Theory for interfacial tension of partially miscible liquids · [11, 12] using the thermodynamic...

Theory for interfacial tension of partially miscible liquids Mohanned-E. Boudh-Hir and G.Ali Mansoori

The University of Illinois at Chicago (M/C 063), Chicago, IL 60607-7052, USA

The aim of this work is to develop a mathematical relation between the interfacial tension ( 'Y,t,) of two partially miscible molecular liquids a and b with their pure liquid state surface tensions ( 'Y,. and rs) with their respective vapours, using statistical mechanics. It is shown here that liquid-liquid interfacial tension can be mathematically correlated to the liquid-vapour surface tension by the following equation:

where !lp,. and /lpb

are the density changes of the components a and b in the two partially miscible liquid phases and !lp �, !lp� are the density changes of the pure liquids a and b with respect to their saturated vapours.

1. Introduction

There is a considerable interest in the understanding of inhomogeneous systems, that is two-phase systems, fluids in contact with a solid or subjected to

an external potential, etc. This is because:

(i) From a fundamental point of view it is of interest to know how to explain

phenomena that are related to the inhomogeneity of systems.

(ii) From a technical point of view this may help in the prediction of the

properties of such systems. Indeed, the most vital fluids in our lives (water,

oils, hydrocarbons, etc.) exist in nature, in reserves or while in transportation

in the state of confined systems. Some of these fluids are partially miscible.

In these cases therefore, the adsorption, the interfacial tension and the

surface tensions are, among others, the important properties characterizing

inhomogeneous systems. Unfortunately, realistic interparticle potentials for

fluids are often unknown. Hence one has to predict the behaviour of such

Physica A, Vol.179, Issue 2, pp. 219-231, March 1991

Email addresses of authors: MEB (meb2553yahoo.com); GAM ([email protected])

DOI: 10.1016/0378-4371(91)90060-P Print ISSN: 03784371

220

systems using the general properties of intermolecular potential functions (i.e.

the magnitude of the potential functions, their range and the manner in which

they decrease). The study of two-phase fluids (i.e. liquid-gas or liquid-liquid) is probably

the most complex one. For this reason few fundamental results are available in this area [1-5]. For instance, to predict the interfacial tension and the surface tensions for two-phase systems, empirical formulae [6-8] are widely used.

(i) MacLeod's formula [6] and its various versions are often applied to thestudies of liquid-vapour systems. (For more details, see, for instance, refs. [9] and [10].)

(ii) In the case of two partially miscible liquids, a and b, there exist two

different equations: (iia) For the relation between the interfacial tension of two liquids to their

respective pure liquid surface tensions the following empirical formula is used:

( )1/2 y = Ya + % - 2K 'Ya% , (1)

where for spherical molecules K is defined by

(2)

is used, where 'Ya

and 'Yb are, respectively, the surface tensions of the pure liquids a and b, and K is a constant depending on the nature of the system [7]. This equation is based on an analysis consisting in the assumption that the

interactions between particles satisfy the so-called Berthelot combining rule. According to this combining rule the potentials, assumed to be universal functions, are such that the magnitude of the interactions between two particles of different species is equal to the geometric mean of the magnitudes of the

two interactions between identical particles.

( iib) For the relation between the interfacial tension and the surface tensions with their common vapour, the following formula is proposed [8]:

(3)

This equation is often exactly true, and this is almost always the case near the

critical solution temperature. Eqs. (1) and (3) as well as the formula of Macleod, in addition to having

very simple analytical forms, give results in good agreement with experiments for several fluid systems. These qualities make them widely used and understanding them is of great interest. The formula of Macleod has been derived

[11, 12] using the thermodynamic equation relating the surface tension, 'Ya, of the system to its superficial internal energy, u

a,

M-E Boudh-Hir and G.A. MansooriTheory for Interfacial Tension of Partially Miscible Liquids

PHYSICA A179(2): 219-231, 1991

221

In this equation, ar denotes the differentiation with respect to the temperature

and S stands for the surface area. Using the statistical mechanical definition of

the surface tension it has been shown that this property is, at the first

approximation, given by Macleod's formula [13]. Concerning eq. (1), some

questions still have to be clarified. Indeed, although it is believed that in the

liquid-vapour and liquid-liquid systems that the one-particle densities have the

same behaviour (i.e. these functions go from their values in the first phase to

their values in the second phase smoothly and within a thin transition region as

shown in fig. 1), we do not know the following.

(i) How can liquids in which the particles interact via analogous potentials be

totally or partially immiscible? (ii) How can interfacial tension of a partially miscible liquid-liquid system

and pure fluid surface tensions really be correlated since they have different

intermolecular reasons for their existence? Indeed, the surface tension arises as

a consequence of the conditions (i.e. pressure and temperature) under which

the system exists; whereas the interfacial tensions results from the dissimilarity

of the two fluids in the system.

p Phase A

a/A

Pa (Z)

--- ---- pa

Pa (Z), Pb (Z)

z

Phase B p b/s

________ P•!s

z

(a)

(b)

Fig. 1. (a) One-particle density in liquid-vapour system. (b) One-particle density in liquid-liquid system.


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222

(iii) What approximation may lead to eq. (1)? Indeed, the interfacial tension

and surface tensions are obtained by the first derivative of the free energy, F,

with respect to the surface area, S. It is therefore impossible to derive the third term in the right-hand side of eq. (1).

(iv) How, using this equation, can the fact be explained that the interfacial

tension of a liquid-liquid system decreases when the mutual miscibilities of the

two liquids increase and that it vanishes rigourously when the two liquids

become totally miscible (i;e. at the critical solution temperature)? Indeed, the

surface tensions of the pure liquids are governed by the conditions under which the systems exist. Therefore they are not necessarily zero at the critical solution

temperature of the system a-b. It follows that even if eq. (1) satisfies the

condition that the interfacial tension is zero at the critical temperature, the physical phenomenon is violated. Indeed, 'Ya

b may vanish at this temperature

provided that the three terms 'Ya, Yb and -2K( 'Ya Yb )112 compensate each other

without being separately zero. This is, of course, not sufficient. Because, when

a and b mix together the mixture is then a homogeneous system and therefore all the contributions to the interfacial tension vanish independently. Conse

quently, y is zero without any need of compensation. This condition is much

more restrictive and can never be satisfied by eq. (1). This last point implies that interfacial tension and surface tensions cannot be

physically correlated and, therefore, any relation between these properties is

necessarily obtained by a mathematical artifact.

The purpose of this paper is therefore to clarify these equations and develop

a new equation relating the interfacial tension and the surface tensions. We start by examining the case of simple liquids. Then we consider more realistic

systems in which the particles may have complex forms and interact via

potentials depending not only on their separations but also on their mutual

orientations. The simplifying assumption that these interactions are additive

pairwise potentials is, however, introduced.

2. Model and theoretical developments

Let us consider a system of particles of species a and b. The pair potentials,

waa

(i, j), wbb(i, j) and w

ab(i, j), in this system denote, respectively, the

interactions between two particles of the same species a or b, and the coupling

potential (i.e. the interaction between two particles of different species). These interactions, in the most general case, can be written as functions of the

orientations, D; and n1, of the particles i and j and the vectors r;1 joining their

centres. We have


PHYSICA A179(2): 219-231, 1991

223

w,.u(i, j) = w

,.u(r;1, fl;, DJ. (5)

Here the subscripts µ,v stand for aa, bb, or ab. The total potential energy, U,

of the system is

(6) i<j i<j i,j

In this equation the first two sums are extended, respectively, to the numbers N. of particles a and N

b of particles b; in the third sum i varies from 1 to N. and

j is contained between 1 and Nb .When two liquids are mixed together, it is well known that: (i) fluids of similar chemical natures tolerate the coexistence of each other

and hence lead to homogeneous solutions, such as the mixture formed by water and ethyl alcohol.

(ii) fluids having very different chemical natures such as water and mercuryor water and lipids are completely immiscible.

This means that the coupling potential plays a central role concerning the miscibility or the immiscibility of the two fluids. Indeed, the existence of preferential interactions between particles of the same natme tends to favour the formation of clusters of particles a-a and b-b, with the consequence that the system splits into two phases. The first, A, is a saturated solution of b in a and the other, B, of a in b. In the limiting case where the difference between the interactions is very important, the two liquids become completely immiscible. In contrast, they are completely miscible when these interactions are comparable. Between the two extreme cases of the totally miscible and immiscible, there exist all the intermediate situations (i.e. fluids partially miscible with compositions varying between zero and one). For systems dilute enough to have the average particle separations larger than the range of the repulsive potential parts, this difference between the behaviour of the particleparticle interactions does not have any consequence concerning the immiscibility of the two fluids. The increase in densities makes the average distances between the particles smaller and smaller and the separation of the system into two phases follows.

From this model the existence of a critical solution temperature beyond which the liquids are completely miscible is easily understood. The increase in temperature leads to the fact that the thermal diffusion effect becomes more important than the mutual interaction force between particles. In mathematical terms the Boltzmann factors associated with the repulsive potential parts tend to the same limit and therefore the formation of a single-phase system follows.

Concerning the attractive potential part, one cannot expect that it may, in any way, generate a phase separation. Consequently, this phenomenon is due


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to the repulsive part of the coupling potential and, therefore, very dissimilar interaction potentials are necessary to split the system into two phases. This conclusion is supported by the explicit calculations made for hard-sphere systems [14-17]. It is true that some differences between idealized and realistic systems exist, but the observed physical phenomena should remain essentially the same. In the previous work [14-17] binary mixtures of hard spheres (a and b) characterized by their hard-core diameters a •• , abb and aab satisfying thecondition:

(7)

have been studied where Dab is the unlike interaction parameter. The following features demonstrated.

( i) There is no phase separation when Dab is equal to zero (mixtures ofadditive hard-core diameters) [14].

( ii) For negative values of Dab the systems have a compound-formingtendency [15, 17].

( iii) For Dab greater than zero (of the order of 10-2), the mixture leads totwo-phase systems at densities depending on this parameter [16]. For very small values of Dab the transition density is in the solid region and the miscible state is frozen before the two-phase transition can occur.

The interfacial tension, 'Yab , is defined as

'Yab = (asF)v,T,µ,

=½I dl d2 P.�(1, 2) [(aswaa(l, 2)]v

+ ½ J dl d2 Pbb(l, 2) [a5wbb(l, 2)]v

+ I dl d2 Pab(l, 2) [(aswab(l, 2)]v

(8)

where, V, T and µ, are, respectively, the volume, temperature and chemical potential characterizing the system under consideration, and Paa(l, 2), Pbb(l, 2) and Pab(l, 2) denote the two-particle densities.

The differentiation with respect to the surface area in this equation may be performed using the variable change [18]

r = (x, y, z) = (S 112x', S u2

y ', Vz'IS), (9)


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with

225

[asw ,,,v(i, j)]v = ½S[r; 1 • a,iiw ,,,Ji, j)]

= (x;1• ax .. , Y;1• ay .. , - 2z;1• a2..)w,,Ji, j),ij ij ij r

(10)

(11)

For potentials depending only on the distance between the particles, eq. (10) isreduced to

(12)

The following features should be noted: (i) For fluids of a similar chemical nature or existing at a temperature higher

than their mixture critical temperature, the mixture is a homogeneous system.Therefore, the three terms, 'Y!, Yt and 'Y:b, become zero independently. Thusthe interfacial tension vanishes, as should be the case, without any need ofcompensation and whatever the values of the surface tensions, 'Y

a and 'Yb, wouldbe when the temperature of the pure liquids is equal to the critical temperatureof their mixture.

(ii) It is clear that the last term, 'Y; b, in this equation does not have theform, (Ya'Yb)112, mentioned in eq. (1). However, the first two terms, 'Y! and 'Yt,are given by an expression analogous to that of the surface tensions, 'Ya and 'Y

b,

of the pure substances a and b, respectively. This analogy is, however, purelyformal. Indeed, we have

'Ya =½ f dl d2 P�.(l, 2) [(aswaa(l, 2)]v

, (13)

which is different from y ! in eq. (8) because the two-particle densities, Paa(l,2) in the partially miscible fluid system a-b are different from their analogue,p �a ( 1, 2), in the liquid-vapour system. The same remark remains valid for y;;

and 'Yb· Nevertheless, since for both of the two-phase systems (liquid-vapour and

liquid-liquid) the transition region is a thin layer and the one-particle densitieshave the same behaviour (see fig. 1), one may find an approximation for 'Y!expressing it as a function of 'Ya and which vanishes when the two liquidsbecomes miscible. We have

'Y! = ½ f dl d2 P�a(l, 2) [aswaa

(l, 2)]v

P�a�

l, 2�Paa

1, 2

= ½ f dl d2 P�a(l, 2) [(aswaa(l, 2)]v<P •• (1, 2) . (14)


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226

In this equation cpaa(l, 2) = Paa(l, 2)/p�a(l, 2). Introducing the average value, ( cpaa(l, 2)), of (/.l

a,/1, 2) over the transition region, (14) becomes

r: = ( cpaa(l, 2))( ½ J dl d2 P:a(l, 2)[aswaa(l, 2)Jv) = ( cpaa(l, 2))y". (15)

By retaining only the lowest order term in the density, ( cpaa(l, 2)) can be approximated by

(16)

where PaiA and PaiB

denote the values of the profile densities of the liquid a in the phases A and B, respectively; whereas p; and p: stand for the values of the profile densities of the pure fluid a in the liquid and vapour phases, respectively.

Eqs. (15) and (16) lead to

(17)

An analogous equation can be written for y �, we have

(18)

Before examining the term y: b

let us assume we separate the two liquids a and b constituting the two-phase system without modifying the manner in which the particles are distributed in the two liquids. It follows that y: and y � in eq. (8) will remain unchanged (because the two-particle densities in a and b are maintained the same as in the two-phase system). However, y !

b will vanish because the distance between the two particles a and b is very large and, therefore, the interaction between the two particles is zero. The energy absorbed by unit area of the system has balanced 'Y!b • Consequently, r!b

is negative. Now this term r!b

can be re-expressed as

r!! = f dl d2 P�a(l, 2) [aswaa(l, 2)]v

x {p.b(l, 2) [aswab(l, 2)]v/p�a(l, 2) [aswaa(l, 2)]v}

x f dl d2 P�b(l, 2) [aswbb(l, 2)]v

x {Pab(l, 2)[asw(l, 2)]vlP�b(l, 2) [aswbb(l, 2)]v}

= f dl d2 P�a(l, 2) [aswaa(l, 2)]vcpaib(l, 2)

x f dl d2 P�b(l, 2) [aswbb(l, 2)]vcpb /a(l, 2). (19)


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227

In this equation

and

Proceeding as for -y ! this equation may be approximated by

(22)

Now, for fluids of spherical particles, using the approximations that: (i) the derivative with respect to the surface area is proportional to the inverse of its soft repulsion parameter, and (ii) the number of neighbours of species b, for instance, that a particle a has is proportional to the local density of the fluid b multiplied by (a,jaab

)3, (<P.;b(l,2)) and (<Pb /a(l, 2)) can be written as

(23)

(24)

Here Pea and Peb

denote the average values of the local densities, in the transition region, of the liquids a and b, respectively. Using eqs. (23) and (24), eq. (22) becomes

which leads to

According to eqs. (8), (17), (18) and (26), this can be approximated by

'}': b =(Lipa/Lip�/%+ (Lipb/Lip�)2%

- 2(Lip.f Lip �)(Lipb/ Lip�)( (Taa (Tbb/ a;b)2( 'Ya %)112

2 /32 2 /3( / 2 )2( )1/2= a 'Ya + % - a aaa abb a ab 'Ya% 2 2 ( )1/2= a 'Ya+ /3 %-2af3K %%

The following features should be noted:

(25)

(26)

(27)


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(i) Two differences exit between this equation and eq. (1) first, in eq. (27),-y: and -y� are associated with the coefficients (Ap.fAp�)2 and (Apb/Ap�)2, respectively. In eq. (1) these two coefficients are simply replaced by one. Second, in eq. (27), the term ( 'Ya %)1 12 is associated with the coefficient-2(<raa <Tbb/<r;b )2(Apa Apb/Ap� Ap�); whereas it is associated with the coefficient -2(<raa<Tbb/<r;b) in eq. (1).

(ii) In addition to the appearance in eq. (27) of the terms Apa and Apb,which guarantee that the three components -y ! , -y � and 'Y ! b go to zero when the temperature of the system tends to its critical value, the approaches leading to these two equations are completely different.

(iii) The factor oab in eq. (7) is very small. Small percentage differences inthis are enough to create vacancies between particles a and b, leading to a split of the system into two phases. This has the following consequences: first, obb

plays an indirect role in the interfacial phenomena, that is, because of the split of the system into two phases, the particles will not be distributed uniformly and the interfacial tension will result. Second, in eq. (27) <Tab can be replaced by (<raa + <rbb)/2. The direct (macroscopic) effect of o can be neglected.

(iv) If the liquid b in the binary mixture is now replaced by a system of nliquids that mix together but are not miscible with the liquid a, eq. (27) becomes

(28)

Because of the approximation used it is expected that the less miscible the liquids are the better this approximation becomes.

We now examine the case of molecular fluids. The particle-particle potential, w(i, j), has the form given by eq. ( 5). To simplify the estimation of the different terms (<1\a(l, 2)), (<Pbb (l, 2)), (<Pa1b (l, 2)) and (<Pb /a(l, 2)), it is convenient to write the molecular pair potential as a sum of simple-fluid potential terms. Such a decomposition can be obtained by introducing the site-site interaction model. The molecular potential becomes

w(i, j) = L v(im j") (29) m,n

where the site-site interaction, v(im, jn ), depends only on the distance between the sites under consideration. It can be written in the form

(30)

where <rm,, is the hard-core parameter associated with these sites.


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229

The use of the site-site interaction model is supported by the good agreement with the results obtained from the molecular dynamics and the experiments [19). The theory related to this model has been recently reformulated and the results obtained are in good agreement with the numerical simulations in both the cases of low and high temperatures [20).

Using the same arguments as in the case of simple fluids eqs. (15), (16) and (18) are automatically extended to the general case. Regarding the approximation of the functions (<Pa

;b(l, 2)) and (<Pb1.(1, 2)) in eq. (22) the results obtained for pure molecular fluids [20) can easily be extended to their mixtures. The molecular pair-distribution functions in the mixture a-b can be written as

g,,v (l, 2) = IT IT g0 (1m , 2n) P,,u {l, 2), (31)

m�l n�1

where the products over m and n are extended to all the sites of the particles 1 of species µ, and 2 of species v, g

,,v (l, 2) stands for the molecular pairdistribution function, g° (lm

, 2,,) are the pair-distribution functions in a mixture of simple liquids whose particles are identical to the sites (denoted by 1) of the particles 1 and 2, and P

,,v (l, 2) is a function that converges rapidly to unity. An estimation for the functions <P.1b (l, 2) and <Pb 1.(l,2) can be obtained by following the steps mentioned above, and ; (i) neglecting the functions P

,,v(l,

2); and (ii), for a given molecule, replacing each site by a uniform spherical distribution of sites. In the case where the liquid a has spherical molecules and b has linear molecules consisting of k identical sites (whose soft repulsion parameter is denoted by <Tss) we can write

( <Pa /b (l, 2)) = (Ap.fAp�)(Apb /Ap�)(PeblPe.)(u • .fuab)3

(u.,Ju.5)(k + 2)/2 , (32)

( <Pb / a (l, 2)) = (Apa/ Ap �)(Apb / Ap�)( Pe) Peb)( <Tbb/ u.b)3 ( u,j <Tas)

leading to

x {(k + 2)/4[1 + (k + 2)2/4)}, (33)

( <Pa /b(l, 2)) ( <Pb / a (l, 2)) = (Ap) Ap�)2 (Apb /Ap�)2 ( <Taa <Tbb/ u;.b)3

X (u •• uju;.5){2[1 + 4/(k + 2)2)} -t, (34)

where ubb is the parameter associated with the spherical molecule of the liquid

b, uss stands for the parameter associated with the interaction between sites of

molecules a and <T88 characterizes the interaction between a site in a molecule a


PHYSICA A179(2): 219-231, 1991

230

and a molecule of the same liquid approximated by a sphere. Eqs. (34) and (25) lead to

'Y: b = -2(Lip.J Lip �)(Lipb/ Lip�)( CTaa CTbb/ cr;b )3

'2( CTaa crj cr;s )112

X {2[1 + 4/(k + 2)2]}

-112( 'Ya %)112

= -2a{3K( 'Ya%) 112• (35)

It follows that the interfacial tension has the same form as in eq. (27), that is

2 2 2 K( )112 'Yab = a 'Ya + f3 'Yb - a/3 'Ya 'Yb , (36)

where the coefficient K is now different from that obtained for simple liquids and is given by eq. (35).

3. Concluding remarks

The purpose of this paper was to examine the problem of the existence of a fundamental relation between the interfacial tension of a system of two partially miscible molecular liquids and their pure state surface tensions. It has been seen that these properties cannot be physically correlated. However, because of the fact that in the two-phase liquid-liquid systems a-b and the one-component liquid vapour systems a and b the transition regions are thin layers and the one-particle densities have the same behaviour, the expression for the interfacial tension can be expressed mathematically in terms of 'Ya and 'Yb· Eq. (36) proposed here is versatile enough to represent the relationship between the interfacial tension of two partially miscible liquids and their surface tensions. It should be pointed out that the three terms on the right-hand-side of eq. (36) vanish independently when the mixture reaches its critical condition. Consequently 'Ya

b becomes zero at the critical point of the

solution as should be the case.

Acknowledgement

The authors would like to thank Dr. S.T. Kim for his valuable comments and discussion. This research is supported by the Gas Research Institute, Contract No. 5090-250-2085.


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