Correlated Rigidity Percolation and Colloidal Gels

Shang Zhang, Leyou Zhang, Mehdi Bouzid, D. Zeb Rocklin, Emanuela Del Gado, and Xiaoming Mao

Phys. Rev. Lett. 123, 058001 – Published 30 July 2019

Abstract

Rigidity percolation (RP) occurs when mechanical stability emerges in disordered networks as constraints or components are added. Here we discuss RP with structural correlations, an effect ignored in classical theories albeit relevant to many liquid-to-amorphous-solid transitions, such as colloidal gelation, which are due to attractive interactions and aggregation. Using a lattice model, we show that structural correlations shift RP to lower volume fractions. Through molecular dynamics simulations, we show that increasing attraction in colloidal gelation increases structural correlation and thus lowers the RP transition, agreeing with experiments. Hence, the emergence of rigidity at colloidal gelation can be understood as a RP transition, but occurs at volume fractions far below values predicted by the classical RP, due to attractive interactions which induce structural correlation.

Received 23 July 2018

DOI:https://doi.org/10.1103/PhysRevLett.123.058001

Physics Subject Headings (PhySH)

Applications of soft matter Phase transitions

Colloidal gel

Statistical Physics & ThermodynamicsCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Shang Zhang ¹, Leyou Zhang¹, Mehdi Bouzid^2,3, D. Zeb Rocklin^1,4, Emanuela Del Gado², and Xiaoming Mao¹

¹Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA
²Department of Physics, Institute for Soft Matter Synthesis and Metrology, Georgetown University, Washington, D.C. 20057, USA
³LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
⁴School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA

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Vol. 123, Iss. 5 — 2 August 2019

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Images

Figure 1
Examples of rigid cluster decomposition of the correlated lattice model ( $ϕ_{l} = 0.6$ ) at different correlation strengths [ $c = 0$ in (a) and $c = 0.6$ in (b)], and the attractive gel model ( $ϕ_{g} = 0.6$ ) at $k_{B} T / ε = 0.4$ in (c) and 0.1 in (d). Red particles belong to the largest rigid cluster, and other particles are colored in gray. In both models, correlation or attraction induces rigidity at volume fractions below the rigidity transition in the uncorrelated or repulsive limit. The rigid clusters percolate in (b) and (d) where there is strong correlation or attraction, but not in (a) and (c). The inset in (a) shows an extreme example where particles are perfectly correlated (on a Warren truss) and exhibit rigidity at $ϕ = 0$ in thermodynamic limit.
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Figure 2
(a) $P (ϕ_{l}, c, L)$ at different $L$ and $c$ (symbols and line styles defined in legends). Inset: $ν$ for different $c$ (blue with error bars), in comparison with average (red line) and standard error (yellow dashed line) of $ν$ in the classical RP (from Ref. [11]). (b) $M (ϕ_{l}, c, L)$ at different $L$ and $c$ . Inset: $d_{f}$ for different $c$ (blue with error bars), in comparison with average (red line) and standard error (yellow dashed line) of $ν$ in the classical RP (from Ref. [11]). In both (a) and (b), curves for different $L$ cross at the same point (marked by red lines), indicating continuous transitions at every $c$ at different $ϕ_{l, c} (c, L = \infty)$ .
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Figure 3
Phase diagram of the correlated lattice model (without and with the strong correlation correction). Calculated phase boundary $ϕ_{l, c} (c, L = \infty)$ are shown as yellow dots (before correction: with black circles around the dots, after correction: without circles. They overlap at small $c$ ). The red dashed line and the black solid line show the phase boundaries before and after the correction by connecting the dots, respectively. The $c \to 0$ limit (classical RP) is shown as the black dashed line. The insets are configurations taken at $c = 0.9, ϕ_{l} = 0.5$ (the yellow star) with and without the strong correlation correction, which avoids the formation of disconnected dense blobs and leads to a percolating rigid cluster.
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Figure 4
(a) Phase diagram of the attractive gel model. Simulated parameters $(ϕ_{g}, k_{B} T / ε)$ are shown as squares colored according to their measured $P_{g} (ϕ_{g}, k_{B} T / ε)$ (color scale shown in legend). Black stars show fitted phase boundary at each $k_{B} T / ε$ and the black line is the phase boundary from fitting these transition points to third order polynomial. The hard sphere limit of the transition is shown as a black dashed line. (b) and (c) show two example configurations with their rigid cluster decomposition, chosen at the two marked points on the phase diagram. The largest rigid cluster percolates in (c) but not in (b), agreeing with the phase boundary.
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