Modeling mechano-chromatic lamellar gels
Wei Hong (洪伟)1, 2* and Xiao Wang (王宵)2
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
*whong@iastate.edu
Abstract
Consisting of alternating swelling and nonswelling polymeric layers (SLs and NLs), lamellar gels are 1D
photonic crystals with tunable optical properties. The lamellar structure induces a constraint between the
SLs and the NLs, resulting in an anisotropic swelling behavior coupled with deformation. The coupling
gives rise to the mechano-chromatic effect, and quantitative understanding of it is the key to many
applications. This letter formulates a nonlinear continuum model for lamellar gels by considering the
constrained swelling of SLs and the anisotropic deformation in both types of layers. A finite-element
method is further developed to simulate the response to non-uniform deformation.
Photonic crystals are materials baring stop-bands in visible or infra-red range of light1. The periodic
microstructure of a photonic crystal intensifies the reflection at the particular wavelength satisfying the
Bragg condition and results in a brilliant appearance. The same mechanism is commonly seen in colorful
biological features such as feathers and scales2-4. Numerous applications could be foreseen: color display,
cosmetics, rapid mechanical and chemical characterization, etc5-9. Polymer-based soft photonic crystals
or photonic gels6,7,9-12, with tunable stop-bands through deformation, have shed light on another
application – full-field measurement and visualization of large deformation.
The recently reported lamellar gels, which consist of aligned nonswelling polymer layers (NLs) and
swelling gel layers (SLs), have been demonstrated to have mechano-chromatic responses9-12. Unlike
regular photonic crystals with three-dimensional lattice structures, a lamellar gel contains unidirectional
layers and exhibits a one-dimensional optical lattice.
Showing a linear correlation between the
characteristic wavelength and the though-thickness strain, a lamellar gel can readily be used to measure
the uniaxial or equi-biaxial strain on a uniformly deformed body10. However, the measurement of a nonuniform strain field cannot be done without a material model. The nonlinearity associated with large
deformation prohibits conventional linear composite theory to accurately predict the coupling behavior of
the material13. This letter aims at formulating a simple yet effective model for the anisotropic property of
lamellar gels, by accounting for both the swelling of the SLs and constraint from the NLs. The discussion
1
in this letter will be limited to cases of the most technical interest: the reversible deformation and swelling
of lamellar gels. If needed, the theoretical framework may be extended to describe damage evolution or
time-dependent behavior of the materials9,11.
Swelling
Layer
λ’3 h
h
Nonswelling
Layer
1
λ3
1
1
λ1
Deformed /
Swollen State
Reference State
(As-synthesized)
λ2
Fig. 1 (Color online) Sketch of the SL-NL structure of a lamellar gel. The as-synthesized state is taken to be the reference, in
which the HPN already contains solvent. The overall stretches are
SLs is
λ1 , λ2 , and λ3 , while the through-thickness stretch of the
λ3′ .
As sketched in Fig. 1, the lamellar structure of the composite gel consists of alternating SLs and NLs.
The deformation and swelling of the soft SLs made from hydrophilic polymer networks (HPN) is
constrained by the stiff NLs, made from glassy polymer or surfactant bilayers. The particular ingredients
and microstructure varies between different systems10,12.
This letter focuses on an idealized
microstructure, in which both SLs and NLs are flat, infinitely large, and perfectly aligned. For simplicity,
the principal directions of deformation are assumed to be either parallel or perpendicular to the lamellae.
In a representative volume much larger than each individual layer yet much smaller than sample size, the
material appears to be homogeneous but anisotropic. The Helmholtz free energy per unit homogenized
volume W is, in general, a function of the principal stretches λi and the number of solvent molecules per
unit reference volume C . The free energy is further divided into three packages to account for the
special microstructure:
W (λi , C ) = Wn (λi ) + Wb (λi ) + Wm (C ) ,
2
(1)
in which Wn is the contribution from stretching the HPN, Wb that from deforming the NLs, and Wm that
from mixing the HPN and the solvent. The interaction energy between the NLs and the solvent hardly
changes during swelling or deformation, and is thus omitted from the free-energy function.
While the choice of reference state is arbitrary, the as-synthesized state is selected to be the reference
for ease of description. The NLs are assumed to be freely extended in this state. The HPN, on the other
hand, is stretched isotropically and homogeneously due to the presence of a small amount of solvent
during synthesis10,12. Let λ0 be the initial stretch of the HPN with respect to the dry state. For simplicity,
the effect of ions is excluded and the HPN is modeled as a neutral polymer. Let the energy of stretching
HPN be neo-Hookean:14
Wn (λi ) = 12 NkT (λ12 λ20 + λ22 λ20 + λ3′2 λ20 − 2 ln λ1λ2 λ3′ ),
(2)
and the energy of mixing be that of a polymer with simple solvent:14
Wm (C ) = −
kT 1 − φD
− (φ −1 − 1)ln (1 − φ ) + χφ .
3
v λ0
[
]
(3)
Here, N is the number of HPN chains per reference volume, kT the temperature in the unit of energy, v
the volume per solvent molecule, and χ the Flory-Huggins parameter. Let λ1 , λ2 , and λ3′ be the
principal stretches in the SLs, and specifically λ3′ be that in the through-thickness direction of a lamella.
Since NLs are usually much stiffer and much thinner than SLs, the through-thickness strain of the NLs is
neglected, and λ3′ is related to the overall stretch λ3 as λ3′ = (λ3 − φD ) (1 − φD ) , where φD is the NL
[
]
volume fraction in the reference state. In Eq. (3), φ = 1 + λ30 vC (1 − φ D )
−1
is the volume fraction of the
HPN in SLs.
Each NL acts as an elastic membrane embedded in the gel. It is assumed that the deformation is
affine and the in-plane stretches of the NLs match the corresponding macroscopic stretches, λ1 and λ2 .
With each NL modeled as an incompressible hyperelastic membrane with modulus µ D , the elastic energy
of deforming the bilayers is
Wb (λi ) = 12 φD µ D (λ12 + λ22 + λ1−2 λ−22 ) .
For simplicity, some constant terms are omitted from the energy functions in Eqs. (2) – (4).
3
(4)
Furthermore, to introduce the coupling between the mechanical deformation and the polymer volume
fraction, we assume molecular incompressibility by insisting that
φD + (1 − φD )λ0−3 + vC = λ1λ2 λ3 ,
(5)
i.e. the total volume of the gel-solvent system is conserved during swelling. Utilizing Eq. (5), the HPN
volume fraction can also be expressed in terms of the stretches, φ = (1 − φD ) (λ1λ2 λ3 − φD )λ30 . The
elastic energies in Eq. (2) and (4) and the energy of mixing in Eq. (3) are coupled through constraint (5).
The calculations here are limited to equilibrium states, in which chemical potential is uniform,
µ = µ0 , and solvent migration has equilibrated. The mathematical representation of such states is
facilitated by introducing an alternative free-energy function through a Legendre transform:15
Wˆ (λi , µ ) = W (λi , C ) − µC .
(6)
With the aid of Ŵ , the equilibrium stress-stretch relation is just like that of a hyperelastic material:
si =
∂Wˆ
.
∂λi
(7)
The solvent chemical potential µ acts as a parameter rather than a field variable. Substitution of Eqs. (1)
– (6) into Eq. (7) yields the explicit constitutive relation of the gel. The in-plane stresses are
[
s1 = NkT (λ20 λ1 − λ1−1 ) + φD µ D (λ1 − λ1−3λ−22 ) − pλ2 λ3 ,
(8)
s2 = NkT (λ20 λ2 − λ2−1 ) + φD µ D (λ2 − λ1−2 λ2−3 ) − pλ1λ3 ,
(9)
]
where p = µ v − kTv ln (1 − φ ) + φ + χφ 2 is the osmotic pressure. The out-of-plane stress is:
 λ − φD
1 
s3 = NkT λ20 3
−
 − pλ1λ 2 .
2
λ
−
φ
(
)
−
φ
1
3
D 
D

(10)
The model for lamellar hydrogels is verified through comparison with existing experiments. Let us
first look at the free swelling of a lamellar hydrogel. Equating the in-plane and out-of-plane stresses (8)(10) to 0, we arrive at the nonlinear equations for the stretches of free-swelling, λi . Take the chemical
4
potential of pure solvent as the reference µ = 0 , and utilizing the symmetry, λ1 = λ2 , we can simplify
the equations as
[ (
NkT (λ20 λ1 − λ1−1 ) + φD µ D (λ1 − λ1−5 ) − pλ1λ3 = 0 ,
(11)
 λ −φ
1 
2
NkT λ20 3 D2 −
 − pλ1 = 0 ,
 (1 − φD ) λ3 − φD 
(12)
)
]
(
)
where p = − kTv ln 1 − φ + φ + χφ 2 and φ = (1 − φD ) λ12 λ3 − φD λ30 are the osmotic pressure and HPN
volume fractions, respectively, in a free-swelling state. Solving the nonlinear algebraic equations (11)
and (12) simultaneously, we can obtain the equilibrium stretches of a free-swelling lamellar gel.
The material parameters are estimated as follows for a specific lamellar gel, in which the SLs consist
of polyacrylamide (PAAm) hydrogel and the NLs are polymeric dodecylglyceryl itaconate (DGI) bilayers,
to enable comparison with the available experimental data10. Taking the room temperature T = 300K
and using the size of a water molecule, we estimate kT v ≈ 100MPa . Taking the molecular weight of
DGI to be 372 and assuming a density similar to that of water, we relate the NL volume fraction φD to
the molar concentration cD of DGI monomer approximately as φD = 0.37mol−1L ⋅ cD . In general, the
stiffness of the HPN NkT , the swelling parameter χ , and the NL shear modulus µ D can be determined
from independent experiments.
Lacking further experimental data, we use these them as fitting
parameters. In the following calculations, we use NkT = 400Pa , χ = 0.5 , and µ D = 1MPa , all within
the range of commonly accepted values for the materials10,16. The resulting free-swelling stretches
parallel ( λ1 ) and perpendicular ( λ3 ) to the lamellae are plotted in Fig. 2a, as functions of the DGI
concentration. The results exhibit good agreement with the experimental measurements10. At low NL
concentration, the lamella structure may be less ordered and the deformation may not be affine due to the
large distance between neighboring bilayers, the relatively large discrepancy is thus expected.
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swelling ratio λ
a
λ3
2.5
2
1.5
λ
1
0
0.05
0.1
0.15
cD (mol/L)
0.2
1
0.25
Modulus E (kPa)
b
100
E1
50
E3
0
0
0.05
0.1
0.15
0.2
DGI concentration c D (mol/L)
0.25
Fig. 2 (Color online) (a) Equilibrium stretches of a free-swelling PAAM-PDGI lamellar gel, paralell ( λ1 ) and perpendicular
( λ3 ) to the lammellae, as functions of the DGI monomer concentration. (b) Tensile moduli of the PAAM-PDGI gel paralell
( E1 ) and perpendicular ( E 3 ) to the lammellae as functions of the DGI concentration. Squares and circles are experimental
results from Ref. 10.
The behavior of the gel under uniaxial tension/compression parallel to the bilayers ( s2 = s3 = 0 ) or
perpendicular to the bilayers ( s1 = s2 = 0 ) can be obtained by solving the nonlinear Eqs. (8) – (10). The
tensile modulus, namely the initial slope of the stress-strain curve in uniaxial tension/compression, can
also be evaluated through linearizing the stress-stretch relation at the small strain limit. Using the same
set of material parameters as in the swelling calculation, we compute the tensile modulus in the direction
of the bilayers, E1 , and that in the perpendicular direction, E3 . The results are plotted in Fig. 2b, and
compared to the available experimental data10. The prediction of the model shows good agreement at a
DGI concentration higher than 0.06 mol/L.
Finally, to demonstrate the mechano-chromatic capability, we simulate the pseudo color map of a
cracked specimen of lamellar hydrogel under tension. The material parameters are the same as those used
6
in previous calculations, and the strain-wavelength relation is directly taken from experimental
measurements10. The numerical results are obtained by implementing the constitutive relations, Eqs. (8)(10), in the finite element software, COMSOL Multiphysics 4.1. Although no experimental result is
available for comparison at the time of writing, we expect our model prediction to deviate from the actual
observation because of the inevitable material damage at the crack tip. However, the difference between
the model prediction and the experimental observation could actually provide valuable insights towards
the fracture process zone of the material. Since the chromatic response is independent of the nonlinearity
in deformation, the highly stretchable lamellar gels are most suitable for measuring the large deformation
of soft materials, on which conventional strain gauges are deemed to fail. The capability of full-field
measurement would also enable the visualization of fracture or localized damage of soft materials and
structures.
Fig. 3 (Color online) Simulated result of a cracked specimen of lammelar hydrogel under tension. The shading indicates the
through-thickness stretch,
λ3 .
It should also be pointed out that the mechano-chromatic mechanism would not be effective when the
lammellar gel is subject to an in-plane compression. The large difference in stiffness between the SL and
the NL makes the composite gel very similar to the structure of a hard film on a soft substrate. When
compressed beyond a critical strain, the deformation will no longer be affine and wrinkles are expected to
form in the hard film or NLs. Taking the modulus ratio between the NL and SL to be ~100, and the
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thickness ratio to be ~0.05, we can estimate the critical compressive strain for wrinkle instability to be
~0.5%, and the wrinkle wavelength to be ~30 times of the NL thickness17. The wavelength of the
wrinkles in the NL is still comparable to that of visible light, but is insensitive to the applied strain. Such
a prediction is awating to be confirmed experimentally.
In summary, in this letter a continuum model has been developed for mechano-chromatic lamellar
gels consisting of alternating SLs and NLs. With the coupling accounted for, the model successfully
recovers the experimental measurements on free-swelling and anisotropic deformation. Upon calibration,
the model can assist experiments in full-field strain measurement of the large deformation on soft
materials. The damage and fracture processes in soft materials may also be visualized with the material
and the supporting theory. The model may well be used on other systems with similar structures. It is
also suggested that the mechano-chromatic mechanism of the lamellar gel would fail if an in-plane
compression is applied, under which wrinkles will form in the stiff NLs.
WH acknowledges the support from the NSF through Grant No. CMMI-0900342, and Prof. J. P.
Gong and Dr. M. A. Haque for providing the experimental details.
1
J. D. Joannopoulos, P. R. Villeneuve and S. Fan, Nature 386 (13), 143-149 (1997).
2
S. Kinoshita, S. Yoshioka and J. Miyazaki, Rep. Prog. Phys. 71, 076401-076431 (2008).
3
S. M. Doucet and M. G. Meadows, J. R. Soc. Interface, 6 S115-S132 (2009).
4
P. Vukusic and J. R. Sambles, Nature 424 (14), 852-855 (2003).
5
J. Ge, J. Goebl, L. He, Z. Lu and Y. Yin, Adv. Mater. 21 (42), 4259-4264 (2009).
6
J. J. Walish, Y. Kang, R. A. Mickiewicz and E. L. Thomas, Adv. Mater. 21 (42), 3078-3081(2009).
7
J. H. Holtz and S. A. Asher, Nature 389 (23), 829-832 (1997).
8
Y. Y. Li, F. Cunin, J. R. Link, T. Gao, R. E. Betts, S. H. Reiver, V. Chin, S. N. Bhatia and M. J. Sailor,
Science 299 (5615), 2045-2047 (2003) .
9
M. A. Haque, T. Kurokawa, G. Kamita, Y. Yue and J. P. Gong, Chem. Mater. 23 (23), 5200-5207 (2011).
8
10
M. A. Haque, G. Kamita, T. Kurokawa, K. Tsujii and J. P. Gong, Adv. Mater. 22 (45), 5110-5114
(2010).
11
M. A. Haque, T. Kurokawa, G. Kamita and J. P. Gong, Macromolecules 44, 8916-8924 (2011).
12
Y. Kang, J. J. Walish, T. Gorishnyy and E. L. Thomas, Nature Mater. 6, 957-960 (2007).
13
K. Okumura and P.-G. de Gennes, Eur. Phys. J. E 4, 121-127 (2001).
14
P. J. Flory, Principles of polymer chemistry. (Cornell University Press, Ithaca, 1953).
15
W. Hong, Z. Liu and Z. Suo, Int. J. Solids. Struct. 46, 3282-3289 (2009).
16
T. Tominaga, V. R. Tirumala, S. Lee, E. K. Lin, J. P. Gong and W. L. Wu, J. Phys. Chem. B 112, 3903-
3909 (2008).
17
Z. Y. Huang, W. Hong, and Z. Suo, J. Mech. Phys. Solids. 53 (9), 2101-2118 (2005).
9