Strain Hardening of Fractal Colloidal Gels

Abstract

We report on experiments on the rheology of gels formed by diffusion-limited aggregation of neutrally buoyant colloidal particles. These gels form very weak solids, with the elastic modulus, G 0 ͑v͒, larger than the loss modulus, G 00 ͑v͒, and with both G 0 ͑v͒ and G 00 ͑v͒ exhibiting only a very weak frequency dependence. Upon small but finite strains g , 0.45 the elastic modulus increases roughly exponentially with g 2 . We explain the observed strain hardening with the highly nonlinear elastic response of the rigid backbone of the gel to elongational deformation. [S0031-9007 (98) Colloidal particles aggregating by attractive interactions form highly disordered clusters; these structures are, on average, self-similar, and the mass of a cluster, M, scales with its radius, R, aswhere a is the size of the colloidal monomer and M 0 is its mass [1]. The fractal mass exponent d f characterizes the ramification of the cluster and varies between d f 2.1 for reactionlimited cluster aggregation and d f 1.8 for diffusionlimited cluster aggregation (DLCA) [2]. In the latter case the distribution of cluster sizes is fairly narrow and the characteristic cluster size grows linearly with time. Aggregation of clusters eventually leads to a spacefilling structure which is no longer fractal on all length scales but which can instead show long-range correlations as revealed by a scattering peak corresponding to a characteristic cluster size R c aw 1͑͞d f 23͒ , where w is the initial volume fraction of monomer particles [3,4]. The clusters themselves are close packed, forming a ramified, tenuous gel structure. This structure should be an elastic material with unique properties, which are determined not only by d f , but also by the connectivity or chemical dimension, d b , which characterizes the scaling of the contour length within the cluster.

The role of structure in the nonlinear mechanics of cross-linked semiflexible polymer networks

“…13,22,37,38 Our results on the role of network structure on γ c show ( Fig. 6(a)) that γ c ∼ R −0.6 cl (which, interestingly, is reminiscent of a relation found previously for F-actin, 24,39 but in terms of R = c cl /c m ).…”

Time-Dependent Strength of Colloidal Gels

“…1 (inset) data obtained on the ISS, for different times after aggregation was initiated. The fractal dimen- , consistent with aggregation that may begin slightly in the reaction-limited regime, but crosses rapidly into diffusion-limited cluster aggregation (DLCA) [14,15]. Roughly 12 h after initialization, the SLS stops evolving in time.…”

“…3 (inset) is remarkably similar to that expected for fractal gels, where the macroscopic modulus is governed by the stiffness of the aggregate network. The elasticity of a fractal aggregate of size R c is determined by the spring constant R c 0 a 2 =NR 2 c of its backbone, where 0 is the interparticle spring constant and N is the number of particles in the chain [15,18]. This expression can be understood in an intuitive way, if one assumes that the primary contributions to elasticity are from bond bending.…”

Micromechanical model for isolated polymer-colloid clusters under tension

“…A detailed description on the geometrical parameters introduced here can be found in Refs. [4,15,51,53,54]. The length of a backbone chain, at any stage of deformation, can be then…”

“…Several models have been developed to calculate the energy of the backbone chain [14,15] by representing the chain by a set of thin elastic rods [16] or using the concept of nodes-links-blobs chains [17,18]. Most recent works on the behavior of PC clusters under deformation are numerical studies based on finite-element analysis of accurate substructures [19][20][21][22][23].…”