Modeling and simulation of liquid-crystal elastomers

We consider a continuum model describing the dynamic behavior of nematic liquid crystal elastomers (LCEs) and implement a numerical scheme to solve the governing equations. In the model, the Helmholtz free energy and Rayleigh dissipation are used, within a Lagrangian framework, to obtain the equations of motion. The free energy consists of both elastic and liquid crystalline contributions, each of which is a function of the material displacement and the orientational order parameter. The model gives dynamics for the material displacement, the scalar order parameter and the nematic director, the latter two of which correspond to the orientational order parameter tensor. Our simulations are carried out by solving the governing equations using an implicit-explicit scheme and the Chebyshev polynomial method. The simulations show that the model can successfully capture the shape changing dynamics of LCEs that have been observed in experiments, and also track the evolution of the order parameter tensor.

Figure 1

The shape evolution of the LCE sample due to an inhomogeneous distribution of temperature. Panel (a) shows the initial state of the LCE sample, panels (b) and (c) are two intermediate states, and panel (d) represents the equilibrium state. Panels (e) and (f) present the shapes of the LCE sample at the equilibrium state [panel (d)] from two different perspectives. In this experiment, the temperature drops linearly from the top of the LCE sample to its bottom and is distributed uniformly on each horizontal slice. The temperature spatial distribution is maintained during the evolution process. This numerical experiment simulates the physical one shown in Fig. 4 in [4].

Figure 2

The order parameter (S) distribution for the top [panel (a)], middle [panel (b)], and bottom [panel (c)] horizontal slices of the LCE sample at the equilibrium state. The order parameter is close to zero on the top slice while it is close to one on the bottom slice. It is the inhomogeneous distribution of the order parameter that leads to internal stress and thus results in the shape changes of the LCE sample. Moreover, the order parameter slightly varies on each of these slices, suggesting the elastic effect on the order parameter.

Figure 3

The nematic direction ($\mathbf{n}$) distribution for the top slice of the LCE sample at the initial and the equilibrium states. Panel (a) represents the nematic direction on the top slice at the initial state. Panels (b), (c), and (d) illustrate the nematic direction on the top slice for different perspectives at the equilibrium state.

Figure 4

The shape evolution of the LCE sample due to an inhomogeneous distribution in temperature. Panel (a) shows the initial state of the LCE sample, panels (b) and (c) are two intermediate states, and panel (d) represents the equilibrium state. Panels (e) and (f) present the shapes of the LCE sample at the equilibrium state [panel (d)] from two different perspectives. This experiment shares the same condition as the previous simulation in Fig. 1 except that a lateral surface of the LCE sample is fixed. This numerical experiment simulates the physical one shown in Fig. 2 in [4].

Figure 5

The order parameter (S) distribution for the top [panel (a)], middle [panel (b)], and bottom [panel (c)] horizontal slices of the LCE sample at the equilibrium state. The order parameter is close to zero on the top slice while it is close to one on the bottom slice. It is the nonhomogeneous distribution of the order parameter that leads to internal stress and thus results in the shape changes of the LCE sample. Moreover, the order parameter slightly varies on each of these slices and oscillates near the fixed lateral surface of the LCE sample, suggesting both the elastic effect and the effect due to the anchored surface on the order parameter.

Figure 6

The nematic direction ($\mathbf{n}$) distribution for the top slice of the LCE sample at the initial and the equilibrium states. Panel (a) represents the nematic direction on the top slice at the initial state. Panels (b), (c), and (d) illustrate the nematic direction on the top slice for different perspectives at the equilibrium state.