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Dynamics of equilibrium linked colloidal gels

Colloids that attractively bond to only a few neighbors (e.g., patchy particles) can form equilibrium gels with distinctive dynamic properties that are stable in time. Here, we use a coarse-grained model to explore the dynamics of linked networks of patchy colloids whose average valence is macroscopically, rather than microscopically, constrained. Simulation results for the model show dynamic hallmarks of equilibrium gel formation and establish that the colloid-colloid bond persistence time controls the characteristic slow relaxation of the self-intermediate scattering function. The model features re-entrant network formation without phase separation as a function of linker concentration, centered at the stoichiometric ratio of linker ends to nanoparticle surface bonding sites. Departures from stoichiometry result in linker-starved or site-starved networks with reduced connectivity and shorter characteristic relaxation times with lower activation energies. Underlying the re-entrant trends, dynamic properties vary monotonically with the number of effective network bonds per colloid, a quantity that can be predicted using Wertheim's thermodynamic perturbation theory. These behaviors suggest macroscopic in situ strategies for tuning the dynamical response of colloidal networks.

Here, we use a coarse-grained model to explore the dynamics of linked networks of patchy colloids whose average valence is macroscopically, rather than microscop- Here, we use simulation to characterize the dynamics of the coarse-grained PolyPatch (Polymer linked, Patchy colloid) model, 16,18 in which the ends of ditopic molecules link neighboring nanoparticles into a network by attractively bonding to patches on their surfaces. Our simulations find dynamic signatures of a re-entrant equilibrium gel as a function of linker concentration, centered at stoichiometric conditions where the number of linker ends matches the number of nanoparticle surface bonding sites. The characteristic slow relaxation time of the self-intermediate scattering function scales with the colloid-colloid bond persistence time of the networks. As predicted from Wertheim theory, 19 deviating from stoichiometry further restricts nanoparticle valence and the number of links per colloid that extend the colloidal network, producing linker-starved or site-starved networks. These structural changes correlate with reduced relaxation time scales and activation energies. Together, the results of this study help clarify how linked colloidal gels relate to patchy particle networks and suggest strategies for tuning the former's properties via macroscopic control parameters, some of which may enable real-time, in situ dynamic control.

A. Model
We adopt the PolyPatch model, a coarse-grained representation of dispersed colloids with surfaces decorated by six patchy sites that can attractively bond to the end beads of bifunctional linker molecules. 16,18,19,23 The linkers are treated as flexible, linear chains of eight beads with diameter σ and mass m. Each colloid is modeled as a spherical particle with diameter σ c = 5σ and mass m c = 125m; its six interaction sites of diameter σ are rigidly fixed in an octahedral arrangement a distance r ≈ 3.12σ from the colloid center.
All particles in the PolyPatch model interact with the hard-sphere-like repulsions of a core-shifted Weeks-Chandler-Andersen potential, 35 i.e., where β = (k B T ) −1 , k B is the Boltzmann constant, r is the center-to-center separation of particles of types i and j, and δ ij = (σ i + σ j )/2 − σ modifies the divergence to reflect the respective diameters, σ i and σ j . This repulsive interaction is truncated at σ * ij = 2 1/6 σ + δ ij . Two chemically bonded monomers of a linker chain also interact by a finitely extensible nonlinear elastic potential, 36 with βk s = 30 σ −2 and r s = 1.5 σ. 37 Attractions between colloidal interaction sites and linker ends can be expressed where ε b is the bond strength.  49 An integration time step of 0.001 τ was chosen, where τ = βmσ 2 is a characteristic time, and temperature was maintained using a Langevin thermostat with friction coefficient 0.1 m/τ applied to each linker bead and colloid.
To study how linker concentration impacts static and dynamic properties, the simulations included N c = 1000 colloids and a range of linker-to-colloid ratios Γ ≡ N l /N c from 1 to 6.
It's helpful to recast the latter as the ratio of linker ends to colloid sites Γ * = Γ/3, which varies from 1/3 to 2. This range encompasses behavior of linker-starved and site-starved state compositions, i.e., conditions below and above the stoichiometric ratio of Γ * = 1, respectively. The dimensions of the periodically replicated cubic simulation cell of volume V were chosen to investigate colloid packing fractions η c = N c πσ 3 c /(6V ) from 0.01 to 0.15. For each choice of η c and Γ * , configurations were thermalized for 0.1 × 10 5 τ in the absence of linker-site attractions (βε b = 0). Next, ε b was increased (at fixed temperature) for 0.1 × 10 5 τ using a linear ramp until ε b = 10. After repeating the thermalization protocol at ε b = 10, ε b was again increased using a linear ramp of 0.8 × 10 −3 ε b /τ . At a given βε b , molecular dynamics simulations were run until the potential energy converged; the required run time varied in the range 1.5 × 10 3 τ -0.1 × 10 7 τ depending on βε b and Γ * . Finally, production simulations were performed to characterize the microstructure and the dynamics, with typical run times in the range 0.2 × 10 5 τ -0.2 × 10 7 τ for the state points considered.
The behavior of the static structure factor at near-zero wavevector was used to estimate state points where macroscopic phase separation occurs. The partial static structure factor of the colloids was computed as 50 Here k = (2π/V 1/3 )n is a wavevector in the cubic box of volume V , n is a vector of integers, and r j is the position of the jth colloid. The angle brackets indicate an ensemble average, which was calculated as a temporal average. Specifically, we averaged S cc (k) for the 22 smallest nonzero wavevector magnitudes k = |k| and extrapolated to k = 0 by fitting the data to a Lorentzian function, 51 where ξ cc is a correlation length. As in previous work, we used S cc (0) > 10 as an approximate indicator of phase separation. 15,16,52 To characterize the colloidal dynamics, we calculated the mean squared displacement of colloidal particles (∆r) 2 (t) ) as follows, where r j (t) is the position vector of the jth colloid at time t. We also computed the colloidal self-intermediate scattering function: For sufficiently strong bonding interaction βε b , F s (k * , t) decayed via a two-step relaxation, where k * is the position of the static structure factor's primary peak. To estimate a characteristic structural relaxation time τ s and the nonergodicity factor A s , we fit the slow relaxation of the self-intermediate scattering to the stretched exponential function, We computed two characteristic times for the bond dynamics: continuous bond lifetime τ b and bond persistence time τ c . [53][54][55][56][57] Here, a bond refers to a linker-mediated connection between two colloids. Continuous bond lifetime was obtained from analyzing the bond survival probability S b (t), which can be expressed as Here, i and j are colloidal indices; s ij (t 0 , t) = 1 if colloids i and j are continuously bonded in the time interval [t 0 , t 0 + t], and s ij (t 0 , t) = 0 otherwise. The bond survival probability data was fit with an exponential function: S b (t) = exp[−(t/τ b )] to obtain an estimate for the bond lifetime τ b (Fig. S2). The more permissive bond persistence time was obtained from the bond correlation function C b (t): Here, h ij (t) = 1 if colloids i and j are bonded at given time t, and h i,j (t) = 0 if the colloids are not bonded at that time (i.e., intermittent bond breaking is allowed). The bond correlation function was fit by a stretched exponential function, Wertheim's perturbation theory decomposes the Helmholtz free energy density, a, into a sum of a bonding portion a bond and a reference contribution a 0 . The former accounts for directional attractions between particles with interaction sites (i.e., linkers or colloids), while the latter accounts for the free energy of the system in the absence of such attractions.
The bonding free energy density is expressed in terms of a series of variables, X Likewise, the fraction of dangling linkers (i.e., with either one, but precisely one, of its ends bonded) is χ dangling = 2(X (l) . We denote the fraction of linkers with both ends bonded The fraction of linkers forming loops χ loop and redundant bonds χ redundant may be solved using relations formed in the derivation of TPT1+ 19 where ν c is the number of interaction site pairs on a colloid capable of participating in a loop or ring, ρ i is the number density of component i, and ∆ 2 and ∆ 4 are bond volumes introduced to account for loop and ring structures. 18 The fraction of effective bonds χ effective can be found with the relation These fractions can be transformed from a per linker basis to a per colloid basis by multiplying by Γ. One quantity of particular interest is n eb = χ effective Γ, the number of effective bonds per colloid, which is related to a colloid's average coordination number Results in all panels are for βε b = 18 and η c = 0. 10. system passes through the stochiometric ratio, where the colloids have the highest average coordination number (Fig. 2b). The peak positions of the distributions closely track their average values CN 1 = 2 n eb , further highlighting that n eb captures key aspects of network connectivity.
To ascertain how much the bonding state of a colloid influences that of its neighbors, we computed CN 2 , the average coordination number of colloids linked to a central colloid with CN 1 (Fig. 2c). To a first approximation, CN 2 ≈ CN 1 , again following the re-entrant trends of n eb in Fig. 2a as Γ * is increased. For each Γ * , there is a weak, positive correlation between CN 2 and CN 1 ; neighbors of a more highly coordinated particle are themselves bonded to slightly more neighbors than the neighbors of low-coordinated colloids. This positive correlation, reflecting network cooperativity, is strongest at the stoichiometric ratio and flattens for Γ * in linker-starved or site-starved regions.

B. Colloidal dynamics
To study how linker concentration influences colloidal relaxation processes in the Poly-Patch model, we computed the mean squared displacement (∆r) 2 and the self-intermediate scattering function F s (k * , t) of the particles, where k * σ = 1 corresponds to the approximate position of the colloidal structure factor's first peak (Fig. 3). shows pronounced increases as Γ * approaches one and the linked-colloidal network becomes more extensive. However, as even more linker is added, creating site-starved conditions, the viscoelastic characteristics of the single-particle dynamics attenuate, and the relaxation processes again approach simple fluid-like behavior.
Linker concentration and strength of attraction between linker ends and colloidal surface sites play distinctive roles in tuning dynamics of colloidal networks (Fig. 4). At fixed βε b , changing Γ * from 0.67 to 1 increases both A s and the characteristic slow relaxation time τ s by over an order of magnitude, dynamic effects that are reversed upon increasing Γ * further to 1.67 ( Fig. 4a and 4b). Underlying this re-entrant behavior is a monotonic change in A s  Fig. 2b and 2c), tuning Γ * toward the stoichiometric ratio also produces coordination-number sensitive relaxation times ( Fig. 4c). Similar coordination-number dependent single-particle dynamics were reported for simulated tetrahedral network forming patchy particles. 63 In both the PolyPatch model at Γ * = 1 and the tetrahedral patchy particle fluid, 63 bond breaking is essential for colloidal translation (Fig. S4a). However, in linked-colloidal systems, modifying Γ * to linker-starved or site-starved conditions with fewer effective bonds per colloid substantially weakens the link between dynamics and coordination number (Fig. 4c), and at Γ * = 1.67 there are no detectable differences in relaxation times of particles with different CN 1 . The reduced connectivity of the colloidal network under these conditions renders it less constrained, and significant colloidal displacements can occur without even rupturing bonds with the neighbors (Fig. S4b).

C. Bond dynamics
A comparison of continuous bond lifetime τ b (i.e., time for a colloid-colloid bond to break), and bond persistence time τ c (i.e., time that a colloid-colloid bond lasts allowing intermittent breaks) for the PolyPatch model provides insight about the microscopic processes underlying differences in structural dynamics as Γ * varies. Both times are found to exhibit Arrhenius temperature dependencies for the full range of conditions examined in this study ( Fig. 5a and   5b). However, the former is found to depend only on βε b and not on Γ * , which determines the connectivity of the network and the constraints it imposes on the colloids. By contrast, the bond persistence time τ c shows a Γ * -dependent activation energy, which is larger at the stoichiometric ratio than under linker-starved or site-starved conditions. This makes  physical sense; at a given βε b , looser, less interconnected networks allow colloids to diffuse away from a bonding partner after a linker end detaches from a colloidal site, while more geometrically constrained networks tend to promote bond reformation between the same pair. A direct comparison of data in Fig. 4b and 5b shows τ c ∝ τ s (Fig. 5c)

DATA AVAILABILITY
The data that support the findings of this study are available from the authors upon reasonable request.