# Numerical Methods in Studies of Liquid Crystal Elastomers

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## Abstract

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## 1. Introduction

## 2. LCE Modeling Approaches

## 3. Finite Element Analysis (FEA)

#### 3.1. Commercial Software

^{®}4. 3a, COMSOL Inc., Burlington, MA, USA, in conjunction with data obtained from stretching experiments, to study the appearance of sub-stripe patterns in sheets of nematic elastomers. The samples used were consisted of side-chain LCEs with a backbone consisting of a siloxane-based polymer and a 3-but-3-enyl-benzoic acid 4-methoxy-phenylester as the mesogenic moiety. The length of the samples was approximately 5 mm and their width was about 3 mm with thickness of about 200 µm. The samples were placed in a test cell equipped with temperature stabilization, a force gauge and a micrometer for controlling both stress and strain. Figure 1 displays the stretched sample, in the nematic phase, in the direction which is perpendicular to the nematic director. It is observed that at the critical strain λc, stripes are formed. The average size of these shear domains is around 10 μm and they show a random pattern of alternating shear and alternating director orientation.

^{®}4.3 software, COMSOL Inc. Burlington, MA, USA, was used to study plane-strain deformations. The authors further developed their model by adding a spatially random temperature fluctuation or “numerical noise”, denoted as DT, to the threshold temperature, denoted by T

_{IN}, to allow for phase transition to happen at different temperature in different points of the sample. The results from numerical modeling for both stretching and cooling experiments were compared with data from experiments [38,40] and good agreement was observed between the two, as can be seen in Figure 2 and Figure 3. Additionally, lateral shear was not observed due to the presence of the clamps, therefore the material develops a micro-structure. The boundary condition of this structure is influenced by the presence of stripes. With this model, unfortunately, complex behavior of LCE cannot be described because of the hypothesis of plane–strain deformation.

^{®}software, Canonsburg, PA, USA, to bridge the gap between elastomer physics and device engineering and design. An empirical model was developed by the authors, for describing the material deformation, using experimental actuation data. The basic concept of the model is as follows: the absorption properties of carbon nano-tubes (CNTs) causes the irradiated light on the sample to be absorbed and consequently, turned into heat. Thus, the CNTs act as internal heat generators. Finally, the opto-mechanical behavior, i.e., the mechanical deformation induced by light as stimuli, of an LCE-CNT system was evaluated and the mechanical response of the said system was optimized. The LCE-CNT system structure which was modeled was an LCE-CNT film in the shape of a cantilever structure with 1 mm length, 0.6 mm-wide, and with a thickness of 0.4 mm. Both 2D and 3D geometries were studied. Effective physical properties were considered for the entire specimen under investigation. Results showed that preliminary results obtained from simulations, using the authors’ proposed model, were in agreement with the experimental data [42]. Another advantage of the developed numerical model was that it could provide the researchers with data that was unattainable via experiment such as, the temperature distribution inside the specimens and also predicting the temperature evolution for any given point in the model geometry.

^{®}5.2a COMSOL Inc., Burlington, MA, USA, was used for carrying out the simulations. This model accounted for the movement of the heat source. Results show a reversible contraction/expansion and relative length changes of 16% was observed at the nematic/isotropic phase transition of the LCE capillary. The practical fabrication and specific actuation mode offer promising prospects for usages including actuators, microrobots, and other devices requiring peristaltic movements. The results showed that as the speed of the heat source increases, so does the peristaltic speed. However, this speed is limited by both the heat exchange and cooling rates, which are integral to the reversible phase transition process. Therefore, the peristaltic crawling speed can be augmented by a more effective heat transfer process.

#### 3.2. In-House Coding

## 4. Monte Carlo Simulations

- The mechanism of a deformation of a thin LCE, under non-uniform illumination of visible light, can be understood in the framework of FG modeling. For such thin LCE, the temperature dependence of physical quantities can be evaluated.
- The second is the deformation of LCE under external electric fields. The variables σ is aligned, either along or vertical to an electric field, and deformation of LCE is expected to be independent of how σ is aligned. Finally, the J-shaped stress–strain diagram of biological materials, e.g., blood vessels and skin, can also be considered in the scope of FG modeling.

## 5. Molecular Dynamics Simulations

- Crosslinking led to a 5 °C increase in the smectic–isotropic temperature.
- In a creep experiment, elastomers in the isotropic phase were observed to have high elasticity.
- When the elastomer undergoes the smectic–isotropic transition, memory effects in LC sample were observed.

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## 6. Other Numerical Methods

## 7. Discussion and Summary Remarks

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**FEM**: Usable for macroscale problems for solving a set of partial differential equations based on the consideration of the continuum medium. A good option for analysis of macroscopic behavior of LCEs, mechanical deformation, bending, and stretching when exposed to external stimuli for macroscale actuators and devices such as beams, pumps, and modules with locomotive motion. Providing an opportunity of monitoring and optimizing their performance whenever needed.- -
**MC**: Creating a connection between molecular structures, microscopic scale and macroscopic properties of LCE phases. Developing the simulation studies in a large time and space scale in comparison to MD. As this method is not deterministic, this feature makes it a good option for simulation of the systems which are in thermodynamic equilibrium state.- -
**MD**: Capability of demonstration of how small changes at molecular scale alter the macroscopic features. A good choice for both equilibrium and time dependent studies. Since this is a deterministic method, the trajectories of molecules can be determined, so it leads this method to be a good option for understanding the interactions between small molecules. The main drawback that can be ascribed to this method is its high computational cost of modeling the behavior of molecules.

## 8. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Stripes at the threshold strain of sample, with the final cross-linking done in the isotropic phase, as seen under the crossed polarized microscope. The orientation of crossed polars is indicated and the length of white lines corresponds to 10μm. Initial director orientation was in the vertical direction, and stretching was performed in the horizontal direction. Close to the transition (just below 77.4 °C) a very fine director modulation can be seen. (

**a**) T = 70.0 °C, (

**b**) T = 75.0 °C, (

**c**) T = 77.0 °C, (

**d**) T = 77.3 °C, (

**e**) T = 77.4 °C [37] (Reprinted with permission from [37]).

**Figure 2.**(

**a**) [40], shows the correlation between r = ${\left(\frac{L}{{L}_{ISO}}\right)}^{3}$ and T °C, and how it depends on the nematic order parameter. Note that L denotes the length of a sample when it is in the nematic phase and LISO denotes the sample’s reference length, when it is 12 °C above threshold temperature of 78 °C. (

**b**) shows the good agreement between numerical results and experimental data close to the transition temperature [39]. DT accounts for the random fluctuations added to the threshold temperature, which fluctuates specially (Reprinted with permission from [39,40]).

**Figure 4.**(

**a**–

**d**), show the deformation, distribution of stress, and contact pressure of the LCE microvalve, at various temperatures. Note that the color intensity represents the intensity of the local stress distribution. (

**e**) illustrates both the contact region and contact pressure between the LCE valve and surrounding silicon wall. Note that the temperature is close to phase transition temperature [25]. Here, X is the distance measured from the central point of the contact and D is the length of the contact area of the LCE beam [25] (Reprinted with permission from [25]).

**Figure 5.**Buckling phase diagram of LCE–PS bilayers with different aspect ratios and temperatures, ranging from 30 °C (reference temperature) 80 °C (Reprinted with permission from [25]).

**Figure 6.**(

**a**) Schematic view of LCE capillary fabrication via PI spin-coating, LC cell assembling, alignment layer rubbing, LCE precursor filling and photo-induced polymerization/crosslinking [43]. (

**b**,

**c**) photographic images of as-prepared LCE capillary with dimensions (Length = 27 mm, Diameter = 0.9 mm and wall thickness = 70 µm); (

**d**) chemical structure of the monomer, crosslinking agent and photoinitiator; (

**e**)chemical structure of polyamic acidused for preparing polyimide alignment layer (Reprinted with permission from [43]).

**Figure 7.**(

**a**–

**c**) illustrate the case studies considered in this study [26] (Reprinted with permission from [26]). (

**a**) FE simulations carried out on an azo-doped nematic elastomer beam which has been anchored by its right edge. When the beam is exposed to light, it automatically bends in an upward. The purpose of this simulation was to numerically investigate an experiment carried out by Camacho-Lopez et al. [57]. Far left: the beam is shown to be completely in the nematic state. Before exposure, the director is oriented horizontally. Middle: a rapid upward bend is induced by the top (red) layer when it switches to isotropic state. Right: six nematic elastomer actuators (illustrated in blue and green) arrayed in a ring at the tube’s base, cause the formation of a red rubber tube. (

**b**) FE simulations—Left: peristaltic motion a nematic elastomer tube. The motion is caused by periodic modulation of the scalar nematic order parameter along the tube’s length, e.g., by temperature or light. Note that The FE mesh nodes are represented as spheres. Right: peristaltic motion in a thin film, which is designed for covering and transporting the contents of a rigid. (

**c**) FE simulations—A nematic elastomer, mimics the crawling motion of an earthworm.

**Figure 11.**Results showing the configuration of LCEs, having rigid cross-linkers, at different stages, during a simulation where the strain-rate is kept constant. (

**a**) Initial state, showing a clear polydomain structure. (

**b**) Alignment begins after applying strain. (

**c**) As the strain is further increased, only small misaligned clusters remain [8] (Reprinted with permission from [8]).

**Figure 12.**Workflow of the multiscale model. At an instance in time (t), a photoisomerization ratio is calculated for given values of light intensity (I

_{o}) and temperature (T); and then is used for carrying out MD simulations, which in turn will provide microscopic information to the nonlinear FEA [103] (Reprinted with permission from [103]).

**Figure 14.**The shape evolution of the LCE when inhomogeneous temperature distribution is applied. (

**a**) the LCE is in its initial state (

**b**,

**c**) both show two transitional states, and (

**d**) demonstrates the sample after reaching steady state condition. (

**e**–

**g**) illustrate the LCE sample of (

**d**) from three different points of view [112] (Reprinted with permission from [112]).

**Figure 15.**Evolution of the LCE sample as a result of inhomogeneous temperature distribution. (

**a**) the LCE is in its initial state (

**b**,

**c**) both show two transitional states, and (

**d**) demonstrates the sample after reaching steady state condition. (

**e**–

**g**) illustrate the LCE sample of (

**d**) from three different points of view [112]. (Reprinted with permission from [112]).

Method | Applications |
---|---|

FEM (software) | (1) studying the Quasi-soft opto-mechanical behavior of a two dimensional (2D) rectangular beam using commercial software, ABAQUS [34] (2) studying the appearance of sub-stripe patterns in sheets of nematic elastomers using COMSOL Multiphysics [37] (3) simulating cooling and stretching experiments to investigate the transitional behavior of nematic elastomers using COMSOL Multiphysics [38] (4) bridging the gap between elastomer physics and device engineering and design in LCE-NTs using ANSYS [41] (5) accurate modeling of non-uniform deformation and instability of a constrained LCE beam, used as a microfluidic valves, in addition to studying a nanoscale poly styrene (PS) film laminated piece of LCE using ABAQUS [25] (6) simulating peristaltic crawling locomotion of earthworms using COMSOL Multiphysics [43] (7) simulating the domain evolution of a square LCE sample clamped on one side and free on all other three sides, performed using FEAP software [24] (8) wrinkling behavior in sheets of stretched nematic elastomer using ABAQUS [45] (9) elastodynamics simulations on dual-phase elastic solids using SALOME [52] (10) evaluation of mechanical reaction of LCE based elements to thermal stimuli using FEM software [53] |

FEM (in house codes) | (1) accurately modeling fine-scale oscillating behavior of nematic elastomers [56] (2) modeling elastodynamics of 3D devices made of nematic elastomers, including azo-doped nematic elastomer beam, peristaltic pumps and soft self-propelled robots [26] (3) studying the physics which govern the dynamic behavior of nematic elastomers by performing 3D elastodynamics simulations [27] (4) Topology optimization for generating reliable folding patterns in monolithic LCEs [60] (5) evaluation of the folding performance of various LCE sample designs [61] (6) studying director-encoded chiral shape actuation in thin-film nematic polymer networks, under the influence of external stimuli, namely via temperature [62] (7) studying the energetics of LCEs, when stress is utilized as a stimulant for reorienting the nematic directors [69] (8) analyzing and optimizing the performance of an LCN multi-legged gripper design [68] (9) study of acoustic properties and behavior of LCNEs [71,72,73] |

MC | (1) performing large-scale computer simulations for studying the effect of electric field as an actuation tool, to clarify the operational behavior of the LCEs on a molecular level [83] (2) developing MD method that could capture and reproduce experimental results obtained from a variety of experiments including stress strain, order, light transmission and thermal effects [85] (3) studying biaxial LCEs and predicting calorimetry data and deuterium magnetic resonance spectra [86] (4) investigating how sample preparation affects the nematic-isotropic behavior in LCE samples with both regular and irregular polymer networks [88] (5) gaining a better understanding of the anisotropic phenomena exhibited by LCEs [90] (6) Investigation of Shape and volume phase transitions of LCEs for both swollen and non-swollen state [97] |

MD | (1) predict the way the mechanical response of LCEs with side-chain architectures depends on their chemical structure [60] (2) studying how phase transition from monodomain to polydomain occurs in LCEs on a molecular scale [8] (3) developed a multiscale framework for studying the photomechanical behavior of photo-responsive polymer networks [103] (4) developed a coarse-grained model consisting of mesogenic molecules and smeared charges for the study of Liquid crystal polymers (LCPs) [107] (5) studying which force field among the three, Deriding, PCFF, and SciPCFF potentials should be applied to LCE modeling system [110] |

Other numerical methods | (1) exploring the dynamical behavior of LCEs [111,112] (2) model thermal–order–mechanical coupling behaviors in LCEs [36] (3) modeling the mechanical procedure by which biological cells exchange information [120] (4) studying the mechanical behavior of Smectic-A LCEs, in response to electromechanical stimuli [124] (5) modeling the electro-elastic behavior of nematic LCEs [128] (6) studying the thermos-mechanical behavior of a soft robot [133] (7) developing a general frame work for studying the interactions between LCs and polymer backbones that constitute a LCE [134] |

Method | Methodology and Formulation |
---|---|

FEM (software) | (1) the authors used an equation proposed by [35] and subsequently derived the linear stress strain relation for soft LCEs[34] (2) By using a non-linear theory developed by Warner and Terentjev [38], in conjunction with stretch test data, the authors were able to derive a linear model for carrying out their numerical analysis on nematic elastomers [37] (3) A modified linear version of the Warner and Terentjev [38] model was used by adding a spatially random temperature fluctuation term [38]. (4) The authors’ developed an empirical model, describing material deformation, using experimental actuation data [41]. (5) user-defined subroutines were developed [25]. (6) a shell conducting model with moving heat source was used [43]. (7) a phase-field model, for continuum-mechanical modeling of nematic LCEs, was implemented into the software [24] (8) A Koiter-type theory was developed, containing two terms: 2D or plane stress alleviation of the relaxed energy of DeSimone and Dolzmann [49] and bending [45] (9) a model first used by [26] was implemented, with two modifications: (a) defining the order parameter and nematic directors in each mesh element (tetrahedral in this case) used for discretizing the sample geometry, (b) insuring that enough number of mesh elements were used so that the directors transitioned smoothly [52]. (10) a statistical definition of microstructure nature of liquid crystal elastomers was presented then a quantitative, physics-based micromechanical model was used [53] |

FEM (in house codes) | (1) a coarse-grain model was presented which used the definition of energy density[56]. (2) Hamiltonian FEM approach was utilized [26]. (3) instead of a linear strain tensor, a rotationally invariant Green-Lagrange strain tensor was used for deriving the Hamiltonian [27] (4) Linear brick elements were used to perform a low fidelity FE analysis [60] (5) the moving asymptotes (MMA) method was utilized [61] (6) Hybrid particle FE elastodynamics simulations, with tetrahedral meshes, were used [62] (7) based on the VWT energy description, two separate contributing terms for the elastic energy were considered, namely deformation of the nematic director and the other, representing elastic component [69] (8) hyper-elastic model was solved using mixed FEM [68] (9) a transient model was developed that could couple elasticity and hydrodynamics of nematic elastomers [71]. A theory for approximating the behavior of acoustic waves in nematic elastomers was developed [72] two approaches were implemented and tested: one was according to the linear continuum theory of nematic rubber elasticity and in the second, a constitutive relation for rubber behavior was obtained using interpolation of experimental master curve of regular material [73] |

MC | (1) A semisoft deformation Monte Carlo method, in which the applied external field was perpendicular to the nematic director was used. all liquid crystal molecules were represented by uniaxial ellipsoids and The total interaction energy was obtained by summing up all non-bonded and bonded intermolecular contributions, using Soft-Core Gay–Berne Interaction [83]. (2) a simple coarse-grained lattice model was proposed. By obtaining the total Hamiltonian for LCE and through conducting constant-force Monte Carlo simulations, the simulations were carried out [85]. (3) a coarse-grain model was developed by summing of pseudo-Hamiltonians describing rubber elasticity, anisotropic interactions between biaxial mesogenic units, and the strain-orientational coupling of the polymeric chains [86] (4) main-chain systems were utilized. soft-core GB potential was used for the simulations. Uniaxial soft-core GB ellipsoids were utilized for modeling mesogenic molecules and for assembling LCE networks, and also to illustrate non-bonded swelling monomers [88] (5) Finsler geometry was used for the simulations [90]. A detailed description of Finsler geometry can be found in [91,92]. (6) the Finsler geometry was expanded by adding an Ising-like variable to account for both swollen and nonswollen LCE states [97] |

MD | (1) The elastomer is formed by crosslinking of the melt in the smectic A phase [60] (2) a coarse grain model was used in which ellipsoidal shapes were used to describe the mesogens particles. In addition, Gay-Berne (GB) potential was used to describe the interactions between mesogens [8] (3) The MD method used, is based on the work of Choi et al. [105] and utilizes energetic relaxation and multi-step crosslinking. In order to fully take into effect, the influence of kinked cis- molecules, Landau expansion was substituted for Maier-Saupe phase transition [6] by using a modified heuristic equation [106]. (4) the intermolecular interaction between ellipsoids is modelled using the SCGB potential. Subsequently, for the calculation of the piezoelectric tensor, point charges were introduced. The simulations were done using a course-grained MD program COGNAC [108,109]. (5) LAMMPS was used to find the most appropriate force field for LCE modeling [110] |

Other numerical methods | (1) non-local continuum model was developed using a novel preconditioner based on Chebyshev spectral collocation method [111,112] (2) total energy was a function of deformation gradient, director direction, order parameter and biaxiality. After deriving equations for mechanical and phase equilibrium, total free energy was considered to be a hybrid of the entropy-induced elastic energy and the Landau– de-Gennes nematic energy [36] (3) soft matter cell model based on Lagrange-type mesh-free Galerkin formulation was developed [120] (4) A free energy density function was obtained by summing contributions from (a) elastic free energy, (b) change in the layer thickness (c) variation in chirality, or equivalently the tilt (or rotation) of the director, (d) energetic contribution of the electrical sources to the total energy and (e) external mechanical loading [124] (5) a continuum-mechanical phase-field approach was adopted. A nemato-electro-elastic model is coupled with the fundamental Landau-de-Gennes theory for isotropic–nematic phase transition [128] (6) transient analytical solution consisting of a one dimensional heat transfer equation (solved using Green’s function) coupled with a bilayer beam model [133] (7) A continuum mechanics model was developed that in addition to taking into account the director or order tensors, along with their respective time derivatives, the deformation gradient and its derivative with respect to time was also considered [134] |

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Soltani, M.; Raahemifar, K.; Nokhosteen, A.; Kashkooli, F.M.; Zoudani, E.L.
Numerical Methods in Studies of Liquid Crystal Elastomers. *Polymers* **2021**, *13*, 1650.
https://doi.org/10.3390/polym13101650

**AMA Style**

Soltani M, Raahemifar K, Nokhosteen A, Kashkooli FM, Zoudani EL.
Numerical Methods in Studies of Liquid Crystal Elastomers. *Polymers*. 2021; 13(10):1650.
https://doi.org/10.3390/polym13101650

**Chicago/Turabian Style**

Soltani, Madjid, Kaamran Raahemifar, Arman Nokhosteen, Farshad Moradi Kashkooli, and Elham L. Zoudani.
2021. "Numerical Methods in Studies of Liquid Crystal Elastomers" *Polymers* 13, no. 10: 1650.
https://doi.org/10.3390/polym13101650