Yield stress fluids and fundamental particle statistics

Yield stress in complex fluids is described by resorting to fundamental statistical mechanics for clusters with different particle occupancy numbers. Probability distribution functions are determined for canonical ensembles of volumes displaced at the incipient motion in three representative states (single, double, and multiple occupancies). The statistical average points out an effective solid fraction by which the yield stress behavior is satisfactorily described in a number of aqueous (Si₃N₄, Ca₃(PO₄)₂, ZrO₂, and TiO₂) and non-aqueous (Al₂O₃/decalin and MWCNT/PC) disperse systems. Interestingly, the only two model coefficients (maximum packing fraction and stiffness parameter) turn out to be correlated with the relevant suspension quantities. The latter relates linearly with (Young’s and bulk) mechanical moduli, whereas the former, once represented versus the Hamaker constant of two particles in a medium, returns a good linear extrapolation of the packing fraction for the simple cubic cell, here recovered within a relative error ≈ 1.3%.

Yield stress in complex fluids is described by resorting to fundamental statistical mechanics for clusters with different particle occupancy numbers.

Yield stress fluids form a particular state of matter,¹ displaying non-linear and novel visco-plasto-elastic flow dynamics upon different boundary conditions. As their name says, they don’t flow until a certain load, the so-called yield stress (or point, τ₀), is applied. This value may be generally interpreted as a shear stress threshold for the breakage of interparticle connectivity.² Furthermore, as it initiates motion in the system, it is connected to mechanical inertia³ and particle settling, i.e. it is a terse summary of buoyancy, dynamic pressure, weight, viscous and yield stress resistances.⁴ For prototype systems such as colloids dispersed in a liquid, yield points sensibly depend on the mechanism by which the solid phase tends to interact or aggregate.^5–8 The macroscopic constitutive equations they obey, such as the Herschel–Bulkley model, were shown to correspond, over a four-decade range of shear rates, to the local rheological response.⁹From the side of an experimenter, however, unambiguously defining a yield stress may not always be straightforward. It can be affected by the experimental procedure adopted, always considering a measurement or some extrapolation technique with the limit of zero shear. Conversely, unyielded domains may be defined by areas where the shear stress second invariant falls below the yield value, plus some small semi-heuristic constant.¹⁰ In addition, theoretically, the meaning of notions like τ₀ and rheological yielding were questioned to be only qualitative or even to stand for an apparent quantity.¹¹ The dependence they generally show on timescales characteristic of the applied (mechanical) disturbance, also suggested an intimate relationship¹² between yield stress and dispersion thixotropy.¹³ On the other hand, assigning a hydrodynamic or mechanical state below the yield point to a material that is not flowing seems not to be scientifically sound. Experimental values are normally obtained by extrapolation of limited data, whereas careful measurements below the yield point would actually imply that flow takes place.¹⁴At any rate, the analysis of properly defined τ₀ concepts forms the subject of interesting investigations and is still a powerful tool in many applications, including macromolecular suspensions,¹⁵ gels, colloidal gels and organogels,^16–18 foams, emulsions and soft glassy materials.¹⁹ It allows for effective comparisons between the resistances which fluids initially oppose to the shear perturbation, somehow specifying a measure of the particle aggregation states taking place in a given dispersant. Electrorheological materials, for instance, exhibit a transition from liquid-like to solid-like behaviors, which is often examined by a yield stress investigation upon a given fluid model (e.g. the Bingham model or the Casson model).^20,21 The combination of yield stress measurements with AFM techniques can be used to well-characterize the nature of weak particle attractions and surface forces at nN scales.⁸ Further issues of a more geometrical nature, which naturally connect to τ₀, are rheological percolation²² and its differences from other connectivity phenomena, such as the onset of electric²³ or elastic percolation.^24,25 In granular fluids, it relates with the theory of jammed states,²⁶ originally pioneered by Edwards.²⁷In nanoscience as well, the stability control and characterization in single and mixed dispersions or melts is an important and complex step.^28,29 Carbon nanotube suspensions,³⁰ for example, can be prepared in association with other molecular systems, like surfactants and polymers^31–33 or by (either covalent or non-covalent) functionalization of their walls with reactive groups, which increases the chemical affinity with dispersing agents.³⁴ As a consequence of large molecular aspect ratios and significant van der Waals’s attractions, the nanotube aggregation is highly enhanced, giving rise to strongly anisotropic systems of crystalline ropes and entangled network bundles, which are difficult to exfoliate, suspend or even characterize.³⁵ Stable CNT dispersions of controlled molecular mass may also exhibit polymeric behavior, and be quantitatively studied by equations taken from the well-established science of macromolecules.^36,37This paper puts forward a basic approach, mostly focused on equilibrium arguments, to devise a yield stress law connected with particle statistics. By conjecturing an ensemble of effective volumes ‘displaced’ at the incipient state of motion, a statistical mechanics picture of τ₀ is proposed. This affords a phenomenological hypothesis that can be developed with reasonable simplicity. The derived relations are applied to typical disperse systems in colloid science and soft matter, such as aqueous and nonaqueous suspensions of ceramic/metal oxides and nanoparticles.