Reversible Aggregation and Colloidal Cluster Morphology: The Importance of the Extended Law of Corresponding States

Néstor E. Valadez-Pérez, Yun Liu, and Ramón Castañeda-Priego

Phys. Rev. Lett. 120, 248004 – Published 15 June 2018

Abstract

Cluster morphology of spherical particles interacting with a short-range attraction has been extensively studied due to its relevance to many applications, such as the large-scale structure in amorphous materials, phase separation, protein aggregation, and organelle formation in cells. Although it was widely accepted that the range of the attraction solely controls the fractal dimension of clusters, recent experimental results challenged this concept by also showing the importance of the strength of attraction. Using Monte Carlo simulations, we conclusively demonstrate that it is possible to reduce the dependence of the cluster morphology to a single variable, namely, the reduced second virial coefficient, $B_{2}^{*}$ , linking the local properties of colloidal systems to the extended law of corresponding states. Furthermore, the cluster size distribution exhibits two well-defined regimes: one identified for small clusters, whose fractal dimension, $d_{f}$ , does not depend on the details of the attraction, i.e., small clusters have the same $d_{f}$ , and another related to large clusters, whose morphology depends exclusively on $B_{2}^{*}$ , i.e., $d_{f}$ of large aggregates follows a master curve, which is only a function of $B_{2}^{*}$ . This physical scenario is confirmed with the reanalysis of experimental results on colloidal-polymer mixtures.

Received 27 July 2017
Revised 14 December 2017

DOI:https://doi.org/10.1103/PhysRevLett.120.248004

Physics Subject Headings (PhySH)

Aggregation Clustering

Polymers & Soft Matter

Authors & Affiliations

Néstor E. Valadez-Pérez^1,2,3, Yun Liu^3,4, and Ramón Castañeda-Priego^1,3,*

¹División de Ciencias e Ingenierías, Campus León, Universidad de Guanajuato, Loma del Bosque 103, 37150 León, Guanajuato, Mexico
²Department of Chemical and Biological Engineering, Illinois Institute of Technology, 3440 S. Dearborn Street, Chicago, Illinois 60616, USA
³NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA
⁴Department of Chemical and Biomolecular Engineering, University of Delaware, Newark, Delaware 19716, USA

^*ramoncp@fisica.ugto.mx

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Vol. 120, Iss. 24 — 15 June 2018

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Images

Figure 1
(a) Phase diagram, $B_{2}^{*}$ vs $ϕ$ , for colloids interacting through a SW potential with $λ = 1.1$ . Solid line represents the fluid-crystal coexistence boundary obtained from Eq. (4) of Ref. [27], diamonds describe the binodal curve calculated with MC simulations [24] and the star is the critical point, inverted triangles indicate the percolation boundary [28]; the line is a guide for the eye. Vertical dotted line indicates the packing fraction (isochoric line) at which the cluster morphology is studied; the open triangle indicates a thermodynamic state below the binodal. Radius of gyration, $R_{g}$ , for clusters made of $s$ particles for SW systems at $ϕ = 0.08$ with (b) $λ = 1.1$ at different $B_{2}^{*}$ values [symbols in (a)] and (c) with variable $λ$ and $B_{2}^{*} = - 1.5$ . Discontinuous lines fit the data to the expression $R_{g} = A s^{1 / d_{f}}$ ; the fits are performed in the intervals $s < 10$ and $10 \leq s \leq 100$ . Vertical arrows indicate the crossover point, located at $s = 10$ , of both fits, which naturally establishes the separation between small and large clusters.
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Figure 2
(a) Phase diagram of colloid-polymer mixtures for small attraction ranges, $ξ = 2 R / σ$ , with $R$ being the polymer radius of gyration. $c_{p}$ and $c_{p}^{*}$ are the polymer and overlap concentrations, respectively. Note that the scale has been inverted to directly compare with Fig. 1. Empty symbols represent a fluid phase, solid symbols the phase separation or gel states, and half solid-empty symbols are the fluid-crystal coexistence. Data correspond to the following experimental systems: $ξ = 0.08$ ( $⊳$ ) [29], $ξ = 0.11$ ( $⊲$ ) [19], $ξ = 0.09$ ( $△$ ) [30], $ξ = 0.078$ ( $⬠$ ) [31], $ξ = 0.026$ ( $▽$ ) [32], $ξ = 0.08$ ( $□$ ) [2], $ξ = 0.02$ , 0.04, 0.15 ( $⬡$ ) [17], and $ξ = 0.09$ ( $⋄$ ) [33]. Colored regions correspond to the gas-liquid separation from Ref. [34] and gelation from Ref. [35] for an Asakura-Oosawa system with $ξ = 0.1$ . Radius of gyration, $R_{g}$ , for clusters made of $s$ particles for experimental systems at (b) different $c_{p} / c_{p}^{*}$ and same $ξ$ taken from Ref. [19] and (c) similar $c_{p} / c_{p}^{*}$ and different $ξ$ taken from Refs. [19] ( $⊲$ ) and [17] ( $⬡$ ). Dashed lines correspond to $R_{g} \propto s^{1 / 1.65}$ ; the fit for small clusters in simulations. For large clusters, we plot the corresponding $R_{g}$ consistent with the $d_{f}$ reported in Ref. [19] and the colored region in (c) denotes the relation $R_{g} \propto s^{1 / 2.0}$ .
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Figure 3
Radius of gyration, $R_{g}$ , for (a) small and (b) large clusters made of $s$ particles interacting with a SW potential for several values of $B_{2}^{*}$ at the same $ϕ = 0.08$ and $λ = 1.1$ . In (a) the $R_{g}$ for clusters in the ground state from Ref. [39] is also shown. (c) Fractal dimension for small and large clusters as a function of $(1 - B_{2}^{*})$ for the SW fluid of variable range, $λ$ , at $ϕ = 0.08$ ; dashed line is a guide for the eye. Colored regions represent the boundary of the gas-liquid and the fluid-crystal phase coexistence displayed in Fig. 1.
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Figure 4
Radius of gyration, $R_{g}$ , for (a) small and (b) large clusters made of $s$ particles in a colloid-polymer mixture, data taken from Refs. [19] (solid symbols) and [17] (empty symbols). Panel (a) shows the fit obtained from panel Fig. 3. In panel (b), solid lines correspond to $d_{f}$ reported in Ref. [19]. (c) Fractal dimension of clusters with less than 100 particles taken from Refs. [17] ( $⬡$ ) and [19] ( $⊲$ ) as a function of $c_{p} / c_{p}^{*}$ . $⋄$ indicate the $d_{f}$ of small clusters from Ref. [19] and $*$ is the fit for $s \leq 100$ for the data in Ref. [17]. Colored region represents the boundary of the gas-liquid phase separation estimated from Fig. 2 and lines are guides for the eye.
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