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A perturbation expansion for the pair distribution function of fluids with non-central forces
Volume 40,kumber
2
A PERTURBATiON
CHEMlCAL
I June 2976
PkIYSICS LETTERS
EXPANSION
FOR THE PAIR DISTRE5UTION OF FLUIDS WITH NON-CENTRAL FOkS
FUTKTION
William R. SMITH Mathematics Department and Department Halifax. Nova Scotia, Canada B3H 37.5
of P!tysioiogy and Biophysics.
Dalhousie University.
Received 23 February IS76
The pair distribution function is calculated’by a perturbation technique for fluids whose molecules interact according to potentials of the form ue + na. where uo is the Lennard-Jones (12,6) potential and U, is either a dipole-dipole, quadrupole-quadrupole, or anisotropic overlap interaction. Agreementwith the computer simulationresultsfor the angle awraged pair distribution function is good up to moderate values of the anisotropy parameters in all cases.
1. Introduction There has been considerable interest recently in developing a simple and accurate computational procedure for predicting the pair distribution functions and thermody%mic properties of fluids whose molecules are not spherically symmetric [l-l 51. Among the most computationally convenient approaches are those which employ a spherically symmetric reference fluid about which the properties of the fiuid of interest are expanded. A form of perturbation expansion for nonspherical fluids suggested over twenty years ago by Cook and Rowlinson [16] was recently tested by Smith et al. [ 151 in the case of hard-core fluids. In the present note we examine the case-when the fluids have a finitely steep repulsive core.
2Theory
“&p
w*, q)
The properties of a fluid whose molecules interact
(1)
= q&2)
f ua(rlp
WI, qk
(74
where IQ(~~~) is a spherically symmetric potential and ua gives the angular dependence. Here we con-
sider the case Uo(‘) = 4E {(cJ/r)l* - (o/#Q.
(3)
Computer simulation results have recently been obtained [17] using eqs. (2) and (3), and for &la,dipoledipole, quadrupole-quadrupo!e, and anisotropic overlap interactions_ The latter are given by ,‘,D = (jI2/&)
via a pair potential u(r12. wl, q), where ~2 is theinterparticle separation and wi is the orientation of molecule i with respect to the interparticle axis, are expanded about those of a spherically symmetric reference fluid with pair potential u*(r&given by [15] u*(r12)= -liTin (exp[-J3ir(r12, ~~‘~2)l)w,,,,¶
where ( ),_. represents an unweighted averaging over the orientation of molecule i. The fluid whose molecules interact via eq. (1) is the reference fluid in a perturbation expansion which recently has been considered by a number of workers [ 12-151. Often the pair potential of the fluid of interest can be written naturally as
[-2cose
1 case 2
f sine1 $ne2cos(*I
-@I,
(4
u~=(~&r;,){1 -5(cos%,+cos%2)-15cos%1cos2e2 + 2[sinfI 1 s~&~cos(~~ -4,) Per a
-4cos01
cos02]2j,(5)
= 466 (o/~~~)’ 2(3 c0s2e I i 3 cos2e 2 - 2).
(6) 313
In-the above, r_cand Q’-aie dipole and qua@rnpole moments r&p&tively, 6 atid Q reprecent the pblaitind azimuthal &@es defined with respect td the inier- -. ._$article axis r12, and 6 is a dimensionless overlap paG&efkr. The pair distribution ftinctiong(rLZ, ol, w2) j18] of.the fluid with potential ~(‘12, (nl, wz), when expanded ;ibout that of the fluid with pair potential given by eq. (l), is given tq Srst order by [ 141
ir(
rife
g(r121w1'"2)=8*!12)[1+~~(13,)1 i:p $ s*W)t@f(W + ~V-(23j),~dr~r
(7)
where (12) denotes r12 and Af(ii) = exp {Lp[tc(v, wj, wi) - zP($)J} - 1.
@I
Because of eq. (I), the angie averaged pair distribution function is g%en to first order by “+&., (9) k(% , aa =g*mIn &is note eq. (9) is considered for potentials defined by eqs. (2)-(6).
LOO
L25
w
r Ie 3. Results and discussion The reference fluid pair potentials for parameter values representing various anisotropic strengths are shown in fig. 1. In all the calculations shown here we used p = kT/c = 0.719 and p* =Nos/V = 0.8, a state considered by Wang et al. [i7] in their compl.:ter simulation studies. It is seen that these potentials iook much like the isotropic (12,6) potential in general shape. Values of go(r) and g”(r) were calculated by a variant of a computational scheme of Andessen et al. [ 191 The only essential difference between our scheme and theirs is that the properties of the hard-core “trial system” [19] were calculated here using the parametrized expressions for the perturbation theory of Barker and Henderson [ZO]. u*(T~z) in eq. (1 j was
calculated by expanding the exponential to 24 terms in the dipole and quadrupole cases. For the anisotropic overlap case, we used u*(r) = u,,(r) LkTIn {sreherf2(@)/12a -.
=_u&j
- f&n {ekF?(j3si)/3a},
.
6 < 0, (101
where &4
), 6 > 0;
-_
Fig. 1. The reference potentials u*(r) given by eqs. (i)-(6) at conditions tvpical of those in this study. Q* = Q/(m5)“2, JL* = j.&~o~)“~, T* = 0.719, p* = 0.8.
(Y= 4 16 !(u/r)‘2/T.
(11)
erf(x) is the error function 2nd S’(x) is Dawson’s integral
F(x) = edX2 j et2 dr.
(12)
0 In figs. 2 and 3 are shown some .results of eq. (9) Lror the potentials considered. In the dipolar case (fig. 2) good agreement with the simulation results was achieved in 3U cases considered (up to reduced dipole moments ~/(eo3)u2 = l-4)_ In the quad_rupolar case (not illustrated) we found good agreement With the computer simulation results below about Q/(E$)~* = 0.8. In the anisotropic overlap case (fig. 3) agreement is reasonable at 6 = 0.25. For 6 = -0.2,
the re~sultswere similar to those at S = 0.25, except for a Iawer height at the first peak of g(r). If the potentiai uo(r) in eq. (3) is used as the reference potential in th,o perturbatiion expansion, the angle averaged pair distribution is, to first order, given by go(‘)_ As was the case for hard-core fluids fl5], the perturbation expansion using the reference :
Volume40,
girl
number
CHEMICAL PIUSI&
2.
LE+FERS
1 June.1976
potential given by eq. (1). seems more rapidly con-
:
vergent.
We are presently examining the predic&ns of the thermodynamic properties implied by our procedure, as well as the angutar dependence ~fg(&~, ol, 02), as given by eq. (7).
Acknowledgement The financial assistance of the National Research Council of Canada is gratefully acknowledged. The author thanks Dr. W. Madden and Dr. D. Fitts for sending tables of integrals for the dipolar and quadrupolar potentials, and Dr. K.E. Gubbins for sending graphs of his computer simulation results.
r/Q Fig. 2. Angle-averaged pair distribution function for the dipolar Lennard-Jofes fluid at T* = 0.719, p* = 0.8. p* = 1.1832 (solid line) and u = 0 (broken line). The open and full circles are the respective computer simulation results of Wang et al. (171.
qt r) 3.0
25
2.0
L5
LO
05
a0 05 Fig. 3. Angle-averaged pair dlstrlbution function for the Lennard-Jones lus anisotropic overlap potential considered in the text at 2 = 0.719, p* = 0.8.6 = 0.25 (solid line) and 6 = 0 (broken line). The open and full circles are the respec tive computer simulation results of Wang et al. [ 171.
References
111K.E. Gubbins and C.G. Gray, Mol. Phys. 23 (1972) 187. I21 G. Stell, J.G. Rasaiah and H. Nanng, Mol. Phys. 23 (1972) 393; 27 (1974) 1393. t31 S. Sung and D. Chandler, J. Chem. Phys. 56 (1972) 4989. I41 D. Chandler and H.C. Andersen, J. Chem. Phys. 57 (1972) 1930. [51 G.S. Rushbrooke. G. Stell and J.S. Hdye, Mol. Phys. 26 (1973) 1199. I61 W-G. Madden and D.D. Fitts, Chem. Phys. Letters 28 (1974) 427. 171 L.J. Lowden and D. Chandler, J. Chem. Phys 59 (1973) 6587;61 (1974) 5228. 181 W-A. Steele and S.I. Sandlcr, J. Chem. Phys. 61 (1974) 1315. PI S-1. Sandier. Nol- Phys. 28 (1974) 1207. [lOI MS. Ananth, K.E. Gubbins and C.G. Gray, Mol. Phys. 28 (1974) 1005. illI K.C. No and K.E. Gubbins, Chem. Phys. Letters 27 (1974) 144; J. Chem. Phys 63 (1975) 1490. [=I J-W. Perram and L.R. White, Nol. Phys. 27 (1974) 527. [1-31 L. Verlet and J-J. Weis, Mol. Phys 28 (1974) 665. 1141 W.R. Smith, Can. J. Phys. 52 (1974) 2022. [ISI W-R. Smith, W.G. Madden and D-D. Fitts, Chem. Phys. Letters 36 (1975) 195. [I61 D. Cook and J.S. Rowlinson, Proc. Roy. Sot. A219 (1953) 405. 1171 S.S. Wang, CC. Gray, P.A. Egelstaff and K.E. Gubbins, Chem. Phys Letters 21 (1973) 123. [ 181 A. Muster. Statistical thermodynamics, Vol. 1 (Springer, Berlin. 1969) ch. S.1. [ 191 H.C. Andersen, D. Chandler and J.D. Weeks, J. Chem. Phys- 56 (1972) 3812. [20] J.A. Barker and D. Henderson, Ann. Rev. Phys. Chem. 23 (1972) 439.
2
A PERTURBATiON
CHEMlCAL
I June 2976
PkIYSICS LETTERS
EXPANSION
FOR THE PAIR DISTRE5UTION OF FLUIDS WITH NON-CENTRAL FOkS
FUTKTION
William R. SMITH Mathematics Department and Department Halifax. Nova Scotia, Canada B3H 37.5
of P!tysioiogy and Biophysics.
Dalhousie University.
Received 23 February IS76
The pair distribution function is calculated’by a perturbation technique for fluids whose molecules interact according to potentials of the form ue + na. where uo is the Lennard-Jones (12,6) potential and U, is either a dipole-dipole, quadrupole-quadrupole, or anisotropic overlap interaction. Agreementwith the computer simulationresultsfor the angle awraged pair distribution function is good up to moderate values of the anisotropy parameters in all cases.
1. Introduction There has been considerable interest recently in developing a simple and accurate computational procedure for predicting the pair distribution functions and thermody%mic properties of fluids whose molecules are not spherically symmetric [l-l 51. Among the most computationally convenient approaches are those which employ a spherically symmetric reference fluid about which the properties of the fiuid of interest are expanded. A form of perturbation expansion for nonspherical fluids suggested over twenty years ago by Cook and Rowlinson [16] was recently tested by Smith et al. [ 151 in the case of hard-core fluids. In the present note we examine the case-when the fluids have a finitely steep repulsive core.
2Theory
“&p
w*, q)
The properties of a fluid whose molecules interact
(1)
= q&2)
f ua(rlp
WI, qk
(74
where IQ(~~~) is a spherically symmetric potential and ua gives the angular dependence. Here we con-
sider the case Uo(‘) = 4E {(cJ/r)l* - (o/#Q.
(3)
Computer simulation results have recently been obtained [17] using eqs. (2) and (3), and for &la,dipoledipole, quadrupole-quadrupo!e, and anisotropic overlap interactions_ The latter are given by ,‘,D = (jI2/&)
via a pair potential u(r12. wl, q), where ~2 is theinterparticle separation and wi is the orientation of molecule i with respect to the interparticle axis, are expanded about those of a spherically symmetric reference fluid with pair potential u*(r&given by [15] u*(r12)= -liTin (exp[-J3ir(r12, ~~‘~2)l)w,,,,¶
where ( ),_. represents an unweighted averaging over the orientation of molecule i. The fluid whose molecules interact via eq. (1) is the reference fluid in a perturbation expansion which recently has been considered by a number of workers [ 12-151. Often the pair potential of the fluid of interest can be written naturally as
[-2cose
1 case 2
f sine1 $ne2cos(*I
-@I,
(4
u~=(~&r;,){1 -5(cos%,+cos%2)-15cos%1cos2e2 + 2[sinfI 1 s~&~cos(~~ -4,) Per a
-4cos01
cos02]2j,(5)
= 466 (o/~~~)’ 2(3 c0s2e I i 3 cos2e 2 - 2).
(6) 313
In-the above, r_cand Q’-aie dipole and qua@rnpole moments r&p&tively, 6 atid Q reprecent the pblaitind azimuthal &@es defined with respect td the inier- -. ._$article axis r12, and 6 is a dimensionless overlap paG&efkr. The pair distribution ftinctiong(rLZ, ol, w2) j18] of.the fluid with potential ~(‘12, (nl, wz), when expanded ;ibout that of the fluid with pair potential given by eq. (l), is given tq Srst order by [ 141
ir(
rife
g(r121w1'"2)=8*!12)[1+~~(13,)1 i:p $ s*W)t@f(W + ~V-(23j),~dr~r
(7)
where (12) denotes r12 and Af(ii) = exp {Lp[tc(v, wj, wi) - zP($)J} - 1.
@I
Because of eq. (I), the angie averaged pair distribution function is g%en to first order by “+&., (9) k(% , aa =g*mIn &is note eq. (9) is considered for potentials defined by eqs. (2)-(6).
LOO
L25
w
r Ie 3. Results and discussion The reference fluid pair potentials for parameter values representing various anisotropic strengths are shown in fig. 1. In all the calculations shown here we used p = kT/c = 0.719 and p* =Nos/V = 0.8, a state considered by Wang et al. [i7] in their compl.:ter simulation studies. It is seen that these potentials iook much like the isotropic (12,6) potential in general shape. Values of go(r) and g”(r) were calculated by a variant of a computational scheme of Andessen et al. [ 191 The only essential difference between our scheme and theirs is that the properties of the hard-core “trial system” [19] were calculated here using the parametrized expressions for the perturbation theory of Barker and Henderson [ZO]. u*(T~z) in eq. (1 j was
calculated by expanding the exponential to 24 terms in the dipole and quadrupole cases. For the anisotropic overlap case, we used u*(r) = u,,(r) LkTIn {sreherf2(@)/12a -.
=_u&j
- f&n {ekF?(j3si)/3a},
.
6 < 0, (101
where &4
), 6 > 0;
-_
Fig. 1. The reference potentials u*(r) given by eqs. (i)-(6) at conditions tvpical of those in this study. Q* = Q/(m5)“2, JL* = j.&~o~)“~, T* = 0.719, p* = 0.8.
(Y= 4 16 !(u/r)‘2/T.
(11)
erf(x) is the error function 2nd S’(x) is Dawson’s integral
F(x) = edX2 j et2 dr.
(12)
0 In figs. 2 and 3 are shown some .results of eq. (9) Lror the potentials considered. In the dipolar case (fig. 2) good agreement with the simulation results was achieved in 3U cases considered (up to reduced dipole moments ~/(eo3)u2 = l-4)_ In the quad_rupolar case (not illustrated) we found good agreement With the computer simulation results below about Q/(E$)~* = 0.8. In the anisotropic overlap case (fig. 3) agreement is reasonable at 6 = 0.25. For 6 = -0.2,
the re~sultswere similar to those at S = 0.25, except for a Iawer height at the first peak of g(r). If the potentiai uo(r) in eq. (3) is used as the reference potential in th,o perturbatiion expansion, the angle averaged pair distribution is, to first order, given by go(‘)_ As was the case for hard-core fluids fl5], the perturbation expansion using the reference :
Volume40,
girl
number
CHEMICAL PIUSI&
2.
LE+FERS
1 June.1976
potential given by eq. (1). seems more rapidly con-
:
vergent.
We are presently examining the predic&ns of the thermodynamic properties implied by our procedure, as well as the angutar dependence ~fg(&~, ol, 02), as given by eq. (7).
Acknowledgement The financial assistance of the National Research Council of Canada is gratefully acknowledged. The author thanks Dr. W. Madden and Dr. D. Fitts for sending tables of integrals for the dipolar and quadrupolar potentials, and Dr. K.E. Gubbins for sending graphs of his computer simulation results.
r/Q Fig. 2. Angle-averaged pair distribution function for the dipolar Lennard-Jofes fluid at T* = 0.719, p* = 0.8. p* = 1.1832 (solid line) and u = 0 (broken line). The open and full circles are the respective computer simulation results of Wang et al. (171.
qt r) 3.0
25
2.0
L5
LO
05
a0 05 Fig. 3. Angle-averaged pair dlstrlbution function for the Lennard-Jones lus anisotropic overlap potential considered in the text at 2 = 0.719, p* = 0.8.6 = 0.25 (solid line) and 6 = 0 (broken line). The open and full circles are the respec tive computer simulation results of Wang et al. [ 171.
References
111K.E. Gubbins and C.G. Gray, Mol. Phys. 23 (1972) 187. I21 G. Stell, J.G. Rasaiah and H. Nanng, Mol. Phys. 23 (1972) 393; 27 (1974) 1393. t31 S. Sung and D. Chandler, J. Chem. Phys. 56 (1972) 4989. I41 D. Chandler and H.C. Andersen, J. Chem. Phys. 57 (1972) 1930. [51 G.S. Rushbrooke. G. Stell and J.S. Hdye, Mol. Phys. 26 (1973) 1199. I61 W-G. Madden and D.D. Fitts, Chem. Phys. Letters 28 (1974) 427. 171 L.J. Lowden and D. Chandler, J. Chem. Phys 59 (1973) 6587;61 (1974) 5228. 181 W-A. Steele and S.I. Sandlcr, J. Chem. Phys. 61 (1974) 1315. PI S-1. Sandier. Nol- Phys. 28 (1974) 1207. [lOI MS. Ananth, K.E. Gubbins and C.G. Gray, Mol. Phys. 28 (1974) 1005. illI K.C. No and K.E. Gubbins, Chem. Phys. Letters 27 (1974) 144; J. Chem. Phys 63 (1975) 1490. [=I J-W. Perram and L.R. White, Nol. Phys. 27 (1974) 527. [1-31 L. Verlet and J-J. Weis, Mol. Phys 28 (1974) 665. 1141 W.R. Smith, Can. J. Phys. 52 (1974) 2022. [ISI W-R. Smith, W.G. Madden and D-D. Fitts, Chem. Phys. Letters 36 (1975) 195. [I61 D. Cook and J.S. Rowlinson, Proc. Roy. Sot. A219 (1953) 405. 1171 S.S. Wang, CC. Gray, P.A. Egelstaff and K.E. Gubbins, Chem. Phys Letters 21 (1973) 123. [ 181 A. Muster. Statistical thermodynamics, Vol. 1 (Springer, Berlin. 1969) ch. S.1. [ 191 H.C. Andersen, D. Chandler and J.D. Weeks, J. Chem. Phys- 56 (1972) 3812. [20] J.A. Barker and D. Henderson, Ann. Rev. Phys. Chem. 23 (1972) 439.