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International Journal of Applied Mechanics
Vol. 5, No. 1 (2013) 1350001 (18 pages)
c Imperial College Press
DOI: 10.1142/S1758825113500014
INHOMOGENEOUS LARGE DEFORMATION KINETICS
OF POLYMERIC GELS
WILLIAM TOH∗ , ZISHUN LIU†,§ , TENG YONG NG∗
and WEI HONG‡
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∗School
of Mechanical and Aerospace Engineering
Nanyang Technological University
50 Nanyang Avenue, Singapore 639798, Singapore
†International Center for Applied Mechanics
State Key Laboratory for Strength and Vibration of
Mechanical Structures, Xi’an Jiaotong University
No. 28, West Xianning Road, Xi’an Shaanxi 710049, P. R. China
‡Department
of Aerospace Engineering, Iowa State University
Ames IA 50014, USA
§zishunliu@mail.xjtu.edu.cn
Received 6 November 2012
Accepted 4 February 2013
Published 27 March 2013
This work examines the dynamics of nonlinear large deformation of polymeric gels, and
the kinetics of gel deformation is carried out through the coupling of existing hyperelastic
theory for gels with kinetic laws for diffusion of small molecules. As finite element (FE)
models for the transient swelling process is not available in commercial FE software,
we develop a customized FE model/methodology which can be used to simulate the
transient swelling process of hydrogels. The method is based on the similarity between
diffusion and heat transfer laws by determining the equivalent thermal properties for gel
kinetics. Several numerical examples are investigated to explore the capabilities of the
present FE model, namely: a cube to study free swelling; one-dimensional constrained
swelling; a rectangular block fixed to a rigid substrate to study swelling under external
constraints; and a thin annulus fixed at the inner core to study buckling phenomena. The
simulation results for the constrained block and one-dimensional constrained swelling
are compared with available experimental data, and these comparisons show a good
degree of similarity. In addition to this work providing a valuable tool to researchers for
the study of gel kinetic deformation in the various applications of soft matter, we also
hope to inspire works to adopt this simplified approach, in particular to kinetic studies
of diffusion-driven mechanisms.
Keywords: Finite element; non-linear large deformation; polymeric gel; swelling kinetics.
1. Introduction
When a cross-linked polymer network imbibes a large amount of solvent, the entropy
associated with mixing results in a swollen state which is more commonly known
§ Corresponding
author.
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W. Toh et al.
as a gel. The cross-linking of polymers can allow for reversible deformation. This
unique property of polymeric gels provides tremendous potential for applications in
diverse areas, such as biomimetic devices, drug delivery, flow control and sensors
[Bashir et al., 2002; Deligkaris et al., 2010; Elvira et al., 2004; Huglin, 1989; Jeong
et al., 2005; Kishida and Ikada, 2001; Li and Kong, 2007; Richter et al., 2008]. The
mechanics of the gel deformation is dependent on the bi-phasic actions from the elastic deformation of strong chemical cross-links and the viscous deformation due to the
diffusion of solvents across the material boundary. Early studies on the mechanics
of gel swelling [Li and Tanaka, 1990; Peters and Candau, 1988] involved decomposing the material into two individual phases for independent analyses, which limits
the analysis of such materials to simple geometries, such as spheres [Tanaka and
Fillmore, 1979], long cylinders and large disks [Li and Tanaka, 1990; Peters and
Candau, 1988].
In the past, the theory of such multi-phasic gel materials was not favorable
for mainstream numerical methods such as the finite element method (FEM),
although early works exist for multi-phasic models. As such, FE analysis of the
large deformation in these gel materials was limited to rubber hyperelasticity, such
as the Arruda-Boyce, Ogden and Mooney-Rivlin models [Marckmann and Verron, 2006]. The introduction of continuum theories treating the different phases
of deformation as single phase [Chester and Anand, 2010; Duda et al., 2010; Hong
et al., 2008] improved the feasibility of FEM implementation for polymeric gels
materials. Following the philosophy of this approach, Hong et al. [2009] developed a hyperelastic model for the large deformation of polymeric gels via the
UHYPER subroutine in the ABAQUS solver. Subsequently, the coupled theory was extended to environmentally sensitive hydrogels, such as pH-sensitive
gels [Marcombe et al., 2010] and temperature-sensitive gels [Cai and Suo, 2011;
Chester and Anand, 2011]. While material models for these models exist, they
are limited to the equilibrium state whereby the chemical potential is homogeneous throughout the gel. Despite the formulation of a user-defined element
for the simulation of gel kinetics [Zhang et al., 2009], it is to the authors’ best
knowledge that the transient swelling kinetics was not implemented in a robust
manner.
In this work, we attempt to utilize FEM to simulate the swelling kinetics of
polymeric gels by drawing an analogy between the diffusion of solvent molecules
and conduction heat transfer within solids. Although similar in the idea of using
of the diffusion-heat transfer analogy with another recent work [Duan et al., 2012],
we present a method which directly defines thermal properties, as opposed to the
definition of the governing equation and its derivatives. This analogy greatly simplifies the procedure for developing full material models by making use of the coupled
temperature-displacement elements which can be found in the ABAQUS element
library.
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Inhomogeneous Large Deformation Kinetics of Polymeric Gels
2. Theory and Formulation
2.1. Kinematics of deformation
In the presence of external forces and/or solvent with a different chemical potential, a polymer network undergoes deformation. We define the initial state as the
reference state and the deformed state as the current state, as shown in Fig. 1.
From the theory of continuum mechanics, the state of deformation is characterized by the deformation gradient, which is defined as:
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FiK (X, t) =
∂xi (X, t)
.
∂XK
(1)
The deformation gradient FiK describes the state of deformation by finding the
partial derivative of the current state xi with respect to the reference state XK at
any time t.
2.2. Thermodynamics of deformation
Adopting the outline of thermodynamics of deformation of gel by Hong et al. [2009],
we will derive the framework for the hyperelastic theory of large gel deformation in
equilibrium under a mechanical load and in contact with a solvent.
Under deformation, there is work done due to external forces and solvent action.
In the equilibrium state, the total work done must be equal to the change in free
energy present in the gel. The amount of work done on the unit element deformed
state with a deformation of δxi (X), defining the concentration of solvent within the
elemental volume as C(X), chemical potential as µ, body force as Bi (X) and traction
as Ti (X), equals to the sum of µδCδV , Bi δxi δV and Ti δxi δA. In this equilibrium
state, the free energy in the unit element is defined by δW (F, C). Integrating over
Fig. 1. (Color online) A dry polymeric gel in the reference state is placed in contact with a solvent
of constant chemical potential and deforms into the current state.
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the entire volume of gel, we arrive at the integral equation
δWdV = Bi δxi dV + Ti δxi dA + µ δCdV .
(2)
In addition, the dependence of free energy on F and C means that a small change
in W is defined as:
δW =
∂W (FiK , C)
∂W (FiK , C)
δC.
δFiK +
∂FiK
∂C
(3)
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From the principle of virtual work for elastic bodies [Kelly, 2008], where change
in virtual potential energy equals external virtual work done,
δU = δWext ,
(4)
defining siK as nominal stress, which is related to the true stress σij by σij =
siK FjK /det F, the virtual potential energy change and external virtual works can
be written as:
∂
(δxi )dV = siK δFiK dV ,
(5)
δU = −δWint = − − siK
∂XK
δWext = Ti δxi dA + Bi δxi dV .
(6)
Substituting Eqs. (5) and (6) into Eq. (4), we obtain
siK δFiK dV = Ti δxi dA + Bi δxi dV .
(7)
Substituting Eqs. (3) and (7) into Eq. (2) and re-arranging, we obtain the conservative form as:
∂W (FiK , C)
∂W (FiK , C)
− µ δCdV = 0.
(8)
− siK δFiK dV +
∂FiK
∂C
Since integration is carried out for arbitrary changes δFiK and δC, the bracketed
terms in the integrands must be equal to zero, therefore the nominal stress and
chemical potential are defined as:
siK =
∂W (FiK , C)
,
∂FiK
(9)
µ=
∂W (FiK , C)
.
∂C
(10)
2.3. Flory-free energy function
The swelling of a polymeric gel depends on two factors: the deformation of the
polymer network and the mixing of solvent molecules into the network. Flory and
Rehner [1943] assumed the free energy function W (F, C) of a polymeric gel to be an
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Inhomogeneous Large Deformation Kinetics of Polymeric Gels
additive decomposition of the free energy function of stretching Ws (F) and mixing
Wm (C)
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W (F, C) = Ws (F) + Wm (C).
(11)
Other works by Flory [1942, 1953] and Huggins [1941] developed explicit forms of
the free energy functions. These explicit forms were adopted by Hong et al., Cai and
Suo, and Kang and Huang in the development of the coupled model. However, the
expressions for the mixing free energy adopted by Hong et al. [2008, 2009] exhibited
a slight difference with the expression adopted by Cai and Suo [2011], and Kang
and Huang [2010]. It was also noted that there is no significant effect on swelling
deformation [Kang and Huang, 2010]. In this paper, the expression adopted by Cai
and Suo, and Kang and Huang is employed. Writing in principal stretches, the free
energy functions are defined as:
1
NkT (λ21 + λ22 + λ23 − 3 − 2 ln(λ1 λ2 λ3 )),
2
kT
1 + νC
χνC
Wm (C) = −
νC ln
+
,
ν
νC
1 + νC
Ws (F) =
(12)
where λ1 , λ2 , λ3 denote the stretch in the principal directions; N is the number of
polymer chains per unit dry polymer volume; kT the temperature in the unit of
energy; ν the volume of a solvent molecule; C the concentration of solvent and χ a
dimensionless measure of mixing enthalpy.
2.4. Chemical potential
The explicit expression for chemical potential is required for the finite element
implementation, and this section provides a derivation of the expression for µ.
One essential assumption is that of molecular incompressibility, which requires
that both polymer network and solvent molecules do not undergo any volumetric
change during the swelling process. This assumption is expressed as:
1 + νC = det(F) = λ1 λ2 λ3 .
(13)
Equations (9) and (10) were derived without the incompressibility constraint. By
imposing Eq. (13) as a constraint through a Lagrangian multiplier Π(X, t) in the
free energy function, Eqs. (9) and (10) are transformed into
siK =
µ=
∂W (FiK , C)
− ΠHiK det F,
∂FiK
(14)
∂W (FiK , C)
+ Πν,
∂C
(15)
where HiK is the transpose of the inverse of deformation gradient.
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Using the Flory–Rehner free energy function, the principal nominal stresses
s1 , s2 , s3 and chemical potential µ are specialized to
s1 = NkT (λ1 − λ−1
1 ) − Πλ2 λ3
s2 = NkT (λ2 − λ−1
2 ) − Πλ1 λ3
(16)
λ−1
3 )
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s3 = NkT (λ3 −
− Πλ1 λ2 ,
νC
χ
1
+
µ = kT ln
+ Πν.
+
1 + νC
1 + νC
(1 + νC)2
Equations (14)–(17) are derived in Hong et al. [2008].
Combining Eqs. (16) and (17) to eliminate Π, we arrive at
νC
χ
1
+
µ̄ = ln
+
1 + νC
1 + νC
(1 + νC)2
¯
I1 (1 + νC)2/3 − 3
1
(σ̄1 + σ̄2 + σ̄3 )
,
+ Nν
−
3
1 + νC
3
(17)
(18)
where σi , i = 1, 2, 3 are the principal true stresses, J = λ1 λ2 λ3 the invariant of
the deformation gradient, I¯1 = (λ21 + λ22 + λ23 )(J −2/3 ) the deviatoric invariant. The
overbars represent non-dimensionalized quantities, µ̄ = µ/kT , σ̄i = σi /(ν/kT ).
2.5. Equivalence to a hyperelastic solid
In this section, we show that the boundary value problem of a deforming polymeric
gel can be reduced into the form of a hyperelastic solid, which may be implemented
into ABAQUS via the use of the user-subroutine UHYPER.
Introducing another free energy function through a Legendre transformation,
(F, µ) = W (F, C) − µC,
W
we convert Eqs. (2), (9) and (10) into the following respective equations
δ WdV
= Bi δxi dV + Ti δxi dA,
siK =
(FiK , µ)
∂W
,
∂FiK
C=−
(FiK , µ)
∂W
.
∂µ
(19)
(20)
(21)
(22)
The equilibrium integral equation (20) now takes the form of a hyperelastic solid.
Combining Eqs. (11)–(13) and (19), we arrive at the free energy function
(F, µ) = 1 NkT (I1 − 3 − 2 ln(J))
W
2
J
kT
χ(J − 1)
µ
−
(J − 1) ln
+
− (J − 1).
ν
J −1
J
ν
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(23)
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Inhomogeneous Large Deformation Kinetics of Polymeric Gels
Equation (23) defines the free energy of the polymeric gel as a hyperelastic
material which defines the state of deformation through the independent variables
I1 , J and µ.
2.6. Transient swelling kinetics
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The swelling kinetics of a polymeric gel is due to the migration of solvent molecules
within and across the material boundary. The chemical potential form of Fick’s
first law is used to represent the chemical potential driven flow of solvent molecules
[Feynman et al., 1963; Hong et al., 2008; Zhang et al., 2009]
ji = −
cD ∂µ
∂ µ̄
= −cD
,
kT ∂xi
∂xi
(24)
where the coefficients, c = C/J represents current solvent concentration and D
represents diffusion coefficient.
Equation (24) describes the state of steady state diffusion. For transient diffusion, the governing equation for diffusion is obtained by the assumption of conservation of solvent molecules, i.e., no chemical reaction takes place. In the reference
state, the governing equation is written as:
∂C(X, t) ∂JK (X, t)
= 0.
+
∂t
∂XK
(25)
Converting it into the current state and integrating throughout, we obtain the weak
form as:
1 ∂C
dV +
jk dS = 0.
(26)
V det(F) ∂t
S
Applying the chain rule on Eq. (18) through the assumption of C = f (µ, I¯1 , σ̄),
we write the material time rate of C as:
∂C ∂ µ̄
∂C ∂ I¯1
∂C ∂ σ̄
∂C
=
+ ¯
+
,
∂t
∂ µ̄ ∂t
∂ σ̄ ∂t
∂ I1 ∂t
(27)
where σ̄ = σ̄1 + σ̄2 + σ̄3 , and the partial derivatives of C are defined as:
∂C
1
9J 3 (J − 1)
=
,
∂ µ̄
ν 9J − 18χ(J − 1) − N ν(9 − I¯1 J 2/3 )(J 2 − J)
9J 3 (J − 1)
Nν 1
∂C
=
−
,
3J ν 9J − 18χ(J − 1) − N ν(9 − I¯1 J 2/3 )(J 2 − J)
∂ I¯1
1 1
∂C
9J 3 (J − 1)
=
.
∂ σ̄
3 ν 9J − 18χ(J − 1) − N ν(9 − I¯1 J 2/3 )(J 2 − J)
(28)
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Combining Eqs. (26) and (27), the governing equation can be expressed as:
1 ∂C ∂ µ̄
1
∂C ∂ I¯1
∂C ∂ σ̄
dV +
+
jk dS = −
dV .
∂ σ̄ ∂t
∂ I¯1 ∂t
V det(F) ∂ µ̄ ∂t
S
V det(F)
(29)
2.7. Heat conduction
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In heat transfer within solids, the temperature gradient forms a driving potential
which transfers energy around the body through heat conduction. Fourier’s law of
heat conduction states that the heat flux is proportional to the temperature gradient
[Bergman et al., 2011; Kakac and Yener, 1985], and can be expressed as:
∂T
.
(30)
∂xi
The transient heat conduction process is described by the heat conduction equation.
In the formulation for the heat transfer elements, the integral form of the heat
conduction equation used takes the following form [Simulia, 2010]
dU
dV +
ρ
qdS =
rdV ,
(31)
dt
V
S
V
qi = −k
where ρ is the density, dU/dt the material time rate of internal thermal energy, q
the heat flux and r the internal heat generation.
By representing chemical potential using temperature, we can see a similarity
between the governing equations of diffusion and heat transfer. This implies that the
transient swelling kinetics takes the same form as transient heat conduction and that
thermal conduction elements can be used to describe the transient swelling kinetics
of polymeric gels.
3. Finite Element Implementation
We assume local equilibrium at all material points within the volume, and subsequently describe the equilibrium condition by applying the hyperelastic gel theory
developed by Hong et al. [2009]. The use of this hyperelastic gel theory allows for the
study of large deformations in polymeric gels subjected to inhomogeneous swelling
caused by external mechanical loads.
For the kinetics of solvent migration, we make use of the similarity between heat
conduction and mass diffusion. Therefore, instead of reformulating an entirely new
user-defined element for this purpose, we are able to make use of the inbuilt coupled
temperature-displacement analysis in ABAQUS. This analogy greatly reduces the
amount of time taken for the finite element formulation of gel swelling kinetics. We
rewrite Eqs. (24) and (29)–(31) in Table 1 for the reader’s convenience.
For a fully coupled temperature-displacement analysis, the thermal properties
required are density, conductivity and specific heat. Using the subroutine USDFLD,
we can directly define these properties as functions of the stretch invariants, as
shown in Eq. (32). In addition, a change in I means a change in volume of the
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Table 1.
Governing equations of gel swelling kinetics and heat transfer.
Flux
Gel diffusion
Z
∂ µ̄
ji = −cD
∂xi
V
Governing equation
Z
jk dS
1
∂C ∂ µ̄
dV +
det(F) ∂ µ̄ ∂t
Z
=−
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Heat transfer
qi = −k
Z
∂T
∂xi
V
ρ
V
„
1
det(F)
dU dT
dV +
dT dt
S
∂C ∂I
∂C ∂ σ̄
+
∂I ∂t
∂ σ̄ ∂t
dV
Z
Z
S
«
qdS =
V
rdV
network, thus results in a change in the number of solvent molecules being present in
the network. Similarly, a change in the stresses present would facilitate the migration
of solvent molecules into the network. Therefore, the effects of I and σ̄ can be likened
to a source of solvent molecules, analogous to a heat source present in internal heat
generation in heat transfer. This internal heat generation is introduced through
subroutine HETVAL, through the definition of Eq. (33). Thereafter, the use of
thermally-coupled elements will account for mass diffusion within the material.
Comparing the equations presented in Table 1, we set
ρ ≡
1
1
= ,
det(F)
J
cp ≡ ν
∂C
,
∂ µ̄
T ≡ µ,
k ≡ νcD =
r ≡ −ν
1
det(F)
(32)
∂C ∂ I¯1
∂C ∂ σ̄
+
¯
∂ σ̄ ∂t
∂ I1 ∂t
J −1
J
≈ −ν
D,
1
det(F)
∂C ∆I¯1
∂C ∆σ̄
+
,
∂ σ̄ ∆t
∂ I¯1 ∆t
(33)
where the partial derivatives of C are found in Eq. (28). Note that cp , k and r are
scaled by a factor of ν as all terms in Eqs. (32) and (33) have a 1/ν term. In the
quantification of the term in Eq. (33), we approximate the time derivatives of I¯1
and σ̄ the terms through finite differencing of each term. ∆I¯1 and ∆σ̄ are obtained
from the difference of results between the current and previous increment, and ∆t
is the time increment in the current increment.
Although stemming from the same heat transfer analogy, the finite element
implementation differs greatly with that of Duan et al. [2012]. The governing
Eq. (26) was directly implemented by Duan et al., whereas we transform Eq. (26)
into Eq. (29) before implementation. This transformation results in two additional
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terms, which arise due to the incompressibility constraint, therefore the effects of
incompressibility on swelling kinetics can also be investigated.
It should be noted that the main function of this heat transfer analogy is simply a numerical means for obtaining the chemical potential distribution within the
gel. This chemical potential distribution, together with the hyperelastic gel theory,
would then determine the stress, displacement and solvent concentration within
the gel.
The implementation of chemical potential boundary condition is instantaneous.
However, an instantaneous increase of chemical potential would lead to computational errors due to excessive distortion in the surface elements. Therefore, instead of
imposing the boundary condition as an instantaneous increase, the chemical potential is imposed as a linear ramp over the duration of an insignificant percentage of
the characteristic time of diffusion, L2 /D. Through this manner of boundary condition imposition, the increase in chemical potential would be virtually-instantaneous,
yet does not result in any computational errors.
To show the dependence of swelling kinetics on the diffusion coefficient, we
normalize time with ζ (= L2 /D) and lengths with L, where ζ is also known as
the Fourier number. Also, D is the diffusion coefficient, and L the characteristic
length.
4. Numerical Examples
In the examples that ensue, the material properties of D = 8 × 10−10 m2 /s, χ = 0.1
and N ν = 0.001 are used. These values are representative of most polymeric gels
[Hong et al., 2008] but do not correspond to any specific polymer. To generalize the
trends which are exhibited, non-dimensionalization is carried out. For a length scale
of L, time is represented by t̂ = t/ζ, and lengths are represented by x/L.
4.1. Free swelling of a cube
It is easy but erroneous to comprehend and visualize that a cube under swelling will
swell homogeneously in all directions during the transition from initial to equilibrium states, i.e., the cube will remain cubic during the swelling process. However,
this is not the case as the corners of the cube possess higher solvent concentration as
compared to the faces of the cube. This would translate into the scenario whereby
swelling is more prominent at the corners of the cube as compared to the faces of
the cube during the transition process. As time progresses, the cube will eventually
reach a state of homogeneous chemical potential, thus giving rise to a perfect cube
at the end of the swelling process.
This example describes a cube initially in equilibrium at µ̄ = −1, but immersed
in a solvent of µ̄ = 0 at t = 0. The transient process of the cube swelling at various
normalized times is shown in Fig. 2. Figure 3 shows the variation of λ1 at three
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Inhomogeneous Large Deformation Kinetics of Polymeric Gels
Fig. 2. (Color online) Displacement contour plot for the non-dimensionalized free swelling cubic
gel at various times. The plots follow the same scale to show the large deformation present.
Fig. 3. (Color online) Time variation of swelling ratio. Discrete points are for a cube with L =
0.25 m, D = 8 × 10−10 m2 /s and L = 8 m, D = 5 × 10−8 m2 /s. Lines are for the normalized
dimensions of L = 1 m, D = 1m2 /s. Inset shows the three selected points for the non-dimensional
study. These points possess the same equilibrium stretch ratio in the x-direction but different
swelling rates.
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points (illustrated in inset) on the y − z surface of two geometrically similar cubes
with different characteristic lengths and diffusion coefficients.
It is evident that while the initial rates of swelling of the points are different, the
eventual uniform distribution of chemical potential within the cube would lead to
uniform swelling at all points. The long term equilibrium swelling ratio computed is
also coherent with analytical solutions for equilibrium swelling conditions presented
by Hong et al. [2009], thus verifying that the end state of this transient analysis
approaches theoretical equilibrium conditions.
In addition, from the overlapping of the two sets of data, we can see the dependence on diffusion coefficient and the square of the characteristic length. This observation shows the usefulness of non-dimensionalization when modeling at a very small
scale, as it allows the user to work at a larger length scale. It also facilitates the
comparison of specimens of different dimensions as will be seen in the succeeding
section.
4.2. One-dimensional constrained swelling
Here, the accuracy of the present kinetic model is evaluated through the case of a
one-dimensional swelling of a hydrogel. This case is an idealization of the case of
swelling of a thin gel bounded to a rigid substrate, as shown in Fig. 4. The present
simulation results are then compared against the experimental data reported by
Yoon et al. [2010].
We adopt similar conditions outlined in Yoon et al. [2010] for our simulation.
The gel, with material properties N ν = 0.09 and χ = 0.46, is initially at a chemical
potential of µ̄0 = −10, is exposed to a chemical potential of µ̄ = 0 at the top of the
layer. From the boundary conditions, the gel has a pre-stretch of λ0 = 1.0000035,
and obtains a final stretch ratio of λ∞ = 1.804056, and giving a net stretch of
about 1.8 times the initial thickness. The transient response of the thickness is
then reported. With different thicknesses for the specimens used in the experiment,
non-dimensionalization was performed to show the trends that are present in the
swelling of the gels.
Noting that there is a difference in definition of the mass flux, the diffusion
coefficient Dexp = 1.5 × 10−11 m2 /s has to be adjusted to fit the nonlinear theory
Fig. 4. (Color online) (a) Thin gel bounded to rigid substrate exposed to solvent. (b) 1D idealization which assumes solvent only contact gel from the top.
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Inhomogeneous Large Deformation Kinetics of Polymeric Gels
used in the finite element model. Bouklas and Huang [2012] compared the linear
poroelastic theory and nonlinear theory, and determined that the adjustment of D
was to be through a scale of ξ(λ∗0 ), i.e.,
D=
Dexp
,
ξ(λ∗0 )
(34)
where
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λ∗0
(λ0 + λ∞ )
,
=
2
2
ξ(λ∗0 )
1
2χ(λ20 λ∗0 − 1)
(λ20 λ∗0 − 1)(λ∗0 + 1)
= 2 ∗4 −
+
N
ν
.
5
4
λ0 λ0
λ40 λ∗0
λ20 λ∗0
Equation (34) gives a value of D = 6.87 × 10−11 m2 /s, which is used in the normalization of experimental data presented by Yoon et al. [2010].
Normalized simulation and experimental results are plotted and presented in
Fig. 5, which shows that simulation results agree very well with the experimental
results in the initial swelling phase, but shows a slower swelling rate in the subsequent times. This observation is due to the nonlinearity of the kinetic theory,
as opposed to the linear poroelastic theory which the experimental results may be
fortuitously fitted to [Bouklas and Huang, 2012].
4.3. Constrained swelling
In practical applications of polymeric gels, the gel will be restricted at some points
in order to secure it in place. The bondage of parts of the gel structure results in
Fig. 5. (Color online) Comparison of simulation results with experimental results of Yoon et al.
[2010]. Solid line represents simulation results, while discrete points represent experimental data
with different gel thicknesses.
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W. Toh et al.
Fig. 6. (Color online) (a)–(f) Shapes of cross-section of gel at t = 0, t = 1.25 h, t = 2.25 h, t = 6 h,
t = 8.5 h, t = 24 h. (b-1) and (c-1) show the polymer concentration at t = 1.25 and 2.25 h [Achilleos
et al., 2000]. Reprinted from Chemical Engineering Science. Achilleos E. C., Prud’homme R. K.,
Christodoulou K. N., Gee K. R. and Kevrekidis I. G. [2000] Dynamic Deformation Visualization
in Swelling of Polymer Gels, Vol. 55, Issue 17, pp. 3335–3340, with permission from Elsevier and
Ioannis G. Kevrekidis (corresponding author).
inhomogeneous swelling subjected to external forces. This section demonstrates the
capability of this finite element model to simulate the kinetics of inhomogeneous
swelling of gels in various cases.
In this example, we attempt to qualitatively compare the present simulation
results with the experimental observations of Achilleos et al. [2000]. The base of
a rectangular block of gel of dimensions 0.44 × 1 × 1 cm was rigidly fixed and the
remaining sides exposed to a constant chemical potential. The experimental data
of Achilleos et al. [2000] and corresponding present simulation results are presented
in Figs. 6 and 7, respectively. We take the characteristic length to be the ratio of
volume to wetted surface area, which gives L = 1.13 × 10−3 m.
It can be observed that while the edge profiles are not exactly the same, there is
a similarity in the shapes and polymer concentration distribution and the discrepancies may be attributed to imperfections in the experimental setup. The polymer
concentration distribution contour plots show similar distribution patterns and gel
shapes. The study of kinetics of constrained swelling is useful for diverse applications of gels. For example, in the modeling of fluid–structure interaction present in
microfluidic valves [Zhang et al., 2012].
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Inhomogeneous Large Deformation Kinetics of Polymeric Gels
Fig. 7. (Color online) Simulation results at various times. The contour plots show polymer concentration at the midplane.
Fig. 8. (Color online) Contour plot of y-displacement of an annulus with height 5 mm, inner
radius 35 mm and outer radius 75 mm. The inner core of the annulus is fixed and the remaining
sides are exposed to external solvent.
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4.4. Swelling-induced instabilities
The onset of instabilities during the swelling process is an interesting phenomena,
and is being studied for possible applications, such as microfluidics, wetting and
tunable adhesion [Yang et al., 2010].
Through studies on the buckling behavior of gels [Liu et al., 2010, 2011, 2012;
Mora and Boudaoud, 2006], we know that buckling occurs for some geometries, such
as a thin film bounded on a substrate and an annulus fixed at the inner wall, see
Fig. 8. These studies assume that chemical potential is homogeneous for all stages
of swelling, which is not reflective of the entire transient swelling process, as it takes
time for the solvent molecules to diffuse into the gel and buckling sets in during the
transient swelling stage, not the equilibrium homogeneous stage.
Upon the onset of buckling, there will be an initial transition between different mode shapes before the gel settles in the equilibrium mode shape and continues to swell in the buckled mode until chemical potential reaches the equilibrium
state. This phenomena can be seen in earlier experiments conducted by Mora and
Boudaoud [2006], where similar gel annuli were immersed in a solution until equilibrium.
5. Conclusions
Nonlinear hyperelastic gel theory is used together with coupled temperaturedisplacement analysis to study the transient swelling response of polymeric gels
by representing chemical potential as a temperature field. This is possible due to
the similarity between the laws of heat conduction and diffusion. Equivalent terms
for making the thermal-diffusion analogy are derived by comparison of the governing equations. This analogy is useful not just for polymeric gels, but can also be
applied to other diffusion-driven mechanical processes.
The transient process of gel undergoing large deformations is investigated with
numerical examples of a free swelling cube, one-dimensional constrained swelling,
rectangular block attached to a rigid substrate and an annulus fixed on the inner
wall. With this analogical FE model, we are able to provide more insight on the transient swelling phenomena using earlier established models in an easy-to-implement
model. Non-dimensionalization allows for the upscaling or downscaling of the modeling subject, thus reducing possible numerical errors. The capability of this model
to simulate constrained swelling and wrinkling is a valuable tool for research in
various potential applications of soft matter.
Acknowledgments
The authors also would like to indicate their appreciation to Nanyang Technological University (NTU) and Institute of High Performance Computing (IHPC)
for providing all the necessary supercomputing resources. Z. S. Liu is grateful
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Inhomogeneous Large Deformation Kinetics of Polymeric Gels
for the support by National Natural Science Foundation of China under Grant
Nos. 11242011 and 11021202.
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