Effect of surface tension on swell-induced surface instability of substrate-confined hydrogel layers

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1 PAPER CREATED USING THE RSC ARTICLE TEMPLATE (VER. 3.) - SEE FOR DETAILS XXXXXXXX 5 5 Effect of surface tension on swell-induced surface instability of substrate-confined ydrogel layers Min K. Kang and Rui Huang* Received (in XXX, XXX) Xt XXXXXXXXX X, Accepted Xt XXXXXXXXX X First publised on te web Xt XXXXXXXXX X DOI:.39/bx Swell-induced surface instability as been observed experimentally in rubbers and gels. Here we present a teoretical model tat predicts te critical condition along wit a caracteristic wavelengt for swell-induced surface instability in substrate-confined ydrogel layers. Te effect of surface tension is found to be critical in suppressing sort-wavelengt modes of instability, wile te substrate confinement suppresses long-wavelengt modes. Togeter, an intermediate wavelengt is selected at a critical swelling ratio for te onset of surface instability. Bot te critical swelling ratio and te caracteristic wavelengt depend on te initial tickness of te ydrogel layer as well as oter material properties of te ydrogel. It is found tat te ydrogel layer becomes increasingly stable as te initial layer tickness decreases. A critical tickness is predicted, below wic te ydrogel layer swells omogeneously and remains stable at te equilibrium state Introduction Subjected to mecanical compression, te surface of a rubber block becomes unstable at a critical strain, beyond wic sarp folds or creases appear on te surface., Similar surface instability as been observed in swelling rubber vulcanizates 3 and polymer gels. 4-7 Suc instability may pose a fundamental limit on load-carrying capacity of rubber or on te degree of swelling for gels. On te oter and, te pysics of surface instability may be arnessed in te design of responsive smart surfaces for novel applications. 8,9 In bot cases, teoretical understanding on te critical condition of surface instability as well as subsequent surface pattern evolution is essential. By considering a alf-space of an incompressible neo- Hooken elastic material under mecanical compression, Biot predicted tat te surface becomes unstable at a critical level of compression. Later, it was found tat Biot s prediction considerably overestimated te critical strain for surface creasing in rubber blocks deformed by bending. A recent paper by Hong et al. presented an alternative analysis of surface ceasing in rubber based on an energetic consideration and suggested tat creasing is a different mode of surface instability in contrast wit Biot s periodic surface wave analysis. Several teoretical models ave also been proposed for swelling induced surface instability in gels. 4, -4 At least two questions remain open for gels: () Wat is te critical condition for swelling induced surface instability? () Wat determines te caracteristic size of surface patterns formed beyond te critical point? For te first question, a critical osmotic pressure was suggested based on a composite beam model. 4 Based on teir experiments wit a ydrogel layer on a rigid substrate, Trujillo et al. 6 suggested a critical linear compressive strain of ~33%, or a critical swelling ratio of ~. Hong et al predicted a critical swelling ratio of ~.4. Experimentally, a wide range of critical swelling ratios were observed for different gel systems, from around to , 5, 6 In a previous study, 4 we performed a linear perturbation analysis for surface instability of substrate-confined ydrogel layers and predicted a range of critical swelling ratios from below.5 to 3.4, depending on te material properties of te ydrogel system. However, two key effects were not considered. First, depending on te kinetics of swelling, te critical swelling ratio for onset of surface instability can potentially be lower due to inomogeneous transient state of swelling. Second, te effect of surface energy or surface tension tends to stabilize te flat surface, wic could lead to larger critical swelling ratios, especially for very tin layers. For te second question, several experimental studies ave reported caracteristic wavelengts proportional to te initial tickness of te gel. 4, 6-8 As discussed in te previous study, 4 wile te substrate confinement indeed suppresses longwavelengt modes of te surface instability, te confinement effect alone does not lead to a finite caracteristic wavelengt since te sort-wavelengt modes are unaffected by te substrate. Two possible mecanisms may be important in addressing tis question. For one, te caracteristic wavelengt may be dynamically determined by te swelling kinetics, similar to wrinkling of an elastic tin film on a viscoelastic substrate. 5-7 Second, te sort-wavelengt modes of surface instability may be suppressed by surface tension of te ydrogel, wic togeter wit te substrate confinement effect would result in a finite wavelengt for te surface instability. Te present study focuses on te effects of surface tension on bot te critical condition and te caracteristic wavelengt. Along wit a separate study on te effects of swelling kinetics, we aim to establis a vigorous teoretical understanding on swell-induced surface instability of substrate-confined ydrogels. Te remainder of tis article is Tis journal is Te Royal Society of Cemistry [year] Journal Name, [year], [vol],

2 χ ˆ μ pv ln Nv λ λ λ λ kbt (.) (a) were te material properties of te ydrogel are represented by two dimensionless parameters (Nv and χ). Te external cemical potential in general is a function of te temperature (T) and pressure (p). Assuming an ideal gas pase (p < p ) and an incompressible liquid pase (p > p ) for te solvent, te 4 cemical potential is given by ˆ μ ( p, T ) ( p p) v, kb T ln ( p / p ), if p > p ; if p < p, (.) (b) (c) Fig. Scematic illustrations of substrate-confined ydrogel layers. (a) Homogeneous swelling; (b) Onset of swell-induced surface instability; (c) formation of surface creases. organized as follows. Section presents an analytical solution for omogeneous swelling of a ydrogel layer on a rigid substrate, based on a specific material model of neutral polymer ydrogels. A linear perturbation analysis is performed in Section 3 to predict onset of surface instability for a omogeneously swollen ydrogel layer. Te effect of surface tension is taken into account troug te boundary condition. Te results are discussed in Section 4. In addition to te critical swelling ratio and te caracteristic wavelengt, a critical tickness is predicted for tin ydrogel layers, wose surface becomes increasingly stable as te initial layer tickness decreases.. Homogeneous swelling of a ydrogel layer Consider omogeneous swelling of a ydrogel layer attaced to a rigid substrate (Fig. a), were a Cartesian coordinate system is set up suc tat x at te ydrogel/substrate interface. At te dry state, te tickness of te ydrogel layer is, wile te oter two dimensions are assumed to be infinite. Confined by te substrate, te ydrogel layer swells only in te tickness direction wit a omogeneous swelling ratio, λ / >, were is te tickness of te swollen ydrogel. Te omogeneous swelling ratio can be determined by minimizing te total free energy of te ydrogel system. 4 Using a particular form of te Flory-Huggins free energy function for te ydrogel, 8- te omogeneous swelling ratio (λ ) is obtained as a function of te cemical potential ( μˆ ) of te external solvent by Eq. (.): were p is te equilibrium vapor pressure of te solvent, v is te volume per solvent molecule, and k B is te Boltzmann constant. At a constant temperature (T), Eq. (.) predicts tat te ydrogel layer swells increasingly as te cemical potential increases until it reaces an equilibrium value at te equilibrium vapor pressure of te solvent (i.e., p p and ˆ μ ). In experiments, owever, te equilibrium swelling 5 ratio may be reaced by immersing te ydrogel in an aqueous solution for a sufficiently long time, wile te transient state of swelling may be inomogeneous. In eiter case, te equilibrium swelling ratio is a function of temperature, depending on tree dimensionless parameters, p pv /( kbt ), Nv, and χ. Te first parameter depends on te solvent only. For water at 5ºC, p ~ 3. kpa, v ~ 3-9 m 3, 5 and tus p ~.3, wic is relatively small and may be approximately taken to be zero. Te second parameter depends on te polymer network of te ydrogel, wit N 6 being te effective number of polymer cains per unit volume of te ydrogel at te dry state. Te polymer network as an initial sear modulus, G Nk B T, wit typical values from. kpa to kpa. 6 Tus, te value of Nv ranges from -6 to - 3. Te tird parameter, χ, is a dimensionless quantity caracterizing te interaction between te solvent molecules and te polymer, wic as been measured and tabulated for many gels. 8 Te value of χ also differentiates a good solvent 5 (χ <.5) from a poor solvent. Taking p.3, Nv -3, and χ.4, we obtain from Eq. (.) te equilibrium 7 swelling ratio, λ Te equilibrium swelling ratio decreases wit increasing values of Nv or χ. Upon swelling, te confinement by te substrate induces a compressive stress in te ydrogel layer. As in te previous studies, 4, te nominal stresses are obtained as a function of 75 te swelling ratio: 8 s B ( λ ) pλ s s Nk T 33 (.3) and s p. Te true (Caucy) stresses at te swollen state are related to te nominal stresses as σ σ 33 s / λ NkBT( λ / λ ) p and σ s p. Tus, te pressure inside te ydrogel is p in 3 ( σ ) σ σ 33 p NkBT λ 3 λ (.4) It is noted tat te internal pressure of te ydrogel is iger Journal Name, [year], [vol], Tis journal is Te Royal Society of Cemistry [year]

3 tan te external pressure (p) of te solvent, analogous to te osmosis penomenon (i.e., te ydrogel is ypertonic and te external solvent is ypotonic). As te surface of te ydrogel layer is assumed to remain 5 flat during te omogeneous swelling (Fig. a), te presence of a surface tension or surface energy does not ave any effect on te one-dimensional (-D) omogeneous swelling of te ydrogel. We note tat tis is not te case for un-constrained tree-dimensional (3-D) swelling of a ydrogel particle in a solvent, for wic surface tension does ave an effect as te surface area increases during swelling, especially for smallscale ydrogel particles. In te present study, we sow tat, despite its no-effect on te omogeneous swelling, surface tension plays a critical role in swell-induced surface 5 instability of substrate-confined ydrogel layers Linear perturbation analysis Previously we ave performed a linear perturbation analysis of te omogeneous solution to predict swell-induced surface instability witout considering te effect of surface tension. 4 Following te same procedure, we present ere a linear perturbation analysis wit te effect of surface tension. As illustarted in Fig., once te surface becomes unstable, it may evolve from a smoot undulation (Fig. b) to form localized foldings and surface creases (Fig. c). For a linear stability analysis, a two-dimensional perturbation is assumed wit small displacements from te omogeneously swollen state, namely u u ( x ) and u ( x ), x u. (3.), x Suc a perturbation leads to an inomogeneous deformation of te ydrogel along wit re-distribution of te solvent concentration inside te gel (Fig. b). As a result, te stress field becomes inomogeneous as well. To te leading order of te perturbation, te nominal stresses in te ydrogel are obtained as follows: were ij B [ FiJ ( λ ξε ) HiJ ] phij s Nk T ~ (3.) ξ u ~ u F H ij u λ u λ (3.3) ~ ~ eijkejklfjkf (3.4) kl ε u u χ Nν λ λ λ λ (3.5) (3.6) In te above equations, F ~ is te deformation gradient tensor 5 relative to te dry state, e ijk is te permutation tensor, and te repeated indices implies summation. Te lowercase indices (i, j, k) refer to te coordinates at te swollen state, wile te uppercase indices (J, K, L) refer to te dry state. Apparently, te linearized stress-strain relationsip for te swollen ydrogel layer, Eq. (3.), is anisotropic, a result due to te anisotropic deformation of te polymer network during -D omogeneous swelling. For te ydrogel to maintain mecanical equilibrium, te stress field must satisfy te equilibrium equations, namely s s s λ x s λ x (3.7) (3.8) In terms of te perturbation displacements, te equilibrium equations become x ( λ ξ ) λ λ ξ x λ x ( ξ λ ) λ ξ (3.9) (3.) In addition, te stress field must satisfy te boundary conditions. Assume a liquid-like surface tension (γ) for te ydrogel. Te perturbed surface as a curvature, κ /, to te first-order approximation. By te classical Young- Laplace equation, te normal stress at te surface of te ydrogel layer is balanced by te capillary pressure due to surface tension and te external pressure, namely, s p γ at x (3.) were λ is te tickness of te ydrogel layer at te swollen state. On te oter and, te sear stress is zero on te surface, i.e., s. At te ydrogel/substrate interface, we assume perfect bonding wit zero displacements, i.e., u u at x. By te metod of Fourier transform we solve te equilibrium equations (3.9)-(3.) and obtain tat 4 ( n) ( x; k) Anu ( qnx ) uˆ exp n 4 ( n) ( x; k) Anu ( qnx ) uˆ exp n were u ˆ ( x; k) and u ˆ ( x; k) displacements, u ( x ) and u ( x ) (3.) (3.3) are te Fourier transforms of te, x, x, wit respect to x, and k is te Fourier wave number. Te four eigenvalues are obtained from te equilibrium equations as k q, ± and q3,4 ± βk, (3.4) λ were β ( λξ )/( λ λξ ). Correspondingly, we ave te eigenvectors, Tis journal is Te Royal Society of Cemistry [year] Journal Name, [year], [vol], 3

4 5 () () (3) (4) u u u u (3.5) () () (3) (4) u u u u iλ iλ iβ iβ We note tat, for eac Fourier component wit a specific wave number k, te perturbation displacement is periodic in te x direction, but varies exponentially in te x direction for eac eigen mode, similar to Biot s analysis for surface instability of a alf-space rubber-like material under mecanical compression. Next applying te boundary conditions, we obtain tat 4 n D (3.6) mn A n were te coefficient matrix is given by (a) λ D p e λ k ( λ p λ kl) e ( λ p λ kl) k β p βkl e λ p βk βe λ βk λ e p e λ k k β p βkl e λ p βk βe λ βk (3.7) wit p p /( NkBT ) and L γ /( NkBT). Te critical condition for swell-induced surface instability of te ydrogel layer is ten obtained by setting te determinant of te matrix D to be 5 zero, namely 5 3 Det L [ D] f k, λ,, Nv, χ, p (3.8) We postpone te discussions on te results of te linear perturbation analysis till next section. Here we note tat te ratio between te surface tension (γ) and te bulk sear modulus ( G NkBT ) of te polymer network defines an intrinsic lengt scale (L). A similar lengt scale appeared in a critical condition tat predicts te maximum pressure for cavitation in ydrogels. Alternatively, a lengt scale can be defined wit respect to te molecular volume of solvent, i.e., L' γv /( kbt), wic is independent of te polymer network. Take te surface tension of te ydrogel to be similar to tat of water. At te room temperature, γ ~. 73 N/m and v ~ 3-9 m 3, we ave L' ~. 53 nm, wile te lengt L can vary over several orders of magnitude (from nanometers to micrometers) depending on te value of Nv. It turns out tat te same lengt scale (L) could result in a size-dependent swelling ratio for nanoscale ydrogel particles. In te present study, we sow tat te presence of suc a lengt scale leads to a tickness-dependent critical condition for swell-induced surface instability of substrate-confined ydrogel layers (b) Fig. (a) Critical swelling ratio and (b) critical cemical potential, versus te perturbation wavelengt for swell-induced surface instability of substrate-confined ydrogel layers wit Nv -3, χ.4, and L.53 μm. Te tick dased lines sow te results witout te effect of surface tension, and te tin dased line in (a) indicates te omogeneous swelling ratio at equilibrium. 4. Results and discussions Te linear perturbation analysis predicts a critical condition, Eq. (3.8), for onset of swell-induced surface instability of substrate-confined ydrogel layers. As plotted in Fig a, te predicted critical swelling ratio ( λ c ) is a function of te normalized perturbation wavelengt, S π /( k ), depending on te initial layer tickness ( ) as well as te material properties (Nv, χ, and p ) of te ydrogel. Figure b plots te corresponding critical cemical potential ( μ c ), wic is related to te critical swelling ratio by Eq. (.). For tese calculations, we assume a specific ydrogel system wit Nv -3 5, χ.4, p.3, and L. 53 µm. For comparison, te tick dased lines in Fig. sow te results from te previous analysis wit no surface effect, wic are independent of te layer tickness. As expected, te surface tension tends to stabilize sort-wavelengt perturbations, leading to increasingly larger critical swelling ratio as te wavelengt decreases. Togeter wit te effect of substrate confinement, wic suppresses te long-wavelengt perturbations, te critical swelling ratio as a minimum ( λ * c ) * * at an intermediate wavelengt ( S ); bot λ * c and S depend on 4 Journal Name, [year], [vol], Tis journal is Te Royal Society of Cemistry [year]

5 (a) Fig. 4 Te critical tickness of substrate-confined ydrogel layers, predicted as a function of Nv for various values of χ. Te lengt scale L is assumed to be a constant (L.53 nm). found tat te caracteristic wavelengt as predicted ere is not exactly proportional to te layer tickness. Instead, it appears to approximately follow a power-law scaling, S * (4.) α L α (b) Fig. 3 (a) Te minimum critical swelling ratio and (b) te corresponding caracteristic wavelengt, versus te initial tickness of te substrateconfined ydrogel layers wit Nv -3 and χ.4. Te lower dased line in (a) sows te tickness-independent critical swelling ratio witout te effect of surface tension, and te upper dased line indicates te omogeneous swelling ratio at equilibrium. Te dased line in (b) sows *.9 te power-law scaling of te caracteristic wavelengt, S ~. te initial tickness ( ) of te ydrogel layer. Figure 3a plots te minimum critical swelling ratio ( λ * c ) as a function of te initial tickness ( ) of te ydrogel layer, and Fig. 3b plots te corresponding wavelengt ( S * ). To illustrate te effect of surface tension, te results are sown for different values of te lengt scale L. As a dimensionless quantity, te minimum critical swelling ratio depends on te ratio between te two lengts, /L. For a relatively tick ydrogel layer, te surface tension as negligible effect, and tus te critical swelling ratio approaces te previous prediction (te lower dased line), wic is independent of te layer tickness. As te layer tickness decreases, te effect of surface tension becomes increasingly important, and te critical swelling ratio increases until it reaces te equilibrium omogeneous swelling ratio (te upper dased line) of te ydrogel layer at a critical tickness ( c ). For a tinner ydrogel layer ( < c ), te omogeneous swelling is stable up to te equilibrium cemical potential ( ˆ μ ). Corresponding to te minimum critical swelling ratio, te caracteristic wavelengt ( S * ) decreases monotonically as te layer tickness decreases (Fig. 3b), in qualitative agreement wit experimental observations. 4, 6-8 However, it is wit a positive exponent α, over a wide range of te layer tickness. As sown by te dased line in Fig. 3b, α. for Nv -3 and χ.4. Te positive exponent (α ) suggests tat te caracteristic wavelengt increases as te surface tension of te ydrogel increases. Remarkably, te exponent is found to be insensitive to te oter material properties of te ydrogel, wit nearly identical value of α for different Nv and χ. It is of interest to note tat, wile te surface tension as no effect on omogeneous swelling of a substrate-confined ydrogel layer, it plays a critical role in stabilizing tin ydrogel layers. In particular, te critical tickness ( c ) for a stable ydrogel layer is linearly proportional to te lengt scale L, wic in turn is proportional to te surface tension γ. Similar critical tickness exists for surface instability of epitaxial crystal tin films. 3 Figure 4 plots te critical tickness ( c ) as a function of Nv for different values of χ, assuming a constant value for L. Te critical tickness depends on Nv troug two competing factors. First, for a constant surface tension (γ) and solvent molecular volume (v), because te lengt scale L decreases wit increasing Nv, te critical tickness tends to decrease as well. On te oter and, since te polymer network of te ydrogel becomes increasingly stiff as Nv increases, te degree of swelling decreases. Wit less swelling, te ydrogel layer is more stable, and te critical tickness tends to increase. Consequently, te ratio between te critical tickness and te lengt scale, c /L, increases monotonically wit increasing Nv. Wen Nv is relatively small, te effect of surface tension dominates, and te critical tickness decreases wit increasing Nv. Te trend is reversed as te elasticity of polymer network becomes significant wit relatively large Nv. On te oter and, te dependence of te critical tickness on χ is simpler. As χ increases, te degree of swelling decreases and te Tis journal is Te Royal Society of Cemistry [year] Journal Name, [year], [vol], 5

6 critical tickness increases. For eac χ, tere exists a maximum value for Nv, beyond wic te critical tickness is essentially infinity. Tis again is attributed to limited degree of swelling, wit wic te ydrogel layer of any tickness would remain stable at te equilibrium state. Terefore, te critical condition for swell-induced surface instability of te substrate-confined ydrogel layer is largely determined by te tree dimensionless parameters: / L, Nv, and χ. As sown in Fig. 4, for typical values of Nv and χ, te critical tickness ranges between nm and µm. As noted in te previous study, 4 te present stability analysis assumes a quasi-statically controlled swelling process, were te ydrogel layer swells omogeneously until te onset of surface instability. As suc, te effect of swelling kinetics as been ignored. In experiments, wen a ydrogel layer is immersed in a solvent, te transient state of swelling is typically inomogeneous and te stability condition depends on te kinetics. 4,4 Tree scenarios may occur. First, te ydrogel layer remains stable and swells omogeneously up to te equilibrium state. Tis was observed for gels wen te degree of swelling is relatively small. 4 Second, te ydrogel layer becomes unstable and develops surface creases during te transient swelling process. Eventually as te ydrogel layer reaces te equilibrium state, te surface creases disappear, and te equilibrium state of omogeneous swelling is stable. 4 Tird, te surface creases develop and evolve during te transient process, and remain at te equilibrium state, 6 suggesting tat te omogeneous swelling is unstable at te equilibrium state. Te critical condition for surface instability as developed in te present study predicts weter te equilibrium state of omogeneous swelling is stable, but does not predict te onset of surface instability during te transient process. It is speculated tat te critical swelling ratio for surface instability could be considerably lower for inomogeneous swelling at te transient state. Te detailed analysis on te effect of swelling kinetics is left for a separate study. Anoter interesting point to note is te effect of temperature on swell-induced surface instability of ydrogels. Witin te present model, we see several possible effects tat depend on temperature. First, in Eq. (.), te omogeneous swelling ratio depends on temperature troug te normalized vapor pressure, p pv /( kbt ), were p itself is a function of temperature. In addition, te oter material properties (N, χ, and v) may all depend on temperature. Experimentally it as been observed tat polymer gels may undergo continuous or discontinuous volume pase transition as te temperature canges, 4 suggesting possible canges in te structure of te polymer network as well as te interaction between te polymer and te solvent molecules. In te stability analysis, as te surface tension may depend on temperature, te predicted critical swelling ratio, te caracteristic wavelengt, and te critical layer tickness all depend on temperature. It is tus possible tat a ydrogel layer is stable at one temperature but becomes unstable at a different temperature. In addition, it is well known tat te kinetics of mass transport and swelling is typically sensitive to temperature. Terefore, te effect of temperature on swell-induced surface instability of ydrogels is in general complicated wit convolution of multiple effects 6 on te material parameters and pysical processes Concluding remarks Tis paper presents a teoretical analysis on swell-induced surface instability of substrate-confined ydrogel layers. In particular, te effect of surface tension is igligted in comparison wit a previous study 4 tat considered te effect of substrate confinement alone. Wit bot surface tension and substrate confinement, we sow tat te stability of a ydrogel layer depends on its initial tickness. A critical tickness is tus predicted, wic is proportional to te surface tension and depends on te oter material parameters of te ydrogel. Te onset of surface instability is predicted at a caracteristic wavelengt wit te minimum critical swelling ratio. An approximate power-law scaling for te caracteristic wavelengt is suggested. Te minimum critical swelling ratio decreases as te layer tickness increases, depending on te ratio between te two lengt scales ( / L ) and approacing a constant at relatively large tickness. Finally we note tat te linear perturbation analysis in te present study assumes a smoot surface perturbation onto a omogeneously swollen ydrogel layer at te onset of surface instability. Tis is in te same spirit as Biot s analysis on surface instability of a alf-space rubber under mecanical compression, but different from te energetic analysis by Hong et al. As sown by numerical simulations in our previous study, 4 te smoot surface perturbation can subsequently evolve to form localized features suc as grooves and creases (Fig. c), as a result of nonlinear postinstability evolution. More studies, bot teoretically and experimentally, are needed to furter elucidate te nonlinear process of swell-induced surface evolution as well as te relationsip between te two types of surface instability patterns. Acknowledgments Te autors gratefully acknowledge financial support by 95 National Science Foundation troug Grant No Notes and references Department of Aerospace Engineering and Engineering Mecanics, University of Texas at Austin, Austin, TX 787. Tel: ; ruiuang@mail.utexas.edu Tis paper is part of a Soft Matter temed issue on Te Pysics of Buckling. Guest Editor: Alfred J. Crosby. A. N. Gent and I. S. Co, Rubber Cemistry and Tecnology, 999, 7, A. Gatak and A.L. Das, Pys. Rev. Lett., 7, 99, E. Soutern and A.G. Tomas, J. Polymer Science A, 9, 3, T. Tanaka, S.-T. Sun, Y. Hirokawa, S. Katayama, J. Kucera, Y. Hirose, T. Amiya, Nature, 987,, H. Tanaka, H. Tomita, A. Takasu, T. Hayasi, T. Nisi, Pys. Rev. Lett., 99, 68, Journal Name, [year], [vol], Tis journal is Te Royal Society of Cemistry [year]

7 V. Trujillo, J. Kim, R. C. Hayward, Soft Matter, 8, 4, M. Guvendiren, S. Yang, J.A. Burdick, Adv. Funct. Mater., 9, 9, E.P. Can, E.J. Smit, R.C. Hayward, A.J. Crosby, Advanced Materials, 8,, J. Kim, J. Yoon, R.C. Hayward, Nature Materials,, 9, M.A. Biot, Appl. Sci. Res. A, 963,, W. Hong, X. Zao, Z. Suo, Appl. Pys. Lett., 9, 95, 9. A. Onuki, Pys. Rev. A, 989, 39, N. Suematsu, N. Sekimoto, K. Kawasaki, Pys. Rev. A, 99, 4, M.K. Kang and R. Huang, J. Mec. Pys. Solids, submitted. Preprint available online at ttp:// 5 R. Huang and Z. Suo, J. Appl. Pys.,, 9, R. Huang, J. Mec. Pys. Solids, 5, 53, R. Huang and S.H. Im, Pys. Rev. E, 6, 74, P.J. Flory, Principles of Polymer Cemistry, Cornell University Press, Itaca, NY, W. Hong, X. Zao, J. Zou, Z. Suo, J. Mec. Pys. Solids 8, 56, M. K. Kang and R. Huang, J. Appl. Mec.,, in press. Preprint available online at ttp:// R. Huang, unpublised work. S. Kundu and A.J. Crosby, Soft Matter, 9, 5, Y. Pang and R. Huang, Pys. Rev. B, 6, 74, Y. Li and T. Tanaka, Annu. Rev. Mater. Sci., 99,, Tis journal is Te Royal Society of Cemistry [year] Journal Name, [year], [vol], 7