Introduction

Broadly speaking, nematic liquid crystalline (LC) solids refers to polymers incorporating rigid nematic liquid crystal molecules within their main-chain or side-chain if not both. Similar to the phase transition of LC materials, which are commonly found in modern display technology [9] the LC polymeric material alternates its shapes upon imposed stimuli. Recently, the materials and their relevant control techniques are envisioned to be game-changers in smart devices such as sensors and actuators [54]. Few examples of the phase behaviors are summarized in Fig. 1, in which primitive deformations such as uniaxial shrinkage and bending as well as complex applications such as micro-scale robotics and fluidic devices are found, demonstrating the versatility of the nematic solids.

Fig. 1
figure 1

Diverse phase behaviors found in literature. Figures are adopted from Refs. [2, 44, 46, 51, 58], with copyright Permission. Copyright (2010) Royal Society of Chemistry. Copyright (2009) WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. Copyright (2010) Royal Society of Chemistry. Copyright (2013) Royal Society of Chemistry. Copyright (2008) WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Thanks to the decades of thorough investigations and development of the materials, the qualitative mechanisms that govern the phase behaviors are mostly known. Long-range interactions between LC molecules predominately determine their molecular arrangements, although they are also affected by short-range molecular interactions that come from the polymeric and LC constituents. Such phase transition, in turn, changes the statistical distribution of crosslinking sites and polymeric shapes alike. The observed macroscopic deformations are then viewed as an outcome of such polymeric shape changes. In this regard, the overall mechanisms are hierarchical, and it is not surprising that any change of the involved agent (e.g., changing LC molecule) greatly alternates the overall behavior. However, the multiscale analysis and design framework, which possibly provides a link between the information of molecular-level and the macroscopic deformation, has received relatively little attention.

In this vein, this report aims to provide the working knowledge toward the phase behavior, as well as to review recent articles covering not only single-scale physics but also the trans-scaled physics found in the nematic solid. The manuscript is organized as follows. Section 2 first discusses phenomena found in the nematic solids, by which not only governing physics but also their correlations are elucidated. Section 3 introducing numerical methods, including recent multiscale analysis models are then followed.

Phase Behaviors of Nematic solids

This section introduces the underlying physics behind the phase behaviors of nematic solids [54]. The overview of mechanisms by which small-scale physics of LC phase transition turns into macroscopic deformation of solids are illustrated in Fig. 2. Note that the illustration assumes thermotropic LC molecules (i.e., reactive to heat), but the phase transition can be induced by other types of stimulus depending on the functionality of mesogenic groups.

Fig. 2
figure 2

Schematics of phase behaviors found in thermotropic liquid crystalline solid: a thermotropic LC phase transition; b polymer conformation change driven by LC phase change (Figure is adopted from Ref. [26] under Creative Commons Attribution License); c macroscopic deformation guided by radially distributed LC alignment. (Figure is adopted from Ref. [12] with copyright permission. Copyright (2012) Nature Publishing Group, a division of Macmillan Publishers Limited.)

The type of orders (e.g., positional, and orientational) of LC molecules differentiate from one phase to the other. For thermotropic LCs, the isotropic phase, in which the molecules lose long-range orders, is found in the high temperature. The nematic phase (orientational symmetry) and smectic phases (orientational and translational symmetries) are found in lower temperatures (Fig. 2a). These configurational changes are vital to understand the shape of polymeric chains. Say the LC networks are created at the nematic phase by crosslinking at a temperature lower than \(T_c\) (nematic-isotropic transition temperature). Mesogens and crosslinking sites are uniaxially oriented to \(\varvec{n}\) and so do the flexible network. Once heated over \(T_c\), the LC molecules lose orientational order, and correspondingly the polymeric network shrinks to \(\varvec{n}\), as described in Fig. 2b. It is worthwhile mentioning that these mesogenic shape change can be spatially different, and is often used to manipulate macro-scale deformation as shown in Fig. 2c; by making \(\varvec{n}\) to be oriented to its radial direction (marked by blue ellipsoids), the flat LC polymer evolves to have a saddle shape due to mechanical frustration.

Microscale LC Behaviors

As noted, the phase behaviors of LC networks are rooted in the microscopic change of the molecules, which determines the natural states of the polymeric conformations. Microscopic interactions can be roughly categorized based on the involving molecules: (1) mesogen-to-mesogen; (2) mesogen-to-polymer chain; and (3) intermolecular interactions between polymeric chains. The first two interactions are particularly important in describing the LC polymer since the third one does not deviate much from the typical non-bonded interaction while the others are peculiar in the materials of interest.

Long- and short-range interactions between mesogens, of which interplay determines the LC phase, are largely affected by the characteristics of the molecules. Since the first liquid crystal polymer synthesized [25] using typical benzoate LC molecule, various types of LC molecules are thenceforth suggested in a way to tune the phase behavior of the material. For instance, the phase transition properties can be tuned by alternating different flexible spacer lengths [18, 19], modulating microscopic irregularity. The crosslinked networks and polymers can also become reactive to light, by using mesogens that bear azobenzene chromophores [31], whose trans-cis isomerization affects overall LC orders. A large number of mesogen types, therefore, can be exploited in a way to modulate the phase behavior of the solid. A more comprehensive overview of the mesogen types can be found in the recent review of Rastogi et al. [42] (Fig. 3).

Fig. 3
figure 3

Classification of LC molecules found in liquid crystal polymer. Figure is adopted from Ref. [42] with copyright permission. Copyright (2019) Elsevier Ltd

The interaction of mesogens and flexible chains also has a significant impact on the structural properties. For instance, mechanical properties such as stiffness are affected by the density of the crosslinking, and spaces between rigid molecules. Strong crosslinking improves mechanical characteristics while restricting molecular orientational change; mild or weak crosslinking demonstrates the opposite effect [32, 43].

Mesoscopic Polymeric Shape Change

In the mesoscopic point of view, the change of polymer conformation (i.e., a shape change of the polymer backbone), and its spatial distributions are driving forces to the diverse phase behaviors. Above all, the polymeric conformation is affected by the LC phase upon crosslinking, by which the LC configurations and symmetry are inscribed to the polymeric system. When it is crosslinked in the deep nematic phase, for example, the change of polymeric conformation incurred by LC transition to isotropic phase, becomes more noticeable, as shown in the various experiments concerning diverse crosslinking temperature [8, 27].

Such spontaneous shrinkage/expansion found in nematic solid upon phase transition is able to be exploited in a way to guide the deformation. Diverse methods, including mechanical force (i.e., the two-step crosslinking method) [25], rubbing [50, 53, 60], optically controlled crosslinking [1, 12, 15, 17, 30], and 3D-printing [24, 52] have been suggested to align the mesogenic director in the designated direction (i.e., locally monodomain). Otherwise, LC molecules are randomly distributed; if the crosslinking occurs at the nematic phase, random distribution of the uniaxial director is created, which is often referred to as the polydomain structures [59] (Fig. 4).

Fig. 4
figure 4

Alignment strategy to assign spatially nonuniform liquid crystal alignment: a selective exposure to polarized light and photomask; b self-alignment using 3D printing technique Figures are adopted from Ref. [15, 24] with copyright permission. Copyright (2012) WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim, and copyright (2018) WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Macroscopic Deformation

The degree and spatially varying principal direction of the mesogenic conformation defines the overall phase behavior of the solid. In macroscopic point of view, these corresponds to the Euclidean metric change of the structure, by which the stress-free state of the solid is determined. When the change of the shape is restrained, such metric change in turn exerts mechanical load in return. For instance, the uniaxial stretch and shrinkage are observed in uniformly aligned thin LC elastomer, of which exerted force is comparable with a human tendon, upon cooling and heating [28]. It is also worth noting that the out-of-plane deformations such as bending and twisting can be realized by tailoring the shrinkage in the thickness direction by imposing uneven stimuli [55, 59], or by varying inscribed directors to the direction [17, 53].

When the induced metric change requires geometrically incompatible deformation, which is often referred to as the mechanical frustration of the structure, exotic shape transition [30] are observed. Such deformation is especially interesting in both theory and experiment since it facilitates usage of the nematic solids in actuation, because of change in Gaussian curvature (e.g., an evolution of the saddle shape) within a relatively short time period [39, 41] (Fig. 5).

Fig. 5
figure 5

Development of non-developable surfaces from flat liquid crystal membrane: a exotic topography (figure is adopted from Ref. [30] with copyright permission. Copyright (2013) WILEY-VCH Verlag GmbH 8 Co. KGaA, Weinheim); and b encoded Gaussian curvature (figure is adopted from Ref. [39] under Creative Commons Attribution License)

Multiscale Analysis of Nematic Solids

As discussed in Sect. 2, phase behaviors of the nematic LC solids are attributed to diverse mechanisms having dissimilar temporal and spatial scales. Although qualitative behaviors and their mechanisms are already well known, it is still true that information at molecular level is necessary especially in sensors and actuators, which are promising applications of the material. In this section, we first visit numerical models to address mechanisms that concern single physics and scale. Then we discuss recent multiscale analysis models accounting for two or multiple scales simultaneously. Note that the present investigation consists primarily in discussing multiscale modeling techniques, and only a limited number of works are addressed when they concern each individual physical phenomenon. Interested readers are kindly referred to a more comprehensive review of numerical methods of liquid crystal polymers [48].

Molecular Dynamics Simulation

Molecular dynamics (MD) simulations, which is based on interatomic potential and equations of motion based on the Newton’s second law, allow thorough investigation of inter- or intra-molecular interactions. Such in-silico experiment is especially pivotal in understanding nematic solids, of which phase behaviors are dependent not only to the bonded interactions between LC molecules and flexible polymer chains but also to their average symmetries.

First of all, all-atom molecular dynamics simulation (AAMD) focuses on detailed atomistic behaviors by considering interactions between all atoms. The method is widely employed to extract essential physical properties, including thermal conduction [45]. The incorporation of inorganic agents [23] within the network also has been discussed. The method has an inherent limitation in describing long-range interactions and ensuing phase evolution likewise. Nonetheless, there are a few recent notable attempts to investigate thermotropic phase behaviors, with and without UV bombardment [3] (Fig. 6).

Fig. 6
figure 6

All-atom molecular dynamic simulation of nematic solid: a thermotropic phase transition assisted by light irradiation (i.e., increase of cis-type molecules), and b increase of modulus anisotropy with the increase of cis-type molecules. Figure is adopted from Ref. [3] with copyright permission. Copyright (2014) AIP Publishing LLC

On the other hand, to understand the phase behaviors of the rigid molecules and the network chains at their vicinity, coarse-grained molecular dynamics simulation (CGMD) has been widely used since the seminal work proposed by Stimson and Wilson [49]. By substituting LC molecules with aspherical molecules and introducing Gay–Berne potential that considers long-range anisotropic interactions, CGMD models are able to capture detailed phase behaviors such as swelling [47], monodomain-polydomain transition [56], and even reversible shape change [20] (Fig. 7).

Fig. 7
figure 7

Coarse-grained molecular dynamic studies of a swelling behavior and b monodomain–polydomain transition, which demonstrates capability of simulating long-range interactions. Figures are adopted from Ref. [47, 56] with copyright permission. Copyright (2011) The Royal Society of Chemistry. Copyright (2013) American Physical Society

Finite Element Method

The finite element (FE) method, inarguably one of the most widely used numerical methods to understand physical systems containing complex boundary conditions, is also used to simulate, and predict the phase behaviors of the nematic solids. Due to its conceptual simplicity as well as an easy introduction to the finite element model, an eigenstrain model, which is a generalized model of a stress-free configuration change, is widely used. Assuming infinitesimal strain, stress within the solid \(\varvec{\sigma }\) reads [11]:

$$\begin{aligned} \varvec{\sigma }_{ij} = \varvec{C}:\left( \varvec{\epsilon } - \varvec{\epsilon }^*\right) , \end{aligned}$$
(1)

where \(\varvec{C}, \varvec{\epsilon }\) indicate elastic modulus of material and strain, respectively. \(\varvec{\epsilon ^*}\) denotes the eigenstrain, which represents polymeric conformation change. Note that Eq. (1) is similar to the thermoelastic relationship, but unlike thermal expansion, the eigenstrain is anisotropic and typically volume-conserving (i.e., \(\det (\varvec{\epsilon }^*) = 1\)).

Many interesting phase behaviors are interpreted in the view of eigenstrain, despite the method’s limitation in reflecting complex physics found in liquid crystalline solids. Dunn and Kurt [10] qualitatively discuss thermo-photomechanical behaviors of the large-deformation of liquid crystalline elastomer by introducing arbitrary volume-conserving eigenstrain.. The model is later utilized in designing a structure that folds in a directed way [13, 57]. A sophisticated version of Eq. (1) is later introduced [29], whereby the eigenstrain is treated as a function of polymer conformation (i.e., shape parameter) as well as temperature condition, both of which correspond to the real experiment. A generalized model utilizing a nonlinear shell is also proposed [5, 7]. Note that the same framework is also employed to understand out-of-plane deformation exhibited by thin LC-incorporating materials [34,35,36], which is later revisited based on plate theory [33].

In contrast to the eigenstrain-based modeling, the complex coupling between LC and material’s elasticity is analyzed through FE-based analysis. The shape change caused by phase transition is obtained by extremizing the action in the presence of dissipation of the system. Primitive deformations such as bending and twisting are first discussed in Zhu et al. [62], whereby the free energy suggested by Verwey [54] is combined with LC energies. Several other works [14, 16] also address complex out-of-plane distortions by simple Hamiltonian-based model. However, it is worth noting that these nemato-elastodynamics models mainly aim to comprehend the coupled behavior of LC solids driven by its nature of nonconvex Lagrangian, in contrast to the eigenstrain-based modeling. The methods thus have limitations in terms of their efficiency, robustness and stability, which are vital in solving inverse problems (Fig. 8).

Fig. 8
figure 8

Macroscopic deformation analyzed by finite elements: a, b eigenstrain-based analysis of thin LC membrane; c, d Nemato-elastodynamics simulation of LC-inscribed nematic solids. Figures are adopted from Ref. [5, 10, 14, 62] with copyright permission. Copyright (2009) Elsevier Ltd. Copyright (2015) American Physical Society. Copyright (2011) American Physical Society. Under Creative Commons Attribution License

Mutlscale Simulation

Micro and macroscopic behaviors should be considered through a multiscale scheme, in order to investigate the effect of small-scale details on the macroscopically observed phase-induced deformation. Chung et al. [6] proposed the upscaling strategy of the photo-mechanical liquid crystal solid for the first time, where the nonlinear finite element model and non-dilute multiscale phase transition model were computed based on the all-atom MD simulation of photo-thermochromic liquid crystal [3] are combined. Yun et al. [61] introduced quantum mechanics-based phase transition theory to the macroscopic elasticity, which is validated through experiment. Despite its straightforwardness, the multiscale simulation reveals that the overall deformation will be underestimated when the dilute model of phase transition is used. Similar schematics are also utilized in liquid crystal networks undergoing a smectic-nematic-isotropic phase transition, which is proposed by Moon et al. [38]. Because of the need of modeling a longer characteristic length than the nematic case, CGMD simulation is utilized to investigate the change of polymeric morphology for the given LC phase transition. Based on classical constitutive modeling with thermodynamic considerations, these multiscale models incorporate several internal variables that reflect the small-scale nature of the material and change its macroscopic behavior. For example, the mesoscopic order parameter, which is available in both micro-and macro-scale theories, is one of these internal variables. However, a straightforward sequential scale-bridging method has limitations similar to the eigenstrain model discussed in Sect. 3.2; the macroscopic change does not affect the phase behavior, and is thus not able to identify the micropolar behavior often found in stretched nematic solid.

It is worth noting that other types of multiscale modeling technique have also been proposed based on the computational homogenization. Several works [4, 37] envisioned a possibility of modulating phase behavior using inclusions such as nanoparticle and carbon nanotube. Nadgir et al. propose the homogenization principle for nematic solid, where local behavior of the represented volume element of the solid is investigated based on the phase-field model [22, 40] (Fig. 9).

Fig. 9
figure 9

Various multiscale simulation schematics: a all atom molecular dynamics simulation-FE coupling (figure is adopted from Ref. [6] under Creative Commons Attribution License); b quantum mechanics-FE coupling (figure is adopted from Ref. [38] with copyright permission. Copyright (2021) American Physical Society); c coarse-grained molecular dynamics simulation-FE coupling (figure is adopted from Ref. [61] with copyright permission. Copyright (2015) Elsevier Ltd)

Summary

Nematic solids are novel synthesized materials that are envisioned to have various applications such as sensors and actuators. Having a wide design space not only in material properties but also macroscopically tailored deformations, the material should be understood in the view of the multiscale ranging from molecular-level details to macroscopic elasticity. The present review first visits the multiscale nature inherent within the system. Experimental studies of both inter- and intra-molecular behaviors are investigated, which greatly affects not only the short-range interactions but to the physics found in higher scales. The governing physics found in meso- and macro- scale are later discussed in terms of polymeric conformation change, and elasticity incorporating such, respectively. Correspondingly, numerical analysis models using molecular dynamics simulation, and finite element are reviewed so to articulate the way the method elucidates the small and large scale behaviors, but not in a coupled way. Lastly, the recent multiscale simulation models are visited and the limitations of the method is discussed. Even with the great potential of the material and decades of numerical investigation, the multiscale method has not been widely employed to understand the material. The authors believe that a deeper understanding of the interesting material will be realized soon by considering novel scale bridging techniques, such as data-driven multiscale [21] that does not assume specific coupling physics and uses data itself.