Boundary curvature effect on the wrinkling of thin suspended films

Wrinkling is an ubiquitous natural phenomenon and has fostered the evolution of tissue that lowers the expenditure of energy.^1,2 Recently, wrinkling has been explored to design efficient devices with self-similar patterns, such as vibration energy harvesters,³ solar cells,⁴ and light gratings.^5,6 However, wrinkling is also known to have adverse effects on the operation of devices such as pressure sensors.⁷ The omnipresence of wrinkling strongly motivates us to understand this phenomenon. Thin films are known to wrinkle because of excess area and geometric compaction,^8,9 and are typically modeled using the Föppl–von Kármán theory of plates in conjunction with scaling arguments and asymptotic analysis. The characteristic wrinkle wavelength λ of a thin film is found by minimizing energy and is provided by the scaling relation

where t is the film thickness and ε is a strain that induces wrinkling.^10–12 Efforts to investigate the effect of the boundary curvature on the wrinkling of thin films began only recently. Experiments and simulations have shown that curvature can significantly affect the wrinkling of cylindrical shells upon stretching.¹³ For spherical^14,15 or tubular^16,17 bilayer systems, similar conclusions have been reached.

Here, we present experiments showing that the boundary curvature of a thin suspended film has a relatively strong impact on azimuthal wrinkling. We provide a model that confirms this finding and can be used to design devices with functional wrinkles. The model hinges on the assumption that the film is composed of a mechanically isotropic material. This assumption is often made to simplify the deformation analysis of 2D materials, amorphous materials, and single crystals, which suggests that the model can be widely applied. We also present a method, which complements those recently reviewed,¹⁸ for measuring the residual compressive strain and film area density from the height profile of a suspended film.

The results are obtained from systematic experiments performed with suspended nanocrystalline diamond (NCD) films of average thickness $t=184 ± 9$ nm, nearly circular boundaries of radius R ranging from 30 to 811 $μ$m, and a compressive prestrain of $ε0≈11×10−4$⁠, which prevails at and beyond the boundaries where the film is bonded to the substrate. A reflecting optical microscope image of such a film is shown in Fig. 1. The films were grown on glass substrates, after which circular portions of the substrates were removed by etching. Each film consists of portions that are bonded to the substrate and free-standing portions. The bonded and free-standing portions of each film are contiguously connected. Due to residual compressive stresses in the bonded portions of the film and the presence of boundary confinement, excess area is formed and geometric compaction occurs in the suspended films. The compressive strain in these films is relaxed radially by buckling and azimuthally by wrinkling. Experimental details are provided hereinafter. In contrast to ultra-thin polymer films,²⁰ NCD films can be regarded to be elastic. The prestrain ε₀ is, therefore, time-invariant so that the wrinkling and underlying strain of the films can be characterized in detail. Another reason for using the suspended NCD films supported by glass substrates is that they are promising candidates for use as robust pressure sensors^7,21 and for the development of 3D micro- and nanodevices.²² 

From the experiments described hereinafter, we find that λ monotonically decreases with boundary curvature $κ=1/R$⁠. To explain this trend, we present a model involving a penetration depth $ΔR$ over which strain in the suspended film relaxes. Only positive compressive strains are considered in the experimental aspects of this work. The setup of our problem is depicted in Fig. 2(a), where $ΔR$ and all other salient geometrical quantities entering our formulation are described. For convenience, the symbols used in this work, and their description, are listed in Table I. We assume that the radial strain ε_rr is independent of the hoop strain $εθθ$⁠; the shear strain $εrθ$ vanishes, the film is made of a mechanically isotropic material, and the portion of the film for which $d≥ΔR$ undergoes pure bending (i.e., is not subject to stretching). Since wrinkling occurs azimuthally with respect to the origin O, we infer that $ε=εθθ$⁠. On this basis, we find that

If the curvature κ of the supporting edge vanishes, as depicted in Fig. 2(b), we find that

For $κ=1/R$⁠, it can be observed from Fig. 2(a) that $l<l0$⁠. This shows that geometric compaction occurs when the curvature κ of the supporting edge is positive. If $R≫ΔR$⁠, we can apply the binomial theorem to expand $(R−ΔR)−1$ and write Eq. (2) as

For a small wrinkle amplitude at $d=ΔR$⁠, the quotient $s/l0$ is practically unity and we may write Eq. (4) as

With Eqs. (1) and (5), we then obtain

For $κ=−1/R$⁠, as depicted in Fig. 2(c), Eq. (6) is also obtained using similar arguments as for $κ=1/R$⁠. Given that $ΔR$ and ε₀ are constant, Eq. (6) predicts that λ decreases monotonically with κ. As expected, Eq. (5) reduces to Eq. (3) for κ = 0. For our problem, Eqs. (1) and (6) are independent of material properties.¹⁰ 

To fabricate our films, we first seeded $10×10×0.2$ mm³ Lotus NXT glass substrates with nanodiamonds.^23,24 Subsequently, a closed film was grown by plasma-assisted chemical vapor deposition in the reactor of an SDS6500X microwave system with 1.5 kW of 2.45 GHz microwaves. During film growth, the substrate temperature was maintained at about 873 K and the reactor was kept at a pressure of 2 kPa with 294  sccm of hydrogen gas and 6  sccm of methane gas. The thicknesses of the NCD films were measured using a Hamamatsu C13027 optical nano gauge. The through holes in the glass substrates and the suspended films were fabricated by etching the substrate locally with hydrofluoric acid using recently described techniques.^21,22 Reflecting optical microscope images and surface profiles of the suspended films were taken using a Keyence VK-X150 confocal laser microscope. We define a surface profile as a collection of heights of a suspended film with reference to plane $p$⁠. Similar to previous work,²² film stresses were obtained by X-ray diffraction (XRD) measurements carried out using a Bruker D8 Discover diffractometer.

To estimate the magnitude of $ΔR$⁠, we use the dimensionless number $d=S/B$⁠, where S and B denote the stretching and bending stiffness, respectively. In so doing, we assume that a portion of suspended film acts as a cantilever of width w, thickness t, and length d. Then, $S=Etw/d$ and $B=Et3w/4d3$⁠, where E denotes Young's modulus, so that $d=4(d/t)2$⁠. For d = t, we see that $d=4$⁠, which indicates that bending and stretching stiffnesses are of the same order. However, for a relatively small value of d, for example, 2 $μ$m, and t = 184 nm, we see that $d=473$⁠, which indicates that bending is strongly favorable over stretching. On this basis, we estimate that $ΔR$ is on the order of micrometers, that $R≫ΔR$⁠, and that $ΔR$ depends on t. Because t is kept nearly constant in our experiments, the latter dependency is not investigated.

From XRD measurements, we infer that the thermal mismatch between the substrate and the film induces a compressive prestrain with a value of $ε0=(11±2)×10−4$⁠, taking into account that, from a Raman spectroscopy study reported previously,²² intrinsic strain is known to be negligible. However, when the material properties of a film are unknown or when dealing with ultra-thin films or amorphous materials, other methods are needed to obtain ε₀. To solve this problem, we demonstrate that ε₀ is estimated accurately by

where A denotes the surface area of the suspended film. The relation $ε0∼ζ$ holds if stretching is negligible, a criterion that is met in our work. The area A of the suspended film is estimated by removing noise from a surface profile with a Gauss filter, creating a surface from that filtered profile by triangulation, calculating the area A_i of each surface triangle T_i, and summing up all A_i that correspond to the suspended film. We find, for our films, that $ε0=(11 ± 2)×10−4$⁠. This value is in excellent agreement with the value obtained from XRD measurements and shows that our approach yields results consistent with a standard strain characterization method. One of the surface profiles used to obtain ε₀ appears in Fig. 3(a), which is for the film shown in Fig. 1. Figure 3(b) depicts $(Ai/Pi)−1$⁠, as obtained from the height profile in Fig. 3(a), plotted with respect to $p$⁠, in which P_i denotes the area of the projection of T_i on $p$⁠. The ratio $Ai/Pi$ represents the scaled film area density under the assumption that stretching is negligible. From the plot, we observe that $Ai/Pi$ is greater at the wrinkled portion than at the remaining area of the suspended film. This shows that in addition to providing a value for ε₀, our strain analysis is useful for investigating the film area density.

To demonstrate that λ decreases monotonically with κ, three surface profiles of the suspended NCD films taken at $d≈6 μ$m are given in Fig. 4(a), with a relatively small offset in height for clarity. Accurate values of λ are found by counting the number of wrinkles of a suspended film and dividing the resulting number by $2πR$⁠. Counting is most easily done with reflecting optical microscope images, of which one appears in Fig. 1. For that image, 39 ± 1 wrinkles were counted, as confirmed from analyzing height profiles such as those given in Fig. 4(a). In Fig. 4(b), the obtained values of λ, scaled by $t1/2$⁠, are plotted vs κ. To fit the data in Fig. 4(b) with our model, using the least squares method, Eq. (6) is written as

in which $ε0=11×10−4, ΔR$ is assumed to be constant, and $ΔR$ and the proportionality constant c act as fitting parameters. Since Eq. (6) is independent of materials properties, so is Eq. (8). The fit confirms that our wrinkling model captures the main physical ingredients that explain our observations. For a confidence interval of 70%, we find that $c=24 ± 1 μ$m^1/2 and $ΔR=0.7 ± 0.2 μ$m. These results support the relation $R≫ΔR$⁠. Since Eq. (8) is free of materials properties, changes in the material properties of NCD, which are often attributed to the grain size,²⁵ are assumed not to affect λ.

Under the assumption that stretching is negligible for $d≥ΔR$⁠, the arc length of the curves in Fig. 4(a) should be similar, in which case the wrinkle amplitude decreases with λ. This is verified by comparing the height axis of the surface profiles in Fig. 4(a). Further analysis of the wrinkle amplitude vs d may be done within the formalism based on wrinklons,¹⁹ which we leave for future work. A wrinklon is the localized transition zone in the merging of two wrinkles. Such a zone is indicated in Fig. 1.

For a leaf that is supported by a stem of radius R, Xu and co-workers²⁶ suggest that residual growth strain induces wrinkling of the leaf. Interestingly, they predict that the associated wrinkle wavelength λ decreases monotonically with R. By modeling such a leaf as a thin suspended film supported by a rigid stem of curvature $κ=−1/R$⁠, as depicted in Fig. 2(c), and attributing ε₀ to growth, our model provides a similar prediction, given that $ΔR$ and ε₀ are approximately constant. Experimentally, the case $κ=−1/R$ may be verified by fabricating micro-disk like structures.^27,28

To finalize our discussion, we underline that due to the presence of the curvature in our experiments, the wrinkle density is practically doubled. It is, therefore, evident that, apart from dynamic wrinkling,²⁹ the boundary curvature also needs to be taken into account when designing devices with functional wrinkles.

We conclude that boundary curvature can strongly influence the wrinkling of thin suspended films. Experimentally, we showed this by growing nanocrystalline diamond films of approximate thickness 184 nm on glass substrates. Due to residual stresses, a compressive strain in the films was introduced. By removing portions of the substrate, suspended, azimuthally wrinkled films with circular boundaries of radius 30–811 $μ$m were fabricated. We found that the wavelength of these wrinkles decreases monotonically with the boundary curvature, leading to a doubling of the wrinkle density. A model that is in line with our experiments is provided. This model can be used to design devices with functional wrinkles, is adapted to explain how the radius of a plant stem affects the wrinkling of the leaf that it supports, and may also be adapted to afford insight into other similar systems. Additionally, taking advantage of the fact that for relatively thin suspended films deformation is dominated by bending, meaning that stretching is negligible, we established a method for measuring residual compressive strain and film area density from height profiles. The method complements standard strain characterization methods.

We gratefully acknowledge the support from the Okinawa Institute of Science and Technology Graduate University with subsidy funding from the Cabinet Office, Government of Japan.

The data that support the findings of this study are available from the corresponding author upon reasonable request.