Collision frequencies and mean collision parameters ...

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Collision frequencies and mean collision parameters in the LennardJones system a

Fernando Del Río & Alejandro Gil-Villegas

a

a

Laboratorio de Termodinámica, Universidad Autónoma Metropolitana, Iztapalapa, Apdo 55-534, México, DF, 09340, México Version of record first published: 22 Aug 2006.

To cite this article: Fernando Del Río & Alejandro Gil-Villegas (1992): Collision frequencies and mean collision parameters in the Lennard-Jones system, Molecular Physics: An International Journal at the Interface Between Chemistry and Physics, 77:2, 223-238 To link to this article: http://dx.doi.org/10.1080/00268979200102411

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MOLECULAR Physics, 1992, VOL. 77, NO. 2, 223-238

Collision frequencies and mean collision parameters in the Lennard-Jones system By F E R N A N D O D E L R I O t and A L E J A N D R O G I L - V I L L E G A S

Downloaded by [Universidad Autonoma Metropolitana] at 10:50 19 April 2013

Laboratorio de Termodin~imica, Universidad A u t 6 n o m a Metropolitana, Iztapalapa, Apdo 55-534, Mrxico, DF, 09340, Mrxico

(Received 24 September 1991; accepted 2 January 1992) Generalized collision frequencies, recently introduced for classical systems with repulsive forces, are developed for fluids with attractive interactions. The kinetics of momentum transfer, and therefore the pressure, depend on two parameters: a mean collision diameter and a mean collision range, characterizing the effects of repulsive and attractive forces, respectively. The theory gives new insight into the various contributions to the pressure of the fluid and is illustrated by calculating the collision frequencies, diameter and range of a Lennard-Jones fluid. The collision diameter and range are shown to be weakly dependent on the thermodynamic state, whereas the density and temperature behaviour of the collision frequencies can be understood in terms of collisions of hard spheres and square wells. 1.

Introduction

In a recent paper [1], collision frequencies were introduced and used to define mean collision parameters characterizing a species of interest. These quantities are state-dependent statistical averages and are related to other thermodynamic properties of a classical fluid. The collision frequencies are generalizations of the expression due to Enskog, which is restricted to hard-sphere fluids [2, 3]. In particular, the case of systems with purely repulsive potentials has been worked out in detail [4]. In the repulsive case, the only relevant parameter is the mean collision diameter (s), which can be used to write equations for the free energy, pressure and other thermodynamic quantities. These equations are based on the known hard-sphere fluid properties. In this work the previous study is extended to analyse systems with more realistic intermolecular potentials. It is shown that when attractive forces are considered, a second mean collision parameter becomes relevant besides the mean collision diameter: the mean collision range (l). The use of the two variables (s) and (l), and the associated collision frequencies VR and VA, allows expression of thermodynamic quantities like the pressure, and gives new insight into the effectis of repulsive and attractive forces. The aim of this work is to calculate the collision frequencies and parameters for a realistic simple system and to discuss their properties and behaviour. An application of these concepts to construct theoretical and analytical equations of state will be addressed in another paper. Here, the theory is developed for systems with spherically symmetric potentials and is applied to the 12-6 LennardJones fluid. t Presently on sabbatical leave at the Departamento de Quimica Fisica, Universidad Complutense de Madrid, 28040 Madrid, Spain. 0026-8976/92 $3.00 9 1992 Taylor & Francis Ltd.


224

F. Del Rio and A. Gil-Villegas

The calculation of v R, uA and the mean parameters (s) and (l) for a fluid in equilibrium requires g(r), the radial distribution function of the system, which can be obtained by any of the statistical mechanical methods developed for the purpose [5]. Using the g(r) obtained from computer simulations, it is shown that, in a simple fluid, the frequency UR behaves like that of hard spheres, whereas uA follows the attractive frequency of a square-well system. It turns out that the parameters (s) and (l) depend weakly on density and very smoothly on temperature, in such a way that their behaviour becomes dominated by their low density limits, which are obtained by simple quadratures. The weak density dependence of the collision parameters is the key property that makes them suitable for writing equations of state. Indeed, virial expansions for (s) and (l) allow the prediction of even high density values with good approximation. It is also shown that the first few temperature-dependent virial coefficients can be accurately parametrized. As in the purely repulsive case [4], the collision parameters are related to similar quantities introduced in perturbation theories using the hard-sphere or the square-well potential as a reference, and are also similar to phenomenological parameters used in corresponding states approaches [6-8]. In section 2 o f this work, the definitions of the mean collision parameters and frequencies are discussed as applied to a system with a continuous potential, and their relation to the pressure o f the fluid is shown. The density expansions of ~s) and (l) and their related frequencies uR and uA are calculated in section 3. Section 4 is devoted to calculating the collision parameters for the 12-6 Lennard-Jones system, both from computer simulated g(r) and from their density expansions, and also the results are discussed. The main conclusions of this work are presented in section 5.

2.

Collision frequencies and parameters

A fluid with an intermolecular potential u(r) is considered, which is typical of simple spherical substances, and resembles that shown in figure 1. The potential is infinite at the interparticle distance r = r0, reaches its minimum - e at r = r m and then falls off as r --+ c~. For most realistic potentials r0 --- 0, but r 0 > 0 for models

U

I

i .............................

i Figure 1. Collision diameter s and collision range l for a simple realistic intermolecular potential u(r).


Collision frequencies in Lennard-Jones fluid

225

with a hard core of the Kihara ~type. In general, two particles collide whenever r is such that they exert a force on each other, and this collision process extends over a finite time. Nevertheless, it has been shown that the effect o f u(r) can be described as an infinite sequence of instantaneous impulsive events [1]. Each of these impulsive collisions can be either elastic or inelastic depending on whether or not there is an instantaneous interchange of potential and kinetic energy [1]. In the rest of this paper reference will only be made to these instantaneous collisions (although, of course, the complete extended collision will always be elastic for structureless particles). A pair of particles will experience a bounce or hard collision at r = s (or l) when their relative kinetic energy vanishes at s (or l). Bounces are the instantaneous elastic collisions and can occur either from the repulsive part of u(r); that is, for r 0 < s < rm, or from its attractive part (l > rm). The kinetic analysis of the momentum transfer in the system Ill, shows that s and l are the distances across which momentum is transferred instantaneously in an elastic collision. Hence, they are parameters involved in expressing the pressure of the system. For the fluid with potential u(r), the differential frequency of bounces dub at distances between s and s + ds has been shown to be given by [1]

dub (s) = -I-Trs2p(vr)y(s)(0 exp ( - f l u ) l o s ) ds (1) where p = N / V is the density, (Vr) = (8kT/Trm) 1/2 is the relative velocity, y(s) is the 'background' correlation function y(s) = g(r) e~u, g(r) is the radial distribution function and/3 -= 1/kT, with k Boltzmann's constant and T the temperature. In equation (1), the plus sign applies to repulsive and the minus sign to attractive parts of u(r). This relation provides the generalization of the well-known collision frequency in hard-sphere systems due to Enskog [2, 3]. The total collision frequency of bounces from the repulsive region is then u R = 7rp(vr)

Ji[ ds s2y(s)(O exp (-~3u)/Os)

(2)

The integral in this equation has the dimensions of an area and is proportional to the mean cross-section for elastic repulsive collisions. For the simple hard-sphere fluid, this integral equals crZgHs(~r+) and equation (2) reduces to the Enskog relation for U~s. For bounces from the attractive region, UA = --Trp(vr)

Vdll2y(l)(O rm

exp

(-~u)/Ol)

(3)

whose meaning is similar to that of equation (1), but for the elastic and attractive collisions. Among the various quantities that can be calculated with the differential frequency in equation (1) for a fluid in equilibrium, the mean collision diameter (s) and the mean collision range (/) are of special importance as average characteristics of the species with potential u(r). These means are calculated over the repulsive and attractive regions of u(r), respectively, i.e.,

(s)(p, T) = (1/t/R)

dUB(S)S ro

jm

= (Trp(Vr)/UR)

dss3y(s)(O exp (-~u)/Os)

o

(4)

226


and

(l)(p, T) = (1/UA)

j

oo

duB(s)s

tm

dss3y(s)(O exp (-flu)/oqs)

= --(~p(Vr)/~'A)

(5)


m

In these equations the p and T dependence of the means (s) and (/) that arises through the correlation function y(s; p, T) is indicated explicitly. To calculate the mean collision parameters using equations (4) and (5), or the total frequencies in equations (2) and (3), knowledge is required of the structure of the fluid; i.e., ofg(r). Kinetically, (s) and (/) are the mean distances over which momentum is transferred instantaneously in elastic collisions and are thus appropriate measures of the sizes of the repulsive and attractive regions of u(r), respectively. In a previous work [1], it was shown that the pressure aand other thermodynamic properties of a purely repulsive system can be written exactly in terms of the single collision parameter (s). Hence, for such systems, (s) or its associated volume b -- 7r(s)3/6 can be rigorously understood as an average size of the molecules. This result is not as trivial as it seems at first if one recalls that inelastic collisions also contribute to the pressure and they are not included in the definition (1). This apparent paradox has been explained elsewhere [1] as a fortunate coincidence that allows the pressure to be written entirely in terms of (s). In the case of the more realistic system being considered here, the presence of attractive forces requires the introduction of the mean range (/) as a second parameter to express the thermodynamic properties and, thus, to characterize the species of interest. In fact, recalling that the pressure P of the system is given by the virial theorem as

9 'Ip =

1 + (2

p/3)

0

dr r3y(r)(O exp (-~u)/Os)

(6)

and using equations (4) and (5), then

t~P/p =

2 1 -~- 3 - ~ r ) (t/R(S) -- VA(I))

(7)

This equation allows the equation of state to be expressed in terms of the mean collision parameters (s) and (l), and allows the separation, in the manner of the van der Waals model, of the contributions to the pressure of the fluid from the repulsive and attractive forces. Before obtaining the collision parameters for a Lennard-Jones system, the density dependence of UR, uA, (S) and (l) will first be analysed.

3.

Virial expansion of the mean collision parameters

To obtain the density dependence of the mean collision parameters, they must be expanded in a virial series, whose first few terms will be shown to dominate the behaviour of the parameters. This series follows from the virial expansion of the background correlation [5] o~

y(r;p, T) = 1 + Z y . ( r ; T)p" n=l

(8)


227

which, after substitution in equations (2-5), leads to + ( s ) 2 ( T ) p 2 q- " "

(9)

(l)(p, T) = (1)o(T) + ( l ) l ( r ) p + (l)2(r)p2 + ''"

(10)

(S)(p, T ) = ( S ) o ( T ) q- ( S ) l ( T ) p


and Density expansions for quantities similar to equations (9) and (10) have been shown to have often a much better convergence than for thermodynamic variables like the pressure. This has been the case for repulsive potentials [4, 6]. For more realistic potentials, this has been shown by perturbation theory [7] and by the corresponding systems approach [8]. The first two virial coefficients of the collision diameter are, from equations (2), (4) and (9), (s)0 = (1/U0R)

I rodss3(O exp (-~u)/Os)

(11)

f;

(12/

ro

(S)l = (1/U0R)

dss3(O exp(-~u)/Os)yl(s) - (s)o(UlR/UOR)

0

and so on for higher orders. The functions UnR(T) in equations (11) and (12) are the density coefficients of the total frequency for repulsive bounces, as obtained from equations (2) and (8),

~'R = 7rp(Vr)[UoR(T) + UlR(T)P

+ / J 2 R ( T ) P 2 "~-'" ']

(13)

where, for example, U0R =

d s s 2 (0

exp (--13u)/Os)

(14)

and rm

ulR =

J

dssZ(O exp (-/3u)/Os)yt(s)

(15)

ro

and so on. For the density coefficients of the mean collision range, (I)o, (l)1, etc., from equations (5), (8) and (10), expresions similar to equations (11) and (12) that involve density coefficients U0A, U~a, and so on, of the attractive total frequency Ua in equation (3) are obtained. 4.

Mean collision parameters of the Lennard-Jones system

4.1. Intermediate and high densities The calculation and behaviour of the mean collision parameters (s) and (l) for a system are illustrated here with the simple and realistic Lennard-Jones potential ULj(r ) ~-- 4c[(o-/r) 12 -- ((7//') 6]

(16)

Given the radial distribution function of the system, gLj(r), the mean quantities of interest are obtained directly from equations (2-5) by integration and with r 0 = 0. For intermediate and high densities, the values of gLJ (r) used here were those obtained by Verlet [9] in a molecular dynamics simulation for 25 states in the ranges 0.45 < p* < 0.88 and 0.591 < T* < 2'934, of reduced density p* -= po3 and temperature T* = kT/e.

228


Table 1. Values of the collision frequencies u~ and u~, (u* = u~r/Tr(vr)),and mean diameter (s*) and range (1"), for the LJ fluid, at some of the states considered previously [9]; Z ca1 is the compressibility factor from equation (6) and Z MD is the molecular dynamics result of Verlet [9].

p* 0-880 0"850


0"750 0"650 0"500 0"450

T*

/2~

/J~

(S*)

(]*)

Z MD

Zcal

1.095 0'936 2"888 2"202 2-845 1-070 0'827 1"827 1'036 0"900 1"360 2-934 t-710 1-552

4"0810 4.2148 2'8065 3"0285 2'0811 2-6323 2"7512 1-6927 1"9429 2"0191 1'1866 0"8252 0-9617 0"9844

1-7601 2-0729 0-6527 0'8579 0"6080 1-6615 2-1727 0"8584 1-5465 1"7865 0"9301 0'3761 0-6653 0'7349

1"0034 1'0092 0-9635 0-9754 0-9675 1-0081 1"0166 0.9881 1'0111 1.0166 1'0031 0.9716 0-9947 0-9982

1-6449 1-6395 1-6511 1"6480 1"6377 1-6180 1"6110 1'6204 1"6079 1"6051 1-6079 1"6244 1-6119 1"6097

3'48 2"78 4'36 4"20 3'10 0"88 -0-53 1-56 -0-10 0"74 0-32 1"38 0-74 0'57

3-51 2'79 4"41 4'23 3"13 0-93 -0'47 1'59 -0"09 -0-71 0-36 1"40 0-76 0-58

In calculating the integrals in equations (2-5), gLj(r) was taken as zero for r < 0'84~r and gLj(r) = 1 for r > 5'0~r, and the smoother background correlation yLj(r) = gtj(r) exp [--r was obtained from Verlet's g(r) in 0-84tr < r < 5.0m Nevertheless, it is well known that pressure is extremely sensitive to values o f g ( r ) for r-~ cr [10], and the same is true for the mean diameter (s). Hence, to check the accuracy of the numerical integrations, the pressures calculated with gLj(r) and the virial theorem (6) are compared with the pressures directly calculated by molecular dynamics, and which are also reported by Verlet for the same states [9]. In 11 out of the 25 states, the deviation of both values of the pressure was greater than 0.05. Therefore, only the values o f (s) and (l) in the remaining 14 states, where gLj(r) is more reliable [10], are reported here. The values of the total collision frequencies for repulsive and attractive bounces, UR and UA, obtained for the Lennard-Jones system are given in table 1 (frequencies are given in reduced units u* = ua/Tc(vr)). In all states, repulsive events are more frequent than attractive ones and therefore the ratio UR/UA > I. Nevertheless, since (l) > rm > (s), m o m e n t u m is transferred across longer distances in attractive collisions and their effect (see equation (7)) can overcome that of the more frequent repulsive collisions. The frequency of repulsive collisions increases with temperature, mostly due to the factor (Vr)or T 1/2. The reduced frequency u~ at constant density, which is constant for hard spheres, has a slight temperature dependence that can be appreciated from figure 2. The decrease of u~ with T* is due to the fact that the 'size' or mean collision cross-section for soft particles diminishes with increasing kinetic energy [4]. Actually, at the highest temperatures shown in figure 2, u~ is smaller than u~js for hard spheres of diameter aLj at the same density. Since the LJ potential does not have a hard core, when T* tends to infinity the frequency u~ tends to zero very slowly. In contrast, the density dependence o f u~ is quite strong: it changes by a factor of 5 in the density range considered. This behaviour is similar to that of u~js itself, and means that the number of bounces from r < rm follows approximately the density behaviour of pgHs(a+).


229

5.0

0.88

4,0

5.0

0.85

>-, 9

~-

0.75

2.0


0.65 \ \

1.0

\ ~'~'-'~--~'-~---~

0.0

p

0.0

L

i

I

10

O. 45

i

I

2.0

5.0

40

T*

Figure 2. Frequency of repulsive elastic collisions, v~ lIRd/Tr('Ur) , for the 12-6 LennardJones fluid. The solid lines are visual interpolation aids drawn through isochores at the densities labelled. The dashed line gives the zeroth-order contribution from equation =

(14) at p* = 0.45.

The contribution to uA arises from pairs with negative energy (the so-called van der Waals dimers), and thus it decreases with temperature because of the relative depopulation of van der Waals dimers with increasing T*. In consequence, u~ against T* has a strong negative slope, as shown in figure 3, and u~, tends to zero faster than u~ when T* tends to infinity. Since, as will be shown shortly, the means (s) and (l) are quite insensitive to changes in temperature, the kinetic mechanism responsible for condensation o f the gas into a liquid at lower temperatures is the

2.5

2.0

1.5 LD

0.65

0 75

CT 1,0

0.5

0.0

I 0.0

I 10

I

i 2.0

t

I 5.0

I

_ _

40

T*

Figure 3.

Frequency of attractive elastic collisions, u~ = UAa/Tr(v,), for the 12-6 LennardJones fluid..The symbols have the same meaning as in figure 2.

230


1.0

I

ooooo(9.5 ': . . . . 0.6 ..... 0.7 * '~ ~ ~' ~ 0 . 8

>-, L) a)


o-0.5

I

0.0

0.0

I

1.0

I

J

I

2.0

I

3.0

I

J

4.0

i

5,0

T*

Figure 4. Frequency of attractive collisions for the square-well system of range R = 1.5e. The lines are drawn through isochores: open circles, p*= 0.5; asterisks, p*= 0-6; squares, p* = 0.7, and diamonds, p* = 0"8. faster increase in v~ over v~ when T* decreases. In contrast to v~,, v~ has a weak density dependence because the integral o f y ( r ; p, T) for r > r m (see equation (3)) is quite insensitive to changes in density. Of course, the behaviour of vA cannot be understood in reference to a hard-sphere system, but its main features appear to be quite independent of the shape of the attractive well. As an example, in figure 4 a graph of v~. for the square-well potential of range R = 1.5e is shown, as obtained from equation (3) using the values o f gsw(r) at r = a and r = R calculated by Henderson et al. [11]. The SW fluid has the simplest interparticle attraction. The temperature dependence of the LJ frequency v~ follows closely that o f the SW case, although in the latter system it is, surprisingly, almost independent of density from moderate to high densities. This strong resemblance of VA for different systems arises from the similarity of behaviour of the correlation y(r; p, T) at r > r,, for simple liquids. The results obtained for the reduced mean diameter ( s * ) = (s)/a%j and the reduced mean range (l*) = (l)/aLj are also shown in table 1, as are the values of the compressibility factor Z = 13P/p obtained directly from equations (6) or (7) with the same gLj(r), and those reported by Verlet [9]. The dependence of the mean diameter on p* and T* is weak, regular and systematic, as can be seen from figure 5. When T* grows at constant density, the higher kinetic energies make the particles bounce at smaller diameters and hence the mean (s*) decreases smoothly. This decrement arises from the softness of the repulsive part of the potential (16). But the temperature dependence of (s) is rather small, as can be appreciated simply by comparing the scales of figures 3 and 5. The softness of the repulsive part of u(r) also introduces a density dependence of (s*), which is even weaker than that on T and is also apparent from figure 5. The fact that (s) varies only slightly with p (as is the case for potentials at least as hard, or steep, as the repulsive part of ULj) is the reason why a density expansion like equation (9) turns out to be quite convergent and why the series can be truncated


231

1.03

1.01


S>~

0.99

0.97

095

~

0.0

I

I

I

1.0

2.0

~

I

3.0

T*

Figure 5. Mean collision diameter (s) for the Lennard-Jones system. The lines are visual interpolation aids drawn through isochores and are labelled as follows: p* = 0.880 (A); p* = 0-850 (B); p* = 0-750 (C); p* = 0-650 (D); p* = 0-500 (E) and p* = 0-450 (F). in practice. The weak dependence of (s) on p and T can be simply understood from its definition, equation (4), where the state dependence of the normalizing factor vR compensates the changes in the numerator. The behaviour of (s) is in general similar to that of the effective hard-sphere diameters obtained by either the perturbation theories of Barker and Henderson [12] and of Weeks et al. [13], or by the method of corresponding systems [6, 8]. Nevertheless, the diameters used in perturbation and corresponding states theories do not have a rigorous interpretation, except in the limit of low densities where the BH and the WCA diameters become equal to collision averages and also allow a kinetic definition [7, 14]. The values of (s*) for the Lennard-Jones system can be compared with those recently obtained for the purely repulsive r -12 potential [4]. The latter is much softer than the repulsive part of ULj(r) and therefore its mean collision diameter has a much stronger density and temperature dependence. The p and T dependence of the LJ mean collision range (l*) is shown in figure 6. As was mentioned in connection to VA, the only molecular events that lead to bounces from the attractive part of u(r) occur in van der Waals dimers. Hence, when T* tends to infinity, the Maxwell-Boltzmann distribution of energies within the attractive well (u(r) < 0) tends to uniformity and (l*) grows to a finite limiting value which only depends on u(r) itself. When the temperature decreases, the probability of finding pairs of particles closer to the bottom of the well grows, particles bounce off more often at smaller distances/, and the mean (l*) decreases. Nevertheless, this is a rather small effect; for instance, along the p* = 0-75 isochore, (l*) decreases only 2% whereas u~ diminishes as much as 72% (compare figures 3 and 6). The density dependence of (l*) is quite a lot stronger than that of (s*) but is also systematic: at higher densities, due to correlations between particles, the number of them at distances close but greater than rm is larger than at smaller densities; hence,

232

F. Del Rio and A. Gil-Villegas 1.66

j Bf 1:64

I*


1.62

I

1.60

0.0

r

I

1.0

i

2.0

I

5.0

T* Figure 6. Mean collision range (l) for the Lennard-Jones system. The lines are visual interpolation aids drawn through isochores and are labelled as in figure 5. the mean collision range (l*) grows with the density. The temperature and density behaviour of (1") is reminiscent of that of the effective attractive range used in square-well perturbation theory [7] and in corresponding systems [8], but, again, the interpretation of the latter does not have a rigorous basis, except at low densities where such ranges become equal to (l*). 4.2. L o w density behaviour The first term in the density expansion of the frequency v R (equation (13)) gives a fair approximation for not too high densities and supercritical temperatures, as can be seen from the dashed line in figure 2, which shows vR = p U0R(T ) at 0-45. At low temperatures, U0R grows very steeply. This figure shows that the density expansion of u~, will converge very slowly. Hence the series in equation (13) with coefficients U~,Rdoes not furnish an adequate expression for u~:, except at quite low densities. Nevertheless, the coefficients UT,R from equations (14) and (15) are needed to calculate the coefficients (S)n in the expansion of (s), given by equations such as equation (12). In contrast, the virial expansion of VA is more convergent, as can be seen from its first-term contribution, u~. = p UA0, represented by the dashed line in figure 3, which is quite close to the full values vA. At low temperatures, UA0 shows a very sharp increase, as did UR0The low density limits of the collision diameter and range, (s*)0 and (1")o (equation (11) and its equivalent for (l)0) are simply obtained by direct integration. These coefficients behave with temperature as shown in figure 7. When T* tends to infinity, (s*)0 tends to zero, but (l*)0 tends to a constant value characteristic of the potential (_~ 1.6629 for the 12-6 Lennard-Jones). When T* tends to zero, the (classical) system has a minimum energy when all interparticle distances r _~ r,, and therefore both (s)0 and (l)0 tend to r m. Comparison of figure 7 with figures 5 and 6 shows that the order of magnitude and the temperature dependence of (s) and (l) is given by the low density coefficients. t~a

*

*

233

Collision frequencies in Lennard-Jones fluid 1.8

1,6

/

Io

1.4


1.2

So 1.0

0,8

i

O0

1.0

i

i

2.0

3.0

4.0

T*

Figure 7. Temperature dependence of the first density coefficients (s*)0 and (l*)0 of the mean collision parameters for the Lennard-Jones fluid. The dashed line shows the asymptotic limit of (t*)0. To generate (s) and (l) from their density expansion at any temperature, the coefficients (s)0 and (l)0 were parameterized as functions of T* in a form that incorporates the above limits (at T* ~ 0 and T* ~ c~) by Soo + Sol T* + S02 T*2 (S*)o = 1 + 803 T* + S04 T*2 + S05T .3

(17)

(l,)0 = Loo + Lol T* + L02 T*e 1 + L03T* + L04 T*2

(18)

where the coefficients S0. and L0. are given in table 2. Equations (17) and (18) reproduce the values of (s*)0 and (l*)0 for 0-8 1-3 differs drastically f r o m lower temperatures. For T * > 0.8, the zeroth-order term (s)0 gives a very good approximation: its deviation from the 'exact' results is less than 0.5% at p* = 0-45 and grows to about 1% for p* = 0.85. Addition of the linear term (s)l produces a further significant improvements at all densities. The effect of the second-order term (s)2 is very small at supercritical temperatures, but at subcritical temperatures is too large and negative. Nevertheless, even at subcritical temperatures the first-order approximation is excellent.

236


1.0.3

0-\ o\ \

1 01

S:~


0.99

0.97

0.95

0.8

I 1.5

I 1.8

2.5

2.'8

T*

Figure 10. Comparison of the first three terms in the virial series for (s*) at p* = 0.65. Zeroth order, solid line; first order, broken line, and second order, dashed line. The circles represent the values of (s) calculated from the gLj(r) of Verlet [9]. Therefore, an approximation for (s) that includes the second-order term (s)2 is very accurate at all densities for supercritical T*. A l t h o u g h it is impossible to use the truncated series to cross the two-phase region to reach the high-density liquid, it is possible for lower T* to use a semiempirical second-order term by s m o o t h i n g out the divergent behaviour o f (s)2 at low temperatures. The behaviour o f the m e a n collision range (l) with density and temperature is m o r e complex, but essential similar to that o f (s), as can be seen from figures 6 and 9. In this case, a density expansion truncated after the O(p 2) term gives the results

1.7

I*

1.5

1.4

I. 5

Figure 11.

0.8

1.

I3

1.8 T*

2)3

2)8

Comparison of the first three terms in the virial series for (1") at p* = 0.45. The symbols have the same meaning as in figure 10.


237

shown in figure 11 at the intermediate density p* = 0-45, which includes the values of (l) obtained from gLj(r) above. At supercritical temperatures, the first-order approximation is excellent. Nevertheless, the second-order term (l)2 seems too large, except at the highest temperatures (T* ~- 3.0). Again, the steep changes of the higher order density coefficients at low temperatures prevents the use of the density series in equation (9) in the subcritical region.


5.

Conclusions

It has been shown here that the pressure in simple classical fluids can be written in terms of the frequencies VR and vA, the mean diameter (s) and the mean range (l), as average parameters characterizing a given species of interest. The repulsive collisions contribute to the pressure of the system through two quantities: the frequency vR and the diameter (s). For a Lennard-Jones system the diameter (s) is almost constant, and takes values very close to the diameter cr. The frequency vR is similar to that of a hard-sphere fluid both in magnitude and density dependence. Hence, as a first estimate of the effect of the repulsive forces, the system of interest could be substituted by a HS fluid with effective diameter ~rEv = cr. As a second approximation the state dependence of the diameter and the temperature dependency of VR should be taken into account by making crEV= (s). Nevertheless, the sharp increase of VR at subcritical temperatures, which is due to the presence of the LJ potential well, cannot be incorporated by any purely repulsive system such as hard spheres. Of the attractive collision parameters, vA and (l), the first behaves very much like the attractive frequency in a square-well system, but the second is far from being constant. The mean range of the attractive forces grows quite noticeably with density although only slightly with temperature. It has also been shown that (s) and (1) have a smooth density and temperature dependence through the whole fluid range and that their low density limits (s)0 and (l)0 dominate their temperature behaviour. The first-order approximation to (s) and (l) is accurate within 0"5% at intermediate (supercritical) densities and supercritical temperatures. Finally, detailed analysis of the elastic collisions in the Lennard-Jones system, and the availability of closed-form equations of state for the square-well system [19], suggests the use of the latter to generate equations of state for realistic simple fuids. From the discussion in section 4 of this paper, it is necessary to make the SW diameter CrEF_~ (S) and its range REF ~ (l) in order to incorporate the main features of more realistic potentials. The authors recognize the hospitality of the Universidad Complutense de Madrid where part of this work was written. Alejandro Gil Villegas was partially supported by a fellowship from the Consejo Nacional de Ciencia y Tecnologia (M6xico) and the Sistema Nacional de Investigadores. References

[1] DEL R~O, F., 1992, Molec. Phys., 76, 21. [2] ENSKO~,D., 1922, Ark. Mat. Astron. Fys., 16, 16. [3] RESmOIS,P., and DE LEENER, M., 1977, Classical Kinetic Theory of Fluids (Oxford) Ch. VI.

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