J. CHEM. SOC. FARADAY TRANS., 1991, 87(20), 3373-3377 3313

Coordination Number and Equation of State of Square-well and Square-shoulder Fluids : Simulation and Quasi-chemical Model

David M. Heyes Department of Chemistry, University of Surrey, Guildford GU2 5XH, UK

Extensive Monte Carlo (MC) computer simulations of square-well fluids with variable well-width have been performed to determine the equation of state and local coordination numbers. An accurate yet simple analyti- cally derived equation of state for square-well fluids is obtained using the quasi-chemical approximation (QCA) and empirically refined using the MC data. This extends the range of earlier models using this approach to arbitrary well width and sign of the well (either attractive well or repulsive shoulder). We also demonstrate, for the first time, that the present QCA formula also reproduces well the coordination number and the equation of state of the square-shoulder (SS) fluid over a wide density, temperature and well width range. The absence of a critical point for the SS fluid enables the simple QCA expressions to be more successful over a wider tem- perature range than those of the attractive square wells.

The hard-sphere fluid has been central to many of the advances in the theory of the liquid state since the original molecular dynamics computations of Alder and Wainwright. Interest in the hard-sphere fluid has continued unabated and the range of its useful applications continues to widen. For example, there have been some recent developments in for- mulating the equation of state,2 devising novel Monte Carlo technique^,^ and interest in the coordination number of hard spheres in 'random packing' and 'loose random packing' with relevance to fluidised beds and quasi-static powder beds4*' Also the rheological properties of the pure hard- sphere fluid have many similarities with those of colloidal and granular materials, although the precise link is still unfolding (see, e.g., ref. 6).

In a number of materials consisting of packed hard-core particles, the excluded volume effects alone are not sufficient to account for material behaviour, because the short-range attractive forces between the particles play an influential role, in addition to the hard core, in determining the macroscopic behaviour. The square-well potential is the simplest inter- action potential that accounts for both of these terms in, admittedly, an idealised form. As such, the square-well (SW) potential has found many applications and been studied with statistical mechanics and applied to a diverse range of parti- culate materials, ranging from, for example, simple molecules in the gaseous and liquid phases, to colloidal particles of ca. pm diameter in dispersions, and more recently to fine powders. The analytic form of this potential is simple enough to enable analytic theory and computer simulation at the microstructural level to provide insights into the molecular origins of physical phenomena. One reason for this usefulness is the formal link between the average local coordination number around a particle and the configurational energy per particle of the bulk material. This facilitates the analytic step from a model for the microstructure directly to the equation of state. In particular, the average coordination number per particle, c, is (apart from a constant) equal to the configu- rational energy per particle. Therefore there is a direct link between the fluid's microstructure (as measured by the coor- dination number) and the thermodynamic state point. Conse- quently the square-well fluid is an ideal molecular model to study single component particulate assemblies and for mix- tures local composition effects in multi-component materials (i.e. where there are deviations in the local mole fraction from the system averages). There are a number of areas where this link could provide a basis for an understanding, or at the very least, a description of other properties of interacting par-

ticles which therefore could also be described in terms of the equation of state. We think, in particular, of the long-range connectivity or percolation of a particulate assembly.

The value of the percolation threshold (the lowest number density of particles at which infinitely ranged connectivity appears) in continuum interacting particulate assemblies is more complicated than in lattice studies because the local geometry (or equivalent lattice coordination number) is state point dependent, and is influenced by the interparticle repul- sive and attractive forces, over a distance range comparable to the particle's diameter. The connection between local coor- dination number and percolation has been largely assumed in percolation-based theories of, for example, the conductivity of conducting particle distributions in an insulating background material,7 the viscosity of concentrated suspensions' and floc- ~ u l a t i o n . ~ To date this is largely based on speculation, as even the link between the equation of state and the local coordination number has not been well characterised (e.g. Janzen7 assumes that the coordination number is proportion- al to the volume fraction of particles, an approximation dating back to van der Waals). The distinct feature of contin- uum systems, as opposed to lattices, is that the particles have a variable coordination number that depends on the ther- modynamic state of the system. Therefore the percolation threshold is linked to the equation of state in an, as yet, largely unresolved manner. It is still not clear, from the studies so far, what is the relationship between the local structure in the fluids (i.e. within the first few coordination shells), the equation of state and the value of the percolation threshold. Although this issue has been resolved in the high- temperature limit. In this limit, i.e. when the well depth --+ 0, Balberg and Binenbaum" have established that c = 1.5 as the interaction shell width + 0 and c = 2.9 when this shell width + 00. The effect of intermediate range attractive or repulsive interactions on the coordination number is still a largely unexplored area.

In this report, we commence this task by linking the coor- dination number and equation of state of the square-well fluid. Ultimately, we aim to establish connections between the equation of state of simple fluids and their percolation properties. In the present study we make a first step in this direction, with an accurate semiempirical formula for the coordination number of square-well and -shoulder fluids in terms of the well width, temperature and number density. This is an extension of the work of Lee and Chao,''"2 Guo et d.,' and Sandler and co-workers,' who only considered the special case of a well width equal to half the diameter of

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3314 J. CHEM. SOC. FARADAY TRANS., 1991, VOL. 87

the hard core (i.e. A = 1.5, see below). While a reasonable choice for molecular fluids, we wish to cover a wider range of well widths that will be of use for other practical applications. For example, A = 1.1 is more realistic for the van der Waals attractions between ca. 1 pm diameter particles.

The Model Consider a square-well potential,

&)= 0; r d a

- - - E ; a < r < A o ( 1 )

= O ; r > A o

where the hard-core diameter of the particle within which no interpenetration can occur is a. For square-well potentials the well strength E > 0 and for square-shoulder potentials E < 0. The finite-energy interaction region for 0 < r < Aa is either attractive ( E positive) or repulsive ( E negative). We define the coordination number to include all the particle centres within the range of the potential interaction I = 10.

The quasi-chemical approximation (QCA) forms the basis of the proposed equation of state. It gives the equation of state from a simple analytic model for the coordination number. The reason for its success is that the coordination number is bounded by an analytic limit as density tends to zero. In the high-density limit the coordination number is dominated by the maximum packing constraint, making it temperature independent. The QCA makes a reasonable interpolation between these two extremes. The QCA approx- imates a liquid by a regular lattice consisting of sites occupied by real particles, denoted by 1 or occupied by spaces, which are denoted by 2. The lattice coordination number or number of nearest-neighbour sites, 2, around any site is

z = n , , + n2, (2)

where the local coordination number, n , , is the average number of occupied sites and n21 is the average number of empty sites (i.e. spaces) about a filled site (i.e. type 1). Further- more if we assume that the two ‘species’ are not distributed randomly but according to the following local composition rule,’

(3)

where /? = l/k, T , E ~ , is the well strength for species 1 and 2, E , , is the well strength between species 1 and 1, N i is the number of species i in volume V . If we define

n21ln11 = N , exPC@,, - Ell)Pl/Nl

Q = exP(&,lfl (4) then

as c 1 2 = 0. Therefore,

Z = n , , + N 2 n l l / N l Q (6)

which upon rearrangement gives,

If we assume that there are N lattice boxes (or equivalently, sites) in volume V then the maximum density of real mol- ecules occurs when all boxes are occupied by real particles (i.e. species 1) rather than vacancies, i.e. N , = N . Let us denote this maximum density by p, = N / V . However, at lower a density: p = N , / V < p m , the number of vacancies can be estimated,

Therefore,

N2/N1= @, - P V / P V = P ~ / P - 1 (9)

Substituting eqn. (9) into eqn. (7) gives

n , , = c = Z/[1 + @,Jp - l)/Q]

= zPQ/bm + P(Q - ‘)I (10)

Lee and Chao chose p, = 1 , 1 2 a prescription we also adopt here. The maximum hard-sphere density in the fluid phase is p = 0.9428, so the chosen value for p, = 1 is reasonable. The equivalent ‘lattice’ coordination number, 2, can be estimated as follows. The coordination number, c, representing the number of molecules located inside the energy well of a central square-well molecule, is formally related to the radial distribution function, g(r). We have

c = 4np g(r)r2 dr (1 1) Lo The pair radial distribution function is in the p -, 0 limit,

g(r) = exPC - 4(r183 (12)

Substitution of eqn. (12) in eqn. (1 1) enables us to define for the QCA the coordination number in the limit p + 0

c = z p n

where Q = exp(&g)

Hence we have

Rewritten in this form we can make a useful comparison with the van der Waals equation of state, which also has the coor- dination number linearly proportional to the density at all densities ([not only in the p -, 0 limit, as for eqn. (15)l.

(16) W / P = 1/(1 - Pb) + PUP

Now the configurational energy per particle is in general,

gives

u = pa (18) but also the van der Waals energy per particle by definition is

u = C&/2 (19) and therefore

Clearly the van der Waals equation of state can therefore be considered to be equivalent to the QCA in the p -, 0 limit.

We have obtained the low-density limiting expression for 2. Eqn. (14), while adequate at low fluid density, fails to account for the important local structural features of the dense fluid near to close-packing. Local composition effects caused by the well-interaction strength ( E ) are most important at low density and are suppressed by the packing constraints in the dense fluid. Nevertheless, eqn. (15) does provide a useful reference formula for the coordination number that can be adjusted in the high-density limit to take into account these additional effects. In recent years, there have appeared a

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J. CHEM. SOC. FARADAY TRANS., 1991, VOL. 87 3375

number of empirical modifications of the quasi-chemical for- mulae [i.e. eqn. (15)] in order to account for this transition to the high-density regime where packing constraints become the dominant factor in determining the coordination number. (e.g. see the summary by Guo et ~ 1 . ) ’ ~ They have been opti- mised for A = 1.5, whereas our interest extends over a wider A range. In response to this need, we propose the following formula developed by a least-squares fit to over lo00 p, /?, A state points from this and previous work (consult the next section).

(21) c = C O s r / [ l + p(i2 - l)]

where

co = 2b(~3 - 1)

x P{1 + eXPC-a,(A - 1)1C(1 - P / P 0 r 2 - 111 (22) where b = 2m3/3 and po is the reference density introduced by Heyes and Woodcock.’6 The term co derives from eqn. (14) but it is modified to account for the increase in the rela- tive local density as A -, 1. In fact as A -+ 1 we find this ratio equal to g(a) the pair distribution function at contact. To a very good approximation, for hard spheres,

(23)

i2 = exp(a&g) (24)

s(4 = 1/(1 - P/Po)2

Now,

where a = 1 - a2p/A

where from the least-squares fit: a1 = 4.3305, a2 = 0.30389 and po = 1.625. The term, a in eqn. (24) attempts to account for the diminishing role of temperature on the coordination number as density increases, and also the effect of A, while ensuring the analytically correct p + O limit for c. In this fitting procedure we used previous MC SW data for a range of A by Guo et u1.,13 Henderson et u1.,17 Lee et ~ 1 . ’ ~ and he ye^",^' for 1.05 < A < 3.0. We also performed additional SW simulations with N = 256 over the entire p, p, 1 phase diagram. This amounted to over lo00 p, /?, A state points. The quasi-chemical approach adopted here is quite different from the many-parameter fits of other work,21*22 in that the number of disposable parameters is only two. These semi- empirical modifications to the QCA are introduced based on firm microstructural considerations.

Additional computations using the method described else- where,l9v2O were conducted at /? = 0 for this work to cover this important limiting case and to ensure that there is a smooth transition between the low- and high-temperature extremes.

Results and Discussion In Table 1 we present the state point dependence of the coor- dination number from MC simulation and from eqn. (21). This illustrates that over a wide range of p, A and T values eqn. (2 1) reproduces the simulation coordination number data to within several percent. It is noteworthy that the present formula accounts exceptionally well for c in the /? + 0 limit for arbitrary A. Table 1 also gives the compressibility factors obtained by MC, the formula given in ref. 17. Using the virial theorem, the equation of state for a square-well or square-shoulder fluid is

where a* = a & 6 and 6 is an infinitesimally small positive number. A cubic extrapolation of g(r) to the various contacts’

‘0’ was used. If Ar is the resolution distance of the pair radial distribution function,

g(a) = [15g(a + Ar/2) - 10g(a + 3Ar/2)

+ 3g(0 + 5Ar/2)]/8 (27)

Note also a x N / ( N - 1) finite-N correction was applied to the ‘raw’ g(r) from the simulation. It is this modified g(r) that has been used in eqn. (26).

The quasi-chemical equation of state is derived from the Helmholtz energy, A, which is related to the configurational energy. In the QCA we have, using the coordination number,

(28) (aapiaa), = u = - 4 2

@/?Pa), = -&Co(P)fi2(a, P)/{l + PCQ(/l, P ) - 11)/2 (29)

where a = A / N

Integration with respect to p yields

up = u”/? - co@)ln[l + p(Q - 1)]/2pa (30)

where uhs is the hard-sphere component to the single-particle Helmholtz energy. Now,

(31) PBlP = z = P(aaP/aP) B

Hence,

z= Z =

F1 =

F , =

F 3 =

A simple accurate hard-sphere compressibility factor expres- sion proposed by Heyes and Woodcock,’ is used for Zhs,

(37) ZhS = 1 + bp/(l - p/p o),

Table 1 compares eqn. (26) and eqn. (33) for the simulation and the model predictions of 2. This semiempirical equation of state is successful in reproducing the simulation Z for most of the phase diagram, except close to the Auid-solid coexistence density p1 z 0.94 and at temperatures below the critical temperature, e.g. T, = 1.41 for A = 1.5, and T, = 0.98 for A = 2.0.23

The coordination number is insensitive to the precise value of p,, values of 0.9 and 1.1 where chosen as measures of this.

The square-shoulder, SS, fluids adhere well to the QCA. The coordination number of the SS is lower than that of the SW fluid at the same p, T and A. This is taken into account in the QCA formula through the sign of E . As density increases the denominator in eqn. (21) causes an increase in coordi- nation number to counteract the effect of i2 in the numerator. Again this brings into the model the dominance of packing constraints at high density.

Clearly, better agreement with the simulation data could have been achieved with more disposable parameters, as have been employed recently.21.22 The objective here was, in con- trast, to limit the number of empirical modifications to the QCA, only introducing those where there is definitely some well characterised microstructural basis. This comes from the behaviour of the hard-sphere fluid [leading to the term with a1 in eqn. (22)] and the suppression of the effect of tem- perature on coordination number as density increases [leading to the term with a,in eqn. (25)].

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3376 J. CHEM. SOC. FARADAY TRANS., 1991, VOL. 87

Table 1 Coordination numbers, c, and compressibility factors, 2, of model square-well and square-shoulder fluids

E Csim CQCA Z s i m

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.85 0.90 0.93

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.85 0.90 0.93

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.85 0.90 0.93

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.93

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.93

0.10 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.93 0.08 0.05 0.03

0.10 0.20 0.30 0.40 0.50 0.60 0.70

10.00 1.10 10.00 1.10 10.00 1.10 10.00 1.10 10.00 1.10 10.00 1.10 10.00 1.10 10.00 1.10 10.00 1.10 10.00 1.10 10.00 1.10

2.00 1.10 2.00 1.10 2.00 1.10 2.00 1.10 2.00 1.10 2.00 1.10 2.00 1.10 2.00 1.10 2.00 1.10 2.00 1.10 2.00 1.10

1.00 1.10 1.00 1.10 1.00 1.10 1.00 1.10 1.00 1.10 1.00 1.10 1.00 1.10 1.00 1.10 1.00 1.10 1.00 1.10 1.00 1.10

10.00 1.20 10.00 1.20 10.00 1.20 10.00 1.20 10.00 1.20 10.00 1.20 10.00 1.20 10.00 1.20 10.00 1.20 10.00 1.20

2.00 1.20 2.00 1.20 2.00 1.20 2.00 1.20 2.00 1.20 2.00 1.20 2.00 1.20 2.00 1.20 2.00 1.20 2.00 1.20

1.00 1.20 1.00 1.20 1.00 1.20 1.00 1.20 1.00 1.20 1.00 1.20 1.00 1.20 1.00 1.20 1.00 1.20 1.00 1.20 1.00 1.20 1.00 1.20

5.00 1.50 5.00 1.50 5.00 1.50 5.00 1.50 5.00 1.50 5.00 1.50 5.00 1.50

- 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1

- 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1

- 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1

- 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1

- 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1

- 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1

- 1 - 1 - 1 - 1 - 1 - 1 - 1

0.18 0.39 0.65 0.99 1.42 1.98 2.66 3.51 4.02 4.61 5.04

0.25 0.51 0.82 1.23 1.72 2.37 3.12 4.03 4.52 5.03 5.44

0.38 0.77 1.20 1.72 2.29 3.04 3.73 4.56 5.09 5.67 5.99

0.48 0.82 1.34 2.01 2.78 3.71 4.75 5.98 7.35 7.72

0.51 1 .OO 1.53 2.33 3.22 4.15 5.16 6.37 7.68 8.05

0.80 2.30 3.10 3.98 4.77 5.78 6.94 8.17 8.5 1 0.60 0.41 0.20

1.23 2.50 3.90 5.40 6.95 8.54

10.09

0.16 0.36 0.59 0.87 1.23 1.70 2.34 3.23 3.82 4.54 5.05

0.23 0.47 0.74 1.06 1.43 1.91 2.54 3.40 3.96 4.65 5.14

0.34 0.65 0.95 1.28 1.66 2.13 2.74 3.57 4.10 4.76 5.22

0.35 0.74 1.18 1.70 2.32 3.09 4.10 5.48 7.44 8.07

0.49 0.98 1.50 2.06 2.70 3.47 4.46 5.77 7.63 8.22

0.73 1.92 2.50 3.13 3.88 4.82 6.06 7.8 1 8.36 0.56 0.39 0.20

1.20 2.39 3.59 4.8 1 6.10 7.49 9.06

1.27 1.55 1.95 2.38 2.94 3.84 5.58 7.43 8.83

10.25 12.05

1.17 1.50 1.77 2.20 2.88 3.34 4.55 6.25 7.14 7.97 8.86

1 .OO 1.14 1.53 1.67 1.83 2.68 3.30 4.2 1 4.74 6.04 6.59

1.29 1.50 2.00 2.37 2.95 3.89 5.24 7.24

10.56 11.58

1.16 1.30 1.43 1.79 2.20 2.68 3.65 5.1 1 7.80 9.04

1.04 0.83 1.19 1.18 1.40 1.88 3.24 4.19 5.51 1.02 0.99 1.03

1.11 1.30 1.52 1.87 2.40 3.45 4.65

1.23 1.52 1.91 2.41 3.08 4.00 5.26 7.07 8.27 9.75

10.79

1.19 1.43 1.74 2.15 2.67 3.35 4.23 5.40 6.12 6.94 7.49

1.12 1.29 1.52 1.80 2.14 2.54 2.98 3.40 3.55 3.61 3.58

1.22 1.50 1.88 2.36 3.01 3.90 5.12 6.85 9.39

10.21

1.13 1.32 1.57 1.90 2.32 2.85 3.51 4.32 5.23 5.46

0.99 1.16 1.30 1.45 1.58 1.60 1.33 0.35

. 0.10 0.99 0.99 0.99

1.13 1.32 1.59 1.98 2.52 3.27 4.3 1

P

0.80 0.90

0.30 0.40 0.50 0.60 0.70 0.80 0.90

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.85 0.90 0.93

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.85 0.90 0.93

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.85 0.90 0.93

0.90 0.93 0.70 0.40 0.30 0.10 0.05

0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.85 0.90 0.93

0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.85

5.00 1.50 5.00 1.50

1.00 1.50 1.00 1.50 1.00 1.50 1.00 1.50 1.00 1.50 1.00 1.50 1.00 1.50

10.00 1.75 10.00 1.75 10.00 1.75 10.00 1.75 10.00 1.75 10.00 1.75 10.00 1.75 10.00 1.75 10.00 1.75 10.00 1.75 10.00 1.75

2.00 1.75 2.00 1.75 2.00 1.75 2.00 1.75 2.00 1.75 2.00 1.75 2.00 1.75 2.00 1.75 2.00 1.75 2.00 1.75 2.00 1.75

1.00 1.75 1.00 1.75 1.00 1.75 1.00 1.75 1.00 1.75 1.00 1.75 1.00 1.75 1.00 1.75 1.00 1.75 1.00 1.75 1.00 1.75

10.00 1.10 10.00 1.10 10.00 1.10 10.00 1.10 10.00 1.10 10.00 1.10 10.00 1.10

4.00 1.10 4.00 1.10 4.00 1.10 4.00 1.10 4.00 1.10 4.00 1.10 4.00 1.10 4.00 1.10 4.00 1.10 4.00 1.10 4.00 1.10 4.00 1.10

1.00 1.10 1.00 1.10 1.00 1.10 1.00 1.10 1.00 1.10 1.00 1.10 1.00 1.10 1.00 1.10 1.00 1.10 1.00 1.10

- 1 - 1

- 1 - 1 - 1 - 1 - 1 - 1 - 1

- 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1

- 1 - 1 - 1 -- 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1

- 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1

1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1

11.50 12.70

6.56 7.46 8.32 9.60

11.09 12.39 13.38

2.4 1 5.99 6.48 8.50

10.65 12.73 14.58 16.29 17.04 17.87 18.36

2.9 1 5.42 7.35 9.25

11.29 13.34 15.24 16.97 17.85 18.57 19.07

8.79 10.74 12.01 12.61 13.60 14.59 16.04 17.83 18.74 19.64 20.1 1

4.36 4.75 2.47 0.9 1 0.56 0.15 0.07

0.06 0.12 0.29 0.50 0.76 1.12 1.63 2.29 3.13 3.63 4.26 4.61

0.03 0.07 0.16 0.3 1 0.5 1 0.74 1.05 1.57 2.39 2.8 1

10.93 13.29

5.50 6.77 7.95 9.14

10.43 11.94 13.86

2.00 3.99 5.96 7.94 9.95

12.01 14.18 16.52 17.79 19.16 20.04

2.82 5.33 7.59 9.68

11.65 13.56 15.47 17.47 18.53 19.68 20.42

4.20 7.32 9.78

11.81 13.58 15.20 16.76 18.36 19.22 20.15 20.75

4.47 5.00 2.23 0.78 0.52 0.14 0.07

0.06 0.12 0.27 0.47 0.72 1.05 1.51 2.14 3.05 3.66 4.41 4.95

0.03 0.06 0.15 0.27 0.44 0.70 1.08 1.66 2.57 3.22

6.61 9.72

- 0.02 - 0.20 - 0.45 - 0.51

0.56 2.79 6.38

1.37 2.68 1.74 1.97 2.69 3.57 5.00 7.07 8.63

10.40 11.47

0.69 0.37 0.5 1 0.52 0.60 1.37 2.50 4.89 5.93 7.95 9.44

- 0.17 0.37

- 0.70 - 0.74 - 1.13 - 1.11 - 0.84

1.46 2.66 4.67 4.45

11.24 12.26 5.9 1 2.63 2.04 1.29 1.11

1.11 1.26 1.55 2.09 2.55 3.60 4.65 5.95 8.93

10.66 12.38 13.43

1.14 1.30 1.67 2.26 2.85 3.72 5.50 8.21

11.91 13.62

5.80 7.96

0.04 - 0.02 - 0.02

0.02 0.02

- 0.16 - 0.84

1.14 1.35 1.65 2.08 2.68 3.52 4.73 6.50 7.68 9.15

10.20

0.68 0.48 0.4 1 0.47 0.69 1.1 1 1.81 2.87 3.58 4.44 5.04

- 0.10 - 0.81 - 1.25 - 1.50 - 1.58 - 1.49 - 1.25 - 0.87 - 0.64 - 0.41 - 0.28

11.21 12.52 5.79 2.54 1.98 1.25 1.12

1.12 1.26 1.59 2.03 2.62 3.43 4.56 6.18 8.59

10.24 12.33 13.85

1.14 1.29 1.69 2.24 3.00 4.09 5.69 8.12

11.96 14.73

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J. CHEM. SOC. FARADAY TRANS., 1991, VOL. 87

Table 1-Continued

3377

p T 1

0.90 0.93

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.85 0.90 0.93

0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.85 0.90 0.93

0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.85 0.90 0.93

1.00 1.10 1.00 1.10

10.00 1.25 10.00 1.25 10.00 1.25 10.00 1.25 10.00 1.25 10.00 1.25 10.00 1.25 10.00 1.25 10.00 1.25 10.00 1.25 10.00 1.25

4.00 1.25 4.00 1.25 4.00 1.25 4.00 1.25 4.00 1.25 4.00 1.25 4.00 1.25 4.00 1.25 4.00 1.25 4.00 1.25 4.00 1.25 4.00 1.25

1.00 1.25 1.00 1.25 1.00 1.25 1.00 1.25 1.00 1.25 1.00 1.25 1.00 1.25 1.00 1.25 1.00 1.25 1.00 1.25 1.00 1.25 1.00 1.25

1 1

1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1

3.42 3.82

0.4 1 0.9 1 1.52 2.26 3.19 4.28 5.49 6.83 7.54 8.24 8.69

0.17 0.36 0.82 1.40 2.09 2.99 4.1 1 5.35 6.67 7.40 8.10 8.5 1

0.08 0.20 0.51 0.98 1.62 2.43 3.47 4.73 6.05 6.76 7.45 7.86

4.05 4.67

0.38 0.8 1 1.31 1.90 2.6 1 3.50 4.65 6.2 1 7.2 1 8.42 9.28

0.16 0.33 0.72 1.19 1.74 2.43 3.31 4.46 6.04 7.07 8.3 1 9.19

0.08 0.17 0.39 0.68 1.07 1.60 2.35 3.44 5.06 6.18 7.59 8.63

16.53 18.38

1.28 1.49 2.06 2.65 3.48 4.44 6.32 8.39 9.97

11.53 12.48

1.14 1.33 1.70 2.05 2.80 3.8 1 5.08 6.9 1 9.15

10.97 12.5 1 13.70

1.18 1.39 2.00 2.79 3.64 5.49 7.64

10.54 13.5 1 15.13 17.74 18.6 1

18.35 21.06

1.26 1.59 2.03 2.60 3.38 4.46 5.99 8.23 9.75

11.66 13.05

1.13 1.29 1.66 2.14 2.79 3.68 4.9 1 6.68 9.34

11.17 13.50 15.20

1.18 1.39 1.91 2.62 3.60 5.00 7.06

10.1 8 15.16 18.77 23.52 27.10

P T A --E Csim CQCA

0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.85 0.90 0.93

0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.85 0.90 0.93

0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.85 0.90 0.93

10.00 1.75 10.00 1.75 10.00 1.75 10.00 1.75 10.00 1.75 10.00 1.75 10.00 1.75 10.00 1.75 10.00 1.75 10.00 1.75 10.00 1.75 10.00 1.75

2.00 1.75 2.00 1.75 2.00 1.75 2.00 1.75 2.00 1.75 2.00 1.75 2.00 1.75 2.00 1.75 2.00 1.75 2.00 1.75 2.00 1.75 2.00 1.75

1.00 1.75 1.00 1.75 1.00 1.75 1.00 1.75 1.00 1.75 1.00 1.75 1.00 1.75 1.00 1.75 1.00 1.75 1.00 1.75 1.00 1.75 1.00 1.75

1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1

0.86 1.80 3.81 5.93 8.10

10.29 12.37 14.26 15.97 16.73 17.42 17.92

0.64 1.43 3.36 5.47 7.62 9.74

11.75 13.56 15.32 16.1 1 16.76 17.19

0.44 1.09 2.90 4.94 6.98 9.00

10.92 12.83 14.58 15.29 16.04 16.42

0.83 1.68 3.42 5.22 7.10 9.08

11.18 13.45 15.96 17.34 18.84 19.8 1

0.57 1.17 2.47 3.92 5.55 7.38 9.46

11.86 14.66 16.27 18.06 19.23

0.35 0.73 1.61 2.65 3.91 5.45 7.34 9.72

12.76 14.62 16.78 18.25

1.11 1.35 1.70 2.25 2.46 3.91 5.18 6.58 8.14

10.15 11.60 12.80

1.32 1.64 2.43 3.42 4.59 5.94 7.30 8.75

10.92 12.58 14.10 15.68

1.52 2.12 3.49 5.01 6.64 8.04

10.13 12.08 14.29 15.71 17.9 1 19.33

1.16 1.33 1.73 2.23 2.86 3.69 4.8 1 6.34 8.55

10.02 11.85 13.16

1.30 1.62 2.35 3.24 4.33 5.69 7.46 9.82

13.13 15.33 18.05 20.0 1

1.41 1.87 2.93 4.24 5.89 7.99

10.73 14.45 19.69 23.19 27.53 30.68

cSim are obtained from the simulation and cWA are obtained from eqn. (21). The simulation and quasi-chemical approximation for the compress- ibility factor, Z, are given by eqn. (26) and eqn. (33), respectively. The SW fluids are denoted by E = 1, and the SS fluids are denoted by E = - 1. The units are: T in -ElkB, p in The standard error of c and 2 are k 3%.

Conclusions A quasi-chemical model has been derived that satisfactorily predicts the coordination number square-well fluids of arbi- trary well width. Expressions for the Helmholtz energy and compressibility factor derived from this model also account well for the machine simulation data, generally to within several percent. Where it does fail to reproduce the machine data (at least for the compressibility factor) no other simple equation of state succeeds at present, either. This lattice model therefore has some merit in accounting reasonably well for the square-well and square-shoulder properties over much of the phase diagram, and should prove useful in future studies where a relationship between the equation of state and the coordination number is required.

The author thanks The Royal Society for the award of a Royal Society 1983 University Research Fellowship and the SERC for the award of computer time on the CRAY-XMP at the University of London Computer Centre.

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1986,25, 31.

Paper 1/02334I; Received 17th M a y , 1991

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