Network topology and the theory of rubber elasticity

The earliest investigations on rubber elasticity, commencing in the 19th century, were necessarily limited to phenomenological interpretations. The realisation that polymers consist of very long molecular chains. commencing c. 1930, gave impetus to the molecular theory of rubber elasticity (1932-). according to which the high deformability of an elastomer, and the elastic force generated by deformation, stem from the configurations accessible to long molecular chains. Theories of rubber elasticity put forward from 1934-1946 relied on the assumption that the junctions of the rubber network undergo displacements that are affine in macroscopic strain. The theory of James and Guth (1947) dispensed with this premise, and demonstrated instead that the mean positions of the junctions of a ‘phantom’ network consisting of Gaussian chains devoid of material properties are affine in the strain. The vital significance of the distinction between the actual distribution of chain vectors in a network and their distribution if the junctions would be fixed at their mean positions went unnoticed for nearly 30 years. Experimental investigations, commencing with the incisive work of Gee in 1946. revealed large departures from the relationship of stress to strain predicted by the theories cited. This discrepancy prompted extensive studies, theoretical and experimental, during succeeding years. Inquiry into the fundamentals of polymer networks, formed for example by interlinking very long polymer molecules, exposed the need to take account of network imperfections, typically consisting of chains attached at only one end to a network junction. Various means were advocated to make corrections for these imperfections. The cycle rank ζ of the network has been shown (1976) to be the fundamental measure of its connectivity, regardless of the junction functionality and pattern of imperfections. Often overlooked is the copious interpenetration of the chains comprising typical elastomeric networks. Theories that attempt to represent such networks on a lattice are incompatible with this universal feature. Moreover, the dense interpenetration of chains may limit the ability of junctions in real networks to accommodate the fluctuations envisaged in the theory of phantom networks. It was suggested in 1975 that departures from the form predicted for the elastic equation of state are due to constraints on the fluctuations of junctions whose effect diminishes with deformation and with dilation. Formulation of a self-consistent theory based on this suggestion required recognition of the non-affine connection between the chain vector distribution function and the macroscopic strain in a real network, which may partake of characteristics of a phantom network in some degree. Implementation of the idea was achieved through postulation of domains of constraint affecting the equilibrium distribution of fluctuations of network junctions from their mean positions. This led in due course to a theory that accounts for the relationship of stress to strain virtually throughout the ranges of strain accessible to measurement. The theory establishes connections between structure and elastic properties. This is achieved with utmost frugality in arbitrary parameters.     

Symmetry-breaking transitions in equilibrium shapes of coherent precipitates: Effect of elastic anisotropy and inhomogeneity

We examine the symmetry-breaking transitions in equilibrium shapes of coherent precipitates in two-dimensional (2-D) systems under a plane-strain condition with the principal misfit strain components ε* _xx and ε* _yy . For systems with cubic elastic moduli, we first show all the shape transitions associated with different values of t=ε* _yy /ε* _xx . We also characterize each of these transitions, by studying its dependence on elastic anisotropy and inhomogeneity. For systems with dilatational misfit (t=1) and those with pure shear misfit (t=−1), the transition is from an equiaxed shape to an elongated shape, resulting in a break in rotational symmetry. For systems with nondilatational misfit (−1<t<1; t ≠ 0), the transition involves a break in mirror symmetries normal to the x- and y-axes. The transition is continuous in all cases, except when 0<t<1. For systems which allow an invariant line (−1≤t<0), the critical size increases with an increase in the particle stiffness. However, for systems which do not allow an invariant line (0<t≤1), the critical size first decreases, reaches a minimum, and then starts increasing with increasing particle stiffness; moreover, the transition is also forbidden when the particle stiffness is greater than a critical value.     

Three ways to implement interfacial techniques: application tomeasurements of chromatic dispersion, birefringence, and nonlinearsusceptibilities

Depending on the spectal width of the source illuminating an interferometer, measurement procedures can utilize either the whole interferogram, or only the fringe envelope, or only the fringe quick oscillations. With an ultraband spectrum source, a simplified adaptation of the methods of Fourier transform spectroscopy yields the variations of the test-fiber propagation constant over the whole wavelength-interval of the source. Chromatic dispersion can then be computed from a single interferogram. With narrower spectrum sources, only the fringe envelopes are utilized and yield measurements of mode delay, with application to chromatic and polarization mode dispersion. In this case, however, interferograms at several wavelengths are necessary. With even narrower spectrum sources, the fringe quick oscillations provide measurements of phase shifts, related to changes in the mode propagation constant, when outside perturbations are applied to the test fiber. A direct method for measuring the third-order nonlinear susceptibilities is discussed. In this case the outside perturbation is an intense pump laser field     

Continuous phase quadrature phase shift keyed signaling techniquewith coding

A continuous phase quadrature phase shift keyed (CPQPSK) modulation technique is presented. This method utilizes a conventional QPSK modulator and a phase trajectory converter to approximate M=4, h=1/4 continuous phase signal and allows low cost, low complexity, and high rate (>1 Gbit/s) CPM modem implementation for bandwidth efficient transmission through nonlinear satellite channels. Using a communications analysis computer program it has been found that CPQPSK has 99 percent out-of-band power of 0.8R (MSK has 99 percent out-of-band power of 1.2 R where R is defined as bit rate), continuous phase trajectories, and nearly constant envelope amplitude. Simulation of realistic hardware designs indicate that the CPQPSK will require an Eb/No of 14 dB to achieve a bit error rate (BER) of 10^-6. Forward error correcting techniques using block codes with an overhead of 10 percent indicate that the Eb/No requirements can be reduced to 11.2 dB for 10^-6 BER