ds 136 (2007) 156–160www.elsevier.com/locate/molliq

Journal of Molecular Liqui

Self-diffusion coefficients of dense fluid for a square-well fluid

Rajat Srivastava, K.N. Khanna ⁎

Department of Physics, V.S.S.D. College, Kanpur, India

Received 15 May 2006; accepted 26 February 2007Available online 1 April 2007

Abstract

Self-diffusion coefficients for a dense fluid of particles interacting with a square-well potential employing high temperature approximationhave been described. Further, the dependence of the diffusion coefficient and shear viscosity on the excess entropy have been analyzed for asquare-well potential. Hence, scaling laws of diffusion coefficients and shear viscosity have been described separately for square-well fluids.© 2007 Elsevier B.V. All rights reserved.

Keywords: Self-diffusion coefficient; Square-well potential; Stokes–Einstein relation

1. Introduction

The knowledge of the diffusion coefficients of dense fluidsin the bulk state has increased dramatically over the past decade[1]. The study of diffusion is very important for the devel-opment of our understanding of molecular motions andinteractions in the dense fluids. The knowledge of the migrationmolecules in dense fluids such as in separation processes,tertiary-oil recovery and the extraction of essential oils fromvegetable matrices etc is an important parameter for designingthese processes.

For monoatomic gases, the diffusion coefficients may becalculated at any temperature using exact kinetic theory [2]while for monoatomic fluids the Enskog kinetic theory for ahard-sphere fluid has been used with reasonable success [3]. Inthis theory the transport properties are calculated through theuse of simple equations relating the particle mass, temperature,density, particle size and radial distribution function at contactdiameter. Several empirical methods [4–6] have been devel-oped to calculate diffusion coefficients for normal real fluidsbased on smooth or rough-hard-sphere systems. Liu et al. [7]have given an extensive review of the methods to calculatediffusion coefficients for simple fluids. In the studies of equilib-

⁎ Corresponding author.E-mail address: knkhanna@rediffmail.com (K.N. Khanna).

0167-7322/$ - see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.molliq.2007.02.004

rium properties, Lennard–Jone (LJ) and hard sphere potentialshave been widely used to study the behaviour of simple andcompound fluids. Yu and Gao [8] developed an equation toestimate self-diffusion coefficients on Lennard–Jone chain (LJC)model. This includes three frictional terms as the temperature-dependent hard-sphere contribution, using an effective hard-sphere diameter suggested by Ben-Amotz and Herschbach [9]and the chain contribution based on MD simulation data ofthe hard-sphere chain (HSC) fluids [10]. Yu et al. [11] calcu-lated the self-diffusion coefficients for a dense fluid ofparticles interacting with a square-well potential by employingChapman–Enskog method which has further been correctedby using molecular dynamics simulation data [12–14]. Theyhave employed the expression of g(σ) that was obtained fromfirst-order perturbation term of the pressure equation. In thepresent work, we have determined gSW(σ) by employing hightemperature expression.

Transport properties account for the coupling betweendiffusive motions and local structural relaxations. Most of theworkers [15–18] have defined this coupling by finding therelationship between diffusion coefficients and excess entropy.Dzugutov [19] has proposed that the normalized diffusioncoefficient d⁎ is proportional to eSex, where Sex is the excessentropy in the unit of kB. Recently, Bretonnet [20] hasinvestigated the exponential behaviour between d⁎ and Sexover a restricted range of densities for hard sphere system. Wehave investigated the behaviour of square-well potential indefining the relation between d⁎ and Sex.

157R. Srivastava, K.N. Khanna / Journal of Molecular Liquids 136 (2007) 156–160

In recent past considerable interest has been shown in ob-taining accurate and reliable values of shear viscosity of squarewell fluid [21]. Hence in the present work, the scaling law ofshear viscosity has also been investigated for square-well fluids.

2. Theory

The square-well fluid is an excellentmodel for a liquid inwhichthe internal degrees of freedom of the individual atoms are notimportant. The pair potential for a square-well fluid is defined as

uðrÞ ¼l rV r

�Є rb rbkr0 rzkr

8<: ð1Þ

where r is the radial coordinate, σ is the diameter of hard core, λσis the diameter of the surrounding well and Є is the magnitude ofattractive part of the potential. The well width is considered asλ=1.5 throughout this paper.

For the dense gases, the well known Enskog equation for ahard-sphere fluid is given by

DE ¼ D0

gHSðrÞ ¼3

8qr2kBTkm

� �1=2 1gHSðrÞ ð2Þ

the radial distribution function at the contact gHS(σ) can becalculated from Carnahan–Starling equation [22]

gHSðrÞ ¼ ð1� 0:5gÞ=ð1� gÞ3 ð3Þwhere η=πρσ3 /6 is the packing density denoting the fractionof the volume occupied by the hard spheres.

For the square-well fluid, Eq. (2) can be modified by employ-ing Chapman–Enskog method. The self-diffusion coefficientobtained by using the Chapman–Enskog method of solution canbe written as [23 24]

D ¼ 38qr2

kBTkm

� �1=2 1

½gSWðrÞ þ k2gSWðkrÞE� ð4Þ

where ρ is the number density of the fluid, m is the mass of theparticle, T is the absolute temperature, kB is the Blotzmann con-stant, gSW(σ) and gSW(λσ) are the radial distribution functionsfor a square-well potential evaluated at the pointsσ+0 and λσ+0respectively. The quantitiesE and gSW(λσ) are defined as [11,25].

E ¼ expЄkBT

� �� Є2kBT

� 2J ð4AÞ

J ¼ 0:5þ 0:28304=T⁎

1þ 0:15360=T⁎ ð4BÞ

and

gSWðkrÞ ¼ gHSðkrÞexp aT⁎ þ

bT⁎2

� �ð4CÞ

Where T⁎¼ kBT=Є; a ¼�0:4317; b¼ �0:1177 and gHS(λσ)=0.99948+0.82404η−3.46976η2.

In the present work, the value of gSW(σ) is determined byemploying high temperature expansion method [26] rather thanfirst order perturbation of pressure equation as employed by Yuet al. [11,27]. In high temperature expansion method gSW(σ)can be determined as follows [26]

gSWðrÞ ¼ gHSðrÞ þ 14T⁎

AaSW1Ag

þ k3

T⁎ gHSðkrÞ ð5Þ

where a1SW is the first order perturbation term associated with

attractive energy −εφ as given below

aSW1 ¼ �4gðk3 � 1Þ 1� geff=2

ð1� geff Þ3( )

ð6Þ

and

geff ¼ c1 þ c2g2 þ c3g

3 ð7Þc1, c2, c3 are the matrices and are given in Ref. [26] and ηeff is

used in the determination of a1SW. Eq. (4) is still not appropriate to

calculate diffusion coefficients at high densities. The correctionfactors due to friction coefficients play an important role in thedetermination of diffusion coefficients. The corrected self-diffusion coefficient for square-well fluid can be written as [11]

D ¼ 38qr2

kBTkm

� �1=2 1

½gSWðrÞ fRðq⁎Þ þ k2gSWðkrÞE fSðq⁎Þ�ð8Þ

The correction factors fR(ρ⁎) and fS(ρ⁎) due to hard corerepulsive and due to attractive part can be written as [11]

fRðq⁎Þ ¼ ð1� gÞ3ð1� 0:5gÞð1� q⁎=1:09Þð1þ 0:4q⁎2 � 0:83q⁎4Þ

ð9Þand

fSðq⁎Þ ¼ 70:771q⁎3 � 58:971q⁎2 þ 19:903q⁎ � 1:3708 ð10Þwhere ρ⁎ is reduced density (=ρσ3).

The reduced diffusion coefficient for a square-well fluid isdefined by Yu et al. [11] as

D⁎ ¼ Dqr2mkBT

� �1=2

ð11Þ

Thus the reduced diffusion coefficient can now be calculated byemploying high temperature expansion approximation.

Diffusion coefficient is related with the structural relaxation,hence there is a coupling between diffusive motions and localstructural relaxations. Rosenfeld [15,16] has related the trans-port coefficients and internal entropy of dilute and dense fluids.Dzugutov [19] has suggested a link between the structure andthe rate of atomic diffusion in condensed matter. Bretonnet [20]has recently presented the analysis of diffusion coefficient ofdense fluids of hard spheres as a function of excess entropy and

Fig. 1. Reduced self-diffusion coefficient for a square-well fluid employing hightemperature expansion method as a function of reduced density, shown by solidlines. Symbols are MD results of Michels and Trappeniers [14]. □T⁎=5.0;○T⁎=2.0 and ▵T⁎=1.316.

158 R. Srivastava, K.N. Khanna / Journal of Molecular Liquids 136 (2007) 156–160

compared with the scaling law proposed by Dzugutov [19]. Itwould be very interesting to know that how far this scaling lawis followed for square-well fluids. Hence, in the present work,we have formulated a relation between the diffusion coefficientof square-well fluid as a function of the excess entropy.

The reduced diffusion coefficient d⁎ (as two authors [11,20]defined the reduced diffusion coefficient in a different way) isrelated with excess entropy Sex by the following relationship[20]

d⁎HS ¼ Dr2Ch

¼ AeSex ð12Þ

where A is a constant and Γh is referred to as the Enskogcollision rate for hard sphere fluid.

Ch ¼ 4ðkkBT=mÞ1=2qr2gHSðrÞ ð13ÞFor square well fluid, we modify the Enskog collision rate Γh

as ΓS:

CS ¼ 4ðkkBT=mÞ1=2qr2½gSWðrÞ fRðq⁎Þ þ k2gSWðkrÞEfSðq⁎Þ�ð14Þ

By substituting the value of D from Eq. (8) and ΓS fromEq. (14) in Eq. (12) we get,

d⁎SW ¼ 332kq⁎2

1

½gSWðrÞfRðq⁎Þ þ k2gSWðkrÞEfSðq⁎Þ�2ð15Þ

The excess entropy is calculated as the configurational entropyin terms of the equation of state and for square-well fluid, it can bederived as

Sex ¼ �Z q

o

dqq

qqkBT

� 1

� �ð16Þ

Sex ¼ Sh þ aSW1T⁎

þ aSW2T⁎2

ð17Þ

where Sh is the configurational entropy of hard sphere system andcan be written as by employing Carnahan–Starling expression

Sh ¼ �gð4� 3gÞð1� gÞ2 ð18Þ

a1SW and a2

SW are the attractive energies. a1SW is defined by Eq. (6)

and a2SW is given by

aSW2 ¼ 12d

gð1� gÞ4ð1þ 4gþ 4g2Þ d

AaSW1Ag

ð19Þ

The value of A (Eq. (12))can be calculated by using followingrelation

A ¼ d⁎SWe�Sex ð20Þ

Thus we can study the variation of diffusion coefficient withexcess entropy for a square-well potential. We can also study thescaling law of diffusion coefficient for a square-well potential.

The Stokes–Einstein (SE) relation between self-diffusioncoefficientD and shear viscosity ηshear of the fluid can be writtenas [21]

D ¼ kBT2kr gshear

ð21Þ

where kBT is the thermal energy and σ is the diameter of the hardspherical particle. Diffusion coefficient is already derived byEq. (8). The shear viscosity for square-well fluid can be obtainedby substituting Eq. (8) into Eq. (21) as

gSWShear ¼43ðkBTmÞ1=2

Mkqr½gSWðrÞfR þ k2fSEg

SWðkrÞ� ð22Þ

In reduced units

g⁎ ¼ gSWShearðkBTmÞ1=2q2=3

¼ 4

3Mkq⁎1=3½gSWðrÞfR þ k2fSEg

SWðkrÞ�

ð23ÞTo find a relationship between transport coefficient and struc-

tural properties, a scaling law is proposed by several workers[16,28]. The relationship between shear viscosity and excessentropy for a one component fluid can be expressed as

g⁎ ¼ g=ððmkBTÞ1=2q2=3Þ ¼ K expð�kSÞ ð24ÞWhere∅ and k are constants depending upon the nature of graphbetween shear viscosity and excess entropy.

3. Results and discussion

The Enskog theory with its various improvements is nowaccessible to calculate diffusion coefficient for more complexfluids in wider range of densities and temperatures. Chapman–

Fig. 2. Reduced self-diffusion coefficient d⁎ of dense square-well fluids as afunction of the excess entropy Sex for T⁎=2. The variation of the quantity A fora square-well fluid is also plotted vs Sex. The dashed line corresponds to resultsproposed by Dzugutov [19]. Comparison is also made between the resultsobtained from a square-well fluid and a hard sphere fluid.

159R. Srivastava, K.N. Khanna / Journal of Molecular Liquids 136 (2007) 156–160

Enskog transport theory has become the backbone to thetransport treatment. In the present work, we have calculated thediffusion coefficient derived on the basis of Chapman–Enskogtheory. The equilibrium radial distribution function gSW(σ) and

Fig. 3. Same as in Fig. 2 for T⁎=5.

gSW(λσ) are the two important factors in the determination ofdiffusion coefficient. In the present work, the radial distributionfunction gSW(σ) is determined by employing high temperatureexpansion method [26]. The self-diffusion coefficient can beobtained from Eq. (8) in association with Eq. (5). A correctedreduced self-diffusion coefficient D⁎ by including the frictioncoefficients (fR and fS) for square well fluid is compared withcomputer simulation results in Fig. 1 for T⁎=1.36, 2 and 5.We find a good agreement between theoretical results by em-ploying gSW(σ) in the high temperature approximation and sim-ulation data. It is important to mention that friction coefficients( fR and fS) play an important role at high densities. It has beenobserved that the friction coefficients become more dominantwith increasing density due to the presence of many-body inter-actions and the diffusion coefficient predicts incorrect results athigh densities without considering friction coefficients. Eq. (4)underestimates the self-diffusion coefficients at lower densitiesand overestimates it at higher densities. However, discrepanciesstill remain present at low temperatures region and high densitiesby including friction coefficients.

A quasi universal scaling lawwith the use of square-well fluidfor which the analytical expression is derived in the present workis shown in Figs. 2 and 3. Reduced diffusion coefficient d⁎SWof

Fig. 4. Diffusivity, in reduced units, as a function of reduced temperature T⁎ atvarious reduced densities ρ⁎ for square-well fluids.

Fig. 5. Plot of the reduced shear viscosity η⁎ vs excess entropy Sex of onecomponent square-well fluids.

160 R. Srivastava, K.N. Khanna / Journal of Molecular Liquids 136 (2007) 156–160

dense fluids determined by employing square well fluid is shownas a function of excess entropy in Figs. 2 and 3 for T⁎=2 and 5respectively. It can be seen that it does not exhibit straight linebut exhibits exponential behaviour as predicted by Bretonnet[20] for hard sphere system.

The hard sphere system and square well potential are alsocompared in Figs. 2 and 3 for T⁎=2 and 5 respectively. It isfound that the square well potential predicts higher values of lnd⁎ and A than hard sphere system for both the temperaturesT⁎=2 and 5. It may be due to that square well potentialincludes attractive forces. It means attractive forces increasediffusion coefficient. Fig. 4 shows the Arrhenius plot ofdiffusivity Eq. (15), in reduced units, as a function of reducedtemperature 1 /T⁎ for square-well potential.

Finding a relationship between the transport coefficients andstructural properties in fact remains one of the most challengingtask in the field of condensed matter dynamics. So far thetheoretical developments of the scaling law were confined to thediffusivity of the fluids, the similar scaling law for shearviscosity is rather scarce. Recently, Rosenfeld [16] and Ali [28]have presented the scaling law for viscosity for one componentL–J fluids. In the present work, we have presented the scalinglaw for square-well one component fluids. The reduced shearviscosity for square-well fluids has been derived from diffusioncoefficients (Eq. (8)) by employing Stokes–Einstein relation(Eq. (21)). The scaling law of shear viscosity has been drawnbetween the reduced shear viscosity (Eq. (23)) and excessentropy (Eq. (17)) for square-well fluids in Fig. 5. We have

computerized the best fitting values of ∅ and k of Eq. (24). Forscaling law curve (Fig. 5) of square-well fluids, we haveobtained ∅=0.42 and k=0.4. Ali [28] has derived these valuesas ∅=0.37 and k=0.7 for L–J potential.

4. Conclusion

The present work concludes that the high temperatureexpansion approximation is a good approach to determine thepair correlation function and can be applied to determine thetransport properties. Further, a relation between the diffusioncoefficient and the structural properties of dense fluids has beeninvestigated for square well fluid.

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