# Gravitational stability of suspensions of attractive colloidal particles

## Contents

## Reference

**Gravitational Stability of Suspension of Attractive Colloidal Particles**

Kim C, Liu Y, Kuhnle A, Hess S, Viereck S, Danner T, Mahadevan L, Weitz DA.

Physical Review Letters **99**: 028303 (2007)

## Depletion Attraction

The attraction between particles in a colloidal suspension must overcome the pull of gravity if the suspension is to be stable. The alternative is gradual sedimentation and phase separation of the particles. There are many applications in which gradual onset of a phase separation severely limits the useful life of a product.

One way to overcome sedimentation is through depletion attraction, in which a polymer added to the suspension increases the attraction between particles. This occurs because the polymer is excluded from regions between particles when the distance between particles less than the size of the polymer. The result is regions between particles which are severely depleted of the polymer. This causes a net osmotic force which pushes the particles towards each other. The range and strength of this attractive force can be varied by changing the size of the polymer and the polymer concentration, respectively.

## Experimental System

The goal was to measure the effect of particle volume fraction <math>\phi</math> on the compressional modulus <math>K(\phi)</math>. The colloidal suspension consisted of a surfactant-stabilized emulsion of paraffin oil in water. The emulsion samples had varying hydrodynamic radii R, but the volume fraction <math>\phi_0</math> and oil-water density mismatch <math>\Delta \rho</math> were constant. Depletion attraction was induced by adding either nonadsorbing polymer polyvinylpyrrolidon (PVP) or a surfactant - which was either Lutensol T08 or Lutensol A8). Time lapse images of the creaming emulsions were collected in order to observe the evolution of the clear fluid phase. The volume fraction at different times <math>\phi(t)</math> was measured by skimming sample off the top 2 mm of the emulsion, then weighing it, drying it, and weighing it again.

## Stress as a Function of Steady-State Volume Fraction

An emulsion with an initial height <math>H_0</math> will cream and eventually approach a steady-state <math>H_\infin</math> that has a corresponding volume fraction <math>\phi_\infin</math> (which, as explained in the experimental section, can be experimentally measured.) Increasing <math>H_0</math> decreases both <math>H_\infin</math> and <math>\phi_\infin</math>. The buoyant stress in an emulsion is:

<math> \sigma = \Delta \rho g \phi_0 H_0 </math>

By measuring <math>\phi_\infin </math> and <math>H_\infin</math> while varying <math>H_0</math>, surfactant concentration <math>c_m</math>, and micelle size <math>r</math>, the <math>\sigma \left (\phi_\infin \right)</math> can be experimentally derived.

The data points can be fit to functions in the form:

<math> \sigma \left (\phi_\infin \right) = -\alpha \frac{\phi_\infin-\phi_g}{\phi_c-\phi_\infin} </math>

where <math>\phi_c = 0.64</math> is the theoretical maximum for random close packing of uniform spheres; <math>\phi_g=0.03</math> is the minimum concentration required for gelation (which is independent of <math>c_m</math> and <math>r</math>); and <math>\alpha</math> is a stiffness parameter that depends on <math>c_m</math> and <math>r</math>.

Which can be related to the compressional modulus K by:

<math> K(\phi) = -\phi \frac{\partial \sigma}{\partial \phi} = \alpha \phi_\infin \frac{\phi_\infin-\phi_g}{\left (\phi_c-\phi_\infin \right)^2} </math>

## Scaling Parameter as a Function of the Hydrodynamic Radius and Micelle Size

Unlike hard sphere suspensions in which K and osmotic pressure scale with thermal energy, the experiment above shows that <math>\sigma(\phi)</math> and <math>K(\phi)</math> scale with the stiffness parameter <math>\alpha</math>. <math>\alpha</math> should reflect the inter-particle attraction, and dimensional analysis shows that it scales with <math>\frac{U}{r R^2}</math>

## References

- Pusey PN, Pirie AD, Poon WCK. Dynamics of colloid-polymer mixtures.
*Physica A***201**: 322-331 (1993).

Second Entry: Nick Chisholm, AP 225, Fall 2009

## General Information

**Authors**: C. Kim, Y. Liu, A. Kuhnle, S. Hess, S. Viereck, T. Danner, L. Mahadevan, and D. Weitz

**Publication**: PRL **99** 028303 (2007)

## Soft Matter Keywords

Compressional Modulus, Colloidal Suspension, Emulsion, Phase Separation

## Summary

The authors present a means by which to stabilize suspensions of attractive colloidal particles against gravitationally-induced sedimentation or creaming. Their idea is to cause a depletion interaction between the colloidal particles by introducing nonadsorbing particles or polymers to the suspension. This causes a weak attraction between the particles, which then results in a solid-like network or gel of the particles that helps support their buoyant weight. Under shear, this network or gel can be made to easily yield, allowing the suspension to flow.

Other means by which to stabilize these suspensions against gravitationally-induced sedimentation or creaming include density matching the particles to the suspending fluid, restricting the size of the particles so that their Brownian motion keeps them suspended, or by increasing the viscosity of the suspending fluid in order to slow down the phase separation. These methods are often not feasible, thus creating the motivation for this paper.

How does this depletion interaction work? Well, the polymers are only able to occupy regions where their size is smaller than the spacing between the colloid particles. This results in regions of lower concentration of polymer, and the osmotic pressure of the polymer pushes the colloid particles together.

## Soft Matter Discussion

In order to quantify the effectiveness of this method, one must measure the compressional modulus. This compressional modulus may be represented as:

<math>K(\phi) = -\phi \partial \sigma / \partial \phi</math>, where <math>\phi</math> is the volume fraction and <math>\sigma</math> is the stress (force per unit area).

Note that one can change the initial height of the emulsion, <math>H_{0}</math> (the emulsion is in a tube; it was found the meniscus does not have any effect on the results). This effectively allows manipulation of the buoyant stress of the emulsion, since the sample is a gel and thus the emulsion at the top feels the full buoyant stress of the suspensions below. As the sample creams, the initial volume fraction at the top, <math>\phi_{0}</math>, increases to the final volume fraction, <math>\phi_{\infin}</math>. Note that <math>H_{0}</math> sets the magnitude of the stress, <math>\sigma = \Delta \rho g \phi_{0} H_{0}</math> (where <math>\Delta \rho</math> is the difference in density between the colloids and the suspending fluid.)

<math>\phi_{\infin}</math> and <math>\sigma_{\infin}</math> are measured experimentally; the specific behavior of <math>\sigma_{\infin}</math> depends on both <math>r</math> (micelle size) and <math>c_{m}</math> (nonadsorbing polymer concentration above the critical micelle concentration), but in every case the data diverges as <math>\phi_{\infin}</math> approaches <math>\phi_{c} \approx 0.64</math>, the maximum value for random close packing of uniform spheres. Thus, the data is fit with functional form:

<math>\sigma(\phi_{\infin}) = -\alpha(r, c_{m}) \frac{\phi_{\infin} - \phi_{g}}{\phi_{c} - \phi_{\infin}}</math>,

where <math>\alpha(r, c_{m})</math> is called the stiffness parameter. Note that <math>\phi_{g} = 0.03</math> is the minimum concentration required for gelation. This equation matches very well with experimental data. See Figure 1.

From the functional form for <math>\sigma(\phi_{\infin})</math>, <math>K(\phi_{\infin})</math> can be determined easily to be:

<math>K(\phi_{\infin}) = \alpha(r, c_{m}) \phi_{\infin} \frac{\phi_{c} - \phi_{g}}{(\phi_{c} - \phi_{\infin})^{2}}</math>.

The stiffness parameter should reflect the attraction between neighboring particles; it has units of stress, so one expect it to be the attractive force divided by the area of the interparticle bond. This would mean:

<math>\alpha(r, c_{m}) \approx U/rR^{2}</math>, where <math>R</math> is the hydrodynamic radii.

This matches well with experiment.

Since we can control <math>\phi_{\infin}</math>, we can control <math>K(\phi_{\infin})</math>, and thus control the stability of the suspension against gravitational creaming; this is what the authors claim.

Personally, I would have liked to see more of an effort on the part of the authors to explain how one can control, exactly, <math>\phi_{\infin}</math> with the addition of polymer. I am not entirely convinced that this is easily possible (at least, I'm not convinced it's as simple as the authors seem to insinuate).

One application of this work is clear: increase in shelf life of commercial products. It would be interesting to consider the commercial products for which this work would be most desirable, and see if a nonadsorbing polymer can be easily found. I wonder if these nonadsorbing polymers would actually have adverse effects on how the emulsion performs when used for the purpose they were original created.

## References

[1] C. Kim, Y. Liu, A. Kuhnle, S. Hess, S. Viereck, T. Danner, L. Mahadevan, and D. Weitz, "Gravitational stability of suspensions of attractive colloidal particles," PRL **99** 028303 (2007).

[2] D. Marenduzzo, K. Finan, and P. R. Cook, "The depletion attraction: an underappreciated force driving cellular organization," The Journal of Cell Biology, Volume 175, Number 5, 681-686 (2006).