## Micro-Rheology of Nanoparticles

### Quick Start

Standard rheology of particle-filled systems is based on the assumption that they are hard spheres. Everything changes if the particles start to self-associate. Here we can mix a small amount of some self-associating particles (e.g. the carbon black in battery manufacture, as per the paper that inspired the app) with large amounts of hard sphere particles such as graphite. Each has distinctive rheological properties so the app helps you disentangle similar issues in your own formulations.

### Micro-Rheology of Nanoparticles

_{n}Nanoparticles

_{f}

_{l}Large Particles

_{0}cP

_{12}*10

^{-20}

The app is based on a paper from the Prasher team at Lawrence Berkeley^{1}, which in turn is based on the micro-rheology analysis of the Mellema team at U Twente^{2}. In addition I've included the shear-dependent hard sphere rheology effects found in the High-Shear Particles page. Though I normally try to describe the science and formulae in these pages, the equations and the detailed logic are too hard to describe here. The default values at startup replicate the results in the Prasher paper for φ=1.2%. Because they happened to use a fixed amount of stabilizing polymer, their data at 0.9% and 3.2% involved stabilizing chain lengths of 17nm and 11nm respectively, compared to the 15nm at 1.2%.

The basic idea is that the particles of radius r self-associate in a manner described in terms of separation distance h by DLVO theory, with a steric barrier set up by a polymer with an effective thickness around the particle of δ Layer. Obviously we are not including charge stabilization. The attraction is described by the Hamaker constant A_{12}. The graph of the DLVO interaction is shown. Clearly this is simplified, but the other parameters of the steric barrier make little difference provided the χ value (discussed in the DLVO app) is less than 0.5. Here it is given the fixed value of 0.1

Via the first and second derivatives of that curve it is possible to find some key parameters that describe how hard it is to break up a flocculated assembly of particles. For any given shear value, the particles will have a flocculated size which has the curious property of acting like a large volume fraction, φ, of normal particles. So for a real fraction φ_{n}=1% of self-associating particles at low shear they might behave like φ=20% of hard spheres and at high shear like φ=5%. Because the relative viscosity of particles depends (to be simple) as (1-φ/φ_{m})^{-2.5φm}, the viscosity of the effective 20% is much higher. Here φ_{m} is the maximum volume fraction, taken here to be 0.61.

Via a complex train of logic and some tricky numerical solvers, the "structural" viscosity of the flocculated particles at any given shear rate can be calculated. The parameter α describes the chances that any collision leads to a contribution to an aggregated chain. The fractal dimension d_{f} varies between, say, 1.3 and 2. There's a parameter k which describes the relationship between the radius of gyration of a clump of particles and the end-to-end distance of the chain making up that clump. As most of us really don't know what α,d_{f} and k should be it's probably a good idea to leave them as the defaults (0.25, 1.6, 1) unless you have good reasons to choose otherwise. Finally, the base viscosity η_{0} (i.e. the fluid without the particles) influences the degree to which shear breaks up the clumps.

If you have a φ_{l} of larger, hard sphere particles, this creates the standard "hydrodynamic" viscosity (calculated as per the High-Shear Particles page from the φ_{total}=φ_{n}+φ_{l}) which gets added to the "structural" viscosity. Note that the hydrodynamic calculation, for simplicity, assumes spherical particles with a critical percolation fraction φ_{c} of 0.28. You get some odd-looking shapes at high volume fractions due to limits of the powerlaw at low shear and the onset of an "infinite shear" limit.

As of March 2020 this is a beta version for comments, suggestions etc. Feedback is especially welcome.

^{1}Fuduo Ma, Yanbao Fu, Vince Battaglia, Ravi Prasher, *Microrheological modeling of lithium ion battery anode slurry*, Journal of Power Sources 438 (2019) 226994

^{2}A. A. Potanin, R. De Rooij, D. Van den Ende, and J. Mellema, *Microrheological modeling of weakly aggregated dispersions*, J. Chem. Phys. 102, 5845-5853 (1995);