Electronic Supplementary Information

Effect of NIPAAm concentration (CNIPAAm) on topography of gel surface in the slow heating regime.

S1. A. Fabrication procedure for hybrid surface. B. Effect of NIPAAm concentration in hydrogel (CNIPAAm) on equilibrium morphologies formed by the surface at different temperatures. Scale bar: 125 µm.

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Electronic Supplementary Material (ESI) for Soft Matter.This journal is © The Royal Society of Chemistry 2019

Setup of finite element simulation

In the framework of the gel lattice spring model (gLSM), the structural evolution of

the local volume fraction of polymer, , is governed by the following continuity equation1,2:

, (1))( )( p

tv

where is the polymer velocity. All equations are presented in the dimensionless form, )( pv

assuming that length and time are measured in the respective units of and (see 0L 0T

below). We assume that it is solely the inter-diffusion of the polymer and solvent that

contributes to the gel dynamics and neglect the net velocity of the polymer-solvent

system.1 The latter assumption is specified through the following equation:

, (2)0)1( )()( sp vv

where is the velocity of the solvent. The dynamics of the polymer network is taken to )(sv

be purely relaxational; 1,2 consequently, the forces acting on the deformed gel are

balanced by the frictional drag due to the motion of the solvent. This force balance is

expressed as1:

(3))()(/)(ˆ )()(0

20

10

spB TLTk vvσ

where is the dimensionless stress tensor (measured in units of ), is the σ TkB10 0

volume of a monomeric unit, and T is temperature. The prefactor in eq. 1 20 0 0( ) /Bk T L T

(3) is used to express this equation in dimensionless form. The polymer-solvent friction coefficient, , is approximated as 3:)(

(4)2/300 )/)(()(

where is the local volume fraction of the polymer in the undeformed gel. This 0

approximation is valid in the semi-dilute and intermediate regimes (i.e., , which is 5.0

2

always satisfied in the following calculations). If the stress tensor is known, then Eqs. σ

(2)-(4) give:

(5)σv ˆ)/)(1( 2/300

)( p

where is the dimensionless kinetic coefficient. 10

20000 )/)(( TLTkB

The stress tensor is obtained from the energy density of the deformed gel, σ

, 4 and is given as ),,( 321 IIIu

(6)12

2/131

2/133322

2/13

ˆ2ˆ2ˆ)(2ˆ BBIσ wIwIIwIwI

Here, is the unit tensor and is the left Cauchy-Green strain tensor. The invariants of I B

this strain tensor, (i=1,2,3), are specified as: iI

. Additionally, . BBBB ˆdet)],ˆtr()ˆtr[(21,ˆtr 3

2221 III ),,( 321 IIIu

Iw

ii

The energy density consists of the elastic energy density associated ),,( 321 IIIu

with the deformation of the gel, , and the polymer-solvent interaction energy density, elu

:FHu

(7))(),( 331el IuIIuu FH

The elastic energy describes the rubber elasticity of the crosslinked polymer network elu

5,6 and is proportional to the crosslink density in the undeformed polymer network : 0c

(8))ln3(2

2/131

00el II

cu

The expression for takes the following Flory-Huggins form: 5FHu

(9))]1(),()1ln()1[(2/13 TIu FHFH

3

Here, is the polymer-solvent interaction parameter. The term ),( TFH 02/1

3 /I

appears in the equation because the energy density is defined by a unit volume in the

undeformed state.1

Substituting Eqs. (7)-(9) into Eq. (6) yields the following relationship between the

stress ( ) and strain ( ) tensors: σ B

(10)BIσ ˆˆ)(ˆ0

00 cP

The pressure is defined as )(P

(11)1000

2 )2(])()1ln([)( cP

where . Here, , with and being the respective 10)( TksThT B/)()(0 h s

changes in the enthalpy and entropy per monomeric unit of the gel. 7 The stress tensor

can be decomposed as the sum of the hydrostatic and deviatoric stress tensors; the latter

terms correspond, respectively, to the volume change and shape distortion of the gel

element. Since the motion in the system is due solely to the inter-diffusion of the solvent

and polymer, the hydrostatic pressure acting on the solvent is related to the hydrostatic

stress on the gel element as

(12)

The above equations are solved numerically using our gLSM (as detailed in

reference 5). In this approach, a 3D deformable gel is represented by a set of general

linear hexahedral elements. Initially, the sample is undeformed and consists of

identical cubic elements, where is the number of nodes in )1()1()1( zyx LLL iL

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the -direction, ; the linear size of the elements in the undistorted state with i zyxi ,,

is set to . 0 0.1286 10 L

The gel is attached to the solid bump, which is modeled by frozen gel nodes where

the node velocities are set to zero and the polymer volume fraction is fixed at . bump 1

The profile of the bump is a truncated circle and is discretized in the simulation as

illustrated in Fig. 3D, where the color bar represents the polymer volume fraction.

We utilize parameter values that are based on the experimental data for poly(N-

isopropylacrylamide) (PNIPAAm), which displays a LCST.7 The polymer volume fraction

in the undeformed gel is set to and the crosslink density in the undeformed 0 0.1286

polymer network is . For the gel-solvent interaction parameters, we set c0 5.76104

, and , where , and . In Eq. (5), 518.01 TksThT

B

)(0

14104.12 h 16107.4 s

we use the dimensionless kinetic coefficient .8,9 We set our units of length and 1000

time to be and , respectively, to ensure that the value of the diffusion L0 5m T0 1.25s

coefficient is .10 The physical size of the system with L02 /T0 210

11cm2s-1 1011132

elements at is . The length of a solid bump is 500m 50m155m W0 500m

and the height . The thickness of the gel on the top of the bump is . h0 145m 10m

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Effect of setpoint temperature on the morphology of transient states in the fast heating regime.

Fig S2. A. Sideview confocal z-stack images show that at a temperature setpoint of 34°C, the gel transitions passes through a double peak state and continues to transition smoothly from non-inverted to inverted. By contrast, at a temperature setpoint of 36°C or 38°C, the surface undergoes a transition through morphology with multiple peak points before full inversion. Scale bar: 125µm. B. Top view images show that a well-defined region of bubbles appear on the surface at 3.5 mins from the start of the temperature ramp during the inversion process for temperature setpoints of 36°C or 38°C. Scale bar: 300µm.

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Effect of temperature ramp direction on the surface inversion trajectory.

Fig S3. A. Comparison between gel surface deformation process during heating and cooling at slow heating rate of R=0.2°C/min. Small differences in the surface morphology are observed between 31 and 36°C, indicating hysteresis (e.g. see arrow). B. Comparison in the surface morphology change as a result of heating vs cooling at 5°C/min. While both processes cause the surface to undergo a double-peak transition, the peak points are offset by half a bump-to-bump periodicity. Height profiles extracted from confocal images (right) provide quantification of this difference. B. Raw confocal data showing the shape change process at critical timepoints. Scale: 125µm. C. The effect of the shape change on the movement of particles. Scale: 200µm.

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Effect of preswollen gel at bump sites (hbump) on the formation of bubble instabilities at bump sites.

Fig S4. Effect of preswollen gel height hbump on the formation of bubble instabilities at the bump sites upon heating to Tsp=36°C at ramp rate of R=5°C/min. (A) Fabrication of sample with linear gradient in preswollen gel height hbump across the length of the sample. (B) Bubbles begin to form at the bump site as hbump increases (see black arrows). Scale: 750µm. (C) The number of bubbles increases with hbump and reaches a plateau at hbump~50µm. (D) Visualization of the top surface of the gel during the contraction process reveals the formation of an ordered array of bumps at the bump sites. Scale: 500µm.

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