## Abstract

A freely floating polymer film, tens of nanometers in thickness, wrinkles under the capillary force exerted by a drop of water placed on its surface. The wrinkling pattern is characterized by the number and length of the wrinkles. The dependence of the number of wrinkles on the elastic properties of the film and on the capillary force exerted by the drop confirms recent theoretical predictions on the selection of a pattern with a well-defined length scale in the wrinkling instability. We combined scaling relations that were developed for the length of the wrinkles with those for the number of wrinkles to construct a metrology for measuring the elasticity and thickness of ultrathin films that relies on no more than a dish of fluid and a low-magnification microscope. We validated this method on polymer films modified by plasticizer. The relaxation of the wrinkles affords a simple method to study the viscoelastic response of ultrathin films.

Thin sheets are much more easily bent than stretched by external forces. Even under purely planar tension, a sheet will often deform out of plane to form wrinkles. This is an everyday phenomenon that can be seen on our skin as it is stretched by smiling, scars, or age; on the film of cream that floats on warm milk; or on the skin of fruit as it dries.

This familiar instability occurs because the elastic energy required to stretch a sheet is reduced by the out-of-plane bending that accompanies wrinkling. Cerda and Mahadevan (*1*, *2*) considered a situation in which a rectangular elastic sheet is clamped at its ends and stretched. Beyond a critical strain, the sheet wrinkles. Minimization of the total elastic energy leads to scaling relationships between the amplitude and wavelength of the wrinkles. Their arguments have been applied to a variety of contexts, including the mechanics of artificial skins (*3*, *4*) and surgical scars (*5*).

We report on a study of wrinkling of films under capillary forces, which has thus far remained relatively unexplored. Because thin films are often immersed in fluid environments, both in biological and in synthetic soft materials, the elastic deformation of films under surface tension is relatively commonplace. Thin polymer films form an ideal experimental setting in which to explore wrinkling phenomena: We study films with very high aspect ratios (the ratio of diameter *D* to thickness *h* is *D/h ∼* 5× 10^{5}), which can be treated accurately in the framework of two-dimensional elasticity.

We used films of polystyrene (PS; atactic, number-average molecular weight *M*_{n} = 91,000, weight-average molecular weight *M*_{w} = 95,500, radius of gyration *R*_{g} ∼ 10 nm) spin-coated onto glass substrates. The film thickness *h* was varied from 31 to 233 nm, as measured by x-ray reflectivity with a precision of ±0.5 nm (*6*, *7*). A circle of diameter *D* = 22.8 mm was scribed onto the film with a sharp edge. When the sub-strate was dipped into a petri dish of distilled, deionized water, a circular piece of the PS film detached from the substrate. Because PS is hydrophobic, the film floated to the surface of the water where it was stretched flat by the surface tension γ of the air-water interface at its perimeter.

Wrinkles were induced in the stretched, floating film by placing a drop of water in the center of the film (Fig. 1), by placing a solid disk in the center of the film (fig. S1A), or by poking the film with a sharp point (fig. S1B) to produce a fixed out-of-plane displacement. All these methods of loading lead to qualitatively similar wrinkling patterns, radiating from the center of the load. We emphasize a crucial difference between loading with a solid and a fluid: The wrinkling induced in Fig. 1 is primarily due not to the weight of the drop, but to the capillary force exerted on the film by the surface tension at the air-water-PS contact line. The radial stress σ_{rr} induced at the edge of the drop is dominated by the surface tension, which for the conditions of Fig. 1 is about 100 times as great as the radial stress developed due to the weight of the drop (*mg/2*π*a*), where *m* is the mass of the drop and *a* its radius. Indeed, a solid object of weight and contact area comparable to those of the drops shown in Fig. 1 would produce no discernible wrinkling. The contact angle of the drop on PS is 88° ± 2°, and thus the geometry of the drop on the film is approximately that of a hemisphere on a flat surface (with perhaps some deformation of the film close to the contact line itself). In view of this attractively simple geometry and the degree of experimental control afforded by loading with a fluid, we focus on wrinkling induced by fluid capillarity as in Fig. 1.

We observe the wrinkling pattern using a digital camera mounted on a low-magnification microscope (Fig. 1). Two obvious quantitative descriptors of the wrinkling patterns are the number of wrinkles *N* and the length of the wrinkle *L* as measured from the edge of the droplet. *N* is determined by counting. Because the terminus of the wrinkle is quite sharply defined and not sensitive to lighting and optical contrast, we are also able to measure *L* directly from the image. The radius of the circle in which the entire wrinkle pattern is inscribed (see top left of Fig. 1) is determined with a precision of 3%.

The central question in understanding this wrinkling pattern is, how are (*N, L*) determined by the elasticity of the sheet (thickness *h*, Young's modulus *E*, and Poisson ratio λ) and the parameters of the loading (surface tension γ and radius of the drop *a*). To study systematically the effect of loading and elasticity, we placed water drops at the center of the film using a micropipette, increasing the mass of the drop in increments of 0.2 mg. As the radius of the drop was increased, both *L* and *N* increased.

We first focus on *N*, which is found to increase as . However, as is evident in Fig. 1, *N* is smaller in thicker films. The combined dependence of *N* on *a* and *h* is correctly captured by the scaling as shown in Fig. 2. To understand this scaling, the arguments of Cerda and Mahadevan (*2*) may be adapted to a radial geometry (*5*, *8*). Because the number of wrinkles remains constant at all radial distances *r* from the center of the pattern, the wavelength of wrinkles varies according to λ = 2π*r*/*N*.

This wavelength can be computed from a minimization of the bending transverse to the folds and the stretching along their length, which leads to (1) where the bending modulus *B* = *Eh*^{3}/12(1–λ^{2}) (*9*). For a circular film with a radial stress due to surface tension γ at the edge of the film and another surface tension γ at the boundary of the droplet, σ_{rr} ∼ γ*a*^{2}/*r*^{2} (*10*). We thus obtain (2) where *C*_{N} is a numerical constant. *C*_{N} may be obtained from an analytical solution of the elastic problem or from an experiment like ours where all relevant parameters are known. Using literature values of *E* = 3.4 GPaand λ = 0.33 for PS (*11*), and γ = 72 ± 0.3 mN/m, we obtain *C*_{N} = 3.62 from the slope of the fit line in Fig. 2.

Before discussing wrinkle length, we make some qualitative remarks regarding the evolution of the wrinkle pattern. The wrinkles shown in the images are purely elastic deformations and can be removed without the formation of irreversible, plastic creases (except possibly at the very center of the pattern). Despite this, the number of wrinkles in the pattern is hysteretic because there is an energy barrier as well as a global rearrangement involved in removing wrinkles. In Fig. 2, the drops are slowly increased in size by gentle addition of increments of water and thus represent our best experimental approximation to the equilibrium number of wrinkles. There is no measurable effect of contact line pinning. Nevertheless, the first droplet added invariably overshoots the equilibrium value of *N*, as may be seen in the slight curvature of individual sets of data in Fig. 2. The length of the ridge shows much less hysteresis because the length can locally increase or decrease continuously. This effect is clearly seen when the wrinkle pattern evolves as the drop is allowed to shrink by evaporation (fig. S2).

The length *L* of the wrinkle increased linearly with *a*, the radius of the drop, as shown in Fig. 3A. A simple argument for a linear increase was presented by Cerda (*5*), where the length of the wrinkle is dictated by the radial distance at which stress due to an out-of-plane force *F* applied at the center of a film decays to the value of the tension τ applied at the distant boundaries. This gives . In our situation *F* = 2π*a*γ and τ = γ, yielding a linear dependence *L* ∼ *a*. However, the data in Fig. 3A clearly show a dependence on thickness *h*, which is not captured by this argument. The dependence on *a* and *h* is reasonably well described by the purely empirical power-law scaling shown in Fig. 3B: *L* ∼ *ah*^{1/2} (as shown in the inset to the figure, an unconstrained fit to a power-law yields a slightly better fit of *L* ∼ *ah*^{0.58}). This scaling is dimensionally incomplete and an additional factor of (length)^{–1/2} needs to be taken into account. In terms of the available physical variables, the only possibility is (E/γ)^{1/2}, leading to (3) where *C*_{L} is a constant. From the fit shown in Fig. 3B, we obtain *C*_{L} = 0.031. *E* and *h* appear in Eq. 3 in the combination *Eh*, which is the stretching modulus of the sheet. This indicates that the length is defined purely by in-plane stresses. However, an attempt to write the radial stresses in a manner that is consistent with Eq. 3 yields an answer for σ_{rr}(*a*) that is independent of surface tension, which is implausible. Thus, the dependence of *L* on *h* and *a* is adequately constrained by the experimental data and is well described by Eq. 3 but does not yet have a definitive explanation.

A measurement of *N* and *L* allows a determination of both *E* and *h* for a film, based on Eqs. 2 and 3. As a demonstration of this technique, we vary the elastic modulus of PS by adding to it varying amounts of di-octylphthalate, a plasticizer. As can be seen in Fig. 4A, we find good agreement with published data (*12*) obtained by other techniques. As a further test of our technique, we note that accompanying the large variation (greater than 300%) in Young'smodulus, there is also a subtle change (of about 10%) in the thickness of the film as a function of the mass fraction, *x*, of plasticizer. The determination of thickness by means of Eqs. 2 and 3 yields a value that is in very close agreement (Fig. 4B) with our x-ray reflectivity measurements of *h*. Thus, measurements of both modulus and thickness can be achieved by a wrinkling assay with comparable or higher precision, and with very basic instrumentation, when compared to the other techniques on display in Fig. 4, each of which involves sophisticated equipment and yields only one of *E* or *h*.

Further, in contrast to the few other methods available for measuring the modulus of extremely thin films, such as nano-indentation (*13*) or stress-induced buckling (*12*), the measurement is performed with the film on a fluid surface, rather than mounted on a solid substrate. This allows the possibility for the film to relax internal mechanical stresses that can develop either in the spin-coating process or during transfer to a solid substrate. Apart from the ability to make measurements on a state that is not pre-stressed, this opens the possibility of measuring bulk relaxational properties of the film without concerns about pinning to a substrate. In Fig. 5A, we show a sequence of images visualizing the time-dependent relaxation of the wrinkle pattern formed by a capillary load. At increasing time, the wrinkles smoothly reduce in length and finally disappear. The strains that develop in response to the capillary load (*14*) can relax due to the viscoelastic response of the PS charged with a large mass fraction of plasticizer. In Fig. 5B, we show the time dependence of wrinkle length, *L*(*t*), for two sets of films with different plasticizer mass fraction, *x. L*(*t*) can be fit with a stretched exponential function *L*_{o} exp[–(*t*/τ)^{β}], where τ decreases with increasing plasticizer concentration, and β = 0.50 ± 0.02, typical of polymer viscoelastic response near the glass transition (*13*, *15*).

Thus, capillary-driven wrinkle formation can be used as the basis for a metrology of both the elastic modulus and the thickness of ultrathin films by means of a very elementary apparatus—a low-magnification microscope and a dish of fluid. This simple technique can also be used to study dynamical relaxation phenomena in ultrathin films.

**Supporting Online Material**

www.sciencemag.org/cgi/content/full/317/5838/650/DC1

Materials and Methods

Figs. S1 and S2