Non-entropic theory of rubber elasticity: flexible chains grafted on a rigid surface

The elastic response is studied of a single flexible chain grafted on a rigid plane and an ensemble of non-interacting tethered chains. It is demonstrated that the entropic theory of rubber elasticity leads to conclusions that disagree with experimental data. A modification of the conventional approach is proposed, where the end-to-end distribution function (treated as the governing parameter) is replaced by the average energy of a chain. It is revealed that this refinement ensures an adequate description of the mechanical behavior of flexible chains. Results of numerical simulation are compared with observations on uniaxial compression of a layer of grafted chains, and an acceptable agreement is shown between the model predictions and the experimental data. Based on the analysis of combined compression and shear, a novel micro-mechanism is proposed for the reduction of friction of polymer melts at rigid walls.Comment: 16 pages, 2 figure

Non-entropic theory of rubber elasticity: flexible chains with weak excluded-volume interactions

Strain energy density is calculated for a network of flexible chains with weak excluded-volume interactions (whose energy is small compared with thermal energy). Constitutive equations are developed for an incompressible network of chains with segment interactions at finite deformations. These relations are applied to the study of uniaxial and equi-biaxial tension (compression), where the stress--strain diagrams are analyzed numerically. It is demonstrated that intra-chain interactions (i) cause an increase in the Young's modulus of the network and (ii) induce the growth of stresses (compared to an appropriate network of Gaussian chains), which becomes substantial at relatively large elongation ratios. The effect of excluded-volume interactions on the elastic response strongly depends on the deformation mode, in particular, it is more pronounced at equi-biaxial tension than at uniaxial elongation.Comment: 21 pages, 3 figure

Stiffness of polymer chains

A formula is derived for stiffness of a polymer chain in terms of the distribution function of end-to-end vectors. This relationship is applied to calculate the stiffness of Gaussian chains (neutral and carrying electric charges at the ends), chains modeled as self-avoiding random walks, as well as semi-flexible (worm-like and Dirac) chains. The effects of persistence length and Bjerrum's length on the chain stiffness are analyzed numerically. An explicit expression is developed for the radial distribution function of a chain with the maximum stiffness.Comment: 21 pages, 6 figure

Scattering function for a self-avoiding polymer chain

An explicit expression is derived for the scattering function of a self-avoiding polymer chain in a $d$-dimensional space. The effect of strength of segment interactions on the shape of the scattering function and the radius of gyration of the chain is studied numerically. Good agreement is demonstrated between experimental data on dilute solutions of several polymers and results of numerical simulation.Comment: 16 pages, 7 figure

The end-to-end distribution function for a flexible chain with weak excluded-volume interactions

An explicit expression is derived for the distribution function of end-to-end vectors and for the mean square end-to-end distance of a flexible chain with excluded-volume interactions. The Hamiltonian for a flexible chain with weak intra-chain interactions is determined by two small parameters: the ratio $\epsilon$ of the energy of interaction between segments (within a sphere whose radius coincides with the cut-off length for the potential) to the thermal energy, and the ratio $\delta$ of the cut-off length to the radius of gyration for a Gaussian chain. Unlike conventional approaches grounded on the mean-field evaluation of the end-to-end distance, the Green function is found explicitly (in the first approximation with respect to $\epsilon$). It is demonstrated that (i) the distribution function depends on $\epsilon$ in a regular way, while its dependence on $\delta$ is singular, and (ii) the leading term in the expression for the mean square end-to-end distance linearly grows with $\epsilon$ and remains independent of $\delta$.Comment: 39 pages, 1 figur

Thermal degradation and viscoelasticity of polypropylene-clay nanocomposites

Results of torsional oscillation tests are reported that were performed at the temperature T=230C on melts of a hybrid nanocomposite consisting of isotactic polypropylene reinforced with 5 wt.% of montmorillonite clay. Prior to mechanical testing, specimens were annealed at temperatures ranging from 250 to 310C for various amounts of time (from 15 to 420 min). Thermal treatment induced degradation of the matrix and a pronounced decrease in its molecular weight. An integro-differential equation is derived for the evolution of molecular weight based on the fragmentation-aggregation concept. This relation involves two adjustable parameters that are found by fitting observations. With reference to the theory of transient networks, constitutive equations are developed for the viscoelastic response of nanocomposite melts. The stress-strain relations are characterized by three material constants (the shear modulus, the average energy for rearrangement of strands and the standard deviation of activation energies) that are determined by matching the dependencies of storage and loss moduli on frequency of oscillations. Good agreement is demonstrated between the experimental data and the results of numerical simulation. It is revealed that the average energy for separation of strands from temporary junctions is independent of molecular weight, whereas the elastic modulus and the standard deviation of activation energies linearly increase with mass-average molecular weight.Comment: 24 pages and 18 figure

Local Energy Velocity of Classical Fields

It is proposed to apply a recently developed concept of local wave velocities to the dynamical field characteristics, especially for the canonical field energy density. It is shown that local energy velocities can be derived from the lagrangian directly. The local velocities of zero- and first- order for energy propagation has been obtained for special cases of scalar and vector fields. Some important special cases of these results are discussed.Comment: 8 Page

A Local Concept of Wave Velocities

The classical characterization of \wp, as a typical concept for far field phenomena, has been successfully applied to many \w phenomena in past decades. The recent reports of superluminal tunnelling times and negative group velocities challenged this concept. A new local approach for the definition of \wvs avoiding these difficulties while including the classical definitions as particular cases is proposed here. This generalisation of the conventional non-local approach can be applied to arbitrary \w forms and propagation media. Some applications of the formalism are presented and basic properties of the concept are summarized.Comment: 18 pages 5 figure

How Long Is a Photon?

An interpretation of an electromagnetic quantum as a single pulse is suggested. In this context the Planck formula is shown to be equivalent to the Heisenberg time-energy uncertainty relation. This allows to treat the photon frequency as an inverse time of emission. Such an ansatz avoids the inherent problems of the conventional approach.Comment: 6 page

Two- and Three-dimensional Generalisation of Lower Order Local Wave Velocities

A general local approach for the definition of velocities and especially phase velocities for waves recently proposed for one-dimensional waves is generalized for 2 and 3 dimensional scalar wave. A geometrically consistent generalization has been found for the local wave velocities of order zero and one.Comment: 5 page