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Dynamics of Patchy Particles in and Out of Equilibrium José M. Tavares, Cristóvão S. Dias, Nuno A. M. Araújo, and Margarida M. Telo da Gama J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.7b10726 • Publication Date (Web): 18 Dec 2017 Downloaded from http://pubs.acs.org on December 21, 2017

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The Journal of Physical Chemistry

Dynamics of Patchy Particles In and Out of Equilibrium J. M. Tavares,∗,†,‡ C. S. Dias,∗,†,¶ N. A. M. Araújo,∗,†,¶ and M. M. Telo da Gama∗,†,¶ †Centro de Física Teórica e Computacional, Universidade de Lisboa, 1749-016 Lisboa, Portugal ‡Instituto Superior de Engenharia de Lisboa, ISEL, Avenida Conselheiro Emídio Navarro, 1 1950-062 Lisboa, Portugal ¶Departamento de Física, Faculdade de Ciências, Universidade de Lisboa, 1749-016 Lisboa, Portugal E-mail: [email protected]; [email protected]; [email protected]; [email protected]

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Abstract We combine particle-based simulations, mean-field rate equations, and Wertheim’s theory to study the dynamics of patchy particles in and out of equilibrium, at different temperatures and densities. We consider an initial random distribution of nonoverlapping three-patch particles, with no bonds, and analyze the time evolution of the breaking and bonding rates of a single bond. We find that the asymptotic (equilibrium) dynamics differs from the initial (out of equilibrium) one. These differences are expected to depend on the initial conditions, temperature, and density.

Introduction The possibility of synthesizing novel materials with enhanced physical properties from the spontaneous self-assembly of colloidal particles is among the most popular challenges of Soft Condensed Matter. 1–6 Functionalized colloidal particles with patches on their surfaces (patchy particles) are promising candidates for the individual constituents, as they allow control of both the valence (number of neighboring particles) and the local structure. 7–17 Equilibrium studies of patchy particle systems revealed rich phase diagrams depending on temperature and density, and in novel ways on the number and type of patches. 12,18,19 However, the potential application of these predictions may be compromised, as the feasibility of assembling the new phases is seriously hampered by the kinetic barriers that emerge from the complex particle-particle correlations. 20–30 Thus, understanding the dynamics towards thermodynamically stable (equilibrium) structures is vital to tackle this challenge. Previous studies attempted to relate the out of equilibrium dynamics to the dynamics at equilibrium. 31–33 The idea is to describe the dynamics as a balance between breaking and forming individual bonds, where the rates for each process are considered time invariant. Since the overall bond probability should converge to that at equilibrium, it is possible to estimate the effective rates from a fit to the time evolution of the bond probability. These studies were restricted to systems where the structures are almost loopless and characterized 2


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by very long chains. In this work, we consider systems of three-patch particles, where branching and loops are significant. We combine particle-based (Langevin) simulations, mean-field rate equations, and Wertheim’s theory to study the dynamics in and out of equilibrium. We show that, while the effective rates obtained from the time evolution of the bond probability describe the out of equilibrium dynamics accurately, the equilibrium dynamics is significantly different and characterized by higher (time invariant) rates. The manuscript is organized in the following way. In the next section we present the model of patchy particles and describe the simulations. In Sec. , we introduce the mean-field approach and compare the equilibrium and out of equilibrium regimes combining Wertheim’s theory and particle-based simulations. Finally, we draw some conclusions in Sec. .

Model and simulations Following previous works, 28,34,35 we model the patchy particles as three-dimensional spheres, with three patches equally distributed on their surfaces. The core-core interaction is described by a repulsive (exponential) potential, VY (r) =

A k

exp (−k [r − σ]), where σ is the

effective diameter of the interacting particles, A/k = 0.25kB T is the interaction strength and k/σ = 0.4 is the inverse screening length. The patch-patch interaction is described by an attractive pairwise potential VG (rp ) = − exp [−(rp /ξ)2 ], where is the interaction strength, ξ = 0.1σ is the size of the patch, and rp the distance between patches. 36 To resolve the individual trajectories of the particles, we perform particle-based simulations using the velocity Verlet scheme of the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). 37 We integrate the Langevin equations of motion for the translational and rotational degrees of freedom, respectively, m m~v˙ (t) = −∇~r V (~r) − ~v (t) + τt

3

r

2mkB T ~ ft (t), τt


(1)


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and ~ − Iω Iω ~˙ (t) = −∇θ~ V (θ) ~ (t) + τr

r

2IkB T ~ fr (t), τr

(2)

where, ~v and ω ~ are the translational and angular velocities, m and I are the mass and moment of inertia of each patchy particle, V (· · · ) is the pairwise potential, τt and τr are the translational and rotational damping times, and f~t (t) and f~r (t) are stochastic terms taken from a random distribution of zero mean. The damping time for the translational motion is τt = m/(6πηR), and the damping time for the rotational motion is τr = I/(8πηR3 ), in line with the Stokes-Einstein-Debye relation for spheres , 38 as obtained from the relation between the translation and rotational diffusion coefficients and the pre-factors of the stochastic terms in Eqs. 1 and 2. Simulations are performed for a three-dimensional system of linear size L = 16, in units of the particle diameter σ, averaged over 20 samples, starting from a random (uniform) distribution of non-overlapping particles. The density ρ is the number of particles per unit 1 1 1 1 1 , 32 , 16 , 8 , 4 } in units of 1/σ 3 , volume. We consider five different densities, namely, ρ = { 64

and four different temperatures: T = {0.075, 0.1, 0.125, 0.15} in units of

. kB

Results and Discussion The dynamics evolves through a sequence of bonding and breaking events. The simplest approach to describe this dynamics is through a (mean-field) rate equation for the bonding probability pb (t), which corresponds to the fraction of bonded patches in the infinite size limit. Accordingly, p˙b (t) = −kbr pb (t) + ρf kbo [1 − pb (t)]2 ,

(3)

where p˙b (t) stands for the time derivative, t for time in units of the Brownian time, f for the valence, ρ for the number density, and kbr , kbo for the breaking and bonding rates of a single bond, respectively. As discussed in Refs., 33,39 Eq. (3) can be derived from a generalized

4


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Figure 1: Time dependence of the bonding probability, pb (t), obtained from the particle1 1 1 based simulations (symbols) at different densities, namely, 64 (stars), 32 (triangles down), 16 1 1 (triangles up), 8 (diamonds), 4 (circles). Different plots are for different temperatures: (a) 0.075; (b) 0.1; (c) 0.125; (d) 0.15. The solid lines are fits to Eq. (4) (details in the text).

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Smoluchowski set of equations, accounting for both aggregation and fragmentation of loopless clusters of particles of functionality f (i.e. f patches), under the assumptions that all patches are identical, the diffusion coefficient is the same for all clusters, and all patches are not bonded at t = 0. If, for simplicity, we consider that kbr and kbo are time invariant and depend on temperature and density only then,

pb (t) = p∞

1 − exp(−Γt) , 1 − p2∞ exp(−Γt)

(4)

where p∞ = limt→∞ pb (t). In the stationary state, p˙b (t) = 0, and the net rates of bonding and breaking are necessarily the same. Thus,

Γ = kbr

1 + p∞ 1 − p2∞ = f ρkbo . 1 − p∞ p∞

(5)

Clearly, p∞ depends only on the ratio kbr /kbo and is simply the bond probability under equilibrium conditions, as discussed below. The value of Γ (and thus the individual values of kbr and kbo ) may be estimated from a fit of Eq. (4) to numerical data for pb (t) obtained from particle-based simulations. Wertheim’s theory is the most successful thermodynamic perturbation theory for associating fluids with short-ranged, anisotropic interactions (see e.g. 40 ). Within this equilibrium theory, the bonding free energy, treated as a perturbation over that of a reference fluid, is found to be a function of the equilibrium bonding probability, pb,eq , which in turn depends on the number density, the functionality f , the interaction potential between the patches VG , and the reference fluid, and it is the solution of

f ρ∆ =

pb,eq , (1 − pb,eq )2

(6)

where 1 ∆= (4π)2

Z

Z d~r

Z dˆ r1

dˆ r2 (exp [−βVG (rp )] − 1) gref (r).

6


(7)

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Figure 2: Density dependence of the equilibrium bonding probability at different temperatures. Symbols are estimated from the asymptotic value of p∞ obtained from the simulations, while the lines correspond to pb,eq obtained from Wertheim’s theory. The behavior at low densities is shown in the inset.

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Here, gref (r) is the pair correlation function of the reference fluid, ~r is the vector between the centers of the two particles participating in the bond, and rˆi is the unit vector that defines the position of the bonded patch on particle i relative to the center of that particle. In order to proceed we replaced the soft-core repulsion VY (r), by the repulsion of hard spheres with a temperature dependent diameter d, obtained through the Barker-Henderson approximation. Thus, gref (r) is given by gHS (r = d, ρ), the contact value of the pair correlation function of a system of hard spheres of diameter d and number density ρ (see e.g., Ref. 34 ). ∆ is now rewritten as, ∆ = gHS (r = d, ρ)G(T, α, δ),

(8)

where α = ξ/σ and δ = (1 − d/σ). G(T, α, δ) is the integral of the Mayer function of the patch-patch interaction VG over the bond volume (see Ref. 34 for further details). Since pb,eq = p∞ , Eqs. (3) and (6), imply that at equilibrium the breaking and bonding rates satisfy, ∆=

kbo , kbr

(9)

consistent with the fact that both Eqs. (3) and (6) may be derived from the Flory-Stockmayer size distribution of clusters. 39,41 It is also known that the equilibrium mean-field relation breaks down in the limit where a strongly connected gel is formed. 31 Note that, both the dynamics and the equilibrium descriptions, neglect patch-patch correlations. In order to assess the validity of the rate equations we compared, in Fig. 1, the time evolution of the bonding probability pb (t) of a single bond, obtained from the particle-based simulations (symbols) and a least squares fit of Eq. (4) (solid lines). Clearly, the rate equation Eq. (4) captures the behavior of pb (t) accurately. In addition, this procedure provides estimates of Γ and p∞ , which through Eq. (5) yield estimates of the breaking and bonding rates, (kbr , kbo ), over a wide range of temperatures and densities. As shown in Fig. 2, we find a remarkable agreement between p∞ (ρ, T ) obtained from the fit and pb,eq (ρ, T ) obtained from Wertheim’s theory. The observed deviations at the lowest simulated temperature may be

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Figure 3: Temperature and density dependence of the breaking and bonding rates of a single bond obtained from the fit to the numerical data in Fig. 1. Rate of breaking of a single bond kbr as a function of (a) the inverse temperature and (b) density, at different densities and temperatures, respectively. The solid line in (a) is aG(T )−1 , where a is a constant adjusted to fit the numerical data. Rate of bonding of a single bond kbo as a function of (c) temperature and (b) density, at different densities and temperatures, respectively. The solid line in (d) is proportional to gHS at r = σ (details in the text).

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Figure 4: Time dependence of the rates of (a) breaking (kbr (t)) and (b) bonding (kbo (t)) of individual bonds, obtained from the particle-based simulations, at ρ = 0.25 and three different temperatures: 0.1; 0.125; 0.15. The horizontal (dashed) lines correspond to the effective values estimated by the fit (details in the text).

due to the approximations required to calculate ∆, as the mapping of the reference system to a system of hard spheres with an effective diameter, or the fact that, at high densities and low temperatures a gel is formed. As discussed above, it is possible to estimate the temperature dependence of the effective rates kbr and kbo , from Γ (Fig. 1) , as shown in Fig. 3. We see that kbr follows an exponential decay with 1/T , as expected for a thermal activated (Arrhenius) process, with no significant dependence on the density. Following Arrhenius’ work, we expect that kbr ∼ hexp(βVG )i, where β = (kB T )−1 and h· · · i stands for an ensemble average over different bonds. From Wertheim’s theory, Eq. (8), G(T ) = hexp (−βVG )i, and we expect kbr ∼ G(T )−1 . The solid line in Fig. 3(a) corresponds to aG(T )−1 , where a is a constant adjusted to fit the numerical data. One sees that the lower the temperature, the better the agreement between the line and the numerical data. This result indicates that kbr = F (T )G(T )−1 , where F (T ) is a decreasing function of T. We note that most other studies of patchy particles consider a square well potential for the patch-patch interaction. In those systems, the energy barrier is equal to the energy of the bond (Eb ) and thus kbr (T ) ∼ exp(βEb ), since Eb is the same for all bonds. 33 10


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Figure 5: Temperature and density dependence of the asymptotic value of the rate of bond ∗ ∗ breaking (kbr ) and bonding (kbo ) of a single bond. Rate of bond breaking of a single bond as a function of (a) the inverse temperature and (b) density, at different densities and temperatures, respectively. Rate of bonding of a single bond as a function of (c) temperature and (d) density, at different densities and temperatures, respectively. The solid lines in (a) and (d), and the symbols in (a) and (c) have the same meaning as in Fig. 3.

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At equilibrium kbo = kbr ∆ and thus kbo ≈ gHS (r = d, ρ)F (T ). Under these conditions, since the dependence on T is only through F (T ), the bonding rate of a single bond should decrease with temperature, as shown in Fig. 3(c). In Fig. 3(d) we plot the dependence of kbo on density at different temperatures. The solid line is proportional to gHS , showing that the density dependence is well described by gHS . Note that, the rate of bonding should depend on the diffusion coefficient of the particles, which sets the time scale for two particles in solution to collide, and thus increases with temperature. However, as time is rescaled by the Brownian time, this effect is not apparent in the results. For simplicity, in line with previous works, we considered that the rates kbr and kbo are indeed constant, i.e. time independent. Particle-based simulations allow us to measure these rates and evaluate this assumption. To that end, we measured the number of new and broken bonds over intervals of 5000 iteration steps (between 0.075 and 0.15 in Brownian time, depending on the temperature) and estimated kbr (t) and kbo (t). Fig. 4 illustrates the time dependence of these rates, at different temperatures and ρ = 0.25. We find that both rates increase with time and saturate at long times. However, while kbr saturates rapidly at times of the order of the Brownian time, kbo takes up to one order of magnitude longer to saturate. The lower the temperature the longer the time it takes to saturate. In the same figure, we also plotted the effective rates estimated from the fit discussed above (horizontal lines). It is clear that the asymptotic values of these rates are significantly different from the effective rates obtained from the fit. In Fig. 5 we plot the asymptotic values of these ∗ ∗ rates, kbr and kbo as a function of temperature and density (computed from an average over

the long-time regime of kbr (t) and kbo (t) - see Fig. 4). While, the qualitative dependence of these rates on T and ρ is consistent with that obtained from the fits (see Fig. 3), the values are significantly different. Nevertheless, their ratio is the same, as expected Wertheim’s theory (6) and (9), with p∞ = pb,eq . Finally, Eq. (4) with Γ obtained using the asymptotic ∗ ∗ values of kbr and kbo from the numerical simulations does not fit the data of Fig. 1 (not

shown). This clearly suggests that the out of equilibrium dynamics is different from that at

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equilibrium, at least in three dimensions, and that the effective rates describing the initial dynamics are different from those that describe the dynamics at equilibrium.

Conclusions We combined Langevin particle-based simulations, mean-field rate equations, and Wertheim’s equilibrium theory to study the dynamics of patchy particles both in and out of equilibrium. To establish a relation between Wertheim’s theory and the approach based on mean-field rate equations, we assumed that the stationary state of the dynamics is the equilibrium state. Since the value of the equilibrium bond probability is very accurately described by Wertheim’s theory, we established a relation between the ratio of breaking and bonding rates at equilibrium and the properties of the patch-patch interaction. From the particle-based simulations, we calculated the dependence of the rate of breaking and forming a single bond, at different temperatures and densities, as a function of time, showing that the dynamics at equilibrium is different from that towards equilibrium (out of equilibrium). In particular, for systems of initially randomly distributed particles with no bonds, both rates are time dependent and systematically higher at equilibrium. Thus, mean-field approaches based on rate equations with constant rates obtained from fitting the time evolution of the bond probability underestimate these equilibrium rates. We have focused on the non-percolative regime (high temperature and low density). At low temperatures and high densities, percolation is expected to occur and thus collective effects will definitely affect the dynamics. 34 Assumptions such as negligible patch-patch correlations and independence of the diffusion coefficient of the cluster size are no longer valid. In that regime, we expect differences between the equilibrium and out of equilibrium dynamics to be more significant and the validity of the mean-field approach compromised. This work highlights the relevance of out of equilibrium studies. It was shown previously that, under certain conditions, the dynamics depends strongly on the initial config-

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uration. 28,34 Here, we have considered only initially not bonded particles. The difference between the equilibrium and out of equilibrium dynamics is expected to depend also on the initial conditions. This difference is expected to impact most strongly on the rate of bonding, as different kinetic pathways will correspond to different mechanisms of bond formation. Notwithstanding, in the limit of very strong patch-patch correlations, even the bond breaking dynamics may be affected by collective effects.

Acknowledgement We acknowledge financial support from the Portuguese Foundation for Science and Technology (FCT) under Contracts nos. EXCL/FIS-NAN/0083/2012, UID/FIS/00618/2013, IF/00255/2013, SFRH/BPD/114839/2016, and FCT/DAAD bilateral project.

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Dynamics of Patchy Particles in and out of Equilibrium - The Journal of

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