# The Role of Polymer Polydispersity in Phase Separation and Gelation in Colloid−Polymer Mixtures

Entry by Emily Redston, AP 225, Fall 2011

Work in progress

## Reference

The Role of Polymer Polydispersity in Phase Separation and Gelation in Colloid−Polymer Mixtures by J. J. Lietor-Santos, C. Kim, M. L. Lynch, A. Fernandez-Nieves, and D. A. Weitz. Langmuir 26 3174–3178 (2010)

## Introduction

Mixtures of non-adsorbing polymer and colloidal particles exhibit a very rich range of morphologies. These microstructures depend on the particle and polymer concentrations as well as the relative size of the particles and polymer. The addition of the polymer to a colloidal suspension leads to a depletion attraction that is capable of inducing a fluid-solid transition (i.e. forming a gel). A gel is defined as a connected network that spans space and can support weak stresses. Gels are extensively used in commercial applications, such as personal care or food products, where they are able to help stabilize the system against sedimentation. In this manner, gels can reduce phase separation, which will increase product shelf life. The nature of the fluid-solid transition depends on the range of the depletion attraction, which, in turn, depends on the ratio of the size of the polymer to the colloidal particle. For short-range interactions, gelation is induced by spinodal decomposition. A gel is formed because, when the system undergoes a gas-liquid phase separation, it is interrupted by the dynamical arrest of the particles in the colloid-rich region. By contrast, for larger ranges of the attraction, phase separation can proceed to completion, without being interrupted by dynamic arrest.

However, in the presence of gravitational effects, the structure may no longer be capable of sustaining its own weight; instead it collapses, disrupting the phase separation process or rupturing the gel that was previously formed. This is obviously an undesirable effect as it can dramatically shorten the shelf life of a commercial product. This suggests that the use of shorter polymers at sufficiently high concentration is of greatest practical interest. However, technological polymers are very rarely monodisperse, and thus the microstructures and their behavior may be drastically modified. However, despite the practical importance, the gravitational behavior of colloid-polymer mixtures using polydisperse polymers has never been investigated. It is not known how the polydispersity of polymers effects the phase behavior of the mixture.

In this paper, the authors investigate the behavior of model colloidal particles mixed with nonadsorbing polymer with a polydisperse size distribution, similar to that often found in commercial samples. Ultimately they find that the presence of even a small amount of large polymer in a distribution of nominally much smaller polymer can drastically modify the behavior.

## Sample Preparation

The authors used an aqueous dispersion of polystyrene particles with a density $\rho = 1.057 g/cm^3$ and an average radius $a = 1.5 \mu m$. They also used polyethyleneglycol (PEG) with an average molecular weight $M_w = 475500 g/mol$, polydispersity index $M_w/ M_n = 2.63$, and mean radius of gyration $r_g = 40 nm$. Here, $M_w$ and $M_n$ are the mass- and number-averaged molecular weights, respectively. Salt was also added to reduce electrostatic interactions.

## Experiments and Results

The rate of sedimentation of the colloidal particles is dependent on the microstructure of the colloid-polymer mixture. Sedimentation is driven by the density mismatch between the solvent and the colloidal particles, Δρ, that results in a body force on the individual particles, F=4/3πΔρg$a^3$. For a particle suspension, the settling rate is hindered by solvent backflow. However, settling in particle gels is more complicated as the gravitational forces act on and are transmitted through the entire network, resulting in a distinctive settling behavior. By monitoring the time-dependent height profile of the sample as it sediments, it is possible to distinguish between the two cases.

At low polymer concentration, the depletion-induced attraction between particles is small and the interface immediately begins to fall, with a constant rate, $v = (1.5 \pm 0.2) 10^{-7} m/s$, as shown in Figure 1 for a polymer concentration $c_p =1 mg/mL$ and a particle volume fraction $\phi$ = 0.1 (closed squares); this value is consistent with hindered settling models for dispersed particles, where

$v = \frac{2}{9} \frac{\Delta \rho g a^2}{\eta} (1-\phi)^{5.5} = 1.8 \times 10^{-7} m/s$

where $\eta$ is the viscosity of the solution. At high polymer concentration, the attraction between particles is large. As a result, the height of the interface evolves in a different fashion, as shown in Figure 1 for $c_p =20 mg/mL$ at the same $\phi$ (closed triangles). The height evolution in this case quantitatively agrees with poroelastic models for the gravitational compression of particle gels. In this case, the height evolution of the interface is determined by the balance of the gravitational and elastic stresses imposed on the network, and by the viscous stress due to solvent backflow through the network as it is compressed.

Surprisingly, at intermediate polymer concentrations, there is a fundamentally different behavior. Initially, the interface slowly moves but, after some time, abruptly and rapidly falls, as shown in Figure 1 for φ=0.1 and cp =7.5 mg/mL (open circles),19 a behavior which is similar to transient gelation,9-11 where a gel forms and subsequently collapses under the influence of gravitational stresses. Using the temporal evolution of the interface as a criterion, we summarize the behavior of our colloid/polymer system in a φ-cp diagram, shown in Figure 2. For low φ or low cp, the system undergoes hindered sedimentation (closed squares), consistent with the absence of gelation. By contrast, at high φ or high c , the p system exhibits a compression behavior (closed triangles) indicat- ing the presence of a particle gel. Between these two behaviors, for intermediate φ and intermediate cp, we find a region in the φ-cp diagram corresponding to a behavior which is reminiscent of transient gelation (open circles). Interestingly, this transient behavior is linked to a coarsening of the structure, as shown in Supporting Information, Movie S2, for cp= 5 mg/mL. This coarsening, which is reminiscent of “curd- ing”, precedes the collapse of the structure. Thus, although collapse must result from gravitational stress, it is unclear whether this external stress is the ultimate cause of the collapse, or whether