- Email: [email protected]
HNC2 and PY2 fifth virial coefficients for hard spheres
Volume 21, n u m b e r 4
HNC2
AND
PHYSIOS
PY2
FIFTH
VIRIAL
LETTERS
COEFFICIENTS
1 June 1966
FOR
HARD
SPHERES
¢
L . O D E N , D. H E N D E R S O N *¢ a n d R. C H E N
Department of Physics, University of Waterloo Waterloo, Ontario, Canada Received 27 April 1966 The HNC2 and PY2 t h e o r i e s a r e used to calculate the fifth virial coefficient for a s y s t e m of hard s p h e r e s . The PY2 values a r e p a r t i c u l a r l y a c c u r a t e .
I t i s w e l l - k n o w n [1] t h a t t h e v i r i a l c o e f f i c i e n t s may be calculated either from the pressure equa-
tion N---k-T = 1 or from the compressibility
1
(r)
rdr
,
~p
(2)
In e q s . (1) a n d (2) p = N / V , u(r) i s t h e i n t e r m o l e cular potential, g(r) is the radial distribution f u n c t i o n , a n d c(r) i s t h e d i r e c t c o r r e l a t i o n f u n c tion which is related to g(r) by h(rl2) = c(rl3) + pfc(r13)h(r23) dr 3 ,
Katsura and Abe
Rowlinson
Present calculations
-1.582
-1.585
-1.5838
+1.328
+1.331
+1.3298
+1.138
+1.141
+1.1394
-0.912
-0.917
-0.9148
(1)
equation
-k-~ (~----~T = 1 - p f c ( r ) dr .
Table 1 I r r e d u c i b l e c l u s t e r integrals *
(3) -1.1877
w h e r e h(r) = g(r) - 1. A n a p p r o x i m a t e t h e o r y f o r g(r) a n d c(r) m a y be obtained by postulating a second equation rel a t i n g c(r) and g(r). R e c e n t l y , V e r l e t [2] h a s p r o p o s e d t h e HNC2 e q u a t i o n :
c(r) = f ( r ) y ( r ) + y ( r ) - 1 - l n y ( r ) + @(r) ,
(4)
where
f ( r ) = exp{-/:3u(r)}- 1 ,
(5)
y ( r ) : exp{~u(r)}g(r)
(6)
,
and ~ = 1 / k T . T h e f u n c t i o n ~I,(r) i s d e t e r m i n e d b y a c o m p l i c a t e d s e t of e q u a t i o n s . H o w e v e r if @(r) i s e x p a n d e d i n p o w e r s of t h e d e n s i t y , t h e f i r s t two terms may be determined fairly easily. The result is:
-1.5032 +0.9867 * In units of b = ~71ff3. where a bar linking the molecules i and j denotes t h e f a c t o r f ( r i j ). E a c h j o i n t b e t w e e n t h e b a r s d e notes a molecule whose coordinates are integrated and each circle represents a molecule whose position remains fixed. In addition, Verlet has proposed the PY2 equation:
c(r) = f ( r ) y ( r ) + ~ ( r ) .
(8)
The function ¢(r) is also determined by a comp l i c a t e d s e t of e q u a t i o n s . If O ( r ) i s e x p a n d e d ¢ This work has been supported by grants from the U . S . D e p a r t m e n t of the I n t e r i o r , Office of Saline Water and the National R e s e a r c h Council of Canada. t ? Alfred P . S l o a n Foundation Fellow.
420
Volume 21, number 4
PHYSICS LETTERS
Table 2 Fifth virial coefficient* B5(c) Exact
B5(P)
0.1103 ~ 0.0005
HNC
0.0493
0.1447
HNC2
0.1230
0.0657
PY PY2
0.1211
0.0859
0.1074
0.1240
In units of b = l y e 3 . in p o w e r s of the density then
~(r)=~(r)+½p2{~(~}+½p312~+4~}. (9) F r o m eqs. (4) and (7) and eqs. (8) and (9) e x p r e s s i o n s m ay be found f o r the v i r i a l c o e f f i c i e n t s . Both the NHC2 and the PY2 give the f i r s t four v i r i a l c o e f f i c i e n t s e x a c t l y but y i e l d a p p r o x i m a t e and d i f f e r e n t e x p r e s s i o n s f o r the fifth v i r i a l c o efficient. T h e r e f o r e , the a c c u r a c y of the HNC2 and PY2 fifth v i r i a l c o e f f i c i e n t s is a useful t e s t of t h e s e t h e o r i e s . R u s h b r o o k e [3] has c a l c u l a t e d the HNC2 and PY2 fifth c o m p r e s s i b i l i t y v i r i a l c o e f f i c i e n t f o r h a r d s p h e r e s . H o w e v e r , no c a l c u l a ti ons have been m a d e f o r the fifth p r e s s u r e v i r i a l coefficient. In this note we r e p o r t the r e s u l t s of our c a l c ula ti on s of the HNC2 and PY2 fifth p r e s s u r e v i r i a l c o e f f i c i e n t f o r h a r d s p h e r e s . Following B a r k e r and Monaghan [4] and H e n d e r s o n and Oden [5] the n e c e s s a r y i n t e g r a l s w e r e computed by expanding the i n t e g r a n d s in t e r m s of L e g e n d r e p o l y n o m i a l s . In table 1 n u m e r i c a l v a l u e s of the n e c e s s a r y i n t e g r a l s a r e l is t e d . A b r o k e n line between m o l e c u l e s i and j denotes the f a c t o r F o r c o m p a r i s o n , the s i m i l a r i n t e g r a l s which involve only solid l i n e s a r e a l s o l i s t e d and c o m p a r e d with the c a l c u l a t i o n s of K a t s u r a and
rijf'(rij).
1 June 1966
Abe [6] and Rowlinson [7]. The o t h er i n t e g r a l s which contribute to the fifth v i r i a l c o e f f i c i e n t a r e tabulated by Rushbrooke and Hutchinson [8]. In table 2 the r e s u l t s of our c a l c u l a t i o n s of the hard s p h e r e fifth v i r i a l c o e f f i c i e n t s a r e c o m p a r e d with the e x a c t value of the fifth v i r i a l c o ef f i ci en t [6, 7,9]. F o r c o m p a r i s o n , the r e s u l t s [1] of the HNC and PY equations a r e a l s o i n cluded. The HNC and PY equations a r e obtained f r o m eqs. (4) and (8), r e s p e c t i v e l y , by putting ~I,(r) and ~(r) equal to z e r o . The PY2 r e s u l t s a r e c o n s i d e r a b l y m o r e a c c u r a t e than t h o se of the o t h er t h e o r i e s . We have a l s o c a l c u l a t e d the HNC2 and PY2 fifth v i r i a l c o e f f i c i e n t s f o r the 6:12 potential and found t h e s e fifth v i r i a l c o e f f i c i e n t s to be v e r y a c c u r a t e f o r this potential. T h e s e r e s u l t s t o g e t h e r with the d e t a i l s of our c a l c u l a t i o n s will be published shortly. The au t h o r s a r e g r a t e f u l to Dr. J. A. B a r k e r and to P r o f e s s o r G. S. Rushbrooke f o r t h e i r v a l uable c o r r e s p o n d e n c e .
R~f87"~CeS 1. J.S. Rowlinson, Rep. Progr. Phys. 28 (1965) 169. 2. L.Verlet, Physica 30 (1964) 95. 3. G.S. Rushbrooke in: Statistical mechanics of equilibrium and non-equilibrium, ed.: J.Meixner (NorthHolland Publ. Comp., Amsterdam, 1965) p. 222. 4. J.A. Barker and J. J. Monaghan, J. Chem. Phys. 36 (1962) 2564. 5. D.Henderson and L.Oden, Mol. Phys. (in press). 6. S.Katsura and Y.Abe, J. Chem. Phys. 39 (1963) 2068. 7. J.S. Rowlinson, Proc. Roy. Soc. (London) A279 (1964) 147. 8. G.S.Rushbrooke and P.Hutchinson, Physica 27 (1961) 647. 9. F.H. Ree and W. G. Hoover, J. Chem. Phys. 40 (1964) 939.
421
HNC2
AND
PHYSIOS
PY2
FIFTH
VIRIAL
LETTERS
COEFFICIENTS
1 June 1966
FOR
HARD
SPHERES
¢
L . O D E N , D. H E N D E R S O N *¢ a n d R. C H E N
Department of Physics, University of Waterloo Waterloo, Ontario, Canada Received 27 April 1966 The HNC2 and PY2 t h e o r i e s a r e used to calculate the fifth virial coefficient for a s y s t e m of hard s p h e r e s . The PY2 values a r e p a r t i c u l a r l y a c c u r a t e .
I t i s w e l l - k n o w n [1] t h a t t h e v i r i a l c o e f f i c i e n t s may be calculated either from the pressure equa-
tion N---k-T = 1 or from the compressibility
1
(r)
rdr
,
~p
(2)
In e q s . (1) a n d (2) p = N / V , u(r) i s t h e i n t e r m o l e cular potential, g(r) is the radial distribution f u n c t i o n , a n d c(r) i s t h e d i r e c t c o r r e l a t i o n f u n c tion which is related to g(r) by h(rl2) = c(rl3) + pfc(r13)h(r23) dr 3 ,
Katsura and Abe
Rowlinson
Present calculations
-1.582
-1.585
-1.5838
+1.328
+1.331
+1.3298
+1.138
+1.141
+1.1394
-0.912
-0.917
-0.9148
(1)
equation
-k-~ (~----~T = 1 - p f c ( r ) dr .
Table 1 I r r e d u c i b l e c l u s t e r integrals *
(3) -1.1877
w h e r e h(r) = g(r) - 1. A n a p p r o x i m a t e t h e o r y f o r g(r) a n d c(r) m a y be obtained by postulating a second equation rel a t i n g c(r) and g(r). R e c e n t l y , V e r l e t [2] h a s p r o p o s e d t h e HNC2 e q u a t i o n :
c(r) = f ( r ) y ( r ) + y ( r ) - 1 - l n y ( r ) + @(r) ,
(4)
where
f ( r ) = exp{-/:3u(r)}- 1 ,
(5)
y ( r ) : exp{~u(r)}g(r)
(6)
,
and ~ = 1 / k T . T h e f u n c t i o n ~I,(r) i s d e t e r m i n e d b y a c o m p l i c a t e d s e t of e q u a t i o n s . H o w e v e r if @(r) i s e x p a n d e d i n p o w e r s of t h e d e n s i t y , t h e f i r s t two terms may be determined fairly easily. The result is:
-1.5032 +0.9867 * In units of b = ~71ff3. where a bar linking the molecules i and j denotes t h e f a c t o r f ( r i j ). E a c h j o i n t b e t w e e n t h e b a r s d e notes a molecule whose coordinates are integrated and each circle represents a molecule whose position remains fixed. In addition, Verlet has proposed the PY2 equation:
c(r) = f ( r ) y ( r ) + ~ ( r ) .
(8)
The function ¢(r) is also determined by a comp l i c a t e d s e t of e q u a t i o n s . If O ( r ) i s e x p a n d e d ¢ This work has been supported by grants from the U . S . D e p a r t m e n t of the I n t e r i o r , Office of Saline Water and the National R e s e a r c h Council of Canada. t ? Alfred P . S l o a n Foundation Fellow.
420
Volume 21, number 4
PHYSICS LETTERS
Table 2 Fifth virial coefficient* B5(c) Exact
B5(P)
0.1103 ~ 0.0005
HNC
0.0493
0.1447
HNC2
0.1230
0.0657
PY PY2
0.1211
0.0859
0.1074
0.1240
In units of b = l y e 3 . in p o w e r s of the density then
~(r)=~(r)+½p2{~(~}+½p312~+4~}. (9) F r o m eqs. (4) and (7) and eqs. (8) and (9) e x p r e s s i o n s m ay be found f o r the v i r i a l c o e f f i c i e n t s . Both the NHC2 and the PY2 give the f i r s t four v i r i a l c o e f f i c i e n t s e x a c t l y but y i e l d a p p r o x i m a t e and d i f f e r e n t e x p r e s s i o n s f o r the fifth v i r i a l c o efficient. T h e r e f o r e , the a c c u r a c y of the HNC2 and PY2 fifth v i r i a l c o e f f i c i e n t s is a useful t e s t of t h e s e t h e o r i e s . R u s h b r o o k e [3] has c a l c u l a t e d the HNC2 and PY2 fifth c o m p r e s s i b i l i t y v i r i a l c o e f f i c i e n t f o r h a r d s p h e r e s . H o w e v e r , no c a l c u l a ti ons have been m a d e f o r the fifth p r e s s u r e v i r i a l coefficient. In this note we r e p o r t the r e s u l t s of our c a l c ula ti on s of the HNC2 and PY2 fifth p r e s s u r e v i r i a l c o e f f i c i e n t f o r h a r d s p h e r e s . Following B a r k e r and Monaghan [4] and H e n d e r s o n and Oden [5] the n e c e s s a r y i n t e g r a l s w e r e computed by expanding the i n t e g r a n d s in t e r m s of L e g e n d r e p o l y n o m i a l s . In table 1 n u m e r i c a l v a l u e s of the n e c e s s a r y i n t e g r a l s a r e l is t e d . A b r o k e n line between m o l e c u l e s i and j denotes the f a c t o r F o r c o m p a r i s o n , the s i m i l a r i n t e g r a l s which involve only solid l i n e s a r e a l s o l i s t e d and c o m p a r e d with the c a l c u l a t i o n s of K a t s u r a and
rijf'(rij).
1 June 1966
Abe [6] and Rowlinson [7]. The o t h er i n t e g r a l s which contribute to the fifth v i r i a l c o e f f i c i e n t a r e tabulated by Rushbrooke and Hutchinson [8]. In table 2 the r e s u l t s of our c a l c u l a t i o n s of the hard s p h e r e fifth v i r i a l c o e f f i c i e n t s a r e c o m p a r e d with the e x a c t value of the fifth v i r i a l c o ef f i ci en t [6, 7,9]. F o r c o m p a r i s o n , the r e s u l t s [1] of the HNC and PY equations a r e a l s o i n cluded. The HNC and PY equations a r e obtained f r o m eqs. (4) and (8), r e s p e c t i v e l y , by putting ~I,(r) and ~(r) equal to z e r o . The PY2 r e s u l t s a r e c o n s i d e r a b l y m o r e a c c u r a t e than t h o se of the o t h er t h e o r i e s . We have a l s o c a l c u l a t e d the HNC2 and PY2 fifth v i r i a l c o e f f i c i e n t s f o r the 6:12 potential and found t h e s e fifth v i r i a l c o e f f i c i e n t s to be v e r y a c c u r a t e f o r this potential. T h e s e r e s u l t s t o g e t h e r with the d e t a i l s of our c a l c u l a t i o n s will be published shortly. The au t h o r s a r e g r a t e f u l to Dr. J. A. B a r k e r and to P r o f e s s o r G. S. Rushbrooke f o r t h e i r v a l uable c o r r e s p o n d e n c e .
R~f87"~CeS 1. J.S. Rowlinson, Rep. Progr. Phys. 28 (1965) 169. 2. L.Verlet, Physica 30 (1964) 95. 3. G.S. Rushbrooke in: Statistical mechanics of equilibrium and non-equilibrium, ed.: J.Meixner (NorthHolland Publ. Comp., Amsterdam, 1965) p. 222. 4. J.A. Barker and J. J. Monaghan, J. Chem. Phys. 36 (1962) 2564. 5. D.Henderson and L.Oden, Mol. Phys. (in press). 6. S.Katsura and Y.Abe, J. Chem. Phys. 39 (1963) 2068. 7. J.S. Rowlinson, Proc. Roy. Soc. (London) A279 (1964) 147. 8. G.S.Rushbrooke and P.Hutchinson, Physica 27 (1961) 647. 9. F.H. Ree and W. G. Hoover, J. Chem. Phys. 40 (1964) 939.
421