Statistical physics of isotropic-genesis nematic elastomers

Isotropic-genesis nematic elastomers (IGNEs) are materials that exhibit fascinating properties such as a remarkable softness in elasticity and do not undergo long-range nematic ordering, even at low temperatures. These materials are formed via the following process. Consider a melt or solution of nematogenic polymers, by which we mean long, flexible polymers carrying rod-like units. These units, which are also known as nematogens, and which give the system the possibility of exhibiting liquid crystallinity, may be integrated along the polymer chain backbones (the main-chain case) or in groups that dangle from the backbone (the side-chain or pendant case). An IGNE is formed when the polymers are permanently and randomly cross-linked to one another in the isotropic state of the nematogens; it is capable of "memorizing" both the positions of the chain segments and the orientations of the nematogen units at the instant of cross-linking. In this thesis, we derive a Landau-type theory of an IGNE, starting from from a microscopic model which consists of dimers that are randomly, permanently, and instantaneously cross-linked via springs. The Landau-type theory involves (a) a nonlocal, network-mediated, nematic-nematic interaction term, and (b) a random-field term that depends on the local, quenched environmental anisotropy, as well as the memorization of the random pattern of nematic fluctuations present at the instant of cross-linking. On the basis of this Landau-type theory, we address the following physical issues associated with the study of IGNEs: (i) The origin, nature, and stability of a certain equilibrium state (known as the polydomain state) of an IGNE. This state is characterized by the presence of a short-ranged, anti-correlated pattern of nematic alignment. By taking the thermal fluctuations of the elastomer medium into account, in addition to those of the nematogen orientations, we are able to predict, for sufficiently strongly cross-linked IGNEs, a novel type of nematic correlation behavior that is both oscillatory and decaying in space. Such oscillatory-decaying behavior is qualitatively consistent with the anti-correlation pattern observed in recent experiments by the Urayama group. By means of a Gaussian variational analysis, we also find that the macroscopically isotropic state of IGNEs (of which the polydomain state is a particular instance) remains stable, at least locally, in the low-temperature regime. Chapters 2, 3, 4, 7 and 8 provide analyses pertaining to issue (i). (ii)~Next comes the issue of capturing aspects of the memorization capability of an IGNE theoretically. We show that the IGNE's memory of the random pattern of thermal nematic fluctuations present at the instant of cross-linking directly influences the pattern of nematic alignment that is subsequently \lq\lq frozen in\rq\rq\ to the IGNE via the cross-linking process. Moreover, we learn that the fidelity of this memorization of the initial fluctuation pattern depends on the strength (i.e., pervasiveness) of cross-linking as well as the temperature at which the cross-linking was performed. Chapter 2 contains a study of this memorization capability of an IGNE. (iii)~The third issue addressed in this thesis pertains to the much softer elastic response of IGNEs, compared to ordinary elastomers, observed experimentally. We propose a physical mechanism for elastic softening, in which the softening is driven by fluctuation correlations of the nematic alignment. Chapters 5 and 6 contain a detailed explanation of this mechanism