Nucleation and critical conditions for stripe domains in monodomain nematic elastomer sheets under uniaxial loading

Highlights

• Strain-induced stripe domains in real-size monodomain nematic elastomers are simulated.

• A stress peak and sharp change in elastic energies are observed when stripe domains nucleate.

• Critical conditions for the formation of stripe domains are obtained.

• Parameter dependence of critical conditions is examined using phase diagrams.

• Minor deviations from the orthogonal loading condition can cause uniform rotations.

Abstract

Liquid crystal elastomers with mobile liquid crystal moieties display unique mechanical instabilities and domain patterns in the strain-induced director reorientation. Based on a continuum mechanical model, we present a complete Lagrangian description of finite element formulation of LCEs by solving the coupled displacement and director fields within the Lagrangian scheme. The nucleation of the stripe domain and critical conditions for its formation are numerically studied for thin monodomain centimeter-sized sheets. Under orthogonal uniaxial loading tests, micrometer-sized stripe domains nucleate after loading to a critical strain at which a stress peak is observed along with a reduction in the neo-classical elastic energy. Further, we observed increase in the semisoft and the Frank elastic energy as indications of strong and heterogeneous director rotations. When the loading axis was not exactly orthogonal to the initial director, we found that the sample morphology was strongly dependent on the material and geometrical parameters. Strain-induced stripe domains could still be observed in the central region of the sample when the loading axis was nearly at right angle to the initial director. If the angle between the loading axis and the initial director was slightly deviated from the right angle, nearly uniform rotations were observed. The critical angle, beyond which strain-induced stripe domains were observed, could be very close to the right angle for samples with rather small shape anisotropy and relatively large length to width ratio. In such cases, the observation of strain-induced stripe domains was highly sensitive to any possible experimental error in orienting the loading axis at right angle, and minor deviations from the orthogonal loading condition can cause uniform rotations.

Introduction

Liquid crystal elastomer (LCE) is a soft solid material with unique combination of rubber elasticity and liquid crystal order (de Gennes, 1975; Finkelmann et al., 1981). It exhibits several unusual properties, such as large spontaneous shape changes in response to external stimulus, transition and instabilities induced by mechanical action, and other abnormal dynamical effects (Broer et al., 2011; de Jeu, 2012; Warner and Terentjev, 2007). Thermal phase transformations can give rise to spontaneous shape change of several hundred percent (Tajbakhsh and Terentjev, 2001). For lightly cross-linked LCEs with mobile liquid crystal moieties, reorientation of the director can be induced by applying mechanical stress or strain (Higaki et al., 2013; Kundler and Finkelmann, 1995, 1998; Küpfer and Finkelmann, 1991, 1994; Mistry et al., 2018; Mitchell et al., 1993; Roberts et al., 1997; Talroze et al., 1999; Urayama et al., 2007; Zubarev et al., 1999). It is now well established that the director of prolate nematic elastomer can reorient itself along the applied uniaxial loading direction to reduce the bulk elastic energy. Remarkably, under some conditions, it can even deform under very low stress and behaves likes entirely soft material, and this phenomenon is called soft or semi-soft elasticity (Bladon et al., 1993; Olmsted, 1994; Verwey and Warner, 1995; Verwey et al., 1996; Warner et al., 1994; Warner and Terentjev, 2007).

As the director rotation is accompanied by both shear and elongation of networks, the two side clamped boundaries in uniaxial stretching experiments can prevent the shear deformation. Thus, textured regions with differently oriented director have been often observed during the strain- induced reorientation process. Especially, when the initial director is nearly orthogonal to the loading axis (orthogonal loading), micrometer-sized parallel stripe domains (SDs) with an alternating sense of director rotation in each domain have been observed after loading to a certain critical strain (Kundler and Finkelmann, 1995). This is attributed to a type of first-order transition under the coupled effect of the semisoft elasticity of the backbone polymers, the Frank energy of the director distortions, and the boundary constraints (Finkelmann et al., 1997; Verwey et al., 1996). The effect of the boundary constraint was experimentally demonstrated by Talroze et al. (1999) and Zubarev et al. (1999) in which SDs were observed only in relatively short samples (smaller length to width ratio or aspect ratio AR = L/H). For longer samples with larger AR, uniform rotations (URs) were reported. However, URs were always observed even for relatively smaller AR in Mitchell et al. (1993) and Roberts et al. (1997). Moreover, URs initiated at some critical strain and jumped to align close to the loading axis. Such an almost discontinuous rotation is similar to the Freedericksz transition in liquid crystals and is called the mechanical Freedericksz transition (MFT) (Mistry et al., 2018). Because the monodomain samples in the above experiments were obtained with two different methods, it was conjectured that the difference between the samples have resulted in different strain-induced reorientation behavior. Zhang et al. (2006) used a third method to prepare the monodomain nematic films and tested samples with the same dimension, which were cut from the same film. Surprisingly, they observed SDs for some samples and URs for others in the uniaxial orthogonal loading tests. Their results clearly indicate that both the director rotation modes (SDs and URs) are possible and cannot be only attributed to the material system, the preparation method, or the geometry of the testing samples. They suggested that it might be due to the unavoidable experimental error in realizing the orthogonal loading condition, i.e., the possible deviation of the angle between the loading axis and the initial director of the samples from right angle. Nevertheless, it is not clear why such high sensitivity to small deviations is only reported for their samples and not for others.

Several theoretical works have focused on the unusual mechanical behavior of LCEs. Bladon et al. (1993) generalized the classical neo-Hooke elastic energy of conventional rubber to LCEs by introducing step length tensors to describe the anisotropy of the network chains. The generalized neo-Hooke (neo-classical) elastic energy could explain the soft elastic response and the URs (Olmsted, 1994; Warner et al., 1994). A semi-soft energy was added to the neo-classical elastic energy to mimic the network constraints on the director rotations (Verwey et al., 1996). Several continuum mechanical models (Calderer et al., 2006; Cesana and DeSimone, 2009; DeSimone and Teresi, 2009; Jin et al., 2010) extended the elastic energies by adding Frank energy (Frank, 1958) and Landau–de Gennes energy (de Gennes et al., 1993) and by considering probable changes in the order parameter (Ericksen, 1991). Some additional constitutive models (Anderson et al., 1999; Brand and Pleiner, 1994; Fried and Sellers, 2004) have been proposed based on statistical and continuum mechanical considerations.

Based on some approximate solutions, the SDs have been studied quite extensively (Finkelmann et al., 1997; Fried and Sellers, 2006; Verwey et al., 1996; Warner and Terentjev, 2007; Weilepp and Brand, 1996). The strong effect of the network anisotropy (step length ratio) on the critical strains for the formation of SDs is considered to be the possible reason that SDs are not observed in some experiments ((Finkelmann et al., 1997; Fried and Sellers, 2006). The possible effect of the spatial inhomogeneities of the initial director field was also proposed as an alternative explanation (Brand et al., 2006; Weilepp and Brand, 1996). However, the boundary constraint imposed by the clamping and the effect of AR (length to width ratio) can only be taken into account by using numerical simulations.

The direct implementation of finite element method (FEM) for the soft elasticity is rather difficult due to the non-convex nature of the neo-classical elastic energy. Conti et al. (2002a, b) were the first to numerically simulate LCEs. They eliminated the orientation field n from the neo-classical model by minimizing the energy for fixed displacement fields. Thus, the resulting energy as a function of displacement field was quasi-convex. They simulated a LCE film stretched perpendicular to the initial director and successfully recovered the soft and semi-soft elasticity as well as the SD pattern. However, the actual geometric features of the SD microstructure (e.g., its width) was not captured in this simulation because it was based on a quasi-convex energy. By adding the approximate interface energy into the neo-classical free energy, Brown and Adams (2013) numerically studied the stretching of monodomain smectic-A elastomer sheets to investigate the effects of the angle between stretching direction and initial orientation and the AR. de Luca et al. (2013) developed a theory for small deformations and large director rotations and performed finite element simulation to study the sub-stripe patterns observed experimentally. Other notable numerical simulations to investigate the coupled dynamics of the director rotation and deformations include the Hamiltonian-based FEM (Mbanga et al., 2010), non-linear FEM with a phase field model (Oates and Wang, 2009), and spectral method with a nonlocal continuum model (Ennis et al., 2006; Zhu et al., 2011).

Although these theoretical and numerical studies have partially explained the mechanism of soft or semi-soft elasticity of LCEs and the appearance of stripe instabilities, a clear understanding of the reorientation process and mechanical instabilities is still lacking in the existing literature, and several questions remain unanswered. For example, what are the conditions to observe SDs instead of URs under uniaxial orthogonal loading? Can we observe SDs under oblique but non-orthogonal loadings? What is the mechanism for the director rotation process to progress gradually or almost discontinuously? Here, we have tried to systematically answer these questions.

One of the major difficulties in the numerical investigation of SDs is the large difference between the dimensions of the real sample (~cm scale) and the stripes (~20 μm). In several earlier studies, much smaller samples were considered due to the high computational cost for resolving the SDs. In a previous work (Zhang et al., 2019), we developed a general continuum theory for the mechanical-order-director coupling behavior of LCEs. Further, we demonstrated that this theory can be easily implemented using a commercial finite element software for effective numerical simulations of large deformations and large director rotations.

In this study, we present a complete Lagrangian description of the governing equations and the finite element formulations for LCEs based on a continuum mechanical model. The rest of this paper is organized as follows. In Section 2, the coupled fields of displacements and directors are solved in the Lagrangian reference frame by implementing a commercial software COMSOL Multiphysics. Semisoft elastic energy and Frank distortion energy are considered. Only plane problems are simulated to reduce the computational cost. The finite element results are presented in Section 3. Firstly, the strain-induced SDs are obtained under orthogonal uniaxial loading, i.e., the loading axis is at right angle to the initial director. The variations in the stress and the elastic energy at nucleation are analyzed. Then, we consider loadings that deviate slightly from the orthogonal condition. The effects of network anisotropy and initial AR on the formation of SDs is systematically investigated. The critical conditions to observe the SDs are obtained and described using phase diagrams. Further, the numerical results are compared with earlier reported experimental results. Finally, the study is concluded in Section 4.