## Abstract

Liquid crystal elastomers (LCEs) are made of liquid crystal molecules integrated with rubber-like polymer networks. An LCE exhibits both the thermotropic property of liquid crystals and the large deformation of elastomers. It can be monodomain or polydomain in the nematic phase and transforms to an isotropic phase at elevated temperature. These features have enabled various new applications of LCEs in robotics and other fields. However, despite substantial research and development in recent years, thermomechanical coupling in polydomain LCEs remains poorly studied, such as their temperature-dependent mechanical response and stretch-influenced isotropic-nematic phase transition. This knowledge gap severely limits the fundamental understanding of the structure-property relationship, as well as future developments of LCEs with precisely controlled material behaviors. Here, we construct a theoretical model to investigate the thermomechanical coupling in polydomain LCEs. The model includes a quasi-convex elastic energy of the polymer network and a free energy of mesogens. We study the working conditions where a polydomain LCE is subjected to various prescribed planar stretches and temperatures. The quasi-convex elastic energy enables a “mechanical phase diagram” that describes the macroscopic effective mechanical response of the material, and the free energy of mesogens governs their first-order nematic-isotropic phase transition. The evolution of the mechanical phase diagram and the order parameter with temperature is predicted and discussed. Unique temperature-dependent mechanical behaviors of the polydomain LCE that have never been reported before are shown in their stress-stretch curves. These results are hoped to motivate future fundamental studies and new applications of thermomechanical LCEs.

## 1 Introduction

Liquid crystal elastomers (LCEs) are soft active materials that combine the orientational order of liquid crystal molecules (mesogens) and the entropic elasticity of rubber-like polymer networks. This inherent coupling at the molecular level provides LCEs with intriguing mechanical responses such as soft elasticity [1], high energy dissipation [2], and programmable anisotropy [3], as well as stimuli-responsive large deformation induced by thermal [3,4], photothermal [5,6], and photochemical [7–9] processes. As a result, LCEs have enabled exciting new applications in mechanics, materials, and robotics. Examples include thermomechanical actuators [10,11], light motors [12], shape morphing structures [13,14], energy dissipaters [2,15], and biomedical devices [16]. Accompanying these new applications is the rapid development of theories for various mechanical responses of LCEs under different loading conditions and/or stimuli, including monotonic load [1], biaxial load [17,18], bending [19,20], heating [21], and light illumination [7,8].

In a typical thermotropic main-chain LCE, LC mesogens are linked in the linear polymer chains of a stretchable amorphous polymer network. The rod-like mesogens form a *nematic* phase with a long-range orientational order at low temperature (e.g., room temperature) and transform to an *isotropic* phase with no orientational order at temperature above a nematic-isotropic transition temperature (e.g., around 60 °C [22]). The low-temperature nematic phase can be either *monodomain* with a uniform mesogen orientation or *polydomain* with many co-existing domains of different mesogen orientations. These different phases and domain formations significantly affect the mechanical (stress-stretch) responses of their corresponding LCEs [23]. In brief, an isotropic LCE at high temperature has isotropic, common rubber-like mechanical responses, a nematic monodomain LCE has anisotropic mechanical responses due to the uniform mesogen order, and a nematic polydomain LCE has macroscopically isotropic mechanical responses due to the co-existing domains at the microscale.

This paper focuses on the elastic responses of nematic polydomain LCEs. Understanding the fundamental mechanics of polydomain LCEs is of great importance to their future research and development for two reasons. First, fundamentally, the mechanical behavior of polydomain LCEs can be considered as a more general case that involves the mesogen director rotation [1,24–26] during deformation, compared to behaviors of monodomain LCEs without involving such mesogen rotation. Second, in practice, a polydomain LCE is the immediate product of the first-step synthesis during the widely used two-step fabrication of LCEs in experiment [27], where the synthesized polydomain sample is subsequently stretched and crosslinked in the second step to form the final monodomain sample.

In studying the important physical parameters and processes in an LCE, temperature is the only parameter that critically affects both the nematic-isotropic phase transition of mesogens and the entropic elasticity of the polymer network. However, despite recent progresses in the theoretical modeling [1,24,25,28], numerical simulation [18], and experimentation [18,29] of polydomain LCEs, thermomechanical coupling in polydomain LCEs, such as the temperature-dependent mechanical response and the stretch-dependent isotropic-nematic phase transition, is not well studied. This knowledge gap severely limits the future development of LCEs with the aim of precisely controlling their behaviors under various temperatures and mechanical loads. This lack of knowledge further hinders the goal of establishing a quantitative relationship between material parameters and the final material property, to replace the current empirical approach in the fabrication of LCEs that involves temperature change and stretching of polydomain samples.

Here, we develop a theoretical model to study the thermomechanical coupling in polydomain LCEs. The continuum model includes a quasi-convex elastic energy of the polymer network and a free energy of mesogens (Sec. 2). Building on this model, we investigate working conditions where a polydomain LCE is subjected to various prescribed planar stretches and temperatures. The quasi-convex elastic energy enables a “mechanical phase diagram” that describes the macroscopic effective mechanical response of the material, and the free energy of mesogens governs the first-order nematic-isotropic phase transition (Sec. 3). Evolution of the mechanical phase diagram and the order parameter with temperature is predicted and discussed (Sec. 4). Unique temperature-dependent mechanical behaviors of the polydomain LCE that have never been reported before are shown in their stress-stretch curves (Sec. 5).

## 2 Theoretical Model

The general theoretical framework is illustrated in Fig. 1. Throughout this paper, we use *T*_{ni} to represent the nematic-isotropic phase transition temperature of mesogens without any additional coupling from mechanical stretch. The high-temperature isotropic phase at *T* > *T*_{ni} is chosen as the reference state (Fig. 1(a)). At a temperature *T* < *T*_{ni} without any mechanical stretch, the LCE stays in the nematic phase with polydomain, where different domains have different mesogen orientations (Fig. 1(b)). In the current state, the LCE is subjected to planar tensile stretches *λ*_{2} and *λ*_{3}, with the corresponding true stresses *s*_{2} and *s*_{3} (Fig. 1(c)). For mathematical convenience, the principal stretches satisfy *λ*_{3} ≥ *λ*_{2} ≥ *λ*_{1} [25,28]. We will investigate various temperatures both below and above *T*_{ni} in the current state.

*Q*is the nematic order parameter (

*Q*= 0 represents the isotropic phase without directional order of mesogens,

*Q*= 1 represents perfect alignment of mesogens, and 0 <

*Q*< 1 represents the general nematic phase),

*a*= (1−

*Q*)/(1 + 2

*Q*),

*N*is the number of polymer chains per unit volume,

*kT*is the temperature in the unit of energy, and

*NkT*is the shear modulus of the elastomer. The three phases “L,” “S,” and “Sm” correspond to different macroscopic mechanical behaviors of the polydomain LCE, which will be discussed in Sec. 3.1. Here we further modify the form of the free energy to make it consistent with the original form derived by Bladon et al. for monodomain LCEs [23,30], to ensure the accurate terms containing

*Q*in the model, which will affect the thermomechanical coupling to be investigated later.

*N*_{n} is the number of mesogens per unit volume, and *J* is the interaction parameter between mesogens in the unit of energy.

*λ*

_{1}

*λ*

_{2}

*λ*

_{3}= 1. Using this to express the out-of-plane stretch

*λ*

_{1}= 1/

*λ*

_{2}

*λ*

_{3}, in thermodynamic equilibrium, we have

*s*

_{2}and

*s*

_{3}are the principal true stresses in the planar directions and

*s*

_{1}= 0. Substituting Eqs. (1) and (5) into Eq. (7), we obtain the specific forms of

*s*

_{2}and

*s*

_{3}as

Finally, we introduce the following dimensionless groups in Table 1. In particular, we choose the dimensionless interaction parameter $J^=4.54$ such that the nematic-isotropic transition takes place at $T^=T/Tni=1$ when no effect of mechanical stretch is considered [8,33] (to be discussed in Sec. 3.2).

Dimensionless group | Expression | Numerical value |
---|---|---|

Average interaction parameter between mesogens | $J^=J/kTni$ | 4.54 |

Number of polymer chains per unit volume of the elastomer | $N^=N/Nn$ | 0.05 |

Temperature | $T^=T/Tni$ | Varying |

True stress | $\sigma ^i=\sigma i/NkTni$ | Varying |

Free energy | $W^=W/NnkTni$ | Varying |

Dimensionless group | Expression | Numerical value |
---|---|---|

Average interaction parameter between mesogens | $J^=J/kTni$ | 4.54 |

Number of polymer chains per unit volume of the elastomer | $N^=N/Nn$ | 0.05 |

Temperature | $T^=T/Tni$ | Varying |

True stress | $\sigma ^i=\sigma i/NkTni$ | Varying |

Free energy | $W^=W/NnkTni$ | Varying |

## 3 Uncoupled Mechanical and Thermal Behaviors

### 3.1 Macroscopic Effective Mechanical Responses.

We first construct a mechanical phase diagram to describe the macroscopic effective mechanical response of a polydomain LCE under different planar stretches, based on the area stretch ratio 1/*λ*_{1} = *λ*_{2}*λ*_{3} and the largest principal stretch *l*_{3} (Fig. 2(a)). To do so, we fix the temperature at $T^=0.8$ and use Eqs. (1)–(6) to solve for *Q* through finding the global minimum of the total free energy *W*(*λ*_{i}, *Q*) at different stretch values (the distributed values of *Q* in the phase diagram will be discussed in Figs. 3 and 4). Afterward, we identify the external and internal boundaries of the phase diagram following the original work of DeSimone et al. [25,26,28].

The phase diagram is bounded by two external boundaries, $1/\lambda 1\u2265\lambda 3$ and $1/\lambda 1\u2264\lambda 32$, corresponding to the uniaxial and equal-biaxial loading paths, respectively. This is a result of incompressibility *λ*_{1}*λ*_{2}*λ*_{3} = 1 and *λ*_{3} ≥ *λ*_{2} ≥ *λ*_{1} in the model setup. In addition, the phase diagram consists of three “mechanical phases,” as discussed in the following.

In *Phase L* (liquid), the solid-state polydomain LCE behaves liquid-like—the material shows zero stress despite finite principal stretches. This is a direct consequence of the infinite ground states of the quasi-convex elastic energy, stemming from the coupling between mesogen rotation and polymer chain stretching during deformation. This phenomenon of zero stress with finite stretches is called *soft elasticity*, which has been widely reported in LCEs before [23,24].

In *Phase Sm* (smectic) upon further biaxial stretches, the polydomain LCE behaves solid-like when being stretched out of plane (along *l*_{1}) but behaves liquid-like when being stretched in plane. The free energy in Phase Sm only depends on the out-of-plane stretch *l*_{1}, such that the LCE always shows identical biaxial true stresses *s*_{2} and *s*_{3} in this mechanical phase (Eq. (8)). This phenomenon has been recently validated in experiment and analyzed in numerical simulation [18].

Further biaxial or uniaxial stretches also lead to *Phase S* (solid), where the LCE behaves like a common rubber. In particular, under the uniaxial stretch state in Phase S, the LCE transforms to monodomain, with all the mesogens aligned along the uniaxial stretch direction *l*_{3} [23].

The boundary between Phase L and Phase Sm is described by 1/*λ*_{1} = *a*^{−1/6} with *a*^{−1/3} ≤ *λ*_{3} ≤ *a*^{−1/12}, and the boundary between Phase Sm and Phase S is described by $1/\lambda 1=a1/2\lambda 32$ [25,26,28], where *a* = (1−*Q*)/(1 + 2*Q*). As a result, the mechanical phase diagram is temperature-dependent, through the thermal-induced change of the order parameter *Q*, as discussed next.

### 3.2 Nematic-Isotropic Phase Transition.

To illustrate the dominant effect of temperature on the order parameter *Q* and the corresponding thermal-induced nematic-isotropic phase transition, we first only consider the Maier–Saupe free energy. This represents a system of only liquid crystal mesogens without any polymer network. The energy landscape as a function of *Q* at various temperatures shows a first-order phase transition (Fig. 2(b)). For temperatures much below the nematic-isotropic transition temperature *T*_{ni} (e.g., $T^=0.6,0.8,and0.9$), the Maier–Saupe free energy has a single-well structure where the free energy is minimized at *Q* between 0 and 1, representing the nematic phase. When the temperature approaches *T*_{ni}, or $T^$ approaches 1, the free energy transforms to a double-well structure with a second minimum at *Q* = 0 in addition to the minimum at a finite *Q*, together with a small but positive energy barrier between the two minima (hardly seen in the plot but confirmed by the simulation), indicating a weak first-order nematic-isotropic phase transition. This is also shown in Fig. 2(c) with the equilibrium order parameter *Q* as a function of temperature. The first-order nematic-isotropic phase transition is indicated by the discontinuity of the curve at $T^=1$.

## 4 Thermomechanical Coupling

We next consider the general case of a system including both the quasi-convex elastic energy of the polymer network and the Maier–Saupe free energy of the mesogen mixture. As will be shown, both temperature and mechanical stretch will affect the mechanical phase boundaries and the corresponding stress-stretch responses.

### 4.1 Mechanical Phase Boundaries at Temperatures Below *T*_{ni}.

Figure 3 illustrates the evolution of mechanical phase boundaries at temperatures below *T*_{ni}, as well as the color map of the equilibrium *Q* at different stretches. As temperature increases (e.g., $T^=0.6,0.8,and0.99$, Figs. 3(a)–3(c)), the areas of Phase L and Phase Sm decrease while the area of Phase S increases, indicating a more pronounced common rubber-like behavior of the LCE. Furthermore, under the same stretch, the order parameter *Q* decreases with increasing temperature, consistent with the trend toward the nematic-isotropic transition shown in Figs. 2(b) and 2(c). The near-uniform color distribution of *Q* at different temperatures indicates its weak dependence on stretch at temperatures below *T*_{ni}. This confirms a long-time adopted assumption in the theory and simulation of LCEs, where the order parameter *Q* or the parameter *a* = (1 − *Q*)/(1 + 2*Q*) is almost always considered as a constant independent of deformation at a fixed temperature below *T*_{ni} [18,24,25,28].

To further show the effect of mechanical stretch on *Q* and phase boundaries, we also plot phase boundaries using the uncoupled equilibrium *Q* when considering only the Maier–Saupe free energy (Fig. 2(c)). The phase boundaries calculated using both free energies and using only the Maier–Saupe free energy are compared in Figs. 3(a)–3(c) right (solid and dashed lines, respectively). Indeed, a negligible difference is found, and the difference becomes larger with increasing temperature. By considering thermomechanical coupling (i.e., including both free energies), the phase boundary between Phases Sm and S shifts toward a higher *l*_{3}. Overall, at temperatures below *T*_{ni}, the thermal effect dominates the equilibrium order parameter *Q*.

### 4.2 Mechanical Phase Boundaries at Temperatures Above *T*_{ni}.

The effect of mechanical stretch on *Q* is much more pronounced at temperatures above *T*_{ni}. Without considering thermomechanical coupling by calculating *Q* using only the Maier–Saupe free energy, we have *Q* = 0 and *a* = 1. In this case, all the phase boundaries degenerate to a single point of *l*_{1} = *l*_{2} = *l*_{3} = 1, leaving only Phase S in the phase diagram. This represents a common rubber-like mechanical behavior. By contrast, with thermomechanical coupling by calculating *Q* using both free energies, an additional finite area of Phase Sm emerges even at temperatures above *T*_{ni} (Fig. 4). The area of Phase Sm decreases with increasing temperature. No Phase L is found at any temperature above *T*_{ni}.

This emergence of Phase Sm at temperature above *T*_{ni} is essentially due to the stretch effect on the equilibrium order parameter *Q* and the nematic-isotropic transition. As shown in Fig. 4, increasing the uniaxial stretch *l*_{3} and area stretch 1/*l*_{1} significantly increases the equilibrium *Q* from nearly 0 to about 0.55. This large stretch-dependence of *Q* at $T^>1$, which dramatically contrasts with the cases at $T^<1$, indicates a possible stretch-induced transition of mesogens from the isotropic phase back to the nematic phase. Effectively, the non-zero uniaxial stretch *l*_{3} or area stretch 1/*l*_{1} increases the nematic-isotropic transition temperature. This behavior is consistent with the previously reported stretch effect on the nematic-isotropic phase transition under the uniaxial mechanical load [7–9,34]. Finally, when the temperature is only slightly higher than *T*_{ni} (e.g., $T^=1.02$, Fig. 4(a)), the large stretch-dependence of *Q* leads to a “nose”-shaped phase boundary between Phases Sm and S.

To examine the evolution of phase boundaries more carefully at temperatures slightly higher than *T*_{ni}, we plot phase boundaries at $T^=1.01,1.02,1.03,and1.04$ in Fig. 5. As the temperature increases, the phase boundary between Phases Sm and S transforms from a deep “nose” shape to a smooth line, with the area of Phase Sm decreasing. We conclude that the emergence of this “nose”-shaped phase boundary results from the three-way coupling between the temperature dependence of *Q*, the stretch dependence of *Q*, and the phase boundary governed by *Q*. To the best of our knowledge, this phenomenon has never been reported before. Our preliminary experiment of biaxial test at a fixed temperature seems to have qualitatively validated this theoretical prediction, and we are currently conducting a systematic experimental study on this unusual phenomenon.

## 5 Stresses-Stretch Responses

The evolution of mechanical phase boundaries due to thermomechanical coupling has a profound effect on the stress-stretch behavior of the polydomain LCE. To show this, we investigate a specific loading path defined by the biaxial stretches *λ*_{2} and *λ*_{3}, as illustrated by the solid black line in Fig. 6(a). At a fixed temperature, a sheet of LCE is initially under equal-biaxial stretch up to *λ*_{2} = *λ*_{3} = 1.05. Afterward, we fix *λ*_{2} = 1.05 and only increase *λ*_{3}. We study three distinct scenarios with temperatures of $T^=0.9,$$1.02,and1.05$. The phase boundaries between Phases Sm and S are plotted as the dashed lines in Fig. 6(a). We calculate the planar true stresses $\sigma ^2$ and $\sigma ^3$ using Eq. (8) and plot them as functions of *l*_{3} in Fig. 6(b).

In the first scenario at $T^=0.9$, the loading path crosses all the three phases one by one through L, Sm, and S. Phase L exists in this case since the LCE is in the nematic phase $(T^<1)$. Both $\sigma ^2$ and $\sigma ^3$ are zero in Phase L, showing the soft elastic behavior [23]. Entering Phase Sm, $\sigma ^2$ and $\sigma ^3$ increase with *l*_{3} while being always identical to each other, as illustrated by Eq. (8) and discussed in Sec. 3.1. Note that this is a non-equal-biaxial stretch state with *λ*_{2} = 1.05 fixed and only *λ*_{3} increased. In other words, in Phase Sm, the polydomain LCE can have non-zero planar shear strain (since *λ*_{2} ≠ *λ*_{3}) but can never have any planar shear stress (since $\sigma ^2=\sigma ^3$ always holds). Finally, in Phase S, $\sigma ^2$ and $\sigma ^3$ deviate from each other and $\sigma ^3$ becomes larger, since the LCE sheet is stretched in the *λ*_{3} direction with *λ*_{2} fixed.

In the second scenario at $T^=1.02$, there is no Phase L. The loading path starts with Phase S, transverses through Phase Sm, and reenters Phase S again. Recall that in general, $\sigma ^2=\sigma ^3$ always holds in Phase Sm, and $\sigma ^3>\sigma ^2$ holds in Phase S due to the specific loading path prescribed here. As a result, after the initial equal-biaxial stretch, $\sigma ^3$ first increases to be larger than $\sigma ^2$ in Phase S, drops back to be identical to $\sigma ^2$ in Phase Sm, and increases to be larger than $\sigma ^2$ again due to the reentrant transformation from Phase Sm to Phase S in the material.

In the third scenario at $T^=1.05$, the loading path is almost entirely in Phase S, and the material behaves like a common rubber with $\sigma ^3>\sigma ^2$.

We further note that while the stress-stretch responses of polydomain LCEs in the first and third scenarios have been well-understood based on previous studies, the second scenario that involves the reentrant transformation due to the “nose”-shaped phase boundary has never been reported before. This again shows the pronounced effect of mechanical stretch on the equilibrium *Q* in thermomechanical coupling at temperatures slightly above *T*_{ni}. Further experimental investigation of this phenomenon is currently under study.

## 6 Conclusion

In this paper, we have constructed a theoretical model to investigate thermomechanical coupling in polydomain LCEs subjected to various planar stretches and temperatures. The thermal-induced change of order parameter and the corresponding nematic-isotropic transition in the LCE greatly affect the mechanical phase boundaries as well as the macroscopic stress-stretch behaviors. On the other hand, mechanical stretch has a negligible effect on the equilibrium order parameter and mechanical phase boundaries at temperatures below the nematic-isotropic transition temperature *T*_{ni}, but has a significant effect at temperatures above *T*_{ni}. When the temperature is slightly higher than *T*_{ni}, thermomechanical coupling leads to the existence of Phase Sm as well as a unique “nose”-shaped phase boundary between Phases Sm and S. The corresponding stress-stretch response shows a non-monotonic behavior at certain loading paths of biaxial stretches. The planar true stresses are always zero in Phase L regardless of the stretch, and are always identical in Phase Sm, only depending on the out-of-plane stretch. The material behaves like a common elastic rubber in Phase S. We hope these theoretical findings can motivate further experiment and model developments on thermomechanical coupling in LCEs, as well as provide guidelines for future designs of complex actuation in LCE-based robots and structures. Effects of non-ideal polymer chains (additional coupling between polymer chains and mesogen directions at the state when the LCE is fabricated) and viscoelasticity on the behaviors reported in this paper deserve further systematic investigation.

## Acknowledgment

This work was supported by the National Science Foundation through grant CMMI-2146409.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.