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Journal of Molecular Liquids 189 (2014) 30–38
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Journal of Molecular Liquids
j ourna l homepage: www.e lsev ie r .com/ locate /mol l iq
Hard convex body fluids in random porous media: Scaled particle theory
Myroslav Holovko ⁎, Volodymyr Shmotolokha, Taras PatsahanInstitute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, 1 Svientsitskii Str., 79011 Lviv, Ukraine
⁎ Corresponding author. Tel.: +380 322760614.
0167-7322/$ – see front matter © 2013 Elsevier B.V. Allhttp://dx.doi.org/10.1016/j.molliq.2013.05.030
a b s t r a c t
a r t i c l e i n f oAvailable online 13 July 2013
Keywords:Hard convex body fluidPorous mediaScaled particle theoryMonte Carlo simulationsPorosityHard spherocylinder fluid
The scaled particle theory (SPT) is applied to describe thermodynamic properties of hard convex body (HCB)fluid in random porous media. To this purpose we extended the SPT2 approach, which has been developedpreviously for a hard sphere fluid in random porous matrices. The analytical expressions for the chemicalpotential and the pressure of an HCB fluid in an HCB or an overlapping hard convex body (OHCB) matrixare obtained and analyzed. Within the SPT2 approach a series of new approximations are proposed. Thegrand canonical Monte Carlo (GCMC) simulations are performed to verify an accuracy of the SPT2 and thedeveloped approaches. The obtained analytical expressions include three types of porosity: the geometricalporosity ϕ0, the probe particle porosity ϕ and the porosity ϕ⁎ defined by the maximum packing fraction ofa fluid in a given matrix. The effect of nonsphericity parameters of matrix and fluid particles on the thermo-dynamic properties of a fluid is discussed.
© 2013 Elsevier B.V. All rights reserved.
1. Introduction
The thermodynamic properties of a fluid adsorbed in porous ma-terials differ to that in a bulk. Mainly it is caused by the excluded vol-ume effect and by a pore wall presence. The pore walls affect a fluidnot only by an attractive interaction with fluid molecules, but alsoby their geometry. Even without any interaction of a fluid with a po-rous material (i.e. hard-core interaction), the pore walls change ther-modynamics of a fluid essentially, since a pore wall surface area, porecurvatures, and distances between pore walls play an important role.Therefore, a geometrical aspect of porous medium confining a fluid isto be studied. And it is useful not only from a practical point of view,but it is also valuable as a fundamental study. During the years nu-merous investigations have been done in this area, including a num-ber of theoretical approaches which have been proposed for amodel of a fluid in pores [1,2]. One of the most popular models is afluid in random porous media [3,4]. Within this model a system ispresented as a set of spherical particles, a part of them can moveand it relates to a fluid, but another part is frozen and depicts solidmaterial, i.e. a matrix. Therefore, a void between matrix particlesforms porous medium, in which fluid particles are confined. A maininterest to this kind of system is provoked by a large class of inorganicand inorganic/organic hybrid materials forming a random network ofmesopores, which are widely used in industry. However, only few ofthe developed theories are able to describe efficiently the thermody-namic properties of fluids confined in a whole variety of such mate-rials, and practically none of them can give accurate pure analytical
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expressions even for the simplest model, such as an HS fluid in anHS matrix.
Recently, we proposed an extension of the scaled particle theory(SPT) [8–10], which enabled us to derive the analytical expressionsof equation of state for a hard sphere fluid (HS fluid) confined in ahard sphere matrix (HS matrix) or in an overlapping hard sphere ma-trix (OHS matrix) [5,6]. This approach is based on a combination ofthe exact treatment of a point scaled particle in an HS fluid with thethermodynamic consideration of a finite size scaled particle. Theexact result for a point scaled particle in an HS fluid in a random po-rous medium was obtained in [5]. A further improvement of the SPThas led us to the SPT2 approach [7]. The expressions obtained in theSPT2 include two types of porosities. One of them is defined by apure geometry of porous medium (geometrical porosity ϕ0), andthe second one is defined by the chemical potential of a fluid in thelimit of infinite dilution (probe particle porosity ϕ). On the basis ofthe SPT2 approach, the SPT2b approximation was proposed, whichreproduced the computer simulation data with a high accuracy atsmall and intermediate fluid densities [7]. However, the expressionsobtained in the SPT2 and in the SPT2b approximations have a diver-gence at densities corresponding to a fluid packing fraction equal tothe probe particle porosity. Consequently, the prediction of thermo-dynamic quantities at high densities of a fluid becomes incorrect.An accuracy of the SPT2 and the SPT2b also can get worse if fluidand matrix particles are of comparable sizes [7]. In a study of aone-dimensional hard rod fluid in random porous media [11,12] thenew approximations SPT2b1, SPT2b2 and SPT2b3 were proposed,and they are free of the mentioned defect. In the SPT2b2 and SPT2b3approximations a third type of porosity ϕ⁎ is introduced, and it isdefined by themaximum value of packing fraction of a fluid in a porousmedium. It was shown that these new approximations essentially
31M. Holovko et al. / Journal of Molecular Liquids 189 (2014) 30–38
improve an accuracy of the SPT2 approach. The application of the SPT2approach for a description of thermodynamic properties of an HS fluidconfined in random porous media was reviewed recently in [13].
A remarkable feature of the SPT theory is a possibility to describesystems of hard convex body particles [14,15]. Therefore, in principlean application of the SPT2 approach is not restricted only to a fluid ofspherical particles. Matrix particles can be also of nonspherical shape,thereby a wider range of random medium morphologies can be in-volved in a description. A purpose of the present study is to extendthe SPT2 approach to the case of a hard convex-body fluid confinedin a hard convex-body matrix. Similarly to our previous studies[5–7,13], we consider two models depicting a porous medium struc-ture. The first model is a hard convex body fluid (HCB fluid) confinedin a hard convex bodymatrix (HCBmatrix). The second one is an HCBfluid in an overlapping hard convex body matrix (OHCB matrix). Theobtained analytical expressions for these models allow us to calcu-late the chemical potential and the pressure of a confined fluidwithin these models. An effect of nonsphericity of fluid and matrixparticles on thermodynamic properties of a fluid is studied. Thenew approximations SPT2b1, SPT2b2 and SPT2b3 are applied andanalyzed. To test the developed theory the computer simulationsare performed for the case of an HCB fluid in an OHCB matrix usingthe method of grand-canonical ensemble Monte Carlo (GCMC).Therefore, the corresponding comparison of the SPT results withthe GCMC data is made in our study.
The paper is organized as follows. A brief review of a non-standardformulation of the SPT for an HCB fluid is presented in Section 2. InSection 3 this formulation is generalized on the case of an HCB fluidin a random porous medium. In Section 4 the SPT will be applied todescribe an HCB fluid in an HCB or an OHCB matrix. Some computersimulation details will be presented in Section 5. In Section 6 wepresent numerical results obtained from the proposed theory usingthe different approximations and make a comparison of the differentapproximations with the simulation data. Also we discuss an effect ofthe nonsphericity parameters of fluid andmatrix particles on thermo-dynamical properties of a fluid in a porous medium. In the last sectionsome conclusions are drawn.
2. SPT theory for HCB fluid
HCB particles are characterized by three functionals—a volume V, asurface area S and the mean curvature R with a factor (1 / 4π). For ex-ample, for a frequently considered case of a system of spherocylindricalrods with a length l and a radius r, these functionals are
V ¼ πr2lþ 43πr3; S ¼ 2πrlþ 4πr2; R ¼ 1
4lþ r: ð1Þ
The basic idea of the SPT approach is an insertion of an addi-tional scaled particle of a variable size into a fluid. With this aimwe use the scaling parameter λs in such a way that the volumeVs, the surface area Ss and the curvature Rs of a scaled particle aremodified as
Vs ¼ λ3s V1; Ss ¼ λ2
s S1; Rs ¼ R1λs; ð2Þ
where V1, S1 and R1 are the volume, the surface area and the meancurvature of fluid particle, respectively.
In the original formulation of the SPT theory for HCB fluids[14–18] the contact value of the pair distribution function of fluidparticles around a scaled particle is a key function of the theory.However, as it was indicated in [5] for an HS fluid such a formulationcannot be applied directly to fluids in random porous media, be-cause there is no direct relation between the pressure and the con-tact distribution function which has been found for a fluid in aporous medium. Due to this in [5] a reformulation of the SPT theory
was presented, which is slightly different from the original one.In this paper we consider a generalization of this approach for anHCB fluid.
The procedure of insertion of the scaled particle into a fluid isequivalent to a creation of cavity, which is free of any other fluidparticles. The key point of considered reformulation of the SPT theoryconsists in a derivation of the excess chemical potential of a scaledparticle μsex, which is equal to a work needed to create a correspond-ing cavity. The expression of excess chemical potential for a smallscaled particle in an HCB fluid is as follows
βμexs ¼ βμs− ln ρ1Λ
31
� �¼ − ln 1−η1 1þ RsS1
V1þ SsR1
V1þ Vs
V1
� �� �¼ 1− ln 1−η1 1þ 3λsα1 þ 3λ2
sα1 þ λ3s
� �h i;
ð3Þ
where β = 1/kT, k is the Boltzmann constant, T is the temperature,η1 = ρ1V1 is the fluid packing fraction, ρ1 is the fluid density, Λ1 is thefluid thermal wave length, α1 ¼ R1S1
3V1is the parameter of nonsphericity
of fluid particles.For a large scaled particle the excess chemical potential μsex is given
by a thermodynamical expression for the work needed to create amacroscopic cavity inside a fluid and it can be presented as
βμexs ¼ w λsð Þ þ βPVs; ð4Þ
where P is the pressure of fluid and Vs is the volume of scaled particle.According to the ansatz of SPT theory [5–14], w(λs) can be
presented in the form of expansion
w λsð Þ ¼ w0 þw1λs þ12w2λ
2s : ð5Þ
The coefficients of this expansion can be found from the continuityof μsex and the corresponding derivatives ∂ μsex/∂ λs and ∂ 2μsex/∂ λs
2 atλs = 0. As a result one derives the following expressions
βw0 ¼ − ln 1−η1� �
; ð6Þ
βw1 ¼ 3α1η11−η1
; ð7Þ
βw2 ¼ 6α1η11−η1
− 9α21η
21
1−η1� �2 : ð8Þ
If one sets λs = 1 Eq. (4) should give the relation between thepressure and the chemical potential of an HCB fluid. Using theGibbs–Duhem equation
∂P∂ρ1
� �T¼ ρ1
∂μ1
∂ρ1
� �T; ð9Þ
we obtain the final expressions for the chemical potential and thepressure of a bulk fluid:
βμex1 ¼ − ln 1−η1
� �þ 1þ 6α1ð Þη11−η1
þ92α1 þ 3� �
α1η21
1−η1� �2 þ 3α2
1η31
1−η1� �3 ; ð10Þ
βPρ1
¼ 11−η1
þ 3α1η11−η1� �2 þ 3α2
1η21
1−η31� � ; ð11Þ
which coincide with the corresponding results obtained in thecommon SPT theory [15].
32 M. Holovko et al. / Journal of Molecular Liquids 189 (2014) 30–38
3. SPT theory for HCB fluid in random porous media
In the case of a porous medium presence the expression for theexcess chemical potential of a small scaled particle in an HCB fluidcan be written in the form
βμexs ¼ βμs− ln ρ1Λ
31
� �¼ − ln p0 λsð Þ−η1 1þ RsS1
V1þ SsR1
V1þ Vs
V1
� �� �
¼ − lnp0 λsð Þ− ln 1− η1p0 λsð Þ 1þ 3λsα1 þ 3λ2
sα1 þ λ3s
� �� ;
ð12Þ
where p0(λs) = exp(−βμs0(λs)) is defined by the excess chemicalpotential μs0(λs) of the scaled particle confined in an empty matrixand it has a meaning of probability to find a cavity created by a scaledparticle in the matrix in the absence of fluid particles. For the caseλs = 0
p0 λs ¼ 0ð Þ ¼ ϕ0; ð13Þ
where ϕ0 is the geometrical porosity, which depends only on astructure of matrix and is related to the volume of void between ma-
trix particles. The term p1=0 λsð Þ ¼ 1− η1p0 λsð Þ 1þ 3λsα1 þ 3λ2
sα1 þ λ3s
� �is the probability to find a cavity created by a scaled particle in afluid-matrix system under a condition that the cavity is locatedentirely inside a pore, in which the scaled particle is located.
For a large scaled particle the excess chemical potential μsex is givenby a thermodynamical expression similar to Eq. (4)
βμexs ¼ w λsð Þ þ βPVs
1p0 λsð Þ : ð14Þ
A difference from the bulk case consists only the presence of mul-tiplier 1/p0(λs) in the last term. This multiplier appears due to an ex-cluded volume occupied by matrix particles. It was introduced in [7],where p0(λs) was considered as a probability to find a cavity in thematrix for a scaled particle of a finite size. This probability is relateddirectly to the second type of matrix porosity, which is defined as
p0 λs ¼ 1ð Þ ¼ Φ ð15Þ
and has a meaning of the thermodynamic porosity. In general thethermodynamic porosity Φ has also a dependence on the fluid densi-ty. A form of this dependence is unknown, however it becomes im-portant at rather high fluid densities and/or under very strongconfinement conditions (low porosity, small matrix particles). As areasonable approximation one can neglect the dependence of Φ onthe fluid density setting Φ equal to the probe particle porosity ϕ in-troduced in [7] as
ϕ ¼ exp −βμ01 λs ¼ 1ð Þ
h i; ð16Þ
where μ10 is the fluid chemical potential in the limit of infinite dilution(ρ1 → 0).
Similar to the bulk case,w(λs) can be presented as Eq. (5) in termsof the coefficients w0, w1 and w2, which can be found using the deriv-atives ∂ μsex/∂ λs and ∂ 2μsex/∂ λs
2 at λs = 0. Consequently, one derivesthe following expressions [13]
w0 ¼ − ln 1−η1=ϕ0� �
;
w1 ¼ η1=ϕ0
1−η1=ϕ03α1−
p′0ϕ0
!;
w2 ¼ η1=ϕ0
1−η1=ϕ06α1−6α1
p′0ϕ0
þ 2p′0ϕ0
!2
−p″0ϕ0
" #þ η1=ϕ0
1−η1=ϕ0
� �23α1−
p′0ϕ0
!2
;
ð17Þ
where p′0 ¼ ∂p0 λsð Þ∂λs
and p″0 ¼ ∂2p0 λsð Þ∂λ2
sat λs = 0.
After setting λs = 1 Eq. (4) gives the relation between the pres-sure P and the excess chemical potential μ1ex of a fluid in a matrix
β μex1 −μ0
1
� �¼ − ln 1−η1=ϕ0
� �þ Aη1=ϕ0
1−η1=ϕ0þ B
η1=ϕ0� �21−η1=ϕ0� �2 þ βP
ρ1
η1Φ
; ð18Þ
where the coefficients A and B define the porous media structure andthe expressions for them are as follows
A ¼ 3α1−p′0ϕ0
þ 12
6α1−6p′0ϕ0
α1 þ 2p′0ϕ0
!2
−p″0ϕ0
" #; B ¼ 1
23α1−
p′0ϕ0
!2
: ð19Þ
Using the Gibbs–Duhem equation (Eq. (9)) one derives the com-pressibility in the form
β∂P∂ρ1
� �T¼ 1
1−η1=Φ� �þ 1þ Að Þ η1=ϕ0
1−η1=Φ� �
1−η1=ϕ0� �
þ Aþ 2Bð Þ η1=ϕ0� �2
1−η1=Φ� �
1−η1=ϕ0� �2 þ 2B
η1=ϕ0� �3
1−η1=Φ� �
1−η1=ϕ0� �3
þ βPη11−η1=Φ
⋅∂ 1Φ
∂ρ1;
ð20Þ
which is a differential equation for the pressure. To solve this differ-ential equation we should know a dependence of the thermodynamicporosity Φ on the fluid density ρ1. At low densities one can make abasic assumption Φ = ϕ. In this case the last term in Eq. (20) van-ishes and the description of a fluid in a matrix reduces to the SPT2 ap-proach [7,12,13]. An integration of Eq. (20) leads us to the excesschemical potential and the pressure of a fluid
β μex1 −μ0
1
� �¼ − ln 1−η1=ϕ
� �þ Aþ 1ð Þ ϕϕ0−ϕ
ln1−η1=ϕ1−η1=ϕ0
þ Aþ 2Bð Þ ϕϕ0−ϕ
η1=ϕ0
1−η1=ϕ0− ϕ
ϕ−ϕ0ln
1−η1=ϕ1−η1=ϕ0
� �
þ2Bϕ
ϕ0−ϕ
�12
η1=ϕ0� �21−η1=ϕ0� �2 − ϕ
ϕ0−ϕη1=ϕ0
1−η1=ϕ0
þ ϕ2
ϕ−ϕ0ð Þ2 ln1−η1=ϕ1−η1=ϕ0
;
ð21Þ
βPρ1
¼ − ϕη1
ln1−η1=ϕ1−η1=ϕ0
þ 1þ Að Þ ϕη1
ϕϕ−ϕ0
ln1−η1=ϕ1−η1=ϕ0
þ Aþ 2Bð Þ ϕϕ−ϕ0
11−η1=ϕ0
− ϕη1
ϕϕ−ϕ0
1−η1=ϕ1−η1=ϕ0
�
þ2Bϕ
ϕ−ϕ0��12
η1=ϕ0
1−η1=ϕ0� �2 −2ϕ−ϕ0
ϕ−ϕ0
11−η1=ϕ0
þ ϕη1
ϕ2
ϕ−ϕ0ð Þ2 ln1−η1ϕ1−η1=ϕ0
:
ð22Þ
It was shown in [12,13] that the obtained expressions reproducecorrectly the second virial coefficient which can be presented as
B2 ¼ V11ϕþ 1þ A
ϕ0
� : ð23Þ
As is seen expressions (21) and (22) have two divergences at highfluid densities, which correspond to η1 = ϕ and η1 = ϕ0 respectively.Since ϕ b ϕ0 the first divergence in η1 = ϕ occurs at lower densitiesthan the second one. It causes a significant overestimation of thechemical potential of a fluid at the high densities. From a geometricalpoint of view such a divergence should appear near the maximumvalue of fluid packing fraction η1max available for a fluid in a givenmatrix. This value is defined by Φ(η1) at η1 = η1max and it is largerthan ϕ and smaller than ϕ0. The divergence in η1 = ϕ is related tothe presence of Φ in the denominator of the last term of Eq. (18)
33M. Holovko et al. / Journal of Molecular Liquids 189 (2014) 30–38
and to the assumption that Φ is constant. On the other hand thisassumption is acceptable at low and intermediate densities.
In our previous works [7,12,13] we proposed an improvement ofthe description of an HS fluid at high densities in the form of the cor-rections for expressions (21) and (22). The same corrections can beeasily extended to the case of an HCB fluid in random porous media.A goal of the modifications to be made is to decrease the role of thedivergence at η1 = ϕ or to shift this divergence to higher densitiesand in the same time not to lose the advantages of expressions (21)and (22) at low densities. The first successful approximation calledSPT2b [7] can be derived if Φ is replaced by ϕ0 everywhere in the ex-pression for the compressibility (20) except for the first term, whereΦ is replaced by ϕ. As a result Eqs. (21) and (22) can be rewritten inthe following form
β μex1 −μ0
1
� �SPT2b ¼ − ln 1−η1=ϕ� �þ 1þ Að Þ η1=ϕ0
1−η1=ϕ0
þ12
Aþ 2Bð Þ η1=ϕ� �2
1−η=ϕ1=ϕ0ð Þ2 þ23B
η1=ϕ0� �31−η1=ϕ0� �3 :
ð24Þ
βPρ1
� �SPT2b¼ − ϕ
η1ln 1− η1
ϕ0
� �þ 11−η1ϕ0
þ A2
η1=ϕ0
1−η1=ϕ0� �2 þ 2B
3η1=ϕ0� �21−η1=ϕ0� �3 ; ð25Þ
The second approximation called SPT2b1 [12,13] corrects SPT2bby avoiding the divergence at η1 = ϕ through an expansion of thelogarithmic term in Eq. (24)
− ln 1−η1=ϕ� � ¼ − ln 1−η1=ϕ0
� �þ η1 ϕ0−ϕð Þϕ0ϕ 1−η1=ϕ0
� � : ð26Þ
Therefore, one obtains the SPT2b1 approximation as follows
β μex1 −μ0
1
� �SPT2b1 ¼ − ln 1−η1=ϕ0� �þ 1þ Að Þ η1=ϕ0
1−η1=ϕ0þ η1 ϕ0−ϕð Þϕ0ϕ 1−η1=ϕ0
� �þ12
Aþ 2Bð Þ η1=ϕ0� �21−η1=ϕ0� �2 þ 2
3B
η1=ϕ0� �31−η1=ϕ0� �3 ;
ð27Þ
βPρ1
� �SPT2b1¼ − 1
1−η1=ϕ1
ϕ0
ϕþ ϕ0
ϕ−1
� �ϕ0
η1ln 1− η1
ϕ0
� �
þA2
η1=ϕ0
1−η1=ϕ0� �2 þ 2B
3η1=ϕ0� �21−η1=ϕ0� �3 :
ð28Þ
Two other approximations called SPT2b2 and SPT2b3 contain thethird type of porosity ϕ∗ defined by the maximum value of packingfraction of a fluid in a given porous medium and they provide amore correct description of thermodynamic properties of a fluid inthe high-density region. To incorporate ϕ∗ in the expression for thechemical potential (Eq. (24)) the logarithmic term is modified inthe following way
− ln 1−η=ϕð Þ ¼ − ln 1−η=ϕ�� �þ η1 ϕ�−ϕð Þϕ�ϕ 1−η=ϕ�ð Þ : ð29Þ
Consequently, the SPT2b2 approximation is derived as
β μex1 −μ0
1
� �SPT2b2 ¼ − ln 1−η1=ϕ�� �þ η1=ϕ0
1−η1=ϕ01þ Að Þ
þ η1 ϕ�−ϕð Þϕ�ϕ 1−η1=ϕ
�� �þ 12
Aþ 2Bð Þ η1=ϕ0� �21−η1=ϕ0� �2
þ23B
η1=ϕ0� �31−η1=ϕ0� �3 ;
ð30Þ
βPρ1
� �SPT2b2¼ −ϕ�
η1ln 1−η1
ϕ�
� �ϕ0
η1ln
1η1
ϕ0
� �þ 11−η1=ϕ0
þϕ�−ϕϕ
ln 1−η1=ϕ�� �þ η1=ϕ
�
1−η1=ϕ�
�
þA2
η1=ϕ0
1−η1=ϕ0� �2 þ 2
3B
η1=ϕ0� �21−η1=ϕ0� �3 :
ð31Þ
Finally, the SPT2b3 approximation can be obtained similarly to Eq.(29) by an expansion of the logarithmic term ln(1 − η1/ϕ∗) in the ex-pression for the chemical potential (Eq. (28)). As a result we obtain
β μex1 −μ0
1
� �SPT2b3 ¼ − ln 1−η1=ϕ0� �þ η1=ϕ
�
1−η1=ϕ0þ η1 ϕ�−ϕð Þϕ�ϕ 1−η1=ϕ
�� �þA
η1=ϕ0
1−η1=ϕ0þ 12
Aþ 2Bð Þ η1=ϕ0� �21−η1=ϕ0� �2
þ23B
η1=ϕ0� �31−η1=ϕ0� �3 ;
ð32Þ
βPρ1
� �SPT2b3¼ ϕ�−ϕ
ϕln 1−η1
ϕ�
� �þ η1=ϕ
�
1−η1=ϕ�
� þ 11−η1=ϕ0
þϕ0−ϕ�
ϕ� ln 1−η1=ϕ0� �þ η1=ϕ0
1−η1=ϕ0
�
þA2
η1=ϕ0
1−η1=ϕ0� �2 þ 2
3B
η1=ϕ0� �2
1−η1=ϕ0� �
x3:
ð33Þ
It is worth noting that all of the presented approximations lead tothe same expression of the second virial coefficient (Eq. (23)).
It was shown recently in [12] by the example of one-dimensionalsystem of a fluid in a random matrix that ϕ∗ is related to ϕ0 and ϕthrough the following equation
1=ϕ� ¼ 1=ϕ−1=ϕ0ð Þ= ln ϕ0=ϕð Þ: ð34Þ
Since Eq. (34) is presented in the general terms and does not de-pend directly on a dimensionality of the system, we extend its appli-cation to the three-dimensional case. A simple analysis of Eq. (34)shows that in a bulk all the porosities are ϕ = ϕ0 = ϕ∗ = 1. There-fore, the expressions for pressure and chemical potential of a bulkHS fluid have a common divergence at η1 = 1 [19,20]. And thesame concerns a bulk HCB fluid. In case of a fluid in a porous mediumone gets an inequality ϕ b ϕ∗ b ϕ0 b 1. Also it should be noticed thatin case of a matrix particle size is essentially larger than a size offluid particles, ϕ tends to ϕ0, and all the considered approximationslead to the same result, which is equivalent to a bulk fluid with an ef-fective density η̂1 ¼ η1=ϕ0.
4. HCB fluid in HCB and OHCB matrices
According to the formalism presented above we apply the SPT the-ory for the description of an HCB fluid in an HCB or an OHCB matrix.We start from concretization of the expressions for the probabilityof a successful insertion of a scaled particle into an empty matrix.The corresponding expressions for p0(λs) are derived under conditionof λs ≤ 0 for the case of an HCB matrix
p0 λsð Þ ¼ 1−η0 1þ λsR1S0V0
þ λ2sS1R0
V0þ λ3
sV1
V0
� �
¼ 1−η0 1þ 3λsα0R1
R0þ 3λ2
sα0S1S0
þ λ3sV1
V0
� �ð35Þ
and for an OHCB matrix
p0 λsð Þ ¼ exp −η0 1þ 3λsα0R1
R0þ 3λ2
sα0S1S0
þ λ3sV1
V0
� �� ; ð36Þ
Table 1Parameters and characteristics of matrices in Systems A, B, C and D.
System η0 r0/R1 l0/r0 R1/R0 S1/S0 α0 ϕ0 ϕ
A 0.282 3 4/3 0.250 0.0667 1.111 0.754 0.556B 0.271 2 3 0.286 0.1000 1.346 0.762 0.493C 0.099 2 5 0.222 0.0714 1.658 0.9052 0.781D 0.167 1 20 0.167 0.0454 4.125 0.846 0.491
Fig. 1. Fluid in overlapping spherocylinder matrix. System D. (Number of particles isreduced for better presentation.)
34 M. Holovko et al. / Journal of Molecular Liquids 189 (2014) 30–38
where η0 = ρ0V0, ρ0 ¼ N0V , N0 is the number of matrix particles, V is
the volume of system, α0 ¼ R0S03V0
is the parameter of nonsphericity of
matrix particles, λs is the scaling parameter, V0, S0 and R0 are thevolume, the surface area and the mean curvature of matrix particles,respectively. Therefore, the geometrical porosity for an HCB matrixhas the form
ϕ0 ¼ p0 λs ¼ 0ð Þ ¼ 1−η0: ð37Þ
And for an OHCB matrix it is
ϕ0 ¼ p0 λs ¼ 0ð Þ ¼ e−η0 : ð38Þ
Using the common SPT theory [15] the following expression forthe probe particle porosity ϕ is derived
ϕ ¼ e−βμ01 ¼ 1−η0
� �exp
�−�
η01−η0
3α0R1
R0þ S1S0
� �þ V1
V0
� �
þ 3α0η20
1−η0� �2 3α0
R21
R20
þ 2V1
V0
!þ 3α2
0η30
1−η0� �3 V1
V0
! ð39Þ
in the case of an HCB matrix and
ϕ ¼ e−βμ01 ¼ exp −η0 1þ 3α0
R1
R0þ 3α0
S1S0
þ V1
V0
� �� ð40Þ
in the case of an OHCB matrix. It can be shown easily that ϕ is alwaysless than ϕ0, except the limit of a point particle fluid, in which ϕ =ϕ0. From Eq. (19) one obtains the coefficients A and B for an HCBfluid in an HCB matrix
A ¼ 3 2α1 þ α0η0
1−η0� �R1
R01þ 3α1ð Þ þ α0
S1S0
η01−η0� �þ 3α2
0R21
R20
η201−η0� �2
" #;
B ¼ 92
α1 þ α0R1
R0
η01−η0
� �2
ð41Þ
and for an HCB fluid in an OHCB matrix
A ¼ 3 2α1 þ α0η0R1
R01þ 3α1ð Þ þ α0
S1S0
η0 þ 3α20R21
R20
η20
" #;
B ¼ 92
α1 þ α0R1
R0η0
� �2:
ð42Þ
An accuracy of the different approximations proposed in thispaper for thermodynamical properties of an HCB fluid in random po-rous media should be tested by a comparison with computer simula-tion data. To this end some computer simulation results are obtainedfor the model of an HS fluid in an OHCB matrix. For the OHCB matrixparticles the prolate spherocylinder particles of the length l0 and theradius r0 were considered. The corresponding functionals V, S and Rcan be calculated using Eq. (1).
5. Simulation details
Computer simulation results are obtained using the grand-canonicalMonte-Carlomethod (GCMC) [21]. To this purpose a few systems de-scribing an HS fluid in a random porous matrix composed of hardconvex body particles are studied. As an example a particular caseof an OHCB matrix is considered, namely, an overlapping hardspherocylinder matrix. Within this model the matrix particles areprolate spherocylinders, their positions and orientations are set ran-domly, hence a void between them forms a random porous mediumwith a specific morphology, which depends on a length l0 and radius r0 ofspherocylinders. Thedifferent ratios l0 to r0 are checked in our simulations(l0/r0 = 4/3, 3, 5 and 20) to see an effect of nonsphericity of matrix parti-cles α0 on the thermodynamic properties of a confined hard sphere fluidand to make a comparison with the SPT results. Therefore, four matrices(Systems A, B, C and D) are considered, in which equilibrium densitiesof a fluid at different chemical potentials are obtained. The parametersof these matrices including the related ratios, the nonsphericity α0 andthe both porosities ϕ0 and ϕ are presented in Table 1. The fluid particlestaken as hard spheres of radius R1 can move in a space, which is notoccupied by matrix particles, while the matrix particles remain at theirfixed positions during a whole simulation run. A snapshot of one of thesimulated systems is presented in Fig. 1 (System D). The simulations areperformed in a cubic box with the periodic boundary conditions applied.A length of the box L is defined by the density of matrix particles ρ0 asL = (N0/ρ0)1/3. Due to a finite size of simulation box all observable quan-tities, related to a fluid, fluctuate depending on a matrix configuration.Thus, the results obtained from simulations should be averaged overdifferent matrix realizations. For each of the systems, eight indepen-dent matrix configurations are chosen. Since a number of matrix par-ticles in a simulation box are taken quite large (N0 = 4000) and theonly quantity of our interest is an equilibrium density of fluid parti-cles it is found to be enough to get the results with an appropriateprecision (statistical error less than 0.5%). To speed up the simula-tions, the linked cell list algorithm was used [21].
6. Results and discussion
In the previous sections we present the analytical expressions forthe chemical potential and the pressure of an HS fluid confined in an
Fig. 2. The probe particle porosity ϕ for an HCB fluid in an HCB or an OHCB matrix in dependence on α0 (left panel) and α1 (right panel). The ratios R1R0
and S1S0
are fixed.
35M. Holovko et al. / Journal of Molecular Liquids 189 (2014) 30–38
HS or an OHS matrix obtained in the different approximations. Theseexpressions are extended to the case of arbitrarily shaped hard con-vex particles, i.e. an HCB fluid in an HCB or an OHCB matrix. Withinthe proposed approximations three types of porosities are introduced,i.e. the geometrical porosity ϕ0, the specific probe particle porosity ϕ
Fig. 3. The excess chemical potential of an HS fluid in a disordered OHCB matrix. A comparresults (symbols).
and the porosity ϕ∗ defined by the maximum packing fraction of afluid in a given matrix. The porosity ϕ∗ used in the approximationsSPT2b2 and SPT2b3 is estimated fromEq. (34). The geometrical porosityϕ0 is defined by the packing fraction of matrix particles η0 and can becalculated from Eq. (37) in the case of an HCB matrix or from Eq. (38)
ison of the SPT2 within the different approximations (lines) and the GCMC simulation
Fig. 4. A comparison of the different approximations for the excess chemical potential of an HS fluid in an HCB matrix.
36 M. Holovko et al. / Journal of Molecular Liquids 189 (2014) 30–38
in the case of an OHCBmatrix. The probe particle porosity ϕ is obtainedfrom Eq. (39) or Eq. (40) for an HCB matrix or for an OHCB matrix,respectively. As is seen the probe particle porosity ϕ, except thepacking fraction of matrix particles η0 also depends on the parameterof nonsphericity of matrix particles α0 ¼ R0S0
3V0and the ratios R1/R0, S1/S0
and V1/V0. Each of these ratios can be expressed through thenonsphericity parameters of fluid and matrix particles, for instance,V1/V0 is
V1
V0¼ R1S1
R0S0
α0
α1: ð43Þ
Therefore, it is observed that for a given nonsphericity parameterof fluid particles α1 and fixed ratios R1
R0and S1
S0the probe particle porosity
ϕ decreases when α0 increases (Fig. 2, left panel). On the other hand,at a constant nonsphericity parameter of matrix particles α0 andthe fixed values of R1
R0and S1
S0, it is observed that ϕ increases slowly
with an increase of a fluid particle nonsphericity (Fig. 2, rightpanel). However, the dependence of ϕ on α1 is very weak in a com-parison with the dependence on α0 (especially for the OHCB case)and it becomes practically unchangeable starting from α1 = 2.0.
To verify an accuracy of the different approximations of the SPT2approach presented in this study we make a comparison with the
Fig. 5. The excess chemical potential of an HS fluid in an HCB and an OHCB matrix. The effectapproximation.
simulation data obtained using the method of grand-canonical en-semble Monte Carlo. The simulation details are given in the previoussection. As an example we consider the case of an HS fluid in an OHCBmatrix. Four sets of the parameters describing an OHCB matrix areused (Table 1). We compare a dependence of the fluid chemicalpotential on the packing fraction η1. In Fig. 3 one can see such a com-parison, where we present the SPT results calculated in the differentapproximations. It is observed that at low densities all approxima-tions are correct, but at intermediate densities the common SPT2 ap-proach overestimates the value of chemical potential in a comparisonwith the GCMC data. On the other hand, the SPT2b approximation im-proves essentially the results for intermediate fluid densities. In themost considered cases the results of the SPT2b and the other approx-imations SPT2b1 SPT2b2 and SPT2b3 coincide and it is comparableto the statistical errors of the simulations, which is less than 0.5 %.However, in the case of a large difference between porosities ϕ andϕ0 and at high fluid densities the approximations SPT2b1, SPT2b2and SPT2b3 give better results than SPT2b approach, as it seen inthe inset of Fig. 3 (System D).
A role of the SPT2b1, SPT2b2 and SPT2b3 approximations becomesmore important for an HCB matrix case. It can be observed in Fig. 4,where the chemical potential of an HS fluid in an HCB matrices isshown. As one can see the SPT2b approximation leads to a divergenceat η1 = ϕ and gives the inaccurate results. At the same time the
of the nonsphericity parameter of matrix particles α0. All curves are calculated in SPT2b
Fig. 6. The excess chemical potential of an HS fluid in an HCB and an OHCB matrix. The effect of the nonsphericity parameter of fluid particles α1. All curves are calculated in SPT2bapproximation.
37M. Holovko et al. / Journal of Molecular Liquids 189 (2014) 30–38
SPT2b1 approximation has a divergence at η1 = ϕ0 and both theSPT2b2 and SPT2b3 approximations have a divergence at η1 = ϕ∗ im-proving the high density region. Unfortunately, computer simulationdata are not available for this case.
To discuss an effect of the nonsphericity parameters α0 and α1 onthe chemical potential of a fluid confined in a random matrix thecorresponding dependencies are obtained. In Fig. 5 one can see thechemical potential of a fluid confined in matrices with variousnonsphericity parameters α0. All curves are calculated in the approx-imation SPT2b. As is seen a value of the chemical potential increaseswith an increase of α0. It can be explained by a decrease of theprobe particle porosity ϕ caused by an increase of α0 (see Fig. 2, leftpanel). Also it is observed in Fig. 5 that the effect of nonsphericity isstronger in the case of an HCB matrix than of an OHCB matrix.
The effect of a nonsphericity of fluid particles α1 on the fluidchemical potential can be seen in Fig. 6 for the HCB and OHCB matrixcases. Similar to Fig. 5 these results are obtained in the approximationSPT2b. At fixed ratios R1/R0, S1/S0 and a constant value of α0 the probeparticle porosity ϕ depends on α1 only through the ratio V1
V0(see Eq.
(39) or (40)). Since V1V0
is proportional to α0α1
(Eq. (43)), an increase ofα1 leads to an increase of ϕ (see Fig. 2, right panel). Therefore, onecan expect the chemical potential decrease, because ϕ affects directly
Fig. 7. The coefficients A and B for the cases of HCB and OHCB m
a contribution to the chemical potential related to the infinite dilutionlimit μ10. However, in Fig. 6 one can observe opposite behavior, whichcan be explained by another contribution to the chemical potential(Eq. (24)) containing the coefficients A and B. According to Eqs. (41)and (42) in Fig. 7 we present the plots of A and B depending on thenonsphericity parameter α1. It is clearly seen that these coefficientsincrease quickly with an increase of α1. Taking into account that thedependence of ϕ on α1 is rather weak (Fig. 2, right panel), the totalchemical potential should increase as well.
7. Conclusions
In order to describe the thermodynamical properties of a hardconvex body fluid confined in a hard convex body porous matrixthe SPT2 approach is extended and the corresponding equations forthe chemical potential and the pressure are derived. The SPT2 approachwas developed previously using the scaled particle theory (SPT) as apowerful tool, which allows one to obtain pure analytical equations ofstate for a system of particles with hard-core interactions, includingsystems of nonspherical particles. Within the SPT2 approach a few ap-proximations (SPT2b, SPTb1, SPT2b2 and SPT3b) have been analyzed.Similar in our previous studies [7,13] it is observed that the SPT2b
atrices in dependence on the nonsphericity parameter α1.
38 M. Holovko et al. / Journal of Molecular Liquids 189 (2014) 30–38
is much better than the basic approximation SPT2. The SPT2 resultsfit the simulation data with a good accuracy (relative error b 2.0 %)at low and intermediate fluid densities. However, the SPT2b gives apoor description for a fluid at high densities due to the divergence inη1 = ϕ leading to an overestimation of the chemical potential in thisregion. As an alternative, we proposed the SPT2b1, SPT2b2 and SPT2b3approximations, which overcome this problem since they take intoaccount different types of porosities. It is worth noting that a newtype of porosity is introduced in this study. In analogy to the one-dimensional system of a fluid in a random matrix considered in [12]we used the porosity ϕ∗, which has a meaning of the maximum fluidpacking fraction available in a given matrix. Numerically a value of thisporosity indicates at which η1 the divergence for the chemical potentialand the pressure should appear. Physically it coincides with the maxi-mum amount of fluid, which can fill a considered matrix. Therefore,it was shown that the best approximation is SPT2b3. However, it stillneeds a more thorough comparison with simulation results at higherfluid densities than those considered in the current study. To this pur-pose additional simulation data for a high-density region are required.
Using the obtained equations the effect of the nonsphericity pa-rameter of fluid and matrix particles on the fluid chemical potentialis studied. The analysis of the SPT results for an HCB fluid in an HCBor an OHCBmatrix shows that an increase of nonsphericity parameterof matrix particles α0 leads to an increase of the chemical potential.The same behavior is observed if to increase a nonsphericity of fluidparticles. In the first case this effect is explained by changes in theprobe particle porosity ϕ, which reduces with an increase of α0.In the second case, an increase of the chemical potential is causedby the coefficients A and B, which enter into the thermodynamicexpressions for a confined fluid and define a porous mediumstructure.
Acknowledgment
The authors thank the National Academy of Science of Ukraine forthe support of this work (the joint NASU-RFFR Program).
References
[1] M.L. Rosinberg, in: C. Caccamo, J.P. Hansen, G. Stell (Eds.), New Approaches toProblems in Liquid State Theory, NATO Science Series C, 529, Kluwer, Dordrecht(Holland), 1999, pp. 245–278.
[2] O. Pizio, in: M. Borowko (Ed.), Computational Methods in Surface and ColloidalScience, Surfactant Science Series, 89, Kluwer, Marcel Dekker, New York, 2000,pp. 293–345.
[3] W.G. Madden, E.D. Glandt, Journal of Statistical Physics 51 (1988) 537–558.[4] J.A. Given, G. Stell, Journal of Chemical Physics 97 (1992) 4573–4574.[5] M. Holovko, W. Dong, Journal of Physical Chemistry B 113 (18) (2009) 6360–6365,
(113 (49) 16091-16091).[6] W. Chen, W. Dong, M. Holovko, X.S. Chen, Journal of Physical Chemistry B 114
(2010) 1225-1225.[7] T. Patsahan, M. Holovko, W. Dong, Journal of Chemical Physics 134 (074503)
(2011) 1–11.[8] H. Reiss, H.L. Frisch, J.L. Lebowitz, Journal of Chemical Physics 31 (1959) 369–380.[9] H. Reiss, H.L. Frisch, E. Helfand, J.L. Lebowitz, Journal of Chemical Physics 32
(1960) 119–123.[10] J.L. Lebowitz, E. Helfand, E. Praestgaard, Journal of Chemical Physics 43 (1965) 774–778.[11] M.F. Holovko, V.I. Shmotolokha, W. Dong, Condensed Matter Physics 13 (23607)
(2010) 1–7.[12] M. Holovko, T. Patsahan,W. Dong, CondensedMatter Physics 15 (23607) (2012) 1–13.[13] M. Holovko, T. Patsahan, W. Dong, Pure and Applied Chemistry 85 (2013)
115–133.[14] R.M. Gibbons, Molecular Physics 17 (1969) 81–86.[15] T. Boublik, Molecular Physics 27 (1974) 1415–1427.[16] M.A. Cotter, D.E. Martire, Journal of Chemical Physics 52 (4) (1970) 1909–1919.[17] M.A. Cotter, Physical Review A10 (1974) 625–636.[18] G. Lasher, Journal of Chemical Physics 53 (1970) 4141–4146.[19] I.R. Yukhnovskyi, M.F. Holovko, Statistical Theory of Classical Equilibrium Systems,
Naukova Dumka, Kyiv, 1980.[20] J.P. Hansen, I.R.Mc. Donald, Theory of Simple Liquids, Academic Press, London, 1986.[21] D. Frenkel, B. Smith, UnderstandingMolecular Simulations, Academic, San Diego, 1995.
