Introduction

Nematic elastomers are novel rubber-like materials with much recent research into their behavior (Anderson et al. 1999; Davis 1993; DeSimone and Dolzmann 2000; Finkelmann, et al. 2001; Warner and Terentjev 1996; Zentel 1989). They are formed by the cross-linking of a polymeric fluid which includes liquid crystalline molecules either as elements of the polymer chains or as pendant side-groups (Anderson et al. 1999). The term ``nematic'' comes from the Greek root ``n$\bar{e}$ma'', and so it describes the thread-like nature of the liquid crystal component particles of the network. Nematic elastomers were first predicted by de Gennes in the mid 1970s (de Gennes 1975) who recognized the connection between the coupling of orientational order (provided by the nematic mesogens) to the rubber elasticity of the underlying polymer network chains and predicted that this would lead to unique material properties (DeSimone and Dolzmann 2000, Bladon et al. 1994). Six years later, Finkelmann and co-workers first reported the successful synthesis of such a material based on the molecule polysiloxane (Finkelmann et al. 1981).

In fact, de Gennes's prediction has been borne out. As reported by Anderson et al., experiments with nematic elastomers have revealed a rich variety of interesting phenomena. These include stress-induced optical switching, stress-induced changes in phase transition temperatures, piezoelectricity, memory effects, and spontaneous macroscopic shape changes during phase transitions (Anderson et al. 1999). In addition, there has been recent work on lasing (Finkelmann, et al. 2001) and artificial muscles (de Gennes et al. 1997). See the following links:

• Artificial muscles [Hebert, M.; Kant, R.; de Gennes, P. G.; J. Phys. I France 7, 1997, 909-919.]
• Mechanical actuators
• Zenithal bistable displays, which do not require power to continuously display an image (actually an LCD variant)

While no technologies based on these materials have arisen yet, there are many interesting ideas, especially the suggestion of Finkelmann's to use nematic elastomers for integrated optical circuits, wherein the changes in optical characteristics due to deformation would be used to create light conducting paths on the surfaces of elastomer films (Anderson et al. 1999; Finkelmann 1988).

In order to proceed with technological applications which take advantage of their unique properties, a thorough theoretical understanding of nematic elastomers is needed. It has been the case that for most materials, continuum level theories have neglected microstructural information without peril. This is because bulk properties are sufficient to adequately predict and describe material behavior. However, the orientational ordering of the nematic molecules in both liquid crystalline and nematic elastomeric materials and its effect on macroscopic quantities, necessitates taking the microstructure into account when developing theories. For nematic fluids, orientational effects have been incorporated by the so-called director theory (Ericksen 1961; Leslie 1968). This has also been done for theories of nematic elastomers where orientational forces are required to satisfy an orientational momentum balance and expend power over the time-rate of the orientation field (Anderson et al. 1999; DeSimone and Dolzmann 2000).

As noted by Fried and Todres, an understanding of defects in traditional liquid crystals is important, and their study has enhanced the study of defects in other media and led to technological advances as well (Fried and Todres 2001). In fact, the liquid crystal zenithal bistable display harnesses defects and doesn't require power to hold an image (Bryan-Brown et al. 1997). (See http://www.zbddisplays.com/.) Similarly, for nematic elastomers, it is expected that investigation into defects will be of fundamental importance and have far-reaching technological consequences. Also, under stress, monodomain nematic elastomers exhibit transitions to striped domains possessing a characteristic size of $\approx15\mu m$ (Verwey et al. 1996, pictures ).

The study of defects and investigation into why stripes form are areas of active and ongoing research (Fried and Korchagin 2001; Verwey et al. 1996). In addition to the nematic director, for defects in conventional liquid crystals, the idea of an order parameter, a scalar field which enters the free-energy density in a way that mollifies the singularity otherwise associated with a defect, was introduced (Ericksen 1991). This concept has been carried over to defects in nematic elastomers. Here a coarse-graining approach is taken in which the agglomeration of polymer chains, the tensor $\boldsymbol{A}$, whose eigenvalues, $q$, describe the asphericity and eigenvectors, $\boldsymbol{n}$, indicate the unit orientation; both $q$ and $\boldsymbol{n}$ have the status of additional kinematical degrees of freedom supplemental to the deformation $\boldsymbol{y}$. For $-1<q<0$, the chains are oblate ellipsoids of revolution about $\boldsymbol{n}$, spheroids for $q=0$, and prolate ellipsoids of revolution about $\boldsymbol{n}$ for $q>0$. This leads naturally to the introduction of aspherical forces which expend power over the time-rate of the asphericity and comply with an aspherical force balance (Fried and Todres 2001).

Much simulation work into polymer networks has occurred (See, for example: Everaers and Kremer 1996; Grest and Kremer 1990 (2); Grest et al. 1993; Grest et al. 2000). In addition, there has been recent molecular simulation work done in the area of defects in conventional nematic liquids (Andrienko and Allen 2000; Denniston 1996; Hudson and Larson 1993). However, with regards to nematic elastomers, which combine aspects of both polymer networks and liquid crystals, no known molecular-scale simulation work has been undertaken. It is this situation and the expected importance of these materials which has led us to attempt work of this kind.