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The plot shows this distribution for all the simulated gels. Clearly, the mean number of nearest neighbors is larger than six for all cases. Therefore, the clusters exceed the Maxwell iso-staticity condition and contain a rigid core that can serve as an elastic sub-unit for the Cauchy-Born theory. It is important to recognize that the l -balanced graph decomposition is not unique.

For example, consider the graph with five vertices connected in a ring. We employ a spectral decomposition method that determines the decomposition through k -means clustering using a stochastically sampled initial condition. Degeneracy of the decomposition should only affect the loose connections between rigid subunits and not the subunits themselves. These uncertainties are smaller than the size of the symbols in Fig.

While it is ultimately desirable to use the experimental confocal image data to perform an identical cluster analysis, the present experimental system sacrifices some particle tracking resolution index mismatch to enable both rheology and optical trapping for direct measurements of the particle interactions by laser tweezers This increases both the static and dynamic contributions to the variation in particle positions. The model data in Fig. The results show that Cauchy-Born theory yields a prediction of the elastic modulus with the correct convexity and in quantitative agreement with experimental measurements and calculations from simulations.

The graph decomposition, which is agnostic to the absolute spatial arrangement of the colloids, has revealed the underlying elastic structure of these heterogeneous gels.

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Similar graph theoretic approaches have proven useful for understanding the mechanical properties of other amorphous solids such as granular packings Because n c is an increasing function of the strength of inter-particle attraction, the internal volume fraction of the clusters is a decreasing function of the same. That is, the elastic subunits contain fewer particles with increasing strength of attraction.

The resultant volume fraction of clusters and their internal volume fraction are also plotted in Fig. The data in Fig. The internal cluster particle concentration. Nearly identical values are found when using the experimentally determined modulus. The square symbols reproduce the data of Ramakrishnan et al. The attractive and repulsive glass lines with a bilinear shift from the mode coupling theory MCT are drawn in solid red 35 , 36 while an extrapolation through the experimental data is indicated with the dashed red line.

## Colloids in Agrochemicals

Fluid-fluid coexistence is indicated by the solid black line and its extrapolation by the dashed black line. The local volume fraction of particle-rich regions shown as open circle-plus is compared with values from ref. It is important to note that glassiness is conventionally defined in terms of long or diverging relaxation times and not merely the density of a phase. We observe in both experiments and simulations that the gel as a whole is arrested. This highlights the importance of work that tracks the morphology and dynamics of individual clusters over longer time scales than studied here, in order to understand how clusters form, relax, and eventually arrest In Fig.

The results are remarkably consistent between depletion gels that differ substantially in size and chemistry, but that otherwise correspond to the same scaled range and strength of the interaction potential. Finally, without a graph-theoretic approach to identify gel clusters, we calculated the local packing fraction of dense particle strands in the network, following the approach of Lu et al.

The Cauchy-Born cluster model of gel rheology accurately describes the elastic modulus of depletion gels based on the bond rigidity, the cluster size, and the cluster density. As the attractive strength increases with depletant concentration, the gel modulus also increases. However, the higher modulus is due mainly to the lower density of particles within clusters, which coincides with the AGL extrapolated into the coexistence region of the phase diagram.

These results are supported by a long line of evidence that there is an intimate connection between the gelation of colloids with short-range attraction and phase separation. For particles with centro-symmetric forces, a thermodynamic driving force in the two-phase region drives aggregation, and locally arrested density fluctuations, clusters that are sparsely connected, give the network elastic properties.

This work raises a number of interesting possibilities for engineering the mechanical strength of colloidal gels, similar to controlling crystallite grain size in metals 41 , By imposing a flow or external field, the cluster size can be altered and clusters can even be made anisotropic. The Cauchy-Born theory has a more primitive tensorial form that can account for such anisotropies.

The l -balanced graph decomposition provides the necessary tool to detect anisotropic clusters from which these anisotropies derive. Finally, the graph theory methods introduced here should also be powerful tools for identifying and characterizing meso-scale structures from real-space position data that dictate the mechanics of other networked colloidal materials, like capillary suspensions 43 and particle networks at interfaces 44 , 45 , Experiments are performed using a recently-developed model colloidal gel that enables the simultaneous measurement of the structure, rheology, and particle interactions The depletion gels consist of poly methylmethacrylate latex PMMA particles dispersed in a mixture of cyclohexane and hexadecane.

With this model system, we measured the bulk shear modulus with a rotational rheometer, the microstructure using confocal microscopy, and the particle interactions with laser tweezers. The material properties of the two model systems are summarized in Table 1. An inverted confocal microscope Nikon A1Rsi equipped with a resonant scanner head and a high-speed piezo stage is used to image the colloidal gels.

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## Encyclopedia of Colloid and Interface Science | Tharwat Tadros | Springer

The top and bottom of the capillaries are glass coverslips suitable for microscopy 0. Confocal image volumes are analyzed using feature finding code based on the particle tracking algorithm of Crocker and Grier 47 , which uses a Gaussian mask to filter out digital noise and that identifies centroids based on their intensity maxima. The particle tracking is modified from the original two-dimensional algorithm to locate the center of a particle within a three-dimensional voxel with sub-voxel resolution.

Errors in centroid location are determined in the xy - and the xz -planes using immobilized samples. Particle position data are used to calculate several measures of the gel structure, including the particle contact number distribution and the number density fluctuations. Particles are considered to be nearest neighbors if they are closer than the distance that defines the first minimum in the radial distribution function, g r. The geometry minimizes confinement effects and a solvent trap is used to prevent the loss of the volatile solvent.

We employ a correction factor for the nonhomogeneous strain rate in the parallel plate geometry The waiting time is chosen to allow direct comparison between the gel microstructure and the rheological measurements on the gels. The average of three independent measurements of the gel rheology are made for each depletant polymer concentration.

Computer simulations are performed using a recently developed a method for rapid calculation of hydrodynamic interactions HI in suspensions of mono-disperse spheres The positively-split Ewald PSE algorithm makes the cost of computing Brownian displacements in simulations of colloidal scale particles with HI comparable to the cost of computing deterministic displacements in freely draining simulations.

The method relies on a new formulation for Ewald summation of the RPY tensor, which guarantees that the real-space and wave-space contributions to the tensor are independently symmetric and positive-definite for all possible particle configurations. Brownian displacements are drawn from a superposition of two independent samples: a wave-space far-field contribution, computed using techniques from fluctuating hydrodynamics and non-uniform fast Fourier transforms; and a real-space contribution, computed using a Krylov subspace method.

The combined computational complexity of drawing these two independent samples scales linearly with the number of particles enabling hydrodynamic simulations with system sizes up to 10 6 particles.

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The short-ranged depletion attraction is modeled with an Asakura-Oosawa form 4 :. As the samples are arrested, the S q , t curves are unchanging at long times. The data that support the findings of this study are available from the corresponding authors upon reasonable request. HOOMD-blue version 1.

Torquato, S. Grant, M. Volume-fraction dependence of elastic moduli and transition temperatures for colloidal silica gels. E 47 , — Verduin, H. Phase diagram of a model adhesive hard-sphere dispersion. Colloid Int. Asakura, S. Interaction between particles suspended in solutions of macromolecules. Vrij, A. Polymers at interfaces and the interactions in colloidal dispersions. Pure Appl. Gast, A. Polymer-induced phase separations in nonaqueous colloidal suspensions.

Lekkerkerker, H. Poon, W. Gelation in colloid-polymer mixtures. Faraday Discuss. Lu, P. Gelation of particles with short-range attraction. Nature , — Ramakrishnan, S. Elasticity and clustering in concentrated depletion gels.