The theory presented starts from the model of gas of statistical segments with potential interactions between them. The fact that the system considered is an elastomer is incorporated into the model by constraints imposed on orientations of segments. Constraints are caused by the elastomer structure. The structure is determined by a connection of segments in linear chains, while chains are connected in the network structure. The statistical and thermodynamic description of the system is obtained by use of the conditional optimization of its free energy (Helmholtz function). The Lagrange multipliers for that optimization are calculated by use of cumulants and the multivariate method. In general, results of the theory presented are formulated for any type of potential interactions.

A system of linear polymer liquid crystal (PLC) macromolecules is considered in which each macromolecule constitutes an alternating copolymer of flexible and LC sequences. The distribution function of the system is factorized so that the Gibbs distribution is used for anisotropically interacting LC sequences while Dirac delta functions represent flexible polymer sequences modeled by linear chains of freely jointed statistical segments. A general formula for the Helmholtz function is derived for arbitrary types of anisotropic interactions between LC sequences; the formula of Maier and Saupe for monomer LCs is obtainable from it as a special case. The phase diagram of the system is obtained in the limit of the mean-field approach. Types and orders of phase transitions that the system can undergo are defined and discussed in terms of the Landau classification; all transitions are of the first order. Formation of cholesteric phases in addition to isotropic and nematic or smectic is predicted without involving additional assumptions such as the biaxiality of the LC interactions.

A network of polymer chains, which is undergoing an external deformation, is considered. Each of the network chains is linear and consists of segments interacting by nematic potential orienting interactions between segments within a chain and segments of other chains. Interactions are considered in calculations within the limit of the Maier and Saupe molecular mean‐field model. Chains of the network are connected at junctions with arbitrary functionality; this is the number of chains issuing from a junction. This parameter is included in calculations. The network free energy is calculated. Then, formulas for the work of the system elongation and the stress‐strain relation are obtained. This is done for the network, which is elongated uniaxially and in an arbitrary direction to the nematic axis. Parameters used in these formulas are as follows: the network junction functionality, the system thermodynamic temperature, the strength of interactions between segments and the average orientation of the segments within the network. Calculations are performed for the limit of long chains. In the isotropic phase, the elongation work is given by the neo‐Hookean constitutive equation and is an invariant of orientation of the elongation axis. In the nematic phase, that work depends strongly on the relative orientation of the system elongation and the nematic field direction. A formula for the stress of the elongated network is obtained. This formula is described by the following parameters: the network topological structure, the relative orientation of the elongation direction and the nematic field, and the intensity of the nematic molecular mean field.

The distribution function of orientations of a segment, which interacts with the orienting field and is within the chain with given the end-to-end distance-vector, is calculated. The number of segments per chain is finite. The Lagrange method of conditional minimization of the chain free energy (the Helmholtz function) functional is used. Constraints for the segment orientations stem from fixed the chain end-to-end distance-vector. Hence, Lagrange multipliers, energy, free energy, and entropy, for the chain with given the end-to-end distancevector, are calculated. Then, the distribution function of values of that vector is obtained. Furthermore, an average free energy per chain inside the polymer network with given a topological structure, the system self-deformation, and modulus of elasticity are calculated and discussed in Gaussian limit, that is, for the number of segments per chain tending to infinity.

Articles you may be interested inPhase and orientational ordering of low molecular weight rod molecules in a quenched liquid crystalline polymer matrix with mobile side chains A system of linear polymer liquid crystal ͑PLC͒ macromolecules is considered. Each macromolecule constitutes an alternating copolymer of flexible and LC sequences. The macromolecules can be either unconnected or else connected into a PLC network. The system is characterized with respect to local orientation. Competition between energetic effects of anisotropic orienting interactions between LC sequences and entropic effects determined mainly by flexible parts is considered. The Maier and Saupe mean-field approach is assumed for the representation of LC interactions. Types and orders of phase transitions that the system can undergo with respect to local order are discussed in terms of the Landau classification. All transitions are found to be of the first order. Thermodynamical and structural parameters of the system at phase transition points are represented by phase diagrams.