Open Access Article

This Open Access Article is licensed under a

Creative Commons Attribution 3.0 Unported Licence

DOI: 10.1039/D0SM01391A
(Paper)
Soft Matter, 2020, Advance Article

Stefano
Iubini
*^{ab},
Marco
Baiesi
^{bc} and
Enzo
Orlandini
^{bc}
^{a}Consiglio Nazionale delle Ricerche, Istituto dei Sistemi Complessi, via Madonna del Piano 10, I-50019 Sesto Fiorentino, Italy. E-mail: stefano.iubini@fi.isc.cnr.it
^{b}Dipartimento di Fisica e Astronomia, Università di Padova, Via Marzolo 8, I-35131 Padova, Italy
^{c}INFN, Sezione di Padova, Via Marzolo 8, I-35131 Padova, Italy

Received
30th July 2020
, Accepted 16th September 2020

First published on 24th September 2020

Microrheology experiments show that viscoelastic media composed by wormlike micellar networks display complex relaxations lasting seconds even at the scale of micrometers. By mapping a model of patchy colloids with suitable mesoscopic elementary motifs to a system of worm-like micelles, we are able to simulate its relaxation dynamics, upon a thermal quench, spanning many decades, from microseconds up to tens of seconds. After mapping the model to real units and to experimental scission energies, we show that the relaxation process develops through a sequence of non-local and energetically challenging arrangements. These adjustments remove undesired structures formed as a temporary energetic solution for stabilizing the thermodynamically unstable free caps of the network. We claim that the observed scale-free nature of this stagnant process may complicate the correct quantification of experimentally relevant time scales as the Weissenberg number.

An important issue is the relation between the macroscopic dynamical response of the system, measured experimentally, and its less accessible local structural rearrangements. From the theoretical side, mean-field theories and microscopic constitutive models have been successfully developed^{3,7–10} to rationalize the outcomes of rheological experiments. However, these studies often dealt with small temperature jumps, which may not lead to the significant disturbance of the micellar networks as, e.g., in microrheological experiments at large Weissenberg numbers where a microbead is moved in the medium at high velocities.^{11,12} At the same time, numerical studies,^{13–18} performed at a coarse-grained molecular level on specific systems (cationic cylindrical micelles and cetyltrimethylammonium chloride micelles) have either looked at the competition between shear flow and internal network dynamics or estimated the strands scission and branching free energies of these systems. However, these models, being still quite detailed and system-specific, cannot have access to the scales spanned by rheological experiments (of the order of seconds and above μm^{3}) or address universal features of the phenomenon.

A generic understanding of the transient behavior of living polymers can be sought by addressing the following questions: which are the dominant local structures (or “microstates”) that compete dynamically in the rearrangement of a relaxing living random network? How does the long-time relaxation depend on the mechanisms of local rewiring and scission of the fibers? Is the time dependence of relevant observables undergoing relaxation characterized by simple exponentials or is it more similar to a dynamical scaling with power laws? Simulations of coarse grained models at mesoscopic level are an ideal tool for answering these questions: they allow a direct investigation both at short/small and long/large scales and they may adopt a well controlled protocol for driving the system out of equilibrium.

In this paper, we introduce a simple, coarse-grained, description of a thermally driven self-assembly process generating a living network of fibers by means of the merging, scission and rewiring of basic units that represent a portion of the tubular structure. These units are described by patchy particles.^{19–33}

By using large-scale Brownian dynamic simulations and analytical arguments to estimate the free-energy barriers between the competing “microstates”, we characterize the underlying microscopic dynamics that leads to the global relaxation towards the formation of the random networks. To go beyond the regime of small perturbations a wide temperature quench is imposed on the system.

The paper is organized as follows. In Section 2 we introduce the coarse-grained model and the numerical set-up. In Section 3 we analyze the thermodynamic and kinetic properties of local motifs and their effect on the large-scale relaxation dynamics of the system upon a temperature quench towards a self-assembled equilibrium state. Section 4 is devoted to a final discussion of the results and to highlight future developments that can be explored within this approach. Finally, the appendix contains the details of the numerical methods employed for this study.

Two main characterizing aspects of our model are worth noticing. First, in most of the patchy colloids models^{19} the branching points of the assembled structures are generated either by introducing a fraction of colloids decorated by three or more attractive patches^{20,21} or by allowing for different types of patch–patch interactions.^{22–25} In either case, a better control of the phase diagram is achieved by satisfying a single-bond-per-patch condition.^{26} In our model only bifunctional patchy beads are considered and the local branching emerges naturally from the formation of multiple contacts between patchy particles. Note that similar systems were studied experimentally^{27} and numerically^{28–30} (see also some more recent work^{31–33}). The second aspect concerns the presence of the tunable branching parameter λ. This has been introduced to mimick the propensity of worm-like micelles to branch as a function of salt concentration (see Section 2.2). In particular, by varying λ ≤ 2 (λ = 2 corresponds to standard surface interactions between patches), one can change the depth of the sticky spots within the repulsive core and hence bias the system towards the formation of either linear strands or branches (see Fig. 1(b) for the simplest case of trimers). More precisely, for large values of λ the units tend to form “vertex” or “3 contact” states (see Fig. 1(b)) rather than the “strands” or “2 contact” states.

This behavior can be explained quantitatively by minimizing analytically the total potential energy of the vertex and strand configurations of a trimer, see Fig. 1(c). As a matter of fact, we find that the formation of strands is energetically advantageous with respect to branch points only for sufficiently small values of λ. Branch points become instead energetically more convenient if λ gets close enough to 2. For example, in the two situations represented in Fig. 1(d), for λ = 1.75 the mutual attraction between sticky spots take place at a distance larger than the interaction range σ, reducing the stability of the trimer, while for λ = 1.95 the three pairs of sticky spots are all at a distance within σ and thus very stable.

The effect of the branching parameter λ can be further rationalized as follows. Let us focus on a linear dimer (see Fig. 1(e)) in its mechanical equilibrium position with bond energy u_{l} < 0 determined by the distance d between unit centers and the distance δ = d − λ between sticky centers. The branched configuration can be obtained by rotating the two reference units by angles of 30° with respect to their centers on the plain containing the trimer (Fig. 1(f)). In this rotated configuration the two sticky spots increase their distance δ′ > δ thus decreasing their attracting force, while the repulsive centers of the units are not modified significantly. Since δ′ > δ the single bond energy u_{b} > u_{l}. However, due to the different number of bonds in branched and linear trimers, the overall energy of the branched trimer E_{b} = 3u_{b} can be lower and thus more stable than the linear trimer configuration with energy E_{l} = 2u_{l}. This takes place if λ is large enough to keep δ′ within the range of attraction of the sticky spots. Upon decreasing λ, one meets a situation in which δ′ is longer than the range of attraction between sticky spots and hence the branched configuration becomes less stable than the linear one.

In this approach an ensemble of surfactants forming a globular micelle or the segment of a wormlike network, as those sketched in Fig. 2, is mapped to a single micellar unit. In this context, one may think of λ as a parameter describing the strength of salt concentration. This is indeed known to shrink the effective size of the hydrophilic heads of the surfactants, thus favoring the appearance of structures with negative surface curvature as hubs in the micellar network.

Fig. 2 Sketch of a wormlike micellar network and of selected portions where each surfactant is depicted as a hydrophilic (red) head and a hydrophobic (blue) tail. The numbering of the network portions follows that of the corresponding patchy particle states of Fig. 1(b). |

Note that the resolution of our coarse grained model cannot describe effects below the size of a globular micelle. In particular, unlike previous models,^{13–18} the dynamics of a single surfactant is not included in our description. On the other hand, the fact that the basic element of our model is not representing a single surfactant but rather a collection (i.e., some dozens) of them, should not affect the large scale dynamics of the network structure that we aim to explore here.

By assuming, for instance, that the core radius of the elementary unit length is of the order of the average width measured in wormlike micelles, i.e., R ≈ 5 nm, the largest box considered in our simulations with dimensions 108R × 108R × 324R maps to a system volume ≈0.5 μm^{3}. Moreover, our runs simulate systems of up to N = 6000 units; this would correspond to a number of molecules of surfactants of the order of 10^{5}.^{37} Finally, since the simulation time unit τ is mapped to 2 μs in the real system (see appendix), the longest simulation time we can reach, t = 10^{7}τ, corresponds approximately to 20 s and is comparable with the duration of recent experiments.^{6,38} Simulations are performed with an integration time step equal to 10^{−2}τ. We have verified that this time step suffices to ensure suitable numerical accuracy in all regimes considered in this study.

We describe the time evolution of the network by monitoring the formation/disappearance of its local motifs, such as dangling ends and branched structures. These are classified by looking at the number of contacts shared by their patches. A sticky spot is defined to share a contact with another one if the distance between their centers is ≤R, see appendix for further details. A dangling end contains a unit with k = 1 contacts, while each branch “3” contains three units each with k = 3 contacts, when they are part of a strand in the network (that is, a chain of elementary units with k = 2) which ends either with dangling ends or branching points. Finally, we define the length L of a strand as the total number of linear bonds between its extrema. For example, a dimer composed of two units with k = 1 has length L = 1, while a sequence on five units with k = 2 between two joints has length L = 6. We assign L = 0 to isolated units (k = 0).

To estimate the free energy difference ΔF_{32} = F_{3} − F_{2} we have carried out Langevin simulations with parallel tempering sampling technique^{39,40} on a system with three units confined within a small fixed volume (see appendix for details). In Fig. 3(a) we report ΔF_{32} as a function of T for four values of λ. It displays a linear growth with T, with an intercept at T = 0 that matches quite well the energy differences found analytically from Fig. 1(c), see open symbols. Entropy differences ΔS_{32} are thereby identified by the (negative) slope ∂ΔF_{32}/∂T, see dashed lines. Note that the combination of the negative entropy gap ΔS_{32} with a λ-dependent energy potential may result in a situation where the most favorable thermodynamic state is T-dependent, as for λ = 1.95, where branch points are stable only at low T. Analogous results involving temperature-dependent effective valence were previously found in particles models with explicit dissimilar patches.^{23} These findings match the prediction^{18} that wormlike micelles may form networks with a variety of features, depending on the inclination of their tubular structures to branch.

Fig. 3 Thermodynamic and kinetic properties of a trimer. (a) Free energy difference ΔF_{32}vs. temperature T for λ = 1.75, 1.79, 1.86 and 1.95. For the same values, −ΔS_{32} = 2.4, 3.2, 3.2 and 5.1 from linear fits, see dashed lines. Open symbols reported on the vertical axis correspond to the energy difference from Fig. 1(c), at T = 0. (b) Average scission time t_{3} for a trimer initially arranged in the configuration “3” and (c) average scission time t_{2} for a strand “2”, as a function of the inverse temperature 1/T. Dashed lines are Arrhenius fits of ΔE^{S}_{13} and ΔE^{S}_{12}. In all panels different symbols refer to different values of the branching parameter λ (see legend in panel (b)). |

To see how local thermodynamic properties affect the dynamics of network formation, we have computed the average scission time either from the strand “2” or from the branch point “3” to the detached dimer-plus-monomer state “1”, see Fig. 1(b). Scission times have been computed from equilibrium simulations for different temperatures T and different values of the branching parameter λ, see appendix for details. They follow a standard Arrhenius law once reported as a function of 1/T. For branched initial configurations (Fig. 3(b)), we find average scission times t_{3} ∼ exp(ΔE^{S}_{13}/T) with energy ΔE^{S}_{13} whose fitted values are ΔE^{S}_{13} ≈ 15, 21, 26, 37 (in units of k_{B}T), respectively for λ = 1.75, 1.79, 1.86, and 1.95. Similarly, the scission times for strand initial configurations is t_{2} ∼ exp(ΔE^{S}_{12}/T) (Fig. 3(c)) and the fitted values of the scission energies ΔE^{S}_{12} are 24, 27, 32, and 29 for the same λ values. Importantly, these values of ΔE^{S}_{12} and ΔE^{S}_{13} are in a range ≈15–35 k_{B}T fully compatible with the current estimates of the scission energies (or enthalpies) of micellar systems.^{18,38,41,42}

The macroscopic relaxation process is further understood by looking at the dynamical evolution of units classified by their contacts. Let N_{k} be the number of units with k contacts and f_{k} = N_{k}/N their fraction. Due to the core repulsion, for λ ≲ 2 a unit may establish at most k = 4 contacts (two per side).

In Fig. 5 we plot the fractions f_{k} vs time for the same system of Fig. 4. After a quick condensation to a network (f_{0} → 0 for t → t_{0}), the network does not contain the equilibrium proportion of network motifs. The fraction f_{1} of units at dangling ends tends slowly to zero, with a scaling compatible with a power law f_{1} ∼ t^{−1}. Eventually, at t_{eq} ≈ 2 × 10^{6}τ, mapped to ≈10 s, the fraction f_{1} reaches its asymptotic value. This very slow reduction of the number of strands with dangling ends highlights a stagnant global relaxation of the network. Also the fraction f_{3} of units meeting at branches slowly decreases as f_{3} ∼ t^{−1} till t_{eq} and then fluctuates around its equilibrium value. We can approximate the asymptotic ratio _{3}/_{2} ≃ _{3} (the bar over f_{k} denotes the long-time value of f_{k}) by the equilibrium expression _{3}/_{2} = exp(−Δf_{32}/T), where Δf_{32} = ΔF_{32}/3 is the free energy difference per unit derived from the single trimer problem, see Section 3.1. For λ = 1.75 and T = 1, this gives _{3}/_{2} ≃ 0.003. This prediction, however, is not fully quantitative, as it neglects the role of trimer interactions (and even more complex many-body interactions) which are present in the macroscopic system. Finally, a very small amount of units with four contacts (k = 4) is temporarily observed during the relaxation process, in correspondence of the maximum concentration of branches f_{3}. This is not surprising, as four-contacts states arise when the branched geometry affects both sides of a unit. Since their energetic cost is essentially twice the cost of a single branch, their asymptotic concentration is expected to be of the order of (_{3})^{2}, a behaviour which, however, was not observed in our simulations due to finite size and finite time effects.

Fig. 5 Evolution of the relative density f_{k} of units with k-contacts for the same relaxation dynamics of Fig. 4 (N = 6000, λ = 1.75 and T = 1). Dashed lines represent a scaling ∼t^{−1}. |

The slow scalings of f_{1}, f_{3}, and 〈L〉 suggest that the dynamics within the network proceeds through a sequence of frustrated rearrangements. For instance, a dangling end joining its free cap to the side of a strand would form a branch. This local state is however thermodynamically less favorable (see Fig. 1(b)), thus leading to a subsequent decay back to a dangling end and a linear strand. A direct merging of two dangling ends would better stabilize the system but this event becomes more and more unlikely with time due to the concomitant decreases of the dangling ends population. From this argument it follows that the time scale of network rearrangements τ_{s} in Fig. 4 is determined by the scission time of branch points. Indeed from Fig. 3(b) we find that t_{3} ≃ 10^{4} ≃ τ_{s} for λ = 1.75 and T = 1.

The variety of micellar structures depends on several parameters (solvent, salt concentration, surfactant chemistry) and includes networks rich in branches. This situation is realized in our model by increasing λ. For example, with λ = 2 and T = 1, the relaxation proceeds through the formation of a metastable network with relatively small average strand length, as shown in Fig. 6(a) (the configuration at the end of the simulation is shown in Fig. 6(b)). Again, there is a very slow convergence of the fraction f_{1} of units at the caps of dangling ends, see Fig. 6(c). In Fig. 6(d), in linear scale on the vertical axis, it is apparent that also the fractions f_{2} and f_{3} are slowly converging toward their asymptotic average values. Here, equilibrium thermodynamics predicts a clear dominance of branches over strands: in the single trimer study in Fig. 3(a), we obtain ΔF_{32} ≃ −8 for the nearby parameters λ = 1.95 and T = 1. Therefore, we conclude that equilibrium is far from being reached and that the configuration reported in Fig. 6(b) is still a metastable one.

The phenomenology described above holds for several particle densities. Indeed, by changing considerably N and keeping the volume V fixed, we have been able to vary ρ_{num} = N/V by a factor of 3 (data not shown) Yet, the scaling of the mean quantities is quite similar to the previous cases. In particular, the average strand length in the initial metastable network is unaffected by the change of density. However, the density determines the time scale t_{0} of the initial network condensation. This is defined as the time needed for the fraction of 0-contacts f_{0} to go below the threshold value f*= 0.4 starting from a homogeneous initial condition of isolated particles (f_{0} = 1) at time t = 0. By scaling arguments, at low concentrations, diffusion leads to t_{0} ∼ ρ_{num}^{−2/3}. At higher concentrations it is rather the “ballistic” expansion of the network core (as in other Brownian coagulation systems^{45}) that determines the time scale t_{0} ∼ ρ_{num}^{−1}. In Fig. 7 we report estimates of ∼t_{0} (up to a constant that is irrelevant for the scaling) which are compatible with the two regimes, see also the appendix for further details. Note that we have verified that f* is irrelevant for the scaling behavior of t_{0} with ρ_{num} in so far as f* is large enough to provide a sampling of the initial relaxation stages.

By combining large-scale simulations and analytical estimates of the free energy barriers between motifs, we rationalize the observed slow convergence of the network to equilibrium in terms of the frustrated dynamics of the local motifs. In particular we argue that the relaxation process develops through a sequence of rearrangements that are not only energetically challenging but also require the solution of topological constraints. Importantly, such constraints do not derive from static functional properties of motifs, but emerge dynamically and depend on local thermodynamic conditions. Moreover, the far-from-equilibrium character of our setup is naturally beyond the perturbative regime, where exponential relaxation laws are expected.^{7,10}

While we have used a standard temperature quench to produce a clean representation of the network relaxation, in practice the considerations we have drawn should be extended to transient regimes of mechanical or chemical origin. For instance, a passage of a micro-bead through a portion of a viscoelastic fluid made of wormlike micellar networks may “scramble” severely the system by generating dangling ends and by distorting the branches. This unbalanced state then should re-equilibrate by following a relaxation dynamics akin to the one we have characterized here. Such scenario is relevant for microbeads of diameter D traveling with mean velocity v at high Weissenberg number Wi = v/v_{r}, where v_{r} = D/τ_{rel} is the velocity scale associated to the relaxation over D with time scale τ_{rel}. Our findings suggest that a proper definition of τ_{rel} should consider the possible emergence of power-law behaviors for strongly disturbed polymer networks. In fact, the relaxation times found here are of the same order of magnitude (seconds) of those measured experimentally in viscoelastic micellar networks.^{5,6} It is thus natural to conjecture that τ_{rel} is the time at which each microscopic motif of the network reaches, through a power-law decay, its typical (equilibrium) fraction. In this respect, it is the ending time of this scale-free aging (not to be confused with chemical aging, as a high-temperature degradation of carbon–carbon double bonds^{46}) that should be considered for evaluating the standard Deborah and Weissenberg numbers used in experiments.^{11,12}

Our findings also raise the question on which of, e.g., self-healing rubber,^{47,48} colloidal systems,^{49–51} or network fluids^{52} do experience a relaxation process characterized by a slow rearrangement of local motifs. In this respect we believe that coarse-grained descriptions based on patchy colloids, as the one presented here, can be highly valuable.

Finally, another perspective of our work is related to the problem of the stability of ring solutions versus network structures in self-assembling systems.^{24,53} Suitable generalizations of our model could be considered in order to bias the aggregation of linear structures towards the formation of closed rings. For instance, instead of units with two antipodal sticky spots, one could consider patchy particles that are non-axisymmetically distributed along the surface.^{24} The interplay between the branching parameter λ here introduced and the propensity to create rings is an open question that will deserve future investigations.

Langevin dynamics.
Let us consider an ensemble of N patchy particles labeled by the index i = 1,…,N and let α = 1, 2, 3 be an additional index that identifies the subunits, where α = 1 is the central repulsive core and α = 2, 3 identify the two sticky spots in Fig. 1(a). In the following we denote by _{iα} the position of the center of mass of the subunit α of unit i and by d^{jβ}_{iα} = |_{iα} − _{jβ}| the distance between two subunits. The repulsive interaction between a pair of central cores of radius R is modeled by a truncated and shifted Lennard-Jones potential

where we have introduced the symbol D_{ij} = d^{j1}_{i1} to simplify the notations. The parameter ε is the typical energy scale of the Lennard-Jones potential and Θ(x) is the Heaviside function.

where the subindices α and β are restricted to the values (2,3). Moreover we set

where U = U_{LJ} + U_{G} + constraints is the total potential energy function and m is the mass of each subunit. The first term in the r.h.s. of eqn (4) is a linear dissipation proportional to the friction coefficient γ, while _{iα}(t) is a Gaussian noise with zero mean and delta-correlated in time. Denoting by η^{()}_{iα} with = 1, 2, 3 the Cartesian component of _{iα} in the direction, the noise satisfies a standard fluctuation–dissipation relation in the form

where k_{B} is the Boltzmann constant and T the temperature. Numerical integration of eqn (4) is performed with LAMMPS engine.^{36} Nonequilibrium simulations of the system relaxation process are realized according to the following protocol. First we evolve the system described by eqn (4) in the absence of the Gaussian attractive potential (U_{G} = 0) until a uniform thermalized state is reached. Then we activate U_{G} and we start monitoring 〈L〉 and f_{k} as discussed in the main text.

(1) |

The attractive interaction between sticky spots is described by a Gaussian potential with amplitude A and range σ, namely

(2) |

(3) |

Subunits within each unit evolve as a rigid body arranged as in Fig. 1(a). The evolution of the system is described in terms of a Langevin equation

(4) |

(5) |

Physical units.
In our simulations the system temperature T is measured in units of ε/k_{B}, where ε is the amplitude of the Lennard-Jones potential, with k_{B} = ε = 1, A = 40ε and σ = 0.4R. Distances are measured in units of monomer radii R. We set m/γ = τ, where is the (unitary) characteristic simulation time. In order to estimate the time τ in physical units, one can proceed as follows. From the Stokes formula of the friction coefficient for spherical beads with radius R, we have γ = 6πη_{sol}R, where η_{sol} is the viscosity of the solvent. By eliminating m in the above formulas for the characteristic time, one has γ = ετ/(2R)^{2}, hence

In the numerical evaluation we consider the nominal water viscosity at room temperature, η_{sol} ≃ 1 mPa s, and we set T = ε/k_{B} = 300 K and R = 5 nm.

For the large-scale simulations of this work, the system is evolved in a prism of sides (d,d,3d) with d = 108R, with periodic boundary conditions. Its volume corresponds to 3d^{3} ≃ 0.5 (μm)^{3}. If N = 6000 units are introduced in this volume, their volume fraction becomes

Contact threshold.
Given a configuration of patchy particles identified by the set of _{iα}, two distinct units i and j ≠ i are considered in contact if there exists a pair of sticky spots α and β such that d^{jβ}_{iα} ≤ θ, where θ is a threshold. To perform an accurate sampling of merging and scission events in the system of particles, θ needs to be of the order of the distance between bound units, in a way that typical thermal fluctuations in their positions of do not produce “spurious” events. To determine a reliable value of θ for the potential U specified in the previous section, we have computed the distributions of sticky spot distances d^{jβ}_{iα} for all i ≠ j in generic networks, see Fig. 8 for an example. The distribution displays a clear gap in the region [0.5R,1.8R] separating the ensemble of bound sticky spots (with d^{jβ}_{iα} ≲ 0.5R ≃ σ) from the ensemble of unbound ones. Accordingly, we have chosen θ = R, well inside the gap. We have verified that this threshold value is reliable in the whole range of λ and T considered in this work.

Fig. 8 Distribution of distances between sticky spots for a system of N = 2000 particles with λ = 1.75, A = 40ε and σ = 0.4R. Particles are prepared in a homogeneous state and evolved at T = 1 according to eqn (4) in a cubic box with side equal to 54R and periodic boundary conditions. The histogram is representative of the configuration obtained at t = 10^{5}τ, see the inset. Black, red, green and blue bars refer respectively to the combination of indices (α,β) = {(2,2),(2,3),(3,2),(3,3)} in the set of distances d^{jβ}_{iα} for i, j = 1,…,N. Their distributions overlap very well, as expected from symmetry reasons. Multiple peaks above the gap region derive from the regular spacing of particles in the strands. |

Parallel tempering simulations.
Free-energy differences between microstates “2” (strands) and “3” (vertices) have been computed through Langevin equilibrium simulations with parallel tempering sampling technique.^{39,40} This method is employed to sample accurately the low-temperature equilibrium distribution of the system, thereby reducing the impact of very long correlation times on computational time.

Simulations have been performed for a system of N = 3 interacting particles confined in a spherical volume with radius R_{0} = 10.85R and volume fraction ϕ ≃ 2 × 10^{−3}, which corresponds to the value of particle density of large-scale simulations with N = 2000. We have verified that in this minimal setup the confining volume is sufficiently larger than the typical particle volume, so that it does not introduce significant finite-size effects. The free energy difference ΔF_{32} is computed via the equilibrium Boltzmann distribution ΔF_{32} = −Tlog(P_{3}/P_{2}), where P_{3} and P_{2} are respectively the occupation probabilities of states “3” and “2” sampled over the equilibrium Brownian dynamics. Parallel tempering evolution is performed for a total time of 5 × 10^{6}τ with a swapping period of 50τ. Temperature swaps are realized between adjacent temperature states within the set of temperatures considered in Fig. 3(a).

Scission times.
Scission times in Fig. 3(b) and (c) have been computed for a system of N = 3 interacting particles confined in a spherical volume with radius R_{0} = 10.85R and evolving according to eqn (4). For each value of temperature T, particles were initially prepared in a “3” configuration (Fig. 3(b)) and in a “2” configuration (Fig. 3(c)) and evolved until a scission event to a dimer-plus-monomer state (configuration “1”) was observed. For each T, the average scission time has been computed by averaging over a sample of 10 independent scission events.

Condensation time scales.
Diffusive and ballistic regimes are further analyzed in Fig. 9, where we compare the average particle distance = ρ_{num}^{−1/3} with the typical size d_{c}(t) of the biggest cluster of particles found in the system at time t. We define d_{c}(t) = 2RN_{c}(t), where N_{c}(t) is the total number of particles in the cluster. We focus on two sizes, namely N = 25 (Fig. 9(a)) and N = 1600 (Fig. 9(b)), which belong respectively to the diffusive and ballistic condensation regime. For N = 25 we find ≫ d_{c}(t) during the whole period in which f_{0} > f* while the case N = 1600 displays a quick growth of d_{c}(t) to values which are of the same order of thus suppressing the diffusive scaling of t_{0}.^{45}

- R. G. Weiss and P. Terech, Molecular Gels, Springer, Nederlands, 2006 Search PubMed.
- M. Jehser and C. N. Likos, Colloid Polym. Sci., 2020, 1–11 Search PubMed.
- M. E. Cates and S. M. Fielding, Adv. Phys., 2006, 55, 799–879 CrossRef CAS.
- R. G. Larson, The structure and rheology of complex fluids, Oxford Univesity Press, New York, 1999 Search PubMed.
- J. R. Gomez-Solano and C. Bechinger, New J. Phys., 2015, 17, 103032 CrossRef.
- J. Berner, B. Müller, J. R. Gomez-Solano, M. Krüger and C. Bechinger, Nat. Commun., 2018, 9, 999 CrossRef.
- M. S. Turner and M. E. Cates, J. Phys., 1990, 51, 307–316 CrossRef.
- T. J. Drye and M. E. Cates, J. Chem. Phys., 1992, 96, 1367 CrossRef CAS.
- R. Granek and M. E. Cates, J. Chem. Phys., 1992, 96, 4758 CrossRef CAS.
- M. S. Turner, C. Marques and M. E. Cates, Langmuir, 1993, 9, 695–701 CrossRef CAS.
- J. M. Dealy, Rheol. Bull., 2010, 79, 14–18 Search PubMed.
- R. J. Poole, Rheol. Bull., 2012, 53, 32–39 Search PubMed.
- J. T. Padding, E. S. Boek and W. J. Briels, J. Phys.: Condens. Matter, 2005, 17, S3347–S3353 CrossRef CAS.
- J. T. Padding, E. S. Boek and W. J. Briels, J. Chem. Phys., 2008, 129, 074903 CrossRef CAS.
- V. Hugouvieux, M. A. V. Axelos and M. Kolb, Soft Matter, 2011, 7, 2580 RSC.
- S. Dhakal and R. Sureshkumar, J. Chem. Phys., 2015, 143, 024905 CrossRef.
- P. Wang, S. Pei, M. Wang, Y. Yan, X. Sun and J. Zhang, J. Colloid Interface Sci., 2017, 494, 47–53 CrossRef CAS.
- T. Mandal, P. H. Koenig and R. G. Larson, Phys. Rev. Lett., 2018, 121, 038001 CrossRef CAS.
- F. Sciortino, C. De Michele, S. Corezzi, J. Russo, E. Zaccarelli and P. Tartaglia, Soft Matter, 2009, 5, 2571–2575 CAS.
- W. Li, H. Palis, R. Mérindol, J. Majimel, S. Ravaine and E. Duguet, Chem. Soc. Rev., 2020, 49, 1955 RSC.
- D. J. Audus, F. W. Starr and J. F. Douglas, Soft Matter, 2018, 14, 1622–1630 RSC.
- J. Russo, J. Tavares, P. Teixeira, M. T. da Gama and F. Sciortino, Phys. Rev. Lett., 2011, 106, 085703 CrossRef CAS.
- J. Russo, J. Tavares, P. Teixeira, M. T. da Gama and F. Sciortino, J. Chem. Phys., 2011, 135, 034501 CrossRef CAS.
- L. Rovigatti, J. M. Tavares and F. Sciortino, Phys. Rev. Lett., 2013, 111, 168302 CrossRef.
- P. Teixeira and J. Tavares, Curr. Opin. Colloid Interface Sci., 2017, 30, 16–24 CrossRef CAS.
- L. Rovigatti, J. Russo and F. Romano, Eur. Phys. J. E: Soft Matter Biol. Phys., 2018, 41, 59 CrossRef.
- Q. Chen, S. Chul Bae and S. Granick, Nature, 2011, 469, 381 CrossRef CAS.
- F. Romano and F. Sciortino, Soft Matter, 2011, 7, 5799 RSC.
- F. Romano and F. Sciortino, Nat. Mater., 2011, 10, 171 CrossRef CAS.
- F. Romano and F. Sciortino, Nat. Commun., 2012, 3, 975 CrossRef.
- N. A. Mahynski, L. Rovigatti, C. N. Likos and A. Z. Panagiotopoulos, ACS Nano, 2016, 10, 5459–5467 CrossRef CAS.
- W. F. Reinhart and A. Z. Panagiotopoulos, J. Chem. Phys., 2016, 145, 094505 CrossRef.
- H. Eslami, N. Khanjari and F. Mueller-Plathe, J. Chem. Theory Comput., 2019, 15, 1345–1354 CrossRef.
- T. Tlusty and S. Safran, J. Phys.: Condens. Matter, 2000, 12, A253 CrossRef CAS.
- A. Zilman, S. Safran, T. Sottmann and R. Strey, Langmuir, 2004, 20, 2199–2207 CrossRef CAS.
- S. Plimpton, J. Comput. Phys., 1995, 117, 1–19 CrossRef CAS.
- S. Hyde, Z. Blum, T. Landh, S. Lidin, B. Ninham, S. Andersson and K. Larsson, The language of shape: the role of curvature in condensed matter: physics, chemistry and biology, Elsevier, 1996 Search PubMed.
- W. Zou, G. Tan, H. Jiang, K. Vogtt, M. Weaver, P. Koenig, G. Beaucage and R. G. Larson, Soft Matter, 2019, 15, 642–655 RSC.
- R. H. Swendsen and J.-S. Wang, Phys. Rev. Lett., 1986, 57, 2607–2609 CrossRef.
- M. Tesi, E. J. Van Rensburg, E. Orlandini and S. Whittington, J. Stat. Phys., 1996, 82, 155–181 CrossRef.
- K. Vogtt, H. Jiang, G. Beaucage and M. Weaver, Langmuir, 2017, 33, 1872–1880 CrossRef CAS.
- H. Jiang, K. Vogtt, J. B. Thomas, G. Beaucage and A. Mulderig, Langmuir, 2018, 34, 13956–13964 CrossRef CAS.
- S. Chandrasekhar, Rev. Mod. Phys., 1943, 15, 1 CrossRef.
- G. Carnevale, Y. Pomeau and W. Young, Phys. Rev. Lett., 1990, 64, 2913 CrossRef.
- M. S. Veshchunov, J. Aerosol Sci., 2010, 41, 895–910 CrossRef CAS.
- Z. Chu, Y. Feng, H. Sun, Z. Li, X. Song, Y. Han and H. Wang, Soft Matter, 2011, 7, 4485 RSC.
- P. Cordier, F. Tournilhac, C. Soulié-Ziakovic and L. Leibler, Nature, 2008, 451, 977 CrossRef CAS.
- T. F. A. de Greef and E. W. Meijer, Nature, 2008, 453, 171 CrossRef CAS.
- E. Del Gado and W. Kob, Europhys. Lett., 2005, 72, 1032–1038 CrossRef CAS.
- E. Zaccarelli, P. J. Lu, F. Ciulla, D. A. Weitz and F. Sciortino, J. Phys.: Condens. Matter, 2008, 20, 494242 CrossRef.
- R. Angelini, E. Zaccarelli, F. A. de Melo Marques, M. Sztucki, A. Fluerasu, G. Ruocco and B. Ruzicka, Nat. Commun., 2014, 5, 4049 CrossRef CAS.
- C. S. Dias, N. A. M. Araújo and M. M. T. da Gama, Adv. Colloid Interface Sci., 2017, 247, 258–263 CrossRef CAS.
- N. G. Almarza, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2012, 86, 030101 CrossRef CAS.

This journal is © The Royal Society of Chemistry 2020 |