Enthalpy recovery in semicrystalline polymers

Constitutive equations are derived for enthalpy recovery in polymeric glasses after thermal jumps. The model is based on the theory of cooperative relaxation in a version of the trapping concept. It is demonstrated that some critical temperature and some critical degree of crystallinity exist above which the energy landscape becomes homogeneous and structural relaxation ceases.Comment: 13 pages, 5 figures, LATE

Modelling structural relaxation in polymeric glasses using the aggregation-fragmentation concept

Governing equations are derived for the kinetics of physical aging in polymeric glasses. An amorphous polymer is treated as an ensemble of cooperatively rearranged regions (CRR). Any CRR is thought of as a string of elementary clusters (EC). Fragmentation of the string may occur at random time at any border between ECs. Two string can aggregate at random time to produce a new string. The processes of aggregation and fragmentation are treated as thermally activated, and the rate of fragmentation is assumed to grow with temperature more rapidly than that for coalescence. This implies that only elementary clusters are stable at the glass transition temperature, whereas below this temperature, CRRs containing several ECs remain stable as well. A nonlinear differential equation is developed for the distribution of CRRs with various numbers of ECs. Adjustable parameters of the model are found by fitting experimental data in calorimetric tests for polycarbonate, poly(methyl methacrylate), polystyrene and poly(vinyl acetate). For all materials, fair agreement is established between observations and results of numerical simulation. For PVAc, the relaxation spectrum found by matching data in a calorimetric test is successfully employed to predict experimental data in a shear relaxation test.Comment: 25 pages, 15 figure

Modeling the viscoelastoplastic response of amorphous glassy polymers

Constitutive equations are derived for the viscoelastoplastic response of amorphous glassy polymers at isothermal loading with small strains. A polymer is treated as an ensemble of cooperatively relaxing regions (CRR) which rearrange at random times as they are thermally agitated. Rearrangement of CRRs reflects the viscoelastic response of the bulk medium. At low stresses, CRRs are connected with each other, which implies that the macro-strain in a specimen coincides with micro-strains in individual relaxing regions. When the average stress exceeds some threshold level, links between CRRs break and relaxing domains begin to slide one with respect to another. Sliding of micro-domains is associated with the viscoplastic behavior of polymers. Kinetic equations are proposed for viscoplastic strains and for the evolution of the threshold stress. These equations are validated by comparison with experimental data in tensile relaxation tests and in tests with constant strain rates. Fair agreement is demonstrated between results of numerical simulation and observations for a polyurethane resin and poly(methyl methacrylate).Comment: 19 pages, 12 figure

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

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

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

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

A tube concept in rubber viscoelasticity

A constitutive model is derived for the time-dependent response of particle-reinforced elastomers at finite strains. An amorphous rubbery polymer is treated as a network of long chains linked by permanent junctions (chemical crosslinks, entanglements and filler particles). A strand between two neighboring junctions is thought of as a sequence of mers whose motion is restricted to some tube by surrounding macromolecules. Unlike the conventional approach that presumes the cross-section of the tube to be constant, we postulate that its radius strongly depends on the longitudinal coordinate. This implies that a strand may be modeled as a sequence of segments whose thermal motion is totally frozen by the environment (bottle-neck points of the tube) bridged by threads of mers which go through all possible configurations during the characteristic time of a test. Thermal fluctuations affect the tube's radius, which results in freezing and activation of regions with high molecular mobility (RHMs). The viscoelastic response of an elastomer is associated with thermally activated changes in the number of RHMs in strands. Stress-strain relations for a rubbery polymer at finite strains and kinetic equations for the concentrations of RHMs are developed by using the laws of thermodynamics. At small strains these relations are reduced to the conventional integral constitutive equation in linear viscoelasticity with a novel scaling law for relaxation times. The governing equation is determined by 5 adjustable parameters which are found by fitting experimental data in tensile dynamic tests on a carbon black filled natural rubber vulcanizate.Comment: 28 pages, 14 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