Modeling kinetics of diffusion-controlled surface wrinkles

Nonlinear wrinkling of a compressed film on a soft substrate in the presence of inhomogeneous swelling actuation strain caused by solvent diffusion is studied. The simulation relies on a continuum model which integrates phase field microelasticity and Föppl–von Kármán plate theory. We show that the wrinkling morphologies developed in the diffusive domain exceeding a critical compression are confined and become shape and size dependent. A rich variety of wrinkling patterns observed in experiments including hexagonal ordered, dimple, or peanut structures, are numerically recovered, depending on the distribution of diffusion-mediated actuation strain. A cascade feature of the diffusion-coupled wrinkle is demonstrated as well: There are two ranges of solvent concentration within which the sequences of wrinkling pattern are different.

Figure 1

A sketch of a surface wrinkle in a film-substrate system with inhomogeneous actuation strain (in-plane expansion ${\varepsilon }_{\alpha \beta }^T$ and eigencurvature $k_{\alpha \beta }^T$) induced by solvent absorption: (a) a flat configuration, (b) a locally wrinkled configuration.

Figure 2

The amplitudes of an equilibrium sinusoidal wrinkle as a function of the reduced eigenstrain ${\varepsilon }_{\mathrm{pre}}/{\varepsilon }_c$ from numerical simulation and analytical solution.

Figure 3

The critical confined wrinkling strain ${\varepsilon }_c^{\prime }/{\varepsilon }_c$ as a function of the reduced domain size $R/{\lambda }_c$.

Figure 4

The simulated wrinkling pattern as a function of the reduced diffusive domain size $R^{\prime }=R/{\lambda }_c$ under ${\varepsilon }_0=3{\varepsilon }_c$, ${\varepsilon }_0^{\prime }=0$, at $t^{\prime }={10}^4$, with the black to white shading mapping the value of $w$ from negative to positive.

Figure 5

Simulated shape-dependent wrinkling pattern at $t^{\prime }={10}^4$ in (a) a square domain $L=3{\lambda }_c$ long, (c) a rectangular domain with $L_x=3{\lambda }_c,L_y=1.2{\lambda }_c$. (b), (d) Visualization of the corresponding profile of the reduced $\left(|{\sigma }_{xx}|-|{\sigma }_{yy}|\right)/{\mu }_s$ in the domains before the onset of wrinkling, respectively, with the darker to lighter shading mapping its value from negative to positive.

Figure 6

Simulated wrinkling patterns in the film obtained at $t^{\prime }={10}^4$ for all cases with $c_0=0.4,{\varepsilon }_0=0.02$, and with additional parameters (a) $M^{*}=0$, ${\varepsilon }_0^{\prime }=0$, (b) ${\varepsilon }_0^{\prime }=0$, (c) ${\varepsilon }_0^{\prime }=0.001$, (d) ${\varepsilon }_0^{\prime }=0.005$, (e) ${\varepsilon }_0^{\prime }=0.01$, (f) ${\varepsilon }_0^{\prime }=0.05$.

Figure 7

(a)–(f) are the corresponding simulated concentration profiles of the cases in Figs. 6a, 6b, 6c, 6d, 6e, 6f, respectively.

Figure 8

Comparison between the observed wrinkling pattern in Ref. [23] with the simulated dimple pattern under the parameters $c_0=0.4$, ${\varepsilon }_0=0.02$, and ${\varepsilon }_0^{\prime }=-0.05$ (reproduced with permission).

Figure 9

Simulated patterns of surface wrinkles and concentration profiles in the film with different levels of solvent absorption. The top and bottom rows indicate the profiles of the out-of-plane displacement and the concentration at $t^{\prime }=2\times {10}^3$, ${\varepsilon }_0=0.03$, ${\varepsilon }_0^{\prime }=0.02$ under (a) $c=0.95$, (b) $c=0.6$, (c) $c=0.4$, respectively.

Figure 10

Typical total energy curves as a function of solvent concentration for the flat and sinusoidally wrinkled configuration in the film-substrate system.

Figure 11

Evolution sequence of the wrinkling pattern (top row) and concentration profile (bottom row) in mode A under the parameters $c=0.4$, ${\varepsilon }_0=0.015$, and ${\varepsilon }_0^{\prime }=0$.

Figure 12

Evolution sequence of the wrinkling pattern (top row) and concentration profile (bottom row) in mode B with the parameters $c=0.4$, ${\varepsilon }_0=0.05$, and ${\varepsilon }_0^{\prime }=0$.