Photo-controlled patterned wrinkling of liquid crystalline polymer films on compliant substrates

Accepted Manuscript

Photo-controlled patterned wrinkling of liquid crystalline polymer films on compliant substrates Chenbo Fu, Fan Xu, Yongzhong Huo PII: DOI: Reference:

S0020-7683(17)30479-1 10.1016/j.ijsolstr.2017.10.018 SAS 9770

To appear in:

International Journal of Solids and Structures

Received date: Revised date: Accepted date:

2 July 2017 29 September 2017 16 October 2017

Please cite this article as: Chenbo Fu, Fan Xu, Yongzhong Huo, Photo-controlled patterned wrinkling of liquid crystalline polymer films on compliant substrates, International Journal of Solids and Structures (2017), doi: 10.1016/j.ijsolstr.2017.10.018

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Highlights • A novel approach to generate surface wrinkles via photo-induced instability was proposed. • Geometric and size effects of photo illuminations on the buckling and post-buckling pattern evolution were carefully explored.

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• Phase diagrams were provided to guide the photo design of surface patterns. • A variety of wrinkling morphologies including wavy shaped, ring-like, checkerboard, stripe, herringbone and hybrid patterns were obtained.

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shielding plates is suggested.

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• A simple and flexible way to design various wrinkling patterns through changing

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Photo-controlled patterned wrinkling of liquid crystalline polymer films on compliant substrates Chenbo Fu, Fan Xu∗ , Yongzhong Huo ∗

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Institute of Mechanics and Computational Engineering, Department of Aeronautics and Astronautics, Fudan University, 220 Handan Road, Shanghai 200433, P.R. China

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Abstract

Photo-chromic liquid crystalline polymer (LCP) is a type of smart materials which are sensitive to light. Here we harness its photo-mechanical response to flexibly control surface patterning, through modeling a film involving homeotropic nematic liquid crystals with director perpendicular to the polymer film attached on a compliant substrate. Theoretical and numerical analyses were conducted to explore the surface instability of such film/substrate systems under both uniform and non-uniform il-

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luminations by ultraviolet (UV) light, respectively. By minimizing energy, the film can buckle into checkerboard patterns at the critical photo load. Fourier spectral method is applied to study the post-

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buckling evolution of various 3D wrinkling patterns including wavy shaped, ring-like, checkerboard, stripe and herringbone patterns. Besides, non-uniform illumination is investigated through square and circular patterned lights, respectively, where the wrinkles can be ordered and controlled remotely and

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flexibly. Furthermore, the geometric and size effects of local illumination are discussed and characterized by some critical dimensionless parameters. Phase diagrams are provided and agree with experimen-

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tal observations, which could be used to guide the photo design of wrinkling morphogenesis that is particularly attractive for remote control applications.

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Keywords: Photo-induced wrinkling; LCP film/substrate; Pattern formation; Post-buckling; Bifurcation.

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Introduction

Surface wrinkles of a stiff thin layer lying on a compliant substrate are abundant both in natural and synthetic systems across length scales such as blooming of hornbeam leaves (Mahadevan and Rica, ∗

Corresponding authors.

E-mail addresses: [email protected] (F. Xu), [email protected] (Y. Huo).

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2005), hierarchical wrinkling of human skins (Efimenko et al., 2005) and fingers (S´aez and Z¨ollner, 2017), differential growth of bacterial biofilms (Zhang et al., 2016), development of human brains (Budday et al., 2014), morphological buckling of fruits and vegetables (Dai and Liu, 2014), and swelling of hydrogels (Basu et al., 2005; Hong et al., 2008; Zhang et al., 2011). These instability phenomena are of growing interest in a number of industrial applications including the fabrication of stretchable electronics (Khang et al., 2006; Li, 2016), micro/nano morphological patterning control (Bowden et al., 1998; Stoop et al., 2015), mechanical property measurement of material characteristics (Howarter and Stafford, 2010), tuning surface adhesion and wettability (Chan et al., 2008; Chung et al., 2007; Lin et al.,

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2008; Lin and Yang, 2009), to the design of moisture-tunable devices (Zeng et al., 2017) and reversible optical functional surface (Zong et al., 2016). In terms of stability analyses, early works are mainly focused on determining the critical conditions of instability threshold (Chen and Hutchinson, 2004; Huang et al., 2005; Song et al., 2008) and it was found that the buckling response depends on material properties (e.g. modulus ratio) and geometric dimensions (e.g. thickness of film). Nevertheless, the

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wrinkling wavelength, wave direction and its amplitude may vary with the increase of applied loads in the post-buckling stage. Due to its well-known complexity, most post-buckling analyses have resorted to numerical approaches since only a limited number of analytical solutions can be obtained in quite simple or simplified cases. During past few years, many efforts have been devoted to 2D or 3D numerical computations of film/substrate instability by using finite element method (Cai et al., 2011; Cao and Hutchinson, 2012; Cao et al., 2012; Chen and Hutchinson, 2004; Jin et al., 2015; Stoop et al., 2015; Sun

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et al., 2012; Xu et al., 2015; Xu and Potier-Ferry, 2016; Xu et al., 2014) or Fourier spectral method (Audoly and Boudaoud, 2008a; Huang et al., 2016, 2015, 2004, 2005; Zong et al., 2016). The former

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is flexible to describe complex geometries and general boundary conditions but more computationally expensive, while the latter actually solves some prescribed differential partial equations (PDEs) with periodic boundary conditions and regular geometries, which can be fairly inexpensive and flexible to

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deal with various loading conditions.

When subjected to in-plane compressions, the film will buckle into various spatial morphologies

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such as sinusoidal, checkerboard, herringbone and hexagonal modes (Audoly and Boudaoud, 2008b,c; Cai et al., 2011; Chen and Hutchinson, 2004; Huang et al., 2005; Xu et al., 2015, 2014), depending on the applied loading, e.g. uniaxial, equi-biaxial or sequent biaxial compressive stresses (Breid and

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Crosby, 2011; Xu et al., 2014). To generate these compressive stresses in the film, three typical ways have been exploited: the first one is mechanical compression (Xu et al., 2014) or pre-stretching of the substrate to induce compression in the film during releasing stage (Cao and Hutchinson, 2012); the second one is thermal heating/cooling residual stress between the film and the substrate (Bowden et al., 1998; Genzer and Groenewold, 2006); and the third one is to introduce a diffusive solvent with a larger molar volume in the film (Chua et al., 2000). Here we propose a more flexible approach to form a wrinkled surface via photo illumination. Light is distinguished as a contactless energy source for micro-scale devices as it can be directed from remote distances, rapidly turned on or off, spatially modulated across length scales, polarized, or varied in intensity (Jeong et al., 2016; Jin et al., 2011; Toh

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et al., 2016). The idea is to use highly cross-linked liquid crystalline polymer (LCP) films containing azobenzene moieties aligned with the nematic host material (Corbett and Warner, 2009; Hayward et al., 2005; Mol et al., 2005; van Oosten et al., 2007; Warner et al., 2010b). The azobenzene moieties are isomerized from the stable trans to the metastable cis state under ultraviolet (UV) illumination, causing the film to contract along and expand perpendicular to the LC director. Besides, these cis molecules will tend to revert to the more stable trans states when they are exposed to longer wavelength radiation. This kind of photo-responsive property is well known in azobenzene-containing polymer systems, e.g. liquid crystalline elastomers (LCEs) and polymer networks (LCNs) that hold high cross-link density

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and elastic modulus (about 1 GPa). In this paper, we focus on these special nematic liquid crystal polymers, especially densely cross-linked LCNs. Photo-sensitive LCPs are a suitable class of materials for many potential applications due to its quick response, reversible shape change and the ability to be controlled remotely (Finkelmann et al., 2001a; Lv et al., 2016; Urayama, 2013; van Oosten et al., 2009; White et al., 2008; Yu et al., 2003). Moreover, LCPs can change their shape reversibly in response to

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temperature (Agrawal et al., 2012; Ohm et al., 2010; Warner et al., 2010a) and electric fields (Kempe et al., 2004) as well. Light-driven LCP actuators, however, offer several advantages over their thermally or electrically driven counterpart, which eliminates the needs for large temperature gradients or electric fields and allows for operation in both wet and dry environments. The potential applications of using light as a viable controlling approach to buckle thin films are only lately being pursued (Fu et al., 2016; Yang and He, 2014, 2015; Zong et al., 2016). A linear stability analysis was conducted in Yang and He

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(2014) to determine the critical wrinkling conditions and buckling mode of glassy nematic films. Then Yang and He (2015) further studied the light-induced wrinkling of glassy twist nematic films attached

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on soft elastic substrates by a kinetics approach and obtained precise wrinkling morphology. Zong et al. (2016) reported a morphological controlling method to tune or erase the surface wrinkles on an azo-containing poly (disperse orange 3) film bonded to a polydimethylsiloxane (PDMS) substrate with

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visible light. The light-induced photoisomerization of azobenzene units in azopolymer films can lead to stress release in the film and consequently to the erasure of wrinkles.

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In the literature on photo-triggered wrinkling of nematic membranes (Yang and He, 2014, 2015), the film is usually exposed to uniform illumination so that the wrinkling morphology remains relatively simple and regular with sinusoidal undulations. Here we consider photo-induced wrinkling of LCP

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films adhered to compliant substrates and investigate the morphological wrinkling and post-buckling evolution under both uniform and non-uniform illuminations with UV light. Fourier spectral method incorporated with photo-generated strain is employed and demonstrates its flexibility for locally patterned photo loading, e.g. periodic circles or squares, to achieve diverse wrinkling patterns. The size

effect of local illumination is characterized by some dimensionless parameters and the corresponding phase diagrams are given. The post-buckling load-displacement curves, energy variations, and pattern evolutions are quantitatively provided as well. Finally several cases of patterned light are performed to demonstrate the flexibility of the proposed photo-controlled approach for obtaining wrinkled surface. Compared with Zong et al. (2016), the method proposed in this paper appears to be more convenient to

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control spatial morphologies. The layout of the paper is arranged as follows. In Section 2, the nonlinear F¨ oppl-von K´ arm´an plate theory with photo strain is adopted to model the film. Then Fourier spectral method is conducted to explore the morphological instability mode transition in the post-buckling stage under both uniform and non-uniform illuminations by UV light. Analytical and numerical results on photo-induced/controlled pattern evolutions are reported in Section 3 and concluding remarks are given in Section 4. Throughout this paper, without special elucidation, a Latin subscript runs from 1 to 3, while a Greek one implies summation over 1 and 2, with repeated subscripts meaning summation. In

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addition, a comma stands for differentiation with suffix coordinate.

Model

2.1

F¨ oppl-von K´ arm´ an plate model with photo strain

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We consider a thin LCP film with the director aligned along the x3 -axis bonded to a compliant thick substrate with periodic boundary conditions, which can buckle under UV light illumination. Geometry of the film/substrate system is shown in Fig. 1. Let x1 and x2 be in-plane coordinates, while x3 is the direction perpendicular to the mean plane of the film/substrate. The LCP film is illuminated normal to the surface by UV light and the director n is along the x3 direction. The width and length of the periodic unit cell are denoted by Lx1 and Lx2 , respectively. The parameters hf , hs and ht represent,

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respectively, the thickness of the film, the substrate and the total thickness of the system. Young’s modulus and Poisson’s ratio of the substrate are respectively denoted by Es and νs , while Ef and νf

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are the corresponding material properties for the film. In the post-buckling stage, the film undergoes large rotations but the strain can be small and thus it is reasonable to assume the film as linear elastic materials (Xu et al., 2014). Thus, we can decompose the total membrane strain into the elastic part

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εeαβ and the spontaneous photo part εsαβ :

εeαβ = εtαβ − εsαβ ,

(1)

εsαβ = εs⊥ δαβ ,

(2)

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in which the photo strain is given by

where δαβ is the Kronecker delta symbol and εs⊥ denotes the spontaneous expansion perpendicular to the director. When the light intensity I0 is low, the light attenuates according to Beer’s law with the penetration depth, and the photo-induced expansion in the film is expressed by (van Oosten et al., 2007) εs⊥ = ϕP⊥ I0 exp [ϕ (x3 − hf /2) /d] ,

(3)

where ϕ is the fraction of azobenzene monomers. P⊥ denotes the photo compliance and d reprents the attenuation distance per azobenzene fraction. According to linear elastic assumption, the stress-strain

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relation of plane stress laws reads 

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 ¯f  {σ} = [C] {εe } = E  νf  0

   εt11 − εs11     0 εt22 − εs22    t (1 − νf )/2  γ12

νf

0

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¯f = Ef /(1 − ν 2 ) and [C] denotes the elastic matrix of the film. where E f

        

,

(4)

When modulus ratio between the film and the substrate is relatively large, e.g. Ef /Es > O(100),

the wavelength is much larger than the film thickness so that the F¨oppl-von K´arm´an nonlinear elastic

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plate theory can adequately describe the light-induced deformation of the film (Huang et al., 2005;

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Landau and Lifshitz, 1959). Let u1 , u2 and w be the displacements of the middle surface (x3 = 0) of

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Figure 1: Periodic unit cell of a thin liquid crystalline polymer film bonded to a soft elastic substrate, illuminated perpendicularly by UV light. The director of homeotropic nematic network follows the x3

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direction.

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the film. The nonzero components of the total strain in the film are written as εtαβ = ε0αβ − x3

∂2w , ∂xα ∂xβ

(5)

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in which the strain components at the reference plane ε0αβ can be defined as   ∂uβ 1 ∂w ∂w 1 ∂uα ε0αβ = + + . 2 ∂xβ ∂xα 2 ∂xα ∂xβ

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The membrane forces and moments of the film can be respectively expressed as     s        N N   11   Z hf /2  11     0  s {N} = {σ} dx3 = = h [C] ε − , N22 N22 f     −hf /2          N   Ns  12 12

(6)

   M  Z hf /2  11 {M} = {σ} x3 dx3 = M22  −hf /2    M 12

        

=−

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       

h3f [C]  12      

∂2w ∂x21 ∂2w ∂x22 2∂ 2 w ∂x1 ∂x2

       

   Ms   11 s − M22         Ms   12 

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        

,

(8)

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s s in which Nαβ and Mαβ are, respectively, the membrane forces and moments caused by the spontaneous

strain, given by s ¯f ε¯s δαβ , Nαβ = (1 + νf )hf E

(9)

s ¯f κs δαβ , Mαβ = (1 + νf ) h2f E

(10)

where the effective average spontaneous strain ε¯s and the effective spontaneous curvature κs are defined as

Z hf /2 1 εs (x3 ) dx3 , hf −hf /2 ⊥ Z hf /2 1 s εs (x3 ) x3 dx3 . κ = 2 hf −hf /2 ⊥ ε¯s =

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(11) (12)

Upon wrinkling, the film elastically buckles to release compressive stresses and remains perfectly bonded to the substrate. At the interface of film/substrate, let T1 and T2 be the shear stresses and T3 be the normal stress imposed on the film by the substrate. The equilibrium requires ∂Nαβ , ∂xβ

T3 =

∂ 2 Mαβ ∂ + ∂xα ∂xβ ∂xβ

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2.2

Tα =

  ∂w Nαβ . ∂xα

(13) (14)

Fourier spectral method

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To solve the PDEs (13) and (14), Fourier spectral method presented in Huang et al. (2005) will be applied to investigate the evolution of various 3D wrinkling patterns in a nematic liquid crystalline

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polymer film/compliant substrate. The Fourier transform of a function g(x1 , x2 ) can be defined as Z ∞Z ∞ gˆ (k1 , k2 ) = F [g(x1 , x2 )] = g (x1 , x2 ) exp (−ik1 x1 − ik2 x2 ) dx1 dx2 , (15) −∞

−∞

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in which the wave numbers k1 and k2 follow the x1 and x2 directions, respectively. In terms of the Fourier component, the elasticity of a half-infinite domain applied on its boundary

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can be solved by elementary methods (Allen, 1969). The Fourier component of the stresses at the interface is given by

Tˆi = Dij u ˆj ,

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in which u ˆj denotes the Fourier component of the displacement vector;    4k 2 (1 − νs ) − k22 /k k1 k2 /k  ¯s (1 − νs )   2  E [D] =  k1 k2 /k 4k (1 − νs ) − k12 /k 6 − 8νs  −2ik1 (1 − 2νs ) −2ik2 (1 − 2νs )

where k =

(16)

2ik1 (1 − 2νs )

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  2ik2 (1 − 2νs )  ,  4k (1 − νs )

(17)

p ¯s = Es /(1−νs2 ). The in-plane equilibrium equation (13) can be reformulated k12 + k22 and E

in the Fourier space as

ˆs Sαβ u ˆβ = ˆbw α − bα ,

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(18)

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in which the stiffness matrix   1¯ 2 Sαβ = Dαβ + E f hf (1 − νf ) k δαβ + (1 + νf ) kα kβ , 2

(19)

ˆs where ˆbw α and bα are the force vectors induced by the deflection and the spontaneous strain, respectively, given by 1¯ ˆbw = −Dα3 w ˆ+ E f hf F [(1 + νf ) w,αγ wγ + (1 − νf ) w,α wγγ ] , α 2   ˆbs = F N s ¯ ˆ¯s . α αβ,β = ikα (1 + νf ) Ef hf ε

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In the Fourier space, the out-of-plane equilibrium equation (14) becomes ! ¯f h3 k 4 E f w ˆ = ikβ F [Nαβ w,α ] − D3α u ˆα + qˆs , D33 + 12

(20) (21)

(22)

in which qˆs is caused by the effective spontaneous curvature κs , given by  s  ¯f h2 (1 + νf ) κ qˆs = F Mαβ,αβ = −k 2 E ˆs . f

(23)

(2005). The iteration procedure reads w ˆ n+1 =

¯f h3 k 4 E f D33 + +ζ 12

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A numerical viscosity is introduced to solve the nonlinear algebraic Eq. (22) as given in Huang et al. !−1

  n n n ikβ F Nαβ w,α − D3α u ˆnα + (ˆ qs ) + ζ w ˆn ,

(24)

where the viscosity is taken as ζ = 0.03Es /hf in computations. Then we update u ˆ1 and u ˆ2 from Eq. (18) as an iteration.

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The mechanical response of film/substrate systems under photo loading is calculated in a periodic square cell in the plane (x1 , x2 ). The computing cell is subdivided into grids and contains 512 grid points

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on each side. The cell must be large enough to accommodate several periodic wrinkles, but the grid spacing must be much smaller than the wrinkling wavelength. Here we choose the cell dimension to be 20 times larger than the wavelength. In computations, we set Poisson’s ratio of the LCP film as νf = 0.3,

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and that of the substrate as νs = 0.4. The film/substrate modulus ratio is set as Ef /Es = 4000. An iterative method with progressively increasing loading is adopted to solve the nonlinear PDEs. A

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random deflection field with magnitude smaller than 0.001hf is prescribed to initiate the iteration and the correction will be stopped when the total potential energy becomes stationary at each incremental

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step.

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Results and discussions

3.1

Uniform illumination

We begin by considering the case where the film is illuminated uniformly by normally incident UV light. In this situation, the photo-induced moment {Ms } is a constant vector in the film and thus does

not contribute to Eq. (14) because of the vanishing derivatives. In addition, ˆbsα and qˆαs are equal to zero in Eqs. (18) and (22), respectively.

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3.1.1

Critical buckling condition for homeotropic LCPs

When the homeotropic nematic liquid crystals hold the director along the x3 direction, the film is isotropic in the x1 -x2 plane, as previously discussed in Fu et al. (2016). According to Eq. (9), one obtains that the spontaneous photo-induced strain of the film is isotropically expansive in both x1 and x2 s s directions, i.e. N ∗ = N11 = N22 > 0. In the pre-buckling stage, the film is under equi-biaxial membrane

compression −N ∗ due to the constraint of the substrate. Chen and Hutchinson (2004) analyzed the critical conditions for the onset of wrinkles based on linearized stability analysis, with the assumption

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on the deflection following a square checkerboard mode expressed as w(x1 , x2 ) = A cos(k1 x1 ) cos(k2 x2 ), where A is the amplitude and k1 , k2 are the wave numbers along the x1 , x2 directions, respectively. By minimizing energy, the analytical solution gives the critical load, amplitude and wave number in functions of the film/substrate modulus ratio, film thickness, or compressive stess. When the substrate is modeled as a semi-infinite 3D elastic solid, the critical membrane force and the corresponding amplitude can be expressed as

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¯f  E ¯f −2/3 hf E , ¯s 4 3E s   8 N A = hf − 1 , 3 + 2νf − νf2 Nc √  ¯f −1/3 2 E 0 kcr = k1 = k2 = . ¯s 2hf 3E Nc =

(25) (26) (27)

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From Eqs. (9) and (25), the critical average spontaneous strain can be obtained:

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ε¯0cr =

1 4 (1 + νf )

 ¯ −2/3 Ef . ¯s 3E

(28)

With the aid of Eqs. (3), (11) and (28), one obtains hf 4d (1 + νf ) [1 − exp (−ϕhf /d)]

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ic0 =

 ¯ −2/3 Ef , ¯s 3E

(29)

where ic0 is the critical dimensionless light intensity and i0 is defined as i0 = P⊥ I0 . Note that the critical

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¯f /E ¯s for given hf /d and ϕ. The smaller the modulus intensity only depends on the modulus ratio E ratio is, the more difficult the film can buckle.

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As an application of the above criterion, we consider a LCN film with an azobenzene-containing monomer adhered to a compliant substrate. For numerical calculations, we choose the related optical parameters as measured in van Oosten et al. (2007): P⊥ = 0.7cm2 W−1 , d = 230nm and ϕ = 2%. ¯f /E ¯s The variations of the critical light intensity ic0 with respect to the film/substrate modulus ratio E

expressed by Eq. (29) is illustrated in Fig. 2a. With the increase of modulus ratio, the critical intensity declines. The critical intensity is strongly affected by the modulus ratio, especially at a relatively ¯f /E ¯s . Besides, for a smaller hf /d, the LCP film is more prone to buckle. The variation of lower E the dimensionless amplitude A/hf with respect to the dimensionless light intensity i0 expressed by Eq. (26) is drawn in Fig. 2b. For hf /d = 0.01, 0.1, 1, 10 and 20, the critical intensities show ic0 = 0.096,

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0.096, 0.097, 0.106, 0.1165, respectively. One can find that the thinner the LCP film is, the smaller the critical buckling intensity is. When the light intensity is smaller than the critical value, the system stays flat with A = 0. When i0 exceeds the instability threshold, it shows supercritical post-buckling bifurcations. Moreover, when hf /d is large, it has a great influence on the critical intensity and the amplitude of wrinkles, while its effect rapidly weakens as hf /d 6 1.

2.5

hf /d = 0.01 hf /d = 0.1 hf /d = 1 hf /d = 10 hf /d = 20

0.3

ic0

0.25 0.2

2

A/hf

0.35

increase hf /d

1.5 1

0.15 0.5

0.05 0

1000

2000

3000

¯f /E ¯s E

increase hf /d

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0.1

hf /d = 0.01 hf /d = 0.1 hf /d = 1 hf /d = 10 hf /d = 20

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0.4

4000

0 0

5000

(a)

0.1

i0

0.2

0.3

(b)

Figure 2: Analytical results of photo-induced wrinkling under uniform illumination: (a) Critical buck-

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¯ f /E ¯s with different dimensionless thicknesses ling light intensity ic0 vs. film/substrate modulus ratio E hf /d = 0.01, 0.1, 1, 10 and 20. (b) The dimensionless amplitude A/hf vs. light intensity i0 under

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different thicknesses. For hf /d = 0.01, 0.1, 1, 10 and 20, the critical intensities read ic0 = 0.096, 0.096,

3.1.2

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0.097, 0.106 and 0.1165, respectively. We set Ef /Es = 4000, νf = 0.3 and νs = 0.4.

Post-buckling evolution with uniform illumination

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To explore the post-buckling morphological evolution of film/substrate systems, the Fourier spectral method (Huang et al., 2016, 2015, 2005) is adopted. The wrinkling pattern evolution of LCP film/substrate illuminated uniformly is depicted in Fig. 3. When the magnitude of the intensity reaches

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the critical value (i.e. ic0 = 0.096), checkerboard patterns occur (see Fig. 3b). The numerical result agrees well with the analytical critical intensity. With the increase of light intensity, the symmetry is quickly broken: the crest bends in one direction. Therefore, the film will buckle into a herringbone mode (see Fig. 3c). In order to make quantitative analyses, we define a root-mean-square (RMS) deflection as RM S =

sP

11

2 wij , N

(30)

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in which N is the total number of the grid points. Upon wrinkling, the potential energy of the film is the sum of membrane part Ufm and bending part Ufb that can be respectively expressed as ZZ 1 m Uf = Nαβ εeαβ dx1 dx2 , 2 Ωf Ufb =

¯f h3 Z Z E f [(1 − νf ) w,αβ w,αβ + νf w,αα w,ββ ] dx1 dx2 . 24 Ωf

(33)

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(b)

(c)

(d)

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(a)

(32)

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The elastic strain energy of substrate Us is given as (Audoly and Boudaoud, 2008a) ZZ q Es (1 − νs ) Us = k12 + k22 w ˆ (k1 , k2 ) w ˆ (−k1 , −k2 ) dk1 dk2 . (1 + νs ) (3 − 4νs ) Ωs

(31)

Figure 3: Post-buckling pattern evolution of the homeotropic nematic LCP film/substrate with the increase of light intensity (dimensionless thickness hf /d = 0.1), presented by the dimensionless deflection w/hf . When the light intensity i0 < ic0 , the film is flat. The checkerboard pattern emerges at the the critical intensity value, and then bifurcates to be herringbone patterns with the increasing load. Post-buckling bifurcation diagram of the homeotropic nematic LCP film/substrate exposed to uniform illumination is shown in Fig. 4. The fundamental solution bifurcates with appearance of checkerboard patterns at the critical value of i0 . The post-buckling range of checkerboard mode is rather narrow

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with respect to light intensity i0 . The second bifurcation appears at i0 = 0.098 with a mode transition from checkerboard to herringbone. According to the variation of energy in Fig. 5, the energy induced by the increase of the spontaneous photo strain is released by the magnification of the amplitude of wrinkles. In the pre-buckling stage, the membrane energy dominates and the system experiences mainly in-plane deformation. When the load reaches the critical value, the bending energy of the film Ufb and the energy of the substrate Us suddenly increases, with the buckling of the film/substrate system. The energy ratio of wrinkle/flat, Utot /Uf lat , however, decreases at the bifurcations (see Fig. 6) where the stretching energy is released with a transition of instability modes. In the post-buckling stage, the

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stretching energy Ufm varies much more slowly than bending energy Ufb . 0.8 0.7

0.15 0.1

0.5 0.4

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RM S/hf

0.6

2nd bifurcation

0.05

0.3 0.2

1st bifurcation 0 0.09350.0950.09650.0980.0995

0 0

0.035

0.07

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0.1

i0

0.105

0.14

0.175

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Figure 4: Bifurcation diagram of the homeotropic nematic LCP film/substrate under uniform illumination: RMS deflection vs. light intensity. Two bifurcations are observed. When the load reaches

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bifurcation points, the RMS amplitude suddenly increases. Each point corresponds to one incremental

3.2

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step.

Patterned light

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Fourier spectral method will be further applied to calculate the buckling and post-buckling of 3D

film/substrate system under a distributed light i0 = i0 (x1 , x2 ). Local illumination is a flexible and frequently used way to deform LCP samples (Cheng et al., 2010; Fu et al., 2016; Jin et al., 2011). Here a patterned shielding plate is set to realize this local loading (see Fig. 7). Let us first analyze the mechanism of size effects of photo loading on wrinkling morphology and then demonstrate various light-controlled wrinkling of LCP films on compliant substrates for pattern design. Local wrinkling patterns are computed in a periodic square cell in the x1 -x2 plane, whose length is denoted as L0 . In the local illuminated square area with width ls (see Fig. 7a) or a circular area with diameter lc (see Fig. 7b), the films are loaded uniformly with dimensionless light intensity i0 . In

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−5 −7

3.5

x 10 6.9

2

3

¯f ) 2U /(hf E

−6

3 x 10 2nd bifurcation

6.8 6.7

1

2.5 2

0 0.0945

1.5 1

Ufm Ufb

0.5

Us Utot

0 0

6.6 0.098 1st bifurcation

0.035

0.07

i0

0.0965

0.105

0.098

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4 x 10

0.14

0.175

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Figure 5: Energy evolutions of the film/substrate system. The membrane, bending and substrate energies are given respectively by Eqs. (31) to (33). Before the load reaches the critical value, the bending and substrate energies Ufb , Us are both equal to 0, i.e. Utot = Ufm . When the load reaches bifurcation points, the bending and substrate energies Ufb , Us suddenly increase. In the post-buckling stage, the stretching energy Ufm varies much more slowly than bending energy Ufb . Each point corresponds to one

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incremental step.

1

PT 1

0.9

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Utot /Uf lat

0.95

1st bifurcation

0.9995

0.85

0.999 0.8 0

2nd bifurcation 0.095 0.095750.09650.09725 0.098 0.035

0.07

i0

0.105

0.14

0.175

Figure 6: Variation of the dimensionless total energy. Uf lat refers to the energy of flat state. When the load reaches bifurcation points, the dimensionless total energy declines. Each point corresponds to one incremental step.

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order to make quantitative analyses, we introduce two dimensionless parameters l/L0 and l/hf , where l = ls for the square illumination and l = lc for the circular illumination. The same modulus ratio

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Ef /Es = 4000 and thickness hf /d = 0.1 are kept as those in Section 3.1.2 for global illumination.

(b)

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(a)

Figure 7: Patterned light generated by shielding plates: (a) Periodic square holes. (b) Periodic circular holes. Light can go through the holes but cannot penetrate the shielding parts.

Critical wrinkling analysis

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3.2.1

Although some key parameters have been identified, it is still quite difficult to have an explicit

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expression for patterned light. In order to promote the application of this model, we give approximate expressions of the critical average photo-induced strain ε¯scr and the number of waves ncr according to numerical results. Inspired by the previous works in the literature (Huang et al., 2015; Li et al., 2012),

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one can assume the following fitting solutions:

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ncr =

l l k0 l = C˜1 0 = C˜1 cr = C1 λcr λcr 2π

ε¯scr

= C˜2 = C2





l hf l hf

a1 a1



l hf

  ¯ −1/3 Ef , ¯s E

(34)

exp (−b1 l/L0 ) ε¯0cr  ¯ −2/3 Ef exp (−b1 l/L0 ) ¯ , Es

(35)

where C1 , C2 , a1 and b1 are constants as yet to be determined. λcr is the critical wavelength under

0 patterned light, while λ0cr , kcr and ε¯0cr respectively denote the critical wavelength, wave number and

average photo strain under uniform illumination, as given in Section 3.1.1. Square illumination To demonstrate that the solutions in Eqs. (34) and (35) work in a wide range of geometric parameters, we vary the geometric parameters ls /L0 in the large range from 0.1 to 0.9 and ls /hf from 200 to 2000. Through linear regression fitting of the numerical data, one can get the approximate values of

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these constants: C1 = 0.19, C2 = 2.61, a1 = −0.12 and b1 = 1.15. The critical number of waves ncr and

photo-induced strain ε¯scr are shown in Figs. 8a and 8b, respectively. Therefore, once the dimensionless

lengths ls /L0 and ls /hf are given, one can easily predict the occurrence of wrinkling and the number of wrinkles through Eqs. (34) and (35). Fig. 9 shows some representative initial wrinkling patterns with different dimensionless lengths ls /L0 and ls /hf , where the color contour represents the dimensionless deflection w/hf . For all the cases, wrinkles emerge around the corners due to stress concentration. The size effects of ls /hf are demonstrated in Figs. 9a-9c, where ls /L0 is fixed as ls /L0 = 0.4. For ls /hf = 400, 800, and 1600, the wavelength remains unchanged but the number of wrinkles shows

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ncr = 5, 10 and 19, respectively. It is found that the number of wrinkles is approximately proportional to the dimensionless length ls /hf . The size effects of ls /L0 are explored in Figs. 9d-9f, where ls /hf is fixed as ls /hf = 700. For ls /L0 = 0.3, 0.5 and 0.7, the number of waves is constantly equal to 9. It can be concluded that the dimensionless length ls /L0 has no influence on the number of waves ncr . −3

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Fitting curve ls /L0 = 0.4 L0 /hf = 2000

x 10

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20

6 5

Fitting curve ls /L0 = 0.4 L0 /hf = 2000 ls /hf = 600

4

ε¯scr

ncr

15

3

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10

0 0

50

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5

100 ¯f −1/3 E ls /hf ( E¯s )

2 1 0.5

150

1

1.5 ¯ E

e−b1 ls /L0 ( hlsf )a1 ( E¯fs )−2/3 (b)

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(a)

2

−3

x 10

Figure 8: Linear fitting curves based on Eqs. (34) and (35) for square illumination: (a) The critical

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¯f /E ¯s )−1/3 . (b) The critical light-induced strain ε¯scr , where wave number ncr , where ncr = 0.19ls /hf (E ¯f /E ¯s )−2/3 . ε¯scr = 2.61 exp(−1.15ls /L0 )(ls /hf )−0.12 (E

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Circular illumination For circular illumination, we also vary the geometric parameters lc /L0 in the large range from

0.1 to 0.9 and lc /hf from 200 to 2000 to fit the solutions in Eqs. (34) and (35). Through linear

regression fitting of the numerical data, one can obtain the approximate values of these constants: C1 = 0.23, C2 = 1.545, a1 = 0.0173 and b1 = 1.523. The critical number of wrinkles ncr and average

photo-induced strain ε¯scr are shown in Figs. 10a and 10b, respectively. The constant a1 is so close to zero that the dimensionless diameter lc /hf has rather limited influence on critical strain ε¯scr . Fig. 11 shows some initial ring-like wrinkling patterns with different dimensionless lengths lc /L0 and lc /hf , also presented by the dimensionless deflection w/hf . The size effects of lc /hf are shown in Figs. 11a-11c

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(b)

(c)

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(d)

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(a)

(e)

(f)

Figure 9: Critical wrinkling patterns with different geometric parameters under square illumination.

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(a)-(c) Size effects of the dimensionless length ls /hf of the illuminated region with ls /L0 = 0.4. For ls /hf = 400, 800, and 1600, the number of waves shows ncr = 5, 10 and 19, respectively. (d)-(f) Effect

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of the non-dimensional length ls /L0 with ls /hf = 700. For ls /L0 = 0.3, 0.5, and 0.7, the number of waves equals to 9.

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with lc /L0 = 0.4. For lc /hf = 400, 800, and 1200, the wavelength remains unchanged but the number of wrinkles reads ncr = 6, 12 and 18, respectively. One finds that the number of wrinkles is approximately proportional to the dimensionless length lc /hf as well as the situation of square illumination. The size effects of lc /L0 are shown in Figs. 11d-11f, where lc /hf is fixed to be 600. For lc /L0 = 0.2, 0.4, and 0.6, the number of waves is fixed to be 9. Namely, the dimensionless length lc /L0 has no influence on the number of waves ncr for the circular illumination as well. Moreover, with the increase of lc /hf or lc /L0 , the boundary effect becomes more and more significant, while its effect is rather limited when

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lc /hf and lc /L0 are small.

−3

7

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Fitting curve lc /L0 = 0.4 L0 /hf = 2000

x 10

6

20

5

ncr

ε¯scr

16

Fitting curve lc /L0 = 0.4 lc /hf = 600

3

8

2

4 20

40

60

80 ¯ E lc /hf ( E¯fs )−1/3

(a)

100

1 1

120

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0 0

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4

12

e

2 −b1 lc /L0

3 4 ¯f −2/3 lc a1 E ( hf ) ( E¯s )

5

−3

x 10

(b)

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Figure 10: Linear fitting curves based on Eqs. (34) and (35) for circular illumination: (a) The critical ¯f /E ¯s )−1/3 . (b) The critical light-induced strain ε¯scr , number of waves ncr , where ncr = 0.23lc /hf (E

Post-buckling evolution with local illumination

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3.2.2

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¯ f /E ¯s )−2/3 . where ε¯scr = 1.545 exp(−1.523lc /L0 )(lc /hf )0.0173 (E

Square illumination

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We examine the effect of irradiation size on post-buckling behavior of the film/substrate system

subjected to square illumination. Let us start with geometric parameters ls /L0 = 0.5 and ls /hf = 1000 as an example. The morphological evolutions with increasing light intensity are plotted in Fig. 12. Wavy wrinkling patterns first appear near the corners of the square and then propagate towards the center of the film along the diagonal directions. In the center of the film where the stress filed is equi-biaxial compression, checkerboard patterns dominate. Only one bifurcation is observed in the load-displacement curve (see Fig. 13). The second example is dedicated to the geometric parameters ls /L0 = 0.7 and ls /hf = 1400. Similar phenomenon is found around the critical intensity. While the intensity is high enough (e.g. i0 = 0.208), a secondary bifurcation appears with stripes around the

18

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(b)

(c)

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(d)

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(a)

(e)

(f)

Figure 11: Critical wrinkling patterns with different geometric parameters under circular illumination.

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(a)-(c) Size effect of the dimensionless length lc /hf of the illuminated region with lc /L0 = 0.4. For lc /hf = 400, 800, and 1200, the number of waves shows ncr = 6, 12 and 18, respectively. (d)-(f) Effect

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of the non-dimensional length lc /L0 with lc /hf = 600. For lc /L0 = 0.2, 0.4, and 0.6, the number of waves equals to 9.

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perimeter of the square and herringbones in the center. This is because along the perimeter of the square, the stress in the normal direction is partially relaxed, while in the center the stress field is similar to the uniform illumination case in Section 3.1.2. The bifurcation diagram is given in Fig. 15. When the light intensity i0 reaches 0.2075, the RMS deflection increases suddenly and the herringbones appear. Overall, pattern selection strongly depends on the geometric size of the illumination region. To ¯f /E ¯s )−1/3 quantitatively reveal the effects of ls /hf and i0 , a phase diagram with respect to ls /hf (E and i0 is provided in Fig. 16. When the light intensity is smaller than the instability threshold, the

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film/substrate system undergoes global bending due to non-uniform illumination. According to Eq. ¯f /E ¯s )−1/3 . It is noticed that the herringbone (34), the number of waves is proportional to ls /hf (E ¯f /E ¯s )−1/3 > 100, with pattern only occurs when the number of waves is large enough, namely ls /hf (E

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the appearance of secondary bifurcations.

(b)

(c)

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(a)

Figure 12: Post-buckling pattern evolution of homeotropic nematic LCP film/substrate system under square illumination with ls /L0 = 0.5 and ls /hf = 1000: (a) Corner mode. (b) and (c) Wavy-

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checkerboard hybrid mode.

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Circular illumination

The post-buckling behavior under circular illumination appears to be more complex compared with the square illumination. Typical wrinkling patterns include ring-like wrinkle, checkerboard, stripe and

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herringbone that are strongly dependent on geometric sizes. When the dimensionless length lc /L0 and lc /hf are both small, e.g. lc /L0 = 0.2 and lc /hf = 400, the ring-like mode is favorable (see Fig. 17).

A bifurcation diagram is illustrated in Fig. 18 and only one bifurcation is found in this case. When the dimensionless length lc /L0 or lc /hf becomes relatively large, e.g. lc /L0 > 0.3 or lc /hf > 500, the wrinkling morphology switches from ring-like wrinkles to checkerboards, then to stripes, and finally transforms to herringbones as shown in Fig. 19. The corresponding bifurcation portrait is given in Fig. 20 where three bifurcations are observed. When the circular illumination becomes rather large to approach the boundary of the computing cell, e.g. lc /L0 = 0.8, the LCP film first buckles into defective rings due to boundary effect and then checkerboard pattern occurs in the center of the circle, and finally

20

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0.4

Fig. 12(c)

0.3

Fig. 12(b)

0.25 0.2 0.15 0.1

Fig. 12(a)

0.05 0.07

0.14

i0

0.21

0.28

0.35

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0 0

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RM S/hf

0.35

Figure 13: Bifurcation curve of homeotropic nematic LCP film/substrate system under square illumination with ls /L0 = 0.5 and ls /hf = 1000: RMS deflection vs. light intensity. Each point corresponds

AC

CE

PT

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to one incremental step.

(a)

(b)

(c)

Figure 14: Post-buckling pattern evolution of homeotropic nematic LCP film/substrate system under square illumination with ls /L0 = 0.7 and ls /hf = 1400. (a) Corner mode. (b) Wavy-checkerboard hybrid mode. (c) Wavy-herringbone hybrid mode.

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0.5

0.3

0.408 0.406

2nd bifurcation

0.404 0.2

0.20720.20740.20760.2078

0.1

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RM S/hf

0.4

1st bifurcation 0.035

0.07

0.105

i0

0.14

0.175

0.21

AN US

0 0

Figure 15: Bifurcation curve of homeotropic nematic LCP film/substrate system under square illumination with ls /L0 = 0.7 and ls /hf = 1400: RMS deflection vs. light intensity. Each point corresponds

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0.35

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to one incremental step.

0.28

CE

i0

III

I

0.21

AC

II

0.14 0

20

40

60

80

100

¯f /E ¯s )−1/3 ls /hf (E

120

140

160

Figure 16: Phase diagram for square illumination. Under different situations, the surface morphology can be global bending (I), wavy-checkerboard hybrid mode (II) or wavy-herringbone hybrid mode (III).

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it transforms to herringbone mode with increasing load as shown in Fig. 21. The bifurcation evolution is plotted in Fig. 22 with three obvious bifurcations. Fig. 23 shows a phase diagram on pattern selection under circular illumination. When the dimensionless length lc /L0 and lc /hf are both small, the wrinkling mode is ring-shaped. Whereas at a large value of lc /L0 or lc /hf , the system may bifurcate into the herringbone mode. Boundary effects will tend to be significant when the circle is approaching the boundary of the film, and the herringbone pattern emerges in the center of the circle. The phase portrait may provide a guideline for the pattern

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design whether the ring-like wrinkling mode or the herringbone mode is preferred.

(a)

(b)

(c)

Figure 17: Post-buckling pattern evolution of homeotropic nematic LCP film/substrate system under

0.25

AC

PT

Fig. 17(c)

0.15

CE

RM S/hf

0.2

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circular illumination with lc /L0 = 0.2 and lc /hf = 400: (a)-(c) Ring-like mode.

Fig. 17(b)

0.1

Fig. 17(a)

0.05

0 0

0.14

0.28

i0

0.42

0.56

0.7

Figure 18: Bifurcation curve of homeotropic nematic LCP film/substrate system under circular illumination with lc /L0 = 0.2 and lc /hf = 400: RMS deflection vs. light intensity. Each point corresponds to one incremental step.

23

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(b)

(c)

(d)

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(a)

Figure 19: Post-buckling pattern evolution of homeotropic nematic LCP film/substrate system under circular illumination with lc /L0 = 0.3 and lc /hf = 600: (a) Ring-like mode. (b) Checkerboard mode. (c) Stripe mode. (d) Herringbone mode.

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0.18

0.1605

0.16 2nd bifurcation

0.12 0.072 0.07 0.1 0.068 0.08 0.066 0.064 0.06

0.1595 3rd bifurcation

0.159

0.3856

0.3115 0.315 0.3185

0.04 0.02 0.14

0.3864

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RM S/hf

0.14

0.16

1st bifurcation 0.175

0.21

0.245

i0

0.28

0.315

0.35

0.385

Figure 20: Bifurcation curve of homeotropic nematic LCP film/substrate system under circular illumi-

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nation with lc /L0 = 0.3 and lc /hf = 600: RMS deflection vs. light intensity. Each point corresponds to one incremental step.

3.2.3

Pattern design

In the last section, we analyzed the shape and size effects of local illumination on the buckling and post-buckling behavior of film/substrate systems. In the present section, we aim to design some

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interesting patterns by patterned light. To manipulate wrinkling patterns, two typical strategies have been exploited: one is to create non-planar substrate topography prior to the deposition of the film

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(Bowden et al., 1998, 1999; Huck et al., 2000; Wang et al., 2015). If the surface of the substrate is structured non-planar at the start of deposition, the wrinkling pattern will be affected by the underlying topography. The other approach is to force a film on a smoothly planar substrate to buckle into a non-

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planar mold that is subsequently removed (Yoo et al., 2002). In both cases the micro structures are usually complex to manipulate and it is difficult to obtain ordered patterns. Here we suggest another

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way to control/tune the wrinkling patterns via patterned light. Compared with the two methods mentioned above, this way appears to be reversible and more flexible for pattern design. Fig. 24 compares qualitatively the diverse wrinkling patterns under patterned light by numerical

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simulation with the experiments in Bowden et al. (1998, 1999). In their experiments, wrinkles were ordered by introducing bas-relief patterns into the PDMS substrate and align perpendicular to the geometric steps on the surface. The rectangles (100µm wide and separated by 500µm), squares (300µm each side) and circles (150µm radius) were elevated by 10 − 20µm to the surface. On cooling, the mismatch of thermal expansion between the PDMS substrate the stiff films generates compressive stresses to buckle the film. No buckling occurs on the plateaux but ordered wrinkling patterns can be observed in the recessed regions (see Figs. 24g, 24h and 24i). Here we demonstrate that the patterned light can replace the bas-relief structure to achieve the same wrinkling patterns by designing some transparent

25

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(b)

(c)

(d)

AC

CE

PT

ED

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(a)

Figure 21: Post-buckling pattern evolution of homeotropic nematic LCP film/substrate system under circular illumination with lc /L0 = 0.8 and lc /hf = 1600: (a) Boundary mode. (b) Circumferential-radial hybrid mode. (c) Boundary-checkerboard hybrid mode. (d) Boundary-herringbone hybrid mode.

26

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0.18

3rd bifurcation 0.162

0.16

0.161 0.1605

0.082

0.1554 0.1556 0.1558

0.08

0.1 0.078 0.076 0.08 0.074 0.14490.14560.1463

1st bifurcation

0.06 0.112

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RM S/hf

0.14 0.12

0.1615

2nd bifurcation

0.119

0.126

0.133

i0

0.14

0.147

0.154

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Figure 22: Bifurcation curve of homeotropic nematic LCP film/substrate system under circular illumination with lc /L0 = 0.8 and lc /hf = 1600. RMS deflection vs. light intensity. Each point corresponds

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to one incremental step.

II

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III

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I

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¯ E

lc /hf ( E¯fs )−1/3

45

ED

60

AC

15

0 0.1

0.2

0.3

0.4

0.5

lc /L0

0.6

0.7

0.8

0.9

Figure 23: Phase diagram for circular illumination. When the dimensionless lengths lc /L0 and lc /hf are

both small, the wrinkling mode is ring-like pattern; whereas at a large lc /L0 or lc /hf , the system may transform into the herringbone mode. The boundary effect becomes significant when lc /L0 is larger than 0.7.

27

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(e)

(f)

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(d)

(c)

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(g)

(h)

(i)

Figure 24: Numerical results regenerate similar patterns observed in the experiments (Bowden et al.,

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1998, 1999): (a)-(c) Transparent sheets with light barriers attached on them. (d)-(f) Computations of wrinkling patterns through patterned light, presented by the dimensionless deflection w/hf . A red spot represents a crest, while a blue spot is a trough. (g)-(i) Experimental observations. Flat rectangles, squares and circles elevated by 10 − 20µm to the surface showed no buckling on the plateaux, but ordered wrinkling patterns appear on the recessed regions. Adapted from Bowden et al. (1998, 1999) with permission.

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sheets with light barriers as shown in Figs. 24a, 24b and 24c. Of course no spontaneous photo strain will be imposed in the shading parts. At the edge of the illumination region, the stress perpendicular to the edge is partially relieved but that parallel to the boundary remains almost unchanged. Due to the same stress distribution, our numerical results in Figs. 24d, 24e and 24f reveal quite similar ordered

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patterns as in the experiments (Bowden et al., 1998, 1999).

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Figure 25: Wrinkling patterns of the letters ‘FDU’ in substrate-bonded LCP films under patterned light. The color contour denotes the dimensionless deflection w/hf : a red spot represents a crest, while a blue spot is a trough.

We can further design complex illuminative regions easily, e.g. the letters ‘FDU’ in Fig. 25. Inside

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the illuminative area, large spontaneous strain is generated in the LCP film, which leads to the local wrinkling of the film. Along the illuminative edge, the stress perpendicular to the boundary is relaxed so that stripes form. Outside the area, the film remains flat. It is expected that such a flexible and smart

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surface patterning controlling method can provide some promising applications especially in situations

Concluding remarks

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4

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where remote operation is desired (van Oosten et al., 2009; White et al., 2008).

This paper proposes a novel method to flexibly form a wrinkled surface via light-induced instability. A LCP film can be reversibly wrinkled under photo illumination. The photo-induced buckling and post-

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buckling behavior of a homeotropic LCP film bonded to a soft elastic substrate has been studied through theoretical analyses and numerical computations. The attention has been focused on the effect of light distribution. Upon uniform illumination with UV light, the system can buckle into checkerboard and herringbone modes with increasing light intensity. The deflection and energy bifurcation diagrams are explored to give quantitative understanding of the post-buckling evolution of pattern formation. When mode transition happens, the total energy of the system declines. Under non-uniform local illumination, a variety of wrinkling morphologies can be obtained, e.g. wavy shaped, ring-like, checkerboard, stripe and herringbone patterns, depending on the light intensity, geometry and size of illumination areas. Approximate analytical expressions are given for critical strain and number of waves. The critical

29

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number of waves is proportional to the dimensionless length l/hf , while the critical photo-induced strain is mainly affected by the dimensionless length l/L0 and decreases with the increase of l/L0 . During post-buckling, the film/substrate system may experience multiple bifurcations with diverse wrinkling patterns and phase diagrams are provided to guide the design of desired instability patterns. The flexibility of photo-induced/controlled surface patterning is highlighted. Under patterned light, wrinkles can be well manipulated and ordered. The results reveal similar patterns observed in the experiments where the structured non-planar substrate must be fabricated a prior (Bowden et al., 1998, 1999). We can further design various wrinkling patterns by flexibly changing the shape of shielding

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plates, which implies a smart way for potential applications especially in situations where remote addressing is desired, e.g. novel biomedical devices (van Oosten et al., 2009), soft robotics (White and Broer, 2015), actuators and sensors (Ohm et al., 2010). More intriguingly, the versatility of LCPs cannot only produce multifarious wrinkling morphologies by flexible light control, but also give a potential way to create more complicated patterns through designing a patterned nematic director distribution,

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e.g. heterogeneous bending of LCP films (White and Broer, 2015).

Acknowledgements

Supports from the National Natural Science Foundation of China (Grant Nos.

11461161008,

11602058 and 11772094), Shanghai Education Development Foundation and Shanghai Municipal Ed-

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ucation Commission (Shanghai Chenguang Program, Grant No. 16CG01), start-up fund from Fudan University, and National Key Research and Development Program of China (2016YFB0700103) are

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acknowledged.

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References Agrawal, A., Luchette, P., Palffy-Muhoray, P., Biswal, S.L., Chapman, W.G., Verduzco, R., 2012. Surface wrinkling in liquid crystal elastomers. Soft Matter 8, 7138–7142. Allen, H.G., 1969. Analysis and Design of Structural Sandwich Panels. Pergamon Press, New York. Audoly, B., Boudaoud, A., 2008a. Buckling of a stiff film bound to a compliant substrate–Part I: Formulation, linear stability of cylindrical patterns, secondary bifurcations. J. Mech. Phys. Solids

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