Electronic Supplementary Material (ESI) for Soft Matter. Thi journal i The Royal Society of Chemitry 15 Online upplementary information Governing Equation For the vicou flow, we aume that the liquid thickne i mall compared to the film length (H L), and by uing the lubrication approximation we obtain: v x p =η x z (S1) where p i the preure in the liquid, v x i the flow velocity in the x-direction, and η i the liquid vicoity; the effect of gravity a a body force i ignored. The vicou hear tre in the liquid i: τ zx vx = η z At the top urface of the liquid layer (z = H) a hear tre i tranferred to the elatic film: (S) T = η v x z=h (S3) At the bottom of the liquid layer, we aume no-lip at the liquid/rubber interface and hence have the boundary condition: vx = ε x at z = (S4) where ε i the train rate of the rubber contraction, with ε < for compreion. The velocity i et to be zero at x = (center of the film) to eliminate rigid-body motion. Auming the preure p i independent of z for the thin liquid layer and integrating Eq. (S1) with the boundary condition in Eq. (S3) and (S4), we obtain: v x (z) = 1 η z(z H) + T z + ε x η (S5) The flow rate in the x-direction i then: H Q x = v x dd = H3 3η + T H + ε xx (S6) η Uing the ma conervation equation for an incompreible fluid: H Q + x = t x (S7) The liquid thickne H i related to the out-of-plane diplacement w of the top urface: H(x) = H + w(x) (S8) Subtituting Eq. (S6) and (S8) into Eq. (S7), we obtain an evolution equation 1
= x = H3 3η H η T ε xx (S9) and the in-plane diplacement u of the top urface i related to the flow velocity in Eq. (S5) a = v x z=h = H η + H T + ε x η (S1) For the elatic film, the equation are imilar to Huang and Suo [13], except that there i no pretrain, but intead there are term correponding to the applied train rate. The in-plane normal tre in the film i: σ = E 1 ν + 1 (S11) where E and ν are Young modulu and Poion ratio of the elatic film, repectively. Subject to the vicou hear tre T acro the film/liquid interface, force balance within the film require that: T = h In addition, a force balance perpendicular to the plane of the film yield: p = D 4 w x 4 σh w T x (S1) (S13) Here D i the flexural rigidity of the elatic film: D = Eh 3 1(1 ν ) The boundary condition to be applied at the end of the film with x = ±L are: No normal tre + 1 = (S14). No bending moment No hear force No preure w x = 3 w x 3 = D 4 w T x4 = (S15) (S16) (S17) Thee are the ame equation a Liang et al, (Acta Mater. 5, 933, ). The above equation can be non-dimenionalized by the caled parameter a decribed in the main text, yielding the dimenionle evolution equation and boundary condition given in the main text (Eq. 1-6).
Linear Perturbation Analyi Uing Eq. and 1 in the main text a perturbation, the diplacement can be written a a um of the bae (hear lag) olution and the perturbation: w = w + w (S18) h p and u = u + u (S19) h p where we denote the hear lag olution with the upercript h. Subtituting thee value for u and w in the governing equation, ubtracting the bae equation i.e. the hear lag olution (ee main text), and neglecting nonlinear term of the perturbation, we obtain the linear evolution equation for the diplacement perturbation. The evolution for w can be written a: 1 p 1 w 1 Ω = + p h 3 h 6 5 B H H w k B H ik B 3 x 1 x 1 3 h h h u u H h 3 w H B H 3 ( ikb) + H w ( k A) + H ( k ia) x x x β x ( ik B ) β B ] (S) Similarly, the correponding evolution of the in-plane diplacement can be written a: Ω = ( H ) p k B p 5 h h h A i Hw H( k A) w ( k A) ibt x 1 (S1) In deriving thee equation, term containing dω /dd have been neglected, which i equivalent to auming that the time-cale over which perturbation grow i very different from the time-cale over which the hear lag olution evolve. The firt derivative of preure i written a: p x p 1 w = ik + N ( ik ) + ( k A ) + T ( k ) 1 3 h 5 h 3 h B B 3 B x w T w + ( k i ) ( ik ) ( k ) h h h A 3 x x B A x (S) and the econd derivative a: p 1 w w 4 h 3 h 6 h 4 h 3 = k B N ( k B) + ( ka) + 3 T ( 4 ik B) + 3( ik A) 3 x 1 x x T w T w + 3 ( ) 3( ) ( ) ( ) x x x x h h h h 3 4 k B k A ik B k ia (S3) 3
Thu Eq. (S41) and (S4) can be repreented a an eigenvalue problem of the form: Ω B A = M 11 M 1 M 1 M B A (S4) where M 11, M 1, M 1, M are complex number, and function of t and k. The eigenvalue of the M matrix are the intantaneou growth rate Ω. The hear lag olution can be regarded a being untable if the real part of the eigenvalue i poitive. The eigenvalue were computed for variou value of k uing MATLAB (Mathwork Inc.). Of the two eigenvalue obtained from the equation, the one with the greater real part i conidered a the dominant growth rate. 4
Quantitative comparion of linear perturbation analyi v experiment Fig. 7 and 11 in the main text repectively howed the fatet growing wavenumber predicted by linear perturbation analyi, and the wavelength oberved experimentally. Fig. S1 compare them directly in non-dimenional form. The ymbol are meaured wavenumber at one pecific liquid thickne, H =.9 mm (H = 36): filled point are ame data a in Fig. 11a, wherea open point are two more experimental run at the ame liquid layer thickne. The olid line i a fit to the k predicted by the linear perturbation analyi, i.e. it i the ame a the olid line in Fig. 7a in the main text. In the experimental range, the linear perturbation analyi typically predict the wavenumber to be roughly twice of what i oberved experimentally. Section 3 in the main text dicued the analogy between the ituation at hand (film being compreed at a pecified rate) and the ituation where a film with a compreive pretrain ε ret on a vicou liquid. Fig. 6 in the main text howed that the diperion relation i imilar in thee two ituation if the pretrain i aigned the value of the intantaneou train. Here we will tet thi analogy againt experiment taking advantage of the fact that in all our experiment, buckle appear at long time (i.e. t τ), when the caled compreive tre ha a value of σ mmm. For the experimental value of H = 36, it i reaonable to take the limit of an infinitely thick liquid layer conidered by Sridhar et al (Appl. Phy. Lett. 78, 48, 1). In that limit, the fatet-growing wave number i predicted to be 4ε (1 + ν). Puruing thi analogy, we replace ε with σ mmm, thu giving the prediction that k = 4 β L (1 + ν). Thi prediction, alo plotted in Fig. S1, i up to 5% higher than the H experiment..6 nordimenional wavenumber k m.5.4.3.9 mm expt thi paper. Sridhar 1 5.E-9 1.E-8 1.5E-8.E-8 β Fig. S1: Dimenionle wave number v dimenionle compreion rate. Solid ymbol are the ame a the.9 mm data in Fig. 11a in the main text. Open point are two more run at the ame liquid layer thickne. Solid line i a fit to the linear perturbation reult, i.e. ame a the olid line in Fig.7a in the main text. For explanation of dahed line, ee above. 5
Numerical imulation A numerical method baed on a finite difference cheme wa adopted, with center difference for pace dicretization and implicit backward difference for time integration. The method i imilar to Liang et al. (Acta Mater. 5, 933, ), uing fictitiou node at each end to atify the boundary condition. For each time tep, the olution wa obtained by iteration uing the Newton-Raphon method. u.5.4.3..1 -.1 -. -.3 -.4 -.5-15 -1-5 5 1 15 x/h w.1.9.8.7.6.5.4.3..1-15 -1-5 5 1 15 x/h x 1-4 8 x 1-7 -1 6 Normal Stre(σ ) - -3-4 -5 Scaled Shear Stre(T ) 4 - -4-6 t =.3τ t =.1τ t =.3τ t = τ t = 3τ -6-15 -1-5 5 1 15 x/h t = 1τ -8-15 -1-5 5 1 15 x/h Fig. S: Comparion of numerical imulation (olid line) with the hear lag olution (ymbol) at hort time (t 1τ ). 6
Wrinkle amplitude (mm).15.1.5 a.1..3.4.5.6.7 t ().5 b -.5-4 - 4.5 c -.5-4 - 4.5 d t =.7 t =.33 t =.6 -.5-4 - 4 x (mm) Fig. S3: Numerical imulation uing the ame parameter a the experiment of Fig. 8 in the main ε =.88 text (L = 75 mm; h = 5 µm; H =.9 mm; ). a. Evolution of wrinkle amplitude with time. The olid line indicate a imple power law: A = k(t t c ).5, with t c =.31. b-d. Simulated wrinkle profile at variou time, correponding to Fig. 8b-d. 1 7
a.5.11-1 -.5-4 - 4.5.9-1 -.5-4 - 4.5.68-1 -.5-4 - 4.5.54-1 -.5-4 - 4.5.43-1 -.5-4 - 4 x (mm) L = 8 mm ; H =.9 mm b.5 8 mm -.5-4 - 4.5 7 mm -.5-4 - 4.5 65 mm -.5-4 - 4.5 6 mm -.5-4 - 4 x (mm).5 H =.9 mm ; ε =.11 1.9 mm -.5-4 - 4.5 1.3 mm -.5-4 - 4.5. mm -.5-4 - 4.5 c.5 mm -.5-4 - 4 x (mm) L = 75 mm ; ε =.9 1 Fig. S4: Simulation approximately matching the condition in the three panel in Fig. 9 in the main text, howing the effect of each of the three parameter on wrinkling. In all graph, the y-axi i the dimenional value (in mm) of the out-of-plane diplacement. Each graph correpond to ε t =.6. 8