The Local and Collective Mobility in Polymer Chains and Networks with Included Rigid Particles Elastically Connected with the Network

Summary: The theory of molecular mobility of a polymer network with included rod-like particles is developed. The case is considered when the length of rods is comparable or greater than the average distance between neighboring cross-links of the primary network. The long-scale dynamics of the network is described by means of a regular cubic “coarse-grained” model. The junctions of this model describe the great network fragments (domains) the sizes of which are near to the average distance between neighboring rods.The quasi-elastic interactions between rods and network fragments lead to a broad relaxation spectrum for included rods as compared with free rods which are characterized by a single relaxation time of rotational diffusion. The frequency dependence of the dielectric loss factor of included rods is calculated for rods with permanent dipole moments directed parallel to the long axes of the rods chaotically distributed in the network. The frequency dependence of dynamic modulus of a polymer network with included rods is obtained. The increment in the dynamic modulus of the relatively short network motions (smaller than the distance between rods) also is taken into account. The broad relaxation spectrum of included rods leads to appearance of several maximums on the frequency dependences of both the dielectric loss factor and dynamical modulus.     

Effects of Nematic Ordering on the Relaxation Spectrum of a Polymer Network with Included rod‐like Particles: Mean‐Field Approximation

Summary: Equilibrium and local dynamic properties of ordered polymer networks with included rod‐like particles are considered using a simplified network model. Lagrange multipliers in the equations of motion of rigid rods are replaced by their averaged values. This approximation corresponds to modelling rod‐like particles by elastic Gaussian springs with mean‐square lengths independent of the orientational order. Nematic‐like interactions between network segments and rods are taken into account in terms of the Maier‐Saupe mean‐field approximation. Nematic ordering of rods induces network segments ordering and changes the relaxation spectrum of the network. The relaxation spectrum of the ordered network splits into two main branches for parallel and perpendicular components of chain segments with respect to the director. Relaxation times of a polymer network are calculated as functions of the wave number for the corresponding normal mode and of the order parameter taking into account both the dynamic factor (determined by friction effects) and the statistical factor (related to mean‐square fluctuations of segment projections). We compare the relaxation spectra of ordered unstretched polymer networks with fixed boundaries with those for polymer networks at free boundaries. A polymer network with free boundaries is stretched along the director. This produces additional fine structure of the two main branches in the relaxation spectrum.

Cell of a three‐chain network model with included rods.     

The relaxation spectra of polymer networks with different types of topology,ordering, heterogeneity

Many characteristic features of the relaxation spectra of the different types of polymer networks (meshlike and tree-like) manifesting in experimental behaviour are determined by manifold types of local and long-range irregularities or inclusions existing even in the simplest network structures. These irregularities in the local topology, in the fluctuations of the local orientational order existing due to stretching of the chains in the bulk elastomers (even in the non-ordered elastomers), also due to possible LC-ordering, the distribution of chain lengths between junctions and possible existence of cross-link agglomerations and domains at random cross-linking and the influence of the position of the chain element relative to junctions lead to variety of relaxation spectra, frequency and time-dependencies. The long-range hydrodynamic effects in bulk network can also lead to drastic variation of relaxation spectra. The inclusion of elongated rigid particles in polymer gels and network leads to the appearance of new branches of relaxation spectra changing and overlapping the relaxation spectra of the primary network system.     

Lattice model of heterogeneous polymer chains in the liquid-crystalline state, 1 statistical properties

A heterogeneous lattice model for main-chain liquid-crystalline polymers is proposed. Phase transition parameters and statistical properties of the chain in the ordered phase are treated in the framework of a mean-field approximation. The order parameters for the mesogen and the spacer bonds, the chain shape anisotropy and the fraction of stretched isomers are obtained. Dependence of the Curie point on the spacer length and flexibility is considered. The “odd-even” effect is shown to increase drastically with the spacer flexibility. A strong dependence of the conformation changes in the ordered state on the spacer properties is obtained.     

Orientational ordering of a polymer chain in a strong dipole field. Tetrahedral lattice model

Orientational and conformational properties have been investigated of a rotational isomeric model of a polar polymer chain on a tetrahedral lattice in a strong electric or mechanical field of dipole symmetry. Two types of dipole moment distribution along the chain are discussed: (A) constant signs of longitudinal components of the dipole moments, and (B) alternating signs of the longitudinal components of the dipole moments. The second case represents polymer chains such as $ \rlap{--} ({\rm CH}_{\rm 2} \hbox{---} {\rm CR}_{\rm 2} {\rm \rlap{--} )}_n $>/UEQN> when the dipole moments are oriented along the bisector of the CR₂-angle, i.e., normal to the extended trans-chain conformation. It is shown that only a discrete most probable orientation of the lattice relative to the field should be considered, namely that coinciding with one of the symmetry axes of the lattices. The average dipole order parameter and dichroic functions (quadrupole order parameter) are calculated in a strong external dipole field for unit vectors with different orientation relative to the chain backbone. The quadrupole order parameter for different unit vectors is obtained also as a function of chain elongation. The polarizability induced by an additional weak dipole field is calculated as a function of the magnitude of the strong external dipole field. For the model considered here the order parameters are more strongly influenced by the external field than those for the freely jointed chain (FJC) model having the same distribution of dipole moments along the chain. The orientational ordering of the chain in a dipole field is higher than in a quadrupole field of the same magnitude.     

Dielectric relaxation of polymer networks built from macromolecules with dipole moment directed along the end-to-end chain vector

The dielectric relaxation properties are considered for polymer networks built from polar macromolecules with the dipole moment directed along the end-to-end chain vector. The viscoeleastic cubic model of a regular network is used. The fixed average volume of a polymer network is ensured by the effective internal pressure. The dynamic models of polymer networks with external and interchain friction are studied. Two cases are considered: (1) polar chains cross-linked in a network at their ends, and (2) a densely cross-linked network with many network junctions per polar chain. The expressions for the autocorrelation functions of the total dipole moment of a network, which determine the dielectric susceptibility, are calculated. The relaxation spectrum of the autocorrelation function consists of two regions: the high-frequency relaxation spectrum of a chain fragment between two neighbouring junctions (intrachain relaxation spectrum) and the lowfrequency interchain relaxation spectrum. The interchain relaxation spectrum is determined by cooperative motions of chains which form a network. The characteristic time of this spectrum for networks of type (1) is the relaxation time of a chain between junctions τ_min. For networks of type (2) a second time scale τ₁ exists, which corresponds to motions inside the volume occupied by a single long polar chain included in a network. It leads to different time behaviour of the autocorrelation functions for both network models. The existence of only interchain friction in the network model leads to a cut-off of the relaxation spectrum at the time τ_max depending on the volume of viscous interchain interactions.     

Theory of relaxation spectra and dielectric relaxation of rigid rodlike particles incorporated in a polymer network

The theory of molecular mobility and relaxation spectra is developed for rodlike particles embedded in a polymer network with allowance for the involvement of the particles in collective network dynamics through topological entanglements with network fragments. A regular cubic coarse-grained network model is used, where the motion of junctions describes the mobility of large fragments (domains) of the initial network with a size equal to the distance between adjacent rodlike particles. The involvement of the rods in collective network dynamics is taken into account by introducing an effective quasi-elastic potential acting between the rods and junctions of the coarse-grained network and preventing long-distance diffusion of the embedded particles. The viscoelastic parameters of the coarse-grained (“renormalized”) network are functions of the viscoelastic characteristics of the initial network. The relaxation time spectra are calculated as well as the frequency dependences of the dielectric loss factor of the embedded particles that possess a permanent dipole moment directed along the major axis of each rod. Depending on the ratio between the viscoelastic characteristics of the rods and the network, the frequency dependence of the dielectric loss factor may have two maxima. The high-frequency maximum corresponds to local orientational movements of particles at fixed junctions of the coarse-grained network, which correspond to the position of the domain centers in the initial network. The low-frequency maximum corresponds to movements of particles involved in large-scale dynamics of network fragments. The dependence of the dielectric loss factor on the ratio between the viscoelastic parameters of the rods and the network is studied.     

Theory of Intrachain Relaxation Spectra for Polymer Networks Possessing a Short- or Long-Scale Ordering. Effects of the Nematic Ordering on the Relaxation Spectrum of a Polymer Network with Included Rods

We discuss the relaxation properties of polymer networks possessing either short-scale ordering caused by rigidity of network strands or long-scale liquid crystalline order. The main topics of the paper are the equilibrium and local dynamic properties of a polymer network ordered due to nematic-like interactions of the network segments with included rod-like particles. A simplified three chain network model is used. Lagrange multipliers in the equations of motion of hard rods are replaced by their averaged values. This approximation corresponds to modelling the rod-like particles by elastic Gaussian springs, their mean-square lengths independent of the ordering. Nematic-like interactions between network segments and rods are taken into account in terms of the Maier-Saupe mean-field approximation. Nematic ordering of rods induces ordering of the network segments. Relaxation spectrum of the ordered network splits into two main branches for the parallel and perpendicular components of the chain segments with respect to the director. We calculate the relaxation times of a polymer network as functions of the wave number. The relaxation spectrum of an isotropic network and that of the ordered network with included rods are compared.     

Rayleigh asymptotics of diffraction problems

The paper presents a general procedure for constructing the complete low-frequency asymptotics of diffractions problems applied to the scalar Helmholtz equation. The scattered fields in the near- and far-field zones for various types of boundary conditions are investigated.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 62, pp. 52–59, 1976.     

Linear superpositions of gap solitons in periodic Kerr media

The existence of a type of soliton in periodic Kerr media constructed as a superposition of noninteracting gap solitons of different kinds (bright, dark, and periodic) is first demonstrated. The periodic modulation of the nonlinearity is used to suppress the cross-phase modulation between component solitons to allow the superimposed beam to propagate for long distances.