On the formation of equilibrium gels via a macroscopic bond limitation

Abstract

Restricting the number of attractive physical "bonds" that can form between particles in a fluid suppresses the usual demixing phase transition to very low particle concentrations, allowing for the formation of open, percolated, and homogeneous states, aptly called equilibrium or "empty" gels. Most demonstrations of this concept have directly limited the microscopic particle valence via anisotropic (patchy) attractions; however, an alternative macroscopic valence limitation would be desirable for greater experimental tunability and responsiveness. One possibility, explored in this paper, is to employ primary particles with attractions mediated via a secondary species of linking particles. In such a system, the linker-to-primary particle ratio serves as a macroscopic control parameter for the average microscopic valence. We show that the phase behavior of such a system predicted by Wertheim's first order perturbation theory is consistent with equilibrium gel formation: the primary particle concentrations corresponding to the two-phase demixing transition are significantly suppressed at both low and high linker-to-primary particle ratios. Extensive molecular dynamics simulations validate these theoretical predictions but also reveal the presence of loops of bonded particles, which are neglected in the theory. Such loops cause densification and inhibit percolation, and hence the range of viable empty gel state conditions is somewhat reduced relative to the Wertheim theory predictions.

Structure and phase behavior of polymer-linked colloidal gels

“…In the thermodynamic limit, S cc (0) should diverge at the spinodal, but in practice, it remains finite in the simulation and S cc (0) > 10 is usually considered to be phase separated. 13,26 We qualitatively confirmed that this criterion held for our model by visually inspecting configurations that it identified as phase separated, finding that they had clear density variations. We compared the measured structure factor to the TPT predictions of phase coexistence for two chain lengths, M = 2 and M = 8, with Γ = 1.5 (Fig.…”

“…In our previous work, 26 we showed how Γ controls the onset of percolation and also the range of colloid volume fraction over which phase separation is expected for colloids joined by spherical linkers. These mixtures exhibited reentrant phase behavior as a function of Γ, as the spinodal initially broadened after percolation with a maximum width at the stoichiometric ratio of linkers with the 6-patch colloids (Γ = 3), before shrinking and eventually vanishing at higher Γ.…”

“…Its predictions are often in close qualitative and quantitative agreement with exact simulations of patchy particle models. 10,12,[14][15][16][17]26 Formally, TPT is derived from a graph-theoretic expansion and a partial re-summation centered around a few key physical assumptions: [51][52][53][54] (1) each patch on a particle can only bond with one other patch, (2) bond formation is statistically uncorrelated between patches on a particle, and (3) bonding occurs in a tree-like structure (i.e., closed loops are not permitted). As a result of these assumptions, TPT has a relatively simple closed-form expression for the free energy, rendering thermodynamic stability and equilibrium phase coexistence calculations computationally inexpensive.…”

“…25 This challenge motivated some of us to study an alternative route to equilibrium gelation that controls the extent of bonding between the (primary) colloids using a (secondary) linking agent. 26 Many examples of such linking systems have been described, including colloid-polymer mixtures, [27][28][29][30][31] mixtures of oppositely charged colloids, 32,33 ion-linked functionalized nanocrystals, [34][35][36][37] DNA-linked nanoparticles, 38,39 and DNA nanostar mixtures. [40][41][42] The degree of bonding between the colloids can be restricted by appropriately tuning the amount of linker added.…”

Binding branched and linear DNA structures: From isolated clusters to fully bonded gels

“…Different from cases in which percolation results as an effect of an irreversible polymerization process (and as such controlled by curing time), here we tune the ratio r between B and A particles to approach the percolation point. At low T , when the lifetime of the bonds is larger than the experimental time, the system is expected to behave as a chemical gel at percolation 45,48,49 . The corresponding DLS results are presented in Fig.…”