Linear Relaxation Modulus Research Articles

Transient flow of o/w sucrose palmitate emulsions

This paper deals with the characterisation and modelling of the nonlinear viscoelasticity properties of concentrated oil-in-water emulsions containing sunflower oil (60–80 wt%), water and a hydrophilic sucrose palmitate (1–5 wt%). With this aim, transient shear flow and nonlinear stress relaxation tests were carried out. Oscillatory shear measurements were also performed to calculate the linear relaxation modulus of the emulsions studied. The transient flow behaviour of all the emulsions studied is qualitatively similar, showing always a stress overshoot followed by a stress decay which tends to a steady-state value. However, at a critical shear rate, which depends on temperature and disperse phase concentration, a stress undershoot may be found. This behaviour has been attributed to optically observed shear-induced microstructural changes. A factorable nonlinear viscoelastic constitutive equation, the Wagner model with Soskey–Winter’s damping function, predicts the transient flow of these emulsions, in a range of shear rates, fairly well. However, a lack of concordance is found as shear rate increases, fact that has been explained on the basis of wall-slip phenomena and shear-induced microstructural changes.

Theoretical derivation of molecular weight scaling for rheological parameters

Recently, a method has been established to determine moments (functionals) of the molecular weight distribution (MWD) of a given polymer directly from measurements of the linear viscoelastic relaxation modulus of that polymer. In part, the need to compute such quantities (functionals) is motivated by the experimentally observed scaling of rheological properties of polymers with respect to moments of their MWD. Although various authors have advanced different ad hoc arguments to derive various molecular weight scaling results for a variety of rheological parameters, such as the zero-shear viscosity, no formal procedure for deriving molecular weight scaling for rheological parameters has been proposed. In this paper, a natural parametric generalization of the reptation based mixing rules is introduced which includes single and double reptation as special cases. For this generalization, it is shown, by invoking the mean value theorem for integrals, how to formalize the derivation of molecular weight scaling for rheological parameters. In particular, from the point of view of choosing practical mixing rules, this paper establishes that when the relaxation function is characterized by a single time constant, the molecular weight scaling is independent of the standard linear mixing rules.

The Linear Viscoelastic Relaxation Modulus Related to the MWD of Linear Homopolymer Blends

This work analyzes the relation between the shear relaxation modulus of entangled, linear and flexible homopolymer blends and molecular weight distribution (MWD). The theory employed is based on the double reptation mixing rule and a law for the relaxation time of chains in a polydisperse matrix. It is shown that chain reptation with contour length fluctuations and tube constraint release are the relevant mechanisms of chain relaxation, when the polydispersity is high. The effect of the last mechanism is to decrease chain relaxation times. Calculations are carried out with rheometric data of linear viscoelasticity for polystyrene, polybutadiene, polypropylene and high density polyethylene. The model predictions are compared with experimental data of gel permeation chromatography (GPC). In homopolymer blends with a wide molecular weight distribution, only chains of relatively low molecular weight relax by reptation with contour length fluctuations. A relaxation law for chains in a polydisperse polymer matrix is suggested and validated with GPC data. The theory verifies that the zero-shear rate viscosity of a polydisperse polymer, is proportional to the mass average molecular weight raised to a power greater than that found for the near monodisperse counterpart.

On the recovery of molecular weight functionals from the double reptation model

Recently, Mead (J. Rheology, 38 (1994) 1797–1827) has shown, for the double reptation model with a monodisperse relaxation function, characterized by a single time constant, how moments for the molecular weight distribution (MWD) can be determined directly from measurements of the linear viscoelastic relaxation modulus. The purpose of this paper is to show that this result is a special case of a more general result which allows linear inner-product functionals defined on the MWD, such as its moments, to be determined directly from the measurements. The applicability of this strategy is verified and validated using synthetic data.

A model with two coupled Maxwell modes

In an effort to quantitatively examine the effect of coupling between multiple relaxation modes, a new model involving two coupled Maxwell modes is developed as a generalization of the upper‐convected Maxwell and the Giesekus models. The model contains, in addition to the parameters inherent to a Maxwell model with two uncoupled modes (i.e., λ1,λ2 and η1≡G1λ1, η2≡G2λ2), a dimensionless coupling coefficient θ that multiplies a quadratic coupling term. In the two characteristic limits θ=0 or (η1,λ1)=(η2,λ2), the Maxwell model with two uncoupled relaxation modes or the Giesekus constitutive model is obtained, respectively. The rheological behavior of the model is investigated in the linear and nonlinear deformation‐rate regimes. Calculation of the linear viscoelastic behavior shows that the linear stress relaxation modulus is the sum of two decaying exponentials with characteristic times and preexponential factors that are quite different from λ1, λ2 and G1, G2, respectively. In slow, slowly varying flows, the zero shear‐rate ratio Ψ02/Ψ01 assumes small negative values when θ takes on small positive values. The nonlinear rheological behavior of the model is examined under the imposition of shear and extensional flow fields, from both a steady‐state and transient perspective. The qualitative behavior observed is remarkably rich in describing the experimental trends seen in polymer melts and Boger fluids for a constant value of θ≊0.1.

Nonlinear rheological behavior of a concentrated spherical silica suspension

Linear and nonlinear viscoelastic properties were examined for a 50 wt% suspension of spherical silica particles (with radius of 40 nm) in a viscous medium, 2.27/1 (wt/wt) ethylene glycol/glycerol mixture. The effective volume fraction of the particles evaluated from zero-shear viscosities of the suspension and medium was 0.53. At a quiescent state the particles had a liquid-like, isotropic spatial distribution in the medium. Dynamic moduli G* obtained for small oscillatory strain (in the linear viscoelastic regime) exhibited a relaxation process that reflected the equilibrium Brownian motion of those particles. In the stress relaxation experiments, the linear relaxation modulus G(t) was obtained for small step strain γ(≤0.2) while the nonlinear relaxation modulus G(t, γ) characterizing strong stress damping behavior was obtained for large γ(>0.2). G(t, γ) obeyed the time-strain separability at long time scales, and the damping function h(γ) (−G(t, γ)/G(t)) was determined. Steady flow measurements revealed shear-thinning of the steady state viscosity η(γ) for small shear rates γ( τ−1). Corresponding changes were observed also for the viscosity growth and decay functions on start up and cessation of flow, η + (t, γ) and η− (t, γ). In the shear-thinning regime, the γ and τ dependence of η+(t,γ) and η−(t,γ) as well as the γ dependence of η(γ) were well described by a BKZ-type constitutive equation using the G(t) and h(γ) data. On the other hand, this equation completely failed in describing the behavior in the shear-thickening regime. These applicabilities of the BKZ equation were utilized to discuss the shearthinning and shear-thickening mechanisms in relation to shear effects on the structure (spatial distribution) and motion of the suspended particles.

Effect of ionic interaction on linear and nonlinear viscoelastic properties of ethylene based ionomer melts

The effect of ionic interaction on linear and nonlinear viscoelastic properties was investigated using poly(ethylene-co-methacrylic acid) (E/MAA) and its ionomers which were partially neutralized by zinc or sodium. Dynamic shear viscosity and step-shear stress relaxation studies were performed. Stress relaxation moduli G(t, y) of the E/MAA and its sodium or zinc ionomers were factorized into linear relaxation moduli G°(t) and damping functions h(y). The relaxation modulus at the smallest strain in each ionomer agreed with the linear relaxation modulus calculated from storage modulus G′ and loss modulus G″. In the linear region, the ionic interaction shifted the relaxation time longer with keeping the same relaxation time distribution as E/MAA. In the nonlinear region, the ionic interaction had no influence on h(y) when the ion content was low. At higher ion content, however, the ion bonding enhanced the strain softening of h(y).

Problems originating from the use of the gordon-schowalter derivative in the johnson-segalman and related models in various shear flow situations

The predictions of the Johnson-Segalman model have been compared to experimental results in some simple flows, namely steady state shearing flow and stress growth in shear, for various molten polyethylenes having different molecular characteristics as far as their molecular weight level, polydispersity index and branching degree are concerned. Attention has been focused on this particular model because it has been shown to be an approximation of the Phan-Thien- Tanner equation in simple shear flows, the later model being frequently used in numerical calculations for complex flows. The Johnson-Segalman model is a codeformational rheological equation of state, which allows non-affine motion of the network junctions by introduction of a single parameter ( a) named the slip factor. This assumption provides a great improvement in the predictions of non-linear response to flows involving large deformations, either in shear or in elongation. However, the determination of the slip parameter from the various experiments has shown that a single value of the slip factor can not be used to describe both tangential stresses (τ 12) and normal stresses (τ 11—τ 22) in polyethylene melts. This has been related to a discrepancy of the model concerning the violation of the Lodge-Meissner rule as a consequence of the use of the Gordon-Schowalter derivative. Similar discrepancy can be expected in the case of the Phan-Thien-Tanner equation. Nevertheless, the slip factor, that can be determined either by fitting tangential stresses or normal stresses, is found to be nearly constant for a particular material, whatever the flow regime is (transient or steady). Moreover, for the various linear polymers (high density or linear low density polyethylenes), these parameters appear to be nearly independent of the molecular weight distribution. In this case, the differences in the non-linear behaviour in shear can only be attributed to differences in the linear relaxation modulus. On the other hand, a highly branched material (low density polyethylene) shows very different values of the slip parameters.

Determination of the relaxation spectrum from oscillatory shear data

In order to use either a linear or nonlinear model of viscoelasticity to calculate the stress response of a material to various deformations, it is usually necessary to have available an explicit equation for the linear relaxation modulus G(t). The most popular procedure is to use the data from a small‐amplitude oscillatory shear experiment to determine the parameters of a generalized Maxwell model. However, this is an ill‐posed problem and is not at all a straightforward curve‐fitting operation. We compare three procedures for determining a set of relaxation times and discrete moduli that can then be used as empirical fitting parameters in fluid mechanics computations. These are linear regression, with and without regularization, and nonlinear regression. Nonlinear regression is found to give a good fit of the data with a minimum number of parameters.

The evolution of viscoelasticity near the gel point of end‐linking poly(dimethylsiloxane)s

AbstractThe linear viscoelasticity of polymers near the gel point can be described by two scaling laws. The material at the gel point has a power‐law linear viscoelastic relaxation modulus, and the relaxation exponent has been found to vary with the composition of the precursor materials, i.e., it is not universal for gelation. A second scaling law describes the evolution of the linear viscoelastic properties through the gel point. The rate of change of the dynamic mechanical modulus/viscosity is observed to scale as a power‐law function of frequency. This power‐law function defines a dynamic critical exponent, and this has been found to be independent of precursor composition for end‐linking poly(dimethylsiloxane) polymers and equal to κ = 0.21 ± 0.02. This exponent may be a universal measure of gelation. The technique of Time Resolved Mechanical Spectroscopy is used to observe the evolution of linear viscoelastic properties of crosslinking polymers in situ in the rheometer. A stretched exponential relaxation modulus describes the evolution of mechanical properties in the vicinity of the gel point very well. The exponents which characterize the divergence of the zero‐shear viscosity and the equilibrium modulus are not universal, since they are related to the relaxation exponent and the dynamic critical exponent.

Nonlinear Shear Relaxation Modulus for a Linear Low‐Density Polyethylene

The response of a polydisperse linear low‐density polyethylene to imposed step shear strains up to strains of 10 were measured. The measured stress is approximately factorable into a product of a time‐dependent term, which is just the linear relaxation modulus, and a strain‐dependent term, the damping function. The damping function can be fit by a simple analytic function derived from a molecular theory with a single ad hoc parameter ξ′; a larger value of ξ′ corresponds to more softening of the nonlinear modulus. A tabulation of ξ′ values from fits to experimental shear damping functions given here and published elsewhere indicates that ξ′ is decreased both by polydispersity and by long chain branching.

The superposition of small deformations on large deformations: Measurements of the incremental relaxation modulus for a polyisobutylene solution

AbstractThe incremental relaxation modulus ΔG(t) for a concentrated solution of polyisobutylene has been determined from step‐shear experiments in which a small deformation Δγ was superimposed on a large deformation γ1; ΔG(t) was found to decrease with increasing γ1 and to increase with the time te after the imposition of the large deformation. It was also observed that the “apparent relaxation specturm” associated with δG(t) narrows and shifts to shorter times when compared to the spectrum associated with the linear viscoelastic relaxation modulus G(t). The results are well described by the nonlinear constitutive equation of the BKZ elastic fluid theory.

A Rheological Model of Nonlinear Viscoplastic Solids

A general rheological model of a nonlinear viscoplastic solid is developed, affording better quantitative prediction of behavior of viscoplastic materials. Rheological properties are quantified in terms of three types of curves obtained by tests, commonly used for description of mechanical material properties: constant rate, force deformation, creep, and relaxation curves. The mathematical model is based on a characteristic nonlinear differential equation describing the mechanical properties of a material. This equation defines the force F acting on a test specimen as composed of a cubic elasticity force K0x+rx3, viscous damping force Cẋ, and internal friction force Ff (sgn ẋ). The elastic parameter K0 quantifies linear elasticity while the strain hardening (or softening when negative) parameter r affords prediction of nonlinear behavior of the material. The viscous and Coulomb damping forces properly account for both rate‐dependent and rate‐independent energy dissipative properties of the material. Local linearization of the characteristic equation permits a closed‐form solution for creep at constant loading force and relaxation at constant deformation. The parameters of the nonlinear model can be read off or computed from practical curves, whereby a realistic link may be established between the mathematical model and real viscoplastic materials. The properties of a nonlinear viscoplastic solid are further clarified in terms of a time‐ and deformation‐dependent relaxation modulus. By setting the nonlinear parameters to zero the deformation‐independent linear relaxation modulus of a Boltzmann solid is obtained. It is believed that this model will afford better quantification of rheological properties of materials and improved interpretation of experimental data.

Modeling of Stresses in Multistep‐Shear Deformation of Polymeric Fluids

The time‐dependent shear and normal stresses arising in polymeric fluids during multistep‐shear deformations have been modeled using the Leonov constitutive theory. An analytical solution of the problem has been obtained. The model parameters for the materials have been determined from the experimental time‐dependent linear relaxation modulus in a single‐step shear. The results based upon the present theory have been compared with the experimental data obtained by Osaki and co‐workers for different polystyrene solution in double‐step‐shear deformation in the nonlinear region and also with theoretical results based upon the Doi‐Edwards and the BKZ theories. The present theory shows good agreement with the experiments, including the strain histories for which the above‐mentioned theories show discrepancies.