1 Supplementary information Effect of the viscoelasticity of substrate: In the main text, we indicated the role of the viscoelasticity of substrate. In all problems involving a coupling of a viscous medium (viscosity μ) to an elastic substrate (elasticity E) there is a natural time scale μ/e. From the time scale we can construct a velocity. Our estimates of this speed (mm/sec based on a viscosity 200 Pa.sec, an elasticity 40,000 Pa, and wavelength 4 microns) however are not the same order of magnitude (20 microns/sec) that we measure and we believe that a deeper understanding of stress relaxation in these systems is needed, which is an open question since the great majority of the literature has discussed statics, not dynamics. Note that our experiment is not one dimensional but rather is two dimensional and so involves stretching in addition to bending, which again is an open question. NATURE MATERIALS www.nature.com/naturematerials 1
2 c Supplementary Figure S1. Schematic showing a multi-layered system having a viscous or viscoelastic film capped with a thin elastic layer under isotropic in-plane compression (σ < 0). 2 NATURE MATERIALS www.nature.com/naturematerials
3 Line profile: A-A Wrinkles A A Supplementary Figure S2. a. AFM image of the initial wrinkle having a sinusoidal wave pattern, wavelength λ ~ 4.5 μm (line profile A-A ). a b c 30 Probability, % 25 20 15 10 5 0 100 μm 0 30 60 90 120 α, degree Supplementary Figure S3. The fold branching at the initial state of the wrinkle-to-fold transition is believed to originate from the local orientation of the wrinkle patterns, i.e. disclination defects (a) so that the branching angle is strongly related to the orientation of parallel wrinkles. In our experiments, we measured the distribution of the angle and observed that the angle peaks near 60 (b, c). Note that we have indicated all of the bifurcations with a gold dot in panel b. NATURE MATERIALS www.nature.com/naturematerials 3
4 120 angle (deg) 100 80 Mechanically linked folds 60 0 40 80 120 Time (sec) Supplementary Figure S4. Examples of evolution of the angle of one segmental branch as a function of time. a Fold1 Fold 2 Fold 3 20 μm Interconnection b Fold 1 Fold 2 Fold 3 20 μm Interconnection Supplementary Figure S5. a,b Examples of interconnections of close folds during growth (a: fold-fold interaction, b: tip-tip interaction). Two different folds (Fold 1 and Fold 2) within a certain distance interact to nucleate a new fold along the perpendicular direction to pre-existing folds. Thus, the network forms closed domains. 4 NATURE MATERIALS www.nature.com/naturematerials
5 a b e 55 50 Isotropic wrinkle field Anisotropic winkkel field 45 Pre-existing folds 40 35 c d counts 30 25 20 15 10 5 0 20 40 60 80 100 120 140 160 angles Supplementary Figure S6, a-b, In case of the randomly oriented wrinkle pattern under an equi-biaxial stress, a site of a nucleation of a fold spreads over the entire surface, while pre-existing folds produce a segmental branches ( T junction) of folds. c-d, Thus, initial domains are irregular polygons under the equi-biaxial stress field rather than having an aligned wrinkle pattern under an anisotropic stress field. e, For a quantitative comparison, we measure an angle distribution form the initially formed domains for each case. The angle of the domains, in the case of the aligned wrinkle field, is near 90 degree, whereas in the case of the randomly oriented wrinkle field, the angles widely distribute in the range 40-120 degree. NATURE MATERIALS www.nature.com/naturematerials 5
6 Supplementary Movies caption Movie S1. A fold propagates on disordered initial wrinkles and relax the wrinkles into folds. The surface was first curved with a disordered wrinkle pattern and then one fold is propagating from the bottom side. While the fold advances, the wrinkle patterns are reoriented and then relax through the fold. Although the wrinkle pattern disappears, its trace remains after folding due to the plastic property of the polymer film. Movie S2. Folds nucleate from wrinkles, grow and form a network. This movie reports the dynamics of the initial network shown in Fig. 2a-c. The network is formed by the connection among the folds. During growth of the folds, they are interacting with neighboring folds since the fold localizes the strain, and reorients the initial wrinkle pattern within a transformed zone. Thus, the fold can connect to other folds, forming closed domains. Movie S3. A Self-regulation process during fold propagation in a domain This movie gives a sequence of the tip-tip interaction in a self-regulated manner shown in Fig. 3b. The propagating elliptical shape tip (tip A), which has parallel wrinkles aligned on either side of the tip and the stagnant bucked shape tip (tip B) are shown. The velocity of tip A eventually decreases at which time tip B begins to propagate and relaxes the local wrinkle field. Movie S4. Subdivision of domain through terminal and segmental branches The movie corresponds to the domain division in Fig. 4a. Initially, the domain has a terminal branching at the center of domain. From this branching, the domain is divided 6 NATURE MATERIALS www.nature.com/naturematerials
7 into three domains. Subsequently, the left and right domains are divided into two domains due to a segmental branching of folds in each domain. In this process, once the branch in the left domain is connected to the boundary, a segmental growth occurs at the same position of the boundary toward the right domain. That is, the domains are physically correlated to the neighboring domains as they share an interfacial boundary. The division thus stimulates the neighboring domain to divide. Movie S5. Network of folds between a few pre-existing folds A few folds were pre-formed in a surface. Such an initial condition plays a crucial role for the entire network. A well-oriented initial wrinkle is observed and then the second fold is perpendicular to the initial folds. The final feature has a reticulated network shown in Fig. 5b. NATURE MATERIALS www.nature.com/naturematerials 7