A Study of Integral Equations for Computing Radial Distribution Functions

Transcription

1 Western Michigan University ScholarWorks at WMU Master's Theses Graduate College A Study of Integral Equations for Computing Radial Distribution Functions Zainuriah Hassan Western Michigan University Follow this and additional works at: Part of the Fluid Dynamics Commons Recommended Citation Hassan, Zainuriah, "A Study of Integral Equations for Computing Radial Distribution Functions" (1985). Master's Theses This Masters Thesis-Open Access is brought to you for free and open access by the Graduate College at ScholarWorks at WMU. It has been accepted for inclusion in Master's Theses by an authorized administrator of ScholarWorks at WMU. For more information, please contact maira.bundza@wmich.edu.

2 A STUDY OF INTEGRAL EQUATIONS FOR COMPUTING RADIAL DISTRIBUTION FUNCTIONS by Z ain u riah Hassan A T hesis Subm itted to the F a c u lty o f The Graduate C ollege in p a r t i a l f u lf illm e n t of th e req u ire m e n ts fo r th e Degree of M aster of A rts D epartm ent of P hysics W estern Michigan U n iv e rsity Kalamazoo, Michigan August 1985

3 A STUDY OF INTEGRAL EQUATIONS FOR COMPUTING RADIAL DISTRIBUTION FUNCTIONS Z ain u riah Hassan, M.A. Western Michigan U n iv e rs ity, 1985 I n te g r a l equation T (a two param eter eq u atio n of the O rn stein -Z ern ik e form) i s stu d ied f o r computing th e r a d i a l d i s t r i b u t i o n fu n c tio n s o f sim ple c l a s s i c a l f lu id s in te r a c tin g p airw ise according to the Lennard-Jones 6-12 p o te n tia l fu n c tio n. N um erical c a lc u la tio n s a re done on a computer fo r th e system in th e gas and liq u id p h a se s. At a high tem peratu r e, equation T r e s u l t s are found to ag ree very w ell w ith equ atio n C when the param eters are chosen in a s im ila r way. At a tem perature s l i g h t l y above th e c r i t i c a l tem p eratu re, r e s u l t s from equ atio n T are found to g iv e good agreem ent w ith r e s u l t s from o th e r so u rces over a wide d e n s ity ran g e. In equation T, by v ary in g two of th e p aram eters, the im portan ce o f th e v a rio u s term s in the power s e r ie s expansion can be d e te c te d. A new one param eter i n t e g r a l eq u atio n is proposed, however more s tu d ie s should be done on th i s equatio n b e fo re i t s u s e fu ln e s s can be acknowledged.

4 ACKNOWLEDGEMENTS I am very g r a te f u l to Dr. David D. C arley fo r h is guidance, advice and p a tie n c e in every a sp e c t of t h i s p r o je c t. A lso, I wish to thank Dr. Soga and Dr. O ppliger f o r expending some of t h e i r busy tim e rev iew in g t h i s t h e s i s. I would also l i k e to thank B etty Hawks fo r bein g very h e lp fu l d u rin g my work h e re. Z ain u riah Hassan

5 INFORMATION TO USERS This reproduction was made from a copy o f a docum ent sent to us for microfilming. While the most advanced technology has been used to photograph and reproduce this docum ent, the quality o f the reproduction is heavily dependent upon the quality o f the material submitted. The following explanation o f techniques is provided to help clarify markings or notations which may appear on this reproduction. 1.T lie sign or target for pages apparently lacking from the document photographed is Missing Page(s). If it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting through an image and duplicating adjacent pages to assure com plete continuity. 2. When an image on the film is obliterated with a round black mark, it is an indication o f either blurred copy because o f m ovem ent during exposure, duplicate copy, or copyrighted materials that should not have been filmed. For blurred pages, a good image o f the page can be found in the adjacent frame. If copyrighted materials were deleted, a target note will appear listing the pages in the adjacent frame. 3. When a map, drawing or chart, etc., is part o f the material being photographed, a definite method o f sectioning the material has been followed. It is customary to begin filming at the upper left hand com er o f a large sheet and to continue from left to right in equal sections with small overlaps. If necessary, sectioning is continued again beginning below the first row and continuing on until com plete. 4. For illustrations that cannot be satisfactorily reproduced by xerographic means, photographic prints can be purchased at additional cost and inserted into your xerographic copy. These prints are available upon request from the Dissertations Customer Services Department. 5. Some pages in any docum ent may have indistinct print. In all cases the best available copy has been film ed. University Microfilms International 300 N. Zeeb Road Ann Arbor, Ml 48106

6

7 H a ssan, Z a in u ria h A STUDY OF INTEGRAL EQUATIONS FOR COMPUTING RADIAL DISTRIBUTION FUNCTIONS Western Michigan University M.A University Microfilms International 300 N. Zeeb Road, Ann Arbor, Ml 48106

8

9 PLEASE NOTE: In all cases this material has been filmed in the best possible w ay from the available copy. Problems encountered with this docum ent have been identified here with a check mark V. 1. Glossy photographs or pages 2. Colored illustrations, paper or print 3. Photographs with dark b ackgro u n d 4. Illustrations are poor copy 5. Pages with black marks, not original copy 6. Print shows through as there is text on both sides of page 7. Indistinct, broken o r small print on several pages 8. Print exceeds m argin requirem ents 9. Tightly bound copy with print lost in spine 10. Computer printout pages with indistinct p rint 11. Page(s) author. lacking when material received, and not available from school or 12. Page(s) seem to be missing in numbering only as text follows. 13. Two pages num bered. Text follows. 14. Curling and wrinkled pages 15. Dissertation contains pages with print at a slant, filmed as received 16. Other University Microfilms International

10

11 TABLE OF CONTENTS ACKNOWLEDGEMENTS... i i LIST CF TABLES..... v LIST OF FIGURES... v ii CHAPTER I. INTRODUCTION I I. THE THEORY OF RADIAL DISTRIBUTION FUNCTIONS... 6 D e fin itio n P ressu re and I n te r n a l Energy from the R adial D is tr ib u tio n F unction Methods of Computing g Monte Carlo... 9 M olecular Dynamics I n te g r a l E quations X -ray and Neutron S c a tte rin g Experim ents. 11 D ensity Expansions III. INTEGRAL EQUATIONS I n te g r a l E quations as P a r t i a l Summations of D ensity Expansions IV. COMPUTATIONAL METHOD C a lcu la tio n of P and U from g D eterm ination of th e I n te g r a l Equation. T P aram eters, bg and b^ V. RESULTS AND COMPARISONS R esu lts in th e Gas Regions R esu lts in th e Liquid Regions i i i

12 Table o f C ontents-c ontinued CHAPTER R adial D is tr ib u tio n F unction R esu lts Dependence o f the P aram eters on T V I. A NEW PARAMETRIC INTEGRAL EQUATION D eriv atio n o f Equation H VII. CONCLUSIONS REFERENCES APPENDICES A. EQUATION T RESULTS B. COMPUTER PROGRAMS BIBLIOGRAPHY i v

13 LIST OF TABLES 1. Comparison o f Equation T to Equation C R e su lts of V i r i a l Expansion a t T = R e la tio n of P and U to th e Param eters of Equation T a t T= C o e ffic ie n ts fo r L east Squares Equation a t T= P re ssu re R e su lts th a t were used fo r Comparison a t T = I n te r n a l E nergyr esults th a t were used f o r Comparison a t T = 1.6 ^ Combination o f Param eters b2 and b which give th e " c o r r e c t v alu e o f P a t T = Equation T R e s u lts u sin g B est F i t Param eters a t T = In te rp o la te d R e su lts a t T = I n te r p o la tio n C o e ffic ie n ts fo r the P re ssu re and I n te r n a l Energy a t Low D e n s itie s (n<.10) fo r T = I n te r p o la tio n C o e ffic ie n ts f o r th e P re ssu re and I n te r n a l Energy a t High D e n s itie s (n >.10) fo r T = R esu lts to E quation T (b j =0) a t T= In te rp o la te d R e su lts fo r E quation T (b7=0) a t T= I n te r p o la tio n C o e ffic ie n ts fo r the P re ssu re and I n te r n a l Energy a t Low D e n s itie s (n <.1 0 ) fo r Equation T (b3=0) a t T = I n te r p o la tio n C o e ffic ie n ts fo r the P re ssu re and I n te r n a l Energy a t High D e n s itie s (n>.10) foe Equation T (b3=0) a t T = v

14 to- 16. R elatio n o f P ^nd U to th e P aram eters of E quation I a t I =1.2, 1.0 and C o e ffic ie n ts f o r L east Squares Equation a t T= 1.2, 1.0 and P ressu re and I n te r n a l Energy R e s u lts th a t were used f o r Comparison a t T =1.2, 1.0 and Combination o f P aram eters b? and b, which give the " c o rre c t" v a lu e of P ax T= 1?2, 1.0 and Equation T R e s u lts u sin g "be t" Combination of P aram eters b^ and b^ a t T =. 1.i Equation T R e s u lts u sin g "begt" Combination of P aram eters b 2 and b j a t T = R adial D is tr ib u tio n F unction R e s u lts fo r n =0.7 and 0.75 a t T = 1.6 and 1.0 r e s p e c tiv e ly Equation T R e s u lts v i

15 LIST OF FIGURES 1. Shape of th e Leonard-Jones P o t e n t i a l R ad ial D is tr ib u tio n Function as a Function of P a r ti c l e S ep aratio n r f o r a F lu id R ad ia l D is tr ib u tio n Function as a Function of P a r ti c l e S ep aratio n r fo r an Id e a l Gas P aram eters b^ v e rsu s b2 T = P v ersu s n a t T = U v ersu s n a t T = P v ersu s n a t T= 1.6 (b^=0) U v e rsu s n a t T= 1.6 (b^=0) The phase Diagram f o r a Lenna^d-Jones F lu id i s shown fo r the n- T plane P aram eters b j v e rsu s b2 a t T = P aram eters b^ v ersu s b2 a t T = P aram eters b^ v e rsu s b2 a t T = P v ersu s n a t T = U v ersu s n a t T= P v ersu s n a t T = U v e rsu s n a t T= Graph of R ad ia l D is trib u tio n F unction as a Function of P a r t i c l e S eparation x fo r n= 0.7 a t T = Graph of R a d ial D is trib u tio n F unction as a Function of P a r t i c l e S eparation x fo r n= 0.75 a t T = Param eter b2 as a Function of T Param eter as a Function o f T vii

16 CHAPTER I INTRODUCTION S t a t i s t i c a l M echanics i s m ainly concerned w ith the stu d y in g o f m acroscopic system s from a m olecular (m icroscop ic ) p o in t of view. In th e l i m i t of low d e n s it i e s, a l l gases approach p e r fe c t- g a s "behavior, in o th e r words th ey obey th e 1 well-known eq u atio n of s t a t e, where P i s th e p re ssu re PV = NkT, (1) V N T i s th e volume i s th e t o t a l number o f p a r t ic le s i s th e a b so lu te tem perature R i s B oltzm ann's c o n s ta n t. However as th e d e n s ity o f a gas i s in c re a se d, th e e q u atio n of s t a t e i s no lo n g er t h a t sim ple. S e v e ra l f a c to r s have to be tak en in to c o n s id e ra tio n such as the average in te rm o le c u la r s e p a ra tio n which h as an e f f e c t on the in te rm o le c u la r p o te n tia l energy. The v i r i a l e q u atio n of s ta t e ex p resses th e d e v ia tio n s 2 from id e a l b eh av io r of gas as, P_ = n + B2(T )n 2 + B5(T) n5... (2) kt where n = N Y ' 1

17 2 I^C T), Bj (T) a re v i r i a l c o e f f ic ie n ts. In t h i s stu d y, sim ple c l a s s i c a l f lu id s in te r a c tin g acco rd in g to th e L ennard-jones 6-12 p o te n tia l fu n c tio n are c o n sid e re d. The term sim ple means th a t th e m olecules can be t r e a te d as p a r tic le s which i n t e r a c t p airw ise w ith p a ir potent i a l e n e rg ie s which depend only upon th e se p a ra tio n d is ta n c e o f p a ir s o f p a r t i c l e s. The term c la s s ic a l means th a t c la s s ic a l m echanics i s s u f f i c i e n t to d e s c rib e the tr a n s l a t i o n a l motion o f m o lecu les, no quantum mechanicsies needed.the term f lu id r e f e r s to a vapor, gas or liq u id. A liq u id has a n o n -c ry s ta l l i n e s tr u c tu r e meaning t h a t th e re are only some s h o rt range c o r r e la tio n s in the p o s itio n s of th e p a r tic le s and no long ran g e c o r r e la tio n s. The c l a s s i c a l H am iltonian (H) fo r a sim ple c la s s ic a l f lu i d can be w ritte n as th e sum of the m olecular k in e c tic energy^k) which is a fu n c tio n of th e g e n e ra liz e d momenta (p) and th e m olecular p o t e n t i a l energy (U) which i s a fu n c tio n o f th e g e n era lize d c o o rd in a te s (q), H = K (3) (4) (5) Where r^jj i s th e d is ta n c e between p a r tic le s i and j 0 i s th e p a ir p o te n tia l energy, m is th e m o lecu lar mass,

18 3 is th e x-com ponent of momentum of p a r t i c l e i. The Lennard-Jones p a ir p o te n tia l has been w idely stu d ie d, 3 The p o te n tia l can be w r itte n in th e form 0 (r) = / - N12 1 ( 6 ) where r is th e s e p a ra tio n d is ta n c e of a p a ir of p a r t ic le s, G,. are the c o n s ta n ts c h a r a c te r is tic of the p a r tic u la r types o f p a r t i c l e s. A sso ciated w ith th e Lennard-Jones p o te n tia l a re reduced tem p eratu res and d e n s itie s d efin ed by, T = kt (7) and h = ntf3. (8) The ty p ic a l shape of th e Lennard-Jones p o te n tia l i s shown in f ig u r e 1. F ig u re 1. Shape o f th e Lennard-Jones P o te n tia l. The r a d i a l d i s t r i b u t i o n fu n ctio n is r e la te d to the p r o b a b ility th a t a second p a r t i c l e w ill be found a t a given

19 4 d is ta n c e from a c e n tr a l p a r t ic le. Once a r a d i a l d is tr ib u tio n fu n c tio n is o b ta in e d, th e in te r n a l energy and p re ssu re can be e a s ily c a lc u la te d. When th ese v alu es are computed over a range o f tem p eratu res and d e n s it i e s, i t is p o s s ib le to compute a l l o th e r e q u ilib riu m thermodynamic fu n c tio n s. I n te g r a l equations can be used to c a lc u la te approxim ate r a d i a l d is tr ib u tio n functi o n s. Equation T i s a r e l a t iv e ly new in te g r a l equation conta in in g two a d ju s ta b le param eters. E quation C i s a one p a ra m eter in te g r a l eq u atio n which has been p re v io u sly stu d ie d q u ite e x te n s iv e ly. This re s e a rc h co n siste d of th e fo llo w in g s te p s : 1) The p aram etric i n t e g r a l equation T was solved n u m erically u sin g a d i g i t a l computer fo r T =10.0 and th e param eters were chosen so as to agree w ith equation C. 2) Equation T was again solved n u m e ric ally fo r the isotherm I = 1,6 a t a h ig h and an in te rm e d ia te d e n s ity. The two param eters were a d ju ste d so t h a t th e eq u atio n g iv es a r e s u l t equal to th e r e s u l t from th e o th e r so u rces th a t were used f o r comparison in t h i s stu d y. The b e s t f i t p aram eters were o b ta in e d, then th e param eters were fix e d and c a lc u la tio n s were done fo r o th e r d e n s itie s along the iso th e rm. R a d ial d is tr ib u tio n fu n c tio n s were c a lc u la te d u sin g th e i n t e g r a l equation and from th e se, the p re ssu re s and i n t e r n a l e n e rg ie s were computed. 3) A computer program was used to o b tain th e in te rp o la tio n c o e f f ic ie n ts which can then be used to fin d th e v alu es of the

20 5 p re ss u re s and i n t e r n a l e n erg ies fo r o th e r d e n s itie s along th e isotherm w ith o u t th e need to go th ro u g h th e long i t e r a t i v e procedure in so lv in g th e in te g r a l eq u atio n. 4) S e ttin g th e second param eter equal to z ero, th e s in g le param eter in te g r a l eq u atio n was solved n u m erically and the param eter a d ju ste d to giv e good agreem ent a t a h ig h density and th e same procedure as b efo re was follow ed. R esu lts o b tain ed were compared w ith v i r i a l expansion r e s u l t s, extra p o la te d and in te rp o la te d V e rle t^ r e s u l t s and th e r e s u l t s c obtain ed by C arley^ u sin g an in te g r a l equation combined w ith p e rtu rb a tio n th e o ry. 5) The equation T was again solved n u m e ric ally b u t t h i s tim e th e tem p eratu res s tu d ie d were T = 1.2, 1.0 and 0.8 and t h i s re p re s e n ts th e system in th e liq u id phase (below th e c r i t i c a l te m p e ra tu re ). 6) The dependence of th e param eters on tem p eratu re were determ ined and shown on g rap h s. 7) A s in g le param eter in te g r a l equ atio n was proposed and h o p e fu lly th is e q u a tio n could give reaso n ab ly good r e s u l t s over a wide range o f tem p eratu res and d e n s itie s w hile a t th e same time..be easy to u se.

21 CHAPTER I I THE THEORY OP RADIAL DISTRIBUTION FUNCTIONS D e fin itio n g ( r a d i a l d i s t r i b u t i o n fu n c tio n ) i s th e f a c to r by which th e number d e n s ity d i f f e r s from th e average number d e n s ity and i s d efin ed by, n ( r) = n g ( r) (9 ) where ti(r) = AN (average number of p a r tic le s AV p er u n it volume a t d ista n c e r from a c e n tr a l m olecule) n = N ( o v e r a ll average number d e n s ity ^ where N i s th e t o t a l number of p a r t ic le s in th e system and V i s th e volume of th e system ) n(r) i s eq u al to ti when th e re a re no fo rc e s between th e p a r t i c l e s. The ty p ic a l shape of th e r a d i a l d i s t r i b u t i o n fu n c tio n i s shown in fig u r e 2.

22 7 F ig u re 2. R a d ial D is tr ib u tio n F unction as a Function of P a r ti c l e S ep aratio n r fo r a F lu id. The shape o f g i s dependent upon th e values of ^ ( p o t e n t i a l ), T (tem p eratu re) and n(density). An ex pression for g in term s o f U(q) is ^ g = z - 1 V2 where Z rj / /?U d r. d rn ( 10> e7^ d r 1 d rn /? = (kt)' The in te g ra tio n i s over th e p o s itio n co o rd in ates of th e p a r t i c l e s. For an id e a l gas U = 0, th e re fo re g = 1. F igure 3 shows g as a fu n c tio n o f r f o r an id e a l g as. g(r) 1.0 F ig u re 3. R ad ial D is tr ib u tio n Function as a Function o f P a r ti c l e S ep aratio n r fo r an Id e a l Go3.

23 8 With th e ex cep tio n o f the case fo r th e id e a l gas, the e x p ressio n f o r g i s v ery d i f f i c u l t to e v a lu a te. However, th e re a re many methods and approxim ations f o r computing g and th e se v a rio u s methods w ill be d isc u sse d l a t e r in th is c h a p te r. P re ss u re and I n te r n a l Energy from the R a d ia l D is tr ib u tio n F unction For a sim ple c l a s s i c a l f lu id, th e e x p re ssio n fo r p re ss u re and i n t e r n a l energy can be w ritte n in term s o f g as follow s ( 11) ( 12) U = 3 NkT, 2 (13) and PV = NkT ( 1 ) T h erefo re, once g i s known f o r a sim ple c l a s s i c a l f l u i d, the p re ss u re s and in te r n a l e n erg ie s can be o b tain e d. When th ese v a lu e s a re computed over a range of tem p eratu res and d e n s itie s, i t i s p o s sib le to compute a l l o th e r e q u i l i brium thermodynamic fu n ctio n s. Thus g c o n ta in s a complete thermodynamic d e s c r ip tio n of th e system fo r sim ple c l a s s i c a l f l u i d s.

24 9 Methods of Computing g 8 Monte C arlo In t h i s method, p e rio d ic boundaries are employed fo r a lim ite d number o f p a r t i c l e s (app ro x im ately 1000). The p a r t i c l e s are moved acco rd in g to p r o b a b ility r u le s. I n i t i a l p o s itio n s a re given to the d is t r i b u t i o n of m olecules in a cu b ic c e l l, then one o f th e m olecules i s randomly chosen and giv en a new p o s itio n. I f t h i s move r e s u l t s in a d ec re ase in th e c o n fig u ra tio n a l energy, then th is move is allow ed. I f th e re is an in c re a s e in s te a d of a d e c re ase in the energy, th e move i s allow ed w ith a p ro b a b ility o f exp (-A u/(kt) ) where Au is the change in p o te n tia l energy betw een the two c o n fig u ra tio n s. A random number from 0 to 1 i s s e le c te d and compared to th e p r o b a b ility. I f th e p r o b a b ility i s g r e a te r than the random number, th e p a r t i c l e i s moved and i f i t l e s s, th e movement i s n o t allow ed. A fte r th e se p ro ce sses a re re p e ate d s e v e ra l tim es, th e thermodynamic p ro p e rtie s of th e system can be d eterm in ed. T his method to g e th e r w ith m olecular dynamics is very d i r e c t, th e re fo re th e se r e s u l t s are o fte n tak en as "exact" r e s u l t s and then used to t e s t o th e r le s s d i r e c t methods such as in te g r a l eq u atio n and p e rtu rb a tio n m ethods.

25 10 Q M olecular Dynamics^ This method i s o fte n considered as th e most d i r e c t method o f computing g. The m otion of th e p a r t ic le s are o btain ed by s o lv in g N ew ton's eq u atio n s of m otion. S im ila r to th e Monte Carlo method, p e rio d ic boundary c o n d itio n s are employed f o r a lim ite d number of p a r tic le s (ap p ro x im ately 1000). A p o s itio n i s s p e c ifie d f o r each p a r t i c l e i n a b a s ic c e l l. There a re o th e r p a r t i c l e s in th e same p o s itio n s in each of th e su rro u n d in g c e l l. As the p a r tic le in th e b a sic c e l l moves, th e same movement w ill be executed by th e o th e r p a r t i c l e s. When th e m otion causes th e p a r t ic le to leav e the b a s ic c e l l, th e o th e r p a r t i c l e from a neig h b o u rin g c e l l would be e n te rin g, th e r e fo r e the number o f p a r t i c l e s in the b a s ic c e l l rem ains c o n s ta n t. The d is ta n c e s between p a irs of p a r t ic le s are c a lc u la te d and a p p ro p ria te averages are comp u ted, th en th e r a d i a l d is t r i b u t i o n fu n c tio n s are o b ta in e d. A lthough th e m o lecu lar dynamics and Monte C arlo methods are o fte n considered a s sta n d a rd s to compare le s s d i r e c t m ethods, they use long computer tim es th ereb y making i n t e g r a l eq u atio n s a t t r a c t i v e as approxim ate methods i n o b ta in in g r a d i a l d i s t r i b u t i o n fu n c tio n s. I n te g r a l E quations There a re many i n t e g r a l eq u atio n s which can be used to compute g. Three d i f f e r e n t in te g r a l eq u atio n s a re : 10 a) PY equatio n (P ercu s-y e v ic k ). R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.

26 11 11 b ) HNC equation (h y p ern etted ch ain ). c) p ara m etric i n t e g r a l equations These eq u atio n s w ill be d iscu ssed in d e t a i l in the next c h a p te r..12 X -ray and Neutron S c a tte r in g E xperim ents. This i s th e o n ly ex p erim en tal method fo r o b ta in in g g. The d isc u s s io n of t h i s method w ill n o t be given h ere. D e n sity Expansions The d e n s ity expansion fo r g can be expressed in terms o f diagram s or l i n e a r g rap h s. A l i n e a r graph i s a c o lle c tio n o f p o in ts w ith lin e s jo in in g c e rta in p a ir s of p o in ts. The Mayer f - f u n c tio n i s given by f (rij) f (i 3) = exp ( (rij')') " 1 There i s a correspondence between i n te g r a ls and diagram s. Examples are 3 -o- _2 V j J f(13)f(32)dq5 = J j, jrf(1 3 )f(3 2 )d x 3dy = j.... f(l3)f(32)f(14)f(42)dq3d^4 ' / f(12)f(24)f(l4)f(34)f(l3)dq1dq2dq3dq4 Note th a t th e re i s a Mayer f-fu n c tio n fo r each l i n e. The in te g r a tio n i s over th e co o rd in ates o f th e open c ir c le s.

27 12 The d e n s ity expansion f o r the r a d i a l d i s t r i b u t i o n fu n ctio n i s given b y ^, ge 1 + n_ 1 j? 2 + n^ 1 1 % i S 1 1 J I = 1 + n o / \ + 1_ 2 (15),14 The v i r i a l expansion f o r p re ssu re i s given by, P = py_ = 1 - n NkT V n P (16)

28 CHAPTER I I I INTEGRAL EQUATIONS The n o n -p aram etric i n t e g r a l eq u atio n s were known to g ive f a i i l y good r e s u l t s a t low d e n s itie s b u t la rg e e rro rs may occur a t in te rm e d ia te and high d e n s it i e s. The "exact" methods much as Monte C arlo and m olecular dynamics which have been d isc u sse d e a r l i e r provide re a so n a b ly dependable r e s u l t s a t th o se d e n s itie s where th e i n t e g r a l eq u atio n s y ie ld la rg e e r r o r s. However i n t e g r a l eq u atio n s are favoured over the 'e x a c t' methods because o f s h o rte r computer tim e req u irem en ts and th e a b i l i t y to determ ine a p a ir p o te n tia l energy given an ex p erim en tal g. A c la s s of approxim ate i n t e g r a l e q u atio n s can be w ritte n in th e fo llo w in g form,-', SC 1 2 ) = 5 J ( g( 13) - 1) (g(32) - s(32) - 1) d3, (17) where ( i j ) = ( t r ^ ^ 1 ), ( r is a p o s itio n v ec to r) d i = d r i = dx^dy^dz^. E quation (17) may be considered as a d e f i n i t i o n o f S. Assuming a r e la tio n s h ip betw een g and S th en g iv es anapproxim ate i n t e g r a l equation f o r g. The i n t e g r a l eq u atio n s of i n t e r e s t here a r e ^. (PY) g = e (1+S), (18) 13

29 (HNC) g = e " ^ es, = e " ^ (1 + S + I S2 + 1 S3 +...), (19) 2 6 (E q u atio n C) g = (1+1 (eas - 1) > a = e ~ ^ (1 + S + 1 a s a 2 S3 +..),(2-0) 2 6 (E quation T) g = e- ^ (1 + S + b 2 S2 + b^s3), (21) where a, b2, b^ are a d ju s ta b le p aram eters. A ll th e se eq u atio n s are o f th e form: g = e ^ ( 1 + S + b2 S2 + > 3 S3 +..,.). (22) By examining th e se eq u atio n s i t can be seen th a t e q u atio n T i s e q u iv a le n t to PY when the v alu es o f the p aram eters b 2 and b^ are eq u al to zero. I f b 2= 1/2 and b^= 1/6, z e q u atio n T i s equal to HNC through the S term. I f b2= a/2 and b3= a2 / 6, eq u atio n T i s e q u iv a le n t to equatio n C through th e S3 term. T h erefo re, i f th e param eters of equation T are o chosen such th a t b2= a/2 and b^= a / 6, equation T should produce r e s u l t s which a re in very clo se agreem ent w ith eq u a tio n C r e s u l t s. The t o t a l c o r r e la tio n fu n c tio n is d efin ed by h ( r) = g ( r) - 1 (23) T his fu n c tio n goes to zero as g (r) goes to 1 ahd is a m easure of th e t o t a l in flu e n c e o f m olecule 1 on m olecule 2 a t a d ista n c e r. The d i r e c t c o r r e la tio n fu n c tio n may be defined as

30 15 C = g - 1 s ( 2 4) With th e se d e f in itio n?, eq u a tio n (17) becomes th e w e llknown O rn stein -Z ern ik e eq u atio n ; h(12) = C(12) + n J,C(13) h(23)d3. (25) This equ atio n shows t h a t th e t o t a l c o r re la tio n between m olecules 1 and 2 i s composed of th e d i r e c t c o r r e la tio n o f m olecule- 1 on m olecule 2 and th e i n d i r e c t c o r r e la tio n which in clu d es the d i r e c t c o r re la tio n of m olecule 1 on m olecule 5 and the t o t a l c o r re la tio n o f m olecule 2 on molecu le 3. The r e la tio n s h ip between the d ir e c t c o r r e la tio n fu n c tio n and th e r a d i a l d i s t r i b u t i o n fu n ctio n f o r th e in te g r a l eq u atio n s are : (PY) C g(1 - e ^ ) (26) (HNC) C = g In ( g e ^ ) (27) (E quation C) C = g In (age r - a + 1) (28) E q u atio n T has no sim p le r e la tio n s h ip between C and g, I n te g r a l E quation as P a r t i a l Summations of D en sity Expansions The in te g r a l e q u a tio n s such as PY and HNC eq u atio n s a re e q u iv a le n t to p a r t i a l summations of term s from the d e n s ity expansion f o r g exp \ ) ^ g exp (0 /kt) = 1 + n ( # B

31 16 + TI3 (...) +... E xact A = 1/2 B = 1/2 HNC A = 1/2 B = 0 py A = 0 B = 0 E quation C A = a /2 B = 0 E quation T A = bg B = 0 A p p ro x im a te'.in teg ral eq u atio n s of th e form of equation ( 1 7 ) should giv e th e f i r s t two term s of th e g ee x p a n sio n e x a c tly. P aram etric i n t e g r a l eq u atio n s should have a p a ram eter or param eters which f i r s t appear in th e th ir d term.

32 CHAPTER IV COMPUTATIONAL METHOD E quation T i s solved n u m e ric a lly on a d i g i t a l com puter. 17 The method of s o lu tio n is e s s e n t i a l l y th a t o f B royles and t h i s in v o lv es an i t e r a t i v e pro ced u re. The fo llo w in g d im en sio n le ss q u a n titie s are in tro d u c e d, x = r (29) <T n = nc^ ( 8 ) T = kt ( 7) P = PV_ (30) NkT U = 2U (31) 3 NkT The computer s o lu tio n s giv e g (x ) f o r a range of x = 0 to x = p o in ts are used to s p e c ify th e fu n c tio n s w ith an i n t e r v a l in x of The tru n c a tio n p o in t i s a t x = » assum ing t h a t g(x) - 1 beyond th e tru n c a tio n p o in t. N um erical s o lu tio n s are e a s ily obtained fo r th e system in th e gas re g io n s b u t i t g e ts very d i f f i c u l t in th e liq u id re g io n s. A la rg e number o f i t e r a t i o n s a re re q u ire d b e fo re a good convergence i s o b ta in e d. 17

33 18 C a lc u la tio n of P and U from g The p re ssu re and in te r n a l energy a re obtained from th e r a d i a l d i s t r i b u t i o n fu n c tio n by P = NkT - 2ttN2 f dj _ g (r) r 5 d r, (12) V 3V2"" -I d r and qo U = 3 NkT + 2rtN2 J ^ ( r ) g ( r ) r 2d r. (11) 2 V o In term s of d im en sio n le ss q u a n titie s, th e p re ssu re and i n t e r n a l energy a re given by P = PV = 1-167^n J (x 10-2x 10 )g(x)dx, ( 3 2 ) NkT T o and oo U = 2U = fin I (x~10 - x")g(x) dx. (33) 3NkT 3T jq There a re two in te r p o la tio n form ulas t h a t can be used to g et s o lu tio n s fo r o th e r d e n s itie s along th e iso th erm w ith o u t th e need to so lv e through th e long i t e r a t iv e procedure f o r g. The in te r p o la tio n form ulas fo r p re ss u re and i n t e r n a l energy a re, f o r c o n sta n t tem p eratu re, P = 1 + a^n + a 2n 2 + a^n^ (34) 1 + a^,n + a^n + a^n J and TT. + >. 2 3 U = 1 + a^n a^n + a^n (35) a^n + a^n + ^ n ^

34 19 F or low d e n s ity, th e se eq u atio n could be transform ed in to th e v i r i a l s e r i e s. p = 1 + d2n + d^n 2 + d^n^ +..., (36) and U = 1 + d 2n + cl^n 2 + d^n^.... (37) D eterm in atio n o f the I n te g r a l E quation T P aram e ters, b 2 and b^ The param eters a re u s u a lly chosen to g ive good h igh d e n s ity agreem ent, then th e param eters a re fix e d and c a l c u la tio n s a re done f o r a wide range of d e n s itie s. The low d e n s ity r e s u l t s u s u a lly a re n o t very much a ffe c te d by the v alu es o f th e p aram eters, th ereb y "good" r e s u lts are ensured a t th e s e d e n s it i e s. To determ ine the p aram eters, re p e a te d re ad ju stm e n ts and re p e a te d c a lc u la tio n s are re q u ire d. S everal s e ts o f param eters a re used to c a lc u la te P and U. Then a computer program i s used to choose th e c 's to give the b e s t l e a s t squares f i t in th e fo llo w in g e q u a tio n s: and P ~ Pq + c^b2+ c 2b j + c^b2 + c^bj + c^b2b^, (3 8 ) 2 2 U ~ ^2 + c2b^+ c^b^ ^ + c^b2b^. (39) A computer program i s then used to fin d v alu e s o f b^ when bj i s g iv en. A graph i s th en p lo te d of b^ versus b 2 to g iv e th e c o rre c t P f o r s e v e ra l d e n s itie s. The range where

35 20 th e curves c ro ss on th e graph of b j v ersu s b 2 determ ine th e b e s t choice of param eters to be used. I n t h i s stu d y, only the r e s u l t s f o r p re ss u re a re used i n d eterm in in g th e b e s t choice of param eters because th e r e s u l t s fo r p re ssu re are more s e n s itiv e to th e changes in the p aram eters.

36 CHAPTER V RESULTS AND COMPARISONS Computer s o lu tio n s to th e in te g r a l equation T were o b tain ed f o r f iv e d i f f e r e n t tem p eratu res. At each temper a tu r e T, r a d i a l d i s t r i b u t i o n fu n c tio n r e s u l t s were obtain ed and from i t P and U were d eterm in ed. R e su lts in the Gas Regions The f i r s t th in g th a t was done was to choose th e param eters 18 o f equation T to g iv e agreem ent w ith equation C r e s u l t s a t T ss This corresponds to a study of a high tem peratu re Lennard-Jones g as. The r e s u l t s o f t h is a re given in Table 1. T able 1 Comparison o f E quation T to E quation G b 2 bj a n T eg.t eg.c

37 22 b 2 b3 a n # T eg.t P eg.c b2 b3 U a n T eg.t eg.c T h erefo re, in re g io n s of tem peratures and d e n s itie s where eq u atio n C works w e ll, we can a lso expect equation T to work w ell provided we choose th e param eters as bg= a / 2 p and t>2 = a / 6. The n e x t tem p eratu re stu d ied i s T =1.6. H ere, th e r e s u l t s th a t were used f o r comparison were the e x tra p o la te d in 20 and in te rp o la te d V e rle t r e s u l t s, r e s u l t s by C arley u sin g i n t e g r a l equation combined w ith p e rtu rb a tio n th eo ry fo r the

38 23 h ig h d e n s itie s and a t low d e n s it i e s, the v i r i a l expansion 21 r e s u l t s were u sed. I t i s th e re fo re n ecessary to determ ine th e low d e n sity r e s u l t s from the v i r i a l expansion., 3 4 P =1+d2n + d^n + d^n + d^n +... (36) Tahle 2 shows th e r e s u l t s of e v a lu a tin g t h i s equation w ith term s in clu d ed up to d 2, then d^ and so on up to d^. Table 2 R e su lts of V ir ia l Expansion a t T = 1. 6 n P r e s u l t s w ith term s included d2 d 3 d4 up to d o C VJ C o e ffic ie n ts d2= d3= d+=1.740 d5= -.795

39 24 In o rd er to determ ine how the param eters in equatio n T a f f e c t th e p re ssu re and energy v a lu e s, i t is n e c e ssa ry to fin d many values r e l a t i n g the p re ssu re and in te r n a l energy to th e param eters in eq u atio n T. These r e s u l t s can he found in ta b le 3. The r e s u l t s given h ere were rounded o ff and only the 297 p o in t r e s u lts are g iv en. More com plete r e s u l t s can be found in the appendix. Table 3 R e la tio n o f P and U to th e Param eters o f E quation T a t T =1.6 b2 b3 n P U o

40 25 b ^ b j n P U The r e s u l t s in ta b le 3 were used w ith the l e a s t sq u are s program to fin d th e c o e f f ic ie n ts in equation (3 8 ). Table 4 g iv es th e v a lu e s of th e se c o e f f ic ie n ts. Table 4 C o e ffic ie n ts f o r L east Squares E quation a t T = 1.6 n C1 c 2 P c 4, ' Table 5 and ta b le 6 show the r e s u l t s th a t were used f o r com parison in t h i s stu d y.

41 26 T able 5 P re ssu re R e su lts th a t were used fo r Comparison a t T =1.6 Source n V i r i a l Expansion R e s u lts E x tra p o la te d and I n te r p o la te d V e r le t R esu lts R e s u lts by Carley u s in g I n te g r a l E quation combined w ith P e r tu r b a tio n Theory :6o

42 Source n Table 6 I n te r n a l Energy R e su lts th a t were used f o r Comparison a t T = 1,6 Source P E x tra p o late d and interp o lated V e rle t R e su lts R e su lts by C arley u sin g I n te g r a l E quation dombined w ith P e rtu rb a tio n Theory

43 28 Source n P o U sing th e r e s u l t s i n ta b le 4 along w ith th e r e s u lts in ta b le 5, equation ( 3 8 ) can be solved f o r the param eters ^ 2 by choosing v ario u s v alu es o f bg and computing th e co rresp o n d in g v alu es fo r b j. The r e s u l t s o f th is can

44 2y b e fo u n d in t a b l e 7. Table 7 Combination of Param eters b 2 fnd bv which give th e " c o rre c t" value of P a t T =1.6 n b ;

45 50 n tig b ^

46 31 # b2 b Graphs o f b^ v ersu s b 2 were drawn f o r n = 0. 7 and n = 0. 9 and th e p o in t ( , ) where th ey crossed g iv es the b e s t f i t param eters a t T =1.60 (see fig u re 4 ) b n = F ig u re 4. P aram eters b^ v ersu s b2 a t T = 1.6.

47 52 Using th ese p aram e ters, th e r a d i a l d is tr ib u tio n fu n c tio n s were r e c a lc u la te d fo r seven d e n s itie s along the iso th erm and from i t, p re ssu re s and in te r n a l en erg ies were o b ta in e d. Table 8 g iv e s th ese re s u lts - A s b e fo re, the r e s u l t s were rounded o ff and only the p o in t r e s u l t s are given. Kore complete r e s u l t s can be found in th e appendix. Table 8 E quation T R e su lts u sin g B est P i t Param eters a t T =1.6 n P U The r e s u l t s in ta b le 8 were used along w ith the in te r p o la tio n form ulas (equations 3 4 and 3 5 ) and the r e s u l t s can be found in ta b le 9.

48 35 T a b le 9 -ft In te rp o la te d R esu lts a t T = R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.

49 n Table 1 0 I n te rp o la tio n C o e ffic ie n ts fo r the P ressu re and I n te r n a l Energy a t Low D e n s itie s (n<c.10) fo r T = 1.i C o e ffic ie n ts U d 2 d 3 d

50 35 T a b le 11 In te rp o la tio n C o e ffic ie n ts fo r Jhe P re ssu re and I n te r n a l Energy a t High D e n s itie s (n >.10) f o r T = 1.6 C o e ffic ie n ts P U a a a a a a ]$ Using r e s u l t s in ta b le 9, graphs of P v e rsu s n and # U v e rsu s n were drawn f o r th e b e s t f i t param eters a t T = 1.6, com paring eq u atio n T r e s u l t s w ith r e s u l t s from v i r i a l expansion, V e rle t and C arley. These can be found in f ig u r e s 5 and 6. Prom the graphs, i t can be seen th a t equatio n T produces re a so n a b ly good r e s u l t s.

51 36 EQUATION T x VIRIAL EXPANSION o PERTURBATION THEORY. YERI3ST n a t T 1.6 EQUATION T 0.5 o PERTURBATION THEORY VERLET 6. U

52 Even though t h i s two param eter in te g r a l equation works w e ll, i t i s very d i f f i c u l t to u se and a l o t o f work had to be done b e fo re th e d e sire d r e s u lts are achieved. Due to t h i s, a one param eter i n t e g r a l eq u atio n is proposed, t h a t is obtain ed by s e tt i n g th e second param eter b j in eq u atio n T to be equal to zero. Then th e equation w ill be of th e form -B p P g = e ( 1+S+b2S ) (40) The param eter b 2 is ad ju ste d u n t i l the d e sire d 37 p re ss u re is reached f o r a high d e n s ity (n = 0.9 ). Then, the p aram eter was fix e d and c a lc u la tio n s were done fo r seven d e n s it i e s along the isotherm T = 1.6. The r e s u lts of t h i s can be found in Table 12. Table 12 R e su lts to Equation T (b^ = 0) a t T = 1.6 b 2 n P U o ' o R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.

53 38 As before, the re s u lts in Table 12 were used along with the in terp o latio n formulas and the re s u lts a fte r th is procedure are given in Table 13. Table 13 fc Interpolated R esults fo r Equation T (bv=0) at T = T

54 n P u i 4 O Table 14 In terp o latio n Energy at Low C oefficients^for the Pressure and D ensities (n <.1 0 ) for Equation In te rn al T(b^=0) a t T C oefficients> P U d d d

55 40 Table 15 In te rp o latio n C oefficients for the Pressure and In tern al Energy at High D ensities (n>.10) fo r Equation T(b^=0) at.t =1.6 C oefficients P U a 1 a2 a3 a b a a Using the r e s u lts in table 13, graphs of P versus n and U versus n were drawn and comparisons were made to r e s u lts from v i r i a l expansion, V erlet and Carley (see fig u res 7 and 8). R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.

56 x o EQUATION T VIRIAL EXPANSION PERTURBATION THEORY VEREET n Figure 7. P versus n at T= 1.6 (bj=0).0 o EQUATION T perturbation theory vereet n # Figure 8. U versus n at T = 1.6 (bj=0)

57 42 From these graphs, i t can be seen th at th is one parameter in te g ra l equation works quite well. However equation T with two param eters is d e fin ite ly b e tte r. Equation 2? T is also b e tter than equation C a t th is temperature. All these computations were done a t T = 1.6. This value for the reduced temperature re fe rs to the low temperature Lennard-Jones gas. R esults in the liq u id Regions Computations were then extended into the liquid regions, th at is a t tem peratures below the c r itic a l temperature. Figure 9 shows the phase diagram for a Lennard- 2-5 Jones flu id. At the liq u id - vapor region and a t the so lid - liq u id region where two phases coexist, computation were not possible. That is why th is part of the study w ill only be concentrating on the system in the liquid region. The liq u id -so lid phase lin e shown on the diagram is ju st approximate because re a l d a ta are not available. In the liquid region, numerical so lu tio n s are very d if f ic u lt to get. A large number of ite r a tio n s are required before a good convergence is obtained.

58 45 T.4 2 ii^uid 0 L I JUID VAPOR.8 Figure 9. The Phase Diagram fg r a Lennard-Jones flu id is shown fo r the n - T plane. Table 16 gives the relatio n of P and U to the X parameters of equation T a t T = 1.2, 1.0 and 0.8. At T = 1.2, computations were performed fo r n = 0.6, 0.75 and n = 0.6 re fe rs to the density.near the liquid-vapor # region, n = 0.85 near the liq u id -so lid region and n = 0.75 is in between. At T = 1.0, computations were performed fo r n =0.7, 0.75 and 0.8. At T = 0.8,computations were only done fo r n =0.82 because of the very narrow region (as can be seen on fig u re 9) where computations are possible. The procedure followed is the same as th at was done a t T =1.6.

59 - T a b le 16 R elation of P and U to the Parameters of.equation T at T = 1.2, 1.0 and 0.8 T b2 b3 n U ,27 -.> i VJI o ' i o

60 45, T b2 ti P U T

61 46 T b2 b3 n P U 1.0 i. VJl o CO Table 17 4t C oefficients fo r Least Squares Equation at T =1.2, 1.0 and 0.8 T n C1 C 2 C E E E V

62 4 7 T n C1 C2 C5 u 4 C5 o oc CM Table 18 gives tne r e s u lts th a t were used fo r comparison fo r tne system in th is region. Table 16 Pressure and In te rn al Energy Results th a t were used for Comparison at T =1.2, 1.0 and 0.8 E xtrapolated and In te rp o lated V erlet "2.12 R esults R esults by Carley using P erturbation Theory Reproduced with permission of the copyright owner. Further reproduction prohibited without permission

63 48 Source T n P U P By p lo ttin g these re s u lts on a graph, the values of a t the d e n sitie s studied can be determined. Using these values together with the le a s t squares co e ffic ie n ts, comb in atio n of parameters b 2 and b^ th a t produce the desired r e s u lts can be obtained. These can be found in tab le 19. Graphs of parameters b^ versus b were drawn fo r each T (see figures 10, 11 and 12). R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.

64 49 T a b le 19 Combination of Parameters b? and b, which give the 'c o rre ct1' Value of P at T =1.2, 1.0 and 0.8 T n b2 b^ vj6

65 5P_ > !!

66 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.

67 52 T n b 2 b.^

68 53 T TV b

69 54: T n b 2 b ^ '

70 55 T n t t> ^

71 io Reproduced with permission of the copyright ow ner Further reproduction prohibited without permission