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 relaxation processes of a wide variety of soft materials frequently contain one or more broad regions of power-law like or stretched exponential relaxation in time and frequency. Fractional constitutive equations have been shown to be excellent models for capturing the linear viscoelastic behavior of such materials, and their relaxation modulus can be quantitatively described very generally in terms of a Mittag–Leffler function. However, these fractional constitutive models cannot describe the nonlinear behavior of such power-law materials. We use the example of Xanthan gum to show how predictions of nonlinear viscometric properties, such as shear-thinning in the viscosity and in the first normal stress coefficient, can be quantitatively described in terms a nonlinear fractional constitutive model. We adopt an integral K-BKZ framework and suitably modify it for power-law materials exhibiting Mittag–Leffler type relaxation dynamics at small strains. Only one additional parameter is needed to predict nonlinear rheology, which is introduced through an experimentally measured damping function. Empirical rules such as the Cox–Merz rule and Gleissle mirror relations are frequently used to estimate the nonlinear response of complex fluids from linear rheological data. We use the fractional model framework to assess the performance of such heuristic rules and quantify the systematic offsets, or shift factors, that can be observed between experimental data and the predicted nonlinear response. We also demonstrate how an appropriate choice of fractional constitutive model and damping function results in a nonlinear viscoelastic constitutive model that predicts a flow curve identical to the elastic Herschel-Bulkley model. This new constitutive equation satisfies the Rutgers-Delaware rule, which is appropriate for yielding materials. This K-BKZ framework can be used to generate canonical three-element mechanical models that provide nonlinear viscoelastic generalizations of other empirical inelastic models such as the Cross model. In addition to describing nonlinear viscometric responses, we are also able to provide accurate expressions for the linear viscoelastic behavior of complex materials that exhibit strongly shear-thinning Cross-type or Carreau-type flow curves. The findings in this work provide a coherent and quantitative way of translating between the linear and nonlinear rheology of multiscale materials, using a constitutive modeling approach that involves only a few material parameters.

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A fractional K-BKZ constitutive formulation for describing the nonlinear rheology of multiscale complex fluids

The effects of wood fibers on the melt rheological behavior of isotactic poly(propylene) (i-PP)/wood fiber (WF) composites have been studied at WF concentrations of 0–32.2 vol % at 493 K. Shear stress–shear rate variations obeyed a power law equation, and the composites exhibited shear thinning, which increased with filler content. At a low shear rate, the apparent melt viscosity increased, while melt elasticity, after an initial decrease, also increased with WF concentration. At a higher shear rate, after an initial decrease, the melt viscosity showed an increase, as did melt elasticity, with increase in filler content. A titanate coupling agent, LICA 38, used to modify the wood fiber surface, modified these rheological parameters by functioning as a plasticizer/lubricant. © 2003 Wiley Periodicals, Inc. J Appl Polym Sci 91: 644–650, 2004

Effect of wood fibers on the rheological properties of i-PP/wood fiber composites

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

This study concerns the shear rheometry of a series of nine silicone fluids, all linear or branched polydimethylsiloxanes (PDMS), with varying mass distributions.

Shear rheometry of polydimethylsiloxanes. Master curves and testing of Gleissle and Yamamoto relations

An attempt is made to incorporate into a quasilinear viscoelastic constitutive equation of the Boltzmann superposition type the two mirror relations of Gleissle, as well as his relation between the steady-state first normal-stress difference and the shear viscosity curve. It is shown that the three relations can hold separately within this constitutive model, but not simultaneously, because they require a different nonlinear strain measure, namelyS12 (γ) =γ − a (γ − 1) (a = 0 forγ ≦ 1,a = 1 forγ ≧ 1) for the mirroring of the viscosities,S12 (γ) =γ − a (γ−k2/γ) (a = 0 forγ ≦k, a = 1 forγ ≧k) for the mirroring of the first normal-stress coefficients, and\(S_{12} (\gamma ) = \frac{1}{2}\gamma \{ 1 - erf [ln (\gamma /2 \sqrt k )/\sqrt {2 ln k} ]\}\) for the third relation. Hereγ denotes the shear strain and erf the error function. Experimental data on melts of a low-density polyethylene, a high-density polyethylene and a polypropylene show that the mirror relations are passable approximations, but that the third relation meets reality surprisingly close if the right value ofk is used.

The mirror relations and nonlinear viscoelasticity of polymer melts

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.

P. J. R. Leblans

Papers

The mirror relations and nonlinear viscoelasticity of polymer melts

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