Networks of Limited-Valency Patchy Particles

P. J. M. Swinkels, R. Sinaasappel, Z. Gong, S. Sacanna, W. V. Meyer, Francesco Sciortino, and P. Schall

Phys. Rev. Lett. 132, 078203 – Published 16 February 2024

Abstract

Equilibrium gels provide physically attractive counterparts of nonequilibrium gels, allowing statistical understanding and design of the equilibrium gel structure. Here, we assemble two-dimensional equilibrium gels from limited-valency “patchy” colloidal particles and follow their evolution at the particle scale to elucidate cluster-size distributions and free energies. By finely adjusting the patch attraction with critical Casimir forces, we let a mixture of two-valent and pseudo-three-valent patchy particles approach the percolated network state through a set of equilibrium states. Comparing this equilibrium route with a deep quench, we find that both routes approach the percolated state via the same equilibrium states, revealing that the network topology is uniquely set by the particle bond angles, independent of the formation history. The limited-valency system follows percolation theory remarkably well, approaching the percolation point with the expected universal exponents.

Received 6 June 2023
Revised 27 November 2023
Accepted 17 January 2024

DOI:https://doi.org/10.1103/PhysRevLett.132.078203

Physics Subject Headings (PhySH)

Fluctuation mediated interactions Self-assembly

Colloids Patchy particles

Optical microscopy

Polymers & Soft Matter

Authors & Affiliations

P. J. M. Swinkels and R. Sinaasappel

Institute of Physics, University of Amsterdam, 1098XH Amsterdam, The Netherlands

Z. Gong and S. Sacanna

Molecular Design Institute, Department of Chemistry, New York University, New York, NY 10003-6688, USA

W. V. Meyer

Universities Space Research Association, with GEARS, NASA Glenn Research Center, 2001 Aerospace Parkway, Brook Park, Ohio 44152, USA

Francesco Sciortino

Department of Physics, University of Rome La Sapienza, 00185 Rome, Italy

P. Schall

Institute of Physics, University of Amsterdam, 1098XH Amsterdam, The Netherlands

Article Text (Subscription Required)

Click to Expand

Supplemental Material (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 132, Iss. 7 — 16 February 2024

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Growth of colloidal network of divalent and pseudo-trivalent particles. (a) Optical microscope image of the network 9.75 h after quench to attractive strength $> 15 k_{B} T$ . Insets show reconstructions of three structural motifs: linear chains of dipatch particles (I), kinked chains (II), and branching point joining three chains (III), linked by a tetrapatch particle. A branching point with four chains is not observed. (b)–(d) Reconstructions showing the evolution of the particle network. Each dot represents a particle, and color indicates the number of bonded neighbors $Z$ (see the legend). With time, a plane-spanning network emerges, where central trivalent hubs connect linear chains. (e)–(g) Radius of gyration $r_{g}$ as a function of cluster size $s$ for the snapshots in (b)–(d) showing the transition from linear chains with $r_{g} \propto s^{1}$ (dotted line) to a branched network with $r_{g} \propto s^{1 / 1.8}$ (dashed line), indicating a transition from fractal dimension $d_{f} = 1$ to fractal dimension $d_{f} = 1.8$ .
Reuse & Permissions
Figure 2
Cluster evolution: quench and near-equilibrium routes. (a),(b) Cluster and bond evolution for the quench (a) and near-equilibrium route (b). Triangles indicate the second moment of the cluster-size distribution $C$ (left $y$ axis), and hexagons indicate the average number of bonds per particle $⟨ Z ⟩$ (right $y$ axis). In the near-equilibrium case (b), the temperature increments are indicated by vertical dashed lines and color grading. (c) Scaling of $C$ with distance to the predicted percolation threshold ( $Z_{c}$ ) for quench (blue dots) and near-equilibrium route (red squares). Lines indicate powers of 2.1 (dashed) and 1 (dotted). Inset: enlarged section enlarging the wobble around $Z_{c} - ⟨ Z ⟩ \sim 0.6$ . (d) Number of loops per particle as a function of $⟨ Z ⟩$ for quench (blue dots) and near-equilibrium route (red squares). Inset: distributions of loop sizes $o$ for increasing $⟨ Z ⟩$ , $⟨ Z ⟩ = 0.17$ (blue), 0.58 (yellow), and 1.13 (red) for quench (circles) and near-equilibrium route (squares). Loops larger than five particles exist only at later stages, i.e., for sufficiently large $⟨ Z ⟩$ ; see [26] for details.
Reuse & Permissions
Figure 3
Cluster-size distribution and free energy. (a) Cluster-size distributions for bond probabilities $⟨ Z ⟩ = 0.17$ (blue), 0.58 (yellow), and 1.13 (red) for the quench route (squares) and the near-equilibrium route (circles). The black dotted line indicates a power law with exponent $τ = 2.16$ . The black solid line shows the cluster-size distribution predicted by Flory-Stockmayer theory at $⟨ Z ⟩ = 1.13$ ; see Supplemental Material [26]. The inset shows the probability of free particles in the equilibrium case (quench omitted for clarity) with a linear $y$ axis. (b) Helmholtz free energy per particle as a function of average number of bonds as determined from cluster-size distributions such as those in (a), for all stages of the experiment. The quench route (blue circles) and near-equilibrium route (red squares) as well as predictions from Flory-Stockmayer (solid black line; see Supplemental Material [26]) are all in good agreement. The free energy is determined using Eq. (1) and is solely dependent on the bonding state of the system.
Reuse & Permissions

Physical Review Letters