Rheol Acta (2008) 47:807–820DOI 10.1007/s00397-008-0281-4

ORIGINAL CONTRIBUTION

Flow–diffusion interface interaction in blendsof immiscible polymers

Ali El Afif

Received: 6 October 2007 / Accepted: 17 March 2008 / Published online: 27 May 2008© Springer-Verlag 2008

Abstract The flow–diffusion interface interaction internary mixtures consisting of a two-component poly-meric immiscible blend and a simple fluid is investi-gated under isothermal conditions. A non-linear 3Dmodel that explicitly couples the interface dynamicsto the Navier–Stokes–Fick governing equations is de-rived. The interface is characterized on the microscopiclevel of description by a distribution function of the unitvector normal to the interface and on the mesoscopiclevel by a second-order tensor. The model shows thatthe morphology of the embedded interface is modifiedby the internal diffusion fluxes as well as by the externalflow, and in return, the deformation of the interfacechanges the behavior of diffusion that may deviatefrom the Fickean mass transport. In a one-dimensionalsetting, the nature of propagation of both linear dis-persive and nonlinear hyperbolic waves is examined,and explicit formulas for the characteristic speed, phasevelocity, and attenuation are provided.

Keywords Interface · Polymeric blend ·Non-Fickean diffusion · Traveling waves

Introduction

Most multicomponent mixtures are immiscible, andtheir immiscibility manifests itself by the presence ofan embedded interface whose morphology might be

A. El Afif (B)Equipe de Modélisation Scientifique (EMS),Faculté Polydisciplinaire de Safi, Université Cadi Ayyad,Route de la Kechla, Sidi Bouzid., B.P. 4162, Safi Maroccoe-mail: alielafif2005@yahoo.fr

as complex as that observed in co-continuous mediaor as simple as that observed in blends consisting ofa matrix and a dispersed phase. The manufacture ofa blend with a prospective tailor-made morphologypresenting a desired rheological behavior is related tothe mechanical as well as physicochemical propertiesof the mixed components, processing conditions, andthe structure of the interface, which generally increasesthe elasticity of the immiscible blend over that of thepure components. However, the probable presence ofinclusions (e.g., surfactants, solvents, compatiblizers),as required in some processing methods, may affect themechanical behavior of the blend.

In the past, pioneering investigations (Taylor 1934;Batchelor 1970) devoted to immiscible fluids have ledto extensive experimental as well as theoretical studies(Doi and Ohta 1991; Lee and Park 1994; Takahashiet al. 1994; Grmela and Ait Kadi 1994; Wagner et al.1999; Wetzel and Tucker 1999; Almusallam et al.2000; Tucker and Moldenaers 2002; Eslami et al. 2007;Grmela et al. 2001), which examined the interactionarising between the flow and the deformation of theinterface. As an important outcome, the average stresswas found out to be explicitly coupled to the interfacialtension, and in return, the structure of the interfa-cial morphology (i.e., size, shape and orientation) wasshown to be strongly dependent on the applied flow.A direct consequence of such a mutual coupling is thatan immiscible multicomponent mixture may exhibitfrom a rheological standpoint, a viscoelastic behavior,even though it is composed of simple Newtonian fluids.The physical reason is mainly attributed to the elasticcontribution of the embedded interface separating thedifferent components that brings about viscoelastic-ity to the dynamics of the whole blend. Thereby, it

808 Rheol Acta (2008) 47:807–820

seems obvious that any physical process that influencesquantitatively and/or qualitatively the morphology ofthe interface is expected to affect the rheology of theimmiscible blend. Indeed, the microstructure can bestabilized by adding surface-active agents or compati-bilizers whose presence generally affects the deforma-tion, breakup, and coalescence of the dispersed phase,as well as the elasticity of the immiscible blend. Forinstance, one well-established method for the fabrica-tion of polymer–clay nanocomposites is the solutionintercalation method consisting in solubilizing the poly-mer in an organic solvent and then dispersing the clayinto the obtained solution. Therefore, to correctly de-scribe and possibly predict the mechanical propertiesof a blend, a proper and comprehensive mathematicalformulation should include the time evolution of theinterface and associated with it that of mass transport(El Afif et al. 2003a, b; Liu and De Kee 2005). Today, afew theoretical studies have investigated the effects ofthe presence of inclusions on the mechanical propertiesof a complex immiscible polymeric blend under (orin the absence of) flow and for which the dynamicchanges of the interface are explicitly incorporated intothe mathematical formulation. On the contrary, theliterature is relatively abundant with models discussingthe (non-Fickean) mass transport into interface-freeviscoelastic polymers (Vrentas et al. 1975; Aifantis1980; Neogi 1983; Durning and Tabor 1986; Wu andPeppas 1993; Kalospiros et al. 1993; De Kee et al. 2000;Hinestroza et al. 2001; El Afif and Grmela 2002;Grmela et al. 2003).

This paper aims at understanding, on two levelsof description, the interaction occurring among masstransport flow and the deformation of the interfaceseparating the immiscible components. A two-levelmodel is then derived using the general equation for thenon-equilibrium reversible and irreversible coupling(General Equation for Non-Equilibrium Reversibleand Irreversible Coupling, GENERIC) that guaran-tees the compatibility of dynamics with thermodynam-ics. The physical system considered in this study is aternary mixture consisting of a compressible blend ofimmiscible Newtonian fluids and a simple fluid (e.g.,a solvent). The process of diffusion exclusively stud-ied is that occurring between the simple fluid andthe polymeric components of the compressible blend(i.e., any mutual diffusion between the components ofthe blend is ignored). The interface is characterized onthe microscopic level of description by an area densitydistribution function and on the mesoscopic level bya symmetric second-order tensor. The interpenetrationprocess is expressed mathematically in the governingequations via the two-fluid standpoint. The resulting

governing and constitutive equations are parameter-ized by the internal free energy density and kineticcoefficients expressing the particularity of the systemunder consideration. Several models derived in theliterature (Batchelor 1970; Doi and Ohta 1991; Leeand Park 1994) are recovered as special cases and newreduced models examining the occurrence of Fickeanand non-Fickean diffusion in the absence of flow arealso addressed. Dimensionless analysis in the absenceof flow leads to the emergence of three groups ofphysical parameters: two Deborah numbers and a con-stant that couples diffusion to the interface dynamicchanges. In a one-dimensional setting, the propagationof linear dispersive and nonlinear hyperbolic travelingwaves is also discussed, and explicit formulas for thecharacteristic speed, phase velocity, and attenuation areprovided.

GENERIC formalism

As a preliminary, we provide a brief recall regardingthe GENERIC formalism (Grmela 1984; Beris andEdwards 1994; Grmela and Ottinger 1997; Ottingerand Grmela 1997; Beris 2003). Under isothermal con-ditions, the GENERIC equation is written in the fol-lowing condensed form:

∂ X∂t

= LδΦ

δX− δ �

δ

(δ�

δX

) , (1)

where the time evolution of the set of the independentstate variables denoted by X(r, t) (t is time and r is theposition vector) and chosen to characterize the systemunder consideration, is generated by the contributionof two competing physical processes. The first, LδΦ/δXexpresses the reversible kinematics, and the second,δ�/δ(δΦ/δX), stands for the irreversible kinematics orrelaxation. In Eq. 1, Φ(X)= E(X) − T0S(X) stands forthe Helmholtz free energy, T0 is the constant tempera-ture, E and S are, respectively, the total energy and theentropy state functionals. The dissipation potential, Ψ ,is a functional of the conjugate variables and expressesrelaxation phenomena. A bracket can be defined fromthe bivector operator L as {A,B}= ⟨ δA

δX , L δBδX

⟩, where

<, > is the scalar product and A and B are smooth real-valued functionals of X. This bracket is called a Poissonbracket if (1) {A, B} = −{B, A}(L = −LT , i.e., L is an-tisymmetric, the superscript T stands for the transposeoperator), (2) the Jacobi identity {{B,C},A}+{{A,B},C}+{{C,A},B} = 0 holds for any regular functionals A, B,and C. The latter identity expressing time invariance ofthe reversible structure may help to single out physi-

Rheol Acta (2008) 47:807–820 809

cally admissible closure approximations (Edwards andOttinger 1997). The dissipation potential Ψ satisfies thefollowing properties: (1) �|equilibrium = 0, (2) Ψ is con-vex in the neighborhood of 0, and (3) Ψ reaches its min-imum at equilibrium. In view of the properties listedabove, the GENERIC equation implies the dissipation

inequality dΦdt = −

⟨δΦδX , δΨ

δ(δΦ/δX)

⟩≤ 0, as by symmetry:

{Φ, Φ} = 0. A derivation of a thermodynamically consis-tent time evolution equation consists in (1) an adequatechoice of the state variables, (2) a determination ofthe reversible kinematics, (3) a determination of theirreversible kinematics (dissipation potential Ψ ), and(4) a specification of the Helmholtz free energy Φ.These steps will be followed systematically in this work.

Model formulation

The system under consideration is a mixture consistingof two interpenetrating media, one is a simple fluid(e.g., a solvent) and the other is a blend of immiscibleNewtonian fluids. We focus on the case where thereis no interpenetration between the blend componentsduring the whole dynamic process. Such a physicalsituation holds far from phase transition regions (crit-ical points) or in case of a large friction between theimmiscible components at the interface (Gunton et al.1983). Therefore, the blend is regarded as a pseudo-one-component medium embedding an interface andcontaining a simple fluid. As the simple fluid (solvent)and the polymeric blend are assumed to be misciblein all proportions, they are regarded as two interpen-etrating media. Thereby, the only mass transport understudy is that resulting from the diffusion of the sol-vent molecules within the different components of theblend and that ignoring any interfacial growth and/ornucleation processes. Following the classical fluid hy-drodynamics viewpoint, the solvent and the blend arecharacterized, respectively, by their apparent mass den-sities ρs and ρb (the subscripts s and b designate,respectively, the solvent and the blend) and their ap-parent linear momentum density vectors us = ρsvs

and ub = ρbvb (vs and vb are the solvent and the blendvelocity vectors, respectively). For convenience, it iswell suited to introduce overall and relative variablesusing the following one-to-one transformation:

ρ = ρs + ρb, u = us + ub, c = ρs

ρs + ρb,

J = ρb

ρs + ρbus − ρs

ρs + ρbub (2)

where ρ is the total mass density, u = ρv is the totalmomentum vector density (v stands for the overallvelocity), c is the mass fraction of the simple fluid, andJ is its relative momentum density vector. The lattercoincides exactly with the mass flux density relative tothe local mass–average velocity. As discussed earlier,the presence of the embedded interface within thepolymeric blend makes the mixture {s + b } to behave asa structured fluid. From the knowledge gained in solv-ing rheological and non-Fickian diffusion problems forcomplex fluids, the classical fields introduced in Eq. 2have proved to be insufficient to characterize the dy-namic behavior of complex media. Indeed, the internalstructure couples to diffusion and brings about inertiaand viscoelasticity to the dynamic motion. Under flowand diffusion, the interface deforms (swells or shrinks)locally, and such morphological changes affect boththe diffusion processes and the flow dynamics. As thetime evolution of the interfacial morphology should beexplicitly expressed into the mathematical formulation,additional (structural) variables are required to directlytrack the local deformation of the interface.

Using the one-on-one transformation (2) and as-suming that the kinetic energy for the mixture can bewritten as a sum of the kinetic energies for the blendand for the simple fluid, one has:

� =∫

d3r

(u2

s

2ρs+ u2

p

2ρp

)+∫

d3rϕ

=∫

d3r(

u2

2ρ+ J2

2ρc (1 − c)

)+∫

d3rϕ (3)

where the first term, u2/2ρ, on the right-hand siderepresents the global kinetic energy; the second term,J2/2ρc(1 −c) stands for the relative kinetic energy; andthe last term, ϕ, is the internal free energy density that isdependent on the global mass density ρ, the mass frac-tion of the simple fluid c, and on the structural variable.The internal free energy involves two contributions:The first is attributed to mixing, and the second is dueto the interface excess energy:

ϕ = ϕmixing + ϕInterface (4)

In this paper, the mixture {solvent + blend} is re-garded as consisting of two components: One is thesolvent, and the other is the two-component blend.Therefore, the mixing part of the free energy for the

810 Rheol Acta (2008) 47:807–820

mixture {s + b} can be well described by the Flory–Huggins mean-field theory (Flory 1953):

ϕmixing = �Ts

(c ln c + (1 − c)

xnln (1 − c) + χsb c (1 − c)

)

(5)

where � is the gas constant, T is the temperature, Ωs isthe molar volume of the solvent, and χ sb stands for theFlory interaction parameter. The second term, ((1 −c)ln (1 −c))/xn, can be ignored in case xn, representing anaverage monomer number in the polymer chains, and isconsidered to be very large. The part of the free energyrelated to the interface excess energy will be specifiedlater for some particular cases.

In this perspective, first and foremost, a kineticmodel will be firstly derived on a microscopic levelof description (to be denoted by the f level), andsecondly, the f -level model will be used as a startingpoint to derive a model on a mesoscopic level (to bedenoted by the N level).

f level of description

On this kinetic level, the interface is characterized byits local and time-dependent area density distributionfunction f (r, n, t) of the outward unit vector n normal tothe interface. Consequently, the set of the independentstate variables on the kinetic level of description isgiven by:

X = (ρ, u, c, J, f)

(6)

The next step in the GENERIC algorithm is to deter-mine the reversible kinematics. The latter is expressedthrough the Poisson bracket, which has the followingform:

{A, B}=∫

dr[ρBuα

∂αAρ +uγ Buα∂αAuγ

+Jγ Buα∂αAJγ

+ Jγ BJα∂αAuγ

−Ac Buα∂αc−Jγ BJα

∂α

× (cAJγ

)+(1−c)(cuγ +Jγ

)BJα

∂αAJγ

+ ρc (1−c) BJα∂α

(Ac

ρ

)

−ρc (1−c) BJα∂α

(uγ AJγ

ρ

)]

+∫

dr∫

dnf[∂α

(Af(Buα

− cBJα

))

+ nβ

(nγ nα − δαγ

) ∂Af

∂nα

∂γ

(Buβ

− cBJβ

)

−nαnβAf∂β

(Buα

− cBJα

)]− A ↔ B (7)

where A and B are arbitrary regular real-valued func-tionals of the state variables. The quantity A ↔ Bstands for the symmetric part (i.e., the roles of A andB interchanged) of the quantities in the integrals. Forthe sake of simplicity, the following notations will beused in the whole paper: ∂α ≡ ∂

/∂rα , α ∈ {1,2,3}, and

Aρ = δ A/δρ represents the functional derivative or theVolterra derivative of A with respect to ρ (idem for u, c,etc.; Beris and Edwards 1994). The Einstein summationconvention for repeated indices is understood. The firsttwo terms with their symmetric counterparts, in the firstintegral in Eq. 7, express the reversible kinematics forthe global mass (continuity) and momentum (Euler)equations. The remaining terms, in the same integral,have been derived in the context of the mutual binarymass transport occurring between simple Newtonianfluids (El Afif et al. 1999). The first and second termsin the second integral in Eq. 7 have been obtained inan interesting formulation (Grmela 1990), examiningthe dynamic behavior of polymeric liquid crystals in theabsence of diffusion (c = 0, J = 0) and for which theunit vector n has been used to characterize the orien-tation of the polymeric chain (i.e., the vector director).However, one should stress out that, here, n stands forthe unit vector normal to the interface. The remainingterms in the integral have been derived in a previousinvestigation (El Afif et al. 2003a).

The dissipation potential is expressed as a quadraticfunctional of the conjugate variables of the statevariables:

� =∫

dr1

4

(η + 1

2

(ηd − 2

3η

)δαβ

) (∂β�uα

+ ∂α�uβ

)2

+1

2�Jρc (1 − c)

(�Jα

)2

+∫

dr∫

dn12�f(f − feq

)(�f)

2 (8)

The first term expresses the effects of shear, η, andbulk, ηd, viscosities as a material response to the gradi-ent of the velocity of the external flow, Φu = u/ρ = v.The second term accounts for dissipation generatedby a simple relaxation of the diffusion mass fluxes (ElAfif and Grmela 2002). The last term in the dissipationpotential (8) arises from the relaxation of the interfacedue to its surface tension. This is also written as aquadratic functional of the conjugate variable δΦ/δ fof the state variable f where the kinetic coefficientΛf is generally a concentration-dependent parameter.By virtue of the properties of the dissipation potentialΨ and also by construction (as Ψ is expressed as aquadratic function of δΦ/δX), the kinetic coefficientsinvolved in its expression (η, ΛJ, and Λf) must be posi-

Rheol Acta (2008) 47:807–820 811

tive parameters to satisfy the second law of thermody-namics (see the last section devoted to traveling waves).Based on the listed assumptions and interests, it is ofimportance to realize that we have ignored any intrinsicdiffusion process between the phases of the immiscibleblend. In view of expressions (1) and (3), scaling showsthat the phenomenological parameter ΛJ is the inverseof the relaxation time (i.e., τ J = (ΛJ)

−1) of the diffu-sion mass flux. In the vicinity of equilibrium, one canexpress this coefficient as a function of the Bearman mi-croscopic friction coefficient (Bearman 1961), ξ ∗

sb, i.e.,

�J = ρN2A

MsMbξ ∗

sb, where NA is Avogadro’s number and Mj

denotes the molecular weight of component j. Finally,the last parameter to be specified is feq, which refersto the area distribution function density at equilibriumand whose expression is given by (Doi and Ohta 1991):

feq (n, r) = f0[det F]2

|F+ · n|4 , (9)

where the second-order tensor F stands for the defor-mation gradient tensor and f0 is an arbitrary reference(initial) state. The presence of the simple fluid withinthe blend may determine the amount of the interfacialarea, which leads to an equilibrium value. Contraryto the diffusion-free problem, the system may nowpossess a characteristic length scale defined here as1/Qeq (where Qeq = ∫ dn feq (n) ) and, thus, an intrinsictimescale inversely proportional to Λf and defined asτf = η

�Qeq(Γ is the interfacial tension).

Using the free energy expression, one easily sees thatδ�δuα

= uα

ρ= vα , which refers to the overall velocity field,

δ�δJα

= Jα

ρc(1−c) , which stands for the relative velocity field

vector and δ�δf = δϕ

δf . If one substitutes into Eq. 1, thepartial specification of the free energy (Eq. 3) and thecontributions of the reversible and irreversible kine-matics (Eqs. 7 and 8), then one will arrive at the fol-lowing governing equations for the set of independentstate variables on the f level.

Total mass

∂ρ

∂t= −∂αuα (10)

overall momentum

∂uα

∂t= −∂β

(uαvβ

)− ∂α p − ∂βσβα (11)

mass fraction

ρ∂c∂t

= −ρvα∂αc − ∂α Jα (12)

diffusion mass flux

∂ Jα

∂t=−∂β

(Jαvβ

)− Jβ∂βvα+c ∂β

(Jα Jβ

ρc (1 − c)

)

−∂β

(Jα Jβ

ρc

)−ρc (1−c) ∂α

(ϕc

ρ

)

+ c∫

dnf∂αϕf−c∂β

[∫dnf[nβ

(nαnγ −δαγ

) ∂ϕf

∂nγ

− (nαnβ − δαβ

)ϕf

]]−ΛJ Jα

(13)

interfacial area density

∂ f∂t

= −(

vα − Jα

ρ (1 − c)

)∂α f

− ∂

∂nα

(nβ

(nαnγ − δαγ

)f ∂γ

(vβ − Jβ

ρ (1 − c)

))

−nαnβ f ∂β

(vα − Jα

ρ (1 − c)

)

−Λ f(

f − feq)ϕ f (14)

The kinetic model deals with five nonlinear cou-pled and time-dependent partial differential equations(PDEs) describing the flow–diffusion interface inter-action occurring in a complex ternary mixture. Thefirst equation, Eq. 10, is the overall mass conservationequation. The second equation, Eq. 11, represents theoverall linear momentum conservation equation andinvolves the hydrodynamic pressure given by

p = −ϕ + ρϕρ +∫

dn f (n, r) ϕf (15)

and the second-order stress tensor

σαβ =−η(∂βvα+∂αvβ

)−(ηd− 2

3η

)(∂γ vγ

)δαβ + Jα Jβ

ρc (1−c)

−∫

dn f[

nβ

(nαnγ −δαγ

) ∂ϕ f

∂nγ

−(nαnβ −δαβ

)ϕ f

]

(16)

The first two terms on the right-hand side of Eq. 16express the effects of shear and bulk viscosities as a re-sponse to the applied flow. The third term, appearing inthe expression of the stress tensor, stems from the mu-tual interactions occurring between the diffusion massfluxes. Note that, even though it is a quadratic term,it may become significant because it generally involvesterms of higher order in gradients (of concentration incase of Fickian diffusion and of structural variables incase of non-Fickean diffusion). For instance, in physicalsituations involving fluctuations or phase transitions,

812 Rheol Acta (2008) 47:807–820

these terms may exhibit very large magnitudes in theneighborhood of critical points. The natural emergenceof this quadratic term in Eq. 16 demonstrates the mu-tual and direct relationship between mass transportand flow. The last term (i.e., the integral) refers to thecontribution of the interface deformation to the extrastress tensor and thus to the flow behavior.

Diffusion is expressed into two equations: Eq. 12 isthe continuity equation for the simple Newtonian fluid(e.g., solvent), and Eq. 13 is the time evolution equationfor the diffusion mass flux vector, J. The latter dependson the flow conditions and incorporates explicitly thecontribution of the deformation of the interface. Thelast equation (Eq. 14) expresses the time evolution ofthe interface where both flow and diffusion directlyinfluence the interfacial size and shape anisotropy. Weshould emphasize that all the constitutive equationsarise naturally from the Poisson bracket without any adhoc assumption.

In the next section, a mesoscopic model (N level) isformulated where its basis foundations spring from thekinetic level derived on the f level of description.

N level of description

On the mesoscopic level of description, the interface ischosen to be characterized by a second-order tensor,N, defined as the second moment of the distributionfunction f (Wetzel and Tucker 1999):

N (r, t) =∫

dn f (n, r, t) n n (17)

where dn is the solid angle. Alternatively, other av-eraged variables can also be used for discerning theinterface morphological changes (Doi and Ohta 1991)and are defined as the trace [i.e., Q(r, t) = Tr(N)] andthe traceless (i.e., q (r, t) = N − 1

3 I Tr(N)) compo-nents of area tensor N. The set of the independent statevariables on the N level of the description is

X = (ρ, u, c, J, N)

(18)

To derive the Poisson bracket on the N level ofdescription, we start from the Poisson bracket given onthe f level by Eq. 7 and use the following chain rule:

δ(·)δf

= nαnβ

δ(·)δNαβ

As a result, one arrives easily at the followingPoisson bracket expression:

{A, B}=∫

d3r[ρBuα

∂αAρ +uγ Buα∂αAuγ

+Jγ Buα∂αAJγ

+ Jγ BJα∂αAuγ

−AcBuα∂αc−Jγ BJα

∂α

× (cAJγ

)+(1 − c)(cuγ +Jγ

)BJα

∂αAJγ

+ ρc (1−c) BJα∂α

(Ac

ρ

)

− ρc (1−c) BJα∂α

(uγ AJγ

ρ

)

+ Nαβ∂γ

(ANαβ

(Buγ

−cBJγ))

− Nαγ ANαβ∂γ

(Buβ

−cBJβ

)− Nβγ ANαβ

∂γ

(Buα

−cBJα

)

+ nαnβnθnνANαβ∂ν

(Buθ

−cBJθ

)−A↔B]

(19)

which involves a fourth-order moment, nnnn =∫dn nnnn f (n, r, t). A closure approximation becomes

then necessary. On this mesoscopic level of description(N level), the dissipation potential can be obtainedby using the same chain rule (defined above) andassuming that the kinetic coefficient �f is exclusivelyconcentration dependent. The result is straightforwardand is given by

� =∫

dr1

4

(η + 1

2

(ηd − 2

3η

)δαβ

) (∂β�uα

+ ∂α�uβ

)2

+1

2ρc (1 − c) �J

(�Jα

)2 + 1

2�ijαβ

�Nij�Nαβ(20)

While the first two terms remain unchanged withrespect to their expression given in Eq. 8, the interfacecontribution to the dissipation is transformed into aquadratic function of the conjugate variable of the areatensor N. It involves the following fourth-order tensor:

�αβγ θ =∫

dn�f(

f − feq)

nαnβnγ nθ (21)

If the kinetic coefficient �f is a constant independentof n, then one will obtain

�αβγθ = �f

(nαnβnγnθ − nαnβnγnθ

∣∣eq

), (22)

where

nnnn|eq = f0

∫dn

[det F]2

|F+ · n|4 nnnn

Using Eqs. 1, 19, and 20 and following the same pro-cedure as before, one arrives at the governing equa-tions for the set of the state variables on the N level.

Rheol Acta (2008) 47:807–820 813

Similarly, the mesoscopic model derived on the N leveldeals with five nonlinear coupled time evolution PDEs.The governing equations for the global mass, the linearmomentum density, and the simple fluid mass conser-vation present the same condensed form as for thoseobtained on the f level except that the constitutiveequations they involve have now different expressionsand are expressed in terms of the mesoscopic variableN instead of the kinetic variable f . The hydrodynamicpressure p is given by

p = −ϕ + ρϕρ + NαβϕNαβ(23)

and the extra stress tensor reads as

σαβ =−η(∂βvα+∂αvβ

)−(

ηd− 2

3η

)(∂γvγ

)δαβ

+ JαJβ

ρc (1−c)+2NαγϕNγβ

−(nαnβnνnγ+Nνγδαβ

)ϕNγν

(24)

The contributions of the viscosities and of the diffusionfluxes to the stress tensor remain the same as on the flevel.

The continuity equation for the simple fluid has thesame form as in Eq. 12 except that the diffusion massflux evolves according to the following time evolutionequation:

∂Jα

∂t=−∂β

(Jαvβ

)−Jβ∂βvα+c∂β

(JαJβ

ρc (1−c)

)

−∂β

(JαJβ

ρc

)−ρc (1−c)∂α

(1

ρ

∂ϕ

∂c

)

+ cNβγ ∂α

(∂ϕ

∂Nβγ

)+c∂β

× [2Nαγ ϕNγβ−(nαnβnνnγ +Nνγ δγβ

)ϕNγν

]−�JJα

(25)

The governing equation for the interface area tensor isgiven by

∂Nαβ

∂t= −

(vγ − Jγ

ρ (1 − c)

)∂γNαβ − Nαγ∂γ

×(

vβ− Jβ

ρ (1−c)

)−Nβγ∂γ

(vα− Jα

ρ (1−c)

)

+ nαnβnθnγ∂γ

(vθ− Jθ

ρ (1−c)

)−�

ϕ

αβγθNγθ

(26)

Similarly, on this mesoscopic level and as was alreadyfound out on the kinetic level, the mass transport of thesimple fluid is strongly influenced by the deformationof the interface as well as by the applied external flow,

and in return, diffusion contributes to the mechanicalchanges occurring at the interface as shown in the lastequation (26). Note that the reversible part in Eq. 26is lower convected by the applied flow and upper con-vected by the internal diffusion mass fluxes. The gov-erning equations on theN level involve a fourth-ordertensor that requires us use of a closure approximation.One simple quadratic closure put forward in Doi andOhta (1991) and has been proved to be physicallyadmissible (Edwards and Ottinger 1997) is

nαnβnνnγ = 1TrN

NαβNνγ (27)

Limiting cases

In this section, we look for some particular cases of theN-level model where some state variables become de-pendent variables. In this reduction, the model derivedhere is shown to encompass well-known models derivedin the past and also new extended formulations wherethe changes of the interface contribute to the mass fluxand vice versa.

Flow and diffusion without inertia in J

An interesting situation to be examined in the contextof the N level is when the relaxation characteristictimescale of the diffusion mass flux, (ΛJ)

−1, becomessmaller than the diffusion characteristic timescale. Inthat case, one can assume that J equilibrates fasterthan the other state variables and consequently dJ

dt ≈ 0.Moreover, ignoring the quadratic terms appearing inthe diffusion mass flux governing Eq. 25, one arrives atthe following expression:

�JJα =−ρc(1−c) ∂α

(ϕc

ρ

)+cNβγ ∂αϕNβγ

+c∂βσ(Interface)αβ

(28)where for simplicity, the symbol

σ Interfaceαβ = 2Nαγ ϕNγβ

− (nαnβnνnγ + Nνγδαβ

)ϕNγν

(29)

is used to denote the interface excess stress tensor.Reformulation of Eq. 28 gives

Jα = −ρD∂αc − ρGβγ∂αNβγ + H∂βσInterfaceαβ , (30)

where the transport coefficients D, G, and E arising inEq. 29 have the following explicit form:

D= c�J

(∂μ

∂c

), Gβγ = c

�J

(∂μ

∂ Nβγ

), H = c

�J, (31)

814 Rheol Acta (2008) 47:807–820

expressed in terms of the chemical potential per unitmass of the simple fluid

ρ(μ − μ0

) = ϕ + (1 − c) ϕc − NαβϕNαβ,

where μ0 is a reference value at the pure state. Thesimple fluid continuity equation is now written as

ρ∂c∂t

=−ρvα∂αc+∂α

(ρD∂αc+ρGβγ∂αNβγ+H∂βσInterface

αβ

)

(32)

and the interface governing equation is given by Eq. 26,provided that the diffusion mass flux is replaced by itsreduced expression (Eq. 30). Fluctuations in concen-tration, phase separation, migration across streamlines,and formation of new morphologies are some of in-teresting physical phenomena that have been detectedin sheared interface-free polymer solutions. Some ofthese observations, whose physical origin seems to bewell understood, have been discussed in the contextof an equation similar in form to Eq. 32. One expectsthen that Eq. 32 comes forward as a good candidateto account and probably predict similar observations tooccur in complex solutions consisting of solvents andimmiscible blends embedding an internal interface.

Flow without mass transport

In this study, we assume that there are no diffusionphenomena involved ( J = 0 and c = 0); that is to saythat the immiscible blend is subjected exclusively toan external flow. It is interesting to realize that theN-level model allows us to recover many approachesas special cases, provided that the free energy is writtenappropriately to express a particular physical situation.To recover the popular model of Doi and Ohta (1991)derived for discussing dynamics of interfaces in theabsence of mass transport, the interface excess freeenergy density is written under the following simpleform:

ϕInterface = � TrN , (33)

where the parameter Γ stands for the interfacial ten-sion and Tr denotes the trace. Therefore, in the absenceof diffusion (c = 0, J = 0) and under the closure approx-imation (Eq. 27), the governing equation for the areatensor becomes

∂Nαβ

∂t= − vγ∂γNαβ − Nαγ∂γvβ − Nβγ∂γvα

+ NαβNγθ

TrN∂γvθ − �f�Nαβ (34)

and the interface excess stress tensor simplifies as

σ(Interface)

αβ = �(Nαβ − (TrN) δαβ

)(35)

with a deviatoric expression equal to Γ q, involvingthe anisotropy tensor and corresponding to the expres-sion derived in a previous work (Batchelor 1970). Oneshould mention that the original paper of Doi and Ohtaassumes complete phase separation [Qeq = Tr(Neq) = 0]and isotropy [qeq = 0].

Another widely discussed model in the literature(Lee and Park 1994) assumes that the relaxation ofthe interface results from the contribution of both theanisotropy and the surface tension. Therefore, the freeenergy density involves a new quadratic term and canbe written as

ϕInterface = �TrN + 1

2α

(N − I

TrN3

):(

N − ITrN

3

),

(36)

where α is a material parameter accounting for theanisotropy of the interface. Assuming complete phaseseparation and isotropy at the final equilibrium state,one arrives at the following time evolution equationderived in the absence of diffusion (c = 0, J = 0):

∂ Nαβ

∂t

∣∣∣∣relaxation

= −�f

(� − α

(N : NTrN

− TrN3

))Nαβ

(37)

whose trace and traceless parts lead to the model de-rived in Lee and Park (1994).

Mass transport without flow (u = 0)

Here, the study will be limited to the following specialsituation corresponding to the case of the absence of anapplied flow:

u = 0, (38)

and under mechanical equilibrium

∂α p + ∂βσβα = 0 , (39)

where p is the hydrodynamic pressure given by Eq. 23and σ is the extra stress tensor whose expressionbecomes σ = JJ

ρc(1−c) + σ(Interface), as v = 0. Underthe requirements of overall incompressibility and theconstraints Eqs. 38 and 39, the diffusion mass fluxgoverning equation becomes

∂Jα

∂t= −∂γ

(JαJγ

ρc

)− �J

(Jα + ρD

(∂αc + Eβγ ∂αNβγ

))

(40)

Rheol Acta (2008) 47:807–820 815

where D=cϕcc/�J is the diffusion coefficient and Eβγ =φcNβγ

/ϕcc is a second-order functional that couples dif-fusion to the interface dynamic changes. Here, themathematical symbol ϕcc denotes the second derivativeof ϕ with respect to c, etc. The governing equation forthe diffusion mass flux, J, depends on the structuralvariable N whose time evolution equation is

∂Nαβ

∂t=(

Jγ

ρ (1 − c)

)∂γNαβ + Nαγ∂γ

(Jβ

ρ (1 − c)

)

+ Nβγ∂γ

(Jα

ρ (1 − c)

)−nαnβnθnγ∂γ

(Jθ

ρ (1 − c)

)

− �αβγθϕNγθ(41)

The reduction to simpler models requires us to elu-cidate, among others, the relevance of inertia in non-standard diffusion processes. The latter depends onthe system under consideration and, in particular, onthe timescales involved. The concept of the Deborahnumber has been applied to mass transfer (Vrentaset al. 1975; Neogi 1983) and has been revealed to be avery useful tool to set up the range of validity of inertialas well as viscoelastic effects. The Deborah numbercompares the characteristic timescale of the relaxationof the internal structure to the characteristic timescaleof diffusion. Following previous analysis (Vrentas et al.1975; Neogi 1983), we define a J-Deborah number asa ratio of the relaxation characteristic time τ J = (�J)

−1

for the diffusion mass flux density to the diffusion char-acteristic timescale τd = L2

0

/D0 (L0 is a characteristic

length scale and D0 is the diffusion coefficient at theinitial state). Therefore,

DeJ = τJ

τd(42)

It is well suited to switch to dimensionless analysisto identify the dimensionless groups of parameters thatare of importance to this dynamic process. Therefore,we introduce the following dimensionless quantities forthe time, space, and the state variables:

θ = t/τd, ∂α = 1

L0∇α, C∗ = c

Ceq,

J∗ = Jτd

ρCeq L0, N∗ = N

Q0(43)

where Q0 = TrN0 the interfacial size density at theinitial state and Ceq is the equilibrium mass fraction.Henceforth and for the sake of clarity, the superscriptstar will be omitted for the dimensionless quantities in

the following analysis. The dimensionless form for themass flux time evolution equation becomes

∂Jα

∂θ= −∇γ

(JαJγ

C

)

− 1

DeJ

(Jα + D

(∇αC + Eβγ ∇αNβγ

))(44)

where D = D/

D0 and E = E/

g0. Several special casesarise as dictated mainly by the magnitude of the dimen-sionless number DeJ. Inertial terms become relevantfor a large J-Deborah number. The analogy of this casewith flows of high Reynolds number is straightforward.For small DeJ, diffusion is mainly governed by theFickian term and the extension provided by the changesoccurring at the interface. Finally, for intermediate val-ues of DeJ, mass transport is influenced by both inertiaand the deformation of the interface. In addition toDeJ, the quantity g0 is a dimensionless group of physicalparameters that emerges into Eq. 44 and is defined as

g0 = �oQos

�T(45)

which involves interfacial tension, Γo, at the initialstate and the gas constant, �. In the following sections,the implications of the magnitudes of this group ondiffusion will be discussed. Note that this group ofnumbers, g0, shows that the behavior of mass transportintimately depends on the physical properties of boththe polymeric blend and the penetrants, as well as onthe experimental conditions.

Let us now focus our attention on the interface timeevolution equations. The treatment of interfacial vari-ables requires particular consideration, as it dependson the nature of the morphology of the interface andthe probable presence of a length scale. Two possiblecases arise related to whether the blend possesses ornot a length scale. If the interface has a length scale, itbecomes possible to define a timescale and, therefore,an N-Deborah number. The relaxation time becomesrelated to the size of the droplets constituting the minorphase. In the case of spherical droplets with an initialradius Ro, the relaxation timescale of the interface isgiven by τN = 3ηoφ

/(�0Q0), where ηo is the matrix

constant viscosity, Qo = 3φ/

Ro refers to the interfacialsize density, and φ is the volume fraction of the minorphase in the blend. If the system is composed of twoco-continuous media, there is no natural length scale inthe system. Let us focus on the case of the existence ofa length scale with a constant timescale (�f = constant).The dimensionless form of the governing equation (41)

816 Rheol Acta (2008) 47:807–820

may be separated as a sum of two parts: One is due todiffusion and the other to relaxation:

∂Nαβ

∂θ= ∂Nαβ

∂θ

∣∣∣∣diffusion

+ ∂Nαβ

∂θ

∣∣∣∣relaxation

(46)

In the relaxation part of the interface governing equa-tions, the interface N-Deborah number defined as

DeN = τN

τd(47)

emerges naturally, and one has

∂Nαβ

∂θ

∣∣∣∣relaxation

= − 1

DeN

((nαnβnγ n

ν− nαnβnγ nν

∣∣eq

)ϕNγν

)(48)

where nnnn|eq is a certain local equilibrium value forthe area tensor and ϕ = ϕ

/(�oQo) is a normalized ex-

pression of the internal free energy density. The pres-ence of these three dimensionless groups (DeJ, DeN, go)

in the model equations leads to several special cases. Inthe following, we shall discuss the relevance of inertia inthe mass flux and derive some particular situations en-compassing the Fickian description and new extendedforms. Such a reduction can be seen as analogous to thereduction of the Navier–Stokes equation to the Stokesequation for low Reynolds number in fluid dynamics.This task necessitates an accurate evaluation of the dif-ferent timescales involved during the dynamic process.We should point out that a precise determination of themagnitude of the relaxation time for the diffusion massflux, J, is required from microscopic considerations thatare still missing at this time. Therefore, the J-Deborahnumber becomes, here, a parameter of the model thathas to be determined from experimental measurementsin a similar way as that has been used in a previousinvestigation (Neogi 1983). As in several media, theinertial terms in the governing equation for J are notrelevant to mass transport. This occurs when the relax-ation characteristic timescale (ΛJ)

−1 becomes smallerthan the diffusion characteristic time scale. The massflux density, J, becomes a dependent state variable. Inthe following, we shall discuss two models derived asspecial cases of the inertia-free formulation where theJ-Deborah number is considered very small comparedto unity.

c-Model

It is well known that diffusion into simple media isadequately described by Fick’s laws, as the changesof the internal structure do not contribute to the dif-fusion process. The Fickian behavior is also expected

to occur in blends of immiscible components in whichthe coupling between diffusion and the changes ofthe interfacial morphology are very weak. Considerfor instance mass transport of methanol into a thinfilm of the immiscible blend [polydimethylsiloxane(PDMS)/polyisobutylene (PIB)] of thickness 10−3 mat room temperature. The methanol has a molar vol-ume of 4.05 × 10−5 and a diffusivity of the order of10−11 m2/s. The blend has a viscosity of 100 Pas, aninterfacial tension of 2.4 × 10−3 N/m, a minor phasevolume fraction of 0.5, and a size density of 3 × 104 m−1.Therefore, the coupling constant g0 is approximately ofthe order of ∼10−6. As g0 << 1, we can assume thatMax(

∣∣Eβγ

∣∣) << 1 in Eq. 43 (i.e., |ϕcN| << |ϕcc|). Theinterface Deborah number DeN = 3ηoφDo

/(Qo�oL2

o

)is of the order of 10−5. Therefore, we can ignore, inaddition to inertial terms, the fourth term on the rightside of Eq. 44 and arrive at the familiar expressionfor the Fickian mass flux: Jα = −ρD∂αc, leading to theclassical diffusion parabolic equation:

∂c∂t

= ∂α

(D∂αc

), (49)

which describes the time evolution of the solvent con-centration into a Fickian immiscible blend.

(c, N) model

If in addition to the situation corresponding to theabsence of inertia (DeJ << 1), one assumes that

g0 ∼ O (1) (50)

Then, the governing equation for the diffusion massflux reduces to

Jα = −ρD(∂αc + Eβγ ∂α Nβγ

)(51)

Clearly, Fick’s second law, (Jα = −ρD∂αc), is now ex-tended by the contribution of an additional term involv-ing the gradient of the area tensor N, which includesthe mechanical changes occurring at the interface. Thesolvent mass fraction balance equation obtained bysubstituting Eq. 51 into Eq. 12 transforms into thefollowing parabolic equation:

∂c∂t

= ∂α

(D(∂αc + Eβγ∂Nβγ

))(52)

Note that, in the (c, N) model, the independent statevariables are the mass fraction, and the surface area

Rheol Acta (2008) 47:807–820 817

tensor N whose time evolution is governed by thefollowing equation:

∂Nαβ

∂t= −D

(∂γc + Eij∂γNij

)(1 − c)

∂γNαβ

−Nαγ∂γ

(D(∂βc + Eij∂βNij

)(1 − c)

)

−Nβγ∂γ

(D(∂αc + Eij∂αNij

)(1 − c)

)+ nαnβnθnγ∂γ

×(

D(∂θc + Eij∂θNij

)(1 − c)

)− �αβγθϕNγθ

(53)

The search for solutions of the governing Eqs. 52and 53 leads to information concerning the profiles ofconcentration, the size, and shape anisotropy of theinterface and also the distribution of stresses (Eq. 29)created by the deformation of the interface. Anotherinteresting situation may occur when the coupling con-stant g0 is greater than unity (g0 >> 1; no significantchanges in the concentration gradients within the sys-tem). From Eq. 51, the diffusion mass flux expressionreduces to Jα = −ρDEβγ ∂αNβγ , and any local changesof the interfacial area (size and shape) may also pro-duce diffusion of the penetrant molecules.

The requirement (50) leads to a size density,Q0∼�T

/�0s, providing an indication of the size of

the droplets constituting the minor phase and also therange of applicability of this reduced model. Equa-tions 44 and 45 also show that an increase of the interfa-cial tension or a change in the experimental conditionssuch as a decrease in temperature may lead to situationsin which mass transport becomes non-Fickian. Indeed,in this case, the magnitude of dimensionless constantg0 becomes significant and strongly couples diffusion tointerfacial changes. Clearly, nanodispersions may alsofall into this category. If one examines the effects ofthe interface Deborah number on mass transport, non-Fickian behavior is expected to occur for values of DeN

in the vicinity of unity.

Traveling nonlinear hyperbolic and lineardispersive waves

Here, we examine the propagation of both hyperbolicand dispersive waves produced by disturbances in themass fraction. To discuss solutions of Eqs. 12, 40, and41, it is useful to rewrite them into the compact form(Mueller and Ruggeri 1998)

∂U∂t

+ M∇U = −PU (54)

where U = (c, vs, N)T is the set of the state variables,and M and P are the matrices that arise by identifyingEq. 54 with Eqs. 12, 29, 40, and 41. We have used thevelocity vs of the front propagation as a state variableinstead of the solvent diffusion mass flux J = ρ c vs.As many experimental observations are unidirectional(sorption, permeation, etc), we derive predictions ofEq. 54 in a one-dimensional setting for which the mix-ture is described by U = (c, vs, N)T (vs is the componentof the velocity in the direction of diffusion). Many ex-perimental observations reported that diffusion in poly-meric fluids is usually accompanied by volume changes(swelling), and therefore, the changes occurring in theinternal structure are assumed to be significant only inthe direction of diffusion chosen here to be the x direc-tion. Therefore, in this simplified picture, the symmetryof the problem reduces the number of the componentsof the tensor N to two independent variables, Nxx andNyy, to be denoted by N1 and N2, respectively (xx ≡ 1and yy ≡ 2). While the off-diagonal components vanish,the remaining diagonal component Nzz coincides withNyy by symmetry. Consequently, the matrices M and Parising in Eq. 54 are given by

M=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

vs c 0 0ϕcc

ρvs

ϕcN1

ρ

ϕcN2

ρ

− r1vs

(1 − c)2 − cr1

1 − c− cvs

1 − c0

r2vs

(1 − c)2 − cr2

(1 − c)0 − cvs

1 − c

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

P =

⎛⎜⎜⎜⎜⎜⎝

0 0 0 0

0 �J 0 0

0 0 ν1 ν12

0 0 ν12 ν2

⎞⎟⎟⎟⎟⎟⎠

(55)

where, for the sake of brevity, we have used the follow-ing notations:

r1 = N21 + 4N1N2

N1 + 2N2, r2 = N1N2

N1 + 2N2(56)

for the parameters arising in the matrix M and

ν1 = �11ϕN1

N1, ν12 = �12ϕN2

N2, ν21 = �21ϕN1

N1, ν2 = �22ϕN2

N2

(57)

for those arising in the matrix P.We use the theory of characteristics (Chorin and

Marsden 1958; Mueller and Ruggeri 1998) to studythe nature of the nonlinear wave propagation dueto the concentration disturbances. The characteristics

818 Rheol Acta (2008) 47:807–820

corresponding to Eq. 54 are the curves τ → (t(τ ),x(τ )) in the (x, t) plane, where τ is a parameter. Thesecharacteristics are generated respectively by

dtdτ

=1,dxdτ

= − cv1 − c

,

dtdτ

=1,dxdτ

= v+[

1

ρ

(cϕcc− c

1−c

(r1ϕcN1 +r2ϕcN2

))]1/2

dtdτ

=1,dxdτ

= v−[

1

ρ

(cϕcc− c

1−c

(r1ϕcN1 +r2ϕcN2

))]1/2

,

(58)

The set of the governing Eq. 54 with Eq. 55 istime-hyperbolic if the eigenvalues of M are real. Thisrequirement leads to the following condition:

κ = 1

ρ

(cϕcc − c

1 − c

(r1ϕcN1 + r2ϕcN2

)) ≥ 0 (59)

where√

κ is the characteristic speed of these nonlinearhyperbolic waves. Moreover, the solutions are stable ifthe real parts of the eigenvalues of P are nonnegative.The requirement of stability leads to

�J ≥ 0,

�11 ≥ 0, �22 ≥ 0, �12 ≤ √�11�22 (60)

Now we turn our attention to linear dispersive waves.These waves are created by small-amplitude perturba-tions to an equilibrium state characterized by Ueq(Ceq,0, Neq) assumed to be time and position independent.Here, we assume that the interface is locked into ananisotropic shape, so qeq is different from zero. Suchwaves can be written as U = Ueq + U exp (i(kx + ωt)),where ω is the real frequency and k refers to a complexwave number. By inserting the expression of U intoEq. 54 and keeping only the terms that are linear in U ,we arrive at the dispersion relation k = k(ω) as a resultof the solution of the following characteristic equation:

det

[I +(

kω

)M −

(iω

)P]

= 0 (61)

where M = M|eq and P = P|eq. By solving Eq. 61, wearrive at(

kω

)2

= X + iY (62)

where

X (ω) = ω2 A1 B1 + A2 B2

ω2 A21 + A2

2

Y (ω) = ω (A1 B2 − A2 B1)

ω2 A21 + A2

2

(63)

written as functions of the following parameters

A1 =[κ + cϕcc

ρω2(ν12ν21 − ν1ν2)

]eq

A2 = −[κ (ν1 + ν2) + c

ρ(1 − c

)

× (r1(ν1 + ν12

)ϕcN1 + r2

(ν2 + ν21

)ϕcN2

)]eq

B1 =[

1 + ν12ν21 − ν1ν2 − �J (ν1 + ν2)

ω2

] ∣∣∣∣eq

B2 = −[� + ν1 + ν2 + �J (ν12ν21 − ν1ν2)

ω2

] ∣∣∣∣eq

(64)

Equation 62 predicts two wave solutions, but one isto be discarded by considering positive values of wavespeed. Therefore, the phase velocity has the followingform:

vph = ω

Re (k)=√

2

X + √X2 + Y2

(65)

and the attenuation of the intensity is given by

α (ω) = −2Im (k) = −ωY

√2

X + √X2 + Y2

(66)

At the high-frequency limit, ω → ∞, we arrive at

vph∞ ∼= √

κeq (67)

α(ω)∞ ∼= (�J + ν1 + ν2)κ + A2

κ√

κ

∣∣∣∣eq

(68)

The phase velocity reduces to the characteristicspeed of the hyperbolic waves, and the attenuation hasa positive value discarding any amplification of suchwaves. Note that these two quantities are frequencyindependent. We recover the results predicted by Fick’slaws by setting in Eqs. 67 and 68, N = 0, νi = 0, andνij = 0 (∀ i, j). We recall that Fickian diffusion presentssome limitations (Jou et al. 1993), such as a predictionof infinite values for the speed and intensity attenuationat small relaxation times (�J → ∞). These limitationsare removed by introducing inertia into the process ofdiffusion, which is expressed both in the equation ofthe mass flux and also by taking the influence of theinterface into account.

At the low frequency limit, ω → 0, we have

vph0 ∼=

√2Dω (69)

α(ω)0 ∼=√

2ω

D(70)

Rheol Acta (2008) 47:807–820 819

where D is the diffusion coefficient defined earlier asD = cϕcc/ΛJ. Both the phase velocity and the atten-uation are directly proportional to the square root ofthe frequency and therefore tend to zero. These resultsagree with previous developments obtained in gases(Jou et al. 1993) and in polymeric fluids (El Afif andGrmela 2002).

Conclusion

This paper examines the flow–diffusion interface in-teraction occurring in a ternary mixture consisting ofone simple fluid (e.g., a solvent) and a blend of twoimmiscible polymers embedding a complex internal in-terface. The main result of this contribution is a familyof models derived on two levels of description: a kineticmodel in which the internal interface is characterized bya distribution function of the unit vector normal to theinterface and a mesoscopic level where the interface isdescribed by a single second-order tensor defined as asecond moment of the distribution function. The non-linear and coupled governing equations for diffusion,flow, and the deformation of the interface are parame-terized by the free energy density and by two kineticcoefficients. These quantities express the individual na-ture of the system under consideration and the par-ticular physical situation involved during the dynamicprocess. The main outcome is that the dynamic changesof the interface are found to be influenced by theapplied flow as well as by the diffusion mass fluxes, and,in return, mass transport becomes intimately depen-dent on the deformation of the interfacial morphology.Several new reduced models are recovered as particularcases and discussed in the light of the N model and pre-vious well-known approaches are obtained as specialcases. We have first examined the case where inertiaof the diffusion mass flux is negligible and treat thecase of a weak coupling with the flow. Such a reducedmodel may be a good candidate to elucidate the flow-induced structure changes in solutions of immiscibleblends. In the absence of flow, the dimensional analysisleads to three groups of physical parameters: two Deb-orah numbers and a constant that couples diffusion tothe deformation of the interface. A careful analysis ofthese groups shows that diffusion may behave as non-Fickian at the nanoscale level (nanodispersions in amatrix). Finally, we have investigated in more detail thenature of traveling of both hyperbolic and dispersivewaves, and provide explicit formulas for the character-istic speed, wave attenuation, and phase velocity. Thestability analysis was used to determine the range ofvalidity of the kinetic coefficients entering the family

of models to satisfy the second law of thermodynamics.These conditions are found to be in good agreementwith the GENERIC requirements.

Appendix

The inequality dSdt =[S H

]>0 is equivalent under iso-

thermal conditions to the inequality d�dt =− [� �

]< 0.

Moreover, as

1.d�

dt= − [�, �

] = −∫

dr�xδ�

δ�x

And the fact that

2. The dissipation potential Ψ is written as a quadraticfunctional of the conjugate variables, which canalso be written in terms of friction matrices as

[�, �

] =∫

dr�x M1�x +∫

dr∇�xM2∇�x

Therefore, it can be easily seen that the condition[Φ, Φ] > 0 implies that the friction matrices M1 and M2

must be definite positive. Consequently, the physicalparameters involved in the dissipation potential shouldbe positive in case they are scalars and semi-definitepositive matrices in case they are tensorial quantities.

On the f level of description, the friction matricesare diagonal:

[�,�] =∫

drη

4(∇�u+TT)2+ 1

2

(ηd − 2

3η

)I (∇ • �u)

2

+�J

2ρc (1−c) (�J)

2+∫

drdn�f

2

(f−feq

)(�f)

2

and the result is straightforward.On the N level of description:

[�,�]=∫

drη

4(∇�u + TT)2 + 1

2

(ηd − 2

3η

)I (∇ • �u)

2

+�J

2ρc (1 − c) (�J)

2 + 1

2�ijαβ

�Nij�Nαβ

The fourth-order submatrix Λ has off-diagonal termsand should also be positive definite. The same conclu-sion is reached by studying stability conditions as isshown in the last section.

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