Understanding thixotropic and antithixotropic behavior of viscoelastic micellar solutions and liquid crystalline dispersions. I. The model

Abstract

A simple model consisting of the Upper Convected Maxwell constitutive equation and a kinetic equation for destruction and construction of structure, first proposed by Fredrickson in 1970, is used here to reproduce the complex rheological behavior of viscoelastic systems that also exhibit thixotropy and rheopexy under shear flow. The model requires five parameters that have physical significance and that can be estimated from rheological measurements. Several steady and unsteady flow situations were analyzed with the model. The model predicts creep behavior, stress relaxation and the presence of thixotropic loops when the sample is subjected to transient stress cycles. Such behavior has been observed with surfactant-based solutions and dispersions. The role of the characteristic time for structure built up, λ, in the extent and shape of the thixotropic loops is demonstrated.

Introduction

Concentrated suspensions, solutions of polyelectrolytes, biological and other complex fluids are known to exhibit thixotropy, antithixotropy (rheopexy) and other complex rheological behavior 1, 2, 3, 4, 5. Recent experimental reports 6, 7, 8, 9, 10 have shown that elongated micellar solutions, dispersions of liquid crystals and lamellar liquid crystalline phases exhibit time-dependent rheological behavior and viscoelasticity. However, very few studies have been devoted to the analysis of thixotropic phenomena in liquid crystals and micellar solutions 11, 12.

In the analysis of thixotropic phenomena, two approaches can be distinguished [5]: a continuum mechanics one, which is phenomenological in nature, and another based in the understanding of the basic processes leading to structural changes as the sample is deformed. In the latter approach, it is usually considered that the instantaneous rheological properties depend on a structural parameter (i.e. molecular entanglements, network junctions, liquid crystalline microdomains, etc.) that is changing with deformation history. Thus, the rheological functions, such as viscosity, depend on the actual level of the fluid structure. When the structure breaks down due to flow, viscosity consequently decreases. Mewis [13], in his review article, points out that the non-linear, time-dependent behavior of the rheological functions is caused by changes in the internal structure of the material, which can be described by a set of two equations. One is a constitutive equation that gives the instantaneous stress τ as a function of the instantaneous kinematics D(r, t) for every possible state of the structure at any position, r. The other is a kinetic equation that describes the rate of change of the degree of structure s(r, t) with the instantaneous kinematics, i.e. the imposed shear rate, $\text{γ}\text{̇}$.

Due to the kinetics of structure breakdown and reformation usually being system-dependent, a unique kinetic expression for these processes is not possible. Thus, several models have been proposed which assign different forms to the basic constitutive equation and to the kinetic equation for the structural parameter. Usually, the structural parameter, which has to be obtained from experimental data, is related to a measurable rheological property. The apparent shear viscosity (η) has been used as a measure of structure 13, 14. This is equivalent to assume that the viscosity is proportional to the instantaneous number of structural points, N(t) (i.e. bonds, links or entanglements). In this approach, it is usually considered that the rate of change of the number of structural points depends on their instantaneous number and on the work done on the system 15, 16, 17, 18, 19. Some time ago, Fredrickson [20] proposed a simple kinetic equation for the destruction and construction of structure coupled to a Newtonian constitutive equation with a time dependent viscosity to predict the thixotropic behavior of inelastic suspensions under shear flow. This model can predict Non-Newtonian behavior and apparent yield stresses (Bingham plastic-like behavior) in steady shear-flow and thixotropic loops under time dependent shear histories. The kinetic equation of Fredrickson has the following form [20]:

$$\text{d}\text{ϕ}\text{d}\text{t}\text{=}\text{ϕ}{}_0\text{−ϕ}\text{λ}\text{+k}\text{ϕ}{}_{\mathrm{\infty }}\text{−ϕ}\text{τ}\text{:}\text{D.}$$

Here, ϕ is the fluidity (≡η⁻¹), ϕ₀ and ϕ_∞ are the fluidities at zero and very high shear rates, respectively, λ is the relaxation time upon the cessation of steady flow and k is a parameter that is related to a critical stress value, below which the material exhibits primary creep. Later we will show that λ is a structural relaxation time, i.e. a structural built up time, whereas k can be interpreted as a kinetic constant for structure breaking down.

In the present work, the kinetic equation of the Fredrickson model (Eq. (1)) is used coupled to the upper-convected Maxwell constitutive equation to account for the Non-Newtonian and the thixotropic and antithixotropic behavior reported for viscoelastic micellar solutions and lamellar liquid crystalline dispersions under shear flow 6, 7, 8, 9, 10. The Maxwell equation can be written as [21]

$$\text{τ+δ}\text{r,}\text{t}\text{τ}\text{∇}\text{=}\text{2D}\text{ϕ}\text{r,}\text{t}\text{,}$$

where $\text{τ}\text{∇}$ is the codeformational derivative of the stress tensor, δ[=(G₀ϕ)⁻¹], is a structure-dependent relaxation time and G₀ is the instantaneous relaxation modulus.