A Thermodynamically Consistent, Microscopically-Based, Model of the Rheology of Aggregating Particles Suspensions

Abstract

In this work, we outline the development of a thermodynamically consistent microscopic model for a suspension of aggregating particles under arbitrary, inertia-less deformation. As a proof-of-concept, we show how the combination of a simplified population-balance-based description of the aggregating particle microstructure along with the use of the single-generator bracket description of nonequilibrium thermodynamics, which leads naturally to the formulation of the model equations. Notable elements of the model are a lognormal distribution for the aggregate size population, a population balance-based model of the aggregation and breakup processes and a conformation tensor-based viscoelastic description of the elastic network of the particle aggregates. The resulting example model is evaluated in steady and transient shear forces and elongational flows and shown to offer predictions that are consistent with observed rheological behavior of typical systems of aggregating particles. Additionally, an expression for the total entropy production is also provided that allows one to judge the thermodynamic consistency and to evaluate the importance of the various dissipative phenomena involved in given flow processes.

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

“…Shear rate scaling in the breakage expression must correspond to the viscous dissipation due to shear flow, because the agglomerate breakage process must be purely dissipative to be thermodynamically admissible. 49 This necessitates that the shear rate involved in the breakage kernel be purely plastic or unrecoverable in nature.…”

“…Yield stress depends on the extent of network formation and hydrodynamic interactions between large and small aggregates. Although the phenomenology of yield stress has been replicated using nonlinear viscoelastic constitutive relationships, 46,49,60 it is difficult to directly associate the parameters with the fractal structure of the agglomerates. Following Shih et al 61 model for effective elastic modulus G, and elastic strain γ e , in the strong link regime, the yield stress can be described as…”

“…This criterion ensures that the resulting equations do not admit thixo-elastic materials which are shown to be non-physical. 10,25,49 In Maxwell-type viscoelastic constitutive equations for stress, Joshi has proven that thermodynamic consistency can be enforced by decomposing the total shear strain rate into its elastic and plastic contributions, ensuring that shear aggregation and breakage depend only on the plastic component of the shear strain rate. 25 Following this development, one can still use the empirically constructed breakage kernels and retain their accuracy without violating the thermodynamic admissibility.…”

“…Viscoelasticity is modeled using a constitutive relationship based on work by Kamani et al 71 where the authors propose a phenomenological equation for modeling yield-stress fluids, given as be replicated through non-linear terms in the viscoelasticity expression. 49 An alternative expression for this constitutive relation is one proposed by Saramito 23 where an explicit distinction between the elastic and visco-plastic behavior is maintained through the use of a discontinuity, max 0,…”

“…21,22,25,46 Alternatively, some approaches also derive constitutive models using non-equilibrium thermodynamic formalisms. [47][48][49] Using the single-generator bracket formalism of non-equilibrium thermodynamics (SGBF-NET), Jariwala et al 49 showed that the energy of mixing associated with the particle size distribution is fundamental to generate aggregation and breakage dynamics required to model thixotropic behavior. The authors also showed that aggregation and breakage depend on the invariants of the internal stresses developed within the agglomerates (modeled via a conformation tensor).…”