Recent Trends in Continuum Modeling of Liquid Crystal Networks: A Mini-Review

Abstract

This work aims to provide a comprehensive review of the continuum models of the phase behaviors of liquid crystal networks (LCNs), novel materials with various engineering applications thanks to their unique composition of polymer and liquid crystal. Two distinct behaviors are primarily considered: soft elasticity and spontaneous deformation found in the material. First, we revisit these characteristic phase behaviors, followed by an introduction of various constitutive models with diverse techniques and fidelities in describing the phase behaviors. We also present finite element models that predict these behaviors, emphasizing the importance of such models in predicting the material's behavior. By disseminating various models essential to understanding the underlying physics of the behavior, we hope to help researchers and engineers harness the material's full potential. Finally, we discuss future research directions necessary to advance our understanding of LCNs further and enable more sophisticated and precise control of their properties. Overall, this review provides a comprehensive understanding of the state-of-the-art techniques and models used to analyze the behavior of LCNs and their potential for various engineering applications.

Keywords: liquid crystal network; phase behavior; phase transition; soft elasticity; spontaneous behavior.

Figure 1

Microscopic description of polymer conformations: (a) spherical shape with isotropic LC phase and (b) prolate shape with nematic LC phase that is oriented with the director vector $R_z$ (Adapted with permission from Ref. [4]. 2018, Sabina W. Ula et al.).

Figure 2

Polydomain–monodomain transition of LCN induced by stretch (a) X-ray scattering patterns of the polydomain (left) and monodomain (right) texture (Adapted with permission from Ref. [31]. 1998, S. M. Clarke); (b) stress–strain and order parameter–strain relationships in polydomain nematic elastomers (PNE), illustrating semi-soft elasticity behavior and the correlation between stress, order parameter (S), and strain (ϵ). (Adapted with permission from Ref. [30]. 2009, Kenji Urayama et al.).

Figure 3

Stretch-induced rotation of monodomain LCN: (a) the stripe domain texture; (b) the scheme of experimental geometry where stretch is imposed to the direction perpendicular to ${\mathit{n}}_{\mathbf{0}}$ (Adapted with permission from Ref. [29]. 1996, Kenji Urayama et al.).

Figure 4

Spontaneous shrinkage of one-dimensional LCN strip: (a) strain induced by heating (Adapted with permission from Ref. [35]. 2010, Antoni Sanchez et al.); (b) order–strain relationship of LCN with flexible siloxane chain (SiF*) reported in [36]. The star and circle markers denote the uniaxial length change, and change of LC orientational symmetry, respectively (Adapted with permission from Ref. [36]. 2001, S. M. Clarke et al.).

Figure 5

Idea of rank-one lamination of the deformation gradient. (a) Example of laminate of compatible gradients A and B with volume fractions $\lambda$, $1-\lambda$, respectively (Adapted with permission with permission from [47]. 2004, Ball, J.M). (b) Upper: demonstration of material continuity between dissimilar deformations ${\mathit{F}}_1, {\mathit{F}}_2$ by rigid body rotation. Bottom: lamination leading to stripe domain shown in nematic monodomains (Adapted with permission from [49]. 2009, Biggins, J.S. et al.).

Figure 6

Phase behaviors predicted by Jin et al. (a) Uniaxial stretch $\lambda$ of a LCN material of nematic director ${\mathit{E}}_{\mathbf{3}}$; (b) spontaneous deformation ${\lambda }_{\mathit{m}}$ induced by LC order collapse $Q_{\mathit{m}}$; (c) the non-monotonic correlation between normalized stress ($\sigma /3\mu$ ), in both stretched (${\sigma }_{xx}$ ) and transverse (${\sigma }_{xz}$ ) directions, and the given stretch (λ) observed during the uniaxial stretch when $\zeta =15°$ (Adapted with permission from Ref. [57]. 2010, Jin, L.).

Figure 7

Heat-induced order collapse and the phase behavior of LCNs observed through molecular dynamics simulations: (a) correlations of the orientational order S and uniaxial shrinkage λ for the given temperature that are modulated via changing isomerization percentage (Adapted permission from Ref. [23]. 2016, Chung, H. et al.); (b) effect of incorporated metallic (Au) nanoparticle to the order collapse, and its dependence on the size of the particle (Adapted with permission from Ref. [79]. 2016, Choi, J. et al.).

Figure 8

Micromechanical model of polymer network incorporating phase-changing molecules (Adapted with permission from [81]. 2019, Brighenti, R. et al.).

Figure 9

Tension-induced instability of the nematic solid: (a) uniaxial tension-induced evolution of the von Mises stress (left) and texture (right), where darker colors denote higher stress concentration and higher index number (Adapted with permission from Ref. [84]. 2002, Conti, S. et al.) (b) under various order parameters (r); changes in the scaled out-of-plane displacement ($u_3^{max}/h$) occur with increased stretch (${ϵ}_{eng})$. The higher the order parameters, the fewer micro-wrinkles observed (Adapted with permission from Ref. [53]. 2017, Plucinsky, P. et al.).

Figure 10

Dynamic texture evolution represented by pseudo-director with colorbar and arrow model, which are expressed according to the local direction. (a) Monodomain to stripe texture due to tension; (b) quenching simulation of heated LCN (left) to cooled LCN (right) (Adapted with permission from Ref. [55]. 1992, Oates, W. S et al.).

Figure 11

Eigenstrain-based modeling of bending deformation: (a) Gaussian curvatures depending on the LC states (Adapted with permission from Ref. [70]. 1992, Dunn. M. L.); (b) prediction of light-induced bending (Adapted with permission from Ref. [83]. 2015, Chung, H. et al.) considering photobleaching effect depending on different thickness of element; (c) numerical model of temperature-induced bending of LCN numerical and comparison with the experiment (Adapted with permission from Ref. [19]. 2021, Brighenti, R et al.).

Figure 12

Structural instability observed in (a) 3D-printed LCN disk (Adapted with permission from Ref. [12]. 2020, Wang, Z et al.) and (b) oblique angle ($\alpha )$ oriented LCN cylinder depending on different $\theta$, which affects shear stress (Adapted with permission from Ref. [97]. 2021, Zhao, S et al.).

Figure 13

Examples of non-Galerkin-based FE model: (a) dynamics of stimuli-induced shape change (Adapted with permission from Ref. [103]. 2021, Zhu, W. et al.), (b) evolution of topography analyzed based on nemato-elasto-dynamics model. With the upper 1$\times$4 patterns (red) 90-degree counterclockwise rotated 3 times, the LC films for the test are generated (4$\times$4) (Adapted with permission from Ref. [37]. 2021, Haan, L.T. et al.).