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1 Electronc Supplementary Materal (ESI) for Soft Matter. Ths journal s The Royal Socety of Chemstry 2015 Supplemental nformaton Modelng of shape memory polymer sheets that self-fold n response to localzed heatng Russell W. Malen a, Yng Lu b, Mchael D. Dckey b, *, Mohammed Zkry a, *, Jan Genzer b, * a. b. Department of Mechancal and Aerospace Engneerng, NC State Unversty. 911 Oval Drve, Ralegh, NC 27695, USA. Department of Chemcal and Bomolecular Engneerng, NC State Unversty, 911 Partners Way, Ralegh NC 27695, USA. Emal: Mchael_dckey@ncsu.edu, zkry@ncsu.edu, jan_genzer@ncsu.edu 1

2 Our model for the self-foldng of pre-straned polymer sheets ss based on the expermental procedure used n our prevous publcaton. 1 A bref descrpton of the procedure follows. Frst, pattern hnges on Shrnky-Dnk sheets usng a desktop nkjet prnter.. Then cut the samples to the desred sze. Place the sample on a hotplate set to 90 C to brng the sample temperature close to the glass transton (T g ) of the polymer. After the temperature of the sample has stablzed, turn on an unfocused IR heat lamp located approxmatelyy 5 cm above the sample. Record the bendng angle usng a vdeo camera and the hnge surface temperature usng an IR camera. The geometrc model, develop prevously, assumes that the top surface of the hnge shrnks completely durng foldng whle the bottom does not shrnk at all and that the axal shrnkage vares lnearly through the depth of the hnge. 2 The geometrc model also assumes that the thcknesss of the polymer does not ncrease durng the foldng process. Ths assumpton does not account for mass conservaton. Fgure S1a offers a slghtly modfed verson of the smple geometrc model thatt accounts for both top and bottom surface shrnkage accordng to: b 2 tan W 2h 1 S 1 S 2, where W s the wdth of the hnge on the pre-straned materal, h s the thckness of the polymer sheet, and S 1 and S2 are the shrnkage of the top andd bottom surfaces of the hnged regon, respectvely. Shrnkage was defned prevously n Equaton 1. Fgure S1b shows the results of ths model for a sample wth a 2 mm hnge wdth. The α b s maxmzed when the top surface shrnks completely and the bottom surface does not shrnk at all. However, expermental results publshed prevously do not maxmze α b (maxmum expermentall bendng angle α b,max = 100 for a 2 mm hnge wdth). 2 The α b s reducedd by shrnkng of the bottom of the hnge so that one α b can be acheved by a combnaton of top and bottom surface shrnkage. If the bottom surface contnues to shrnk after the top has shrunk completely, the sample unfolds as seen expermentally. (S1) Fgure S1. Alternate geometrc model that accounts for shrnkage of top and bottom hnge surfaces. a) Geometrc defntons. b) Bendng angle ( b ) predcted by alternate geometrc model for a range of top and bottom surface shrnkage values. To model the shrnkage and foldng behavor of thee pre-straned polystyrene polymer sheet, we frst evaluate the vscoelastc propertes of the materal. A TA Instruments AR2000 rheometer wth a torson-bar clamp fxture measured the tme and temperature dependent shear modulus. Pre-shrunk samples were mounted n the fxture and subjected to an oscllatory stran of 0.2% across a frequency sweep from 0.03 Hz Hz. Ths frequency sweep was repeated at 2

3 5 ncrements rangng from 95 to 120C. The reacton force was obtaned as a functon of stran and frequency, and the materal shear storage modulus (G ), shear loss modulus (G ), and phase angle (tan(δ) = G /G ) were calculated. Fgures 4a,c n the paper plot the storage modulus and phase angle as measured by the rheometer, respectvely. A vertcal shft of T g /T (temperatures n Kelvn) has been appled to the storage and loss modulus data used to obtan these plots. These curves can be shfted horzontally n frequency accordng to the tme-temperature superposton prncple to allow the evaluaton of vscoelastc materal propertes at frequences or temperatures that are otherwse not measurable. Once shfted, the sothermal curves make up a vscoelastc master curve at the selected reference temperature. Assumng that the materal s thermo-rheologcally smple, the sothermal curves can be shfted n frequency accordng to the Wllams-Landel-Ferry (WLF) equaton: 3 log a t C T T 1 ref, (S2) C2 T Tref where C 1 and C 2 are emprcal parameters, T s the expermental temperature, and T ref s typcally taken as the glass transton temperature, T g. Reasonable algnment of the shear modulus curves was obtaned wth the standard WLF shft factors of C 1 = and C 2 = and a reference temperature of T ref = T g = 103C. After obtanng the master curve, the tme-temperature dependent behavor at any temperature above the glass transton temperature T g can be determned through further applcaton of the tme-temperature superposton prncple. The vscoelastc master curve obtaned by the tme-temperature superposton prncple can be modeled usng a Prony seres to represent a generalzed Maxwell model 4,5. To do ths, a seres of sutable dmensonless relaxaton modul g and relaxaton tmes τ were calculated to ft a Prony seres to G and G usng a bounded search algorthm mplemented n Matlab. The Prony seres for the storage and loss modul of a vscoelastc materal can be mplemented n ABAQUS 5 accordng to: G' ( f ) G 0 G"( f ) G 1 n 0 1 n 1 g 1 g G 2f 2f 2 n 0 1 g 1 2f 2f 2 2, (S3), (S4) where G 0 s the nstantaneous shear modulus and s the frequency n Hz. The generalzed Maxwell model s valdated by the ft of the model to the expermental storage modulus data (cf. Fgure 4b), and the model s further valdated by the ft of the model to the phase angle data as shown n Fgure 4d. As seen n these plots, the ft of the model begns to deterorate as the materal enters the rubbery plateau and termnal zones of the data (low frequency). Ths s attrbuted to the fact that polystyrene s a thermoplastc that wll flow at hgher temperatures or low frequences (long tmes) even though the molecular weght of Shrnky-Dnks exceeds the entanglement molecular weght of polystyrene. The calculated Prony seres coeffcents are summarzed n Table I. Other materal propertes used as nputs to the fnte element model are lsted n Table II. 3

4 Table I. Prony seres coeffcents for Shrnky-Dnk materal # g τ (s) Table II. Materal propertes used n the fnte element model Property Value Reference Thermo-Rheologcally Smple WLF Parameters Instantaneous Elastc Modulus, E Posson s Rato, ν C 1 = [3] C 2 = 51.6 [3] T ref = T g = 103 C ths work 1.78 x 10 9 Pa ths work 0.33 (T < T g ) [6] (T > T g ) Thermal Conductvty, k 0.14 W/mK [7] Densty, ρ 1050 kg/m 3 [8] Thermal Expanson 2.09 x 10-4 K 1 (T < T g ) [6] 5.65 x 10-4 K 1 (T > T g ) [6] Specfc Heat 1300 J/kgK ths work The n-plane shrnkage model makes use of two planes of symmetry (cf. Fgure S2c). The dmensons of the model pror to shrnkng were 10 mm (length) 10 mm (wdth) 0.3 mm (thckness). Followng the pre-stranng sequence, the model was subjected to a specfed temperature boundary condton on all surfaces that were not planes of symmetry as shown n Fgure S2b. Mechancal dsplacement normal to each plane of symmetry was prevented n addton to a preventon of out-of-plane (Z-drecton) dsplacement of one node at the ntersecton of the two planes of symmetry. 4

5 Fgure S2. In-plane shrnkage model. a) Mechancal boundary condtons durng programmng sequence. b) Mechancal and thermal boundary condton off n-plane shrnkage model durng shrnkng process. c) In-plane foldng model showng model symmetry and mesh. Labels specfy the number of elements that span the ndcated regon. X=10 elements. Y=10 elements. Z=4 elements. We evaluate the effect that coolng rate durng materal prestranng has on the Shrnky-Dnrate) reduced the materal recovery. Increasng the tme to cool the sample (.e., a lower coolng ntal rate of recovery of the sample as seen n Fguree S3. Ths effect s most notceable for sothermal recovery near T g. In the comparson of recovery model results to expermental data, the expermental process of placng the sample n the hot stage whch then must re-establsh the preset temperature then takng the sample out of the hot stage for measurement ntroduces uncertanty, whch has not been quantfed and may account for some of the dscrepances n the comparsons. Addtonally, although the samples nomnally shrnk ~55% n both n-plane drectons, ths value may vary by a few percent from sample to sample as well wth some varaton wthn each sample. Fgure S3. Comparson of effect of short and long coolng tmes on sothermal recovery behavor. Samples programmed by short coolng tme shown as sold lnes. Samples programmed by long coolng tme are shown as dashed lnes. Expermental data are shownn as symbols. 5

6 The out-of-plane foldng model wth one plane of symmetry was shown n Fgure 2. Followng the pre-stranng sequence, the model was subjected to convectve boundary condtons on all surfaces that were not the plane of symmetry. An addtonal thermal boundary condton was appled to the hnge surface to model heatng of the hnge surface. Mechancal dsplacement normal to the plane of symmetry was prevented n addton to fxed dsplacements at specfc locatons on the model as shown n Fgure 2b. Intally, a model for the foldng of a sample wth a 1 mm hnge wdth was developed. To account for the non-unform thermal boundary condtons, a convectve heat transfer coeffcent of h = 5 W/mK and snk temperature of 90C was appled to all surfaces except the face n contact wth the hot plate and the back of the hnge. For the face n contact wth the hot plate, a convectve heat transfer coeffcent of h = 2000 W/mK and snk temperature of 90C was appled to model the thermal contact conductance between the hot plate and the polymer. For the back of the hnge, whch lfts away from the hot plate through the foldng process, a heat transfer coeffcent that transtons from h = 2000 W/mK to h = 5 W/mK across the wdth of the hnge was appled. These thermal boundary condtons were valdated by applyng the expermentally measured hnge temperature to the hnge regon of the model. The model results provde a very reasonable comparson of bendng angle to the expermental results as seen n Fgure S4. For the larger hnge wdths, the same length of transton for the heat transfer coeffcent on the back sde of the hnge as that for the 1 mm hnge wdth model was used, although the locaton of ths transton on the back of the hnge was shfted toward the edge of the hnge nearest the hot-plate. Fgure S4. Bendng angle results obtaned by specfyng expermentally measured hnge temperature for 1 mm hnge wdth model. The sold lne represents model results. Symbols represent expermental results. A sutable model for the IR heat flux was developed based on reported values for the IR heat flux measured by a thermople and comparson of modeled temperature results to the measured hnge temperature. The IR flux of the lght n the expermental setup s ~1,000 mw/cm 2 (=10,000 W/m 2 ); 1 ths value was ntally appled as a constant heat flux on the hnge surface. A model of the 1 mm hnge wdth sample usng ths constant surface heat flux underpredcts the maxmum bendng angle of the hnge. Addtonally, the average temperature of the hnge heated by a surface heat flux (because the temperature s non-unform when heated by a surface heat flux) s 6

7 lower than the expermentally measured temperature. Ths expermentally measured temperature s somewhat msleadng becausee the software for the IR camera smooths the expermental data, and the IR mage s pxelated due to necessary aspects of the expermental setup. Because the bendng angle and temperature were underpredcted, a range of constant surface heat flux values were evaluated. As seen n Fgure S5, no sngle, constant surface heat flux accurately reproduced the bendng angle and average hnge temperature. Instead, a non-constant heat flux was used that rapdly ncreases when the lght s turned on and eventually approaches some constant value. Ths heat flux varaton s representatvee of the mechansm of lght producton n an IR lght where electrcal current flowng through the flament n the lghtbulb ntally causes a rapd heatng of the flament, and after a whle, the flament approaches a constant temperature that yelds a constant heat flux. The tme to reach constant heat flux s long compared to the duraton of the experment. Fgure S5. Constant IR flux results for 1 mm hnge. a) Average hnge temperature results. b) Bendng angle results. Sold lnes represent model results. The symbols represent expermental results. The local, axal shrnkage feld n Fgure 8a was calculated by determnng the shrnkage between two adjacent nodes n the model, and then assgnng that shrnkage to the mdpont between those nodes. Ths axal shrnkage feld movess and rotatess wth the hnge locally as t 7

8 folds. Addtonally, nformaton for the shrnkage profle through the thckness along the foldng angle bsecton lne was extracted from the model by frst selectng two nodes each on the top and bottom of the lfted and non-lfted face of the foldng polymer away from the hnged regon (8 nodes total). The two ntersecton ponts of vectors defned by these sets of nodes are used to defne the foldng angle bsecton vector. These vectors are depcted as gray lnes n Fgure 8a. References 1 Y. Lu, J.K. Boyles, J. Genzer, M.D. Dckey, Soft Matter, 2012, 8, Y. Lu, R. Malen, Y. Zhu, M.D. Dckey, J. Genzer, Phys. Rev. E, 2014, 89, M.L. Wllams, R.F. Landel, J.D. Ferry, J. Am. Chem. Soc., 1955, 77, H.F. Brnson, Polymer engneerng scence and vscoelastcty : an ntroducton, Sprnger, New York, Analyss User's Manual ABAQUS. Dassault Systèmes. 6 Polymer handbook (J. Brandrup, E.H. Immergut, E.A. Grulke, Edtors), Wley- Interscence, New York, N. Demarse, TA Instruments, personal communcaton, G. Natta, J. Polym. Sc., 1955, 16,