Light-induced deformation in a liquid crystal elastomer photonic crystal

Author's Accepted Manuscript

Light-induced deformation in a liquid crystal elastomer photonic crystal D. Krishnan, H.T. Johnson

www.elsevier.com/locate/jmps

PII: DOI: Reference:

S0022-5096(13)00166-X http://dx.doi.org/10.1016/j.jmps.2013.08.013 MPS2344

To appear in:

Journal of the Mechanics and Physics of Solids

Received date: 1 April 2013 Revised date: 11 July 2013 Accepted date: 23 August 2013 Cite this article as: D. Krishnan, H.T. Johnson, Light-induced deformation in a liquid crystal elastomer photonic crystal, Journal of the Mechanics and Physics of Solids, http://dx.doi.org/10.1016/j.jmps.2013.08.013 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Light-induced deformation in a liquid crystal elastomer photonic crystal D. Krishnan, H. T. Johnsona) Department of Mechanical Science and Engineering, 1206 W. Green Street, Urbana, Illinois 61801 University of Illinois, Urbana – Champaign, Illinois, USA a) [email protected], Ph: +1-217-265-5468, Fax: +1-217-244-6534

Abstract Elastomer materials can undergo large, reversible elastic deformation, and offer novel possibilities for coupled optomechanical behavior when light itself is used to induce that deformation. This phenomenology is especially interesting to consider when photonic bandstructure effects and mechanical instabilities are present over the same length scales. Here we investigate a novel, coupled optomechanical material behavior whereby complex deformation, with the potential to occur cyclically, occurs in a soft photonic crystal structure due to a mechanical instability, as a result of constant, uniform illumination by normally incident light. We suppose that the base material for the structure is a material that responds to light by undergoing a microstructural change. Such a behavior is observed, for example, in a liquid crystal elastomer containing azobenzene moieties attached to the liquid crystal main-chains;(Finkelmann et al, 2001) transformational strain generated by the effect of localized light energy on the isomerization of the azobenzene moieties can be calculated from an order-parameter based model.(Hogan et al, 2002) Under uniform exposure to constant illumination, the interaction between the light, the material, and the deforming structure lead to a complex, reversible deformation sequence. We analyze the electromagnetic energy distribution inside this photonic crystal structure by solving Maxwell’s equations for the electromagnetic problem of light transmittance using finite element analysis. First, upon contraction of the structure due to isomerization in the uniformly illuminated material, the photonic bandstructure shifts, thereby significantly reducing the average illumination of material within the structure. The locally reduced illumination allows for a relaxation of the strain in some parts of the structure, due to the reversible isomerization at room temperature. Then, as a result of this relaxation, the structure is subjected to uniaxial stress, leading to a mechanical instability that triggers a geometrical pattern transformation. This in turn produces a second contractile deformation, as a result of the buckling-like deformation in the structure. Finally, the highly nonuniform local strain field that results generates a dramatic change in the photonic bandstructure of the system, leading to a new localization of the light that tends to reverse the effect of pattern transformation. This completes the transformation sequence, demonstrating the potential for cyclical deformation induced simply by uniform illumination. The

coupled optomechanical material/structure behavior observed here could lead to applications in optically sensors, energy harvesters, or other reversible optomechanically active structures.

1 Introduction Structures consisting of soft materials such as elastomers and hydrogels can be made to undergo large, reversible deformation due to the effects of various external stimuli, and they have use in a variety of applications including bio-MEMS devices, electro-active polymer actuators, and chemical or optical sensors. Numerous recent studies have explored the mechanics of soft materials in periodic structures that undergo buckling-like instabilities leading to symmetry transformations. Mullin and co-workers (2007) showed using finite element analysis, for example, that periodic two-dimensional elastomeric structures can be forced into symmetry-breaking buckling-like deformation patterns.(Mullin et al, 2007) This work was followed by several studies by Bertoldi and co-workers on the compressive deformation response of such structures.(Bertoldi and Boyce, 2008; Bertoldi et al 2008) Lee and co-workers and later Zhang et al reported analogous 3D and 2D pattern transformations under large, reversible deformation conditions in hydrogel materials.(Lee et al, 2006; Zhang et al 2008) Most of the recent work on large deformation of soft materials in periodic structures has focused on the effects of mechanical loads applied directly to the boundaries, or on chemical or thermal swelling of the base material.(Zhu et al, 2012) Recent progress in theoretical and experimental studies of dielectric elastomers has demonstrated the possibility of electrostatic actuation of periodic structures with instabilities(Li et al, 2013; Bertoldi and Gei, 2012; Zhao and Suo, 2007; Suo, 2010; Li and Landis, 2012) The primary objective of these studies has been to accurately predict the deformation through appropriate constitutive modeling and robust finite deformation computational algorithms. In a few other recent studies, however, the focus has been to predict the effects of symmetry-breaking deformation modes on acoustic (phononic) or optical (photonic) wave propagation through the material.(Robillard et al, 2009; Krishnan and Johnson, 2009) Our own study showed that photonic bandstructure can be significantly modified by buckling-like deformation modes. In the present work we consider a novel extension to this concept that introduces a new aspect to the mechanics of periodic soft structures: that of optical actuation of the buckling-like deformation modes, in the presence of photonic bandgaps. Such a configuration presents the possibility of a complex deformation behavior mediated by both structural characteristics (e.g. manufactured periodicity) and material characteristics (e.g. light-induced strain), in a regime with two-way optical-mechanical coupling.

The potential applications of such soft structures include sensors, actuators, and even energy harvesters. Building on the recent work on deformation modes of soft structures, the challenges in designing such new configurations lie in appropriately selecting and accurately modeling the material, and in coupling the material and structural models. One particularly interesting soft material to consider for use in large-deformation applications is a liquid crystal elastomer (LCE) that can be functionalized with azobenzene, a widely used compound that photoisomerizes from a trans- to cis- configuration under optical irradiation, producing macroscopic strains and deformation. (Blair et al, 1980) This forward reaction from a stable configuration can be reversed either optically or thermally, as the cis-configuration is considered metastable. The behavior of liquid crystal polymers with azobenzene side-chain moieties depends on several factors including the light intensity incident on the material,(Wu et al, 1998) the dynamics of the alignment process of the azobenzene moieties(Wu et al, 1999a) and the azobenzene chemistry itself. (Wu et al, 1999b) Moreover the material is sensitive to the polarization of the incident light as observed by Harvey and Terentjev in their study on nematic elastomers.(Harvey and Terentjev, 2007) It is also known that the direction of contraction of a sheet of an azobenzene liquid crystal elastomer is parallel to the polarization of incident light,(Yu et al, 2003) which may be useful in numerous applications such as actuators,(van Oosten et al, 2008; White et al, 2008; White et al, 2009) artificial muscles and shape memory materials,(Ikeda et al 2007; Barrett et al 2007) heterojunctions,(Hwang et al, 2005) diffraction gratings(Bai and Zhao, 2002) and as optically stimulated mechanical devices.(Koerner et al, 2008; Tajbakhsh and Terentjev, 2001; Dunn and Maute, 2009) Finkelmann et al. report on the first LCE that displays a large macroscopic contraction of 22% due to the incident light.(Finkelmann et al, 2001) To predict the compressive strain, they map the optically induced change in the nematic order as an effective shift in the nematic-isotropic transition temperature of the system. The temperature-strain effect is studied to validate the use of order transition theories in the mapping process.(Tajbakhsh and Terentjev, 2001) Detailed investigations on the coupling of liquid crystals to the main chain of the elastomer with the orientational behavior indicate that the properties depend on the type of the liquid crystal mesogens and the Young’s modulus of the elastomer. (Krause et al, 2009) The micromechanics and kinetics of liquid crystal elastomers are well understood; the notion of soft elasticity is used to represent orientation change of liquid crystal mesogens due to bulk deformation.(Warner and Terentjev, 2003) Speeding up the dynamics of the isomerization process, however, is difficult because of two competing effects that must be tuned with respect to the kinetics of the problem: higher elastic stiffness of the elastomer increases the coupling constant but slows the

orientation dynamics, while reducing the material stiffness increases the dynamics but reduces the coupling.(Cviklinski et al, 2002) In the present work we first develop a coupled rigorous model based on large deformation finite element methods (FEM) and analytical models of phase transition in liquid crystalline materials to model the optomechanical response of structures made from an optically sensitive material. Using such a material as a basis for a photonic crystal, which undergoes a mechanical phase transformation at the appropriate level of macroscopic strain, we demonstrate the possibility for a unique and highly sensitive structure that can be made to deform cyclically under constant intensity, uniform incident light. In section 2 the constitutive model for the LCE material is presented, followed by the simulation method for the optomechanical FE analysis of the photonic crystal in section 3. Then, through the use of a simple computational example, we discuss the design and analysis of this unique optomechanically coupled structure in section 4. 2 Material Model The elastomer material under consideration is made optically sensitive by the presence of the azobenzene molecules, which act as liquid crystal mesogens. The bulk material response to light is modeled as a coupled electromagnetics-mechanics problem where the strain experienced by the material is obtained from the field distribution of incident light energy in the material. The kinetics of the transcis-trans photochemical reactions that give rise to the light-induced transformational strain are mapped onto a material constitutive behavior following the work of Hogan et al (2002). In this model, the light induced deformation is imposed as a volumetric transformational strain. This transformational strain is a function of the population of isomers in the trans- state, through a quantity described in Hogan et al (2002) as a “strain parameter”. This model is adapted to the present application, in which the light intensity is nonuniform due to the material microstructure, and then coupled to a standard isotropic hyperelastic neo-Hookean constitutive law. The model for the kinetics of the light-induced transformational strain is introduced in the following paragraphs. First, the rate of population change of the trans- isomers (denoted by n) in the photonic crystal domain at a point (x, y) is given by

∂ 1 1 (n0 − n ) n(t ( x, y )) = −ηn − n+ ∂t τ tc τ ct

(1)

where t(x,y) is the duration of exposure to light energy of material point (x, y), η is the irradiation rate, and τ ct is the time required for cis- to isomerize into trans-.(Hogan et al, 2002) The relaxation time τ tc for trans- to isomerize into cis- is taken to be sufficiently large that the second term in Eqn. (1) can be reasonably neglected. With an initial condition of n(t(x, y) = 0 ) = n0 the solution to (1), (the “forward population equation”) is

n(t ( x,y )) =

with τ eff = ⎛⎜ η + 1

⎝

n0 ⎡ ⎛ 1 + τ ct ηexp⎜ − t(x,y) ⎢ τ eff 1 + τ ct η ⎣ ⎝

⎞⎤ ⎟⎥ ⎠⎦

(2)

−1

⎞ . The reverse reaction, is due to a thermally driven flux of cis- into the transτ ct ⎟⎠

ground state. Assuming a fully saturated final state, where ncis = n0 − n∞ = n0 τ ct

η

(1 + τ ct η) , the

population of trans- isomers (the “reverse population equation”) is

⎤ ⎡ τ η n(t ( x,y )) = n0 ⎢1 − ct exp⎛⎜ − t ( x,y ) ⎞⎟⎥ τ ct ⎠ ⎝ ⎦ ⎣ 1 + τ ct η

(3)

The effect of the trans-cis transformation is a shift in the critical strain parameter Tni, which, in notation of Hogan et al. (2002), corresponds to the threshold for the isotropic-nematic (or nematicisotropic) transformation of the LCE. In a first approximation, Tni is taken to be a linear function of the cis- isomer population, and is given as

Tni (n ) = Tni(0) − β [n0 − n(t ( x, y ))]

(4)

where β is a dimensionless constant. The shift in Tni is defined as the strain parameter difference

δT j ( x, y ) given by δT

j

(x, y ) = Tni (n ) − Tni(0)

(5)

where j=a or b when using n(t(x, y)) from Eqns. (2) or (3), the forward and reverse population equations, respectively. The critical strain parameter Tni and the strain parameter difference δTj are both scalars. Computing the strain parameter difference, δTj, a function of material parameters and the time of exposure to light, makes it possible to determine the swelling or volumetric transformational strain.

In an application such as a photonic crystal where the LCE is used as the matrix material, the distribution of light energy within the material is in general highly nonuniform, because the structural features of the system, e.g. holes or inclusions, are nonuniform at a length scale that is commensurate with the wavelength of light. Furthermore, as the material deforms, some regions in the structure receive more or less exposure to light, due to scattering, diffraction, absorption, etc. Thus, it is important to keep track of the effective total duration of light exposure of point (x,y), denoted by t(x,y). We introduce this new quantity and refer to it as local time. The local time is less than or equal to the total or global time t’ over which light is incident on the external boundary of the structure. Any regions in the interior of the structure that experience reduced light intensity due to scattering, diffraction, absorption, etc., consequently experience a local time that is smaller than, or retarded relative to, the global time. The global and local times would be the same only in regions in which the material experiences full, uniform illumination intensity. For convenience in tracking the evolution of the material state and the behavior of the structure, a discrete time-stepping scheme is introduced. With i=1,2,…, tfinal indexing discrete time steps up to the total time tfinal of illumination, globally, the swelling or volumetric transformational strain at local time

t i ( x, y ) is given by

⎤ ⎡ (1 − H ( x, y )) ⋅ δT a ( x, y ) + H ( x, y ) ⋅ δT b ( x, y ) ξ i i i i ⎥ ⎢ ε i ( x, y ) = α ⋅ ξ⎥ ⎢ a b ⎢⎣− (1 − H i −1 ( x, y )) ⋅ δTi -1 ( x, y ) + H i −1 ( x, y ) ⋅ δTi -1 ( x, y ) ⎥⎦

(6)

⎧1 t i −1 ( x, y ) ≥ t c is the ⎩0 t i −1 ( x, y ) < t c

where α is the strain coefficient, ξ is a curve fitting parameter and H i −1 (x, y ) = ⎨

Heaviside function that compares the local time ti(x,y) to the parameter tc , which is the elapsed local time by which an assumed designated steady state value of trans- population has isomerized into cisconfiguration. This nominal time to complete transformation is associated with, for example, 80% or greater transformation. Thus, the transformational strain in Eqn. (6) is considered to be analogous, for example, to that of thermal strain, which is typically written in the form εTH = κ ⋅ Δθ , where κ is the coefficient of thermal expansion (corresponding to α here) and Δθ is the change in temperature (corresponding here to the expression in the square brackets, which is the difference between the strain parameter in the present and previous time increments). The Heaviside function allows the function to take strain parameter δT a ( x, y ) before local time tc and δT b ( x, y ) after it in Eqn. (6).

The time associated with the global time scale follows the scheme t i' = t i' −1 + Δt . This time scale is distinguished from the concept of local time, which is described in discrete form by the variable ti(x,y), associated with position (x,y), and incremented as

ti ( x,y ) = ti −1 (x,y ) + Δt ⋅ ψ i (x,y ) ⋅ (1 − H i −1 ( x,y )) + (1 − ψ i ( x,y )) ⋅ H i −1 ( x,y ) where ψ i ( x, y ) =

(7)

Ei (x, y ) − E min (x, y ) is a parameter measuring the relative local intensity of light; E max ( x, y ) − E min ( x, y )

Emax and Emin are the maximum and minimum values of the time averaged total energy density Ei ( x, y ) , respectively, from the electromagnetic analysis. Thus, for each incremental unit of global time, the local time at each point is incremented by some lesser or equal amount according to the normalized intensity of the light energy at that point, such that the second term on the right hand side of Eqn. (7) follows the constraint Δt ⋅ ψ i ( x,y ) ⋅ (1 − H i −1 ( x,y )) + (1 − ψ i ( x,y )) ⋅ H i −1 ( x,y ) ≤ Δt . This discrete time scale, the local time, makes it possible to track the time over which the material at a point has been exposed to light; this effective light exposure induces the transformational strain expressed in Eqn. (6). Material properties for an azobenzene LCE, in the form of parameters τ ct , η, etc., can then be taken from the literature. Using data given by Hogan et al (2002), for example, Figs.1a and 1b show the strain parameter in the material as a function of illumination time and intensity. Once direct illumination of the material leads to a nominally complete transformation, no further strain is imposed on the material as long as the illumination is maintained. If the illumination is removed, thermal relaxation from cis- to trans- leads to reversal of the transformational strain.

(a)

(b)

Strain Parameter δT(x,y)

0 −0.5 −1 −1.5

Ψ(x,y)=1 t(x,y)tc

−2 −2.5

Ψ(x,y)=1 t(x,y)tc

−3 −3.5 0

2000

4000

6000

Global Time t’ (s)

8000

10000

Figure 1. (a) Effect of light intensity parameter ψ i ( x, y ) on strain parameter, with azobenzene material parameters taken from Hogan et al (2002). Under full illumination, the strain parameter increases in magnitude until global time tc. After tc if light remains localized at full intensity at position (x,y) then there is no change in the strain parameter (dotted line). If that position is no longer illuminated locally, then the strain parameter relaxes to zero (solid line). (b) Effect of two different intensity parameters as a function of global time t’, with the same local characteristic time tc. If the intensity parameter is reduced by a factor of two (dashed line) it will require twice the global time to reach the same strain state as in the full intensity case (solid line) (inset shows the strain parameter as a function of local time).

Having prescribed the transformational strain based on local illumination, the stress in the material is readily obtained using an isotropic hyperelastic neo-Hookean constitutive law which is a good constitutive model for elastomeric materials at low to moderate strain levels.(Boyce and Arruda, 2001) For simplicity, in the remaining examples considered here, the Young’s modulus and Poisson’s ratio are taken as 2.0 GPa and 0.35, respectively, which are typical values for elastomeric materials. Furthermore, it is assumed here that the elastic properties are spatially homogeneous, though this need not be the case in general. One additional material effect that must be considered is the effect of the alignment of the liquid crystal mesogens on the bulk optical properties of the medium. The refractive index of the material in the direction of polarization decreases because the incident polarized electromagnetic radiation causes alignment of the liquid crystal mesogens orthogonal to the direction of polarization. (White et al, 2009) As an example, the initial value of the refractive index r0 is taken as 1.57, which is typical for a polymer material. The birefringence (Δr) in the case of fully aligned liquid crystal mesogens, i.e. when the isomerization of trans- to cis- has reached steady state, is assumed to be 0.06.(Cimrova et al, 2002) The birefringence depends on the aspect ratio and type of the liquid crystal (side or main chain).(Warner and Terentjev, 2003) The modified refractive indices (in both x and y directions), affected by the birefringence of the liquid crystal mesogens that isomerize due to incident light linearly polarized in the x direction, when

t i −1 ( x, y ) ≤ t c , are given by t (x, y ) ⎞ rxx (t i (x, y)) = r0 − 0.5 Δr (1 − H i −1 ( x, y ))⎛⎜ i −1 t c ⎟⎠ ⎝ t ( x, y ) ⎞ ryy (t i (x, y)) = r0 + 0.5 Δr (1 − H i −1 ( x, y ))⎛⎜ i −1 t c ⎟⎠ ⎝

(8)

The refractive indices relax to an isotropic configuration with rxx=ryy=r0 for t c < t i −1 (x, y ) < t final following a similar update to Eqn. (8). The effect of the liquid crystal mesogen alignment change on the mechanical properties of the material can be reasonably neglected in this work.

3 Simulation Methodology Coupled electromagnetics and mechanics simulations are then carried out based on the material model described in section 2. In order to demonstrate possible coupled optical and mechanical effects of periodic geometries such as those described in the introduction, we consider the structure shown in Fig 2. The photonic crystal structure under consideration consists of a 2-D slab of material containing a square array of circular air holes. Such a structure comprises thin ligaments of material, referred to as veins, connecting larger bulk-like regions of material at the hole interstices, referred to as spots. The periodicity of the holes is characterized by the lattice parameter a, and the diameter of each hole is d. The subdomain properties and boundary conditions for the electromagnetics transmittance analysis are shown in Fig 2a. An air domain is incorporated on the left of the structure so that the normal incidence plane wave boundary condition for the incident light on the left is not specified directly on the material/air interface. Another such domain is included on the right side to provide an interface for a boundary condition to absorb waves transmitted through the structure.(Jin, 2002) Periodic boundary conditions are specified on the top and bottom faces. A full 2x2 matrix is prescribed to account for the birefringence of the material via the dielectric tensor ε ( d ) (x, y) . The Maxwell’s equations for TE polarization (electric field transverse to the plane of the photonic crystal) are solved to obtain the intensity distribution of the incident plane wave field in the structure. The total energy density is calculated as the time-averaged quantity of the combined electric and magnetic energy field intensities. For the mechanics model shown in Fig 2b, the deformable photonic crystal is modeled using the constitutive description specified in the previous section. The air region is not included in the stress analysis. Boundary conditions are prescribed such that on the left, top, and bottom edges of the structure, only transverse displacements are allowed. On the right face of the structure a traction free boundary condition is prescribed. The mechanics and electromagnetics simulation methodology follows our previous work on modeling two dimensional deforming photonic crystal structures.(Krishnan and Johnson, 2009)

a

b

Y 

X  Figure 2. (a) Boundary conditions and subdomain properties for the electromagnetics model of light transmittance through the structure from left to right, as shown by the arrow. The top and bottom edges are governed by periodic boundary conditions. The local refractive indices of the liquid crystal elastomer are computed from the strain parameter in each global time step t’. Veins are ligament-like regions in the structure and spots are bulk-like regions, or interstices in the periodic array of holes. (b) Subdomain types and boundary conditions for the mechanics model used to obtain deformation and stress from the transformation strain applied according to Eqn. (6). The top and bottom edges are constrained against normal displacements, while the voids (or air holes in the electromagnetics models) are allowed to deform freely.

The reflectance spectrum of the square array photonic crystal with d/2a=0.44 and a=1 (dimensionless), is shown in Fig. 3a, for a plane wave incident on the left face of the domain with periodic boundary conditions on the top and bottom faces. The reflectance is obtained as the ratio of the power reflected from the incident face to the input power of the incident plane wave. In this analysis absorption is not considered, so the sum of reflectance and transmittance is understood to be unity. The energy density field E(x, y) mapped onto the photonic crystal domain is shown in Fig. 3b. This energy density field induces the transformational strain (equation 6) that causes both the refractive index of the material to change (equation 8) and the geometry of the photonic crystal to change.

a

b

Y 

X 

Figure 3. (a) Reflectance spectrum of undeformed 9x9 photonic crystal (b) Total light energy distribution E(x,y) at λ/a=1.57 (shown by arrow in Fig a) with darker regions having higher values.

The simulation proceeds according to the following algorithm for each increment in the global time t’, with the transformational strain obtained from Eqn. (6): 1. Preprocessing: Import undeformed geometry Ω0 in the initial iteration or the deformed geometry Ωi −1 in subsequent iterations and create a new finite element mesh over the domain. In the example shown here, the mesh is constructed using second order triangular elements. 2. Model setup: a. FEM Electromagnetics analysis: Compute the electromagnetics boundary value problem for light transmittance through the structure from Step 1 with appropriate boundary conditions and updated dielectric properties using a static solver. Obtain the total energy density distribution Ei(x,y) (and hence ψ i ( x, y ) ) to calculate the parameter distribution δT(x,y) at time iteration number i. b. FEM stress and deformation analysis: Carry out a stress and deformation analysis using a static solver to obtain deformed configuration due to the transformational strain ε i (x, y) . 3. Store updated values of the dielectric property ε ( d ) (x, y) and the local time distribution ti(x, y) to calculate Hi (x, y), stress components σx,i (x, y),σy,i (x, y), σx,y,i (x, y) from Step 2b, and the deformed structure Ωi 4. Increment the global and local times following the time scheme in Eqn. (7) and repeat steps 1 to 3 until t i' =tfinal It is noted that the convergence of the two analyses in steps 2a and 2b are not related in any way to the global and local time steps. The coupling of the mechanics to the electromagnetics is enforced by porting the updated deformed configurations Ω and thus in turn the dielectric distribution ε ( d ) (x, y) to the electromagnetics equations. The coupling of the electromagnetics to the mechanics analysis is implemented by considering the total energy distribution in Ω and imposing the corresponding transformational strain ε i (x, y) . Values of the parameters used in the example presented here are provided in the table below. Table

Parameter

Definition

Value

Units -4

η

Irradiation rate (Hogan, 2002)

2.2 x 10

s-1

τct

cis to trans Transition time (Hogan, 2002)

1480

s

tc

Characteristic time

3000

s

tfinal

Total time

12000

s

βn0

Measure of trans isomer concentration (Hogan, 2002)

15

-

ξ

Rate parameter (Hogan, 2002)

0.19

-

α

Strain coefficient

0.044

-

r0

Initial refractive index

1.57

-

Δr

Birefringence in liquid crystals (Cimrova, 2002)

0.06

-

4 Simulation Results Fig 4a shows the macroscopic strain in the photonic crystal (defined in the inset figure as the average strain across the structure in the direction parallel to the incident light) as the structure deforms due to the effect of light. Fig. 4b shows the deformed structure at several representative time points identified on the continuous strain-time plot in Fig. 4a. Due to the evolution of the strain parameter, shown in Fig. 1, the structure undergoes macroscopic compression as the enforced transformational strain is compressive in nature. This deformation phase is globally homogeneous, or affine, with the circular features becoming elliptical as seen in the deformed structures in Fig. 4b at time points 2000s and 5000s. It is induced by localization of the light energy in the spot regions (shown in red in Fig. 4b) producing contractile transformational strain uniformly across those regions. This deformation changes the photonic bandstructure by shifting the features of the reflectance spectrum, shown in Fig 3a for the undeformed structure, to shorter wavelengths.(Krishnan and Johnson, 2009) As a result, even though light is incident with constant intensity on the left side of the structure, beyond 5000s the local energy density goes down, causing some regions in the structure begin to relax. This allows the structure to expand between 5000s and 8000s. Relaxation occurs locally at different rates depending on the intensity parameter and rate of change of the slope of the strain parameter (as shown in Fig. 1). This expansion leads to a mechanical instability that triggers the pattern transformation from circular holes to alternately oriented ellipses; such a pattern transformation is well-documented in recent work. The transformation is associated with buckling of the vein regions and rotation of the spot regions; globally it produces some macroscopic contraction of the structure between the ~8000-12000s time period. This is because the buckling deformation draws the structure inwards, thereby reducing the contained free volume within it. By 12000s the buckled structure has sufficient disorder that the light intensity becomes relatively

nonuniform. Beyond 12000s the curve turns up as the compressive strain decreases spontaneously as a result of a relaxation of the buckled structure. The vein regions straighten as the spot regions rotate back toward their initial orientation; this accounts for the reversal that leads to the cyclic nature of the deformation. Then, once the buckling deformation fully relaxes and the structure returns to a configuration comparable to the 8000s case, the strain induced by light incident on the LCE material will again induce a contraction, repeating the process seen between 2000s and 5000s.

a

b

t'=2000

5000s

8000s

12000s

Y  X  Figure 4. (a) Macroscopic extensional strain in x-direction as a result of the light induced transformational strain. The structure undergoes compression and when the cutoff time is reached in some regions inside the structure due to changes in the photonic bandstructure, it undergoes relaxation and expansion. After expanding uniformly, the structure undergoes a pattern transformation and accommodates more macroscopic compressive strain. Finally the structure relaxes as the pattern transformation reverses, reducing the overall macroscopic compressive strain. This process occurs spontaneously under uniform and constant illumination. (b) Deformed structures at four representative times (indicated by black points in (a)). Light energy density distribution E(x, y) is shown over the deformed structure.

5 Discussion

Previous work on pattern transformation in two-dimensional soft photonic crystals shows that the nonuniform deformation affects the transmittance and bandstructure properties in both the primary (lower-frequency) and secondary (higher-frequency; λ/a=1-1.5 range) peaks, as the change in structure leads to changes in the way that light is localized.(Krishnan and Johnson, 2009) However, in that work, the effect of light on the material is ignored because the material is taken to be a simple elastomer. In the model developed in the present work, however, the incident light interacts with the liquid crystal elastomer and itself induces a transformational strain in the material. This transformational strain depends significantly on the intensity parameter and the distribution of light energy within the structure. The dependence of the structural reflectance on the light-induced strain enables a unique coupling phenomenon observed here for the first time. This phenomenology is analogous to the widely studied electromagnetic-induced transparency (EIT) possible in some quantum optics and photonic crystal systems, in that an electromagnetic field itself is used to change the optical transmittance of the material.(Boller, 1990) Here, the change in transmittance due to the electromagnetic field is mediated by a purely mechanical effect, as opposed to an electronic structure change. In order to track this coupling, two time scales are introduced in our analysis: the global time, or the time over which the structure is illuminated; and the local time, or accumulated time over which each material point in the structure experiences a non-zero intensity parameter. The local time is nonuniform, and strictly less than or equal to the global time. The relationship between these time scales is shown in detail in Fig. 1b. The potential for cyclical deformation in such a material is sensitively related to the tuning of the incident wavelength of light, the structural length scale of the material, the boundary conditions, and the time scales for the forward and reverse microstructural transformations. The value of the wavelength chosen for the incident light in the example presented here is such that most of the light energy is localized in the spot regions of the structure. There are other values above λ/a=1.57 at which similar localization occurs and similar optomechanical behavior may be observed. For some wavelengths smaller than λ/a=1.57, localization of light energy also occurs in the spot regions. However, a requirement for the cyclical deformation behavior observed in the example presented here is that the contraction of the structure should cause the periodicity to change such that the light energy localizes more arbitrarily (i.e. less homogeneously), without maintaining the initial energy distribution. This requirement is not met for smaller wavelength incident light, and consequently the pattern transformation is not induced. For the material properties in the example considered here, the pattern transformation of the initial square array of holes into alternating ellipsoidal holes occurs between 8000-12000s when the structure displays a compressive behavior. This pattern change causes the structure to experience a reduction in macroscopic strain because of the buckling of the vein regions and rotation of the spot

region. Once this buckling occurs, there may be sufficient disorder in the structure to cause some localization of the light intensity; this localization is analogous to Anderson localization. Moreover, it can also be observed that the structure starts to expand and the pattern transformation reverses after 12000s due to two local relaxation effects: 1) spontaneous isomerization back to the trans state in some regions of the structure that have already undergone the trans-cis transformation and experience low values of light intensity parameter, and 2) stresses present from prior time steps in some regions of the structure that presently experience low transformational strains. The mechanism that is primarily responsible for the slow cyclic deformation time scale is the cistrans transition time, τct, which is simply a property of the azobenzene chemistry itself. The time dependence of the mechanical constitutive behavior, and the time scale for the onset and reversal of the mechanical instability, need not significantly limit the rate of the deformation cycle. The process observed in the example is too slow for many applications, but it demonstrates the potential for an unusual cyclical deformation mechanism that can be induced by uniform illumination of a soft, periodic structure. 6 Conclusion In this work we introduce a coupled optomechanical framework that captures the mechanics and kinetics of complex, light-induced cyclical deformation in a photonic crystal structure made from optically sensitive liquid crystal elastomer material. The model takes into account the effect of incident light on the material (as a transformational strain) and calculates the localization of incident light inside the deforming structure. Simulation results show that for a certain wavelength of the incident light and hence for a certain profile of the initial state of localized light energy inside the structure, it is possible to optically induce a symmetry-reducing pattern transformation in a two-dimensional photonic crystal made of square holes in a liquid crystal elastomer matrix. This optical actuation of the pattern transformation is the first reported example of such coupled behavior. The pattern transformation leads to a change in the optical transmittance of the material, and thus a reduction in the transformational strain. The cyclical nature of the deformation, so noted because the structure eventually returns to an intermediate deformation state even while under constant illumination, and without any change in the mechanical boundary conditions,is then seen to begin after an elapsed time of ~12000s. The unique coupled optomechanical behavior may be useful for a variety of sensing, actuating, and energy harvesting applications.

Acknowledgments This work is supported by the Department of Energy, through grant no. DE-FG02-07ER46471 at the Frederick Seitz Materials Research Laboratory at the University of Illinois. DK would like to thank Dr. Andrey V. Semichaevsky for reading through the manuscript and offering helpful suggestions.

References Bai, S., and Y. Zhao, Macromolecules 35, 9657 (2002). Barrett, C. J., J. Mamiya, K. G. Yager, and T. Ikeda, Soft Matter 3, 1249 (2007). Bertoldi, K., M. C. Boyce, S. Deschanel, S. M. Prange, and T. Mullin, J. Mech. Phys. Solids 56, 2642 (2008). Bertoldi, K. and M. C. Boyce, Physical Review B (Condensed Matter and Materials Physics) 77, 052105 (2008). Bertoldi, K., and M. Gei, J. Mech. Phys. Solids 59, 1, 18-42, (2011). Blair, H. S., H. I. Pague, and J. E. Riordan, Polymer 21, 1195 (1980). Boller, K. J., A. Imamolu, and S. E. Harris, Phys. Rev. Lett. 66, 2593–2596, (1991). Boyce, M. C., and E. M. Arruda, Mathematics and Mechanics of Solids 6, 641 (2001). Cimrova, V., D. Neher, R. Hildebrandt, M. Hegelich, d. L. von, G. Marowsky, R. Hagen, S. Kostromine, and T. Bieringer, Appl. Phys. Lett. 81, 1228 (2002). Cviklinski, J., A. R. Tajbakhsh, and E. M. Terentjev, European Physical Journal E 9, 427 (2002). Dunn, M. L., and K. Maute, Mech. Mater. 41, 1083 (2009). Finkelmann, H., E. Nishikawa, G. G. Pereira, and M. Warner, Phys. Rev. Lett.87, 015501 (2001). Harvey, C. L. M., and E. M. Terentjev, The European Physical Journal E: Soft Matter and Biological Physics 23, 185 (2007). Hogan, P. M., A. R. Tajbakhsh, and E. M. Terentjev, Physical Review E 65, 041720 (2002). Hwang, J., M. Song, B. Park, S. Nishimura, T. Toyooka, J. W. Wu, Y. Takanishi, K. Ishikawa, and H. Takezoe, Nat Mater 4, 383 (2005). Ikeda, T., J. Mamiya, and Y. Yu, Angewandte Chemie International Edition 46, 506 (2007). Jin, J., The Finite Element Method in Electromagnetics (John Wiley & Sons, Inc., New York, NY, 2002). Koerner, H., T. J. White, N. V. Tabiryan, T. J. Bunning, and R. A. Vaia, Materials Today 11, 34 (2008). Krause, S., F. Zander, G. Bergmann, H. Brandt, H. Wertmer, and H. Finkelmann, Comptes Rendus Chimie 12, 85 (2009).

Krishnan, D., and H. T. Johnson, J. Mech. Phys. Solids 57, 1500 (2009). Lee, Y. J., C. E. Heitzman, W. R. Frei, H. T. Johnson, and P. V. Braun, J. Phys. Chem. B 110, 19300 (2006). Li, W., and C. M. Landis, Smart Materials and Structures 21, 9, 094006, (2012). Li, T., C. Keplinger, R. Baumgartner, S. Bauer, W. Yang, and Z. Suo, J. Mech. Phys. Solids, 61, 2, 611628, (2013). Mullin, T., S. Deschanel, K. Bertoldi, and M. C. Boyce, Phys. Rev. Lett. 99, 084301 (2007) Robillard, J. F., O. B. Matar, J. O. Vasseur, P. A. Deymier, M. Stippinger, A. C. Hlady-Hennion, Y. Pennec, B. Djafari-Rouhani, Appl Phys. Lett. 95, 12, 124104, (2009). Suo, Z., Acta Mech Sin, 23, 6, 549-578, (2010). Tajbakhsh, A. R., and E. M. Terentjev, The European Physical Journal E: Soft Matter and Biological Physics 6, 181 (2001). van Oosten, C. L., D. Corbett, D. Davies, M. Warner, C. W. M. Bastiaansen, and D. J. Broer, Macromolecules 41, 8592 (2008). Warner, M., and E. M. Terentjev, Liquid Crystal Elastomers (Clarendon Press, Oxford, 2003). White, T. J., N. V. Tabiryan, S. V. Serak, U. A. Hrozhyk, V. P. Tondiglia, H. Koerner, R. A. Vaia, and T. J. Bunning, Soft Matter 4, 1796 (2008). White, T. J., S. V. Serak, N. V. Tabiryan, R. A. Vaia, and T. J. Bunning, Journal of Material Chemistry 19, 1080 (2009). Wu, Y., Y. Demachi, O. Tsutsumi, A. Kanazawa, T. Shiono, and T. Ikeda, Macromolecules 31, 349 (1998). Wu, Y., A. Kanazawa, T. Shiono, T. Ikeda, and Q. Zhang, Polymer 40, 4787 (1999). Wu, Y., Q. Zhang, A. Kanazawa, T. Shiono, T. Ikeda, and Y. Nagase, Macromolecules 32, 3951 (1999). Yu, Y., M. Nakano, and T. Ikeda, Nature 425, 145 (2003). Zhao, X., and Z. Suo, Appl. Phys. Letters, 91, 6, 061921, (2007). Zhu, X., G. Wu, R. Dong, C.-M. Chen, and S. Yang, Soft Matter, 8, 8088-8093, (2012).