## 2-Constraints Particle Rheology

### Quick Start

*This is a rough prototype for discussion purposes only*

Particle-filled systems can show many strange behaviours at different shear rates/stresses. Often there is a built-in contradiction - increasing shear can lead to more particles bashing into each other, causing one kind of behaviour, it can equally rip particles apart, causing another type.

In this app, based on a paper by Guy et al^{1} at U Edinburgh, nothing is said about the physics of the specific effects; instead we just note that there can be two competing effects and depending on their relative sizes, onsets and stress dependencies, we see what behaviour is obtained. This allows us to look at any real-world behaviour and find the set of generic parameters that creates it. This then lets us think about, and quantify, the relevant physics in that system.

### 2-Constraints Particle Rheology

_{AB}

_{aB}

_{Ab}

_{ab}

_{A}

_{B}

_{0}

Typically we think of particles in dispersion being dominated by a single interaction such as friction. Shear forces can overcome friction so we see a big reduction in viscosity at higher shear. But this single-factor model cannot possibly describe some of the complex behaviours seen with real dispersions

By introducing a second factor, such as adhesion, with the *opposite* behaviour to the first factor, then a large variety of complex behaviours is readily calculated. By opposite is meant that if shear reduces viscosity for the first factor then it increases viscosity for the second.

This complex behaviour can be simulated via 8 parameters, though because they conform to common sense, there is an order and logic to them which makes them a powerful way to look at these issues. In the paper the opposite of a factor such as A is shown with a bar across it. HTML doesn't do a good job with that so the opposite is shown in lower case. We start with φ_{AB} which is a volume fraction where the particles behave as we expect, by jamming because A is unconstrained sliding (i.e. low friction induced by shear) and B is constrained adhesion. So a typical φ_{AB}=0.64. The fraction where A is constrained, φ_{aB} is the case of random loose packing of 0.55. φ_{Ab} is often similar to φ_{aB}, around 0.35 and φ_{ab} is the lowest of all, say 0.2

We need to know the fractions of A and a and of B and b. These are calculated as follows and plotted in the graph. Obviously f_{A}=1-f_{a}:

`f_a = exp[-(σ_A/σ)^α]`

`f_b = 1-exp[-(σ_B/σ)^β]`

The stress values, σ_{A} and σ_{B} are the onsets of the (un)constrained behaviours and α and β determine how swiftly the changes take place.

From these factors a jamming fraction, φ_{J} is calculated as:

`φ_J = (1-f_a)(1-f_b)φ_"AB" + φ_a(1-f_b)φ_"aB"+(1-f_a)f_bφ_"Ab"+f_af_bφ_"ab"`

And finally the viscosity is calculated via the usual method:

`η = η_0(1-φ/φ_J)^(-2)`

Obviously this is only a bare bones explanation for this draft version.

^{1}B. M. Guy, J. A. Richards, D. J. M. Hodgson, E. Blanco, and W. C. K. Poon,
*Constraint-Based Approach to Granular Dispersion Rheology*, PHYSICAL REVIEW LETTERS 121, 128001 (2018)