The Maxwell–Stefan equations for diffusion in multiphase systems with intersecting dividing surfaces

Physica A 254 (1998) 365–376

The Maxwell–Stefan equations for diusion in multiphase systems with intersecting dividing surfaces Leonard M.C. Sagis ∗ Department of Chemical Engineering, University of Amsterdam, Nieuwe Achtergracht 166, 1018 WV Amsterdam, The Netherlands Received 17 September 1997

Abstract In this paper, the Maxwell–Stefan theory for diusion is extended to multiphase systems with intersecting dividing surfaces. The general Maxwell–Stefan equation is separated into equations for bulk diusion, surface diusion and (common) line diusion. Expressions for the driving forces in these equations are derived using the jump entropy inequality and the entropy inequality at the common line. The resulting equations for surface and line diusion are more general than previously reported extensions of Fick’s law. They incorporate contributions from forced diusion, and diusion resulting from surface tension and line tension gradients into the total c 1998 Elsevier Science B.V. All rights reserved diusive ux. Keywords: Maxwell–Stefan theory; Dividing surface; Three-phase line; Surface diusion; Line diusion

1. Introduction In multiphase ows, the properties of the phase interfaces are often important factors in determining the mass and heat transfer between the various phases in the system. A phase interface is de ned as a three-dimensional region of nite thickness, separating two adjoining bulk phases. The material properties in this region change rapidly but continuously across the region, from their value in one bulk phase to their respective value in the other bulk phase. ∗

Present address: Department of Physical and Macromolecular Chemistry, Leiden Institute of Chemistry, Leiden University, The Netherlands. Tel.: +31 71 527 4542/4523; fax: +31 71 527 4397; e-mail: [email protected]. c 1998 Elsevier Science B.V. All rights reserved 0378-4371/98/$19.00 Copyright PII S 0 3 7 8 - 4 3 7 1 ( 9 8 ) 0 0 0 8 9 - 2

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In continuum mechanics a phase interface is usually modelled as a two-dimensional surface, or dividing surface [1]. The properties of the adjoining bulk phases are extrapolated up to the dividing surface. The dierence between the extrapolated and the actual elds is accounted for by associating excess properties, like a surface mass density, surface momentum, surface energy, and surface entropy, with the dividing surface. The incorporation of these excess properties into the general conservation principles for the system, leads to a set of balance equations for the dividing surfaces of a system, in addition to the well-known balance equations for the bulk elds. These surface balance equations are the jump mass balance [2] (p. 706), the jump momentum balance [2] (p. 710), the jump energy balance [2] (p. 719), and the jump entropy balance [2] (p. 731). Systems with three-phase con uence zones are modelled in a similar manner. A three-phase con uence zone is de ned as a three-dimensional region of nite size, where three bulk phases meet. Like in the interfacial regions, the material properties of the system change rapidly but continuously across the three-phase con uence zone. The con uence zones are usually modelled by extrapolating the dividing surfaces into the three-phase zone, to their intersecting space curve, the three-phase line of contact or common line. The bulk and excess surface elds are extrapolated up to this common line, and the dierence between the actual and extrapolated elds is accounted for by associating line excess properties, like a line mass density, line momentum, line energy, and line entropy, with the common line. The incorporation of these line excess properties into the conservation principles leads to yet an additional set of balance equations: the mass balance at the common line [3] (Eq. (74)), the momentum balance at the common line [3] (Eq. (91)), the energy balance at the common line [3] (Eq. (100)), and the entropy balance at the common line [3] (Eq. (110)). The set of surface and line balance equations must be complemented with a set of constitutive equations for the surface and line stress tensors, mass ux vectors, and heat ux vectors. This paper will focus on the constitutive equations for the surface s l and j(A) , de ned by and line mass ux vectors j (A) s s s j (A) ≡ (A) (v(A) − v s)

and

l l l j (A) ≡ (A) (v(A) − v l) ;

(1)

i i (i = s; l) are the surface and line mass densities of species A, v(A) the surface where (A) i and line velocities of species A, and v the mass averaged surface and line velocities of the N -component mixture. The simplest constitutive equation that can be proposed for the mass ux vectors in Eq. (1) is an extension of Fick’s law. According to Fick’s law, the equation for the surface and line mass ux vectors for a binary mixture is given by i i i = − i D(AB) ∇i !(A) ; j (A)

(2)

i the surface and line binary where  i are the total surface and line mass densities, D(AB) i the surface diusion coecients, ∇i the surface and line tangential gradients, and !(A) and line mass fractions. Several authors have used an equation of this type in their description of surface diusion. Djabbarah and Wasan [4], Ting et al. [5], and Feng

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[6] used Eq. (2) in their studies on the dynamic properties of the air–solution interface of surfactant solutions. Edwards and Oron [7] used it to study the stability of a thin non-wetting lm. Ha and Yang [8] used this type of equation to investigate the eects of surfactant on the deformation and stability of a drop in an electric eld. Finally, Tsai and Miksis [9] used the surface equivalent of Fick’s law to determine the eects of surfactant on the dynamics of bubble snap-o. It is a well known fact that the Fick formulation for diusion has its limitations. It is valid only for dilute solutions, and in the absence of body force elds, like electrostatic, magnetic, or centrifugal elds. In addition, a constitutive equation of the type of Eq. (2) is unable to describe diusion as a result of surface and line tension gradients. The Maxwell–Stefan formulation of diusion provides a much more general description of the diusion process. This formulation eliminates many of the objections raised against the Fick formulation. An extensive review of the Maxwell–Stefan theory for bulk diusion is given by Ref. [10]. The aim of this paper is the extension of the Maxwell–Stefan theory to a description for surface and line diusion. In Section 2 the separation of actual elds in terms of bulk, surface excess, and line excess elds is demonstrated. In Section 3 this separation procedure is applied to the Maxwell–Stefan formulation for diusion, resulting in the Maxwell–Stefan equations for surface and line diusion. In Section 4 we will use the jump entropy balance and the entropy balance at the common line to derive expressions for the driving forces that appear in these Maxwell–Stefan equations. 2. De nition of surface and line excess properties In this section we will illustrate the separation of an actual eld (scalar, vector, or tensor) into bulk, interfacial and line excess contributions. Let us consider an actual eld  de ned on a bounded domain R. This domain consist of an arbitrary number of bulk phases Rib , a number of interfacial regions, and several three-phase con uence zones. The interfacial region between bulk phase Rib and Rjb is denoted by Rijs . The l three-phase con uence zone between bulk phases i, j, and k is denoted by Rijk . Hence we have   ) ( ) (  [ X [ X X l Rib Rijs Rijk : (3) R=   i

i; j

i; j; k

The dividing surface inside the interfacial region Rijs is denoted by ijol . The extrapol is denoted by lation of this dividing surface into the three-phase con uence zone Rijk l ij . We de ne the surface ij as ij ≡ ijol ∪ ijl :

(4)

l we will use the For the common line contained in the three-phase con uence zone Rijk symbol Cijk .

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Let us now consider the integral of the quantity , over the spatial domain R. We can write this integral as Z Z Z b  dV =  dV + ( −  b ) dV ; (5) R

R

R

where  b is a piecewise continuous function satisfying  b = ib

for x ∈ Rib

(6)

in which ib is the value of  in bulk phase Rib . In view of Eq. (3) we can rewrite Eq. (5) as Z Z XZ XZ b b  dV =  dV + ( −  ) dV + ( −  b ) dV : (7) R

i; j R s ij

R

i; j; k

l Rijk

We will rst consider the interfacial contributions in this expression. We de ne the set {ijol ()} as the set of all interfaces parallel to the dividing surface ijol . Here  denotes the distance of ijol () to the dividing surface, measured along the unit normal to this surface. The set of parallel surfaces is assumed to span the entire interfacial region Rijs . We can use this set to rewrite the volume integral over the interfacial regions as an integral over  and a surface integral over the set of surfaces {ijol ()}. We nd that +

Z

Z

b

Z

( −  ) dV = Rijs

( −  b ) dA d

− ijol () +

Z Z

( −  b )J (ijol (); ijol ) d dA ;

=

(8)

ijol −

where ± denote the boundaries of the interfacial region, and J is the Jacobian for the change of integration over ijol () to integration over ijol . The particular form of this Jacobian can be found in Ref. [2] (p. 70). The last line of Eq. (8) suggests that we de ne the interfacial excess of the quantity , associated with the dividing surface ijol , as +

ijs ≡

Z

(() −  b )J (ijol (); ijol ) d :

(9)

−

Substituting Eqs. (8) and (9) in Eq. (7) we nd (using Eq. (4)) Z

Z  dV =

R

R

 b dV +

XZ i; j

ijol

ijs dA +

XZ i; j; k

l Rijk

( −  b ) dV

L.M.C. Sagis / Physica A 254 (1998) 365–376

Z =

 b dV +

XZ i; j  ij

R

ijs dA +

XZ i; j; k

( −  b ) dV −

l Rijk

369

XZ i; j

ijs dA :

ijl

(10) l in terms We will now rewrite the integral over the three-phase con uence zone Rijk of an integral over the common line Cijk . With Sijk we denote a surface perpendicular ∗ )} to the common line, located at an arbitrary point of this line. The set {Cijk (dSijk is de ned as the set of all lines, parallel to the common line Cijk . This set spans l ∗ ∗ the region Rijk . The intersection of this set with Sijk is denoted by Sijk . dSijk is an ∗ in nitesimal surface element of this surface, and Cijk (dSijk ) is the element of the set ∗ ∗ of parallel lines, whose point of intersection with Sijk lies in the surface element dSijk . These de nitions allow us to write Z Z Z b ( −  ) dV = ( −  b ) dL dA ∗ ∗ ) Sijk Cijk (dSijk

l Rijk

Z Z =

∗ ( −  b ) J (Cijk (dSijk ); Cijk ) dA dL ;

(11)

∗ Cijk Sijk

∗ where dL denotes integration over a line element, and J (Cijk (dSijk ); Cijk ) denotes the ∗ Jacobian for a change of integration over Cijk (dSijk ) to integration over Cijk . Eq. (11) suggests that we de ne the contribution from the bulk elds to the line excess as Z lb ∗ ijk ≡ ( −  b )J (Cijk (dSijk ); Cijk ) dA : (12) ∗ Sijk

The last term to be considered is the contribution from the surface excess elds to the line excess. Let us denote the line of intersection of Sijk and ij by Lij , and let dLij be an in nitesimal line element of this line. Let {Cijk (dLij )} be the set of lines parallel to Cijk , and inside ij . The set spans the entire surface ijl . Cijk (dLij ) denotes the element of the set of lines, whose point of intersection with Lij lies in dLij . These de nitions allow us to write Z ∗ Z ∗ Z X X ijs dA = ijs dL1 dL2 i; j

i; j L ij Cijk (dLij )

ijl

=

Z X ∗ Z

ijs J (Cijk (dLij ); Cijk ) dL2 dL1 :

(13)

Cijk i; j Lij

The asterisk in the sum denotes the fact that the sum is only over those surfaces and lines that are contained in the particular three-phase zone being considered. The Jacobian J (Cijk (dLij ); Cijk ) is the Jacobian for a change of integration over the line

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Cijk (dLij ) to integration over the line Cijk . Eq. (13) suggests we de ne the contribution of the surface elds to the line excess as ∗ Z X ls ≡ ijs J (Cijk (dLij ); Cijk ) dL : (14) ijk i; j L ij

The total line excess associated with the common line Cijk is given by l lb ls ≡ ijk + ijk : ijk

(15)

Substituting these results into Eq. (10) gives us Z Z XZ XZ l  dV =  b dV + ijs dA + ijk ds : R

R

i; j  ij

(16)

i; j; k C ijk

If we now de ne  as the union of all dividing surfaces ij , and C as the union of all common lines Cijk , and we introduce the piecewise continuous functions  s and  l satisfying  s ≡ ijs

for x ∈ ij

and

l  l ≡ ijk for x ∈ Cijk ;

we can simplify Eq. (16) to Z Z Z Z  dV =  b dV +  s dA +  l ds : R

R



(17)

(18)

C

This completes the separation of an arbitrary eld into bulk, surface, and line excess contributions. 3. Derivation of the Maxwell–Stefan equations In this section we will derive the Maxwell–Stefan equations for the dividing surfaces and common lines of an arbitrary multiphase body R. We will assume that the system consists of N distinct species. The Maxwell–Stefan theory assumes that the driving force for diusion of species A in a N -component mixture, is balanced by the friction forces exerted on species A by all the other species. This balance can be expressed as d (A) =

N X

f(AB) ;

(19)

B=1 B6= A

where d (A) denotes the driving force for diusion of species A, and f(AB) the friction force exerted on the material particles of species A by the material particles of species B. If we now integrate both sides of Eq. (19) over the spatial domain R, and

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apply the separation procedure of Section 2 to the integrals we nd Z

b d (A) dV +

Z

R

s d (A) dA +



=

Z X N B=1 B6= A

R

Z

l d (A) ds

C

b f(AB)

dV +

Z X N B=1  B6= A

s f(AB)

dA +

Z X N B=1 C B6= A

l f(AB) ds ;

(20)

s s and surface friction force f(AB) are de ned as where the surface driving force d (A) +

s d (A)

Z

b (d (A) () − d (A) )J (ijol (); ijol ) d ;

≡

(21)

− +

Z

s f(AB) ≡

b (f(AB) () − f(AB) )J (ijol (); ijol ) d :

(22)

− l and line friction force f(AB) are given by The line driving force d (A) l ≡ d (A)

Z

b ∗ (d (A) − d (A) )J (Cijk (dSijk ); Cijk ) dA

∗ Sijk

+

∗ Z X

s d (A) J (Cijk (dLij ); Cijk ) dL ;

(23)

i; j L ij

l ≡ f(AB)

Z

b ∗ (f(AB) − f(AB) )J (Cijk (dSijk ); Cijk ) dA

∗ Sijk

+

∗ Z X

s f(AB) J (Cijk (dLij ); Cijk ) dL :

(24)

i; j L ij

Since R, , and C were chosen arbitrarily, Eq. (20) implies that b = d (A)

s d (A) =

N X B=1 B6= A N X B=1 B6= A

b f(AB) ;

(25)

s f(AB) ;

(26)

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L.M.C. Sagis / Physica A 254 (1998) 365–376

l d (A) =

N X B=1 B6= A

l f(AB) :

(27)

b s l The expressions for d (A) , d (A) , and d (A) will be derived in the next section. Here we will focus on the expressions for the (excess) friction forces. In the Maxwell–Stefan theory, the friction force between species A and B is assumed to be linearly proportional to the dierence in velocity of the two species, and linearly proportional to the “number” b be expressed as concentrations of each species. This suggests that f(AB) b f(AB) =

b b x(B) x(A) b − D(AB)

b b (v(A) − v(B) );

(28)

b b where x(A) is the molar concentration of species A in the bulk phase, and − D(AB) is the Maxwell–Stefan diusivity. The surface friction force can be treated in a similar manner. The surface friction force is assumed to be linearly proportional to the dierence in surface velocities and linearly proportional to each of the surface excess molar concentrations. Likewise, the line friction force is assumed to be linearly proportional to the dierence in line velocities, and each of the line excess molar concentrations. Hence the expressions for s l and f(AB) are given by f(AB) i = f(AB)

i i x(B) x(A) i − D(AB)

i i (v(A) − v(B) );

i = s; l :

(29)

4. Expressions for the driving forces In this section we will derive expressions for the driving forces in the Maxwell– Stefan equations. We will use the entropy inequality to derive these expressions. In the entropy inequality the uxes are coupled to their respective driving forces. This inequality can therefore give us information on the speci c form of these driving forces. b has been given elsewhere (see Ref. [10] and The driving force for bulk diusion d (A) references therein). We will therefore focus only on the expressions for driving forces in the surface and line Maxwell–Stefan equations. We will consider only isothermal systems, and we will neglect any contributions to the diusive ux resulting from thermal diusion. For an isothermal system (T b = T s = T l = T = a constant) the jump entropy inequality is given by [2] (p. 794) Ss : Ds +

N X

s s s (b(A) − ∇s  (A) ) · j (A) −

A=1

"" +

N X A=1

N X

s s  (A) r(A)

A=1 s b  (A) (j(A)

· +

b (A) (v b

## s

− v ) · )

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373

+ < b [T Sˆ b − Uˆ b ](v b − v s ) ·  + (e b − q b ) ·  = − < 12 |v b − v s |2 (v b − v s ) ·  − (v b − v s ) · T b ·  =¿0 ;

(30)

where S s is the surface extra stress tensor, D s the surface rate of deformation tensor, s s the surface force per unit mass acting on species A,  (A) the surface chemical b(A) s potential per unit mass, and r(A) the rate of production or conversion per unit mass for species A (by surface reactions). The vector  is the unit normal to the dividing surface, Sˆ b is the bulk entropy per unit mass, and Uˆ b the bulk internal energy per unit mass. The vectors e b and q b are the bulk thermal energy and bulk energy ux vectors. Finally, T b is the bulk stress tensor. The open brackets in Eq. (30) denote the jump terms for the dividing surface, de ned by < = ≡ i ij + j ji = (i − j )ij ;

(31)

where ij is the unit normal to ij , pointing into bulk phase Rib . The second term of s can be expressed as Eq. (30) suggest that the surface driving force d (A) s ≡ d (A)

s (A)

c s RT

s s (b(A) − ∇s  (A) );

(32)

where c s is the total surface molar concentration and R the gas constant. The surface s chemical potential depends on the set of variables {T s ; ; !(B) (B 6= A)}, where is the surface tension. For an isothermal system we can expand the surface gradient in Eq. (32) as !  s  s X @ (A) @ (A) s s ∇s  (A) = ∇s !(B) + ∇s s @!(B) @ T; ! s s B6=A

T; ; !(C) (C6=A; B)

(A)

=

s − A(A) ∇s ; = ∇ ;s T  (A)

(33)

=

where A(A) is the partial area per unit mass for species A in the dividing surface. Substituting Eq. (33) into Eq. (32) gives s ≡ d (A)

s (A)

c s RT

=

s s (−∇ ;s T  (A) + A(A) ∇s + b(A) ):

(34)

The jump momentum balance for the dividing surfaces is given by [2] (p. 237) s

N

X ds v s s s − divs S s − ∇s − 2H  − (C) b(C) dt C=1

+ < b (v b − v s )(v b − v s ) ·  − (S b − PI) ·  = = 0 ;

(35)

where H is the mean curvature of the interface, P the bulk pressure, and I the 3-dimensional unit tensor. In most diusion problems hydrodynamic equilibrium is

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L.M.C. Sagis / Physica A 254 (1998) 365–376

obtained much faster than thermodynamic equilibrium. When hydrodynamic equilibrium is obtained we have s

ds v s − divs S s + < b (v b − v s )(v b − v s ) ·  − S b ·  = = 0 ; dt

2H  −
(36) (37)

This last expression is the familiar Laplace equation. With Eqs. (36) and (37) the jump momentum balance reduces to ∇s +

N X

s s (C) b(C) =0 :

(38)

C=1

Dividing this last equation by  s and subtracting the result from Eq. (34) gives us " #!   N s X (A) = 1 s

; T s s s s −∇s  (A) + A(A) − s ∇s + b(A) − !(C) b(C) : (39) d (A) ≡ s c RT  C=1

From this expression for the driving force for surface diusion we see that in an isothermal system there are three distinct contributions to the diusive ux. The rst contribution results from a gradient in the surface chemical potential (which is related to the surface concentration) along the dividing surface. This type of diusion is generally referred to as ordinary diusion. The second term in Eq. (39) describes the contribution to the diusive ux resulting from surface tension gradients. Finally, the third term represents contributions to the diusive ux resulting from force elds (forced diusion). The expression for the driving force for line diusion can be derived from the entropy inequality at the common line [4]. For an isothermal system this inequality is given by Sl : Dl +

N X

l l (b(A) − ∇l  (A) ) · j l(A) −

A=1

+

N X

N X

l l  (A) r(A)

A=1 l s  (A) (j (A)

· +

s (A) (v s

! l

− v ) · )

A=1

+ (  s [T Sˆ s − Uˆ s ](v s − v l ) ·  + (e s − q s ) · ) − ( 12 |v s − v l |2 (v s − v l ) ·  − (v s − v l ) · T s · )¿0 ;

(40)

l where S l is the line extra stress tensor, D l the line rate of deformation tensor, b(A) l the line force per unit mass acting on species A,  (A) the line chemical potential per l the rate of production or conversion per unit mass for species A unit mass, and r(A) (by reactions at the common line). Sˆ s is the surface entropy per unit mass, and Uˆ s the surface internal energy per unit mass. The vectors e s and q s are the surface thermal energy and surface energy ux vectors. Finally, T s is the surface stress tensor. The

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boldface parentheses in Eq. (40) denote the jump terms for the common line, de ned by

( s ) ≡ ijs ij + jks jk + iks ik ;

(41)

where ij is the unit vector normal to the common line Cijk and tangent to the dividing l can be expressed as surface ij . Eq. (40) suggest that the line driving force d (A) l ≡ d (A)

l (A)

c l RT

l l (b(A) − ∇l  (A) );

(42)

where c l is the total line molar concentration. The set of independent variables that l l depends on is given by {T l ; ; !(B) (B 6= A)}, where  is the line tension [11]. For  (A) an isothermal system we can expand the line gradient in Eq. (42) as ! ! l l X @ (A) @ (A) l l ∇l !(B) + ∇l  ∇l  (A) = l @ @!(B) l l B6=A

T; ; !(C) (C6=A; B)

T; !(A)

=

T l = ∇; l  (A) − L(A) ∇l  :

(43)

Substituting Eq. (43) into Eq. (42) gives l ≡ d (A)

l (A)

c l RT

=

T l l (−∇; l  (A) + L(A) ∇l  + b(A) ) :

(44)

If we again assume hydrodynamic equilibrium the momentum balance at the common line [3,4] reduces to ∇l  +

N X

l l (C) b(C) =0 :

(45)

C=1

Dividing this last equation by  l and subtracting the result from Eq. (44), we nd " #!   N l X (A) = 1 ; T l l l l l −∇l  (A) + L(A) − l ∇l  + b(A) − !(C) b(C) : (46) d (A) ≡ l c RT  C=1

In this expression for the driving force for line diusion we can again distinguish three contributions: ordinary diusion (resulting from a gradient in the line chemical potential), diusion resulting from line tension gradients, and forced diusion (resulting from external force elds). 5. Conclusions In this paper we derived the Maxwell–Stefan equations for a multiphase system with intersecting dividing surfaces. We separated the general Maxwell–Stefan equation into

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equations for bulk, surface, and line diusion. The Maxwell–Stefan equation for surface diusion is given by (combining Eqs. (39) and (29)) " #!   N s X (A) = 1

; T s s s s −∇s  (A) + A(A) − s ∇s + b(A) − !(C) b(C) c s RT  C=1

=

N s s X x(A) x(B) B=1 B6= A

s − D(AB)

s s (v(A) − v(B) ):

(47)

The Maxwell–Stefan equation for line diusion is given by (combining Eqs. (46) and (29)) #! "   N l X (A) = 1 ; T l l l l −∇l  (A) + L(A) − l ∇l  + b(A) − !(C) b(C) c l RT  C=1

=

N X B=1 B6= A

l l x(A) x(B) l − D(AB)

l l (v(A) − v(B) ):

(48)

Unlike previously reported extensions of Fick’s law (like Eq. (2)), these equations incorporate the contributions from forced diusion and diusion resulting from gradients in surface and line tension. References [1] J.W. Gibbs, The Collected Works of J.W. Gibbs, vol. 1, Yale University Press, New Haven, CT, 1928. [2] J.C. Slattery, Interfacial Transport Phenomena, Springer, Berlin, 1990. [3] L.M.C. Sagis, J.C. Slattery, Incorporation of line quantities in the continuum description for multiphase, multicomponent bodies with intersecting dividing surfaces, I. J. Colloid Interface Sci. 176 (1995) 150 –164. [4] N.F. Djabbarah, D.T. Wasan, Dilatational viscoelastic properties of uid interfaces – III. Mixed surfactant systems, Chem. Eng. Sci. 37 (1982) 175 –184. [5] L. Ting, D.T. Wasan, K. Miyano, S.-Q. Xu, Longitudinal surface waves for the study of dynamic properties of surfactant systems II. Air-solution interface J. Colloid Interface Sci. 102 (1984) 248 –259. [6] S.-S. Feng, Determination of diusion coecient and rheological properties of surface layers from surface trough diusion, J. Colloid Interface Sci. 160 (1993) 449 – 458. [7] D.A. Edwards, A. Oron, Instability of a non-wetting lm with interfacial viscous stress, J. Fluid Mech. 298 (1995) 287–309. [8] J.-W. Ha, S.-M. Yang, Eects of surfactant on the deformation and stability of a drop in a viscous

uid in an electric eld, J. Colloid Interface Sci. 175 (1995) 369 –385. [9] T.M. Tsai, M.J. Miksis, The eects of surfactant on the dynamics of bubble snap-o, J. Fluid Mech. 337 (1997) 381– 410. [10] R. Krishna, J.A. Wesselingh, The Maxwell–Stefan approach to mass transfer, Chem. Eng. Sci. 52 (1997) 861– 911. [11] L.M.C. Sagis, J.C. Slattery, Incorporation of line quantities in the continuum description for multiphase, multicomponent bodies with intersecting dividing surfaces, II. J. Colloid Interface Sci. 176 (1995) 165–172.