#### 2.2. Model

We next introduce a simple model for the buckling that is observed in our experiments. Consider the coating of paint to be an elastic thin film that is attached adhesively to a solid substrate (

Figure 5). Because of the absorption of the organic solvent, the elastic film swells, producing compression stress. Suppose that the elastic film exists on a flat substrate, whose surface corresponds to the

${x}_{1}$-

${x}_{2}$ plane, and let the

${x}_{3}$ axis be normal to the surface of the substrate. The total energy

${F}_{\mathrm{tot}}$ of this system consists of the elastic strain energy in the film and the interfacial traction energy between the film and substrate:

where

${f}_{\mathrm{film}}$ and

${f}_{\mathrm{int}}$ are the energies per unit area of the film and the interface, respectively.

According to the Föppl–von Kármán plate theory, the elastic energy per unit area in a film of thickness

h is given by [

8,

9,

10,

11,

12]

Greek subscripts refer to in-plane coordinates

${x}_{1}$ or

${x}_{2}$, and repeated Greek subscripts indicate summation over indices 1 and 2. The parameters

$\mu $ and

$\nu $ are the shear modulus and Poisson ratio of the film, respectively. Mid-plane displacements in the in-plane and

${x}_{3}$ directions are denoted by

${u}_{\alpha}$ and

w, respectively. Supposing the film to be under equibiaxial stress, we take the initial in-plane strain to be

${\epsilon}_{0}{\delta}_{\alpha \beta}$. Then,

To express the interfacial traction energy between the film and substrate, we use the cohesive zone model [

13,

14,

15]. The interfacial energy per unit area is then

where

$\zeta $ is the distance between the substrate and film, and

${T}_{\mathrm{n}}$ is the normal traction. When the film thickness is constant,

$\zeta =w$. We represent the normal traction as

where

${\mathsf{\Gamma}}_{\mathrm{n}}\equiv {\gamma}_{\mathrm{n}}/{\delta}_{\mathrm{n}}^{2}$. The parameters

${\gamma}_{\mathrm{n}}$ and

${\delta}_{\mathrm{n}}$ are the normal interfacial toughness and the characteristic length of a normal displacement jump, respectively.

The total energy

${F}_{\mathrm{tot}}$ is thus expressed in terms of the displacements

${u}_{\alpha}$ and

w. Equilibrium states must satisfy

$\delta {F}_{\mathrm{tot}}/\delta w=0$ and

$\delta {F}_{\mathrm{tot}}/\delta {u}_{\alpha}=0$. However, instead of solving

$\delta {F}_{\mathrm{tot}}/\delta w=0$, we employ the time-dependent Ginzburg–Landau equation, which is often used in dynamical systems,

where

$\eta $ is a constant related to the characteristic relaxation time. Scaling all lengths by

h, times by

$h/\left(\mu \eta \right)$, the nondimensional equation and

${\gamma}_{\mathrm{n}}$ by

$h\mu $ in Equation (

10), we obtain

where the variables are dimensionless,

${T}_{\mathrm{n}}^{\prime}$ is the nondimensional form of Equation (

9), and

The in-plane displacements

${u}_{\alpha}$ included in

${e}_{\alpha \beta}$ are obtained from the equation

$\delta {F}_{\mathrm{tot}}/\delta {u}_{\alpha}=0$.

#### 2.3. Linear Stability Analysis

A linear stability analysis of Equation (

11) provides some insight into the condition of buckling. Linearizing Equation (

11) around

$w=0$, and taking the Fourier transform of the linearized equation, we obtain

where

$\tilde{w}$ is the Fourier transform of

w, and

k is the wavenumber. The linear growth rate

g is given by

Unstable modes, which cause deformations in the layer, appear when

$g\left(k\right)>0$; in other words,

This equation shows that wrinkles emerge above a certain threshold of stress. The existence of the threshold is consistent with the experimental results shown in

Figure 2, which indicate that buckles emerge under an upper limit of

T, since

${\mathsf{\Gamma}}_{\mathrm{n}}$ and

${\epsilon}_{0}$ depend on

T. Equation (

15) shows that the wavenumber of the fastest-growing mode is

The growth rate of the fastest-growing mode is inversely proportional to the timescale,

#### 2.4. Numerical Simulations

Numerical simulation is useful for demonstrating that our simple model does reproduce buckling of the film. Simulated patterns of the displacement

w are shown in

Figure 6. In the initial states, we take

$w=0$ plus a small amount of noise, and we impose periodic boundary conditions on a

$256\times 256$ grid system. The length of a side corresponds to about 6.8 mm for

$h=0.13$ mm. The values of the side length and

h are close to experimental ones. In

Figure 6, we take

${\epsilon}_{0}=1.2$,

${\mathsf{\Gamma}}_{\mathrm{n}}=1.4$. The other parameters used in the following simulations are

$\nu =0.3$ and

${\delta}_{\mathrm{n}}=0.5$.

Some characteristics of the snapshots in

Figure 6 look similar to those of the experiments in

Figure 1c–f. Small bumps appear at an early stage (

Figure 6a). The amplitudes of the bumps grow with time, and some bumps coalesce with others (

Figure 6b,c). However, the amplitudes continue to grow in the simulations (

Figure 6d), which is significantly different from the experiments. This indicates that our model is not yet adequate to explain the nonlinear effects in the actual experiments.

The phase diagram shown in

Figure 7 illustrates the numerical verification of the wrinkle-emerging condition. The linear stability analysis suggests that wrinkles should appear above the solid curve, which is given by Equation (

16). Numerical data shown as symbols demonstrate that the analysis is sufficiently valid.

Figure 8a shows the time evolution of the root-mean-square (RMS) of

w,

where

L is the system size. The parameter values except for

${\epsilon}_{0}$ are taken as the same as

Figure 6. This is also interpreted as the time evolution of the average amplitude

A of wrinkles, since

${W}_{\mathrm{RMS}}=\sqrt{1/2}A$, where we suppose a stripe form of

$w=Asin\left({k}_{\mathrm{f}}{x}_{1}\right)$. The arrows in

Figure 8a indicate the initial-growth time

$\tau $ which is the transition time from the initial-growth regime to the coarsening one. Let us here estimate the amplitude at the initial-growth time. In the coarsening regime, we can assume that the interfacial energy becomes negligible and that the time evolution slows down significantly. Thus, setting

$\partial w/\partial t={T}_{\mathrm{n}}^{\prime}=0$ in Equation (

11), we have

On the other hand, calculating the spacial average of

${\sigma}_{11}$ given by Equation (13) leads to

Equating Equtaions (

20) and (

21), we have

$A=\sqrt{1/3}$. The initial-growth time

$\tau $ is determined as the time when

$A=\sqrt{1/3}$, and thus,

${W}_{\mathrm{RMS}}=1/\sqrt{6}$.

The average wavelength of wrinkles is

$\lambda =2\pi /\overline{k}$, where

Here,

$\tilde{w}$ is the Fourier transform of

$w-\overline{w}$, where

$\overline{w}$ is the spacial average of

w. The time evolution of the average wavelength is shown in

Figure 8b. In the initial-growth regime, the average wavelength is approximately equal to (or slightly larger than) that of the fastest-growing mode, which is indicated by the arrows in

Figure 8b. The wavelength

${\lambda}_{\mathrm{f}}$ of the fastest-growing mode is estimated from the linear stability analysis and evaluated as

${\lambda}_{\mathrm{f}}=2\pi /{k}_{\mathrm{f}}$, where

${k}_{\mathrm{f}}$ is given by Equation (

17).

#### 2.5. Correspondence between Experimental and Numerical Results

We here rewrite the time scale

${\tau}_{\mathrm{f}}$ in other forms to examine experimental and numerical results by means of the linear stability analysis. Suppose that the growth rate of a certain unstable mode

${k}_{1}$ is

$g\left({k}_{1}\right)=C$, where

C is a positive constant. Using Equations (

15), (

17) and (

18), we have

where

${k}_{1}<{k}_{\mathrm{f}}$. Equation (

23) leads to

where

${k}_{\mathrm{f}}=2\pi /{\lambda}_{\mathrm{f}}$,

${k}_{1}=2\pi /{\lambda}_{1}$, and

${C}^{\prime}$ is a constant. For the minimum wavenumber

${k}_{\mathrm{min}}$ with a non-negative growing rate,

$g\left({k}_{\mathrm{min}}\right)=0$, and thus,

$C={C}^{\prime}=0$. Then, Equtaions (

23) and (

24) turn to be

where

Equations (

24) and (

25) are useful to examine experimental and numerical results, respectively.

We first examine experimental results, using Equation (

24). We assume that

${\lambda}_{\mathrm{f}}$ and

${\lambda}_{1}$ in Equation (

24) correspond to

${\lambda}_{\mathrm{s}}$ and

${\lambda}_{\mathrm{w}}$ in

Figure 3, respectively. This assumption implies that structures with wavelengths

${\lambda}_{\mathrm{s}}$ and

${\lambda}_{\mathrm{w}}$ appear in the linear-instability region and that

${\lambda}_{\mathrm{s}}$ and

${\lambda}_{\mathrm{w}}$ correspond to unstable modes of pattern formation. We also assume that

${\tau}_{\mathrm{ex}}\propto {\tau}_{\mathrm{f}}$ [

8].

The closed circles in

Figure 9 show experimental data about the relation between

${\tau}_{\mathrm{ex}}$ and

$(1/{\lambda}_{\mathrm{s}}^{2})-(1/{\lambda}_{\mathrm{w}}^{2})$. Values of

${\lambda}_{\mathrm{s}}$ and

${\lambda}_{\mathrm{w}}$ are obtained using Equations (

1) and (

2). The experimental data agree reasonably closely with the line given by Equation (

24), which is shown as a solid line. The exponent of the fitted line is

$-2$, and

${C}^{\prime}$ of Equation (

24) nearly vanishes for the fitted line.

Figure 10 shows the numerical counterparts. The initial-growth time

$\tau $ is plotted as a function of

${k}_{\mathrm{f}}^{2}-{k}_{min}^{2}$. The initial-growth time is defined as the time when

${W}_{\mathrm{RMS}}$ reaches

$1/\sqrt{2}$ (See

Figure 8), and its value is obtained from numerical simulations with different combinations of

${\epsilon}_{0}$ and

${\mathsf{\Gamma}}_{\mathrm{n}}$. The parameters used in the simulation gives

${k}_{\mathrm{f}}^{2}-{k}_{min}^{2}$ through Equation (

26). Assuming that the initial-growth time

$\tau $ is proportional to

${\tau}_{\mathrm{f}}$ of Equation (

25), we fit the line given by Equation (

25) to the numerical data. The fitted line reasonably agrees with the numerical data.