Copyright WILEY-VCH Verlag GmbH & Co. KGaA, 69469 Weinheim, Germany, 2014. Supporting Information for Adv. Mater., DOI: 10.1002/adma.201403510 Programming Reversibly Self-Folding Origami with Micropatterned Photo-Crosslinkable Polymer Trilayers Jun-Hee Na, Arthur A. Evans, Jinhye Bae, Maria C. Chiappelli, Christian D. Santangelo, Robert J. Lang, Thomas C. Hull, and Ryan C. Hayward*
Supporting information for: Programming reversibly self-folding origami with micro-patterned photo-crosslinkable polymer trilayers By Jun-Hee Na 1, Arthur A. Evans 2, Jinhye Bae 1, Maria C. Chiappelli 1, Christian D. Santangelo 2, Robert J. Lang 3, Thomas Hull 4, and Ryan C. Hayward 1, 1 Department of Polymer Science and Engineering, University of Massachusetts Amherst, Amherst, 01003 MA, United States. 2 Department of Physics, University of Massachusetts Amherst, Amherst, 01003 MA, United States. 3 Lang Origami, Alamo, 94507 CA, United States. 4 Department of Mathematics, Western New England University, Springfield, 01119 MA, United States. Keywards: self-folding, microscale, programmable matter, origami Correspondence Prof. Ryan C. Hayward (email: hayward@umass.edu) 1 Temperature dependent swelling of PNIPAM hydrogel layers To better understand the swelling-driven deformation of patterned bilayers and trilayers, we first measured the temperature dependent swelling behavior of homogeneously crosslinked disks of PNIPAM, as summarized in Fig. S1. A circular disk of unswelled diameter d 0 was first allowed to swell to its unconstrained equilibrium extent (diameter d) in an aqueous buffer at 22 C, and the swelling quantified in terms of γ = d/d 0, the linear increase in size. Subsequently, the temperature was increased in 2 C increments up to 50 C, with at least 30 min at each temperature to ensure that both the temperature and degree of swelling of the gel had reached equilibrium at each step. 1
Figure S 1: The temperature dependent linear swelling ratio γ of fully crosslinked hydrogel disks. 2 Bilayer bending mechanics In the design of self-folding structures, we consider the swelling-induced bending of PNIPAM/PpMS bilayer films, as summarized in Figure S2. In the absence of the PpMS layer, the PNIPAM film would swell isotropically by a linear extent of γ =1.40 ± 0.06 in each dimension, relative to the dry state, at 22 C (Figure S1). However, bonding to the rigid PpMS layer almost completely prevents in-plane expansion of the PNIPAM layer, leading to a compressive mismatch strain of ε m in the PNIPAM layer. This mismatch strain is partially alleviated by bending into a nearly cylindrical configuration with radius of curvature R. To determine the degree of bending achieved in our materials, we prepared uniform PpMS/PNIPAM bilayer films in the shape of squares or rectangles with in-plane dimensions of 15-200 µm and measured how they deformed upon swelling. After fully swelling to equilibrium in aqueous buffer at room temperature, bent PpMS/PNIPAM bilayers were imaged using LSCM, and the radius of curvature R in the middle of the PNIPAM film was 2
a b hn hp c 50µm Figure S 2: Bilayer bending mechanics. a-b, A schematic illustration of swelling-induced bending of a PNIPAM/PpMS bilayer with respective layer thicknesses hn and hp. c, The normalized curvature R/hN is plotted against the ratio of layer thicknesses (hp /hn ) for four different values of hn, along with the best-fit to Equation 2 (red line), and the zero free-parameter prediction from morpho-elasticity (dotted black line). The inset shows an optical micrograph of the rolled shape adopted by a PNIPAM/PpMS bilayer (originally a 200 200 µm square in the dry state) upon swelling at room temperature in aqueous buffer. extracted after correcting for refractive index mismatch. The normalized radius of curvature R/hN is plotted in Fig. S2c as a function of hp /hn for four values of hn from 1.0-5.5 µm. With this scaling, the data appear to collapse to a single curve, as expected, since the bilayer bending mechanics should not be sensitive to the overall size-scale of the materials (at least not in this 3
regime). A simple model for this behavior is provided by Timoshenko s solution for a bilayer of two linear elastic materials [1], or equivalently by the generalized version of Stoney s equation [2 4], which has been widely applied for modeling the bending of bilayers based on swellable polymer layers connected to rigid films [5 11] and can be written as R = 1 + 4ηξ + 6η2 ξ + 4η 3 ξ + η 4 ξ 2, (1) h N 6ε m ηξ(1 + η) where η = h P /h N is the contrast in thickness and ξ = E P /E N the contrast in plane-strain modulus between the layers. As described in the main text, for the material system used here, E P 4 GPa, while E N 0.8 MPa, so we estimate ξ 5000. On the other hand, η = h P /h N 0.02 0.5 in our experiments. Thus, we always have that ηξ >> 1. This means that the stretching modulus of the PpMS layer is much larger than that of the PNIPAM layer, i.e., so that there is essentially no strain induced in the PpMS film and the PpMS/PNIPAM interface remains almost constant in length. If we consider the limit of small η (but still with ηξ >> 1), then considering the numerator of Equation 1, we see that the terms 1, 6η 2 ξ, and 4η 3 ξ can be ignored compared to 4ηξ. In the denominator, we have (1 + η) 1, leaving us with the simplified approximate form: R = 4ηξ + η4 ξ 2 h N 6ɛηξ = 4 + η3 ξ 6ε m. (2) 4
This form allows for a rather good fit to the data, with best-fit values of ξ = 470 ± 60 and ε m = 0.26 ± 0.02 (red curve in Fig. S2c). However, because the tight curvatures achieved (R/h N 3 at small h P /h N ) violate the assumption of small strains used in deriving this model, and because the gel layer cannot be adequately modeled as a linear elastic material at such large deformations, these fit parameters do not accurately represent the real material properties. A more realistic description is provided by a morpho-elastic model (see below for details) that assumes a multiplicative decomposition of the deformation tensor into contributions from swelling and elastic deformation, and can appropriately capture the nonlinearities of such large deformations and tight bending. With the independently measured values of ξ = 5000 and γ = 1.40, morpho-elasticity yields the dotted black curve in Fig. S2c, which describes the data reasonably well, especially considering that this is a zero free-parameter prediction. However, the data at small η show tighter curvatures than predicted by this model. The use of numerical approaches and more complex constitutive equations for the gel layer [12, 13] to both quantitatively capture the data in Figure S2, and match the independently measured material properties, is an interesting topic for study, but would not provide any further qualitative understanding for the purposes of the current report. Notably, the normalized radii of curvature in Fig. S2c appear to approach a limiting value for small h P /h N, as also predicted by both models. This can be qualitatively understood by considering the form of Equation 2, noting that the term η 3 ξ = (E P h 3 P )/(E Nh 3 N ) represents the ratio of the bending stiffnesses of the two layers. Small values of η 3 ξ correspond to the case where the rigid PpMS layer offers negligible bending resistance, instead simply preventing in-plane expansion along one surface of the PNIPAM film, and hence the radius adopted is determined entirely 5
by how strains are distributed within the PNIPAM film. Thus, R/h N attains a limiting value when η 3 ξ << 1, which according to Equation 2 is simply 2/(3ε m ). For a given thickness h P, this also means that an optimum radius of curvature given by Equation 2 as R = h P ε m ( ) 1/3 ξ, (3) 2 is achieved when η = (2/ξ) 1/3. 3 Morpho-elastic model For a bilayer gel with PpMS thickness h P, PNIPAM thickness h N, and length L, swelling of the PNIPAM forces the initial slab geometry to bend into a cylindrical shape. The stretching modulus of PpMS is very large, and if we assume that it does not change its length after bending, the geometry of the cylinder is given by L F L R rh N Figure S 3: Schematic of bilayer swelling. 6
Rθ 0 = L, (4) (R + h p + λ r h N )θ 0 = λ θ L, (5) where R is the radius of curvature corresponding to the middle surface of the PNIPAM layer, θ 0 the angle swept out by the section, and λ r, λ θ are the principal stretches of the PNIPAM. Using the fundamental assumption of morpho-elasticity [14], we decompose the deformation tensor F into an elastic strain tensor A and a growth tensor G such that: F = G A = γ 0 0 γ α r 0 0 α θ. (6) We write an expression for the curvature of the PpMS layer: C p = λ θ 1 h p + λ r h N. (7) Modeling the elasticity of the PNIPAM as an incompressible Neo-Hookean solid [15], we find that α r α = 1/α θ ; assuming isotropic swelling by a linear factor of γ, the principal stretches are 7
then given by λ θ = γ α, (8) λ r = γα. (9) Since the stretching modulus E p h p of the PpMS layer is so large, the elastic energy is given approximately by plate bending, so that the total energy density of the bilayer system is E = E p h 3 p 24(1 νp) 2 C2 p + µ Nh N 2 (α 2 + 1α 2 2 ). (10) If both materials are incompressible, µ N = E N /3 and ν p = 1/2, such that ( E = 1 E N h N 72 ξη3 1 η + γ α γα2 γ α ) 2 + 1 (α 2 + 1α ) 6 2. (11) 2 By minimizing this energy with respect to the elastic stretch α, we find the optimal radius of curvature set by the balance of swelling and elastic strain. Using the independently determined values of ξ = 5000 and γ = 1.4 we find good qualitative agreement with the data, as shown in Figure S2. While capturing the behavior of bilayers in full quantitative detail most likely requires a more complex constitutive equation for the PNIPAM layer, this simple morpho-elastic approach captures the nonlinear elasticity of the system while qualitatively retaining the volumetric change associated with swelling of the polymer gel. [1] S. Timoshenko, J. Opt. Soc. Am. 1925, 11, 233. 8
[2] G. G. Stoney, Proc. R. Soc. Lond. A 1909, 82, 172. [3] L. B. Freund, J. A. Floro, E. Chason, Appl. Phys. Lett. 1999, 74, 1987. [4] L. B. Freund, S. Suresh, Thin Film Materials: Stress, Defect Formation and Surface Evolution, Cambridge University Press, 2003. [5] E. Palleau, D. Morales, M. D. Dickey, O. D. Velev, Nat. Commun. 2013, 4, 2257. [6] R. M. Erb, J. S. Sander, R. Grisch, A. R. Studart, Nat. Commun. 2013, 4, 1712. [7] W. Guo, M. Li, J. Zhou, Smart Mater. Struct. 2013, 22, 115028. [8] Q. Pei, O. Inganäs, J. Phys. Chem. 1992, 96, 10507. [9] Q. Pei, O. Inganäs, I. Lundström, Smart Mater. Struct. 1993, 2, 1. [10] K. Kumar, V. Luchnikov, B. Nandan, V. Senkovskyy, M. Stamm, Eur. Polym. J. 2008, 44, 4115. [11] W. Li, G. Huang, H. Yan, J. Wang, Y. Yu, X. Hu, X. Wu, Y. Mei, Soft Matter 2012, 8, 7103. [12] S. Alben, B. Balakrisnan, E. Smela, Nano Lett. 2011, 11, 2280. [13] D. S. VanBlarcom, N. A. Peppas, Biomed Microdevices 2011, 13, 829. [14] A. Goriely, M. Ben Amar, Phys. Rev. Lett. 2005, 94, 198103. [15] R. W. Ogden, Nonlinear elastic deformations, Courier Dover Publications, 1997. 9