Dey, Tarun K

Karki, Bimal P

Sinha, Suresh K

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Indian J. Phys. 75A (1). 93-94 (2001)U P  Aan international journalA perturbation theory of classical simple fluidsT aru n  K  D eyDepartment of Physics, Government IMahila Inter College, Purnia-854 201, Bihar^ IndiaB im a l P  K a rk i an d  S u r c s |  K  S in h a ’*'Department of Physics, L. $ College,B. B A Bihar University, Muzijffaipur*842 001, Bihar, IndiaAbstract The Weeks - Chandler - Andersen (W C A) perturbation theory of classical simple fluid is leexamined using the 'modified' Born -C'ircen - Yvan expression (or the (unction Y(r) of the hard sphere fluid We calculate the thermodynamic piopemes of the Leiinard ~ Jones (12-6) fluid I hey are found to be in good agreement with the Verlet -  Weis, Boublik and Simulation resultsKeywords Perturbation theory, free energy, equation of statePACS Nos. 05 70 Ce, 64 10 +h. 47 K)4^ gL IntroductionW eeks -  C h a n d le r  ~  A n d e rs e n  (W C A ) f 1 ] p e r tu rb a t io n  th e o ry  w ith th e  se m i e m p ir ic a l  e x p re s s io n  fo r  th e  h a rd  *- sp h e re  ra d ia l d is tr ib u tio n  fu n c tio n  (R D F )  [2 ,3 J is  in  e r r o r  in  so m ea p p lic a tio n s . T h is  d if f ic u l ty  c a n  b e  a v o id e d  by u s in g  a  b e tte r  e x p re ss io n  fo rIn th e  p re s e n t  w o rk , w e  a rc  p r im a r i ly  c o n c e rn e d  w ith  th e  W C A  th e o ry  a n d  th e  ’m o d if ie d ' B o rn  -  G re e n  -  Y v an  (B G Y M ) e x p re ss io n  fo r  fo r  r <  o b ta in e d  b y  C h a e , R e e  a n d  R e e  14], F hese  v a lu e s  fo r  (r) a re  in  m u c h  b e tte r  a g re e m e n t th a n  those  o f  P Y  ~  v a lu e s  [5 ).2, Theoretical formulationW c c o n s id e r  a  s y s te m , w h o s e  m o le c u le s  in te ra c t  v ia  th e  L J (12-6) p o te n tia lM(r) = 4€[(<T/r)'^-«T/r)®], 0 )U sin g  th e  d iv is io n  o f  p o ten tia l a c c o rd in g  to  th e  W C A  sch em e [ 1J, th e  e x c e s s  f re e  e n e rg y  p e r  p a r t ic le  is g iv e n  by/  =  . / o + / ,  » (2)w h e r e i s  th e  e x c e s s  free  e n e rg y  p e r  p a r tic le  o f  th e  re fe re n c e  sy s te m  a n d  is th e  f irs t o rd e r  p e r tu rb a t io n  c o rre c tio n  to  it. T h u s/ ,  =2jipjg^(r)Ui,{r)r^dr,w h e re  ( r )  is th e  RDf" o f  th e  r e fe re n c e  sy s te m .(3)w h e re  e  a n d  a  a re  c o n s ta n ts  w ith  u n its  o f  e n e rg y  a n d  le n g th , re sp ec tiv e ly .Corresponding AuthorPor correspondence . Ramani Mohan Garden, Kalambag Road Chowk, Muzaffarpur-842 002, Bihar, India3. Reference systemT h e  fre e  e n e rg y  o f  th e  re fe re n c e  sy s te m  is e x p re s s e d  in  te rm s  o f  th a t o f  th e  h a rd  sp h e re s  o f  d ia m e te r  rf, w h ic h  c a n  b e  d e te rm in e d  b y  th e  V erle t-W e is  m e th o d  [2 ,3 ) .In  th e  p r e s e n t  w o rk , to  e v a lu a te  d, w e  u.sc th e  B G Y M  e x p re s s io n  fo r  Yjj^(rld), g iv e n  as  (4]Y„s(r/d)  =  gus^d) e x p  ^npd' 'g„^{d)  { (I /1 2 )  {r! d)^- ( r / r f )  +  ( l l / 1 2 ) } ] l 'o r  rS^d,  (4)©2001 lACS94Substituting eq. (16) in eq, (3), we obtainTarun K  Dey, B im al P  Karki and  Suresh K  Sinhawhere gff^d) is the hard sphere RDF at the contact and given by[6Jg „ ,(rf) = ( l - n / 2 ) / ( I - r j ) \  (5)where 77 = /  6 is the packing fraction. Then d is given by/^= </«[! +(a,/2 (To >4 (6)wheredg = j ( l - e x p [ - /3 M o (» |) r / / - ,0O5 = | ( ( r / ( / « ) - l ) ‘ (<//(/r)(exp(-/JMo(r)J)(/r(7)(8)and/ ,  = 27tpjY„s(r/d)Ui, ( r ) r d r  + (X5). (17)(T, / 2 a„  = ( l - (l l /2)»7 + (17/4)i7^ + q ’) / ( l - t7 ) \ (9 )Only <Ti /  2(To differs from that derived by Verlet and Weis (VW).The virial equation o f stale for the reference system is given by [2.31^ / » o / p = Z o = Z „ 5 + 4 5 4 Z ,  (10)where is the hard sphere compressibility factor and given by [6]2 / f f  =  ( l +  ^  +  t / ^ + / ? ’ ) / ( l - 0 ) ’ 0 1 )and 4 Z  is derived using the BGYM expression for F„^(r/</)AZ = -2 7 ] U l - r j /2 ) ' / ( l -n f  (12)which differs from that given by Verlet and Weis [2,3].With the help of eq. (10), wc obtain an expression for the free energy per particle for the reference system/3/o =  / 3 / m + 4 5 / 3 4 / .  (13)where [6]^ / /w  = 0 ( 4 - 3 q ) / ( l - r / ) '  (14)is the excess free energy o f the hard sphere system and^ 4 /  = ( l /3 0 ) ( r 7 ^ / ( l - 7 j ) ’ ) - ( l / 8 ) ( n - / ( l - n ) ' )- ( l / 1 0 ) ( r j / ( l - r ? ) ') + ( 1 5 / 1 6 ) ( t ] - / ( l - J 7 ) '’) (15)4. First order perturbation termIn the W CA theory, the RDF g^(r) o f the reference system is approximated as (1 ]In the present calculation, we use the M C (2 ,3 ] and M D [2,3] values o f g^^fr) for r>d.5. Results and discussionWc compare our results o f /3 F  /  p  and / N for the U  (12-6) fluid with VW  |2], Boublik [7) and MC |2] values in a range ol reduced density p  * at T* = 1.15 in Table 1. The agreement i.s quite good. At low density. Boublik theory [7 | is superior to the pre.senl theory due to the second order perturbation terms.Tabic 1. Valuc.s of p P / p  and P U l ' N  for the U (12-6) fluid at T* = 1 15P* Exact Present VW Boublilpp/p 0.30 0 12 0 05 0 05 0 OS0.50 -^ 0 n -4) 28 -0 27 -0 190 65 0 31 0 17 0 17 0 230 75 1.17 1 12 1 10 I 09PUIN 0 30  ^ 1 05 -1 60 1 60 -1 8^0 50 -3 02 -2 86 -2 86 -3 020 65 87 -3 89 83 3 900 75 -4 46 -4 42 4 43 -4 45g o ( r ) »  e x p [- /3 H „ (r ) ]y ,„  (r/d) ( 16)Thus wc come to the conclusion that the WCA perturbation theory, using the BGYM integral equation for Y^5 (r/d) for »■ < d . can be employed to calculate the equilibrium properties of simple fluid.AcioiowledgmentsW e a c k n o w le d g e  th e  f in a n c ia l  s u p p o r ts  o f  th e  U n iv e rs ity  G ran ts  Commission, New Delhi.References(II J I) Weeks, U ('handler and M C Andersen J  C h e m .P h y s . 54 52,t7 (1971)(2) L Verlet and J J Weis P h \ s .  K ev A5 989 (1972)[.7) L Verlet and J J Weis M o l  P h y s 24 101.7 (1972)(41 DC Chae. F H Rcc and T Rce J  C lie m  P h y s  50 1581 (1969)(5] Y ,S Sainger, S K Sinha and Y Singh J. C h e m  P h y s  6 8  6252 (1987)(61 N F Carnahan and K E Starling J  C h e m  P h y s . 51 675 (1969) (7] T Boubhk J . C h e m . P h y s .  8 7  1751 (1987)

A Perturbation Theory of Classical Simple Fluids

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A Perturbation Theory of Classical Simple Fluids

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