A clue to the deformation mechanism came from a passage from Porter and Easterling's “Phase Transformations in Metals and Alloys” and the accompanying illustration:

“Note that the vertical component of γ_SL remains unbalanced. Given time this force would pull the mould surface upwards until the surface tension forces balance in all directions. Therefore Equation 4.14 only gives the optimum embryo shape on the condition that the mould walls remain planar.”

In this case, the substrate, the membrane, wasn't strong enough to remain planar. The features therefore seem to be caused by the out-of-plane component of the surface tension of drying droplets. Attempts to reproduce the phenomenon in the lab were only partly successful; experiments with drying fluids led to features, but never those as varied as the examples shown above. We still don’t fully understand the details of the mechanism. Fortunately, the features do not affect the operation of the films. If necessary, they could be eliminated by engineering the membrane for higher stiffness or by incorporating a preexisting tensile stress to offset the buckling.

Since first observing the features, I have collected other examples of thin film patterns in the literature. The variety is fantastic, but I have not found any that perfectly match the original features. Most buckled films display very regular, periodic patterns that do not resemble the blotchy, irregular arrangement of features shown in optical and AFM micrographs above. The closest patterns are those of drying liquids as they dewet substrates. The random arrangement of droplets bears a much stronger resemblance to the original features than the folded films. It may simply be that nobody has combined dewetting experiments with extremely compliant free-standing films before and published the results.

This note summarizes these findings for the interest of others. Mounds, bumps, wrinkles, dimples... There are as many descriptions as images. The original nomenclature of the authors is the best accompaniment.

Thin film buckling

We begin with the compressive buckling of thin films. Suspended square membranes undergo at least two buckling transitions. The first is characterized by a domed shape that continues into an X, or saltire, under higher stress. The second resembles an augmented tetraskelion whose folded features become more prominent with increasing stress or membrane area. I have often seen these patterns on ceramic and metal membranes, and can usually distinguish them from the random bumps that seem to be related to drying fluids.

(Ziebert et al, “Strongly buckled square micromachined membranes,” J MEMS 8 423-432, 1999)

Toyoichi Tanaka at MIT studied gels of acrylamide and the effect of osmotic swelling in water. As the outer portion of the gel swells due to a phase transition, the unaffected interior region transmits a compressive stress to the surface.

“When a polymer gel undergoes an extensive swelling, a beautiful, regular pattern appears on the surface... At the beginning, the pattern is extremely fine, having a texture similar to that of a frosted glass. As time goes on, the units of pattern coalesce, doubling their characteristic size. When the unit size becomes comparable to the size of the gel, the pattern gradually disappears.”

Tanaka also describes the patterns as having “cusps” and “thorns.”

(Tanaka et al, “Mechanical instability of gels at the phase transition,” Nature 325 396-398, 1987)

“Skeletal wrinkles” appear when the hexamine solvent is evaporated during zinc oxide sol-gel preparation:

(Kwon et al, “Wrinkling of a sol-gel-derived thin film,” Physical Review E 71 011604, 2005)

Annie Viallet at Joseph Fourier University investigated the shapes created in lipid vesicles during osmotic deswelling of the interior.

“When the reduced volume of partially deflated vesicles varies, a great variety of shapes (pears, peanuts, starfish. . .) is observed.

A great variety of vesicle shapes were obtained in the chamber at the end of the shrinkage process: ellipsoids, dumbbells, stomatocytes, discocytes, pears, stars, etc.

As in the case for slow shrinkage, we observed various shapes...the formation of multiple small invaginations of the membrane pointed toward the interior of the vesicles, which looked like raspberries...the formation of lipid pearl necklaces.”

The breaking of axial symmetry in the glassy skin of drying polysaccharide dextran leads to a pentagonal arrangement of radial wrinkles:

(Pauchard et al, “Stable and unstable surface evolution during the drying of a polymer solution drop,” Phys Rev E 68 052801, 2003)

Drying droplets of colloidal polystyrene become buckled and folded (photographs are on top, simulated shapes below):

(Tsapis et al, “Onset of buckling in drying droplets of colloidal suspensions,” Phys Rev Lett 94 018302, 2005)

And blisters appear as a polymer film swells in water:

(Sharp et al, “Swelling-induced morphology in ultrathin supported films of poly(d,l-lactide),” Phys Rev E 66 011801, 2002)

Simulations of tethered membranes produce a “wavelike pattern of mounds:”

(Moldovan et al, “Tethered membranes far from equilibrium: buckling dynamics,” Phys Rev E 60 4377, 1999)

Another group's analysis predicts “ridges,” “cones,” “mounds,” and “basins:”

(Uchida, “Orientation order in buckling elastic membranes,” Physica D 205 267-274, 2005)

Some of these simulated shapes have been observed in a different context. A elastomer substrate whose top layer undergoes volumetric expansion from oxygen plasma exposure displays “complex yet periodic wavy structures”:

(Chua et al, “Spontaneous formation of complex and ordered structures on oxygen-plasma-treated elastomeric polydimethylsiloxane,” Applied Physics Letters 76 721-723, 2000)

“Herringbone” shapes appear when a gold film is cooled after deposition on PDMS:

(Chen et al, “A family of herringbone patterns in thin films” Scripta Materialia 50 797-801, 2004)

Finally, permanent wrinkles are seen in a free-standing boron nitride film after its salt support is dissolved away:

(Coupeau et al, “Evidence of plastic damage in thin films around buckling structures” Thin Solid Films 469-470 221-226, 2004)

Thin Film Dewetting

We now turn to the dewetting of liquids, sometimes modeled as a decomposition into wet and dry phases. Anshutosh Sharma of the Indian Institute of Technology describes dewetting morphology:

“As the film thickness is increased, the initial pathway of evolution changes from the formation of small spherical droplets, to long mesas (parapets) and islands, to circular holes, all of which eventually resolve by ripening into a collection of round pancakes at equilibrium. However, beyond a certain transition thickness, a novel metastable honeycombed morphology, resembling a membrane or a slice of Swiss cheese, is uncovered, which is produced by an abrupt “freezing” of the evolution during hole growth.”

(Sharma et al, “Pattern formation and dewetting in thin films of liquids showing complete macroscale wetting: from ‘pancakes’ to ‘swiss cheese’,” Langmuir 20 10337-10345, 2004)

Leonard Schwarz at the University of Delaware dried a thermosetting polymer on an inked aluminum sheet. His group observed “particular patterns of holes, ridges, filaments, and, ultimately, droplets” both in experiments (left) and in simulations (right):

(Schwartz et al, “Dewetting patterns in a drying liquid film,” J Colloid Interface Sci 234 363-374, 2001)

Observers of drying water films report the appearance of “star-like formations” and “parabolic dendrites” as dry patches nucleate:

(Samid-Merzel et al, “Pattern formation in drying water films,” Physica A 257 413-418, 1998)

And “fingering instabilities” are seen as viscous polystyrene dewets from an octadecyltrichlorosilane-coated silicon wafer during annealing:

(Reiter, “Unstable thin polymer films: rupture and dewetting process,” Langmuir 9 1344-1351, 1993)

Thin Film Deposition

When silver is sputtered onto a layer of silicone oil, the initial sputtered atoms are driven into the top layer of the oil. At some percolation threshold, a “branched structure” consisting of silver clusters nucleates:

(Ye et al, “Structural and electrical properties of a metallic rough-thin-film system deposited on liquid substrates,” Phys Rev B 54 14754-14757, 1996)
(Feng et al, “Growth behavior and surface morphology of Ag rough films deposited on silicone oil surfaces,” Thin Solid Films 342 30-34, 1999)

I’ve included these silver features because of their visual appeal; however, it’s important to note that their origins are more complex than the other patterns shown previously. All of the earlier patterns are caused by imposed or internal compressive stress (in the case of the buckled films) or surface-volume energy tradeoffs (in the case of the dewetting films). In contrast, the sputtered silver film morphology is attributed to mixing of the silver atoms with the silicone oil layer.

The large number of reports in the literature and the frequency of rediscovery of these patterns shown above (and there are many more, similar, patterns not included here) suggest that an atlas of patterns would be helpful to researchers. Because of the evocative descriptions found so far, I am awaiting the discovery of additional reports of crinkles, creases, nodules, buttons, indents, grooves, channels, fingers, furrows...scallops... corrugations... crenelations... undulations...

A final note: please see the original journal articles for the highest quality images and the context of their appearance. I believe the use of the images above falls under the guidelines of fair use as outlined in 17 U.S.C. § 107. If you are the author of one of the papers listed and you would prefer it not be used, please contact me.