to the smallness of  to H*  .  90 F i g u r e 17.  Evaporation  of a P e r t u b a t i o n  91 We  f e e l t h a t they do not  have any. s e r i o u s  s i n c e the temperature s t a b i l i z e s the c e n t e r . The  of the  itself  e f f e c t on the r e s u l t s  as one moves toward  droplet.  growth r a t e s change from p o s i t i v e to n e g a t i v e  as the s u p e r c o o l i n g supercooling  r a t i o i s lowered.  r a t i o we  should  shows the s i t u a t i o n at achieving  Therefore  at some  find a stable droplet.  T,^.  3,5  a D = 2.35  and  a smooth shape, the d r o p l e t f a i l s  values  Figure .  18  After  to grow, i n d i c a t i n g  an e q u i l i b r i u m s i t u a t i o n o f a l i q u i d d r o p l e t surrounded by i t s vapor. ature  T h i s i s f u r t h e r j u s t i f i e d by i s constant.  obtained  the f a c t that the  I t i s to be noted that the e q u i l i b r i u m i s  with a degree of s u p e r c o o l i n g  T h i s i s due  temper-  not  equal to  one.  to the f a c t t h a t , f o r small d r o p l e t s , the  t e n s i o n r a i s e s the  e q u i l i b r i u m vapor p r e s s u r e  surface  considerably.  This  e f f e c t decreases as the r a d i u s of the drop becomes l a r g e r  and,  indeed, i t was  found t h a t f o r the  degree of s u p e r c o o l i n g Finally,  i n Figure  l e t s with i n i t i a l eratures  and  same temperature  a-drop w i t h a bigger, radius would grow. 19>  we  p l o t the growth r a t e s f o r drop-  s i z e s as i n F i g u r e 15*  degrees of s u p e r c o o l i n g .  degrees of s u p e r c o o l i n g  would expect l a r g e r growth r a t e s . I t must be  but with v a r i o u s  I n a l l cases the  decrease with a decrease i n s u p e r c o o l i n g . tures, higher  and  temp-  rates  For lower tempera-  are p o s s i b l e , so  one  This behaviour i s i n d i c a t e d .  s t a t e d that these r a t e s are not very p r e c i s e , s i n c e  they are c a l c u l a t e d from curves l i k e the where i n some .cases i t was t i o n on as l o n g as i * = 3 o . o  ones i n F i g u r e  not p o s s i b l e to c a r r y the due  15,  integra-  to the l e n g t h of c a l c u l a t i o n .  92 Figure 1 8 .  Equilibrium Distribution  F i g u r e 19.  Growth Rates  I t i s i n t e r e s t i n g t o see what happens i f the system i s on the p o s i t i v e s l o p e r e g i o n of the isotherm. what happens i n t h i s case. density  = 0.I7Z  rapidly.  F i g u r e 20 shows  At a temperature T  a disturbance  of height  This i s a very s m a l l d i s t u r b a n c e  c o n t a i n s o n l y twelve p a r t i c l e s  D  = 0.15  and a  0.200 grows  s i n c e f o r Argon I t  i n excess o f the uniform  number.  Note t h a t the growth i s much more r a p i d and c h a o t i c than the o r d e r l y growth shown i n F i g u r e 16, when the d e n s i t y i s i n the negative  s l o p e r e g i o n of the isotherm.  s l o p e r e g i o n o f the i s o t h e r m  i s unstable  Since  the p o s i t i v e  even f o r s m a l l  dis-  turbances  one would expect that a s m a l l d i s t u r b a n c e  would grow  rapidly.  I t i s i n t e r e s t i n g t o see how q u i c k l y the whole den-  s i t y r e g i o n changes i n t o a s t a t e with d e n s i t i e s c l o s e t o the equilibrium l i q u i d  and vapor d e n s i t i e s .  This would  probably  be even more r a p i d i f the d e n s i t y and temperature were not f i x e d a t the boundary values  a t )r *= 12. , 0  I n p r i n c i p l e the approximations and numerical used i n c a l c u l a t i n g have e v a p o r a t i o n  the previous  of a l i q u i d  curves  should  procedure  work when we  i n t o a vapor d r o p l e t ( i . e . , the  s i t u a t i o n where the d e n s i t y at the c e n t e r i s lower than the d e n s i t y outside, which i s superheated). able to o b t a i n any r e s u l t s i n t h i s case, procedure d e s c r i b e d  However, we were never s i n c e the numerical  i n S e c t i o n l l ; always became u n s t a b l e f o r  a l l values  of A"b  completely  r e w r i t t e n and a d i f f e r e n t method of s o l u t i o n was  used.  .  T h i s was t r u e even when the program was  There could be s e v e r a l reasons why t h i s case was un-  s t a b l e ; f i r s t , perhaps the process than the ones presented  here,  proceeds much more r a p i d l y  so that n e g l e c t of the terms  i n v o l v i n g the l o c a l v e l o c i t y i n the moment equations 'justified; liquid  second,  i s not  a l a r g e r p o r t i o n of the system i s i n the  r e g i o n of the isotherm where i t s s l o p e i s l a r g e , and  t h i s may have some.effect  on the n u m e r i c a l procedure  of i n t e -  g r a t i n g the equations; t h i r d , i t i s w e l l known from the num e r i c a l s o l u t i o n of n o n - l i n e a r p a r t i a l d i f f e r e n t i a l t h a t some i n i t i a l procedure  equations  c o n d i t i o n s are u n s t a b l e f o r a g i v e n numerical  while others are not; i f t h i s  numerical approximation  Is the case, a new  may produce r e s u l t s .  The f a c t t h a t  the vapor d r o p l e t case was n u m e r i c a l l y u n s t a b l e was q u i t e d i s t u r b i n g , s i n c e a p r e v i o u s approximation t i o n and temperature yield results  i n which heat  conduc-  dependent c o n d u c t i v i t y were n e g l e c t e d d i d  i n both the cases.  T h i s approximation,  although  y i e l d i n g the same q u a l i t a t i v e behavior, d i d g i v e q u i t e d i f f e r ent q u a n t i t a t i v e r e s u l t s from the ones g i v e n here, that heat  c o n d u c t i o n i s a v e r y important  indicating  aspect o f the problem.  From the r e s u l t s o f t h i s s e c t i o n , we see t h a t numerical i n t e g r a t i o n of the moment equations  (li|.10) and ( l i 4 _ . l l ) do  g i v e the c o r r e c t q u a l i t a t i v e b e h a v i o r o f a d i s t u r b a n c e i n a metastable  state.  The m e t a s t a b l e  s t a t e s i n the supercooled  p o r t i o n o f the isotherm are, indeed, u n s t a b l e with r e s p e c t t o large scale pertubations.  It i s significant  that a single s e t  of equations d e s c r i b e s the s i t u a t i o n i n the whole of the P-V plane and that d i f f e r e n t  assumptions need not be i n t r o d u c e d f o r  each of the r e g i o n s (e.g., e v a p o r a t i o n , condensation the c r i t i c a l p o i n t ) .  We f u r t h e r o b t a i n s p e c i f i c v a l u e s f o r the  growth r a t e s which depend on only two parameters, diameter  and above  and the range o f the long-range  force.  the hard It i s  core  "unfortunate t h a t ( t o - t h e author's-knowledge) no other theory or experiments e x i s t w i t h which to compare these  results.  CHAPTER IV. 16.  CONCLUSION AND  CONCLUSION AND  DISCUSSION  DISCUSSION  From the r e s u l t s presented  i n t h i s work i t i s apparent  t h a t the i d e a of s e p a r a t i n g the p o t e n t i a l i n t o a s h o r t range and  long range p a r t i s a f r u i t f u l  s t a t i s t i c a l mechanical theory one  idea.  a l s o needs to i n t r o d u c e  i d e a of a space dependent d e n s i t y . f o r the k i n e t i c theory but In the simple Van  a Van der Waals gas Sobrino  who  This i s a natural  appears to have been  der Waals theory due  the work of Van Kampen.  obtained  was  In the e q u i l i b r i u m  Apparently,  the  concept  overlooked,  to O r n s t e i n , u n t i l  the k i n e t i c  theory  of  not worked out u n t i l r e c e n t l y by  the same e q u i l i b r i u m equation  as  de  Van  Kampen. In t h i s work, b e s i d e s  r e d e r i v i n g many of Van Kampen  r e s u l t s by d i f f e r e n t methods and u s i n g fewer we have shown how  the e x i s t e n c e of a c r i t i c a l p o i n t i s r e l a t e d  to the mathematical s t r u c t u r e of Van i n t e g r a l equation.  approximations,  An obvious  i n v e s t i g a t e phase t r a n s i t i o n s  Kampen's n o n - l i n e a r  e x t e n s i o n of t h i s i d e a i s to i n g e n e r a l from the p o i n t of  view of the b i f u r c a t i o n of a n o n - l i n e a r o p e r a t o r .  One  could  study phase t r a n s i t i o n s from t h i s p o i n t of view i n other systems, the superconducting  phase t r a n s i t i o n f o r example.  Some work towards a g e n e r a l i z a t i o n of the i d e a to i n c l u d e n o n - e q u i l i b r i u m s t a t i s t i c a l mechanics and the study of S p e c if ial  systems ha-s a l r e a d y been c a r r i e d out by Grmela.  a l s o be u s e f u l to see how  I t would  our d e f i n i t i o n of a phase t r a n s i t i o n  as a b i f u r c a t i o n p o i n t of a n o n - l i n e a r o p e r a t o r i s r e l a t e d to  conventional  d e f i n i t i o n s such as those  Z7  o f Yang and L e e ,  6 4 Lebowitz and Penrose, and Kac , Uhlenbeck, and Hemmer. I t would be i n t e r e s t i n g t o see how our d e f i n i t i o n o f a phase t r a n s i t i o n i s r e l a t e d to the number of dimensions o f the system by s o l v i n g a one dimensional  system, such as t h e one s t u d i e d  by Ksc, Uhlenbeck, and Hemmer, u s i n g our methods. The its  c a l c u l a t i o n o f t h e s u r f a c e t e n s i o n i n t h i s work and  agreement w i t h  experiment i n d i c a t e s that the s e p a r a t i o n o f  the p o t e n t i a l i n t o l o n g and s h o r t range p a r t s and the subsequent n e g l e c t of c e r t a i n terms i n the e v a l u a t i o n of the partition  f u n c t i o n i s a s u r p r i s i n g l y good p h y s i c a l approximation  even when a r e a l i s t i c Kampen equation.  a t t r a c t i v e p o t e n t i a l i s used i n the Van  However, i t would be i n t e r e s t i n g  to compute  the s u r f a c e t e n s i o n t a k i n g i n t o account the l o n g range c o r r e l a t i o n s t o see i f even b e t t e r agreement with experiment c o u l d be obtained  near the c r i t i c a l  point.  We have not s t u d i e d , i n d e t a i l , the nature ing  of the branch-  i n a s m a l l neighborhood o f c r i t i c a l p o i n t because c o r r e -  l a t i o n s due to the l o n g range p o t e n t i a l are known to be important equation.  t h e r e , and these have been n e g l e c t e d I t would be o f i n t e r e s t , then,  theory o f de Sobrino t i o n s t o study  which Includes  i n Van Kampen's  t o use the k i n e t i c  the long range  correla-  the b i f u r c a t i o n in. a r e g i o n o f the c r i t i c a l  point. F i n a l l y , we have shown by n u m e r i c a l  i n t e g r a t i o n o f the  moment equations t h a t l a r g e s c a l e p e r t u b a t i o n s s t a b l e states,- a t l e a s t on the supercooled therm, are u n s t a b l e  o f the meta-  s i d e of the i s o -  and have c a l c u l a t e d some growth r a t e s .  I t would be d e s i r a b l e to o b t a i n s i m i l a r r e s u l t s f o r the superheated p o r t i o n o f the Isotherm,  and to r e a r r a n g e the numerical  c a l c u l a t i o n s to study the growth o f l a r g e drops and compare w i t h experiment. A l s o an a n a l y t i c  i n v e s t i g a t i o n o f the moment  equations i n some simple cases may h e l p to understand  the na-  t u r e o f the processes i n v o l v e d i n the growth o f p e r t u b a t i o n s .  101 BIBLIOGRAPHY'  •.'"  1.  E.G.D. Cohen, Ed.; Fundamental Problems i n S t a t i s t i c a l Mechanics, I I ; N o r t h H o l l a n d (I960), Chapters 1 , 2 , 3 .  2.  J.D. Van der Waals; D i s s e r t a t i o n , L e i d e n  3.  L.S. O r n s t e i n ; D i s s e r t a t i o n , L e i d e n  I4..  M. Kac, G.E. Uhlenbeck, and P.C. Heramer; J . Math. Phys.., h_, 216 ( 1 9 6 3 ) .  5.  N.G.  6.  .J.L. L e b o w i t z and 0 . Penrose; J . -Math. Phys. 7 ,  7.  L. de S o b r i n o ; Can. J . Phys. l£,  8.  L . Tonks; Phys. Rev. 5 q ,  9. 10.  (1908).  Van Kampen; Phys. Rev. 13 5A, 362  955  (1873).  363  (I96J4.). 98  (1966).  (1967).  (1936).  H. Grad; P r i n c i p l e s of the k i n e t i c t h e o r y o f gases, . Handbook der Physik, v o l . X I I ; S p r i n g e r V e r l a g ( 1 9 5 8 ) .  ,  S. Chapman and T.G. Cowling; The Mathematical Theory o f Non-uniform Gases; Cambridge (1958 ) . .*  ;  11.  H.L. F r i s c h and J.L. L e b o w i t z , Ed.; C l a s s i c a l Benjamin (196I4.), page I I 3 7 .  12.  F.H. Ree and W.G.  13.  B.J. A l d e r and T.E. Wainwright; Phys. Rev. 1 2 7 ,  II4..  M. Grmela; to be p u b l i s h e d .  15.  A. 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