# Gelation of particles with short-range attraction

## Contents

# Additional Entry: Xu Zhang, AP225 Fall 2009

## Reference

Peter J. Lu, Emanuela Zaccarelli, Fabio Ciulla, Andrew B. Schofield, Francesco Sciortino, and David A. Weitz,**Gelation of particles with short-range attraction** , Nature 453, 499-504 (2008).

## Keywords

gelation, DLCA, phase separation, spinodal decomposition

## Summary

This paper shows that gelation of spherical particles with isotropic, short-range attractions is initiated by spinodal decompositions, a kind of thermodynamic instability that triggers the formation of density fluctuations, leading to spanning clusters that dynamically arrest to create a gel.Moreover, their experimental results show that gelation ,which was often considered a purely kinetic phenomenon, is in fact a direct consequence of equilibrium liquid-gas phase separation.

A colloid-polymer system is used for the experiment where <math>U/k_BT</math> and ξ are controlled by the polymer size and free-volume concentration <math>c_p</math>. Samples with fixed <math>\phi=0.045\pm 0.005</math> and ξ=0.059 and various <math>c_p</math> are mixed. Figure 1a and 1b summarize the samples studied by plotting their values of <math>c_p</math>,normalized by polymer overlap concentration <math>c_p^*</math> in the phase diagrams.Two phases are observed. In samples with low <math>c_p</math>, below the experimental region boundary <math>c_p^g</math>, a fluid of many clusters that is stable for day is observed. A full three-dimensional image of these clusters in the fluid phase for a sample with <math>c_p=3.20\pm 0.03mg ml^{-1}</math> is shown in Fig.1c. By contrast, in samples with <math>c_p>c_p^g</math>, particles aggregate immediately into clusters, which in turn form a network that spans the macroscopic sample.This network subsequently arrests to create a gel and a sample with <math>c_p=3.31\pm 0.03mg ml^{-1}</math> is shown in Fig.1d.These two phases are separated by a very sharp boundary: the gel and fluid illustrated differ in <math>c_p</math> by only a few percent.

The development of a peak in the static structure factor S(q) at finite scattering vector q is a distinctive hallmark of spinodal decomposition. In fluid samples with <math>\phi</math>=0.045, ξ=0.059 and <math>c_p<c_p^g</math>, S(q) shows only a slight rise at low q; however, increasing <math>c_p</math> by just a few percent across<math>c_p^g</math> increases the height of the peak in S(q) by two orders of magnitude(Fig.2a). Further distinguishing characteristics of spinodal decomposition occur in the temporal evolution of S(q), where the peak narrows and moves towards lower q, and in its first moment <math>q_1(t)</math>, which exhibits a power law dependence. The gel samples demonstrate these features: at the earliest times, the peak in S(q) narrows and moves to lower q(Fig.2b)Moreover, <math>q_1(t)</math> scales as <math>t^{-1/6}</math>(Fig.2c), exactly as in molecular spinodal decomposition.

## Soft Matter Connection

Nanoscale or colloidal particles can dramatically change the properties of materials, imparting solid-like behavior to a wide variety of complex fluids. This behavior arises when particles aggregate to form mesoscopic clusters and networks. The essential component leading to aggregation is an interparticle attraction. In the limit of irreversible aggregation, infinitely strong interparticle bonds lead to diffusion-limited cluster aggregation (DLCA). This is understood as a purely kinetic phenomenon that can form solid-like gels at low particle volume fraction. However,the systems with weaker interactions, where gel formation requires higher volume fractions remained unknown. This report gave an explanation to this question: gelation is in fact a direct consequence of equilibrium liquid-gas phase separation and it is phase separation, not percolation, that corresponds to gelation in models for attractive spheres.

Moreover, this report sheds light on non-equilibrium behavior in technological systems. Because of the onset of non-equilibrium behavior is in fact governed by equilibrium phase separation, thermodynamic calculations may facilitate quantitative prediction of prediction of product stability, a critically important problem in the formulation and manufacture of commercial complex fluids.

Peter J. Lu, Emanuela Zaccarelli, Fabio Ciulla, Andrew B. Schofield, Francesco Sciortino & David A. Weitz *Nature* **453**, 499-504 (2008).

### Soft Matter Keywords

gel, gelation, spinodal decomposition, PMMA

## Summary

This paper examines the phenomenon of gelation, or the process of forming a gel. Nanoscale or colloidal particles change the properties of materials, imparting solid-like behaviour to a wide variety of complex fluids. This behaviour arises when particles aggregate to form mesoscopic clusters and networks. Numerous scenarios for gelation have been proposed, but no definitive agreement in the community has been reached. The authors report experiments showing that gelation of spherical particles with isotropic, short-range attractions is initiated by spinodal decomposition; this thermodynamic instability triggers the formation of density fluctuations, leading to spanning clusters that dynamically arrest to create a gel. This simple picture of gelation should apply to any particle system with short-range attractions.

## Experiment Details

The authors use polymethylmethacrylate (PMMA) colloidal spheres of radius a = 560 nm in solution as a model colloid–polymer system. They control U/k_{B}T and ξ by the polymer size and free-volume concentration c_{p}. They fix the particle volume fraction (φ = 0.045) and ξ = 0.059, and mix samples at various c_{p}. Gravitational sedimentation is eliminated by matching the colloid and solvent densities to within <10-4. The particle aggregates are then broken up by shearing, and the system is then observed with a high-speed confocal microscope.

## Results

Two phases are observed, which are distinguished by differing free volume concentrations (c_{p}). In samples with low c_{p}, below the experimental gelation boundary c_{pg}, a fluid of many clusters is observed that is stable for days (Figure 1 c). In samples withc_{p} > c_{pg}, particles are found to aggregate immediately into clusters, which in turn form a network that spans the macroscopic sample. This network subsequently arrests to create a gel (Figure 1 d). The gel undergoes no major structural rearrangement for days, even though it exchanges particles with a dilute gas. These phases are separated by a very sharp boundary: the gel and fluid illustrated differ in c_{p} by only a few per cent. The observation of only these two dramatically different phases contrasts findings of more complex phase behaviour in non- buoyancy-matched systems, where sedimentation can shift or obscure the observed phase boundaries.

## Conclusions

The results suggest that gelation—often considered a purely kinetic phenomenon is in fact a direct consequence of equilibrium liquid– gas phase separation. Gelation is observed in all samples predicted by theory and simulation to phase- separate; suggesting that it is phase separation, not percolation, that corresponds to gelation in models for attractive spheres.

--Cassidy 16:56, 12 September 2009 (UTC)