Pattern formation in soft elastic films cast on periodically corrugated surfaces—a linear stability and finite element analysis

Abstract

Length scales of instabilities exhibited by soft, thin elastic films cast on smooth and sharp corrugated surfaces and in adhesive contact with an external contactor are investigated by means of linear stability analysis (LSA) and non-linear finite element analysis. The instability length scales are found to decrease with either an increase in the amplitude β or a decrease in the wavelength λp of the substrate pattern. For same substrate parameters, a step-patterned surface with higher RMS roughness is generally found to engender smaller length scales. The linear stability analysis with sinusoidally patterned substrates shows a reduced scaling of critical wavelength with substrate amplitude: λc = 2.96(1 − β)h, where h is the mean film thickness. The largest substrate amplitudes explored with finite element analysis are limited to β = 0.7, which is found to beget instability length scales of 0.89h, that are much smaller than the 3h length scale that is obtained with flat substrates. This suggests that an increase in substrate roughness via patterning can prove to be an attractive route for the production of miniaturized patterns.

Miniaturized Pattern Formation in Elastic Films Cast on Sinusoidally Patterned Substrates

“…As β increases, the dominant length scale of instability decreases (as shown in the Figure 2), visible from the fact that the quarter ring in the inset shifts farther away from the origin. Thus, the dominant length 44 Figure 2A represents the results for the case of a a nearly flat substrate. Though the instability length scales decrease as the substrate pattern amplitudes β increase, there is no long-range order present in the patterns formed.…”

“…It is reported in the prior works on elastic films cast on patterned substrates that the instability length scales are mainly influenced by the substrate pattern amplitude β p and are virtually independent of the substrate pattern wavelength λ p (except in some pockets of λ p ). 43,44 Hence, we assume a long-wave limit of the substrate patterning (i.e., k p → 0) for simplifying the stress-equilibrium equations of eqs A4−A6 in the computational plane. The stress equilibrium equations, thus, simplified are given in the Supporting Information.…”

“…42−44 Recent works on using a patterned substrate have suggested that patterned substrates yield reduced length scale patterns due to the lateral variation in film thickness which results in lateral variation in elastic film stiffnesses. 43,44 If the effects of van der Waals interactions are made negligible by increasing the air gap d between the film and the contactor and electric field is introduced instead, the strength of the destabilizing interactions varies laterally 23,30,45−55 because of the lateral variation in thicknesses during evolution of such films. Exposing the films cast on flat substrates to electric field instead of van der Waals interactions had proved to produce patterns which still scaled as 3h but were highly ordered compared to the labyrinthine patterns obtained from van der Waals interactions.…”

“…A two-dimensional (2D) nonlinear finite element analysis of elastic films cast on step and sinusoidally patterned substrates was performed earlier which provided valuable insights into understanding of length scales formed for adhesive films undergoing deformations due to van der Waals interactions. 44 It, however, does not prove to be useful in predicting the exact morphological phases in the threedismensional (3D) space, more so for electric fields which now have a strong lateral variation in the interaction strength even before evolution. In the present work, we study, with the help of 3D nonlinear energy minimization simulations, the patterns formed on elastic films in contact proximity with an external contactor and under the influence of van der Waals interactions at critical gap distance as well as electrostatic interactions at different voltages for fixed gap distances.…”

“…Extensive literature exists in the area of morphological patterning of soft elastic surfaces in adhesive contact or in peeling configuration in close proximity with an external contactor where patterns formed are of length scales 3 h or 4 h depending on the contactor-film configuration and are independent of the nature of interactions present between the film and the contactor and the elastic modulus of the film. − In yet another work on hydrogel sandwiched between rigid plates, length scales 7 h have been observed, and it is proposed that these large length scales can affect the fracture modes as well as the adhesive strength of the hydrogel . There has been a lot of interest lately in forming miniaturized elastic film patterns of length scales less than ∼3 h by various methods like elastic bilayers − and using topographically patterned substrates. − Recent works on using a patterned substrate have suggested that patterned substrates yield reduced length scale patterns due to the lateral variation in film thickness which results in lateral variation in elastic film stiffnesses. , …”