The damping function has been a concept introduced in rheology since more than 30 years ago, and although a similar concept was already earlier implemented in studying rubber materials, its implementation in the modeling of polymer melts was an essential step forward in the classification and understanding of nonlinear viscoelasticity phenomena. It is the objective of this contribution to give an overview on the theoretical background and physical interpretation of the concept of the damping function for different types of deformation, as well as a review on the experimental results including the experimental artefacts to be considered. Besides homopolymers, a summary is given on different investigations of other types of systems, where the concept of the damping function has also been applied, for example, rubbers, rubber-like materials, block copolymers, polymer composites, liquid crystals, polymer blends, suspensions, emulsions, micellar systems, and in food rheology.

The damping function in rheology

The Doi-Edwards model with segmental stretch and a non-linear finitely extensible spring law is described and examined in simple flow situations where analytic results are derivable; namely oscillatory flow and steady state flow at high deformation rates. The model is shown to be consistent with the Bueche-Ferry hypothesis in fast large strain unidirectional flows but to violate this rule in small strain reversing flows. The discrepancy is identified with a preaveraging approximation used to describe the relative tube-chain velocity. Experimentally verifiable scaling rule for the birefringence as a universal function of a planar flow-type parameter and deformation rate are identified. Sensitivity to the extensional flow character, absent in the original tube model, manifests itself with the introduction of segmental stretch. Although the model generates a non-separable memory function kernel the deformation dependence of the memory function is quantitatively shown to have negligible impact on the predicted theological properties relative to the original Doi-Edwards model. With this simplification, relatively uncomplicated approximations to the segmental stretch model can be deduced.

The reptation model with segmental stretch. I: Basic equations and general properties

The state of the art in the rheology of polymer fluids (polymer solutions and melts) and filled composites is reviewed. This review includes two parts: analysis of the basic principles for the construction of rheological constitutive equations in terms of the continuum mechanics and finding correlations between the rheological characteristics and molecular structure of polymers on the basis of molecular models. Possible approaches to the formulation of constitutive equations are discussed. Special attention is focused on the correct selection of the form of the elastic potential for rubbery deformations induced under the flow of polymer fluids. The use of a power-law potential leads to the best results. To gain unequivocal results and minimize the number of free constants, viscoelastic characteristics of polymer fluids should be described in terms of a continuous relaxation spectrum as a power-law function limited by the maximum relaxation time. To solve the boundary problems by the selected constitutive equation, analysis of the dynamic stability is required, because the combination of viscosity and elasticity controls the limits of flow upon shear and tensile. Deformation can also lead to changes in the phase state of a polymer system. Furthermore, correct formulation of the boundary conditions is necessary because, in many cases, polymer fluids and, in particular, filled materials tend to efficient slip along walls. The existing molecular models adequately describe the characteristics of monodisperse polymers; however, on passing to polydisperse polymers, the additional use of semiempirical approaches is required. The modern level of experimental studies allows test measurements over a wide range of deformation rates, frequencies, and temperatures. However, in this field, the mainstream tendency in experimental studies is concerned with hybrid methods, which combine direct rheological measurements with optical observations of local structure and its evolution in the material. In this case, various physical principles of measurements are applied. In recent years, much interest has been focused on studying polymer compositions containing nanosized fillers, which are able to produce their structures in melt.

The state of the art in the rheology of polymers: Achievements and challenges

The uniaxial extensional flow at 150°C of 11 linear low density polyethylenes (LLDPE) and one low density polyethylene was measured in a Rheometrics Extensional Rheometer. The presence of silicone oil did not affect the results. However, large effects of the molding time were observed. For specimens molded for 14 min, strain hardening was not observed for any gas-phase polymerized LLDPE. As the molding time was increased to 40 min, the strain hardening was quite apparent, the elongational viscosity nearly doubled, the equilibrium plateau vanished, and the maximum strain at break Increased by about 20 percent. Explanation for the molding time effects can be found in the concept of low entanglement density in the virgin gas-phase resins. The entanglement increases with time at temperatures above the melting point. The specimens molded for longer time show strain hardening.

Linear low density polyethylenes and their blends: Part 3. Extensional flow of LLDPE's

In this paper we describe two numerical methods by which the relaxation modulus and memory function of a polymer can be determined from a set of discrete linear viscoelastic storage and loss moduli measurements. The first method involves expressing the linear viscoelastic data by means of a rational function (with certain constraints) and then taking the Laplace transform of that function. The second method is the Fourier transform. We apply the different methods to samples of commercial grade polyethylene and liquid crystal polymers and conclude that the Fourier transform route appears to perform best, although the Laplace transform method is adequate for highly polydisperse materials of modest molecular weight. The form of the relaxation modulus and memory functions obtained by these two routes is compared with those obtained from other rheological tests on the materials and the results discussed in terms of application to integral constitutive equations.

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The determination of polymer relaxation moduli and memory functions using integral transforms

It is shown that a single integral constitutive equation with an integrand factorable into a time-dependent and a strain-dependent function gives a satisfactory description of the rheological properties of a LDPE melt in uniaxial extensional flow. The time-dependent memory function obtained in simple shear has been inserted into the equation and the tensorial strain functions deduced from different types of tensile experiments coincide reasonably.

Rheological properties of some polyolefine melts in transient uniaxial elongational flow, described with a special type of constitutive equation

In melts of polydisperse linear polymers like high-density polyethylene (HDPE) the dynamics of a given macromolecule are affected by the renewal of the confining tube owing to movements of surrounding molecules, the more so according as the matrix contains more shorter molecules. At large strains, the relaxation of the tube due to the retraction of matrix molecules also contributes to stress relaxation. For long-chain branched molecules in a polydisperse melt like that of low-density polyethylene (LDPE), the dynamics are dominated by tube renewal and tube relaxation effects. It is argued that for melts of HDPE as well as LDPE the combination of these ideas leads to a rheological constitutive equation consisting of the sum of two integrals. Both integrands are factored into the product of a strain-independent time function and a time-independent nonlinear strain tensor. Using the relaxation spectrum from oscillation experiments and the nonlinear strain measure from first normal stress growth data, this equation satisfactorily describes the decay of the stresses after step strain or after cessation of steady shear flow for melts of two commercial HDPEs and of two LDPEs. For the HDPE samples one integral suffices, but for the LDPE melts the additional time and strain dependences of the integral relating to tube relaxation are indispensable for a good fit of the experimental curves.

Ideas about reptation, tube renewal and tube relaxation applied to the description of the non-linear viscoelastic behaviour in shear of some polydisperse polyethylene melts

H.C. Booij

Papers

Rheological properties of some polyolefine melts in transient uniaxial elongational flow, described with a special type of constitutive equation

Ideas about reptation, tube renewal and tube relaxation applied to the description of the non-linear viscoelastic behaviour in shear of some polydisperse polyethylene melts