# 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

Figure 1. Experimental height profiles at different polymer concentrations: (\blacksquare) cp = 1 mg/mL, ($\circ$) cp = 7.5 mg/mL, and (▲) cp = 20 mg/mL. We measure the height from the bottom of the cell, as shown in the schematic, where h0 = (1.90 ± 0.05) cm is the initial height. (Upper inset) Expanded details of the initial evolution for a time window of 5 h.
Figure 2. Diagram for the behavior of mixtures of polystyrene colloidal particles and polyethylene glycol polymer as a function of the particle volume fraction and the polymer concentration cp: () sedimentation; () collapse; (▲) compression.
Figure 3. (a) Snapshots at different stages during the evolution of samples prepared with monodisperse polymer. The volume fraction of colloidal particles is = 0.05 in all samples. (b) Experimental height profiles at two polymer concentrations: () cp = 20 mg/mL, () cp = 30 mg/mL. The line represents the linear fit to the experimental points.

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 because 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 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 $\phi$ = 0.1 and $c_p = 7.5 mg/mL$(open circles). This behavior is similar to transient gelation, where a gel forms and subsequently collapses under the influence of gravitational stresses.

Using the temporal evolution of the interface as a criterion, the authors summarize the behavior of the colloid/polymer system in a $\phi - c_p$ diagram, shown in Figure 2. For low $\phi$ or low $c_p$, the system undergoes hindered sedimentation (closed squares), consistent with the absence of gelation. By contrast, at high $\phi$ or high $c_p$, the system exhibits a compression behavior (closed triangles) indicating the presence of a particle gel. Between these two behaviors, for intermediate $\phi$ and intermediate $c_p$, there a region in the $\phi - c_p$ diagram corresponding to a behavior that 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 $c_p= 5 mg/mL$. This coarsening precedes the collapse of the structure. To investigate this phenomena, the authors prepared samples at intermediate $\phi$ and $c_p$ and varied both the initial height and the density difference between the particles and the solvent while keeping both $\phi$ and $c_p$ constant. They found that the structure of the colloid/polymer mixtures always coarsens to some extent before it actually collapses due to the gravitational stress. Furthermore, the time for this coarsening to become appreciable in our images does not change with the total gravitationally induced stress (σg) and always remains equal to 15 min.

As a result, the authors concluded that the coarsening behavior is not dependent on the gravitational stress, thereby implying that the presence of this stress is ultimately not the mechanism responsible for the temporal evolution of the samples. To further investigate this, they followed the time evolution of a sample using a CCD camera and a high magnification objective, focused on a small region in the middle. They do not observe any collapse of the structure within the experimental time-window since σg is so small. However, as time proceeds, the system separates into colloid-poor and colloid-rich regions, as shown in Supporting Information, Movie S3, where darker regions correspond to colloid-poor regions and brighter regions correspond to colloid-rich regions.

Further experiments confirmed their hypothesis that this coarsening behavior is due to spinodal decomposition. However, if the densities of the colloids and solvent are not matched, spinodal decomposition is interrupted by the presence of gravitational stresses. Therefore the authors propose that the nonadsorbing polymer induces the requisite attraction to initiate spinodal decomposition of the mixtures. The initially homogeneous mixture then evolves into a bicontinuous network of colloid-poor and colloid-rich domains. The structure continues to coarsen and eventually looses mechanical strength; it fractures and collapses under the presence of the gravitational stress.

The authors were initially puzzled because the typical range of the attractive interaction in these experiments was much lower than those that are typically required to induce phase separation. However, this value is based on the average molecular weight of the polymer, $M_w$. In these experiments, because of the polydispersity of the polymer, there is a sizable fraction of chains that are several orders of magnitude larger than the mean. Despite the relatively smaller concentration of these larger chains, the authors believe that they nevertheless determine both the magnitude and width of the attraction, resulting in a weaker and wider attraction than would be expected from the peak of the $M_w$ distribution, and causing the mixtures to exhibit a transient gel behavior.

To test this hypothesis, the authors performed the same experiments with monodisperse polymer. In these mixtures, the coarsening behavior is not observed. Instead, they only observed hindered sedimentation at low polymer concentrations and gel compression at high polymer concentrations, as shown in Figure 6 for = 0.05 and cp ranging from 5 mg/mL (right vial) to 60 mg/mL (left vial). The transition between these two behaviors occurs between cp = 20 mg/mL and cp = 30 mg/mL; for cp = 20 mg/mL, the system exhibits hindered sedimentation, as shown by the squares in Figure 6b, while for cp = 30 mg/mL, the system exhibits gravitational compression, as shown by the circles in Figure 6b. This behavior is in striking contrast to that observed with the polydisperse polymer, emphasizing the important role of the largest polymer chains on the gravitational stability of colloid/polymer mixtures when the polymer is polydisperse, and confirming our hypothesis that it is these larger polymer chains that determine the behavior of the mixture. However, by comparing the behavior presented in Figure 6 with that of the polydisperse case (see Figure 2, for = 0.05), we observe that also in this case a polymer concentration of cp ≈ 20 mg/mL marks the formation of a gel that compresses in time. Thus, at high polymer concentrations the overall behavior of the system is mainly controlled by the smaller rather than by the larger polymer chains. It is thus apparent that to properly account for the behavior of colloid−polymer mixtures prepared with polydisperse polymer, the full nature of the polymer distribution must be considered. This is of particular importance in practical applications of these mixtures.

## Conclusion

We find that polymer polydispersity can play a crucial role in the phase behavior of colloid−polymer mixtures. For broad polymer size distributions, the large polymers can effectively increase the range of the depletion attraction giving rise to spinodal decomposition rather than fully dynamically arrested gelation. By contrast, for monodisperse polymer, spinodal decompostion is dynamically arrested resulting in gelation. The effect of gravitational stresses can be dramatically different depending on whether the suspension phase separates or forms a gel. While for our phase separating samples, the system exhibits a sudden collapse, for our gelling samples, only a slow compression of the gel is observed. These results demonstrate that the stability of colloidal structures formed by the inclusion of nonadsorbing polymer can depend on the details of the polymer polydispersity. This is of particular importance for applications based on colloid−polymer mixtures, where not only the colloidal suspension but also the polymer can be polydisperse. In particular, these results are important for commercial products, where the coarsening and collapse behavior can severely limit the shelf life and utility of the final product.