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Edited by: Oliver Bäumchen, Max-Planck-Institute for Dynamics and Self-Organization, Germany

Reviewed by: Dong Chen, Zhejiang University, China; Shashi Thutupalli, Princeton University, USA; Aurora Hernandez-Machado, Universitat de Barcelona, Spain

Specialty section: This article was submitted to Biomaterials, a section of the journal Frontiers in Materials

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Liquid crystal elastomers represent a novel class of programmable shape-transforming materials whose shape-change trajectory is encoded in the material’s nematic director field. Using three-dimensional non-linear finite element elastodynamics simulation, we model a variety of different actuation geometries and device designs: thin films containing topological defects, patterns that induce formation of folds and twists, and a bas-relief structure. The inclusion of finite bending energy in the simulation model reveals features of actuation trajectory that may be absent when bending energy is neglected. We examine geometries with a director pattern uniform through the film thickness encoding multiple regions of positive Gaussian curvature. Simulations indicate that heating such a system uniformly produces a disordered state with curved regions emerging randomly in both directions due to the film’s up/down symmetry. In contrast, applying a thermal gradient by heating the material first on one side breaks up/down symmetry and results in a deterministic trajectory, producing a more ordered final shape. We demonstrate that a folding zone design containing cut-out areas accommodates transverse displacements without warping or buckling, and demonstrate that bas-relief and more complex bent/twisted structures can be assembled by combining simple design motifs.

Liquid crystal elastomers (LCEs) undergo reversible shape transformations under any stimulus that changes their degree of nematic order, including heat, illumination, or change of chemical environment (White and Broer,

Complex shape transformation trajectories may be encoded in the material by patterning of the nematic director when the polymer is cross-linked (Liu et al.,

In previous work, we collaborated with experimenters to explore how even relatively simple director patterns give rise to complex actuation behavior. For instance, a flat, straight LCE ribbon with director twist through the thickness can form either helicoid or spiral structures, depending on sample properties including director twist geometry and sample aspect ratio; remarkably, these chiral structures undergo transitions from right- to left-handed with temperature (Sawa et al.,

Here, we explore a variety of programmed shape transformations in LCEs with more complex blueprinted director fields, inspired in many cases by recent experiments. An explanation of simulation methods is provided below in the final section.

Accurate simulation tools are needed for design/engineering of future devices, to explore dependence of shape transformation geometries on blueprinted microstructure, and to test how known design motifs combine. Simulations can in particular provide insight into observed differences between analytical predictions – which often neglect bending energy – and experimental observations. Here, we examine several geometries that have been studied experimentally and go on to explore several novel designs.

Modes et al. (

Figure

To further investigate effects of finite bending energy, we carry out another simulation of a film containing a +1 defect using a disk-shaped sample; its shape evolution is shown in Figure

We also investigate shape transformations in LCE films containing higher order defects. Shown in Figure

Figure

In order to achieve deterministic rather than random shape evolution, it is necessary to break this up–down symmetry. Instead of applying a thermal gradient, a better option would be to introduce a material gradient. For instance, the cross-link density may be varied through the thickness of the film, by introducing a UV absorbing component in the mixture and cross-link

“Auto-origami” refers to spontaneous deformation of a programmable material whose microstructure encodes a sequence of folds, stretches, and shears to produce a trajectory of motion from one stable conformation to another, triggered by an external stimulus such as a change of temperature. A key goal in design of shape-changing auto-origami devices is the formation of folds, actuated, e.g., on heating, induced by a localized gradient in the nematic director field between the top and bottom surfaces. For instance, director twist through the thickness of the material can be used to induce a fold (Ware et al.,

To localize the stresses associated with bending deformation to the hinge region, we designed a structure containing holes. In Figure

We also demonstrate combining two different types of director gradients along a beam in order to construct a more complex shape evolution. Figure

Actuators in the form of bas-relief or raised patterns are desirable for applications including haptic displays and for dynamic control of surface texture. A variety of director motifs have been proposed to pattern surfaces to achieve desired surface structures including director fields with (McConney et al.,

Here, we examine a simple design motif to drive emergence of a bas-relief pattern on heating. To create a raised region in the shape of a valentine heart, we model a square LCE film with different anchoring conditions on the top and bottom surfaces. On the top surface, we impose homeotropic anchoring inside a heart-shaped region, and uniform planar anchoring outside, and on the bottom surface, we reverse the two anchoring conditions, as shown in Figure

To create a three-dimensional non-linear finite element model of our system, we start by meshing our sample in an unstructured tetrahedral mesh using the open-source application Salome (available at

Here, the first term in the sum over elements is the elastic strain energy, where _{ij}_{ijkl}_{ij}_{n}_{p}_{p}

When the sample is heated, we assume that the blueprinted nematic director, described as piecewise constant within each element, remains fixed in the body reference frame but the scalar order parameter drops. Thus the nematic order tensor within each element takes the form _{i}_{ij}

The system evolves forward in time

The force calculation proceeds as follows. The displacement field is linear interpolated within each tetrahedral element using the positions of the corner nodes in the current state and in the element’s undeformed, stress-free reference state, which we take to be the shape of the element when the material is cross-linked. From the gradients of the displacement field, we calculate the components of the Green-Lagrange strain tensor and from that, calculate the potential energy. In this manner, we write the potential energy of the system as a function of the node positions. We calculate the effective force on each of the four nodes by taking the negative derivative of the potential energy within each element with respect to the node positions. The effective force on each node receives a contribution from each element of which it is a member.

Interestingly, our finite element algorithm closely resembles a conventional molecular dynamics simulation (Rapaport,

Within this model, if two parts of the sample intersect, they pass through each other, ghostlike, without any collision. To avoid this unphysical behavior in any simulation, we add short-range pairwise repulsive interactions between the surface nodes using the Weeks–Chandler–Andersen potential, which is the Lennard-Jones potential truncated at its minimum and shifted such that the potential vanishes at the cutoff (Ahmed and Sadus,

While less computationally efficient than a two-dimensional shell model (Chung et al., ^{5} elements. More details of the model may be found in Mbanga (

Devices engineered using patterned LCEs have a somewhat limited design space as the nematic director field may be controlled only

In contrast, other materials with anisotropic shape evolution have recently been introduced (Gladman et al.,

Finite element simulation is a valuable design tool to explore a wide variety of device designs for materials undergoing programmable shape change. The three-dimensional nature of our model allows study of systems where bending energy plays an important role, and allows modeling of samples where the nematic director is not uniform through the film thickness.

While quasistatic relaxation is sufficient to model a slow shape evolution process, some shape transitions involve mechanical instabilities and may show rapid snap-through behavior. These rapid shape transitions can be modeled

RS in the primary investigator of this work and developed the simulation method used in this paper. VG-P implemented the simulation for several different director structures, and performed data analysis and visualization of the results. AK implemented the simulation algorithm in the CUDA programming language for execution on a GPU-enabled computer, producing dramatically faster computational speed, and carried out design and shape-change simulations for several of the designs listed here. He also developed improved data visualization methods.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The authors thank Tim White, Mike McConney, Qihuo Wei, and Badel Mbanga for helpful discussions. Work supported by the National Science Foundation (NSF-DMR 1106014, NSF-DMR-1409658, and NSF-CMMI 1436565). Computer resources provided by the Ohio Supercomputer Center and by the Kent State College of Arts and Sciences.