ESI for Suppression of Self-Stratification in Colloidal Mixtures with High Peclet Numbers by Schulz et al in Soft Matter

Model and comparison of model with experimental findings for stratification

by Malin Schulz, Richard Brinkhuis, Carol Crean, Richard P. Sear and Joseph L. Keddie

This Jupyter notebook has the Python code for a model that predicts when stratification will and will not occur, in the parameter space of the volume fraction of small colloid, $\phi_S$ and Peclet number for the small colloid, Pe$_S$. It also briefly summarises the equations, see main text for more details.

It relies on reading in the experimental data from a csv file in the same directory.

The model is a simple extension of that in Sear, J. Chem. Phys 148, 134909 (2018). Please see there for details. NB There are minor differences in notation between that work in JCP and this. We will highlight the novel part in this work.

Jamming layer from Sear (J Chem Phys 2018)

Treatment of jamming is just as in Sear, J. Chem. Phys (2018)

Volume fraction of small colloid at descending interface, before the onset of jamming is

$$ \mbox{volume fraction of small colloid at descending air/water interface}=\phi_S(1+\mbox{Pe}_St^*) $$

for Peclet number Pe$_S=v_{ev}H/D_S$ and reduced time $t^*=v_{ev}t/H$. Dropping 1 inside () and requiring that jamming occur at the latest at $t^*=1$ we have

$$ \phi_S\mbox{Pe_S}=\phi_{jam} $$

and

$$ \phi_S>\phi_{jam}/\mbox{Pe_S}~~~~\mbox{Equation (1)} $$

for jamming to occur during drying. This is equation (11) of Sear (2018).

Accumulation gradient from Sear (J Chem Phys 2018)

From Equation A1 in the Appendix of Sear (2018) we have that immediately below a descending jamming front (of small colloidal particles) the volume fraction of small particles is

$$ \phi_S(\Delta z)=\phi_S+(\phi_{jam}-\phi_S)\exp[-\Delta z/\lambda_G] $$

for $\Delta z$ the distance below the (moving) jamming front, and $\lambda_G=D_s/v_{jam}$. So the derivative

$$ \frac{\partial \phi_S}{\partial z} = -\frac{\phi_{jam}-\phi_S}{\lambda_G}\exp[-\Delta z/\lambda_G] $$

Diffusiophoresis from Sear (2018)

The diffusiophoretic speed $U$ of the large colloid is given by Equation (18) of Sear (2018)

$$ U=-(9/4)D_S(\partial \phi_S/\partial z) $$

Using above equation for the gradient

$$ U=-\frac{9}{4}D_s\frac{\phi_{jam}-\phi_S}{\lambda_G}\exp[-\Delta z/\lambda_G]=(-9/4)v_{jam}(\phi_{jam}-\phi_S)\exp[-\Delta z/\lambda_G] $$

as $\lambda_G=D_s/v_{jam}$.

In the 2018 we considered a large colloid very close to the jamming front, i.e., $\Delta z=0$, and then required that the large colloid's diffusiophoretic speed be at least equal to $v_{jam}$ - the speed at which the jamming front descends. Then

$$ U = v_{jam}\frac{9}{4}(\phi_{jam}-\phi_S) \ge v_{jam} $$

or

$$ \frac{9}{4}(\phi_{jam}-\phi_S) \ge 1 $$$$ \phi_S \le \phi_{jam}-\frac{4}{9}\simeq 0.64-\frac{4}{9}\simeq 0.196~~~~\mbox{Equation (2)} $$

In summary: Equation (1) is a lower limit to the volume fraction of small colloid at which stratification can occur--- below (in either $\phi_S$ or Pe) that limit there is not enough small colloid to generate a jammed layer early enough in drying. Equation (2) is an upper limit to the volume fraction of small colloid at which stratification can occur --- above that limit the gradients in concentration of the small colloid are too small to drive diffusiophoresis of the large colloid that is fast enough to outrun the descending jammed layer of small colloid. Thus stratification can only occur between the values set by equations (1) and (2).

Also, note that both equations rely on numerous approximations, some of which are discussed in Sear and Warren (2017). Equation (1) is just one formulation of the fact that you need enough small colloid, while equation (2) is one formulation of the fact that if the initial concentration of small colloids is high then diffusiophoresis is weak to due weak gradients (eg in limit that $\phi_S=\phi_{jam}$ then no gradients at all).

Extension of Sear (2018)

In Sear 2018, the phoretic velocity $U$ at $\Delta z=0$ was used. This assumes that the large colloids are much smaller than the width of the accumulation zone, $R_L\ll \lambda_G$. Here we relax this constraint but still assume that we can use the phoretic speed $U$ at the centre of the large colloid. In reality the diffusiophoretic stresses will vary over the surface of the large colloid when $R_L$ and $\lambda_G$ are comparable but we assume that we can just approximate $U$ by that at the large colloid centre.

Then the maximum velocity occurs at the closest point to the jamming front, that a large colloid can approach, this is $\Delta z=R_L$. So the

$$ \mbox{maximum diffusiophoretic speed}=(9/4)v_{jam}(\phi_{jam}-\phi_S)\exp[-R_L/\lambda_G] $$

and if our condition for stratification is that this maximum value exceed the downward speed of the jamming front, $v_{jam}$, we have as a condition for stratification

$$ (9/4)v_{jam}(\phi_{jam}-\phi_S)\exp[-R_L/\lambda_G]>v_{jam} $$

or

$$ (9/4)(\phi_{jam}-\phi_S)>\exp[R_L/\lambda_G] $$

This is new to this work.

We want to work with the variables $\phi_S$ and Pe$_S$ so we use that

$$ \lambda_G=\frac{D_S}{v_{jam}}=\frac{D_S(1-\phi_S/\phi_{jam})}{v_{ev}}=\frac{H(1-\phi_S/\phi_{jam})}{\mbox{Pe}_S} $$

where we used Equation (12) of Sear (2018) for $v_{jam}=v_{ev}/(1-\phi_S/\phi_{jam})$.

So we can rewrite our condition for stratification as

$$ (9/4)(\phi_{jam}-\phi_S)>\exp\left[\frac{R_L\mbox{Pe}_S}{H(1-\phi_S/\phi_{jam})}\right] $$

for our systems, we have that approximately

$$ \frac{H}{R_L}=3000 $$

and then our inequality is a just a function of $\phi_S$ and Pe$_S$. We set it to an equality, and solve for Pe$_S$ as a function of $\phi_S$.

Note that this is perhaps just as much the upper limit of validity of the model, as much as it is a prediction of an upper limit on stratification.

It should be borne in mind that the above model is very crude, in particular, it is based on the initial Pe$_S$, which uses the diffusion constant $D_S$ at the start of drying. As the viscosity increases dramatically during drying then although we assume $\lambda_G$ is a constant during, in fact it will decrease due $D_S$ decreasing as the viscosity increases. However, the model is simple and gives a physically reasonable upper bound on stratification, above which stratification cannot occur due to the width of the accumulation zone of the small colloids (below a jammed layer of small colloids) being comparable to or smaller than the size of the large colloids. Above this limit the model breaks down and we do not predict stratification.

Plotting model predictions and experimental data on same plot

We now want to compare model predictions with experimental data. First, we read in experimental data from .csv file

Functions for viscosity - from fit to experimental data at pH=12 - and for diffusion constant and Peclet number

Note that we calculate viscosity at the concentration of the thickener (ASE - alkali-swellable emulsion - our thickener) in the continuous phase, i.e., $\phi_{CP}$. This is

$$ \phi_{CP}=\phi_T/(1-\phi_{TOT})=\phi_T/0.7 $$

for $\phi_{TOT}=0.3$ the total volume fraction of colloid in the suspension (both small and large colloid). In other words, we measure viscosity in ASE solutions without colloids and then need to use these values in suspensions with 30% solids. So we scale up ASE concentrations by a factor of $1/0.7$.

just some calculations to check

Make plot of experimental data plus model predictions

The shaded area above is where there are enough small colloids combined with a high enough Peclet number, to create a jammed layer of small particles, but where the accumulation zone in front of this jammed layer is wide relative to the size of the large colloidal particles.

NB Dashed curve position does depend a bit on $H$ and experimental data for range of $H$ values, but this affect is small on the log axis for Pe$_S$.