A phase-field model for systems with coupled large deformation and mass
transport
Wei Hong1,2* and Xiao Wang2
1
Department of Aerospace Engineering, Iowa State University, Ames, IA 50011, USA
2
Department of Materials Science and Engineering, Iowa State University, Ames, IA 50011, USA
Abstract
Phase-field model is a powerful tool for studying the microstructure evolution of materials. This
paper seeks to introduce phase-field modeling to the field of soft materials, especially for studying
polymeric gels. A general framework for the field theory of coupled large deformation and mass
transport is established, and two specific models of diffuse interface are proposed. The ideal liquid-like
interface has a deformation-independent energy and gives rise to a constant surface tension, and a nonideal interface would result in a strain-dependent surface stress. Either model gives a stress field
consistent with the effect of interface line force. The field theory is implemented into a finite-element
code, and several numerical examples are calculated with representative material models in which
deformation is weakly or strongly coupled with mass transport. The numerical models demonstrate the
versatility of the phase-field methodology, and reveal some interesting phenomena due to the coupling.
For example, the composition of a separated phase is significantly affected by the kinematic constraint,
and varies during coarsening.
Keywords: phase-field model, finite strain, interface model
*whong@iastate.edu
1. Introduction
Phase field is a versatile tool for modeling phase transition and morphological and
microstructure evolution in condensed matter, and has undergone continuous development for various
applications (Boettinger et al., 2002; Chen, 2002; Emmerich, 2008). In this paper, we seek to advance
the theory to incorporate concurrent finite deformation and mass transportation. Besides the classic
problem of diffusional solid-state transition in alloys (Cahn, 1961; Onuki, 1989a; Onuki and Furukawa,
2001), a typical problem requiring such considerations is the volumetric phase separation of polymeric
gels (Shabayama and Tanaka, 1993; Matsuo and Tanaka, 1988; Shibayama and Nagai, 1999; Hu et al.,
2001). A swollen gel consists of a crosslinked polymer network and a solvent. Under certain conditions,
a homogeneous gel may separate into shrunk and swollen phases characterized by the drastic
differences in both solvent concentration and the stretch of polymer network (Tanaka, 1979; Hochberg
and Tanaka, 1979). During the phase separation of a gel, the growth of one phase is enabled by
absorbing swelling liquid from its neighbors, as well as spatially replacing other phases (Shabayama and
Tanaka, 1993). Understanding the physics of this model system will provide further insights towards the
role of elasticity in pattern formation in soft materials during more complicate processes, such as
gelation (Bansil et al., 1991; Matsuo et al., 1993; Hong and Chou, 2000), transient gel formation during
1
viscoelastic phase separation (Tanaka 2000), phase dynamics in polymer-stabilized liquid crystals
(Lapena et al., 1999), vesicle dynamics (Du et al., 2004; Biben et al., 2005;), and even the growth of
biological tissues (Lappa, 2004). Unfortunately, most established theoretical analysis are limited to the
small-strain region (Cahn, 1961; Tanaka et al., 1985; Onuki ,1988, 1989a, b; Onuki and Furukawa 2001;
Garcke, 2003) or homogeneous deformation (Hirotsu and Onuki 1989; Cai and Suo 2011), whereas the
inhomogeneous stress plays an dominant role in the morphology (Nishimori and Onuki, 1990; Onuki
1999; Onuki and Furukawa, 2001) and the kinetics of pattern growth (Tanaka, 2000), and the typical
strain involved is several hundred percent, for soft materials in particular. A few efforts have been made
to consider finite deformation and its coupling with swelling, and interesting structures similar to
experimental observation have been revealed (Onuki, 1999; Uchida, 2002; Zhou, 2010). The effect of
introducing both kinematic and physical nonlinearity is yet to be discussed, and the detailed phase
separation processes in these systems are far from being fully understood.
Phase-field models are distinguished by two principle characteristics: a continuous phase field to
differentiate domains of dissimilar microstructures (Laudau & Khalatikow, 1963), and a diffuse interface
across which physical properties smoothly transition from one phase to another (van der Waals, 1894).
Depending on the nature of the problem, the phase field could be an auxiliary variable (Collins and
Levine 1985; Karma et al., 2001), or a physical parameter (Cahn & Hilliard, 1958; Laudau & Khalatikow,
1963; Cahn and Allen, 1977). In either case, the diffuse interface is associated with an excess free
energy, which is often written as a function of the spatial gradient of the phase field. However, in a
large-deformation context, it is unclear whether the gradient should be taken with respect to the
deformed configuration or the reference configuration. It has recently been shown that by using the
gradient in the deformed configuration, the interface energy function could recover the behavior of
surface tension (Levitas and Samani, 2011). As will be shown in the current paper, such an interface
model represents a liquid-like interface, on which the molecules are capable of rearranging themselves.
The corresponding interface energy per unit current area is independent of the state of deformation.
More generally, the interface energy may vary with deformation. For example, for a solid-like interface
that preserves its atomic structure during deformation, the surface tension may be a function of strain,
and is often referred to as the surface stress (Shuttleworth, 1950; Gurtin et al, 1998). Although the
effect of surface tension/stress on deformation or stress field could often be neglected on macroscopic
structures, it is of importance to nanomaterials (Levitas and Samani, 2011) in which the interfacial layer
occupies a significant portion of the total volume, and at a much larger length scale to soft materials
which deform under very low forces. In this work, we aim at formulating a general framework for the
phase-field model of coupled deformation and mass transport, and developing interface models which
describe the behaviors of liquid-like and solid-like interfaces.
Thermodynamic theories of coupled mass transport and elastic deformation date back at least
to Gibbs (1878), who formulated the equilibrium theory of a solid that absorbs liquid. The more general
non-equilibrium theory that describes the transport processes is known as poroelasticity (Biot, 1941;
Rice and Cleary, 1976). Specifically for polymeric gels, existing models include nonlinear poroelasticity
(e.g. Srinivasa 2004; Hong et al., 2008; Duda et al., 2010; Chester and Anand, 2010), and multi-phasic
2
models (e.g. Doi 2009; Rajagopal, 2003). However, with these continuum field theories alone, it is
difficult to deal with problems like phase separation, in which a boundary between domains could
spontaneously emerge, migrate, and vanish. In this paper, we extend the capability of nonlinear
poroelasticity by combining it with a phase-field model. In Section 2, by introducing the gradient energy
terms, the basic non-equilibrium thermodynamic theory is modified to incorporation the interface
contribution. Two interface models and their coupling with elastic deformation is introduced in Section
3. The theory is then specialized with a detailed material model and tested with finite-element
calculations in Section 4. It is shown both analytically and numerically that the liquid-like interface
model gives a thermodynamically consistent contribution to the stress field. The coupling behavior of
elastic deformation and mass-transport-enabled phase separation is illustrated through several
numerical examples. The effect of strong coupling and elastic misfit between separated phases is
demonstrated through a numerical model by assuming the molecular incompressibility. The effect of
strain-dependent interface stress is exemplified through a model with stress-induced interface
anisotropy.
2. Non-equilibrium thermodynamics
The microstructure evolution of a multicomponent material system usually involves mass
transportation and structure reconfiguration. To track the deformation, we introduce the reference by
assuming that at least some material particles do not change relative positions and the aggregate
remains to be a continuum during the evolution process. An example of the background continuum
frame is the polymer network of a permanently crosslinked gel. Imagine attaching to these material
particles a set of markers, with coordinates X in the reference state. We will associate the properties
of a material particle to the local marker, by writing the physical fields as functions of X and time t .
The coordinates of a material particle at time t , for example, is written as xX, t  . The field of
deformation gradient,
FiK X, t  
xi X, t 
,
X K
(1)
measures the deformation of the continuum part of the material.
Just as all phase-field models in a small-deformation context, we differentiate dissimilar phases
using a continuous phase field, and write it also as a function of the reference coordinates and time,
C X, t  . The phase field may be conservative, such as the concentration of a species, or non-
conservative, such the internal variable characterizing the state of damage. While the general
framework is applicable to both cases, in the current paper, we will illustrate it through the conservative
case, in which C represents the number concentration of a mobile species. In the current paper, we
will refer to the mobile species simply as the solvent. To simplify description, we assume the
homogeneous state of a material particle to be fully determined by F and C . The theory could easily
be extended when more physical fields are present. Let the free energy stored in a material cell with
unit volume in the reference state be
3
W F, C, C  .
(2)
The explicit dependence of the free energy on the spatial gradient of phase field C is essential in a
phase-field model. A material point on an interface differs from one in a homogenous bulk in the abrupt
but continuous change of the phase field. The excess energy associated with C is the interface
contribution. Although whether to take the gradient with respect to the current coordinates or the
reference coordinates is a question yet to be discussed, the general form of (2) should always be valid,
given the obvious relation between these two gradients:


.
 FiK
X K
xi
(3)
In this paper, we use  to represent the gradient operator with respect to the reference configuration,
and  x that with respect to the current configuration.
To simplify presentation, we neglect all body forces and body sources. On a boundary, the
potential of traction t is  t  x , while that of a solvent reservoir is  μI , where  is the chemical
potential of the species in the reservoir and I is the total number of molecules entering the material
from the reservoir. The material, the traction, and the reservoir constitute a thermodynamic system. In
equilibrium, the total free energy of the system
x, C   W F, C, C dV   ti xi dA   IdA
(4)
is minimized. Utilizing the variational principle and the divergence theorem, we can write the
minimization problem as


X K
 W

 FiK
 W


xi dV   


 C X K
 W

 C
 ,K

CdV




W
W
   ti 
N K xi dA   N K
CdA   IdA  0
FiK
C, K


,
(5)
for arbitrary test fields x and C , where C, K represents the components of C , and N K those of
the unit normal vector on a surface. In equilibrium, the nominal stress s  W F satisfies
siK
0
X K
(6)
siK N K  ti
(7)
in the volume, and
4


on a surface where traction t is prescribed. Noticing the conservation of species, CdV  IdA , we
can deduce the equilibrium chemical potential of solvent

W


C X K
 W

 C
 ,K

.


(8)
The equilibrium morphology of the material depends on the energy landscape in term of the
phase field C . If W has a single minimum, a homogeneous phase given by   W C is stable, and
the dependence of W on C does not affect the equilibrium state. If W has multiple minima, the
homogeneous material may simultaneously separate into coexisting phases, each corresponding to a
local minimum of W . Inside the diffuse interface, the continuous phase field changes abruptly, where
the local state of material differs from that in the bulk, due to the non-zero contribution from the last
term of Eq. (8).
The theoretical framework can be extended to model a close-to-equilibrium transient process.
As phase separation is enabled by mass transport, the interface propagation is much slower than sound
speed, and the inertia of the material could safely be neglected, with the partial mechanical equilibrium
holds as in Eqs. (6) and (7). On the other hand, if we assume the linear kinetics on solvent migration, the
nominal flux (the number of particles going through a material surface of unit area in the reference state
per unit time) is related to the spatial gradient of chemical potential as
J K   M KL

,
X L
(9)
where M is the mobility tensor. The chemical potential  is still given by Eq. (8) if we insist local
equilibrium at every material point. Further substituting Eq. (9) into the conservation equation of the
solvent, we arrive at the evolution equation of the phase field C
C


t X K

 
 M KL
.
X L 

(10)
Equations (6) and (10) consist a PDE system of the time-varying fields xX, t  and C X, t  , and
can be integrated numerically when initial and boundary conditions are properly prescribed. The nature
of this thermodynamic framework is that the system is always driven towards a state of lower total
energy, and the long-term solution of a closed system would be at (or close to) equilibrium, as given by
Eqs. (6-8). The actual equilibrium morphology and the process of evolution is determined by the
detailed form of the free-energy function W F, C, C  and the mobility tensor M . In the following
sections, we will illustrate the framework through some representative free-energy functions, and the
different dependences on C . On the kinetics of mass transportation in finite-deformation bodies,
simple models are available for materials that are isotropic in the current configuration (Hong et al,.
5
2008) or in the reference configuration (Chester and Anand, 2010). In the current paper, we will assume
the kinetic properties of the material to be isotropic in the current state, and independent of
deformation, so that the true flux measure with respect to the current area
ji  
cD 
,
kT xi
(11)
where c  C det F is the true concentration of the solvent, D the coefficient of diffusion, and kT the
temperature in the unit of energy. Using the geometric relation between corresponding quantities in
the reference and current configurations, Eq. (11) can be written equivalently into the form of Eq. (9),
with the mobility tensor
M KL 
DC
H iK H iL ,
kT
(12)
where H is the inverse tensor of the deformation gradient F . As this paper focuses on the general
formulation of the coupled deformation and phase transition, especially on the effect of phase
boundaries, the difference between various types of kinetic laws and the dependence of mobility on
deformation will not be studied.
In some cases, the migration of solvent and the deformation are coupled, such that the
concentration C and the deformation gradient F are not independent. For example, if molecular
incompressibility is assumed, such as in a swelling polymeric gel, the volumetric deformation det F is
directly related to the concentration C , and consequently the phase transition is coupled with the
heterogeneous deformation. In this strongly coupled case, the general constraint, det F  f C  , could
be enforced numerically by adding to the free-energy function W ,

2
det F  f C 2 ,
(13)
with a large bulk modulus  . The constraint enters the expression of nominal stress as
siK 
W
  det F  f C det FH Ki ,
FiK
(14)
and chemical potential as

W

  det F  f C  f C  
C
X K
 W

 C
 ,K

.


(15)
Alternatively, the incompressibility constraint may also be enforced by employing a Lagrange multiplier
(Hong et al,. 2008).
6
The incorporation of Eqs. (14) and (15) into Eqs. (6) and (10), together with the constraint
det F  f C  , forms a differential-algebraic system for the unknown fields xX, t  and C X, t  .
3. Interface models
Following the usual approach of phase-field modeling, we assume the free-energy density to be
the sum of three contributions:
W F, C, C   We F  Wm C   Wi F, C, C  .
(16)
Here, We is elastic energy, and is assumed to be a function of only the deformation gradient. Wm is the
energy of mixing, which is assumed to be deformation independent. Wi is the contribution from the
coherent interface between dissimilar phases, and it must be an even function of C . Different from
regular phase-field models in the context of small deformation, the interface energy is deformation
dependent through F . Typical interfacial models that give rises to specific forms of Wi will be
discussed in the following subsections. For clarity in description, we will focus on the effect of interface
and leave the coupling between elasticity and mixing (i.e. that between Wm and Wi ) to the discussion
on specific examples in Section 4.
3.1 Ideal interface
In conventional phase-field models, a deformation-independent interface energy,
Wi 
 C C
2 X K X K
,
(17)
is often assumed, where  is a parameter controlling the magnitude of interface energy between two
phases distinguished by different solvent concentrations. Such an approach assumes a phase field
totally independent of deformation. For example, the equilibrium thickness of the interface between
coexisting phases, measured in the reference configuration, is a constant, although the true thickness of
a boundary will change with deformation in the current state. The stress field and deformation, on the
other hand, is never influenced by the presence of an interface. In the small-strain limit, this treatment
recovers the interface model commonly used for hard materials (Cahn 1961; Onuki, 1988, 1989a; Onuki
and Furukawa 2001).
Recently, Levitas and Samani (2002) propose to use the gradient (of a non-conservative phase
field) with respect to the current spatial coordinates, xC , in the interface-energy function. Following
their approach, we assume the interface energy of a compressible material
Wi 
 (C )
2
det F
C C  (C )
C C

det FH Ki
H Li .
xi xi
2
X K X L
7
(18)
Substituting the energy functions in Eqs. (16) and (18) into the equations of state, (14) and (15), we have
the nominal stress
siK 
We
C C

C C
  det FH Mj
H Li H Ki  det FH Mj
H Lj H Ki ,
FiK
X M X L
2
X M X L
(19)
and the chemical potential

dWm 1 
C C
 
C
 det FH Li H Ki

det FH Ki
H Li  
dC 2 C
X K X L
X L 
X K

 .

(20)
The physical significance of the added terms in the stress expression can be made clear when written in
the current configuration. Utilizing the simple geometric relation, σ det F  s  F , we have the true
stress
 ij 
FjK We
C C  C C


 ij .
det F FiK
xi x j 2 xn xn
(21)
Compared to the through-thickness tress (in the direction of C ), the hoop stress along the interface is
higher by  xC . As shown in Appendix, in equilibrium, the total excess energy per unit current
2
volume of a diffuse interface equals   xC . Therefore, the added hoop stress equals the excessive
2
energy per unit volume in the current state. By integrating through the thickness of an interface, the
resultant line force, i.e. the interface tension, equals the interface energy per unit area in the current
configuration.
Through this analysis, it is clear that the interface-energy function, Eq. (18), results in an ideal
interface, namely a liquid-like interface of which neither the actual thickness in the current state nor the
surface tension depends on deformation. The inclusion of this energy function provides a
thermodynamically consistent means of introducing the effect of interface tension to a phase-field
model with coupled mechanical deformation.
3.2 Non-ideal interface
In this section, we will construct an interface-energy function for the mechanical behavior of a
non-ideal interface. The interface or surface of a solid differs from that of a liquid when the solid is
subject to deformation. Due to the permanent structure of a solid interface, the surface stress, namely
the resultant force per unit length on the interface is dependent on the local deformation state (Gurtin
1998; Fischer et al,. 2008). In the limit of small deformation, surface stress is often treated to be linearly
proportional to the in-plane strain of an interface. Although the choice of the energy function for an
ideal interface is quite limited, that for a non-ideal interface is more or less arbitrary – almost any
functional of the concentration gradient and the deformation gradient other than Eq. (18) would give
8
non-ideal behaviors. Inspired by the fact that the presence of surface stress is a consequence of the
different solid structure on the interface between two materials, we take a modified form of the elastic
energy density as the interface contribution:
Wi 
 C C
2 X K X K
We F .
(22)
Here, the parameter  measures the stiffness of the interface relative to the bulk. The corresponding
nominal stress reads
siK 
We
FiK
  C C 
1 
 ,
2

X

X
L
L 

(23)
and the chemical potential

dWm
 
C 

We F
.
dC
X K 
X K 
(24)
Along the interface, where the non-vanishing gradient of phase field C exists, the stiffness of
the material is effectively increased by

2
C . The resultant force per unit length of the interface, i.e.
2
the interface stress, is dependent on the local deformation through We F . In the small-deformation
limit, the interface stress is also linear in the strain tangential to the interface.
4. Numerical examples
In this section, the theoretical framework of coupled deformation and phase field, together with
the two interface models will be illustrated through some representative examples. As the examples are
created for the illustration purpose, we will not take specialized material models. Instead, we use a
compressible neo-Hookean model for the elastic energy of deformation:
We 
G
FiK FiK  3  2 ln det F   det F  12 ,
2
2
(25)
and the regular solution model for the energy of mixing:
Wm 
kT
det F ln   1    ln 1      1    ,

(26)
where G is the initial shear modulus,  the bulk modulus, and   c the volume fraction of the
solvent, with  being the volume occupied by each particle. The first two terms in the bracket of Eq.
(26) is the entropy of mixing, while the third represents the enthalpy of mixing. Consider the simple
case when the elastic deformation and phase field variable are decoupled, the phase stability is
9
determined entirely by the energy of mixing, and the interaction parameter  controls the
thermodynamics stability of system. The system has a critical point at   2 . When   2 , the energy
of mixing has a single well, and a homogeneous phase is stable; when   2 , the energy of mixing has
double wells, and a mixture with initial composition satisfying  2Wm  2  0 separates into two
phases spontaneously.
Besides  , the system has two more dimensionless parameters: the dimensionless stiffness
G kT , and the relative compressibility G  . To demonstrate the coupling effect, we consider a
relatively soft material with dimensionless stiffness G kT  0.1 (note that the dimensionless energy
of mixing Wm kT is on the order of 0.1 for intermediate values of  ). The competition between the
energy of mixing and the interface contribution gives rise to an intrinsic length L0   / kT . In the
case of a phase transition physically decoupled from deformation, the thickness of an equilibrium phase
boundary scales with L0 , which may be a means of estimating the actual value of parameter  . The
following calculations are carried out with dimensionless variables, by normalizing all lengths with L0 ,
all energies with kT , and time with L20 D .
We implement the coupled large deformation and phase-field model, specifically the equation
system (6), (10), (14) and (15), into a finite-element code through the commercial software COMSOL
Multiphysics 4.2a. The concentration C and chemical potential  are interpolated using linear
elements, while the displacement field is interpolated with quadratic Lagrange elements. To ensure
accurate representation of the interface, a uniform mesh of element size no larger than L0 3 is used.
In the fully coupled deformation and phase transition model, the deformation is often unstable,
involving various types of mechanical instabilities, such as the coalescence of two more swollen phases.
In the general formulation described in Section 2, as the volumetric strain is coupled with the field of
concentration by the incompressibility constraint (13), the dilatational deformation of the material is
limited by the mass transportation process given by Eq. (10). The deviatoric part of the deformation,
however, is undamped. A local pure shear deformation, which involves no solvent migration, takes
place instantaneously whenever it is energetically more favorable. Numerically, an instantaneous
deformation may cause convergence issues in the nonlinear solver. To stabilize the numerical
procedures, we slightly damp the system by introduce a viscous dissipation term
to the energy functional (Wang and Hong 2012), where d ij 
1
2
v
i

2

det F dij dij  13 dii 
x j  v j xi  is the strain rate
2

tensor, and  is the artificial viscosity. The characteristic time for viscous deformation is set to be much
shorter than the diffusion time of a single element to avoid numerical artifact.
4.1 Weakly coupled phase transition and deformation
We will first consider the special case, when the phase field variable is decoupled from elastic
deformation, to verify the numerical method and the analysis of surface tension and surface stress in
10
Section 3. Here, the decoupled physics indicates that the state of mixing does not contribute to
deformation, i.e. the material does not swell upon solvent intake, and thus there is not elastic misfit
between phases. However, the stress field is still influenced by the structure of coexisting phases, due
to the presence of the interface tension. We therefore refer to this case as the weakly coupled system.
To enable phase separation, we set   2.5 , so that the energy of mixing, Eq. (26), has double wells at
  0.145 and   0.855 . By assuming the material to be nearly incompressible, i. e. det F  1, the
two energy minima correspond to the concentrations of the coexisting phases, C  0.145 and
C  0.855 . As the elastic deformation is physically decoupled from the phase transition behavior, no
numerical damping is necessary in this case.
We will examine the structure and mechanical contribution of an equilibrium interface through
two numerical examples: 1) a large bi-phase domain divided by a straight interface, and 2) a circular
interface surrounding an island of dissimilar phase in a large domain.
In the first example, a square sample much larger than the interface thickness ( 50 L0  50L0 )
with a phase boundary in the middle is calculated. As initial conditions, the concentration in the left half
domain is C  0.145 while that on the right is C  0.855 , and a smooth transition is prescribed in
between. Mechanically, the sample is subject to uniaxial tension in the vertical direction, and is free to
deform laterally, as sketched in Fig. 1a. The system is then given ample time for relaxation to
equilibrium. The solvent transportation is subject to a periodical condition on the top and bottom
boundaries, and a zero-flux condition on the side boundaries. The additional normal stress due to the
presence of the straight interface (  y minus its value far from the interface) is plotted in Fig. 1b. The
integration of this stress across the interface region would result in the surface tension,
  0.184 L0kT  for the material parameters chosen. The numerical results also show that this
stress equals the excessive free energy Wm  Wi compared to the background homogeneous phases,
just as proven in Appendix. Fig. 1b also shows that the structure of the interface, as well as the
excessive stress due to the interface, is independent of the deformation in the current configuration,
within the error of the numerical calculation. The deformation independent interface behavior is a
direct consequence of the physically decoupled model assumption.
11
Excessive stress   /kT
y
Phase 2
Interface
b
Phase 1
a
0.08
 y (=0)
 y (y =10%)
0.06
 y (y =20%)
W m+W i
0.04
0.02
0
-10
-5
0
5
Deformed coordinates x/L0
10
Fig. 1. (a) Sketch of the bi-phase sample calculated. (b) Equilibrium distribution of the
excessive stress and free energy near an ideal interface in a large, weakly coupled system.
The stress under different strain values are plotted, all in the deformed configuration, as
functions of the dimensionless coordinate x L0 .
Using the same setup, we also test the effect of interface stress by using the non-ideal interface
model described in Section 3. To demonstrate the surface-stress effect, we pick a (maybe non-physically)
high dimensionless parameter   50 . The same interface energy, Eq. (18), as in the ideal case, is
retained to enable phase separation in absence of strain. The same values of all other material
parameters as in the previous case are taken. The resulting distribution of the hoop stress across a
straight interface is plotted in Fig. 2a. To show the effect of the interface, the background stress, i.e.
that in a homogeneous state is deducted. At zero strain, the interface effect is identical to the idealinterface case. When the material is stretched, as the change in interface energy is relatively small, the
weak coupling still results in an almost constant interface thickness in the deformed state, as shown by
Fig. 2a. We then integrate the excessive stress through the interface thickness, and plot in Fig. 2b the
resultant line force as a function of the strain along the interface. As a positive value of  is used, the
interface tension f is a monotonically increasing function of strain. In the small-strain limit, the
relation is linear, with the dimensionless interface stiffness (i.e. the slope of the interface tension-strain
curve) f GL0  0.18 . In the model presented here, the interface stiffness slightly decreases at a
relative large strain. Such an effect is due to the specific form of the interface energy function (22), as
the concentration gradient with respect to the reference configuration would decrease when the
material shrinks laterally. If a specific form of the surface stress-strain dependence is needed, Eq. (22)
may be modified accordingly.
12
b
0.1
0.08
0.06
=0
=0.06
=0.12
=0.2
0.04
0.02
0
0.12
Interface force f  /L kT
Excessive stress   /kT
y
a
0
0.25
0.2
0.15
0.1
0.05
0
-5
0
5
Deformed coordinates x/L0
0
0.05
0.1
Strain 
0.15
0.2
Fig. 2. Behavior of a non-ideal interface. (a) The equilibrium distribution of the excessive
stress near a straight interface in a physically decoupled system. The stress under different
strain values are plotted, all in the deformed configuration. (b) The resultant interface force,
plotted as a function of the strain along the interface.
In the second example, we study the equilibrium interface structure of a circular island of
dissimilar phase. The computational domain is shown by Fig. 3a. Symmetric boundary conditions are
taken on the left and bottom edges, while the top and right edges are left traction free. Due to the
presence of interface tension, the pressure inside the circular island is higher than that outside. For
clarity, in this example, we only use the ideal interface model, Eq. (18). Fig. 3a plots the equilibrium
pressure distribution for an island of radius 10L0 . We then vary the radius of the island r and plot the
pressure drop across the interface as a function of the normalized curvature L0 r in Fig. 3b, and
compared it to the Laplace-Young equation, p   r (with the numerical value of surface tension
  0.184 L0kT  obtained from the previous example). The good agreement with the Laplace-Young
equation numerically verifies the consistency of the interface-energy formulation. The results in Fig. 3b
also shows that this formulation tends to overestimate the interface effect when the size of the domain
is comparable with the interface thickness. Since the distribution of fields through the thickness of an
interface is given by the artificially prescribed interface energy function, the phase-field method is thus
incapable of predicting the behaviors taking place at a length scale comparable to the interface
thickness. To ensure numerical accuracy, domain size (or radius of curvature) needs to be at least one
order of magnitude larger the interface thickness.
13
a
Pressure drop p /kT
b
p
kT
0.02
0.015
0.01
0.005
0
0.02
p=/r
numerical results
0.04
0.06
0.08
Normalized curvature L0/r
0.1
Fig. 3. (a) Distribution of the normalized pressure p kT in a system with a circular island of
radius 10L0 . (b) The pressure drop across the interface, plotted as a function of the
normalized curvature L0 r , and compared with the Laplace-Young equation, p   r .
4.2 Strongly coupled phase transition and deformation
We will now illustrate the model through an example of strongly coupled phase field and elastic
deformation. In this example, the molecular incompressibility of solvent is assumed, so that the
substrate expands upon intake of the solvent. The elastic energy of deformation is slightly modified to
incorporate this coupling:
We 
G
FiK FiK  3  2 ln det F    det F  1  C 2 .
2
2
(27)
In the limit when the mixture is incompressible, the large bulk modulus  enforces the constraint
commonly invoked in modeling polymeric gels (Hong et al,. 2008; Doi 2009),
C  det F  1 .
(28)
In the numerical calculations, we take a relatively large bulk modulus   103 G , so that the constraint
is approximately satisfied.
Due to the strong coupling, the condition for phase separation is different from that given the
energy of mixing only. Using constraint (28), the dilatational part of the elastic energy can be written as
a function of C , which in turn is related to the volume fraction as 1  C  1 1    . The phase
separation behavior is coupled with elastic deformation not only through the coupled free energy, but
also via the kinematic compatibility between dissimilar phases that are highly distorted. To demonstrate
these two levels of coupling, let us first look at the contribution from the coupled free energy. By
neglecting the kinematic compatibility, we imagine the scenario in which both phases deform
homogeneously and freely. In a plane-strain deformation state with the through-thickness stretch 0 ,
14
the isotropic in-plane stretch  
1  C  0 , and the free energy per unit current volume of the
material

Wm  We
G2
 kT ln   1    ln 1      1       20  3 1     G1    ln 1    .(29)
det F
2  0



The dependence on 0 is not very strong. For illustration, we take 0  1.1 and plot Eq. (29) in Fig. 4
against the non-coupled energy of mixing. Even with the mechanical constraints between dissimilar
phases neglected, the phase separation behavior given by Eq. (29) is already very different from that in
the decoupled case. Using the same dimensionless parameter   2.5 , the energy landscape is still
non-convex and phase separation is possible, but the volume fractions for the coexisting states have
shifted to 1  0.168 and 2  0.815 , which correspond to the normalized concentrations of
C  0.201 and C  4.41 , respectively. It is noteworthy that a misfit of approximately 4.21 in
relative volume expansion is present between the two phases. The large misfit would essentially
invalidate the homogeneous assumption.
Free energy W  /kT
0.1
Decoupled
Coupled no constraint
0.05
0
2
-0.05
1
-0.1
0
0.2
0.4
0.6
Volume fraction ( )
0.8
1
Fig. 4. The free-energy profile in a homogeneous system with coupled deformation and phase field
variable compared to that of a decoupled system.
15
t=2×103
t=500
t=1×104
C
t=5×104
t=2×105
t=2×103
t=1×104
t=5×104
t=2×105
t=2×103
t=1×104
t=1×106
Low concentration
0  0.3
t=500
Mid concentration
t=1×106
0  0.43
t=500
High concentration
t=5×104
t=2×105
t=1×106
0  0.55
Fig. 5. Evolution processes in a system of strongly coupled phase transition and deformation.
Snapshots at various times during the phase-separation processes are presented, with the
dimensionless times indicated on each plot. The deformation is shown by distorted mesh (drawn to
scale), and the normalized solvent concentration is shown by the color scale.
16
Now let us include the effect of kinematic constraints between coexisting phases. Using the
same set of material parameters, we numerically solve the phase-field equations in a square unit cell
measured 300L0 in width. On all four boundaries, periodical boundary conditions are prescribed for
both the displacement of the matrix material, and the concentration C and chemical potential  of
the solvent. The initial condition is taken to be a close-to-uniform distribution of solvent, with large
enough perturbation to enable the nucleation of dissimilar phases. The simulation is started from three
initial concentrations, with corresponding volume fractions of solvent  0 being 0.3, 0.43, and 0.55.
Snapshots of the evolution processes are shown in Fig. 5.
Just as in regular spinodal decomposition, the amplitude of initial fluctuation grows with time,
followed by the coarsening of phases to reduce interface energy. Depending on the initial concentration,
the coexisting phases exhibit the morphologies of solvent-rich islands (at a relatively low concentration),
labyrinth pattern (at an intermediate concentration), and solvent-poor islands (at a relatively high
concentration). Due to the large volumetric mismatch, the separated phases are distorted, i.e. the
deformation is highly non-uniform, especially near the phase boundaries. The large distortion, together
with the interface tension, gives rise to a non-uniform stress field, even in a close-to-equilibrium state.
We plot the distribution of hydrostatic pressure in different phases in Fig. 6. The states plotted here
correspond to the final states after long-time evolution shown in Fig. 5. The representative
dimensionless pressure p kT  0.5 has a non-negligible contribution to the chemical potential of
solvent. Due to the large volume expansion, the solvent-rich phases always bare higher pressure. The
numerical results demonstrate that the effect of mechanical mismatch dominate over that of interfacial
tension, especially for coarser phases. Due to the different dependencies of the elastic strain energy
and the interface energy on the size of an island of dissimilar phase (the former scales with the area
while the latter scales with the circumferential length), for large enough islands, the elastic contribution
always wins.
p
kT
Fig. 6. Distribution of hydrostatic pressure at close-to-equilibrium states of
phase separation with coupled elastic deformation. The three images
correspond to the last stages of the three evolution processes in Fig. 5. The
solvent-rich phase is always under higher compression.
In contrast to the behavior of a liquid solution, the presence of the kinematic-constraint-induced
stress causes the composition of each phase to be dependent on its size during the coarsening process.
17
Even at equilibrium, the solvent concentrations in the coexisting phases differ significantly from the
prediction of the homogenous model by neglecting the constraints. As shown by Fig. 5, the difference is
more significant in the solvent-rich phases, where the actual concentration C  2.8 is much lower
than that given by Fig. 4, C  4.41. The size-dependent composition change in the coarsening stage
may lead to non-self-similar growth, a phenomenon previously solely explained by the dynamical
asymmetry of individual phases (Tanaka 2000). The material-specific details of the difference between
these two processes are not well studied, but the discussion is beyond the scope of the current paper.
To illustrate the effect of an non-ideal interface, we add to the numerical model the
contribution from Eq. (22) and take the dimensionless parameter   10 . The initial boundary
conditions are taken to be the same as that in the low-concentration case of the ideal-interface
simulations (Fig. 7). All parameters are taken to be the same, except that an initial strain  0  10% is
added to the unit cell of computation in the horizontal direction, as sketched in Fig. 7. Snapshots at
different stages during the evolution process are also shown in Fig. 7. Due to the presence of straindependent interface stress and the applied strain, the microstructure evolution is no longer isotropic.
As interface energy increases with strain, forming a horizontal phase boundary is energetically more
expensive than forming a vertical one. As a result, the separated solve-rich phase takes the shape of an
ellipse, with the longer axis perpendicular to the direction of applied strain. In other words, the phase
boundaries exhibit higher curvature (larger tension) on horizontal segments than vertical segments.
Although the results are not shown here, similar phenomena could also been observed on simulations
under other solvent concentrations.
ε0
ε0
t=2×103
t=500
t=1×104
C
t=5×104
t=1×106
t=2×105
Fig. 7. Evolution process in a system of strongly coupled phase transition and elastic
deformation. The unit cell is under horizontal tension of 10% strain. A non-ideal
interface model is adopted so that the interface stress increases with strain. The
deformation is shown by distorted mesh, and the normalized concentration is shown by
the color scale.
18
C
t = 3×106
t = 4×106
t = 5×106
0  0.87
t = 1.2×107
t = 8×106
t = 1.5×107
Fig. 8. Snapshots during the phase-separation processes of a PNIPAM gel, with the dimensionless
times indicated on each plot. The deformation is shown by distorted mesh (drawn to scale), and
the normalized solvent concentration is shown by the color scale.
By adopting material-specific free-energy functions, our theory can be readily applied to the
phase separation of polymeric gels. As the last example, we consider the thermo-sensitive Poly(Nisopropylacrylamide) (PNIPAM) hydrogel by replacing the energy of mixing in Eq. (26) with the FloryHuggins-Staverman model:
Wm 
kT
det F1    ln 1      1    .

(30)
The enthalpy of mixing depends on temperature and concentration as    0  1 (1   ) ,
 0  12.95  0.04496T , 1  17.92  0.05690T (Afroze et al, 2000). Here for simplicity,
temperature T is assumed to be uniform. PNIPAM gel is known to be swollen at lower temperature
and collapse with a phase transformation at T  307 K , depending on network stiffness. In the
numerical example, we test a initially homogenous gel with swelling ratio det F  8 ( 0  0.87 ), and
overheat it to an unstable temperature T  314K . The interface between swollen and shrunken
phases is assumed to be liquid like. We assume the gel to be large enough, and only look at a
representative block in the middle by neglecting the solvent loss from free surfaces. As shown by Fig. 8,
a sponge structure gradually forms with highly swollen domains embedded in a network of shrunken
domain, similar to experimental observations (e.g. Hu et al. 2001). Details of this process, however, fall
out of the focus of the current paper. Nevertheless, comparison between the simulated morphology
and that observed may provide another means of estimating the magnitude of some physical
19
parameters. For example, by comparing the results shown in Fig. 8 to the experimental observation (Hu
et al. 2001), we estimate the characteristic length L0 ~ 2nm and an interface energy of ~ 0.08 J m 2 .
5. Conclusions
In this paper, the general framework for a phase-field model with concurrent finite deformation
and mass transportation is constituted. On top of this framework, several diffuse interface models have
been proposed and their properties studied theoretically and numerically. It has been shown that under
the finite-deformation context, the interface models give rise to a stress field consistent with the line
force exerted by the interface, i.e. they recover the classic Laplace-Young equation. Specifically, an ideal
interface, which has liquid-like properties, would induce a deformation-independent surface tension. A
solid-like non-ideal interface, on the other hand, can give rise to the more general strain-dependent
surface stress.
The coupled phase-field model is further implemented in to a finite-element code to enable
numerical simulations for both weakly coupled and strongly coupled systems. In the weakly coupled
system, the stress and elastic deformation is influenced by phase transition through the interface effect,
but both mass transport and phase transition are deformation-dependent. In the strongly coupled
system, on the other hand, the phase transition and elastic deformation are coupled through molecular
incompressibility. Due to the kinematic constraint between dissimilar phases with drastically different
strains and the resultant inhomogeneous stress fields, the compositions in the coexisting phases differ
from that predicted by a homogeneous model. In contrast to the commonly accepted view of spinodal
decomposition, the composition varies with time as the island size of each phase coarsens. The effect of
strain-dependent surface stress on phase transition is demonstrated through a numerical example, in
which anisotropy is introduced by a uniaxial stretch on the originally isotropic material. As a practical
example, we apply the framework to study of phase-separation process of a thermo-sensitive gel and
the calculation reveals the formation of a sponge-like structure.
By introducing finite-deformation phase-field modeling to the mechanics of soft materials, it is
foreseeable that the theoretical framework and numerical procedures can be used to study various
intriguing phenomena, such as gelation, microstructure evolution, damage initiation and propagation.
Appendix
The purpose of this appendix is to show that the excess energy per unit volume on an ideal
diffuse interface, as defined by Eq. (18) in Section 3, is  det F xC . Consider the spatial gradient of
2
the energy of mixing Wm in an equilibrium state:
Wm 
dWm
C .
dC
Assume the size of an equilibrium phase to be much larger than the characteristic thickness of the
interface, so that the phase boundary can be regarded as close to straight and infinitely long when
20
(A.1)
studying the through-thickness change of the fields. Let us now construct a local coordinate system with
one axis,  , along the normal direction of the interface in the reference configuration (i.e. the direction
of C ). Equation (A.1) can be rewritten in the local coordinates as
We  Wm  We  dWm C
.



 
dC 
(A.2)
If we further neglect the shear deformation along a phase boundary, the normal of the surface would
coincide with a principal direction of the deformation gradient. Let    be the stretch along the  axis. Utilizing Eq. (20), we further write the gradient of Wm as
2
Wm
1  det F  C 
  det F C  C

  


.

2 

2 C    
  2   
(A.3)
Here, without losing generality, we have taken the equilibrium state to be the reference for the
chemical potential,   0 .
2
 We  Wm  We    det F  C   



 


    2 2     


 det F  C 2  

 
.

2 2     


(A.4)
The normal stress across the interface is continuous
2
We   det F  C  
s


 .

   2 2    


(A.5)
Integrating Eq. (A4) along  yields
2
 det F  C 
We  Wm  s  W  W  s  

 ,
2 2   
0
e
0
m
0
(A.6)
where the quantities with superscript 0 represent those in a homogeneous bulk phase. Adding the
contributions from We and Wm in Eq. (A.6) to Eq. (18), we obtain the total excess energy per unit
reference volume, due to the presence of an interface,


We  Wm  s  We0  Wm0  s0  Wi   det F xC .
2
Equivalently, the excess energy per unit current volume is  det F xC .
2
21
(A.7)
This derivation is also applicable to the case when molecular incompressibility is assumed, in
which the concentration and deformation gradient are related as det F  f C  . Using the equilibrium
condition
2
dWm 1  det F  C 
  det F C 

  pf C   

  0,

2 
dC
2 C    
  2  
(A.8)
it could be shown that the gradient of the modified free energy
We  Wm  p f  det F 



 det F  C 2 

,

   s

2 

 2     
(A.9)
where the nominal stress normal to the interface now takes the form
We
det F 
s
p




 det F  C 2 

  .

2 
 2     
(A.10)
Integrating Eq. (A9) over  from a homogeneous state, we will arrive at the same equality as in Eq. (A.6).
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