# Spontaneous formation of ordered structures in thin films of metals supported on an elastomeric polymer

Original entry: Tony Orth, APPHY 225, Fall 2009

Ned Bowden, Scott Brittain, Anthony G. Evans, John W. Hutchinson, George M. Whitesides. Nature vol. 393 p.146 [1]

## Soft Matter Keywords

buckling, wrinkling, self-assembly, continuum mechanics

### Abstract

Thin films of metals on compliant elastomeric substrates are made to self assemble into ordered or disordered buckling patterns. The topography of the buckling patterns in sinusoidal with a well defined wavelength.

### Paper Summary

Figure 1

The authors describe a fabrication process for self-assembling sinusoidal topographical buckling patterns on a piece of polydimethylsiloxane (PDMS) coated with a thin film of gold on the order of 50nm thick. At the heart of this phenomenon is the mismatch between the Young's modulus of the thick substrate $E_p$ (PDMS, ie. the elastomeric polymer), and that of thin film covering it $E_m$. Suppose the substrate is pre-strained in some manner, before it is coated with the stiff covering. A surface of the substrate is then coated with a stiff, thin layer of other material (in this case, gold). The gold is under no stress nor strain at this point. Now, when the strain applied to the substrate is released, the compliant thin coating is subject to a stress. However, because the stiff layer has a Young's modulus $E_m \approx 10^3 E_p$, it takes far too much energy to compress the this layer than the pre-strained substrate can provide. The equilibrium state turns out to be a buckled state. The stiff surface layer is deflected upwards from the surface in roughly sinusoidal fashion (see figure 1).

The procedure to create these buckled surfaces (depicted in figures 1,2) is as follows. A piece of PDMS roughly 1cm thick is placed in an e-beam evaporator. The PDMS is heated to roughly 300 degrees Celsius. Once heated, the PDMS is in a thermally expanded state (the expansion here is isotropic, and this temperature corresponds to roughly a few percent expansion). A thin layer of gold is then evaporated onto the surface of the PDMS in the e-beam evaporator. Typical thickness of the gold layer was on the order of 10s of nm. Once the deposition is finished, the PDMS/surface layer composite is removed from the e-beam evaporator and the heat source. Consequently, the PDMS begins to contract and "pulls" the stiff gold into buckling. On an unstructured surface, the buckling pattern looks like that in figure 2a. That is, there is no preferred orientation for the buckles since the thermal strain was isotropic.

Figure 2

The ordering comes from the structuring of the PDMS substrate. This is done readily using soft lithography techniques. Consider a step edge on the substrate, which has been thermally (isotropically) expanded. Once it is let to cool, there is an equal stress is all directions far away from the step edge. At the step edge, the stress in the direction normal to the edge (the x-direction in figure 3) vanishes. Roughly speaking, this is because once expanded, the extent of the step in 'x' is the same as it is when cooled. The extent of the step doesn't change since it is a discontinuity in the first place; the surface is oriented in the YZ plane at the step. Consequently a key boundary condition in this problem is vanishing stress in the x-direction at the step. Therefore, near the step-edge, the buckles orient in the 'y' direction since they only need relieve stress in the 'y' direction. This is why the undulations in the gold in figures 2b-f are oriented straight across throughs, or radiating out from pillars.

Figure 3

The scaling of the wavelength $\lambda$ of the undulations is unfortunately not trivial to arrive at. A relatively straight-forward energy minimizing method is outlined in [2]. It amounts to writing down an integral representing the energy of the system (taking into account most importantly the bending energies). The Euler-Lagrange equations (ie. variational principle) then give differential equation corresponding to energy extrema. Upon substituting a periodic solution for the height of the gold, you can arrive at a scaling $\lambda \propto t\left(E_m / E_p\right)^{1/3}$ where $t$ is the thickness of the gold. In this paper, a typical length scale of the undulations works out to about $4 \mu m$ which is about 9 times less than observed. The authors note that a more sophisticated model with a slightly stiff substrate layer near the gold explains this discrepancy. Nevertheless, this scaling explains observations in a similar more simple system [3].