Rheological modeling of concentrated colloidal suspensions

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Abstract

The use of concentrated colloidal suspensions is common in several industries such as paints, foodstuffs and pulp and paper. These suspensions are generally composed of strongly interactive particles. If the attractive forces dominate the repulsion and Brownian forces, the particles aggregate to form a three-dimensional network yielding a gel structure. Under flow, the micro-structure of suspensions can be drastically modified and the rheological properties are then governed by structure breakdown and build-up. In this work, we propose a structural network model based on a modified upper convected Jeffreys model with a single relaxation time and a kinetic equation to describe the flow-induced micro-structure evolution. Three distinct kinetic equations are tested for this purpose. The proposed model describes yield and thixotropic phenomena, nonlinear viscoelastic behavior and output signal distortions observed for relatively small strain amplitude during oscillatory measurements, and overshoots observed in stress growth experiments. A comparison of model predictions and experimental data for fumed silica and coating colors is also presented. However, different model parameters must be used to correctly predict the different flow properties indicating that a more versatile or generalized kinetic equation must be proposed.

Introduction

Concentrated suspensions of colloidal particles exhibit a very wide range of rheological behavior depending on the nature and magnitude of the particle interactions [1]. These interactions depend on factors such as the physico-chemical properties of particles, the suspending medium nature 2, 3, the concentration of added stabilizing agent 4, 5 and the temperature [6]. When the attractive forces dominate, the suspended particles aggregate and the material becomes highly non-Newtonian. If the forces are sufficiently important, these aggregates grow to form a network. These suspensions exhibit two important properties which pose a lot of practical and theoretical problems related to yield stress and thixotropy. The rheological behavior is governed by the evolution of the induced micro-structure and, therefore, by the competition between breakdown under shearing and build-up at rest. The material characteristic time depends on the magnitude of the interactions in the suspension. The breakdown is related to shear flow and the bonding forces between particles whereas the build-up is essentially due to the Brownian motion and the collision probability between two particles or between one particle and a cluster. Particular experimental studies of interest are those of Ackerson and Clark [7] and Laun et al. [8], who used light scattering and neutron scattering techniques respectively.

The yield stress is generally characterized by the forces needed to break down the microstructure. It is often described by simple models using a yield criterion introduced by Bingham [9]. Extending this concept, Oldroyd [10] considered that the fluid behaved like a linear Hookean elastic material before yielding and like a fluid after yieldingσ=Gγ,|σ|<σ0,σ=σ0|γ̇|γ̇,|σ|>σ0,where G is the elastic modulus, σ0 the yield stress and η the viscosity after yielding. Yoshimura and Prud'homme [11] used this equation with a Newtonian viscosity to model the response of oil-in-water emulsions to oscillatory deformations in the nonlinear regime. The Oldroyd equation was also used by Doraiswamy et al. [12] with a power-law viscosity to describe the rheological properties of a suspension of silicon particles in polyethylene. They correlated the steady shear and the complex viscosities. The Oldroyd equation allows a good qualitative prediction of the rheological behavior of systems with yield stress in dynamic measurements [2], but as it does not include time-dependent effects, it cannot describe thixotropic fluids.

Thixotropy was extensively studied for various industrial materials (suspensions, liquid crystals, elastomers etc.). Mewis [13], and more recently Barnes [14], have presented extensive reviews of the subject. Thixotropy is characterized by a decrease in micro-structure with time under flow, followed by a recovery when the shear stress or shear rate is set equal to zero. Thixotropic behavior is generally described using a kinetic equation, analogous to the kinetics of reversible chemical reactions. A structural parameter, ξ, is used where ξ = 1 for completely built-up structure and ξ = 0 for completely broken-down structure. The structural parameter is proportional to the total number of bonds and the evolution is described by a kinetic equation of the following form given by Barnes [14]:∂ξ∂t=a(1−ξ)b+cξγ̇d,where a, b, c and d are characteristic parameters of the material. The rate of change of micro-structure is the sum of one build-up and one breakup terms. The creation process is assumed to be due to only the Brownian motion. This build-up term depends on the number of available bonds assumed to be proportional to (1  ξ), with ξ = 1 corresponding to the bonding number at equilibrium. The structure breakup is induced by the shear rate and depends on the number of bounds. The parameter c can be positive or negative: c > 0 allows for the description of shear-induced structure as proposed by Cheng and Evans [15] and more recently generalized by De Kee and Chang Man Fong [16]. The case of c < 0 corresponds to structure breakup.

Eq. (2)has been used to describe thixotropic phenomena in purely viscous fluids, for blood by Quemada [17] and for various food systems by De Kee et al. [18]. Few authors have used such a kinetic equation to account for viscoelastic effects. Leonov [19] used a similar concept to describe the rheological behavior of highly filled polymers containing small interacting particles. Coussot et al. [20] applied a modified Leonov model to describe the rheological properties of concentrated suspensions in a Newtonian solvent. They assumed that the total stress contains two main contributions, a viscoelastic contribution, σe, from interactions between particles and a viscous one from the suspending medium:σ=σemγ̇,where ηm is the suspension viscosity in absence of interactions.

The viscoelastic contribution is described by a Maxwell-type equation:1G∂σe∂t+σeη(ξ)=γ̇,where G is the elastic modulus of the structure and η(ξ) the viscous term which depends on the structural factor ξ. This factor (proportional to the total number of bonds) can be determined from the following kinetic equation:χ(ξ,γ̇)∂ξ∂t+1−ξθγ̇γc,where γc is the yield strain, θ a characteristic time and χ a kinetic function. The viscosity and the structural factor are related by the following empirical relations:η(ξ)=ηpf(ξ),withf(ξ)=ξ1−n−1nwhere n is an empirical parameter. For steady state conditions, with n = 0, , , reduce to the Bingham model. This model provides only a qualitative description of the dynamic behavior of colloidal suspensions as shown by Yziquel et al. [2]. If the model is shown to correctly predict the decrease of the elastic modulus with increasing strain amplitude above a critical strain, the values of the loss modulus are considerably overestimated.

Similar kinetic structural equations have also been used to predict nonlinear viscoelastic effects observed in polymer melts and solutions for large amplitude oscillatory shear and for stress growth and relaxation. The theories are based on the transient network models developed by Green and Tobolsky [21], Lodge [22] and Yamamoto [23]. The evolution of the structure is not described in terms of breakdown and build-up of the micro-structure, but in terms of generation and destruction of polymeric chain entanglements (or junctions) where the structural parameters, ξi, indicate how far the internal structure is from equilibrium. Marrucci et al. [24] introduced a kinetic rate equation combined with the upper convected Maxwell model:σ=iσi,σiGiiδδtσiGiiγ̇,Gi=G0iξi,λi0iξ1.4i,where σ is the extra stress tensor, σi the ith spectral component and γ̇ the rate deformation tensor. G0i and λ0i are the equilibrium (no flow) values. The dependence of λi is chosen so that the zero shear viscosity is proportional to c3.4, where c is the polymer concentration. The contravariant (upper) convected derivative is defined byδσiδt=dσidtv·σiσi·(v)T.

The kinetic equation is given by∂ξi∂t=1−ξiλiaλiGiiIIσi|1/2,where ξi is the structural parameter, ranging from 0 to 1 and a is a dimensionless parameter obtained by fitting steady shear viscosity data and IIσi the second invariant of the extra stress tensor σ. The first term of the kinetic equation is related to the generation of entanglements due to the Brownian motion and the second to the destruction of entanglements induced by stress.

Different kinetic equations have been proposed. Acierno et al. [25] introduced the first invariant of the extra stress tensor, Iσ, in the kinetic equation which is given by∂ξi∂t=1−ξiλiiλiIσi2Gi1/2.

Acierno et al. [25] reported good agreement between the network model predictions and experimental data in shear and elongational stress growth for a low density polyethylene melt (LDPE). Mewis and Denn [26] proposed a modified expression of the Acierno kinetic equation:∂ξi∂t=k11−ξiλmik2ξiλmiIσi2Gim/2,where k1 and k2 are kinetic constants characterizing, respectively, the entanglement generation due to the Brownian motion and the entanglement destruction induced by the flow and m is a dimensionless parameter. According to Giacomin and Oakley [27], this equation coupled with the upper convected Maxwell model allows a good description of a LDPE under large amplitude oscillation shear flow.

Liu et al. [28] suggested that the destruction of entanglements depends on the second invariant of the rate of deformation, IIγ̇, in the following way:∂ξi∂t=k11−ξiλmik2ξiλmiIIγ̇4m/2.They obtained good agreement between the model predictions and experimental data in shear and extensional flows for a polyisobutylene in decalin.

The use of the second invariant of the rate of deformation tensor is often criticized. Indeed, as the amplitude of the shear rate (γ0ω, product of the strain and frequency) in oscillatory flow increases with frequency, the strain amplitude limit for the linear behavior decreases with increasing frequency (see for example [29]). This is not verified experimentally for homogeneous polymer systems. For this reason, the so-called rate dependent constitutive equations are often considered inadmissible [30].

The objective of this paper is to develop a model based on the transient network theories to describe the rheological behavior of suspensions which are thixotropic and have an apparent yield stress. In the first part, a structure-dependent model and three different kinetic equations are proposed to characterize the evolution of the micro-structure. The model is used to describe the rheological behavior of two distinct suspensions: fumed silica particles dispersed in paraffin oil, and coating colors like those used in paper industries. The results of experimental data and theoretical predictions are then compared and discussed.

Section snippets

Structure-dependent model

We propose a kinetic network model based on ideas of Marrucci et al. [24] and Coussot et al. [20] to describe the nonlinear behavior of concentrated suspensions composed of interactive particles. We assume that the flow properties are controlled by the simultaneous breakdown and the build-up of the suspension microstructure. The stress is described by a modified upper convected Jeffreys model with a single relaxation time:δδtσG(ξ)+ση(ξ)=1+ηη(ξ)γ̇δδtγ̇G(ξ),withG(ξ)=G0ξ+Gwhere η and G are

Analysis

As we will see, several rheological tests including shear flow have been used. In shear flow, Eq. (14)reduces to∂σ11∂t+G(ξ)f(ξ)η0G0G(ξ)∂ξ∂tσ11−2γ̇σ12=0,∂σ12∂t+G(ξ)f(ξ)η0G0G(ξ)∂ξ∂tσ12=G(ξ)1+ηf(ξ)η0−ηG0G(ξ)∂ξ∂tγ̇γ̇∂t,∂σ22∂t+G(ξ)f(ξ)η0G0G(ξ)∂ξ∂tσ22=0,σ33=0.

As Eq. (26)also holds for steady shear flow, σ22 is equal to zero. For the kinetic equations, the first invariant of the stress tensor and the second invariant of the rate-of-strain simplify. Hence , , becomeλ0k1∂ξ∂t=(1−ξ)−k2k1ξ(λ0γ̇)2,λ

Experimental

Two distinct systems which are thixotropic and have a solid-like behavior at low strain are chosen to illustrate this study. The first system is a suspension of fumed silica particles in paraffin oil with different mass fractions and the second is a coating color similar to coating colors used for paper offset printing applications. The rheological behavior of these two systems is discussed elsewhere by Yziquel et al. 2, 33 and Yziquel [34].

Fumed silica particles Aerosil A200 (Degussa

Results and discussion

The model predictions for the three proposed kinetic equations are compared with the experimental data obtained for two different suspensions. These suspensions differ mainly in the nature of the structure. The coating color is a concentrated suspension of 34 vol.% mineral pigment stabilized sterically by a water-soluble polymer adsorbed on the pigment surface. Their nonlinear behavior can be attributed to the motion and restoration of the particle equilibrium position. The structure formed by

Concluding remarks

A model describing the rheological behavior of suspension which is controlled by structural changes was proposed. This model consists of a modified upper convected Jeffreys model with a single relaxation time and by a kinetic equation which describes the evolution of the microstructure with flow. The proposed model describes the nonlinear phenomena observed with the suspension. Three kinetic equations were proposed to predict the nonlinear viscoelastic behavior. The first equation depends on

Acknowledgements

The authors acknowledge financial support received from PAPRICAN and NSERC.

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    On leave from Laboratoire Mécanique et Matériaux, Université de Brest, France

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