- Email: [email protected]
Equation of state for systems of hard non-spherical molecules
Volume 20, number S
EQUATiON
1 Ju!y 1973
CHEMICAL PHYSlCS LETTERS
OF STATE FOR SYSTEMS OF HARD ~~~-SP~~~&A~
h~OLECULES
Re&ved 2 April 1973
The equation of state for systems of hard pro&e spherocyliaders has been obtained usinp a SfonteCarIo simuIation method. The results in the fluid region are compared with the predictions of s&cd particle theory, and are Found to agree quite well. The equation of state is sipniticantly different f;om that for hard sphere moIecuIes, and it is concluded that the effects of non-spherical shape must be esplicitly dealt with.
2. Method In recent years considerable progress has been made towards the de~~e~opmentof a satisfactory theory of the liquid state for systems of simple molecules [I]. Various forms of perturbatiqn theory, all based ulti; mateiy on the properties of ‘model systems of hard sphere molecules, have been shown to reproduce adequately the equilibrium properties of monoatomic fluids, and have also provided useful simp~ifyj~g concepts about the role of the repulsive and attractive intermolecular forces in determining the structure of dense fluids [a]. However, for systems of polyatomic moiecules the development has been less complete, as aresuit of the more complicated non-spherical nature of the intermolecufar interactions and the additional complexities arising from internal degrees of freedom. Some clarification of this situation might resr?Jt from the study of model systems of non-spherical molecules, analogous to hard sphere systems. We have’
accordingly made an inves~gation of the equation of state of systems of hard sphero+cylindricaI moiecu!es, hoping that these will provide some insight into the effects of molecular shape, i.e., non-spherically symmetric repulsive interactions, on the bulk properties. The virial expansion for such systems was the subject
of an earlier ~nvestigatjon [3], and in this note we report the results of Monte Carlo simulation studies.
The equation of state of systems of spherocylinders has been obtained using a Metrorjoiis type Monte Carlo simulation method 141. In using this approach the equation of state is conveniently obtained via the virial theorem. For hard molecuies interacting without any attractive forces, the virial function is zero fcr al: pair separations other than those corresponding to contact between the molecules. For hard sphere molecules the equation of state is obtained by determirting the value of the pair distribution function, dr), at the point of contact, r = 0. The c~m~ress~~~~ty then calculated from the equation
P~~~~~T- 1 = ~2~i~~~3 v) &(o)
factor is
.
For non-spherical hard molecules this procedure must be modified, since g(r) is now aIso a Function of the relative orientations of the molecules as well as their separation, and contact can occur over a range of vafucs of the separation of the centres, z-. One possible approach to this problem, which has been explored by Steele and coworkers [S].. is to txpand the orientation dependent pair distribution function as a series, using spherical harmonics. This metliod could be used in a Monte Carlo calculation, but does not see& to be ideal, since it is uncertain in any given case how many, terms must be taken in the se-
.‘,
433
‘. ”
CHEMICALPH’I’SICSLEmERS
Volume 20, number 5
ties. However,‘this method does offer great promise in the detailed interpretation of orientational correlation effects. An alternative, simpler, approach has been used in this work, which leads more directly to the equation of state, but giveS less information about orientationa: effects. ti some ways our approach resembles that used by VieiIIard-Baron [6] in his recent Monte Carlo study of systems of two-dimensional hard ellipses. For a pair of convex hard molecules we denot: the shortest. distance between the hard cores as p, and define a distribution function, G(p), analogous to the conventional pair distribution function. Thus for a system of 121molecules in volume V the number of molecular pairs whose cores lie between p and p + dp is given by N2G(p) F(p) dp/2 V. F(p) dp is the volume element available to the centres of molecuies at p. In general this will depend on the shapes of the molecules, and on the degree Qf’ orientational order in the system. For convex hard moiecules without angular correlations, Kihara [7] has shown that F(p) may be defined in terms of parameters characteristic of the molecular shape, and can be written F(p) = 2s t Sd2
+ I6n&
t 4np2 ,
where S is the surface area and R is the mean radius. We tiay also define a mean intermolecular separation, y(p), which will in general also depend on the orienta-
tional order of the system, In a Monte Carlo si.zulation it is possible to determine values of F(p) and G(p)I;Tp), as average values over a small range of p. The virial function is zero for all values ofp # 0, and on application of the virial theorem we obtain an expression for the equation of state in terms of the limiting values of G, F, 7, as p tends to zero. PV/NkT-
prolate
spherocylinder
model.
AA’ is the axis of
rotation. cells were needed in order to use potentially close packed structures for the initial configurations. The present set of results are for systems of 120 molecules, of prolate spherocylindrical shape (fig. 1) in which R* = 1. Further calculations for other values of R” are planned. Typical Monte Carlo runs were of
IO5 to 5 X lo5 steps.
3. Results The data are presented in table 1, and are shown graphicall~~in fig. 2. The results are seen to fall on two curves, with a discontinuous region in the density range near to 0.5. The lower density region appears to correspond to 3 fluid phase, and the higher densities to a solid. The tie line joining the equilibrium solid and fluid phases has not been located. The
lowest density point on the solid branch (d = 0.525) appears to be stable, and shows no tendency to melt. In comparison, the hard sphere system has a minimum stable solid density of d= 0.54 [8], and it appears that
Equation
of state
d
Table 1 for hard spherocylinders, -PVjNkT
l~=(A!~6~~ojG(o)~(o).
Values of F(p) G(p) and r(p) were obtained from the simulation for a range of values of p and the results extrapolated to p = 0 in order to obtain the compressibility factor. The-Monte Carlo calculations followed conventional procedures [4]. The orientations of the molecules were.defined in terms of two integers between I and 100, which determined the two polar angles needed for the.& axially symmetric mo!ecules. Some care was needed in the choice of initial lattice configurations, especially at the higher densities. Non-cubic basic 434 . .
Fig. 1. The
I July 1973
,.
:
-,
0.1
0.64
0.2
1.73 3.69 7.75
0.5 0.4 0.45 0.475 0.5 ,0.525
0.55 0.5 75 0.6 0.65 0.7
10.78
12.55 14.58 10.53 11.50
12.28 14.10 18.04
27.76
R* = 1 -- 1
CHEhfICAL PHYSICS LETTERS
Volume 20, number 5
1 July 1973
where 7 = &T/v and d = NV/V: It has been shown that the virkl coefficients derived from this model are in good agreement with direct!y calculated values [3], and it is seen in fig. 2 that the equation of state is in good agreement with the simulation data in the fluid region. (For the spherocylinders studied here, 7 = 3.6.) It should be no ted
0 24 1
that despite the relativaly small deviation from spher-’ ical shape shown by these molecules, the equation of state differs significantly from the hard sphere equation. At d = 0.45, the hard sphere pressure is about
.I
-2
.4
.3
.5
.6
20% smaller than that for the spherocylinders. Furthermore, it was pointed out by Gibbons that the properties of convex non-spherical molecules cannot be adequately reproduced using an effective spherical model. Any successful treatment of non-spherical molecules must include explicitly the effects of nonspherica! repulsive forces.
.7
d Fig. 2. Equation
of state for hard spherocylinders,
R* =
1. d
=
Nv/V. X fluid phase; o solid phase; - - - scaled particle the-
ory.
the freezing density for our spherocylinder molecules is somewhat lower than that for hard spheres. This result seems intuitively reasonable, and is also consistent with the findings of Vieillard-Baron in two dimensions. A de tailed investigation of this density region, with careful study of orientational correlations is in progress. Preliminary results show a pronounced increase in orientational order accompanying the freezing process. Also shown in fig. 2 is the equation of state given by the scaled particle theory for convex h‘ard molecules obtained by Gibbons [9]. According to this theory, the equation of,state for convex hard molecules is a function of the density, d, and of a shape factor, y, determined by the mean radius, E, surface area, S, and volume per molecule, v. Pv -= NCT
3+d(3y-66)+d2(3-3y+72) 3(1 -d)3
>
References
[ 1 J J.S. Rowlinson, Liquids and Liquid mixtures, 2nd Ed. (Butterworths, London, 1969) ch. 8. [2] J.A. Barker and D. Henderson, J. Chem. Phys. 47 (1967) 4714; G.A. Mansoori and F.B. Canfield, 3. Chem. Phys. 51 (1968) 4958; J.D. Weeks, D. Chandler and H.C. Anderson, J. Chem. Phys. 54 (1971) 5237. [3] hl. Rigby, J. Chem. Phys. 53 (1970) 1021. [4] W.W. Wood, in: Physics of simple liquids, ec!s. H.N.V. Temperloy, J.S. Rowlinson and G.S. Rushbroake (Nortkb Holland, Amsterdam, 1368). [5] J.R. Sweet and W.A. Steele, J. Chem. Phyr 47 CL967j 3029. [6] J. Vieillard-Baron, J. Chem. Fhys. 56 (1572) 4729. [7] T. Kihara, Advan. Chem. Phys. 5 (1963) 147. [8] W.G. Hoover and F.H. Ree, J. C_Wm. Phys. 49 (1968) . 3609. [9] R.M. Gibbons, Mol. Phys 17 (1969) 81.
435
EQUATiON
1 Ju!y 1973
CHEMICAL PHYSlCS LETTERS
OF STATE FOR SYSTEMS OF HARD ~~~-SP~~~&A~
h~OLECULES
Re&ved 2 April 1973
The equation of state for systems of hard pro&e spherocyliaders has been obtained usinp a SfonteCarIo simuIation method. The results in the fluid region are compared with the predictions of s&cd particle theory, and are Found to agree quite well. The equation of state is sipniticantly different f;om that for hard sphere moIecuIes, and it is concluded that the effects of non-spherical shape must be esplicitly dealt with.
2. Method In recent years considerable progress has been made towards the de~~e~opmentof a satisfactory theory of the liquid state for systems of simple molecules [I]. Various forms of perturbatiqn theory, all based ulti; mateiy on the properties of ‘model systems of hard sphere molecules, have been shown to reproduce adequately the equilibrium properties of monoatomic fluids, and have also provided useful simp~ifyj~g concepts about the role of the repulsive and attractive intermolecular forces in determining the structure of dense fluids [a]. However, for systems of polyatomic moiecules the development has been less complete, as aresuit of the more complicated non-spherical nature of the intermolecufar interactions and the additional complexities arising from internal degrees of freedom. Some clarification of this situation might resr?Jt from the study of model systems of non-spherical molecules, analogous to hard sphere systems. We have’
accordingly made an inves~gation of the equation of state of systems of hard sphero+cylindricaI moiecu!es, hoping that these will provide some insight into the effects of molecular shape, i.e., non-spherically symmetric repulsive interactions, on the bulk properties. The virial expansion for such systems was the subject
of an earlier ~nvestigatjon [3], and in this note we report the results of Monte Carlo simulation studies.
The equation of state of systems of spherocylinders has been obtained using a Metrorjoiis type Monte Carlo simulation method 141. In using this approach the equation of state is conveniently obtained via the virial theorem. For hard molecuies interacting without any attractive forces, the virial function is zero fcr al: pair separations other than those corresponding to contact between the molecules. For hard sphere molecules the equation of state is obtained by determirting the value of the pair distribution function, dr), at the point of contact, r = 0. The c~m~ress~~~~ty then calculated from the equation
P~~~~~T- 1 = ~2~i~~~3 v) &(o)
factor is
.
For non-spherical hard molecules this procedure must be modified, since g(r) is now aIso a Function of the relative orientations of the molecules as well as their separation, and contact can occur over a range of vafucs of the separation of the centres, z-. One possible approach to this problem, which has been explored by Steele and coworkers [S].. is to txpand the orientation dependent pair distribution function as a series, using spherical harmonics. This metliod could be used in a Monte Carlo calculation, but does not see& to be ideal, since it is uncertain in any given case how many, terms must be taken in the se-
.‘,
433
‘. ”
CHEMICALPH’I’SICSLEmERS
Volume 20, number 5
ties. However,‘this method does offer great promise in the detailed interpretation of orientational correlation effects. An alternative, simpler, approach has been used in this work, which leads more directly to the equation of state, but giveS less information about orientationa: effects. ti some ways our approach resembles that used by VieiIIard-Baron [6] in his recent Monte Carlo study of systems of two-dimensional hard ellipses. For a pair of convex hard molecules we denot: the shortest. distance between the hard cores as p, and define a distribution function, G(p), analogous to the conventional pair distribution function. Thus for a system of 121molecules in volume V the number of molecular pairs whose cores lie between p and p + dp is given by N2G(p) F(p) dp/2 V. F(p) dp is the volume element available to the centres of molecuies at p. In general this will depend on the shapes of the molecules, and on the degree Qf’ orientational order in the system. For convex hard moiecules without angular correlations, Kihara [7] has shown that F(p) may be defined in terms of parameters characteristic of the molecular shape, and can be written F(p) = 2s t Sd2
+ I6n&
t 4np2 ,
where S is the surface area and R is the mean radius. We tiay also define a mean intermolecular separation, y(p), which will in general also depend on the orienta-
tional order of the system, In a Monte Carlo si.zulation it is possible to determine values of F(p) and G(p)I;Tp), as average values over a small range of p. The virial function is zero for all values ofp # 0, and on application of the virial theorem we obtain an expression for the equation of state in terms of the limiting values of G, F, 7, as p tends to zero. PV/NkT-
prolate
spherocylinder
model.
AA’ is the axis of
rotation. cells were needed in order to use potentially close packed structures for the initial configurations. The present set of results are for systems of 120 molecules, of prolate spherocylindrical shape (fig. 1) in which R* = 1. Further calculations for other values of R” are planned. Typical Monte Carlo runs were of
IO5 to 5 X lo5 steps.
3. Results The data are presented in table 1, and are shown graphicall~~in fig. 2. The results are seen to fall on two curves, with a discontinuous region in the density range near to 0.5. The lower density region appears to correspond to 3 fluid phase, and the higher densities to a solid. The tie line joining the equilibrium solid and fluid phases has not been located. The
lowest density point on the solid branch (d = 0.525) appears to be stable, and shows no tendency to melt. In comparison, the hard sphere system has a minimum stable solid density of d= 0.54 [8], and it appears that
Equation
of state
d
Table 1 for hard spherocylinders, -PVjNkT
l~=(A!~6~~ojG(o)~(o).
Values of F(p) G(p) and r(p) were obtained from the simulation for a range of values of p and the results extrapolated to p = 0 in order to obtain the compressibility factor. The-Monte Carlo calculations followed conventional procedures [4]. The orientations of the molecules were.defined in terms of two integers between I and 100, which determined the two polar angles needed for the.& axially symmetric mo!ecules. Some care was needed in the choice of initial lattice configurations, especially at the higher densities. Non-cubic basic 434 . .
Fig. 1. The
I July 1973
,.
:
-,
0.1
0.64
0.2
1.73 3.69 7.75
0.5 0.4 0.45 0.475 0.5 ,0.525
0.55 0.5 75 0.6 0.65 0.7
10.78
12.55 14.58 10.53 11.50
12.28 14.10 18.04
27.76
R* = 1 -- 1
CHEhfICAL PHYSICS LETTERS
Volume 20, number 5
1 July 1973
where 7 = &T/v and d = NV/V: It has been shown that the virkl coefficients derived from this model are in good agreement with direct!y calculated values [3], and it is seen in fig. 2 that the equation of state is in good agreement with the simulation data in the fluid region. (For the spherocylinders studied here, 7 = 3.6.) It should be no ted
0 24 1
that despite the relativaly small deviation from spher-’ ical shape shown by these molecules, the equation of state differs significantly from the hard sphere equation. At d = 0.45, the hard sphere pressure is about
.I
-2
.4
.3
.5
.6
20% smaller than that for the spherocylinders. Furthermore, it was pointed out by Gibbons that the properties of convex non-spherical molecules cannot be adequately reproduced using an effective spherical model. Any successful treatment of non-spherical molecules must include explicitly the effects of nonspherica! repulsive forces.
.7
d Fig. 2. Equation
of state for hard spherocylinders,
R* =
1. d
=
Nv/V. X fluid phase; o solid phase; - - - scaled particle the-
ory.
the freezing density for our spherocylinder molecules is somewhat lower than that for hard spheres. This result seems intuitively reasonable, and is also consistent with the findings of Vieillard-Baron in two dimensions. A de tailed investigation of this density region, with careful study of orientational correlations is in progress. Preliminary results show a pronounced increase in orientational order accompanying the freezing process. Also shown in fig. 2 is the equation of state given by the scaled particle theory for convex h‘ard molecules obtained by Gibbons [9]. According to this theory, the equation of,state for convex hard molecules is a function of the density, d, and of a shape factor, y, determined by the mean radius, E, surface area, S, and volume per molecule, v. Pv -= NCT
3+d(3y-66)+d2(3-3y+72) 3(1 -d)3
>
References
[ 1 J J.S. Rowlinson, Liquids and Liquid mixtures, 2nd Ed. (Butterworths, London, 1969) ch. 8. [2] J.A. Barker and D. Henderson, J. Chem. Phys. 47 (1967) 4714; G.A. Mansoori and F.B. Canfield, 3. Chem. Phys. 51 (1968) 4958; J.D. Weeks, D. Chandler and H.C. Anderson, J. Chem. Phys. 54 (1971) 5237. [3] hl. Rigby, J. Chem. Phys. 53 (1970) 1021. [4] W.W. Wood, in: Physics of simple liquids, ec!s. H.N.V. Temperloy, J.S. Rowlinson and G.S. Rushbroake (Nortkb Holland, Amsterdam, 1368). [5] J.R. Sweet and W.A. Steele, J. Chem. Phyr 47 CL967j 3029. [6] J. Vieillard-Baron, J. Chem. Fhys. 56 (1572) 4729. [7] T. Kihara, Advan. Chem. Phys. 5 (1963) 147. [8] W.G. Hoover and F.H. Ree, J. C_Wm. Phys. 49 (1968) . 3609. [9] R.M. Gibbons, Mol. Phys 17 (1969) 81.
435