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41
2Nonequilibrium
Thermodynamics of Interfaces
Leonard M.C. Sagis
2.1 INTRODUCTION
Chapter 1 of this book discussed the thermodynamics of interfaces, in equilibrium with their adjoin-
ing bulk phases. Multiphase systems such as foams and emulsions are often produced or processed
under conditions far from equilibrium, where they are subjected to high deformation rates and sig-
nicant temperature gradients. Their response to these gradients can be highly nonlinear as a result
of an intricate coupling of dynamic interfacial processes and subphase processes. For example,
inhomogeneities in the ow along an interface can cause compositional gradients, where regions
that are compressed coexist with expanded regions (Figure 2.1). The difference in concentration
along the interface drives diffusion of mass along the interface (surface diffusion) and diffusive
exchange between the interface and the subphase. But the concentration gradients also induce in-
plane gradients in the surface rheological properties (such as the surface shear viscosity, or surface
dilatational modulus), leading to a nonlinear rheological response, which affects the deformation
of the interface.
Similar couplings of transfer modes can also be observed when temperature gradients are pres-
ent in the system. These drive energy transfer along and across the interface but, in view of the tem-
perature dependence of surface rheological properties, also affect the deformation of the interface.
Moreover, the temperature may also induce mass diffusion along and across the interface (thermo-
phoresis), an effect that is often negligible in single-phase bulk systems but can be highly important
in describing mass transfer across interfaces [1–3]. Subphase mass transfer can also induce transi-
tions in behavior of an interface. For example, when an interface is stabilized by negatively charged
rodlike particles (e.g., protein brils), adsorption of divalent ions like calcium or magnesium to the
interface can cross-link the brils [4], inducing a transition from liquid-like to solid-like behavior
(Figure 2.2). Rodlike particle stabilized interfaces can also undergo transitions by an applied defor-
mation (Figure 2.3): by compressing the interface its state can change from isotropic to nematic [5].
Whether and when this occurs depends on the degree of mass transfer of the particles between bulk
CONTENTS
2.1 Introduction ............................................................................................................................ 41
2.2 Conservation Principles .......................................................................................................... 44
2.2.1 Conservation of Mass .................................................................................................44
2.2.2 Conservation of Momentum and Moment of Momentum .......................................... 46
2.2.3 Conservation of Energy ..............................................................................................48
2.2.4 Entropy Balance .........................................................................................................49
2.3 Constitutive Modeling ............................................................................................................49
2.3.1 Simple Interfaces ........................................................................................................ 50
2.3.2 Complex Interfaces ..................................................................................................... 53
2.4 Conclusions and Outlook ........................................................................................................ 56
References ........................................................................................................................................ 57
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42 Computational Methods for Complex Liquid–Fluid Interfaces
Convection
Surface
compression
Surface
extension
Desorption
High T Bulk diffusion and heat conduction
Surface diffusion
and heat conduction
Adsorption
Low T
FIGURE 2.1 Coupled mass, momentum, and energy transfer in multiphase systems, in the bulk, along the
interface, and between bulk and interface.
Liquid-like behavior
Divalent cations
Negatively charged
rodlike particles
Solid-like behavior
FIGURE 2.2 Effect of mass transfer between bulk and interface on interfacial behavior: adsorption of posi-
tively charged divalent ions cross-links adsorbed protein brils, inducing a change from liquid-like to solid-
like (2D gel) behavior.
Isotropic Nematic
FIGURE 2.3 Example of a deformation-induced phase transition in an interface stabilized by rodlike
particles: upon compression the state of the interface changes from isotropic to 2D nematic.
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43Nonequilibrium Thermodynamics of Interfaces
and interface. If they are soluble in the subphase, the compression may cause the particles to desorb
from the interface, prior to reaching the surface concentration at which the transition occurs.
To describe the behavior and time evolution of multiphase systems under dynamic conditions
and account for the coupling of mass, momentum, and energy transfer, we need to resort to a
generalization of classical equilibrium thermodynamics, generally referred to as nonequilibrium
thermodynamics (NET) [6].
There are several NET frameworks currently available to describe the dynamics of multiphase
systems [7]. A rst classication of these frameworks can be made based on the description of the
interface. The two principal methods to describe interfaces in multiphase ows are the diffuse
interface model [8] and the Gibbs dividing surface model [9]. In the former, the interface is mod-
eled as a thin 3D region, in which densities and material properties change rapidly but continuously
from their value in one bulk phase to their value in the adjoining bulk phase [8]. In the Gibbs divid-
ing surface model, the interface is modeled as a 2D surface, placed sensibly within the interfacial
region [9]. Bulk properties are extrapolated up to the surface, and the difference between actual
and extrapolated elds is accounted for by associating excess properties (e.g., a surface excess den-
sity or surface excess energy) with this dividing surface. The Gibbs dividing surface model nds
application mainly in macroscopic ow problems, where the interfacial thickness d ≪ L, where L
is a characteristic dimension of the ow problem. The diffuse interface model is typically used to
describe problems where d ≃ L, such as droplet coalescence, phase separation in immiscible blends,
or nano- and microuidic ows.
In this chapter, we will focus only on NET frameworks employing the Gibbs dividing surface
model. At the core of all Gibbs-based NET frameworks is the derivation of a set of partial differen-
tial equations for the (coupled) time rate of change of mass, momentum, and energy of a multiphase
system, and the derivation of expressions for the uxes that appear in these balances, which together
with an equation of state and a set of appropriate boundary conditions, provide a complete descrip-
tion of the dynamics of a multiphase system.
One of the rst NET frameworks to be generalized to multiphase systems was classical irre-
versible thermodynamics (CIT) [10–15]. This framework is particularly suited for describing
multiphase systems near equilibrium, with interfaces with a v iscous behavior (Section 2.3.1).
By introducing internal (or structural) variables [16], the CIT framework can also be used to
derive descriptions for (nonlinear) viscoelastic interfaces (Section 2.3.2). A second framework
we will review here is extended irreversible thermodynamics (EIT) [17–19]. In this framework,
the surface uxes for mass, momentum, and energy are treated as additional independent vari-
ables. EIT is also suitable for constructing constitutive models for viscoelastic interfaces (Section
2.3.2). Other frameworks capable of describing multiphase systems exist, such as rational ther-
modynamics [7,20], extended rational thermodynamics [7,21], or GENERIC [7,22–27], but these
are outside the scope of this chapter. See the recent articles [21–27] for discussions of these
frameworks.
The rst, and one of the most important steps in all NET frameworks, is to select a comprehen-
sive set of system variables, capable of describing the dynamics of the system we wish to study. For
a single-phase system without a complex microstructure, the classical choice of system variables is
the bulk density, ρ; the momentum density, m = ρv (where v is the bulk velocity eld); the internal
energy density,
UU=rˆ
(where
ˆ
U
is the internal energy per unit mass); and, in case we are deal-
ing with an N-component system, the densities of the individual components, ρ(J) (J = 1,…,N−1).
For a multiphase system with excesses mass, momentum, and energy, associated with the dividing
surface, the most straightforward choice for the surface variables would be to introduce an excess
variable for each of the aforementioned bulk variables. This means we assume that our relevant
surface variables are the surface mass density, ρs; the surface momentum density, ms = ρsvs (where
vs is the surface velocity eld); the surface internal energy per unit area,
UU
sss
=r ˆ
(where
ˆ
Us
is
the surface internal energy per unit mass); and the surface densities of the individual components,
r()J
s
(J = 1,…,N − 1).
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44 Computational Methods for Complex Liquid–Fluid Interfaces
In Section 2.2, we will show that the inclusion of these surface excess variables in the conserva-
tion principles for a multiphase system allows us to derive balance equations for these variables
[7,10–15,19–21,25–27]. These balance equations contain contributions from uxes, and in Section
2.3.1, we will show how CIT can be used to derive constitutive equations for these uxes, which are
consistent with the second law of thermodynamics, for simple viscous interfaces without a complex
microstructure. In Section 2.3.2, we show how, with the introduction of internal variables [16],
the CIT framework can also be used to derive constitutive equations for interfaces with a complex
microstructure, with (nonlinear) viscoelastic behavior. We also discuss how such equations can be
derived using the EIT framework.
2.2 CONSERVATION PRINCIPLES
In this section, we will discuss the conservation of mass, momentum, moment of momentum,
energy, and entropy, for multiphase systems with excess variables associated with their dividing
surfaces. We will show that the inclusion of excess variables in the conservation principles leads
to a set of partial differential equations for these excess variables, describing their time evolution
as a result of in-plane convective and diffusion processes, and convective and diffusive exchange
between the dividing surface and the adjoining bulk phases [7,10–15,19–21,25–27]. These bal-
ances are often referred to as jump balances [7,20], and together with a set of boundary condi-
tions, they couple the motion and deformation of the dividing surfaces to the time evolution of
the bulk phases.
2.2.1 coNSerVatioN of MaSS
The principle of conservation of mass states that the mass of a multiphase system should be constant
in time. For a multiphase system with mass associated with its dividing surfaces, we can express
this principle as [7,11,15,20]
d
d
dd
s
tV
R
rr
òò
+
é
ë
ê
ê
ù
û
ú
ú=W
S
0, (2.1)
where
R denotes the domain of the bulk phases
Σ denotes the domain of the dividing surfaces
dV denotes a volume integration
dΩ denotes an area integration
Evaluating the time derivative on the left-hand side of Equation 2.1, we nd that the principle of
conservation of mass requires that at any point in the bulk phase R
d
d
b
r
r
t
=- Ñ×v, (2.2)
which is generally referred to as the equation of continuity [7,11,15,20]. Here, ∇ is the 3D gradient
operator. In addition to this expression, we nd that conservation of mass also implies that at any
point on the dividing surface Σ
d
d
ssss
ss
rrr
t
=- Ñ× -
()
×-vv
vn
. (2.3)
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45Nonequilibrium Thermodynamics of Interfaces
This equation is referred to as the jump overall mass balance [7,20]. At every point on the divid-
ing surface, it describes the time rate of change of the overall surface mass density, as a result of
in-plane convective processes (rst term on the right-hand side), and exchange of mass with the
adjoining bulk phases (second term on the right-hand side). The operator ∇s is the surface gradient
operator [2,15]. The material derivatives appearing in Equations 2.2 and 2.3 are dened as
d
d
d
d
bs
ss
ss
yy yyy y
tt tt
=¶
¶
+Ñ
()
×=
¶
¶
+Ñ
()
×
vy
. (2.4)
The velocity
y
is the intrinsic surface velocity, dened as
yv uº-
s,
where u is the speed of dis-
placement of the interface [20]. The double square brackets in Equation 2.3, describing the exchange
of mass between the interface and the adjoining bulk phases, are dened as [20]
yy ynnn
=+
II II II, (2.5)
where
ψI and ψII are, respectively, the value of ψ in bulk phases I and II, evaluated at the dividing
surface
The vector nI is the unit vector normal to the dividing surface, pointing in the direction of phase I
(and hence, nI = −nII)
When our system is a m ulticomponent system, we can also explore what the principle of con-
servation of mass implies for each individual component in the mixture. For each component in
the mixture, mass conservation implies that its time rate of change is equal to the rate at which
the component is converted by chemical reactions in the bulk phase and at the dividing surfaces.
Mathematically, we can express this as [7,11,15,20]
d
ddd
dd
ss s
t
Vr
Vr
J
R
JJ
R
J
rw rw
() () () () ,
òò
òò
+
é
ë
ê
ê
ù
û
ú
ú=+
WW
SS
(2.6)
where
ω(J) and
w()J
s
are, respectively, the bulk and surface mass fraction of component J (J = 1,…, N)
r(J) is the rate at which mass of component J is converted per unit volume, by reactions in the
bulk phase
rJ()
s
is the rate at which J is converted per unit area, by reactions in the dividing surface
Evaluating the time derivative on the left-hand side of Equation 2.6, we nd that in the bulk phases,
rwd
d
b()()
()
,
J
JJ
t
r=-Ñ× +j (2.7)
which is the familiar differential mass balance for component J in the mixture. At any point on the
dividing surface, Equation 2.6 requires
rwrw w
sss
sss ss
d
d
() () () () () ()
JJJ JJ J
t
r=-Ñ× -
()
-
()
×+ ×+-
jv
vn
jn
. (2.8)
This balance is referred to as the jump species mass balance [7,20] and describes the time rate of
change of the surface excess concentration of component J, as a function of (in order of appearance,
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46 Computational Methods for Complex Liquid–Fluid Interfaces
on the right-hand side of the equation) in-plane diffusion, chemical reactions, convective transport
between the interface and bulk phases, and diffusive transfer between the interface and bulk phases.
The mass ux vectors appearing in Equations 2.7 and 2.8 are dened as
jvvj
vv
() () () () () ()
()
,(
).
JJJJJJ
=-
=-rr
ss
ss
(2.9)
The jump species mass balance is an important balance. When interfaces are deformed, surface
concentrations typically change, and in-plane gradients may develop, which drive mass transfer
along the interface (Gibbs–Marangoni effect) and exchange of mass with the adjoining bulk phases.
These mass-transfer processes affect the composition of the interface and hence its local material
properties, such as surface tension or surface viscosities. The jump mass balance then needs to be
solved simultaneously with the jump momentum balance, which we will discuss next.
2.2.2 coNSerVatioN of MoMeNtuM aNd MoMeNt of MoMeNtuM
The principle of conservation of momentum states that the time rate of change of momentum of a
system is equal to the sum of the body forces acting on the material in the bulk phases and dividing
surfaces and the stresses (or contact forces) acting on the system through its outer boundaries. For
a multiphase system with excess properties associated with its dividing surface, we can formulate
this principle as [7,11,15,20]
d
ddd dd
dd
ss ss s
tVV
TT
RR
SC
òò òò
òò
+
é
ë
ê
ê
ù
û
ú
ú=+ +× +×rrWr rW W
SS
vv bb nmLL, (2.10)
where
b and bs are the body forces per unit mass acting on, respectively, material in the bulk phase and
material in the dividing surfaces
T is the stress tensor in the bulk phase
Ts is the surface stress tensor
S and C domains are the outer bounding surface of the system and the curve formed by the inter-
section of this surface with the dividing surface Σ
μ is the unit vector, perpendicular to C, tangent to Σ, and pointing outward from the system
dL denotes a line integration
Evaluating the time derivative on the left-hand side of Equation 2.10 and using the divergence and
surface divergence theorems, we nd that the principle of conservation of momentum implies that
at each point in the bulk phase, we must satisfy the differential momentum balance:
rr
d
d
bv
Tb
t
=Ñ×+ . (2.11)
In addition, on each point on the dividing surface Σ, we must satisfy
rrr
sss
ssssss
d
d
vTb vv vv nTn
t
=Ñ ×-
()
-
()
×-×+-
. (2.12)
The latter equation is often referred to as the jump momentum balance [7,20]. Often, we will nd it
convenient to split the stress tensors in these two equations in a hydrostatic and deviatoric contribu-
tion, according to
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47Nonequilibrium Thermodynamics of Interfaces
TI TP=+ =+
-g
ps
ss
s
,,
ss
(2.13)
where
p is the thermodynamic pressure
σ is the extra stress tensor in the bulk phase
γ is the surface tension
σs is the surface extra stress tensor
I and P are, respectively, the 3D and 2D unit tensors [7,20]
Substituting Equations 2.13 in 2.12, we obtain
rsss
ss
sssss
d
d
vnbvv vv nn n
t
HP
=Ñ ++Ñ× +
()()
×+ ×gg r-r-
--
2s
ss
s
. (2.14)
Here, H is the mean curvature of the dividing surface. This expression allows us to calculate the
time rate of change of momentum associated with the dividing surface, as a result of (in order of
appearance on the right-hand side of the equation) surface tension gradients (Marangoni stresses),
curvature-induced stresses, deviatoric stresses, body forces, and stresses exerted on the interface by
the adjoining bulk phases (respectively, inertial, hydrostatic, and deviatoric stresses). Equation 2.14
is a generalized form of the Young–Laplace equation, which is often used in the analysis of droplet-
based tensiometry methods, such as prole analysis tensiometry or bubble pressure tensiometry
[7,28]. For slow surface deformations, the term on the left-hand side of the equation will be negligi-
bly small. If in addition the deformation is uniform and deviatoric stresses and external force elds
are negligible, as well as the inertial and deviatoric stresses exerted on the interface by the bulk
phases, then Equation 2.14 reduces to
20
gHnn
-=
p
. (2.15)
And for a spherical bubble with radius R, this can be written as
g
Rpp=-
II I, (2.16)
where pII is the pressure in the interior of the bubble. This is the familiar Laplace equation for the
pressure difference across the interface of a spherical bubble [20]. In tensiometry experiments on
interfaces with a complex microstructure, the deviatoric contributions to (2.14) are often not negli-
gible, which may lead to a strain and droplet size dependence of the dilatational moduli [29].
The principle of conservation of moment of momentum states that the time rate of change of the
total moment of momentum of a system is equal to the total torque applied on the system. If we
assume that the body forces and stresses we previously introduced are the only contributions to the
applied torque, we can formulate this principle as [20]
d
ddd
dd
ss ss
t
VV
RR
S
òò òò
ò
´+´
é
ë
ê
ê
ù
û
ú
ú=´ +´
+´×
rrWr rW
SS
rv rv rb rb
rT(nnrT)(
),dd
s
W+ ´×
ò
C
Lmm
(2.17)
where r is the position vector. Evaluating the time derivative on the left-hand side of (2.17), using
the divergence and surface divergence theorems, and invoking the differential and jump momentum
balance, we nd that [20]
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48 Computational Methods for Complex Liquid–Fluid Interfaces
ee
:,
:,TT==00
s (2.18)
which implies that both stress tensors are symmetric. The tensor ε is the skew-symmetric Levi-
Civita tensor [20]. Note that when there are additional sources of moment of momentum, beyond
those resulting from the body and contact forces (such as those induced by a suitable rotating elec-
tric eld), the stress tensors may no longer be symmetric. In the remainder of this chapter, we will
assume that both the bulk and surface stress tensors are symmetric.
2.2.3 coNSerVatioN of eNergy
The principle of conservation of energy states that the time rate of change of the sum of the internal
and kinetic energy of a system is equal to the work done on the system by the sum of all body and
contact forces, plus the energy transmitted to the interior of the system by radiation, plus the energy
transmitted to the system through its outer boundaries. For a multiphase system with mass, momen-
tum, and energy associated with its dividing surface, we can formulate this principle as [7,11,15,20]
d
ddd
ss s
tUvVUv
RR
J
òò òå
+
æ
è
çö
ø
÷++
æ
è
çö
ø
÷
é
ë
ê
ê
ù
û
ú
ú=rrW
S
ˆˆ
[]
1
2
1
2
22
rr
rW W
S
()() ()
()() () () (
JJ J
J
JJ J
SC
Vbv
bv vTnv
×
+×+××+×
òåòò
d
dd
ss
ss
TT
qn q
s
ss s
d
dddd
×
++ +× +×
òò òò
m
m
)
,
ˆˆ
L
QV
QL
RSC
rrWW
S
(2.19)
where
ˆ
Q
is the rate of energy transfer per unit mass to the material in the bulk phase by radiation
ˆ
Qs
is the rate of radiative energy transfer per unit mass to the material in the dividing surface
q is the energy ux vector
qs is the surface energy ux vector and v2 = v ⋅ v
In the rst two terms on the right-hand side of this equation, we have allowed for the fact that the
body forces may vary for different components. Evaluating the time derivative on the left-hand side
of the equation, using the divergence and surface divergence theorems, the differential and jump
mass balances, and the differential and jump momentum balances, we nd that the principle of
conservation of energy requires that at each point in the bulk phase
r-
-r
d
d
b
J
JJ
U
t
pQ
ˆ
ˆ
:.
() ()
=Ñ Ñ× +×Ñ× +
å
ss vvjb q (2.20)
In addition, we must require at each point on the dividing surface
rg -r
-r
sss
ssssssssss
d
d
U
tQ
Up
J
N
JJ
ˆ
ˆ
ˆ
=Ñ+Ñ×+ ×Ñ×+
+
=
å
ss :() ()
vvjb q
1
VVU
ˆˆ
-----
ss
ss
+
æ
è
çö
ø
÷×+×
()
××
1
2
2
vv vv nqnv
vn
() .ss
(2.21)
These balances are generally referred to as, respectively, the differential energy balance and the
jump energy balance [7,20]. In Equation 2.21,
ˆ
/
V
=1
r
denotes the volume per unit mass, and the
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49Nonequilibrium Thermodynamics of Interfaces
colon in the rst term on the right-hand side denotes a double contraction between the two tensors.
The jump energy balance allows us to calculate the time evolution of the surface internal energy
per unit mass, as a result of in-plane viscous dissipation, the work done by surface tension and body
forces, in-plane conduction, radiative heat transfer, convective and conductive exchange with the
bulk phases, and viscous friction between the interface and the bulk phases.
2.2.4 eNtropy balaNce
The nal principle we will consider in this section is the entropy balance. This principle states that
for any system the time rate of change of its total entropy must be equal to the entropy production in
its interior, plus the entropy transmitted to the system through its outer boundaries. Mathematically,
we can formulate this as [7,11,15]
d
ddddd d
ss ss
tSV SEVE
RRS
S
C
òò òò
òò
+
é
ë
ê
ê
ù
û
ú
ú=+ ×rrWr rW-W-
SS
ˆˆ ˆˆ
jn jjSL
sd×mm, (2.22)
where
ˆ
E
is the rate of entropy production per unit mass in the bulk phase
ˆ
Es
is the rate of entropy production per unit mass in the dividing surface
jS is the bulk entropy ux vector
jS
s
is the surface entropy ux vector
Evaluating the time derivative on the left-hand side of this equation and using the divergence and
surface divergence theorems, we nd that at any point in the bulk phase, we must satisfy
rr
d
d
bˆˆ.
S
t
E
S
=-Ñ× +j (2.23)
In addition, at every point on the dividing surface, we must require
rrr
ssssssss
d
d
ˆˆˆˆ
() .
S
t
ESS
SS
s=-Ñ× +--
()
-×+×
jv
vn
jn
(2.24)
These equations are, respectively, the differential and jump entropy balances [7,20]. To satisfy the
second law of thermodynamics, we must impose
ˆ
,
ˆ
,
EE³³00
s (2.25)
where the entropy production rates are zero for reversible processes, and greater than zero for irre-
versible processes. The entropy balances provide an important tool in the construction of admissible
constitutive equations, which do not violate the second law of thermodynamics. We will illustrate
this in the next section.
2.3 CONSTITUTIVE MODELING
When we examine the balance equations we derived in the previous section more closely, we observe
that we still need to supply some additional information before we can solve them: (1) we need to
choose appropriate constitutive equations for the uxes that appear in these balances, and (2) we
need to construct the boundary conditions that couple the bulk dynamics to the time evolution of
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50 Computational Methods for Complex Liquid–Fluid Interfaces
the dividing surface. The various NET frameworks differ in how they construct these equations.
Here, we will primarily follow the procedures used in the CIT framework [6]. In Section 2.3.1, we
will rst discuss constitutive modeling for simple interfaces, and in Section 2.3.2, we briey discuss
modeling the behavior of interfaces with a complex microstructure.
2.3.1 SiMple iNterfaceS
In the CIT framework, the entropy balance is used to guide us i n the construction of constitu-
tive equations for the uxes and the boundary conditions that couple the bulk and jump equations
[6,7,11,15]. We will illustrate this procedure here for the surface uxes, using the jump entropy bal-
ance. The derivation of constitutive equations for bulk uxes is described in detail elsewhere [6] and
is very similar to the development we present here.
To construct equations for the surface uxes, we rst need to choose a functional dependence
for the surface entropy per unit mass [6,7,11,15]. In the CIT framework, it is typical to assume local
equilibrium, that is, although the system as a whole is not in a global state of equilibrium, locally, at
any point in the system, the material is assumed to be in equilibrium with the material in its imme-
diate neighborhood [6]. This means that locally, the surface entropy per unit mass depends on the
same set of variables, as the entropy of an interface in global equilibrium:
ˆˆ
ˆˆ
,, ).(,,()
()
SSUA
JN
ssss s
=-
ww
…1 (2.26)
Here,
ˆ/A=1rs
is the surface area per unit mass. If we now take the surface material time derivative
of
ˆ
Ss
and invoke (2.26), we obtain using the chain rule of differentiation
rrgrr
mw
ssss
s
sss
s
ss
s
sss
d
d
d
d
d
d
d
d
ˆˆˆ,
() ()
S
tT
U
tT
A
tT t
J
J
N
J
=--
=
å
1
(2.27)
where
mJ
s
is the chemical potential of component J
T s is the surface temperature
Next, we substitute the jump energy balance (2.21), the jump mass balance (2.3), and the jump com-
ponent mass balance (2.8) in this expression and subsequently introduce the result into the jump
entropy balance (2.24), to nd an expression for the surface rate of entropy production per unit mass:
r---
ss
s
ss s
s
s
s
ss
s
s
tr tr
ˆ:()
() ()
ETTTT
J
N
JJ
J
=+ ×
==
å
111
1
2
1
ss ss
DD jd q
NN
JJ
J
N
T
T
UpVTT
TT
å
å
æ
è
ç
ç
ö
ø
÷
÷×Ñ
+
()
é
ë
êù
û
ú+
=
m
-
r-
()()
ˆˆ
ss ss
s
ss
j
1
1
mm -mw-
r-
˜() ˜() () ()
()
JJ
J
TT
vv
s
s
s
s
æ
è
ç
ç
ç
ö
ø
÷
÷
÷
æ
è
ç
ç
ç
ö
ø
÷
÷
÷×
+
vv n
1
2
22 22
+
æ
è
çö
ø
÷×
+× é
ë
êù
û
ú
()
××+
vT vT
T
TT
TT
ss s
sss
--
--- s
() ()vv n
qn vv n
JJ
N
JJJ
TT
=
å×æ
è
çö
ø
÷
1
jn
() () ()
mm
-s
s
≥0.
(2.28)
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51Nonequilibrium Thermodynamics of Interfaces
The bar over the tensors σs and Ds denotes the symmetric traceless part of these tensors, so,
ssssss
s=ss
tr-12P()
and DD PD
s
=-
ss
tr12
()
.
The tensor Ds is the surface rate of deformation tensor, equal to
12/.
[]Pv
vP×Ñ Ñ×
()
+
s
s
s
sT
The chemical potentials
µ()J
and
µ()J
s
are, respectively, the velocity modied bulk and surface
chemical potentials of component J, equal to
mm
() ()JJ v=-12 2
/
and
mm
() () (.)
JJ v
ss s
/=-12 2
T is the temperature in the bulk phase, and the vector
db
() () ()
.
JJJ
ssss
ºÑ -m
In arriving at t his result, we have assumed that the reaction rates
rJ()
s
and the rate of radiative
energy transmission per unit mass
ˆ
Qs
are both zero.
Equation 2.28 is a bilinear form consisting of products of uxes and driving forces. To ensure
Equation 2.28 is satised, we assume that the uxes that appear in this expression depend linearly
on all driving forces of equal tensorial order [6,7,11,15]. For the traceless part of the surface extra
stress tensor, and its trace, this implies, we choose
ssss
sssss
==
() ,
DD
2e (2.29)
tr tr tr tr
sssds
ssss
==
() ,
DD
e (2.30)
where
εs is the surface shear viscosity
εd is the surface dilatational viscosity
Both these viscosities may depend on temperature and composition of the interface, but not on the
rate of deformation. Combining these two expressions, we obtain
sssds sss
tr=
()
()
+e-
ee
DP D
2,
(2.31)
which is the linear Boussinesq model [30–32], the surface equivalent of the Newtonian uid model.
For the mass and heat ux vectors, Equation 2.28 suggests the following functional dependence:
jj
dd
() () () ()() ()
(,
)ln,
JJJJKK J
K
TD T
ssssssss
ss
=Ñ Ñ=- -
å
a (2.32)
qjjj
dd
ssssssss sssss
-==Ñ
=Ñ--
=
å
ma
l
()() () () ()
(, )
J
J
N
JSSJ JJ
TT T
1
ss s
ln ,T
J
å (2.33)
where
the coefcients
DJK()
s
denote the components of the N × N diffusion matrix
a()J
s
is the surface thermal diffusion coefcient for component J
λs is the surface thermal conductivity
Using the fact that
m() () ()
,
JJJ
HTS
sss
=-
where
HJ()
s
and
SJ()
s
are the partial surface enthalpy and entropy
of component J, the left-hand side of Equation 2.33 can be written as
εqj j
s
()
s
1
()
ss s
()
s
()
s
1
,
-+
º
==
åå
mJ
J
N
JJ
J
J
N
TS (2.34)
where
ε
ss
ss
==
å
qj
-H
JJ
J
N
()
()
1 is often referred to as the measurable heat ux.
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52 Computational Methods for Complex Liquid–Fluid Interfaces
Note that Equations 2.32 and 2.33 contain couplings between mass and energy transfer, which
are the surface equivalents of the Soret effect (mass transfer driven by gradients in temperature,
also known as thermodiffusion or thermophoresis) and Dufour effect (the energy ux driven by
concentration gradients). These effects are typically negligible in the bulk phase, and for in-plane
mass and energy transfer, but are often highly relevant in the description of mass and energy transfer
across interfaces [1–3]. When the Soret effect and contributions from forced diffusion are negli-
gible, Equation 2.32 reduces to
j() ()
()
,
JJKK
K
D
ss
ss
=- Ñ
å
m (2.35)
which is the surface equivalent of Fick’s law. For an interface with uniform surface composition, the
Dufour effect is negligible, and Equation 2.33 reduces to
qss
sss
=- Ñ
l
T
T. (2.36)
This expression is the surface equivalent of Fourier’s law.
Apart from the constitutive expressions for the in-plane uxes, Equation 2.28 allows us also to
construct constitutive expressions for the uxes describing exchange between the interface and the
adjoining bulk phases. For this, we have to focus on the double bracket term in this expression.
Following along the lines we used for deriving the constitutive equations for the surface uxes, we
obtain (M,N = I,II)
ssMM MM MM
NI
II
MN N
N
T
TT
×
()
×= ×æ
è
çö
ø
÷
å
nvvvn
vv
-r -z-
ss
s
s
=
,, (2.37)
qn vvnv n
MM MM MM MM MM
MM
M
UpV
TT
R
×+ ++
é
ë
êù
û
ú
()
××
×
=
r-
-
--
ˆˆ
()
1
2
2
vs
s
ss
KK
M
J
J
TM MJ
M
M
J
TT TT
--
åæ
è
çö
ø
÷
Lmm
() () () ,
ss
s
(2.38)
jn vvn
() () () () () (JMJ
MJJ
M
JJ
MMMM
M
TT
×+ -
()
×- -
æ
è
çö
ø
÷-=r mm
ss
s
LL
)) ,
TM M
TT
-
()
s (2.39)
where
ζM,N are the friction tensors, quantifying the exchange of momentum between the bulk phases
and the dividing surface
RK
M
is the Kapitza coefcient, which is the resistance against energy transfer between bulk and
interface
LL()J
M
is the mass-transfer coefcient for exchange of component J between bulk and interface,
driven by differences in chemical potential
L()J
TM
is the mass-transfer coefcient for the transfer of component J between bulk and interface,
driven by temperature differences
These equations are not only constitutive equations for the exchange of mass, momentum, and
energy between the bulk phases and the dividing surface. They also act as boundary conditions,
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53Nonequilibrium Thermodynamics of Interfaces
coupling the differential equations for the bulk density, velocity, and energy elds, with their respec-
tive jump balances.
The entropy balance allows us also to make a statement about the sign of the coefcients we
introduced in the constitutive equations [6]. For a system without exchange between the bulk phase
and the dividing surface, Equation 2.28 reduces to (substituting Equations 2.29, 2.30, 2.32, and 2.33)
ee
a
s
ss
d
ssss
ss s
trDD
Dd
d
d
:()
ln
()() ()
() ()
++
+
×
×Ñ
åå
å
2
2
DJK
JK
JJ
KJ
J
T
TT
ss
ss
+³
Ñl (ln) .
20
(2.40)
For this quadratic expression to be satised for any arbitrary rate of deformation, concentration
gradient, or temperature gradient, we must require
ee l
sd
ss
³³
³³
00
00
,,
,.
()
DJK (2.41)
In a s imilar manner, we nd that the coefcients quantifying the transfer of mass and energy
between the bulk phase and the dividing surface must satisfy
RK
MJ
MJ
TM
³³³00
0
,,
() ()
LL
, (2.42)
and that the tensors ζM,N are positive semidenite tensors.
The set of bulk and jump balances, coupled by the boundary conditions Equations 2.37 through
2.39, is not yet complete. This is perhaps most obvious when we substitute (2.31) in the jump
momentum balance to nd
rgge-e e-e
sss
sdssdsss
ss
d
dtr tr
tr
vnD
nD
D
tHH=Ñ ++
+Ñ
+Ñ
22()()()
()
(ee-ee e
r-r- --
ds ss sss s
ss ss
)()+Ñ×+Ñ×
+
()()
×+ ×
22
DD
bvvvvn nnpss
.
(2.43)
To solve this equation, we need to supply an equation of state that links the surface tension to surface
composition and temperature of the interface (see Chapter 1). We also need to know the concentra-
tion and temperature dependence of the surface viscosities. The dependence of material properties
on surface composition and temperature cannot be obtained from macroscopic NET frameworks.
For this we have to resort either to microscopic theories (and simulations), or alternatively, we can
determine them experimentally.
2.3.2 coMplex iNterfaceS
The constitutive equations we have derived in the previous section are meaningful only for small
departures from equilibrium and interfaces with a purely viscous behavior. However, many
interfaces actually display viscoelastic behavior and may have a h ighly nonlinear response to
applied deformations or temperature and concentration gradients. This is particularly true for
interfaces in which the surface-active components tend to self-organize into complex micro-
structures, after adsorption into the interface. A wide range of interfacial microstructures can
be observed in multiphase systems. For example, proteins [33,34], protein aggregates, colloidal
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54 Computational Methods for Complex Liquid–Fluid Interfaces
particles [35,36], surface-active polymers [37], and some low-molecular-weight surfactants [38]
can all form (quasi) 2D gels. Proteins and low-molecular-weight surfactants can also form 2D
glasses [39,40]. Protein brils [5,41,42], semiexible and rodlike polymers, or anisotropic col-
loidal particles can all form 2D liquid crystalline phases. Mixtures of immiscible surface-active
components can phase-separate after adsorption, forming interfaces with a heterogeneous struc-
ture, such as 2D dispersions or 2D emulsions [43]. Mixtures of proteins and charged polysac-
charides can form interfaces with a composite structure [44–46]. Whereas for simple interfaces,
the surface tension is often the only interfacial parameter with a signicant effect on the macro-
scopic behavior of a multiphase systems, for complex interfaces, rheological properties like the
shear and dilatational loss and storage moduli or the interfacial bending rigidity often have a
nonnegligible impact. For example, the latter has a signicant effect on the dynamics of disper-
sions of vesicles, cells, and droplets in phase-separated biopolymer solutions [47–54], whereas
the shear and dilatational properties appear to have a signicant effect on stability and dynamics
of foam and emulsions [55,56].
Complex interfaces have a nonlinear response to deformations because the applied deforma-
tion induces changes in the microstructure of the interface. The most common way to capture
these changes in the microstructure, and their effect on overall macroscopic behavior, is to include
additional variables, referred to as structural or internal variables, in the set of independent system
variables [16]. The set of surface excess variables may, for example, be extended to
{,
,,,,,, },,,,,,,,
() ()
rrr
ssss ssss ss s
mc
cC CUcc
Nm
np
1111 1
……
……
- (2.44)
where
cm
s
are the scalar structural variables
cn
s
are the vectorial structural variables
Cp
s
denote the structural variables of a tensorial nature
Examples of variables of a scalar nature are the segment density of an adsorbed polymer or the
surface fraction of particles in a Pickering stabilized interface. An example of a vectorial structural
variable would be the director eld of a 2D liquid crystalline interface. Examples of tensorial vari-
ables are the second moment of a particle orientation or polymer segment orientation distribution
function, which describes deformation-induced changes in orientation (and in the case of polymers,
also degree of stretching) of adsorbed components.
The incorporation of such variables in the set of independent surface variables leads to an addi-
tional set of partial differential equations, describing the time evolution of these variables as a result
of in-plane convective, diffusive, and relaxation processes and through exchange with the adjoining
bulk phases [58,59]. This time evolution is combined with an expression for the surface stress tensor
in terms of the structural variables and is closed with a constitutive model for the surface congu-
rational Helmholtz energy, in terms of these same variables. For example, for an interface with a
microstructure that can be described by a single unconstrained tensor, Cs, the constitutive model for
the surface extra stress tensor could take the form
ss
ss
s
=
nn
C, (2.45)
d
d
sssssT
sss ss
ss
s
CCv vC DC
CC
t-×Ñ
()
Ñ
()
×= ×
--
-211
12
btt . (2.46)
Here, νs, βs, τ1, and τ2 are the scalar coefcients, which may depend on composition and tem-
perature of the interface. The derivative on the left-hand side of (2.46) is the upper-convected
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55Nonequilibrium Thermodynamics of Interfaces
surface derivative. The rst term on the right-hand side of this expression, proportional to the rate of
deformation tensor, drives the microstructure of the interface out of equilibrium. The last two terms
describe linear and nonlinear relaxation processes, which drive the structure back to equilibrium. A
model such as the one in Equation 2.46 can predict strain-thinning behavior [58].
To construct structural models capable of describing surface rheological data, we need to deter-
mine the time evolution of the microstructure experimentally (or computationally). Such informa-
tion is not always available, and when this is the case, we can resort to the extended thermodynamics
framework. In this framework, the uxes are included in the set of system variables, and hence
(2.44) reduces to [7,19]
{,
,, ,, ,,,,
,,
}.
() () () ()
rrr
--
ssss ssss
ss
tr
mq
jjU
NN
11
11
¼¼
s
ss
s (2.47)
The procedure to construct constitutive equations is similar to that used in the CIT framework. We
choose
ˆˆ
ˆˆ
,, ,, ,,,,
,,
() () () ()
SSUA
NN
ssss ssss
ss
tr=¼ ¼
()
ww
--
11
11
s
ss
sqj j (2.48)
and use this functional to construct an expression for the surface rate of entropy production. This
bilinear form is then again used to guide us in the construction of the constitutive equations. When
we limit ourselves to linear models, the EIT approach leads to the following expressions for the
surface extra stress tensor [7,19]:
d
d
ss
s
ss
s
s
ss ss
t+=
12
t
e
tD, (2.49)
dtr
dtr tr
ss
d
s
d
s
d
ss ss
t+=
12
t
e
tD. (2.50)
These expressions are the surface equivalent of the linear single-mode Maxwell model. The
parameter τs is the surface shear relaxation time, and τd is the surface dilatational relaxation time.
Multimode variations of this model can simply be derived by replacing (2.48) by
ˆˆ
ˆˆ
,, ,, ,,,,,,,
() () () (
SSUA NkN
ssss sssss
tr tr=¼¼¼ww
-1111
ssssqj j--11),,
,,
sss
s
ss
s¼
()
k (2.51)
where
tr s
ssk
and
ssk
s
represent the k-th mode of the surface extra stress tensor. For each of these
modes, expressions of the form of Equations 2.49 and 2.50 are then obtained. It is also straightfor-
ward to generate nonlinear models in this framework. By switching to convective derivatives and
incorporating higher order terms in the stress tensor, we obtain
d
d
ssssssss
s
ss
s
ss s
s
s
ss ssssssssss
t
T
-- t
a
e
e
t
×Ñ
()
Ñ
()
×+ +×=vv D
1
2,
(2.52)
dtr
dtr tr tr
ss
d
sd
d
sd
d
s
ss ssss
t++
()
=
12
2
t
a
e
e
tD. (2.53)
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56 Computational Methods for Complex Liquid–Fluid Interfaces
These two equations are the surface equivalent of the single-mode Giesekus model [60]. The coef-
cients αs and αd are the surface shear and dilatational mobility, and to satisfy the entropy balance,
we must require these coefcients to be nonnegative. The Giesekus model is just one example of
a constitutive model for the surface extra stress tensor we can construct using EIT. We can also
derive surface equivalents of the upper- and lower-convected Maxwell model, the Jeffreys model, or
the Oldroyd-B model [61,62]. However, for modeling bulk rheological behavior, such models have
all shown their limitations. Whenever possible, preference should be given to the development of
structural models, in view of the direct coupling between stress signals and the time evolution of
the interfacial microstructure.
Maxwell-type equations such as those in Equations 2.49 and 2.50 can also be obtained for the
other uxes. For example, for the surface energy ux vector, we nd (again limiting ourselves to
linear expressions)
d
d
ssss
sss
qq
tT
T+=
-Ñ
1
t
l
t
ll
. (2.54)
This is the surface equivalent of the Maxwell–Cattaneo equation for the energy ux vector [63]. The
coefcient τλ is the relaxation time associated with surface energy transfer. Note that in the limit
τλ→0, this expression reduces to Equation 2.36. An attractive property of constitutive equations of
the form of Equation 2.54 is that upon substitution in the jump energy balance, they yield a hyper-
bolic partial differential equation, which predicts a nite speed of propagation of thermal signals
along the interface [17]. This is in contrast to Fourier’s law, Equation 2.36, which yields a parabolic
equation, predicting innite speeds of propagation.
2.4 CONCLUSIONS AND OUTLOOK
In this chapter, we have shown how NET (and in particular CIT and EIT) can be used to derive
descriptions for multiphase systems away from equilibrium. In the Gibbs dividing surface model,
we can assign excess variables to an interface, and we have discussed how the inclusion of these
surface variables in the conservation principles for mass, momentum, and energy leads to a set of
time-evolution equations for these variables. We have also discussed the derivation of thermody-
namically consistent constitutive equations for the uxes appearing in these time-evolution equa-
tions and the derivation of boundary conditions that couple the time-evolution equations to the
balances for the bulk variables. When this set of balance and constitutive equations is combined
with an equation of state for the surface tension and the bulk pressure and expressions for the
concentration and temperature dependence of the bulk and surface material properties, we have a
complete set of coupled equations for the dynamics of a multiphase system.
The balance and constitutive equations we have presented here are suited to describe the behavior
of interfaces with a complex microstructure, with (nonlinear) viscoelastic responses to a perturba-
tion. So far, the nonlinear structural models that have been discussed here, but also the phenom-
enological ones, such as the single-mode Giesekus model, have not been used widely to analyze
experimental surface rheology data or in computational studies of multiphase ows. Linear and
quasilinear models have been incorporated in simulations of multiphase ows [64–73], but more
work is needed to nd efcient solvers that can handle nonlinear structural models. Structural mod-
els have been very successful in modeling the behavior of complex bulk phases, such as particle dis-
persions, polymer blends, liquid crystalline phases, and even food biopolymer gels [61,74]. In view
of these successes, structural models for interfacial behavior are preferred over phenomenological
ones, and more effort should be invested in deriving such models.
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57Nonequilibrium Thermodynamics of Interfaces
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