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# Swell induced surface instability of confined hydrogel layers

A previous work suggested a critical condition to form surface creases in elastomers and gels. For elastomers, the critical condition seems to have closed a gap between experimental observations (e.g., by bending a rubber block) and the classical instability analysis by Biot. For gels, however, experiments have observed a wide range of critical swelling ratios, from around 2 to 3.7. Here we present a linear perturbation analysis for swollen hydrogels confined on a rigid substrate, which predicts critical swelling ratios in a similar range.

The critical condition for swell induced surface instability, which may be expressed by different critical quantities (chemical potential, external pressure, compressive stress, or linear strain), depends on the material properties of the hyrogel systems in general. Using a nonlinear finite element method, we show by numerical simulations that an initial smooth perturbation can evolve to form localized grooves and creases on the hydrogel surface. In addition, pressure induced surface instability is predicted with a critical pressure greater than the equilibrium vapor pressure of the solvent. While the present study assumes a quasistatic swelling process, we believe that kinetics plays a critical role in most experiments, which will be considered in a future study.

The attached manuscript has been submitted for review. Comments and discussions are welcome.

Update on July 15, 2010: The revised manuscript has been accepted for publication in Journal of the Mechanics and Physics of Solids. DOI: 10.1016/j.jmps.2010.07.008.

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## Comments

## correction

A revised version of the manuscript, with a correction to the boundary condition, has been uploaded to replace the earlier version.

RH

## Swell induced surface instability of confined hydrogel layers

Dear Professor Rui. I have been studying your paper for several months (and I have to say that I have enjoyed it) I have some questions in regards to the Finite Element Model.

1) What was the size of the domain?

2) How many quad elements did you use? How many different meshes did you use for the convergence analysis?

3) What was the wave amplitude you used for the perturbation? Was it smaller than the element size?

4) Did you use any stablization technique to traverse the crease formation?

5) Why a hyperelastic material subroutine is not able to handle an anisotropic homogeneous initial state? Your explanation in the paper is not clear to me, yet.

On the other hand, Does Figure 4(a) show that the critical swelling ratio is indepent of the perturbation wave number?

Cordially,

Mario Juha

## my answers to Mario

Dear Mario,

Thanks for your interest in our work. I am sorry that the paper is not clear enough for you. Since the student has graduated and left, let me try to answer your questions as follows.

1) Supposing you mean the computational domain in Fig. 8, it should be clear from the figure that shows the entire domain. Since the problem does not have any length scale other than the initial thickness (h_0), all lengths are normalized by h_0. In ABAQUS, set h_0 = 1 and the length is 10 as you can see more clearly from Fig. 9. So the domain size at the dry state is 10x1.

2) I do not know the exact number of elements in the model. I recall that the mesh was finer near the surface and increasingly coarse away from the surface. For a rough idea, the final crease depth you see in Fig. 8e is about 3-5 element size of the surface. For the convergence study, we tried a few different meshes, but I do not remember exactly how many.

3) The perturbation amplitude can be seen more clearly in Fig. 9a. I believe it is smaller than the element size.

4) Yes. Numerical stabilization has to be used to continue the simulation with creasing. The physical instability causes numerical instability that we have not found a better way to deal with.

5) This is due to the way ABAQUS implements UHYPE subroutine, which assumes an isotropic initial state. We found it the hard way as we tried to use UHYPE at the beginning of this study.

6) Fig. 4a shows that the critical swelling ratio is independent of the wave number as long as the wave number is not too small. The curve bends up at the end of small wave number (long wavelength) because of the substrate confinement effect. In a subsequent paper , by including the effect of surface tension, the curve bends up at the other end as well, resulting in a finite wavelength for the minimum critical swelling ratio.

RH