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The ram perturbation theory and the hard dumbbell fluid
Voknne 64, number 1
CHEbUCAL
PHYSICS
15 June 1979
LETTERS
--_ THE RAM PERTURBATION
THEORY
AND THE HARb DUMBBELL
FLUID
Ivo NEZBEDA
Wlhm
R. SMITH
The RAM perturbation theom is ekxluated in the use of the hard dumbbell fluid and compared with computer simuhtion rcsuIts. The zeroth-order theory lieIds good agreement nith the centres pair correhtion functiongooo(r) even up to Iage dumbbell sphere sep;lr;ttions. thus sirowing that the basic idea of the theory is sound. Some possible reasons for the failure of 3 rehted perturbation theory are discussed.
.I. Introduction There has been much recent interest in the hard dumbbell fluid [I -71. both IS ;I model of real fluid behaviour and as an exxt fluid modct against which theories may be tested. Monte Carlo simulations have been performed by Streett and TiidesIey [I] for dumbbelk with L”. the ratio of centre-to-centre sphere distance to sphere dismeter, equal to G-2,0.4 and 0.6. Freasier et al_ [?-Sl considered L* = 0.6 an3 1.0. Theoretical approaches hzve been the approshnate numerical solution of the Percus-Yevick (PY) equation [a]_ the RIS%I equation
[S]. and forms of pertur-
batbn theory [6.71_ Both the PY and RISM approaches have serious disadvantages a! present- The PY equation is very diffcult to solve and the RISM equation &es only qualitative agreement with the exact siniulrition results [5]_ The molecukr biip-function (BF) perturbation theory was considered by Steele and Sandier [61_ Streett and Tildesky [I f found it yielded poor results for the ceIltfeS p3ir correlation functiongo&-) for L* = Oh_ The reference-system average Mayer-function (RAM) perturbation theory. due to Smith [SI and to Permm and White [91 shows considerable promise [7] _ it has 146
been shown to yield accurate results for the centres pair correlation function of dipolar and quadrupoku hard-sphere fluids [IO] and of other molecular fluids [IIIThe theory has not been tested before for a hard moIecular fluid model. It is the purpose of this note to present some prelhninxy tests of the RAhl theory in the case of the hard dumbbeli fluid.
3. Theory The RAM theory bas been described in det3iI by Smith et AL_[7$ j and we only briefly summarize the pertinent details here_ We expand the properties of the molecular fluid about an arbitr3ry spherically symmetric reference fluid with pair potential u&I-,) using an arbitrary expansion functional of tile pair poientiai rl(i2). Annulling the first-order term in the expansion of the He!mholtz free energy yields the reference fluid pair potential u&&=--kTln(elp
[-_PLt(13_)])W,rW2,
(I)
where < lWi denotes an unweighted averaging over the orientation of molecu!e i and 0 = I/kT_ Choosing the
Voiume 64, number 1
Mayerf-function as expansion functional yieids the RAM theory resirlts for the pair distribution function
(3
(3) and A_til;?r(~) is a coefficient nic expansion of Af(l2)
= exp [-Dn(l2)]
15 June 1979
CHEWCAL PHYSICS LETTERS
in the spherical harmo-
- exp [+u&,,)]
_
(4)
~~(12) = exp [~u0(12)]g,-,(12),P~ is the Legendre polynomial, Bi is the angle between the axis of molecule i and the line of molecular centres, h0 = go - 1 and subscript 0 denotes the reference system with pair potential uo@L2) defined by eq. (I)_ The RAM theory is closely related to the BF expansion [6]. It is readily shown that Steele and Sandier’s [6] BF results can be obtained from eqs. (2) and (3) by replacing the reference fluid defined by eq. (1) with a hard-sphere fluid whose diameter is determined by annulling the-first-order term in the BF free energy expansion- The RAM theory can thus be regarded as a resummation of certain terms of the BF expansion resulting in a new reference system. We may also view the zeroth-order BF theory for molecular fluids as equivalent to the following two-step procedure_ The first step uses the zeroth-order RAM theory, expanding the molecular_fluid properties about those of the reference fluid with pair potential given by eq_ (1). The second step is to use the zeroth-order BF theory for a simppiefluid [S] to expand the properties of the RAM theory reference fluid about those of a hardsphere fluid.
3. Results and discussion In order to efficiently implement the RAM theory, one must be abIe to compute the properties of the reference system with pair potential Uo(rt2)- This is a soft repulsive hard-core potential, which is shown in fig. 1 for several dumbbell elongations_ In order to remove the effects of inaccuracies in this calculation on the accuracy of the RAM theory itself, we have performed computer simulations for the pair potentials ln fig. 1 at several state points corresponding to Streett andTildesley’s [l] hard dumbbell systems- The zeroth-order RAM theory yields the result from eq (2) that the centres pair correlation functiongOOO@) is given by go(r), the radial distrrbution function of the reference fluid with pair potential defined by eq. (1). Comparison of go(r) from these simulations with the dumbbell centres pair correlation functiong,-,ou@) thus provides an indication of the potential accuracy of the RAM theory_ The usual blonte Carlo procedure of &Metropolis et
I
t 1.0
I
I
I
1.2
1.4
l-6
Fig_ 1. The RALl theory reference system potential defmed by eq. (1) as a function of distance for various dumbbell elongations L*. r is measured in units of the dumbbell sphere diameter cr.
147
VoIume 64. number
15 Juoe 1979
CHEMICAL PHYSICS LET’ERS
1
0.2 but become progressively worse as the elongation increases. JZspeciaIIy at L* = 0.6, the fist peak in go,&) is considerably overestimatedThe basic approach of using a reference system defmed by eq. (I) thus seems sound. The success of the RAM theory seems qualitatively to be a result of the “blending in” to the reference system of some of the vlguIar dependence of u(12) via eq_ (I)_ In order to predict the details of the angular dependence of the dumbbeIIg(l2) it is probable that (at least) the firstorder term y1 (I 2) will be required.
aL 1131 was used for the potentials of fig- I- We started 10s atoms from a hexagonal lattice in a cubic boxAfter stabie fluid behaviour was exhibited we calculated go(r) by averaging over 4 X 105 configurations. Our resuks are shown in figs_ 24_ Ako shown in figs. 24 are the exact hard-dumbbell results for gooo(r) and the zeroth-order BF prediction of this quantity_ It is seen that go(r) is an excellent approximation to goO&), even at the largest dumbbell elongation considered_ The BF resuits are good at L*=
/ 1.0
1.5
2.0
I FJ% 2 The Center p;rir cureistion functiongoo&) of the hard dumbbell fluid (solid Ike) and its prediction by the zeroth-order RAM (point@ and BF (dashed line) ~eories- r is measured in tmits of the sphere diameter cr. p* is the reduced density jVa3[V_ go&r) is obtained from ref. [I I_ _
o.oI
fi
1.0
t I-5
t 2.0
r
Fig_ 3_ See caption 14s
to fs_ Z~~:OM)(~) was kindly provided
by Streett and TiIdesIey (private communication).
Volume
64, number 1
CHEMICAL
PHYSICS
LETTERS
15 June 1979
Fig. 4. See caption to fii_ 3.
The zeroth-order BF theory apparently does not take sufficient account of the angular dependence of ~(12). The fact that the BF result is too high near the first peak (at L* = 0.6) indicates, at least qualitatively, that go&r) is “seein$ an effective potential which is qualitatively more like the potentials of fig. 1 than a pure hard-sphere repulsion as is used in the zerothorder BF theory. Another reason for the failure of the zeroth-order BF theory can be deduced from the description of that theory given at the end of section 2. Since it is well-known [S] that the simple fluid version of the zeroth-order BF theory gives poor results for a simple fluid whose pair potential e_xhibits soft repuision, the BF theory fails because it cannot accurately predict the properties of the fluid with pair potential u&2) ofeq- (1% We are presently examining the theoretical prediction ofg&) as well as the angular dependence ofg( 12) predicted by eq. (2), which results will be reported in due course.
References
[ 11 W.B.
Streett and D-J- Tildesley. Proc- Roy. Sot. A 348 (1976) 485. [2] B.C. Freasier, Chem. Phys. Letters 35 (1975) 280[3I B.C. Freasier, D. Jolly and R-J. Bear&, hfol. Phys. 31 (1976) 255. [4] Y-D. Chen and W-A. Steele, J. Chem. Phys. 50 (1968)
151 161 [7] [S] 191 [lo] [ll] [12]
1428.
D. Chandler, C S. Hsu and W-B. Streett, J. Chem. Phys. 66 (1977) 5231. W-A. Steele and S-1. Sandier, J. Chem. Phys. 61 (1974) 1315. W-R. Smith, L Nezbeda, T-V- Melnyk and D-D. Fitts, Faradz~y Discussions 65, to be published. W-R. Smith.Can. J. Phys. 52 (1974) 2022. J.W. Perram and L R. White, Mol. Phqs. 27 (1974) 527. W-R. Smith, WC. Madden and D.D. Fitts, Chem. Phys. Letters 36 (1975) 195. W.R. Smith, Chem. Phys. Letters 40 (1976) 313. N. hletropolis, hf-N. Rosenbluth, A-W- Rosenbluth, A-H. Teller and E. Teller, J. Chem. Phys. 21 (1953)
1087.
149
CHEbUCAL
PHYSICS
15 June 1979
LETTERS
--_ THE RAM PERTURBATION
THEORY
AND THE HARb DUMBBELL
FLUID
Ivo NEZBEDA
Wlhm
R. SMITH
The RAM perturbation theom is ekxluated in the use of the hard dumbbell fluid and compared with computer simuhtion rcsuIts. The zeroth-order theory lieIds good agreement nith the centres pair correhtion functiongooo(r) even up to Iage dumbbell sphere sep;lr;ttions. thus sirowing that the basic idea of the theory is sound. Some possible reasons for the failure of 3 rehted perturbation theory are discussed.
.I. Introduction There has been much recent interest in the hard dumbbell fluid [I -71. both IS ;I model of real fluid behaviour and as an exxt fluid modct against which theories may be tested. Monte Carlo simulations have been performed by Streett and TiidesIey [I] for dumbbelk with L”. the ratio of centre-to-centre sphere distance to sphere dismeter, equal to G-2,0.4 and 0.6. Freasier et al_ [?-Sl considered L* = 0.6 an3 1.0. Theoretical approaches hzve been the approshnate numerical solution of the Percus-Yevick (PY) equation [a]_ the RIS%I equation
[S]. and forms of pertur-
batbn theory [6.71_ Both the PY and RISM approaches have serious disadvantages a! present- The PY equation is very diffcult to solve and the RISM equation &es only qualitative agreement with the exact siniulrition results [5]_ The molecukr biip-function (BF) perturbation theory was considered by Steele and Sandier [61_ Streett and Tildesky [I f found it yielded poor results for the ceIltfeS p3ir correlation functiongo&-) for L* = Oh_ The reference-system average Mayer-function (RAM) perturbation theory. due to Smith [SI and to Permm and White [91 shows considerable promise [7] _ it has 146
been shown to yield accurate results for the centres pair correlation function of dipolar and quadrupoku hard-sphere fluids [IO] and of other molecular fluids [IIIThe theory has not been tested before for a hard moIecular fluid model. It is the purpose of this note to present some prelhninxy tests of the RAhl theory in the case of the hard dumbbeli fluid.
3. Theory The RAM theory bas been described in det3iI by Smith et AL_[7$ j and we only briefly summarize the pertinent details here_ We expand the properties of the molecular fluid about an arbitr3ry spherically symmetric reference fluid with pair potential u&I-,) using an arbitrary expansion functional of tile pair poientiai rl(i2). Annulling the first-order term in the expansion of the He!mholtz free energy yields the reference fluid pair potential u&&=--kTln(elp
[-_PLt(13_)])W,rW2,
(I)
where < lWi denotes an unweighted averaging over the orientation of molecu!e i and 0 = I/kT_ Choosing the
Voiume 64, number 1
Mayerf-function as expansion functional yieids the RAM theory resirlts for the pair distribution function
(3
(3) and A_til;?r(~) is a coefficient nic expansion of Af(l2)
= exp [-Dn(l2)]
15 June 1979
CHEWCAL PHYSICS LETTERS
in the spherical harmo-
- exp [+u&,,)]
_
(4)
~~(12) = exp [~u0(12)]g,-,(12),P~ is the Legendre polynomial, Bi is the angle between the axis of molecule i and the line of molecular centres, h0 = go - 1 and subscript 0 denotes the reference system with pair potential uo@L2) defined by eq. (I)_ The RAM theory is closely related to the BF expansion [6]. It is readily shown that Steele and Sandier’s [6] BF results can be obtained from eqs. (2) and (3) by replacing the reference fluid defined by eq. (1) with a hard-sphere fluid whose diameter is determined by annulling the-first-order term in the BF free energy expansion- The RAM theory can thus be regarded as a resummation of certain terms of the BF expansion resulting in a new reference system. We may also view the zeroth-order BF theory for molecular fluids as equivalent to the following two-step procedure_ The first step uses the zeroth-order RAM theory, expanding the molecular_fluid properties about those of the reference fluid with pair potential given by eq_ (1). The second step is to use the zeroth-order BF theory for a simppiefluid [S] to expand the properties of the RAM theory reference fluid about those of a hardsphere fluid.
3. Results and discussion In order to efficiently implement the RAM theory, one must be abIe to compute the properties of the reference system with pair potential Uo(rt2)- This is a soft repulsive hard-core potential, which is shown in fig. 1 for several dumbbell elongations_ In order to remove the effects of inaccuracies in this calculation on the accuracy of the RAM theory itself, we have performed computer simulations for the pair potentials ln fig. 1 at several state points corresponding to Streett andTildesley’s [l] hard dumbbell systems- The zeroth-order RAM theory yields the result from eq (2) that the centres pair correlation functiongOOO@) is given by go(r), the radial distrrbution function of the reference fluid with pair potential defined by eq. (1). Comparison of go(r) from these simulations with the dumbbell centres pair correlation functiong,-,ou@) thus provides an indication of the potential accuracy of the RAM theory_ The usual blonte Carlo procedure of &Metropolis et
I
t 1.0
I
I
I
1.2
1.4
l-6
Fig_ 1. The RALl theory reference system potential defmed by eq. (1) as a function of distance for various dumbbell elongations L*. r is measured in units of the dumbbell sphere diameter cr.
147
VoIume 64. number
15 Juoe 1979
CHEMICAL PHYSICS LET’ERS
1
0.2 but become progressively worse as the elongation increases. JZspeciaIIy at L* = 0.6, the fist peak in go,&) is considerably overestimatedThe basic approach of using a reference system defmed by eq. (I) thus seems sound. The success of the RAM theory seems qualitatively to be a result of the “blending in” to the reference system of some of the vlguIar dependence of u(12) via eq_ (I)_ In order to predict the details of the angular dependence of the dumbbeIIg(l2) it is probable that (at least) the firstorder term y1 (I 2) will be required.
aL 1131 was used for the potentials of fig- I- We started 10s atoms from a hexagonal lattice in a cubic boxAfter stabie fluid behaviour was exhibited we calculated go(r) by averaging over 4 X 105 configurations. Our resuks are shown in figs_ 24_ Ako shown in figs. 24 are the exact hard-dumbbell results for gooo(r) and the zeroth-order BF prediction of this quantity_ It is seen that go(r) is an excellent approximation to goO&), even at the largest dumbbell elongation considered_ The BF resuits are good at L*=
/ 1.0
1.5
2.0
I FJ% 2 The Center p;rir cureistion functiongoo&) of the hard dumbbell fluid (solid Ike) and its prediction by the zeroth-order RAM (point@ and BF (dashed line) ~eories- r is measured in tmits of the sphere diameter cr. p* is the reduced density jVa3[V_ go&r) is obtained from ref. [I I_ _
o.oI
fi
1.0
t I-5
t 2.0
r
Fig_ 3_ See caption 14s
to fs_ Z~~:OM)(~) was kindly provided
by Streett and TiIdesIey (private communication).
Volume
64, number 1
CHEMICAL
PHYSICS
LETTERS
15 June 1979
Fig. 4. See caption to fii_ 3.
The zeroth-order BF theory apparently does not take sufficient account of the angular dependence of ~(12). The fact that the BF result is too high near the first peak (at L* = 0.6) indicates, at least qualitatively, that go&r) is “seein$ an effective potential which is qualitatively more like the potentials of fig. 1 than a pure hard-sphere repulsion as is used in the zerothorder BF theory. Another reason for the failure of the zeroth-order BF theory can be deduced from the description of that theory given at the end of section 2. Since it is well-known [S] that the simple fluid version of the zeroth-order BF theory gives poor results for a simple fluid whose pair potential e_xhibits soft repuision, the BF theory fails because it cannot accurately predict the properties of the fluid with pair potential u&2) ofeq- (1% We are presently examining the theoretical prediction ofg&) as well as the angular dependence ofg( 12) predicted by eq. (2), which results will be reported in due course.
References
[ 11 W.B.
Streett and D-J- Tildesley. Proc- Roy. Sot. A 348 (1976) 485. [2] B.C. Freasier, Chem. Phys. Letters 35 (1975) 280[3I B.C. Freasier, D. Jolly and R-J. Bear&, hfol. Phys. 31 (1976) 255. [4] Y-D. Chen and W-A. Steele, J. Chem. Phys. 50 (1968)
151 161 [7] [S] 191 [lo] [ll] [12]
1428.
D. Chandler, C S. Hsu and W-B. Streett, J. Chem. Phys. 66 (1977) 5231. W-A. Steele and S-1. Sandier, J. Chem. Phys. 61 (1974) 1315. W-R. Smith, L Nezbeda, T-V- Melnyk and D-D. Fitts, Faradz~y Discussions 65, to be published. W-R. Smith.Can. J. Phys. 52 (1974) 2022. J.W. Perram and L R. White, Mol. Phqs. 27 (1974) 527. W-R. Smith, WC. Madden and D.D. Fitts, Chem. Phys. Letters 36 (1975) 195. W.R. Smith, Chem. Phys. Letters 40 (1976) 313. N. hletropolis, hf-N. Rosenbluth, A-W- Rosenbluth, A-H. Teller and E. Teller, J. Chem. Phys. 21 (1953)
1087.
149