International Journal of Solids and Structures 51 (2014) 4440–4451

Contents lists available at ScienceDirect

International Journal of Solids and Structures

journal homepage: www.elsevier .com/locate / i jsolst r

Mechanics of inhomogeneous large deformation of photo-thermalsensitive hydrogels

http://dx.doi.org/10.1016/j.ijsolstr.2014.09.0140020-7683/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail address: zishunliu@mail.xjtu.edu.cn (Z. Liu).

William Toh a, Teng Yong Ng a, Jianying Hu b, Zishun Liu b,⇑a School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singaporeb International Center for Applied Mechanics, State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong University, Xi’an 710049, People’s Republicof China

a r t i c l e i n f o a b s t r a c t

Article history:Received 26 April 2014Received in revised form 21 August 2014Available online 5 October 2014

Keywords:Photo-thermal sensitive hydrogelHyperelasticityLarge deformationPhase transition

A polymer network can imbibe copious amounts of solvent and swell, the resulting state is known as agel. Depending on its constituents, a gel is able to deform under the influence of various external stimuli,such as temperature, pH-value and light. In this work, we investigate the photo-thermal mechanics ofdeformation of temperature sensitive hydrogels impregnated with light-absorbing nano-particles. Thefield theory of photo-thermal sensitive gels is developed by incorporating effects of photochemical heat-ing into the thermodynamic theory of neutral and temperature sensitive hydrogels. This is achieved byconsidering the equilibrium thermodynamics of a swelling gel through a variational approach. The phasetransition phenomenon of these gels, and the factors affecting their deformations, are studied. To facili-tate the simulation of large inhomogeneous deformations subjected to geometrical constraints, a finiteelement model is developed using a user-defined subroutine in ABAQUS, and by modeling the gel as ahyperelastic material. This numerical approach is validated through case studies involving gels undergo-ing phase coexistence and buckling when exposed to irradiation of varying intensities, and as a microv-alve in microfluidic application.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

A three dimensional polymer network is formed by the covalentcrosslinking of polymer chains. When immersed in an aqueoussolution, the network swells due to absorption of the solution.The resultant swollen state is known as a hydrogel. Light-sensitivehydrogels are gaining popularity amongst researchers due to theattractive properties of light being able to stimulate the gel remo-tely and precisely. Potential applications of light-sensitive gelsinclude, but are not limited to, photo-actuators (Mahimwallaet al., 2012), photo-robotics (Mahimwalla et al., 2012), photo-patterning (Kim et al., 2012) and artificial muscles (Takashimaet al., 2012).

Depending on the type of photo-sensitive particle (photo-chrome) present, there will be different types of mechanismscausing the deformation of the gel. The mechanisms of photo-induced deformation include pericyclic reaction (Yang, 2008),cis–trans isomerization (Finkelmann et al., 2001; Jiang et al.,2006; Yang, 2008), intramolecular hydrogen transfer (Yang,

2008), intramolecular group transfer (Yang, 2008), dissociationprocesses (Jiang et al., 2006; Yang, 2008) and change in crosslinkdensity (Jiang et al., 2006; Lendlein et al., 2005). Swelling of gelstypically takes place through the migration of solvent (Hong et al.,2008; Toh et al., 2013) or ions (Hong et al., 2010; Toh et al., 2013)through the polymer network, which constitutes a slow process.However, it was reported that by incorporating light-absorbingnano-particles (NPs) into a temperature sensitive hydrogel, photo-induced heating is able to achieve a much faster swelling rate(Jiang et al., 2006; Suzuki and Tanaka, 1990; Yoon et al., 2012).

The phase transition of a temperature-sensitive hydrogel hasbeen widely observed and studied (Cai and Suo, 2011; Ding et al.,2013). It is well accepted that hydrogels made of the same mono-mer, with the same crosslink density, will possess a unique transi-tion temperature. However, it was shown that the phase transitiontemperature of a temperature-sensitive hydrogel can be altered byirradiation of light of varying intensities (Suzuki and Tanaka,1990). It has also been reported that at a fixed temperature belowthe transition temperature, the gel is able to undergo phase transi-tion by simply changing the intensity of light being irradiated onthe gel. Experimental findings have proven the viability of photo-thermal sensitive hydrogels as a good material for remote control-ling of devices in applications such as drug delivery (Sershen et al.,

W. Toh et al. / International Journal of Solids and Structures 51 (2014) 4440–4451 4441

2000), microfluidics (Lo et al., 2011) and optical switching (Yoonet al., 2012).

The photo-mechanics of light responsive polymer systems havebeen extensively studied (Dunn and Maute, 2009; Dunn, 2007;Long et al., 2011, 2009a,b, 2013; Mahimwalla et al., 2012;Saphiannikova et al., 2009, 2011; Toshchevikov et al., 2012,2011), but mainly as shape-memory polymers or liquid crystalelastomer (LCE) systems. There is acutely limited literature onthe modeling of the mechanics and thermodynamics of photo-sensitive hydrogel systems.

Recent theories developed for hydrogels originate from tradi-tional thermodynamics framework of energy balance and entropyimbalance (Bowen, 1989; Truesdell, 1969). In view of the vastliterature available for the thermodynamics of deformation of aswelling polymer network (Baek and Srinivasa, 2004; Chesterand Anand, 2010, 2011; Duda et al., 2010; Hong et al., 2008), wewill not repeat the highly involved classical derivations here, butinstead utilize the earlier results of these rigorous derivations toadvance the present theoretical model.

In the present study, we will develop a theory to model thedeformation of hydrogels due to photo-thermal effects by buildingon earlier works on neutral gels (Hong et al., 2009; Toh et al.,2013a,b) and temperature-sensitive gels (Ding et al., 2013). Thistype of deformation mechanism is made possible through theincorporation of light absorbing NPs into the temperature sensitivepolymers.

In what follows, Section 2 describes the thermodynamicframework by expressing the stress and chemical potential interms of the free energy function. Section 3 specializes the freeenergy function using the Flory–Rehner theory and describesthe physical constraints experienced by the gel. Section 4describes the process of finite element (FE) implementation.Section 5 investigates some potential applications of the presenttheory and FE model, which include phenomena such as coexis-tent phases within the gel, its buckling during the deswellingprocess, and its contact with a membrane in microfluidicapplications.

2. Theoretical development

Here we shall describe the mechanism of a hydrogel by consid-ering the change in temperature due to light irradiation, as well asthe inhomogeneous large deformation theory for thermally-sensitive hydrogels.

Fig. 1. A hydrogel domain subjected to external weight P, exposed to an externalsolvent of fixed chemical potential ls, and irradiated with light.

2.1. Light irradiation induced temperature change

Due to the addition of light-absorbing NPs in the gel network,light energy is converted into heat energy and transferred to thepolymer through local heat conduction. The use of different NPswill result in different wavelengths required for absorption ofirradiation, with copper chlorophyllin absorbing best at 488 nm(Suzuki and Tanaka, 1990), gold–gold sulfide at 1064 nm(Sershen et al., 2000), gold nano-rods at 810 nm (Gorelikov et al.,2004), graphene oxide at 808 nm (Zhu et al., 2012) and iron oxideat 470 nm (Yoon et al., 2012).

Under irradiation, the temperature rise Dh is proportional to theintensity I0 and polymer volume fraction /, as shown in the phe-nomenological Eq. (1) (Richardson et al., 2009; Suzuki andTanaka, 1990),

Dh ¼ aI0/ ð1Þ

where a is the proportionality constant related to the heat capacityof the gel.

2.2. Inhomogeneous large deformation of a temperature sensitive gel

With reference to Fig. 1, consider a gel subjected to geometricalconstraints, with an external weight of P hanging on the gel andmaintained in immersion with a solvent of chemical potential ls.Concurrently, a monochromatic light is being irradiated onto thegel. Using a thermodynamic approach on the irradiation, the lightpossesses a non-zero chemical potential lp, produced by photo-chemical reactions (Ries and McEvoy, 1991; Herrmann andWurfel, 2005; Wurfel, 1982; Kelly, 1981; Haught, 1984).

In equilibrium, the weight is displaced by a distance dl, thechemical potential is maintained by pumping dNs solvent mole-cules into the gel and the irradiation causes dNp photochemicalreactions at chemical potential lp. The work done on the gel wouldbe equal to the sum of the work done by the external weight, thechemical potential pump and the irradiation, i.e. the sum of Pdl, ls-

dNs and lpdNp.In a deformed state at any time t, a point originally at position

with reference coordinates XK relocates to the current positionwith coordinates xi, we denote the deformation gradient byEq. (2), given by

FiK ¼@xiðXK ; tÞ@XK

ð2Þ

Thermodynamically, the change in free energy density of the gelW, is balanced by the work done on the gelZ

dWdV ¼Z

BidxidV þZ

TidxidSþ ls

ZdCsdV þ lp

ZdCpdV

ð3Þ

where Cs and Cp represent the concentration of solvent and photo-chemical reactions in the reference state respectively.

Assuming that the free energy density in the current state isdependent on the state of deformation, concentration of solventand photochemical reactions within the gel, i.e. W = W(FiK, Cs, Cp),a small change in the free energy density can be written as

dW ¼ @W@FiK

dFiK þ@W@Cs

dCs þ@W@Cp

dCp ð4Þ

Combining Eqs. (3) and (4), and applying the divergencetheorem, the equilibrium equation is re-written as Eq. (5),Z

@W@FiK

� siK

� �dFiK þ

@W@Cs� ls

� �dCs þ

@W@Cp� lp

� �dCp

� �dV ¼ 0

ð5Þ

4442 W. Toh et al. / International Journal of Solids and Structures 51 (2014) 4440–4451

where siK is the nominal stress. This equation holds for arbitrary val-ues of dFiK, dCs and dCp, which reduces the conservative form intothe differential form

siK ¼@W@FiK

ls ¼@W@Cs

lp ¼@W@Cp

ð6Þ

3. Free energy function

From the approach of earlier works (Hong et al., 2009, 2008; Caiand Suo, 2011; Ding et al., 2013), we assume that the free energydensity of the gel is the sum of the individual free energies of net-work stretch and mixing. For a photo-thermal sensitive gel, weintroduce an additional free energy density term due to photo-chemical reactions Wp, into the free energy density function of atemperature sensitive hydrogel

WðF;Cs;CpÞ ¼WsðFÞ þWmðCsÞ þWpðCpÞ ð7Þ

3.1. Free energies of network stretch and mixing

The free energy density of network stretch is taken to be (Flory,1953)

Ws ¼12

Nkh½FiK FiK � 3� 2 ln ðdet FÞ� ð8Þ

where N is the number of crosslinks per unit reference volume ofthe polymer network, kh the temperature in the unit of energy,and det F the determinant of the deformation gradient.

The free energy density of mixing is assumed to take the Flory–Huggins form (Flory, 1942; Huggins, 1941)

Wm ¼ kh Cs lnmCs

1þ mCsþ vCs

1þ mCs

� �ð9Þ

where m the volume of a solvent molecule, v is the Flory–Hugginsinteraction parameter, which is dependent on polymer density /and temperature. This relation is given by (Huggins, 1964)

v h;/ð Þ ¼ v0 þ v1/ ð10Þ

where v0 = A0 + B0h and v1 = A1 + B1h, while A0, B0, A1 and B1 areconstants specific to the monomer. The temperature used in the cal-culation of the Flory–Huggins interaction parameters is the realtemperature of the gel consisting of two parts, given by the equa-tion h = h0 + aI0/, where h0 is the ambient temperature and aI0/ isthe temperature increment given in Eq. (1). In subsequent analyses,all Flory–Huggins interaction parameters are calculated based onthe real temperature, thus implicitly containing the temperatureincrement due to irradiation.

3.2. Free energy of photochemical reaction

As the nano-particles absorb UV irradiation, molecular excita-tion occurs, promoting an electron to the next lowest unoccupiedorbital, with an energy jump corresponding to the product of thePlanck’s constant and the frequency of the UV. The excited particledecays back into the ground state immediately, thus releasing theenergy harvested during photon absorption.

For gold NPs (and other NP used for the purpose of photo-heating), it has been shown that the quantum yield is of the orderof 10�6 (Dulkeith et al., 2004), resulting in a near unity conversionof light energy into heat energy (Richardson et al., 2009).

This electronic transition can be represented as a series of twoconsecutive chemical reactions, with the reactants being theground-state nano-particle (X) and a photon of relevant frequency(hf), the intermediate product being the excited particle (Y), andthe product being the ground-state nano-particle and heat energy.This series of chemical reactions is illustrated in Fig. 2 and writtenas Eq. (11).

X þ hf ! Y

Y ! X þ heatð11Þ

Under steady state irradiation, X and Y will have concentrations[X] and [Y] respectively. The affinity (free energy) associated withone such chemical reaction, A, is given by (Parson, 1978)

A ¼ hf þ kh ln½Y�½X�

� �ð12Þ

where hf corresponds to energy transition of the electron into theexcited state and the logarithmic term corresponds to the entropychange associated with the transition.

The free energy density of the entire gel, Wp, resulting fromphotochemical reactions is thus expressed as

Wp ¼ Cp hf þ kh lnCp

Cg;0

� �� �ð13Þ

where Cp = [Y] and Cg,0 = [X] is taken to be a nominal referencevalue given that the concentration of excited particles is low(Cp/Cg � 10�10) (Lavergne and Joliot, 2000).

3.3. Legendre transformation

As a standard approach to model hydrogels as hyperelasticmaterials (Ding et al., 2013; Hong et al., 2009; Marcombe et al.,2010), we introduce a new free energy density function W , whichis obtained by a Legendre transformation of the free energy densityfunction W

W ¼W � lsCs � lpCp ð14Þ

where a small change in this free energy density function will yieldthe following relation

dW ¼ siKdFiK � Csdls � Cpdlp ð15Þ

Eq. (15) transforms the equilibrium Eq. (3) intoZdWdV ¼

ZBidxidV þ

ZTidxidS ð16Þ

Eq. (16) is the equilibrium equation assumed by a hyperelasticsolid.

3.4. Constraint equations

During the deformation process, the stress induced is verysmall, thus it is insignificant in relation to causing deformationsto the individual polymer and solvent molecules. Therefore thevolume of the swollen state is assumed to be equal to the sum ofthe individual volumes of polymer and solvent components ofthe gel. This assumption can be written as

1þ mCs ¼ det F ð17Þ

Whilst the constituents of a temperature sensitive hydrogelundergoes temperature changes and experience volumetricchanges due to thermal expansion/contraction, we note from theorder of magnitude of the coefficients of thermal expansion (seeTable 1) that the volumetric expansion of individual particles isvery small as compared to the volumetric change experienced dur-ing phase transition and therefore we assume that the effects from

Fig. 2. Excitation and decay of an atom absorbing a photon.

W. Toh et al. / International Journal of Solids and Structures 51 (2014) 4440–4451 4443

thermal expansion are negligible. This practice is also seen inanalyses of temperature sensitive gels (Birgersson et al., 2008;Chester and Anand, 2011).

In photochemical reactions, the number of photons and chemi-cal reactions is conserved, as dictated by the Stark–Einstein equa-tion, or second law of photochemistry (Rohatgi, 2006). The lightintensity is proportional to the amount of light energy absorbed.In the irradiation of a material, the spatial distribution of photonswithin a material follows the radiative transfer equation (Longet al., 2013). However, we assume that at sub-millimeter dimen-sions, the spatial distribution of photochemical reactions takingplace within the gel is evenly distributed. This assumption can beinferred from Eq. (1), as proportionality to / imply that the gel isnot optically thin, and the gel undergoing uniform temperaturechange implies that there is no spatial distribution of photochem-ical reactions for gels of sub-millimeter dimensions.

The concentration of photochemical reactions is obtained byconsidering conservation of energy, where all light energy is trans-formed into the rise in temperature of the system (Walker, 2009).

The amount of photonic energy released is the sum of energiesof all photons undergoing photo-excitation. For a monochromaticlight source of uniform frequency, this energy is equal to the prod-uct of the energy of a single photon with the number of photo-excited photons

E0 ¼ hf � Cp ð18Þ

where E0 is a measure of the energy absorbed per unit referencevolume, h the Planck constant and f the frequency of irradiation.

Table 1List of material parameters used in analysis.

Parameter Value Reference

A0 �12.947 Afroze et al. (2000)B0 0.04496 K�1 Afroze et al. (2000)A1 17.92 Afroze et al. (2000)B1 �0.0569 K�1. Afroze et al. (2000)a 0.6 Km2 W�1 Suzuki and Tanaka (1990)f 488 nm Suzuki and Tanaka (1990)c 2.998 � 108 ms�1 Serway and Jewett (2013)h 6.626 � 10�34 m2 kg s�1 Barrow (2002)m 10�28 m3 Hong et al. (2008)c(polymer) 4.18 � 106 Jm�3 K�1 Birgersson et al. (2008)c(w) 2.4 � 107 Jm�3 K�1 Birgersson et al. (2008)c(cu) 3.44 � 106 Jm�3 K�1 Serway and Jewett (2013)a(p) 77 � 10�6K�1 Chester and Anand (2011)a(w) 67 � 10�6 K�1 Chester and Anand (2011)a(Cu) 17 � 10�6 K�1 Serway and Jewett (2013)

The product hf measures the amount of energy possessed by asingle photon. The amount of thermal energy per unit referencevolume generated by the temperature rise is the product of mass,specific heat and temperature change of the system, i.e.

E0 ¼ qgel cgelDh ð19Þ

Substituting the relations mgel = qgelJ and Dh = aI0/J into Eq. (19),the thermal energy can be re-written as

E0 ¼ cðgÞaI0

Jð20Þ

where c(g) = qgelcgel is the volumetric heat capacity, and J thedeterminant of the deformation gradient, J = det F.

The volumetric heat capacity of the gel is taken to be thevolume weighted average of the constituents, i.e.

cðgÞ ¼ fcðcuÞ þ ð1� fÞcðpolymerÞ� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}cðnetworkÞ

/þ cðwÞð1� /Þ ð21Þ

where c(cu), c(polymer) and c(w) are the volumetric heat capacitiesof copper, polymer and water, respectively. The volumetric heatcapacity of the dry network, c(network) is calculated using the ruleof mixtures consisting polymer molecules and copper chlorophyl-lin molecules with volume fractions (1 � f) and f respectively.

Combining the two equations above, we obtain an expressionfor Cp in terms of swelling ratio and light intensity

Cp ¼cðwÞ

Jþ cðnetworkÞ � cðwÞ

J2

" #aI0

hfð22Þ

Substituting free energy functions into Eq. (14), we arrive at

W ¼ 12

Nkh J23�I1 � 3� 2 lnðJÞ

h iþ kh

mðJ � 1Þ ln

J � 1J

� �þ v0

Jþ v1

J2

!" #

� khaI0

hfcðwÞ

Jþ cðnetworkÞ � cðwÞ

J2

" #� ls

J � 1m

� �ð23Þ

where �I1 is the first deviatoric invariant of the deformation gradient.With the free energy density function, we are able to obtain anexplicit form of the nominal stress

siK ¼ NkhðFiK � HiKÞ þkhm

lnJ � 1

J

� �þ 1

Jþ v0 � v1

J2 þ 2v1

J3

" #JHiK

þ khaI0

hf2ðcðnetworkÞ � cðwÞÞ

J3 þ cðwÞ

J2

" #JHiK �

ls

mJHiK ð24Þ

Fig. 3. Gel in (a) dry state; and (b) reference state.

Fig. 4. Comparison of experimental (Suzuki, 1993) and analytical normalizeddiameters of a free swelling gel. The discrete points correspond to experimentaldata of gels with different amounts of copper chlorophyllin (Naming of exper-imental data set follows results presented by Suzuki (1993)), while the curves arethe analytical solutions of Eq. (27).

4444 W. Toh et al. / International Journal of Solids and Structures 51 (2014) 4440–4451

where HiK is the transpose of the inverse of the deformationgradient, HiKFjK = dij. To convert the nominal stress into Cauchystress, we use the following relation

rij ¼siK FjK

det Fð25Þ

The Cauchy stress is thus given by

rij ¼Nkh

JðFiK FjK � dijÞ þ

khm

lnJ � 1

J

� �þ 1

Jþ v0 � v1

J2 þ 2v1

J3

" #dij

þ khaI0

hfðcðnetworkÞ � cðwÞÞ 2

J3 þ cðwÞ1J2

!" #dij �

ls

mdij ð26Þ

3.5. Homogeneous state of free swelling

Consider a dry cubic gel without any solvent imbibed in thepolymer network, the volume ratio of the gel is unity, J = 1. Thisgives rise to a singularity in the free energy function. Therefore, asecond reference state of a freely swelling gel with mCs > 0 isrequired, see Fig. 3. In the free swelling state, there is no stresspresent (rik = 0) and the gel experiences stress free isotropicstretches with k1 ¼ k2 ¼ k3. Taking this free swelling state as thereference, we denote the isotropic stretches as k0, occurring atthe reference solvent chemical potential ls,0 and temperature h0

and exposed to light at intensity I0. In this configuration, Eq. (24)becomes

ls;0

kh0¼ Nm

k30

ðk20 � 1Þ þ ln 1� 1

k30

!þ 1

k30

þ v0 � v1

k60

þ 2v1

k90

þ maI0

hf2 cðpÞ � cðwÞ

k90

þ cðwÞ

k60

" #ð27Þ

where v0 = A0 + B0h0, v1 = A1 + B1h0.This homogeneous state of equilibrium serves as an initial con-

dition during the FE analysis to circumvent the singularity in thedry state. The variables in Eq. (27) are k0, h0, I0 and ls,0. Givenany three variables, we can find the fourth using this equation, thusproviding the initial condition.

4. Finite element (FE) implementation

To facilitate the modeling and simulation of gel deformationwhen subjected to external stimuli and constraints, we utilizethe powerful commercial FE numerical solver ABAQUS. Based onearlier works which implement hydrogel properties as hyperelasticmaterials (Ding et al., 2013; Hong et al., 2009; Marcombe et al.,2010), we develop the user-defined subroutine UHYPER to carryout FE analysis on the gel deformation. The definition of materialproperties is based on the free energy function derived earlier inEq. (23) and its respective derivatives with respect to the invariants

of the deformation gradient. Details of all required formulae in theUHYPER subroutine are given in Appendix A.

4.1. Phase transition of a photo-thermal sensitive hydrogel

The phase transition of temperature-sensitive hydrogels is acommonly observed phenomenon, and has been widely studied(Ding et al., 2013; Kondo et al., 1993; Suzuki and Ishii, 1999;Suzuki and Kojima, 1994; Suzuki and Tanaka, 1990; Tanaka,1978; Wang, 2007). The phase transition temperature is dependenton several factors, including the crosslink density of the polymernetwork (Harmon et al., 2003), the chemical potential of the sol-vent (Pastoor and Rice, 2012), and the intensity of light irradiation.In the present study, we investigate the effects of varying lightintensity on the free swelling state of poly(N-isopropylacrylamide)(PNIPAm) using the parameters listed in Table 1.

The values of A0, A1, B0 and B1 are material dependent and areobtained by experimental fitting (Afroze et al., 2000). Experimentalresults have shown that for the same monomer, the addition ofchromophores will alter the swelling properties of the gel(Suzuki, 1993), thus implying a change in the Flory–Huggins inter-action parameter (Caykara et al., 2006).

We define a new Flory–Huggins interaction parameter by intro-ducing a factor e. to the experimental values obtained for a polymernetwork without chromophores. Under this definition, we have

v0;new ¼ ev0 ¼ A0eþ B0ehv1;new ¼ ev1 ¼ A1eþ B1eh

ð28Þ

Performing a curve fitting process on the experimental resultsreported, the Flory–Huggins interaction parameters of the gel con-taining 5% and 8% copper chlorophyllin are obtained by multiply-ing the values A0, B0, A1 and B1 by a factor e = 0.78 and 0.85,respectively, and the crosslink density corresponds to Nm = 0.004.The change in transition profile is shown in Fig. 4, which shows acomparison of experimental results with analytical results. With-out any chromophores, the gel undergoes a discontinuous phasetransition as temperature is varied; and this is indicated by thetwo turning points in the curve. The discontinuous phase transitionexhibits hysteresis phenomena which will be elaborated subse-quently. Upon chromophore addition, it is evident that the two

W. Toh et al. / International Journal of Solids and Structures 51 (2014) 4440–4451 4445

stationary points are not present, indicating that the gel undergoesa continuous phase transition. In addition to stabilizing the phasetransition process, the addition of chromophores also increasesthe phase transition temperature, and this is corroborated by thehigher curves Fig. 4 for higher chromophore concentration.

4.2. Effects of varying light intensity

In photo-thermal gels, the primary function of irradiation is tocreate a temperature change in the temperature sensitive hydro-gel. It might not appear obvious that light intensity plays a directrole in the phase transition process but it does in fact affect thetransition. Experiments have shown that increasing the light inten-sity lowers the transition temperature and increases the tendencyfor the gel to undergo a discontinuous phase transition (Suzuki andTanaka, 1990). Here we will first study the effects of light intensityon the free swelling. To verify that the FE subroutine developed isnumerically correct, we compare the simulation results with ana-lytical results of free swelling by solving Eq. (27).

Fig. 5(a) shows that increasing intensity decreases the phasetransition temperature of the gel. In addition, the change in lightintensity may trigger a change in the mechanism of the phase tran-sition from a continuous transition into a discontinuous one. Forcontinuous phase transitions, the FE analysis is able to fully simu-late the entire transition process. In the case of discontinuousphase transition, the simulations conclude when the turning pointsare reached as the Newton method is not adapted to capture thefull resolution of the turning feature. When held at a constant tem-perature, phase transition takes place as we vary the intensity ofirradiation onto the gel. In Fig. 5(b), we hold temperature constantand vary the light intensity from 0 to 300 mW. The curves obtainedshow that an increase in light intensity decreases the transitiontemperature and also increases the tendency for the gel to undergoa discontinuous phase transition. These computed result showssimilar trends with experimental observations (Suzuki et al.,1996; Suzuki and Tanaka, 1990; Suzuki, 1993).

The change in transition temperature and the existence of tran-sition intensity suggests that in this photo-thermal deformationprocess, irradiation does not simply act as a heating agent toinitiate temperature sensitive deformation, but also has a secondfunction to alter the phase transition pathway.

Fig. 5. Parametric study of effects of changing light intensity on the phase transition p488 nm. The curves show the analytical results, while discrete points show the numtemperature of the gel is varied from 293 to 313 K; and (b) Temperature of gel is kept300 mW.

4.3. Hysteresis

Depending on values of the various factors discussed in thepreceding sections, the swelling–deswelling process of a hydrogelmay undergo hysteresis (Suzuki et al., 1996). From the plotspresented in Fig. 6, we see two different types of curves, one thatis monotonic, as shown in Fig. 6(a), and one that possesses station-ary points, as shown in Fig. 6(b). The former monotonic curverepresents a smooth transition between the different phases,whereas the latter curve suggests signs of hysteresis during theswelling–deswelling process.

It was demonstrated that deformation caused by one stimulus(light or heat) can be reversed by application of the other stimulus(Suzuki et al., 1996). This property provides useful applications inthe area of optical switching where one stimulus can be used toactivate the switch, and the other to deactivate the switch.

4.4. Internal stress in a constrained gel

We now consider a gel, initially at a stress-free state of k0 whichis constrained on both lateral sides and allowed to deform only inthe longitudinal direction. Deformation is triggered by varyinglight intensity at an isothermal condition. In the deformed state,the stretches are k1 ¼ k2 ¼ k0, whereas k3 is a variable. The in-planestresses are r1 = r2 = r while r3 = 0 as the gel is allowed to swellfreely in the 3-direction. With these definitions, Eq. (27) becomes

0 ¼ Nmk2

0k3ðk2

3 � 1Þ þ n ð29aÞ

r ¼ Nmk2

0k3ðk2

0 � 1Þ þ n ð29bÞ

where n ¼ ln 1� 1k2

0k3

!þ 1

k20k3þ v0 � v1

ðk20k3Þ

2 þ2v1

ðk20k3Þ

3

þ maI0

hf2 cðpÞ � cðwÞ ðk2

0k3Þ3 þ cðwÞ

ðk20k3Þ

2

" #� ls

kh

Eq. (29a) gives the longitudinal stretch while Eq. (29b) gives thein-plane stress present in the gel at various light intensities.

ath. The gels of crosslink density Nm = 0.01 are irradiated with light of wavelengtherical results. (a) Light intensity is kept constant at I0 = 0, 40 and 80 mW whileconstant at h0 = 300, 302, 304, 306 and 308 K as light intensity is varied from 0 to

Fig. 6. (a) Continuous phase transition; (b) Discontinuous phase transition exhibiting hysteresis. The arrows above the curve represent the direction of deswelling and arrowsbelow the curve represent the direction of swelling.

4446 W. Toh et al. / International Journal of Solids and Structures 51 (2014) 4440–4451

From Fig. 7, we can see that a photothermal gel at fixed defor-mation experiences a tensile internal stress when exposed to irra-diation. As compared to the case of free swelling, the gel deswellsmore in the case of constrained swelling.

5. Numerical examples

5.1. Coexistent phases of gel deformation

During the deformation process of a gel under external con-straints, a part of the gel may undergo phase transition while otherparts remain in the original phase. This is known as the coexistentphase where the collapsed gel is able to exist with the uncollapsedgel. Although an analytical approach is possible by solving the gov-erning equations (Eshelby, 1956) for temperature sensitive hydro-gels (Cai and Suo, 2011), the phenomenon of coexistence within agel is a complex one and often requires numerical methods to pre-dict the overall self-consistent behavior.

Here we will attempt to employ the developed FE formulationto study this phenomenon. Consider a hydrogel rod immersed inwater (ls,0 = 0), fixed at both ends and originally unexposed to

Fig. 7. A gel in contact with solvent with chemical potential ls,0 = 0 and maintained atintensities. The gel experiences a (a) longitudinal stretch and (b) in-plane stress.

irradiation. As light is being irradiated on the middle portion ofthe rod, the exposed portion starts to undergo phase transition.The process takes place under an isothermal condition of 308 K.Axisymmetric four-node CAX4 elements were used for the mesh.Fig. 8 shows the simulation results which successfully capturesthe state of deformation when the gel is subjected to light withdifferent intensities. Under increasing intensities, the extent ofdeswelling of the exposed rod increases. As the middle portionshrinks, the interface between the two phases experiences tension,which changes the proportion of the radial dimensions of the twophases.

It should be noted, however, that the parameters used in thissimulation causes continuous phase transition, thus allowing thesimulation to be completed. Certain combinations of parameterscause discontinuous phase transition and does not allow finite ele-ment analysis beyond the instability points.

5.2. Buckling due to deswelling phase transition

The wrinkling of hydrogel layers has been studied intensively,mainly in the context of the swelling of neutral hydrogels and

h0 = 300, 302, 304, 306 and 308 K undergoing uni-axial deformation at various light

Fig. 8. Cross-section view of a gel (Nm = 0.005, e = 0.78) subjected to irradiation of different intensities at 308 K. The contour plots show the deformation in the radialdirection.

Fig. 9. Contour plot of vertical displacement, l3of gel rings of (a) Ri/H = 2; (b) Ri/H = 5; and (c) Ri/H = 10. Different wrinkling mode shapes are exhibited for variousradius-to-height ratios. Parameters used are Nm = 0.005, e = 0.78 subjected toirradiation at 308 K.

W. Toh et al. / International Journal of Solids and Structures 51 (2014) 4440–4451 4447

temperature sensitive hydrogels. Moreover, most of earlier studiesare based on the swelling induced instabilities, with most studiesfocusing on thin hydrogel films (Hu et al., 2014; Liu et al., 2011,2010; Mora and Boudaoud, 2006; Yang et al., 2010; Zhang et al.,2014; Toh et al. 2014), surface instabilities (Huang and Suo,2002; Huang et al., 2004; Wu et al., 2013; Zalachas et al., 2013)and bifurcation (Ding et al., 2013; Okumura et al., 2014). In aphoto-thermal gel, a deswelling is more likely to take place asthe gel is suddenly being irradiated with light. There is currentlya lack of literature focusing on the wrinkling of gel layers due todeswelling action. Thin gel discs have been demonstrated to beable to transit into programmable configurations of buckling modeshapes during unconstrained deswelling (Klein et al., 2007).However, the tests were conducted on gels prescribed with

non-homogeneous polymer concentration. In this study, we studythe instabilities that arise in a constrained homogenous gel discand try to bridge this gap for the buckling of a deswelling gel withhomogeneous polymer concentration. Specifically, we will investi-gate the wrinkling induced when a photo-thermal sensitive gel isbeing irradiated.

A thin annulus ring with variable inner radius is studied toinvestigate wrinkling phenomenon due to gel deswelling. Thethickness of the gel is fixed at H and the outer radius of the annulusis fixed at 30H, as shown in Fig. 9. The inner radius is then varied atRi/H = 2, 5 and 10 to investigate the effects of geometry on the sta-bility and wrinkling pattern of the gel. Eight-node brick elementsC3D8 were used in the simulations. To initiate the instability, weintroduce a small perturbation to the mesh in the form of a smalldisplacement to a random node.

With varying values of Ri/H, we observe that the thin annulusbuckles into different mode shapes, as shown in Fig. 9. This showsthat the buckling of gel sheets can be controlled simply by geomet-rical constraints, in addition to prescription of non-homogeneouspolymer concentrations. The presence of instabilities, when well-understood, provides a basis for potential utilization of the variousbuckling modes in a diverse array of applications, such as micro-fluidics, thin film metrology, tunable wetting and particle sorting(Chen and Yang, 2012).

5.3. Application in microvalves

The large deformation present during irradiation of light to aphoto-thermal gel makes it a useful material as a remotely con-trolled flow regulator. Without irradiation exposure, the flow pas-sage is blocked by the gel. Upon irradiation, the gel shrinks thusallowing the liquid to flow across the channel. We study the defor-mation process of such an application using the FE model presentlydeveloped. In this example, we investigate a hybrid hydrogel-membrane microfluidic valve (He et al., 2012; Moore et al.,2000), where a membrane in contact with a hydrogel is used tocontrol flow across an orifice located directly below center of thegel, see Fig. 10.

In this application of the photo-thermal gel, the gel is initiallynot exposed to irradiation, and the gel is in a swollen state which

Fig. 10. (a) Configuration of valve when gel is not being irradiated, gel is swollen and blocks the orifice, restricting flow; (b) Configuration of valve when gel is irradiated, geldeswells and thus allowing flow across orifice.

Fig. 11. Contour plot of the normal displacement of gel with parameters Nm = 0.005 and h0 = 308 K.

4448 W. Toh et al. / International Journal of Solids and Structures 51 (2014) 4440–4451

prevents fluid flow across the orifice. However, as the gel is beingexposed to irradiation, the gel shrinks and thus removing theblockage in the orifice and allows flow. This action of shrinkingand swelling is reversible and thus would be a suitable materialas a flow control valve. In the valve, a gel originally in the dry stateis placed in contact with the membrane. There are no deformationsin the membrane. Next, the gel is being exposed to solvent ofchemical potential of l = 0 to reach the swollen state. This swollenstate inhibits flow in the channel. As the gel is being irradiated,shrinkage occurs and unblocks the orifice, thus allowing for flowacross the channel.

Using axisymmetric four-node elements, Fig. 11 illustrates thesimulation results of the deformation of the gel as it is being irra-diated. The capability of presently developed FE model to capturethe gel deformations across various light intensities is clearly evi-dent. Fig. 11(a) shows the valve being prepared by first setting adry (l0 = �0.5) cylindrical gel in contact with the membrane,while Fig. 11(b) shows the swollen gel at equilibrium with thesolvent of chemical potential l = 0. The swollen gel causes themembrane to block the opening of the orifice, thus restrictingflow. In Fig. 11(c), the gel deswells under 15 mW irradiation,and in Fig. 11(d), it deswells further under 60 mW irradiation, andis restored to a strain-free state. Although similar mechanisms

have been applied using pH-sensitive hydrogels, the slowresponse mechanism of pH-sensitive hydrogels limits their appli-cation, whereas these photo-thermal gels would be a more viableoption as a microfluidic valve.

6. Concluding remarks

This paper proposes a field theory of photo-thermal sensitivegels by incorporating effects of photochemical heating into thethermodynamic theory of neutral and temperature sensitivehydrogels. The phase transition of the gel and its dependence onlight intensity are studied. The field theory is implementedthrough the development of a user-defined subroutine in a finiteelement solver by modeling the material as a hyperelastic material.Verification of the finite element formulation and source code isperformed by comparison with analytical results for the case offree swelling of cubic gel. However, the finite element analysis can-not be completed for cases with discontinuous phase transitions asthe solution method is unable to overcome the turning point of theinstability. Furthermore, through several examples, we demon-strate that the developed field theory and the finite element for-mulation can accurately and efficiently predict the deformation

W. Toh et al. / International Journal of Solids and Structures 51 (2014) 4440–4451 4449

of photo-thermal sensitive gels with complex geometries. This isshown through the analysis of deformation of the gel in coexistentphases, buckling due to the deswelling action and a hybrid hydro-gel-membrane light actuated microfluidic valve.

Acknowledgments

The authors also would like to express their appreciation toNanyang Technological University (NTU) and Institute of HighPerformance Computing (IHPC) for providing all the necessarysupercomputing resources. The authors are also grateful for thesupport from the National Natural Science Foundation of Chinathrough grant numbers 11372236 and 11321062.

Appendix A

Supplementary information for details of the free energy func-tion and its derivatives in the finite element implementation.

Due to singularity in the dry state, we assume an initial freeswelling condition of

F0 ¼k0 0 00 k0 00 0 k0

264375

Relative to the initial free swelling state, the current deforma-tion of the gel is, as measured by ABAQUS, F0. To obtain the actualdeformation gradient of the gel, we write F = F0F0.

Therefore, in all subsequent implementation in ABAQUS, wewrite J ¼ J0k3

0; where J denotes actual swelling ratio and J0 denotesswelling ratio used in ABAQUS. The non-dimensionalized freeenergy density is thus re-written as

mWk¼ h

(12

Nm k20J0

23�I1 � 3� 2 ln k3

0J0 h i

þ k30J0 � 1

� lnk3

0J0 � 1k3

0J0þ v0

k30J0þ v1

k30J0

2

0@ 1A24 35� m

aI0

hfcðpÞ � cðwÞ 1

k60J02

!þ cðwÞ

1k3

0J0

!" #� ~ls k3

0J0 � 1 )

First derivatives

@fW@�I1¼ 1

2Nmk2

0J023

� �h

@fW@�I2¼ 0

@fW@J0¼ 1

2Nm

23

k20J0�

13�I1 �

2J0

� ��

þ k30 ln

k30J0 � 1k3

0J0þ 1

J0þ v0 � v1

k30J02

þ 2v1

k60J03

" #� l�

k30

þ maI0

hfcðpÞ � cðwÞ 2

k60J03

!þ cðwÞ

1k3

0J02

!" #)h

Second derivatives

@2fW@�I2

1

¼ 0

@2fW@�I2

2

¼ 0

@2fW@J02 ¼ �1

9Nmk2

0J0�43�I1 þ

NmJ02

!"

þ k30

J0ðk30J0 � 1Þ

� 1J02� 2ðv0 � v1Þ

k30J03

� 6v1

k60J04

!

� maI0

hfcðpÞ � cðwÞ 6

k60J04

!þ cðwÞ

2k3

0J03

!" ##h

@2fW@�I1@�I2

¼ 0

@2fW@�I1@J

¼ 13

Nmk20J0�

13

� �h

@2fW@�I2@J

¼ 0

Third derivatives

@3fW@�I2

1@J¼ 0

@3fW@�I2

2@J¼ 0

@3fW@�I1@�I2@J

¼ 0

@3fW@�I1@J2 ¼ Nm �1

9k2

0J0�43

� �h

@3fW@�I2

2@J¼ 0

@3fW@J3 ¼

427

Nmk20J0�

73�I1 � 2

NmJ03

� ��þ � k3

0ð2k30J0 � 1Þ

J02ðk30J0 � 1Þ2

þ 2J03þ 6ðv0 � v1Þ

k30J04

þ 24v1

k60J05

!

þ maI0

hfcðpÞ � cðwÞ 24

k60J05

!þ cðwÞ

6k3

0J04

!" ##h

For the initial condition of k0, we solve Eq. (27) at initial tempera-ture h0, initial light intensity I0 and chemical potential l0.

l0

kh¼ Nm

k30

ðk20 � 1Þ þ ln 1� 1

k30

!þ 1

k30

þ v0 � v1

k60

þ 2v1

k90

þ maI0

hfcðpÞ � cðwÞ 2

J3 þ cðwÞ1J2

!" #Due to limitations of the UHYPER subroutine, only one loading

parameter can be applied in the form of a temperature field. There-fore simulations are made by varying one field (either temperature,light intensity or chemical potential) and keeping the other twofixed.

The required loading parameter is thus represented by temper-ature in Abaqus. For example, to simulate effects of change inchemical potential at fixed temperature and intensity, we repre-sent chemical potential as a temperature field, with the fixedvalues of temperature and intensity as material inputs, togetherwith the crosslink density.

4450 W. Toh et al. / International Journal of Solids and Structures 51 (2014) 4440–4451

References

Afroze, F., Nies, E., Berghmans, H., 2000. Phase transitions in the system poly(N-isopropylacrylamide)/water and swelling behaviour of the correspondingnetworks. J. Mol. Struct. 554 (1), 55–68.

Baek, S., Srinivasa, A.R., 2004. Diffusion of a fluid through an elastic solid undergoinglarge deformation. Int. J. Non-Linear Mech. 39 (2), 201–218.

Barrow, J.D., 2002. The Constants of Nature: From Alpha to Omega – The Numbersthat Encode the Deepest Secrets of the Universe. Pantheon Books.

Birgersson, E., Li, H., Wu, S., 2008. Transient analysis of temperature-sensitiveneutral hydrogels. J. Mech. Phys. Solids 56 (2), 444–466.

Bowen, R.M., 1989. An Introduction to Continuum Mechanics for Engineers, vol. 39.Plenum Press, New York.

Cai, S., Suo, Z., 2011. Mechanics and chemical thermodynamics of phase transitionin temperature-sensitive hydrogels. J. Mech. Phys. Solids 59 (11), 2259–2278.

Caykara, T., Kiper, S., Demirel, G., 2006. Network parameters and volume phasetransition behavior of poly(N-isopropylacrylamide) hydrogels. J. Appl. Polym.Sci. 101 (3), 1756–1762.

Chen, C.-M., Yang, S., 2012. Wrinkling instabilities in polymer films and theirapplications. Polym. Int. 61 (7), 1041–1047.

Chester, S.A., Anand, L., 2010. A coupled theory of fluid permeation and largedeformations for elastomeric materials. J. Mech. Phys. Solids 58 (11), 1879–1906.

Chester, S.A., Anand, L., 2011. A thermo-mechanically coupled theory for fluidpermeation in elastomeric materials: application to thermally responsive gels. J.Mech. Phys. Solids 59 (10), 1978–2006.

Ding, Z., Liu, Z.S., Hu, J., Swaddiwudhipong, S., Yang, Z., 2013. Inhomogeneous largedeformation study of temperature-sensitive hydrogel. Int. J. Solids Struct. 50(16–17), 2610–2619.

Duda, F.P., Souza, A.C., Fried, E., 2010. A theory for species migration in a finitelystrained solid with application to polymer network swelling. J. Mech. Phys.Solids 58 (4), 515–529.

Dulkeith, E., Niedereichholz, T., Klar, T.A., Feldmann, J., von Plessen, G., Gittins, D.I.,Mayya, K.S., Caruso, F., 2004. Plasmon emission in photoexcited goldnanoparticles. Phys. Rev. B 70 (20), 205424.

Dunn, M.L., 2007. Photomechanics of mono- and polydomain liquid crystalelastomer films. J. Appl. Phys. 102 (1).

Dunn, M.L., Maute, K., 2009. Photomechanics of blanket and patterned liquid crystalelastomer films. Mech. Mater. 41 (10), 1083–1089.

Eshelby, J.D., 1956. The continuum theory of lattice defects. In: Seitz, F., Turnbull, D.(Eds.), Solid State Physics, vol. 3. Academic Press, ISBN 9780126077032, pp. 79–144. http://dx.doi.org/10.1016/S0081-1947(08)60132-0.

Finkelmann, H., Nishikawa, E., Pereira, G., Warner, M., 2001. A new opto-mechanicaleffect in solids. Phys. Rev. Lett. 87 (1).

Flory, P.J., 1942. Thermodynamics of high polymer solutions. J. Chem. Phys. 10 (1),51–61.

Flory, P.J., 1953. Principles of Polymer Chemistry. Cornell University Press, Ithaca.Gorelikov, I., Field, L.M., Kumacheva, E., 2004. Hybrid microgels photoresponsive

in the near-infrared spectral range. J. Am. Chem. Soc. 126 (49), 15938–15939.Harmon, M.E., Kuckling, D., Pareek, P., Frank, C.W., 2003. Photo-cross-linkable

PNIPAAm Copolymers. 4. Effects of Copolymerization and cross-linking on thevolume-phase transition in constrained hydrogel layers. Langmuir 2 (19),10947–10956.

Haught, A.F., 1984. Physics considerations of solar energy conversion. J. Sol. EnergyEng. 106 (1), 3–15.

He, T., Li, M., Zhou, J., 2012. Modeling deformation and contacts of pH sensitivehydrogels for microfluidic flow control. Soft Matter. 8 (11), 3083–3089.

Herrmann, F., Wurfel, P., 2005. Light with nonzero chemical potential. Am. J. Phys.73 (8), 717–721.

Hong, W., Liu, Z.S., Suo, Z., 2009. Inhomogeneous swelling of a gel inequilibrium with a solvent and mechanical load. Int. J. Solids Struct. 46 (17),3282–3289.

Hong, W., Zhao, X., Suo, Z., 2010. Large deformation and electrochemistry ofpolyelectrolyte gels. J. Mech. Phys. Solids 58 (4), 558–577.

Hong, W., Zhao, X., Zhou, J., Suo, Z., 2008. A theory of coupled diffusion and largedeformation in polymeric gels. J. Mech. Phys. Solids 56 (5), 1779–1793.

Hu, J.Y., He, Y.H., Lei, J.C., Liu, Z.S., Swaddiwudhipong, S., 2014. Mechanical behaviorof composite gel periodic structures with the pattern transformation. Struct.Eng. Mech. 50 (5), 605–616.

Huang, R., Suo, Z., 2002. Instability of a compressed elastic film on a viscous layer.Int. J. Solids Struct. 39 (7), 1791–1802.

Huang, Z., Hong, W., Suo, Z., 2004. Evolution of wrinkles in hard films on softsubstrates. Phys. Rev. E 70 (3), 30601.

Huggins, M.L., 1941. Solutions of long chain compounds. J. Chem. Phys. 9 (5), 440.Huggins, M.L., 1964. A revised theory of high polymer solutions. J. Am. Chem. Soc.

86 (17), 3535–3540.Jiang, H.Y., Kelch, S., Lendlein, A., 2006. Polymers move in response to light. Adv.

Mater. 18 (11), 1471–1475.Kelly, R.E., 1981. Thermodynamics of blackbody radiation. Am. J. Phys. 49 (8),

714–719.Kim, J., Hanna, J.A., Byun, M., Santangelo, C.D., Hayward, R.C., 2012. Designing

responsive buckled surfaces by halftone gel lithography. Science 335 (6073),1201–1205 (New York, N.Y.).

Klein, Y., Efrati, E., Sharon, E., 2007. Shaping of elastic sheets by prescription of non-euclidean metrics. Science 315 (5815), 1116–1120.

Kondo, O., Suzuki, A., Shimazaki, T., 1993. Volume phase transition in polymer gelsinduced by uniaxial stress. J. Intell. Mater. Syst. Struct. 4 (2), 279–282.

Lavergne, J., Joliot, P., 2000. Thermodynamics of the excited states ofphotosynthesis. In: Cramer, W.A. (Ed.), Energy Transduction in Membranes.Biophysical Society.

Lendlein, A., Jiang, H., Junger, O., Langer, R., 2005. Light-induced shape-memorypolymers. Nature 434 (7035), 879–882.

Liu, Z.S., Hong, W., Suo, Z., Swaddiwudhipong, S., Zhang, Y., 2010. Modeling andsimulation of buckling of polymeric membrane thin film gel. Comput. Mater.Sci. 49 (1), S60–S64.

Liu, Z.S., Swaddiwudhipong, S., Cui, F.S., Hong, W., Suo, Z., Zhang, Y.W., 2011.Analytical solutions of polymeric gel structures under buckling and wrinkle. Int.J. Appl. Mech. 03 (02), 235.

Lo, C.-W., Zhu, D., Jiang, H., 2011. An infrared-light responsive graphene–oxideincorporated poly(N-isopropylacrylamide) hydrogel nanocomposite. SoftMatter. 7 (12), 5604–5609.

Long, K.N., Scott, T.F., Dunn, M.L., Jerry Qi, H., 2011. Photo-induced deformation ofactive polymer films: Single spot irradiation. Int. J. Solids Struct. 48 (14–15),2089–2101.

Long, K.N., Scott, T.F., Jerry Qi, H., Bowman, C.N., Dunn, M.L., 2009a. Photomechanicsof light-activated polymers. J. Mech. Phys. Solids 57 (7), 1103–1121.

Long, K.N., Scott, T.F., Qi, H.J., Dunn, M.L., Bowman, C.N., 2009b. Constitutive modelfor photo-mechanical behaviors of photo-induced shape memory polymers7290 (pp. 72900N–72900N–10).

Long, R., Qi, H.J., Dunn, M.L., 2013. Thermodynamics and mechanics ofphotochemcially reacting polymers. J. Mech. Phys. Solids 61 (11), 2212–2239.

Mahimwalla, Z., Yager, K.G., Mamiya, J., Shishido, A., Priimagi, A., Barrett, C.J., 2012.Azobenzene photomechanics: prospects and potential applications. Polym. Bull.69 (8), 967–1006.

Marcombe, R., Cai, S., Hong, W., Zhao, X., Lapusta, Y., Suo, Z., 2010. A theory ofconstrained swelling of a pH-sensitive hydrogel. Soft Matter 6 (4), 784.

Moore, J.S., Bauer, J.M., Yu, Q., Liu, R.H., Devadoss, C., Jo, B., Beebe, D.J., 2000.Functional hydrogel structures for autonomous flow control inside microfluidicchannels. Nature 404 (6778), 588–590.

Mora, T., Boudaoud, A., 2006. Buckling of swelling gels. Eur. Phys. J. E, Soft Matter. 20(2), 119–124.

Okumura, D., Kuwayama, T., Ohno, N., 2014. Effect of geometrical imperfections onswelling-induced buckling patterns in gel films with a square lattice of holes.Int. J. Solids Struct. 51 (1), 154–163.

Parson, W.W., 1978. Thermodynamics of the primary reactions of photosynthesis.Photochem. Photobiol. 28 (3), 389–393.

Pastoor, K.J., Rice, C.V., 2012. Anion effects on the phase transition of N-isopropylacrylamide hydrogels. J. Polym. Sci., Part A: Polym. Chem. 50 (7),1374–1382.

Richardson, H.H., Carlson, M.T., Tandler, P.J., Hernandez, P., Govorov, A.O., 2009.Experimental and theoretical studies of light-to-heat conversion and collectiveheating effects in metal nanoparticle solutions. Nano Lett. 9 (3), 1139–1146.

Ries, H., McEvoy, A.J., 1991. Chemical potential and temperature of light. J.Photochem. Photobiol., A 59 (1), 11–18.

Rohatgi, K.K., 2006. Fundamentals of Photochemistry. New Age InternationalPublishers, New Delhi.

Saphiannikova, M., Toshchevikov, V., Ilnytskyi, J., 2009. Photoinduced deformationsin azobenzene polymer films. Nonlinear Opt. Quantum Opt. 41, 27–57.

Saphiannikova, M., Toshchevikov, V., Ilnytskyi, J., Heinrich, G., 2011. Photo-induceddeformation of azobenzene polymers: theory and simulations. In: SPIEProceedings, 8189, pp. 10–15.

Sershen, S.R., Westcott, S.L., Halas, N.J., West, J.L., 2000. Temperature-sensitivepolymer–nanoshell composites for photothermally modulated drug delivery. J.Biomed. Mater. Res. 51 (3), 293–298.

Serway, R.A., Jewett, J.W., 2013. Physics for Scientists and Engineers, eighth ed.Brooks/Cole Cengage Learning.

Suzuki, A., 1993. Phase transition in gels of sub-millimeter size induced byinteraction with stimuli. In: Dušek, K. (Ed.), Responsive Gels: VolumeTransitions II SE - 9, Vol. 110. Springer, Berlin Heidelberg, pp. 199–240.

Suzuki, A., Ishii, T., 1999. Phase coexistence of neutral polymer gels undermechanical constraint. J. Chem. Phys. 110 (4), 2289–2296.

Suzuki, A., Ishii, T., Maruyama, Y., 1996. Optical switching in polymer. J. Appl. Phys.80 (1), 131–136.

Suzuki, A., Kojima, S., 1994. Phase transition in constrained polymer gel. J. Chem.Phys. 101, 10003–10007.

Suzuki, A., Tanaka, T., 1990. Phase transition in polymer gels induced by visiblelight. Nature 346 (6282), 345–347.

Takashima, Y., Hatanaka, S., Otsubo, M., Nakahata, M., Kakuta, T., Hashidzume, A.,Harada, A., 2012. Expansion–contraction of photoresponsive artificial muscleregulated by host-guest interactions. Nature Commun. 3, 1270.

Tanaka, T., 1978. Collapse of gels and the critical endpoint. Phys. Rev. Lett. 40 (12),820–823.

Toh, W., Liu, Z.S., Ng, T.Y., 2013a. Non-linear large deformation kinetics of polymericgel. Key Eng. Mater. 535–536, 338–341.

Toh, W., Liu, Z.S., Ng, T.Y., Hong, W., 2013b. Inhomogeneous large deformationkinetics of polymeric gels. Int. J. Appl. Mech. 05 (01), 1350001.

Toh, W., Ng, T.Y., Liu, Z.S., Hu, J.Y., 2014. The deformation kinetics of pH-sensitivehydrogels. Polym. Int. 63 (9), 1578–1583.

Toshchevikov, V.P., Saphiannikova, M., Heinrich, G., 2012. Theory of light-induceddeformation of azobenzene elastomers: influence of network structure. J. Chem.Phys. 137 (2), 24903–24913.

W. Toh et al. / International Journal of Solids and Structures 51 (2014) 4440–4451 4451

Toshchevikov, V., Saphiannikova, M., Heinrich, G., 2011. Light-induced deformationof azobenzene elastomers: a regular cubic network model. J. Phys. Chem. B 116(3), 913–924.

Truesdell, C., 1969. Rational Thermodynamics. McGraw-Hill.Walker, J.S., 2009. Physics, fourth ed. Prentice Hall.Wang, X., 2007. Modeling the nonlinear large deformation kinetics of volume phase

transition for the neutral thermosensitive hydrogels. J. Chem. Phys. 127(174705), 1–8.

Wu, Z., Bouklas, N., Huang, R., 2013. Swell-induced surface instability of hydrogellayers with material properties varying in thickness direction. Int. J. SolidsStruct. 50 (3–4), 578–587.

Wurfel, P., 1982. The chemical potential of radiation. J. Phys. C: Solid State PhysicsEmail Alert RSS Feed 15 (18), 3967–3985.

Yang, S., 2008. Photosensitive hydrogels. In: Li, D. (Ed.), Encyclopedia ofMicrofluidics and Nanofluidics SE - 1227. Springer, US, pp. 1643–1647.

Yang, S., Khare, K., Lin, P.-C., 2010. Harnessing surface wrinkle patterns in softmatter. Adv. Funct. Mater. 20 (16), 2550–2564.

Yoon, J., Bian, P., Kim, J., McCarthy, T.J., Hayward, R.C., 2012. Local switching ofchemical patterns through light-triggered unfolding of creased hydrogelsurfaces. Angew. Chem. Int. Ed. 51 (29), 7146–7149.

Zalachas, N., Cai, S., Suo, Z., Lapusta, Y., 2013. Crease in a ring of a pH-sensitivehydrogel swelling under constraint. Int. J. Solids Struct. 50 (6), 920–927.

Zhang, Y., Chen, L., Swaddiwudhipong, S., Liu, Z.S., 2014. Buckling deformation ofannular plates describing natural forms. Int. J. Struct. Stab. Dyn. 14 (1),1350054.

Zhu, C.-H., Lu, Y., Peng, J., Chen, J.-F., Yu, S.-H., 2012. Photothermally sensitivePoly(N-isopropylacrylamide)/graphene oxide nanocomposite hydrogels asremote light-controlled liquid microvalves. Adv. Funct. Mater. 22 (19), 4017–4022.