The non-isothermal elastic dumbbell: A model for the thermal conductivity of a polymer solution

In a flowing polymeric liquid, molecular orientation will give rise to anisotropic conduction of heat. In this paper, a theory is presented relating the thermal conductivity tensor to the deformation history of the fluid. The basis of this theory is formed by the Hookean dumbbell. It is shown that the anisotropy of the thermal conductivity is proportional to the polymer contribution to the extra-stress tensor. This stress-thermal law makes it relatively simple to incorporate anisotropic heat conduction into the numerical simulation of a flowing polymeric liquid.     

Anisotropic conduction of heat in a flowing polymeric material

In this paper a theory is presented which relates the thermal conductivity tensor of an amorphous polymeric material to the history of deformation of the material. The basis of the theory is formed by the network theory for polymeric materials. It will be shown that the results obtained here are in good agreement with experimental results on rubber. The effect of anisotropic heat conduction on the flow of a polymeric material will be demonstrated by the simple example of viscous heating in shear flow.Presented at the Golden Jubilee Conference of the British Society of Rheology and Third European Rheology Conference, Edinburgh, 3–7 September, 1990.     

The molecular theory of viscoelasticity for thermoplastic elastomer SBS (SIS) at large deformations

A microstructure model for SBS and SIS triblock copolymers with hard domains as multifunctional reinforcing fillers is proposed. Based on this model and proposed mechanism of large deformations, the probability distribution function of the end-to-end vector for each constituent chain and the free energy of deformation for the total networks was calculated by the combination of statistical thermodynamics and kinetics. A new molecular theory of non-linear visco-elasticity for SBS and SIS at large deformations is presented. It is successful in relating the viscoelastic state to molecular constitution by three important parameters (C ₁₀₀,C ₀₂₀, andC ₂₀₀) of the networks. The relations of stress to strain for four types of deformation, the elastic modulus and the constitutive equation for the stress relaxation were derived from this theory. It provides a theoretical foundation for studying the relationships of multiphase network structures and mechanical properties at large deformations. An excellent agreement between the theoretical relationships and experimental data from the experiments and the reference was obtained.Project supported by the National Natural Foundation of China     

Rubber-like elasticity by boundary element method: Finite deformation of a circular elastic slice

We briefly review the phenomenological theory of rubber-like elasticity and report a microstructural model that leads us to eventually adopt a particular constitutive equation, which includes the Neo-Hookean and the Mooney materials. A numerical implementation of the Boundary Element method for solving a general two-dimensional or axisymmetric finite deformation problem is described and tested with some simple deformations. The resulting program is used to analyse the finite deformation of a circular elastic slice perfectly bonded to two parallel rigid end plates; the bottom plate is stationary and the top plate is given a constant displacement. The problem has a free surface which must be found as part of the solution. The results indicate that the Boundary Element method can be an efficient tool for stress-strain analyses with rubber-like materials.     

Extended thermodynamics and the Jeffrey's constitutive equation

Extended irreversible thermodynamics provides an evolution equation for the viscous pressure tensor which reduces to the Jeffrey's constitutive equation in the long-wave limit. in contrast with Jeffrey's equation, the equation obtained in extended irreversible thermodynamics leads to finite speed of propagation for shear pulses. The nonlocal effects are included into the theory by allowing the entropy to depend on higher-order fluxes, instead of spatial gradients. The use of the former ones is clearly advantageous in the high-frequency domain.     

A theoretical model for the prediction of thermal expansion behaviour of particulate composites

The thermal expansion coefficient of particle-reinforced polymers was evaluated using a theoretical model which takes into account the adhesion efficiency between the inclusions and the matrix — an important factor affecting the thermomechanical properties of a composite. To measure the adhesion efficiency a boundary interphase, i.e. a layer between the matrix and the fillers having a structure and properties different from those of the constituent phases, was considered. This layer is assumed to have varying properties.To obtain information concerning the properties and extent of the interphase, an experimental study of the thermal behaviour of aluminium-epoxy composites was undertaken. Differential Scanning Calorimetry (DSC) measurements were performed to evaluate heat capacity with respect to temperature. In addition, the effects of different factors, such as heating rate and filler concentration on the glass transition temperature of the composite, were examined. The sudden changes in heat capacity values in the glass transition region were used to estimate the extent of the boundary interphase according to an existing theory.Finally, the values of the thermal expansion coefficient, predicted by this model, were compared with theoretical results obtained by other authors and with experimental results.     

Temperature effects in the Rayleigh problem for a convected Jeffreys fluid

We use a thermodynamic approach to non-linear viscoelasticity which predicts a coupling between the stress tensor and the heat flux, to generalize the Rayleigh problem for an Oldroyd B model of a fluid. For a special choice of the external temperature gradient, we show that thermal effects do alter the hydrodynamic velocity field as well as the first and second normal stress differences. These quantities depend on renormalized parameters wich explicitly depend on temperature and on the strength of the coupling. In particular we find that in contrast to the isothermal situation, the second normal stress difference does not vanish. The possibility of an experimental verification of our theoretical predictions is also mentioned.On sabbatical leave from Laboratorio de Energia Solar, IIM-UNAM, Zona Cultural Xochicalco, Temixco, Mor. 62580, MéxicoOn leave from Departamento de Polimeros, IIM-UNAM, México, D.F. 04510, México     

Nonlinear viscoelasticity of polymer melts in different types of flow

The nonlinear viscoelastic properties of a fairly large class of polymeric fluids can be described with the factorable single integral constitutive equation. For this class of fluids, a connection between the rheological behaviour in different flow geometries can be defined if the strain tensor (or the damping function) is expressed as a function of the invariants of a tensor which describes the macroscopic strain, such as the Finger tensor. A number of these expressions, proposed in the literature, are tested on the basis of the measuring data for a low-density polyethylene melt. In the factorable BKZ constitutive equation the strain-energy function must be expressed as a function of the invariants of the Finger tensor. The paper demonstrates that the strain-energy function can be calculated from the simple shear and simple elongation strain measures, if it is assumed to be of the shape proposed by Valanis and Landel. The measuring data for the LDPE melt indicate that the Valanis-Landel hypothesis concerning the shape of the strainenergy function is probably not valid for polymer melts.     

The motion of rodlike particles in the pressure-driven flow between two flat plates

The motion of freely suspended rodlike particles has been observed in the pressure-driven flow between the two flat plates of a Hele Shaw flow cell at low Reynolds numbers. Data are reported for rodlike particles with aspect ratios of 12.0 suspended in a Newtonian fluid for gap thickness to particle length ratios of 3, 6, and 20; and for rodlike particles with aspect ratios between 5 and 8 in a non-Newtonian fluid (79.25 wt.% water, 20.2 wt.% glycerine, and 0.55 wt.% polyacrylamide). For the Newtonian fluid, the time-dependent orientation of the particles near and far from walls was shown to be in quantitative agreement with Jeffery's theory for ellipsoids suspended in a simple shear flow if an effective aspect ratio is calculated from the experimental period of rotation. Particles aligned with the flow direction and less than a particle half-length from a wall interacted irreversibly with the wall. For the non-Newtonian fluid, the timedependent orientation far from a wall was shown to be in qualitative agreement with Leal's theory for a second-order fluid; however, particles that were aligned with the flow direction and were near walls did not rotate.     

Rheological properties of dilute polymer solutions: An extended thermodynamic approach

It is shown that extended irreversible thermodynamics can be used to account for the shear rate and frequency dependences of several material functions like shear viscosity, first and second normal stress coefficients, dynamic viscosity and storage modulus. Comparison with experimental data on steady shearing and small oscillatory shearing flows is performed. A good agreement between the model and experiment is reached in a wide scale of variation of the shear rate and the frequency of oscillations. The relation between the present model which includes quadratic terms in the pressure tensor and the Giesekus model is also examined.     

On the derivation of the constitutive equation of the simple fluid from that of a simple material

It is shown, by the use of the symmetry group of a fluid, that the stress in an elastic fluid depends on the current value of the deformation gradient through the absolute value of the determinant of the deformation gradient, whereas in a simple fluid it depends on this value of the determinant and the history of the relative deformation gradient and nothing else. This derivation is then used to show some of the errors in the criticisms made about the work of Noll. Additionally, a lacuna in Noll's derivation is removed to overcome the only valid criticism made about this theory of simple fluids.     

The behaviour of macromolecules in the flow through a sudden planar contraction

Analytic steady-state results for FENE-P model macromolecules, in the nearly coiled-up and nearly stretched state respectively, in general two-dimensional flow fields are derived. These results are utilized in the flow through a sudden planar contraction. Special emphasis is devoted to the structure tensor R R, which furnishes, among other things, the mean square extension and the average orientation of the macromolecules.     

Statistical mechanics of temporary polymer networks

The statistical theory of temporary polymer networks developed in parts I and II of this work is closed by a discussion on dynamical effects, including comparison with experiments. After a short review of the theory developed in parts I and II we discuss the situation in which a prescribed velocity gradient is imposed on the physical system formed by the temporary network and the solvent. The dynamical equation for the 2nd moments (derived from a generalized bead-spring model) contains the configuration-dependent transition probability, which depends on the second moments in a complex way. An approximate solution of these equations is obtained from a computer program. It shows the experimentally observed behavior, in particular the stress overshoot maximum after a sudden start of the flow.The derivation of the above equations contains simplifying assumptions which, however, leave the essential physics intact. The most important assumptions concern the relaxation time approach and the decay and formation processes. These were supposed to be dilute enough to break up the many-junction processes into one-junction processes. Both assumptions are analogous to frequently used assumptions in Boltzmann's kinetic theory.     

Linear viscoelastic behavior of narrow molecular weight distribution 1,4 polybutadiene: Comparison with Doi-Edwards theory of reptation

Linear viscoelastic behavior of narrow molecular weight distribution 1,4 polybutadiene samples with molecular weights between 42500 and 779000 has been correlated with molecular structure using a simple modification of the Doi-Edwards theory of reptation. The entire GPC curve is required for the calculations of viscoelastic behavior.The plateau modulus obtained from the experimental data is comparable to literature values, while the equilibrium compliance (which is indicative of polydispersity) is greater than values reported in the literature for nearly monodisperse polybutadienes. Reasonable agreement between theory and experiment is obtained over the entire molecular weight range. The agreement between theory and experiment using the GPC curve is better than that obtained by assuming the polymer to be monodisperse or by using the Doi fluctuation model. The model appears to break down for a more polydisperse sample . This study indicates that it may be possible to use the Doi-Edwards theory to explain the viscoelastic behavior of narrow MWD polybutadienes without introducing any new concepts into the theory (fluctuations, constraints release, etc.).     

Thermal expansivity of particulate composites: Interlayer versus molecular model

We continue the comparison of the results of an interlayer model, based on the theory of elastic continua, and a molecular model, derived from a theory of mixtures, previously presented in terms of bulk moduli K. We now derive expressions for the dependence of the thermal expansivity _c on the volume fraction _f of the filler, at low and elevated values of _f . Correspondencies between the characteristic parameters, viz. adhesion and repulsion ratios on the one hand, and interlayer content and thermal properties of matrix, filler, and layer, on the other, are examined. Since in the molecular theory both andK are derived from an equation of state, the identical set of parameters determines both functions and suggests correlations between them.     

The effect of molecular weight distribution on the elongational properties of linear polymers

The elongational properties of a series of six polypropylene and two polystyrene samples have been studied at constant rate of strain. A Wagner-type constitutive equation has been used to fit the experimental data, and the shape of the damping function has been correlated with the polydispersity index of the samples. As the memory function or relaxation function of linear viscoelasticity may be derived from the molecular-weight distribution using either molecular or phenomenological models, it is therefore possible to calculate the stress growth function of a linear polymer in elongation from its molecular-weight distribution.     

Finite shear strain behavior of a crosslinking polydimethylsiloxane near its gel point

A power law distribution of relaxation times, large normal stress differences, and physical rupture of molecular network strands dominate the shear behavior of polymers at the gel point (critical gels). This is shown in a series of well-defined experiments with increasing magnitude of shear on a model-network polymer system consisting of a linear, telechelic, vinyl-terminated poly-dimethylsiloxane (PDMS) and a four-functional siloxane crosslinker. Stable samples were prepared by stopping the crosslinking reaction at different extents of reaction in the vicinity of the gel point (GP). The Gel Equation has been shown to be valid up to strains of about 2 when using a finite strain tensor. Larger strains have been found to disrupt the network structure of the crosslinking polymer, and introduce a mechanical delay to the gel point. A sample that was crosslinked beyond the gel point (p>p _c ) can be reduced from the solid state to a critical gel, or even to a viscoelastic liquid, depending on the magnitude of shear strain. As a consequence, the relaxation exponent of a critical gel created under the influence of shear is less than that of a quiescently crosslinked critical gel.     

Elastic recovery and polymer-polymer interactions

i) Elastic recovery in polymeric liquids is a cooperative phenomenon in the sense that individual polymer molecules undergoing retraction must interact with one another in order to generate recovery. Stress generated by polymer molecules under an externally imposed flow field may or may not be a cooperative phenomenon. We suggest that the ability to describe the large elastic recovery exhibited by many polymeric liquids furnishes a crucial test of the validity of methods used to model the interaction of a given polymer molecule with its neighbors. Temporary-junction network models appear to be capable of explaining observed recoveries. Elastic recovery cannot be explained by single-molecule-in-a-mean-field theories which involve no calculation of the effect of the single molecule on the mean field. ii) A Gaussian network theory equation for the change of volume with elongation for a cross-linked elastomer is generalized in order to allow the bulk compliance to depend on elongation. iii) It is proved that two classes of flow history, namely shear-free and shear, are constitutively independent in the sense that, for a given viscoelastic liquid of unknown constitutive equation, the behavior in one class cannot be predicted from rheological measurements (however extensive) made solely in the other class.Dedicated to Prof. Dr. J. Meissner on the occasion of his 60th birthday.     

On the cone and plate deformation of a rubber-like material: Instability

In this report it is shown that the place boundary-value problem, for a small deformation superimposed on the large cone-and-plate deformation of a Mooney-Rivlin material, has no unique solution at some displacement angles of the cone. The kinematics of the small deformation are chosen such that the problem is reduced to a set of ordinary differential equations. With respect to this restricted class of kinematics the cone-and-plate deformation is unstable.