When the brush, which consists of regularly spaced, flexible lamellae, is progressively withdrawn from the bath of liquid, a cascade of successive sticking events leads to a hierarchical bundling pattern (Fig. 1a). We studied one such elementary sticking event for two lamellae separated by a distance d. If the strips were rigid, the perfectly wetting liquid would rise up to Jurin's height, hJ=2LC2/d, where LC=(γ/ρg)1/2 is the capillary length; γ and ρ are the liquid's surface tension and density, respectively. When the strips are flexible, capillary suction bends the lamellae and the liquid rises higher in this more confined environment. As two lamellae are withdrawn to height L (Fig. 1b), a capillary rise to height hJ (or to the top when L<hJ) precedes the sticking together of the strips, which happens when L becomes large.

Figure 1: Flexible lamellae stick together after wetting.
figure 1

a, Lamellae in a wetted model brush after a sequence of sticking or unsticking events that cause aggregation (viewing from top to bottom) or fragmentation (from bottom to top), respectively. b, Height of rise, Lwet, of liquid between a lamella pair is plotted against the withdrawal height, L, showing the transition from the capillary rise (dashed line; Lwet=hJ) to the sticking regime (full green line; Ldry=LLwet is constant). Polyester strips separated by d=1 mm (width, 25 mm; thickness, e=100 µm; bending rigidity, κ=5.1×10−4 N m) were dipped into silicon oil (density, ρ=950 kg m−3 and surface tension γ=20.6 mN m−1, leading to hJ=4.3 mm). Inset, sticking regime. Non-dimensional dry length, Ldry/LEC, is plotted against non-dimensional separation, d/LEC; LEC=(κ/γ)1/2, which is the elastocapillary length (red circles: e=50 µm, LEC=47 mm; blue diamonds: e=100 µm, LEC=150 mm; green triangles: e=170 µm, LEC=370 mm); line: comparison with theory (equation (1); no adjustable parameter). c, Aggregation of multiple lamellae into bundles (e=50 µm, d=1 mm). Number of lamellae per bundle is plotted against dry length. Blue crosses, raw data; red circles, averaging of data; line, comparison with theory (equation (2); no adjustable parameter).

Surprisingly, the height of rise Lwet increases linearly with L in this last regime, whereas Ldry=LLwet remains constant. In fact, Ldry is prescribed by a balance between capillarity and elasticity. The capillary energy (per unit width) is −2γLwet, whereas the elastic energy is proportional to the square of the typical curvature, d/Ldry2, and reads exactly 3κd2/Ldry3 in this geometry, where κ is the bending stiffness of the strips. Minimizing the sum of the two energies (gravity becomes negligible in this regime) with respect to Ldry yields

where LEC=(κ/g)1/2 is the elastocapillary length, in agreement with measurements made over several orders of magnitude (Fig. 1b, inset). An identical energy formulation is found in fracture theory9, in which the capillary energy is replaced by the material fracture energy. Whereas LEC gives the typical curvature induced by capillarity10, Ldry is the critical length above which lamellar structures collapse. When the dimensions of a structure are scaled down by a factor λ, both LEC and Ldry (scaling as λ3/2 and λ5/4, respectively) eventually become smaller than the structure size, an effect that is responsible for damaging microsystem structures2,3,4,5,6,7,8.

To generalize equation (1) to multiple lamellae, we assume that a cluster of N lamellae behaves as a single lamella that is N times more rigid (we neglect solid friction as the wetting liquid lubricates the strips). On average, such a cluster results from the self-similar aggregation of two bundles of size N/2, clamped at a distance Nd/2. The dry length (above the junction of the two clumps) becomes

which is in good agreement with experiment (Fig. 1c). The maximum size Nmax of clusters in a brush with lamellae of length L is given by equation (2), with Ldry=L. However, smaller bundles are also seen if their aggregation with a neighbour leads to a size exceeding Nmax. The broad distribution of clump sizes results from random initial imperfections, and requires statistical analysis. A derivation based on Smoluchowski's coalescence process11 leads to an original self-similar distribution that predicts an average cluster size of 0.67Nmax (A. B. et al., manuscript in preparation).

Our results, once scaled down, could help to improve the design of micro-electromechanical systems. The self-similar aggregation process described here should extend to different geometries (such as those of fibrous materials) and to similar systems involving coalescence or fragmentation.