A constitutive equation for thixotropic suspensions with yield stress by coarse‐graining a population balance model

Abstract

A population balance model appropriate for the shear flow of thixotropic colloidal suspensions with yield stress is derived and tested against experimental data on a model system available in the literature. Modifications are made to account for dynamic arrest at the onset of the yield stress, in addition to enforcing a minimum particle size below which breakage is not feasible. The resulting constitutive model also incorporates a structural based relaxation time, unlike existing phenomenological models that use the inverse of the material shear rate as the relaxation time. The model provides a reasonable representation of experimental data for a model thixotropic suspension in the literature, capturing the important thixotropic timescales. When compared to prevalent structure kinetics models, the coarse‐grained population balance equation is shown to be distinct, emphasizing the novelty of utilizing population balances as a basis for thixotropic suspension modeling. © 2016 American Institute of Chemical Engineers AIChE J, 63: 517–531, 2017

Distinguishing thixotropy from viscoelasticity

“…1(d). There are many reports available in the literature, which confirm the above protocol for various thixotropic systems such as bentonite-water suspension [31], crude oil [32], fumed silica suspension [30,[33][34][35], carbon black suspension [36], Carbopol dispersion [37], human blood [38], theoretical models [39,40], which suggest this distinction to be useful. Recently Choi et al [41] discussed the role of elasticity during step down shear rate experiments using 2.9 vol % fumed silica suspension.…”

A polydisperse model for thixotropic elasto‐viscoplastic suspensions of aggregating particles using population balances

“…μ s is the suspending medium viscosity, and η r is the relative suspension viscosity which depends on the volume fraction of particles in the suspension. Following Mwasame et al, 26 the medium viscosity term in the original Smoluchowski formulation of the aggregation kernel is replaced by the suspension viscosity to extend the applicability of the expression which was initially derived for dilute suspensions.…”

“…Foundational work by Mwasame et al 26 demonstrated that population balance modeling can be incorporated into a robust, efficient, and accurate rheological constitutive model. In that work, the model presented (hereafter referred to as PBEMono) used a monodisperse closure to develop the moment version of the population balance equation as a coarse-graining assumption.…”

“…In that work, the model presented (hereafter referred to as PBEMono) used a monodisperse closure to develop the moment version of the population balance equation as a coarse-graining assumption. The use of a closure assumption for the moments of the size distribution enables reducing the inherent integro-differential nature of population balance equations to a finite system of simultaneous ordinary differential equations in time, similar in complexity, but distinct in form to the typical structure kinetics models (for a more detailed discussion, see Mwasame et al 26 ). While successfully applied to model a range of thixotropic and viscoelastic behavior in suspensions of fumed silica and human blood, 36 the oversimplistic nature of the monodisperse assumption fails to adequately represent the agglomerate distribution and significantly overpredicts the agglomerate sizes when compared with independent microstructural measurements.…”

Towards the Development of a Strategy to Characterize and Model the Rheological Behavior of Filled, Uncured Rubber Compounds

“…Population Balance Models (PBMs) take into account the flow-induced changes of the microstructure and describe how these changes influence the bulk rheology of the material. In the last years, there has been a growing interest in developing thixotropic constitutive models from PBMs [ 33 , 34 ].…”