Wrinkling instabilities of swelling hydrogels

We investigate the formation of wrinkling instabilities at the interface between layers of hydrogel and water, which arise to relieve horizontal compressive stresses caused by either differential swelling or confinement. Modelling the gel using a linear-elastic-nonlinear-swelling approach, we determine both a criterion for marginal stability and the growth rates of normal modes. Furthermore, our formalism allows us to understand the influence of differential swelling on the stability of hydrogels brought into contact with water, and we find three distinct phases of the instability. Initially, when only a thin skin layer of gel has swollen, buckles grow rapidly and the gel deforms as an incompressible material. A balance between normal elastic stress and pore pressure selects a wavelength for these buckles that increases with the square root of time. At late times, when the gel approaches a uniformly swollen state, buckles can only grow by differential swelling on much slower timescales determined by solvent transport. At intermediate times, growth is driven by the same fluid transport process as at late times but gradients in fluid pressure in the gel as it swells destabilize the interface, driving faster growth of wrinkles. We also explain why some instabilities can be transient, “healing” as time progresses, while others must remain for all time.

Figure 1

Three examples of wrinkling in swelling hydrogel materials shortly after contact with water. (a) A thin hydrogel layer (of dry thickness $\sim 1$ mm) anchored to an inextensible plastic base forms a reticulation pattern minutes after immersion in water. (b) Initially spherical hydrogel beads form lobelike wrinkles when swelling commences. (c) A cube of hydrogel forms a crumpled shape. In the latter two of these three cases, further swelling “heals” the wrinkles.

Figure 2

(a) The isotropic, partially swollen, initial state considered in this model, with uniform polymer fraction ${\mathrm{\Phi }}^{*}$ and a Cauchy strain tensor $\mathsf{e}=(1-{{\mathrm{\Phi }}^{*}}^{1/2})\mathsf{I}$ with respect to the fully swollen state $\mathrm{\Phi }\equiv 1$. (b) The state of the gel after being brought into contact with water. Notice that horizontal strains do not change but the interface swells to polymer fraction ${\mathrm{\Phi }}_1$.

Figure 3

(a) A plot of the evolution of $\mathrm{\Phi }$ in time, from Eq. (8), with $\mathcal{M}=10$ and initial polymer fraction ${\mathrm{\Phi }}^{*}=4.6225$ (chosen so that ${\mathrm{\Phi }}_1=4$). The arrow points in the direction of increasing time. (b) A plot of the thickness of the gel scaled with the steady-state thickness $a_1$, showing the diffusive growth of the layer on the swelling timescale, governed by Eq. (13).

Figure 4

Plots of the marginally stable value of ${\mathrm{\Phi }}_1$, determined implicitly in Eq. (47), at which wrinkles neither grow nor shrink, against wave number $\alpha$, showing that instability is triggered first at infinite wave number $\alpha$, with, in general, stabilization for small $\alpha$ and a critical value ${\mathrm{\Phi }}_1\ge {\mathrm{\Phi }}_{\text{crit}}=(3+\sqrt{5})/2$ for instability.

Figure 5

Plots of the critical compressive strain $-{\mathsf{e}}_{xx}$ to trigger an instability of any wave number [i.e., to have ${\mathrm{\Phi }}_1\ge (3+\sqrt{5})/2]$ in the uniformly swollen steady state for different values of $\mathcal{M}$. The dashed line shows the large-$\mathcal{M}$ limit for ${\mathsf{e}}_{xx}$, equal to $(\sqrt{5}-1)/2\approx 0.618$.

Figure 6

Plots of the dispersion relation (49) for the growth rate $s$ as a function of wave number $\alpha$ and polymer fraction ${\mathrm{\Phi }}_1$ when $\mathcal{M}=1$, showing faster growth for larger ${\mathrm{\Phi }}_1$ or $\alpha$.

Figure 7

Plots of the perturbation eigenfunctions, normalized by the value of $\eta$ at $y=1$, for a variety of wave numbers when $\mathcal{M}=1$ and ${\mathrm{\Phi }}_1=4$, as derived in Eq. (pp1). In these examples, all but $\alpha =1$ are unstable. Notice that $p_y<0$ at $y=1$ in all the unstable cases and $p_y>0$ in the stable case. Also note the decreasing decay lengthscale as $\alpha$ increases.

Figure 8

Contour plots of the perturbation polymer fraction field $\tilde{\phi }$ (A3a) alongside arrows indicating the direction of the interstitial fluid flow in each case, underlining the growth or shrinkage mechanism. When the polymer fraction is low, fluid is driven out of peaks and into troughs, stabilizing the interface, while the opposite happens in unstable cases. In both cases, $\mathcal{M}=1$ and $\alpha =5$.

Figure 9

Plots of the dispersion relation for a swelling gel at various times $\tau$ when $\mathcal{M}=10$ and ${\mathrm{\Phi }}^{*}=4.6225$ (and thus ${\mathrm{\Phi }}_1=4$), with a logarithmic scale for $s$. (a) The dispersion relation for early times, showing a distinct peak at a finite wave number ${\alpha }^{*}$. (b) The peaks seen at early times disappear beyond $\tau \approx 5.5\times {10}^{-5}$ and the growth rates decrease, approaching the steady-state value derived in Sec. 4 with ${\mathrm{\Phi }}_1\equiv 4$.

Figure 10

Plots of the dispersion relation at early times (a) and later times (b) when ${\mathrm{\Phi }}^{*}=2.75$ and $\mathcal{M}=10$. In this case, ${\mathrm{\Phi }}_1<{\mathrm{\Phi }}_{\text{crit}}$ and thus the final steady state is stable to perturbations of all wave numbers. However, as $\tau \to 0$, some wave numbers are unstable. Note the logarithmic scale in (a) and the linear scale in (b).

Figure 11

Plots of the most unstable mode ${\alpha }^{*}$ where the growth rate approaches infinity at early times, with $\mathcal{M}=10$, for different values of ${\mathrm{\Phi }}^{*}$, showing that ${\alpha }^{*}\propto {\tau }^{-1/2}$ at early times.

Figure 12

(a) A plot of the marginally stable modes for different values of ${\mathrm{\Phi }}_y$ when $\mathcal{M}=1$ for different values of ${\mathrm{\Phi }}_y$, showing how the transient swelling state destabilizes some wave numbers $\alpha$ where $\overline{\mathrm{\Phi }}<{\mathrm{\Phi }}_{\text{crit}}$. (b) A plot of the critical value of $\overline{\mathrm{\Phi }}$ above which instabilities of some wave numbers are seen, corresponding to ${\mathrm{\Phi }}_{\text{crit}}$ in the fully swollen case. Notice how, at early times when ${\mathrm{\Phi }}_y\to -\infty$, the criterion for an instability is weaker.

Figure 13

Comparisons of the early-time approximation for ${\alpha }^{*}$ (dashed curves) from Eq. (54) with the intermediate-time solutions (70) that allow for some compressibility (solid curves). In all cases, ${\mathrm{\Phi }}^{*}=5$.

Figure 14

(a) Plots of ${\delta }_c$ for different values of $\overline{\mathrm{\Phi }}$ and $\mathcal{M}$, showing the critical swollen layer thickness beyond which buckling instabilities no longer occur. (b) Plots of ${\alpha }_{\infty }$, the final value of ${\alpha }^{*}$ when $\delta ={\delta }_c$, again shown for different $\mathcal{M}$ and ${\mathrm{\Phi }}^{*}$.

Figure 15

Plots of the different stability properties for a gel where $\mathcal{M}=10$ and how these depend on time $\tau$ and initial polymer fraction ${\mathrm{\Phi }}^{*}$. To produce this plot, a full numerical solution has been used to deduce $\overline{\mathrm{\Phi }}$, so that the solution remains valid for larger values of $\tau$.