Phase Separation Kinetics of Dynamically Self-Assembling

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Phase separation kinetics of dynamically selfassembling nanoparticles with toggled interactions Zachary M. Sherman, Helen Rosenthal, and James W. Swan Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b02902 • Publication Date (Web): 19 Sep 2017 Downloaded from http://pubs.acs.org on September 26, 2017

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Phase separation kinetics of dynamically self-assembling nanoparticles with toggled interactions Zachary M. Sherman, Helen Rosenthal, and James W. Swan∗ Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139 E-mail: [email protected]

Abstract

describe and predict the self-assembly rates for two predominant kinetic mechanisms. The first model describes the coarsening of percolated, gel-like networks, and the second describes the nucleation and growth of dense phases.

Ordered materials passively self-assembled from dispersions of nanoparticles with steady interactions are subject to thermodynamic constraints on their phase separation kinetics forcing a tradeoff between throughput and quality. Dynamically self-assembling dispersions whose interactions vary in a controlled way with time do not have these constraints and can rapidly form ordered structures while avoiding kinetic arrest. These out-of-equilibrium processes cannot be understood in terms of equilibrium thermodynamics or kinetic models derived from equilibrium thermodynamics, so new theories must be developed before dynamic selfassembly can be used to reliably fabricate nanomaterials. Here, we use dynamic simulation and theory to study the self-assembly kinetics of a monodisperse suspension of spherical nanoparticles interacting with a short-ranged, isotropic attraction that is toggled on and off cyclically in time. The rate of phase separation, local and global quality of the selfassembled structures, and range of tunable parameters leading to acceptable self-assembly are all enhanced with toggled attractions compared to steady attractions. The kinetic mechanism and rate of assembly can be easily controlled with the temporal toggling parameters. We develop simple phenomenological expressions to ∗

Introduction Structured materials self-assembled from dispersions of nanoparticles and colloids have numerous demonstrated applications as functional materials 1–3 including wave guides for optical computing, 4,5 photonic sensors for disease management and detection, 6 novel display technologies, 7 microlens arrays for energy harvesting, 8,9 high performance thermal barrier coatings, 10 and hierarchical porous electrodes for energy storage. 11 Typically, these nanomaterials are synthesized by engineering the interactions between particles to drive a disorder-toorder thermodynamic phase transition. Unfortunately, the envelope in phase space of experimentally achievable ordered states, the crystallization slot, is quite small. 12 The phase transition often arrests, kinetically trapping the dispersion in a defective or disordered metastable state. The performance of these materials is related to the quality of their microstructures, with ordered, crystalline structures outperforming defective or disordered ones, so kinetically arrested phases are undesired. For each self-assembled material, the particle inter-

To whom correspondence should be addressed

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actions must be tuned to find the crystallization slot and then carefully controlled to remain inside. Within the crystallization slot, the selfassembly kinetics are necessarily slow so as not to induce defects leading to arrest. This leaves the fabrication process difficult, inflexible, and expensive, prohibiting scalable production. Kinetic arrest can be avoided if the particle interactions are toggled on and off cyclically in time via an external control. 13,14 This was first demonstrated in experiments on suspensions of paramagentic colloidal particles whose dipolar interactions were controlled with an externally applied magnetic field. Toggling the magnetic field on and off cyclically over time avoided arrest and rapidly drove the particles into low energy, crystalline configurations, even when the particle interactions were orders of magnitude larger than thermal forces and far outside the crystallization slot. 13–18 While the field was off, thermal motion allowed arrested configurations to relax, annealing defects during every on-off cycle. Table 1 lists systems in which particle interactions can be controlled temporally via external stimuli and therefore are candidates for enhanced self-assembly using toggled interactions. These experiments have inspired simulations of toggled self-assembly in a number of different model systems, with the aim of fundamentally understanding these processes at finer length and time resolutions. 19–25 Toggling implies instantaneous switching between an “on" state for a duration ton and an “off" state for a duration toff . The off switching state does not have to be one in which particles are noninteracting (for example, the particles could have attractions in the on state and repulsions in the off state), but we use the on/off notation for convenience. Both the kinetics and the terminal structure of the self-assembling dispersion can be controlled with the toggling parameters, ton and toff , rather than by adjusting physio-chemical parameters like surface chemistry or particle concentration. 16,25 These extra temporal parameters greatly broaden the crystallization slot, allowing for simple control over self-assembly. Additionally, because the toggling protocol is an inherently out-ofequilibrium process, the dispersion can organize

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into nonequilibrium structures that cannot be stabilized with steady interactions. 22,25 Because of the cyclic driving force, structures formed by toggled self-assembly are not true, time-independent equilibrium states but rather periodic-steady-states that are unchanged from cycle to cycle but vary within a toggle period. Periodic-steady-states are energy dissipating, so we cannot use equilibrium thermodynamics to describe them. Kinetic theories that rely on equilibrium thermodynamic principles, such as classical nucleation theory 26 and Cahn-Hilliard kinetics, 27,28 are similarly invalid. This lack of understanding is the main obstacle to implementing toggled selfassembly for nanomaterials fabriation. In a previous work, we developed and validated a first-principles, nonequilibrium thermodynamic theory to describe phase separation of particles interacting via toggled isotropic, short-ranged attractions. 25 We showed that two coexisting phases in periodic-steady-state have equal timeaveraged chemical potential and time-averaged pressure, and their coexisting densities could be calculated from well-known equations of state valid for steady interactions. While useful to describe terminal structures, our theory offers no insight into the phaseseparation kinetics in dispersions with toggled interactions. In this paper, we investigate the self-assembly kinetics in simulations of hard, spherical nanoparticles interacting via a toggled isotropic, short-ranged attraction. First, we compare the kinetics of suspensions selfassembling with steady attractions versus that with toggled attractions, showing that toggling enhances the crystallization rate and crystal quality. Then, we develop models to explain two different kinetic mechanisms we observe in the self-assembling dispersions. The first model describes the coarsening of crystalline strands in percolated, gel-like networks. The second model describes nucleation and growth of large crystals via one-step or two-step mechanisms. These models are an important step to understanding the complicated kinetics involved in dynamically self-assembling systems.

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Table 1: Examples of stimulus-chemistry pairs promoting switchable interparticle interactions in nanoparticle systems. stimulus light

chemistry photo-sensitive ligand photochromic solute

interaction van der Waals 29,30 electrostatic 31,32

temperature

thermally responsive polymer depletion 33 DNA functionalization hybridization 34

pH

pH-responsive ligand

electrostatic, 32 hydrogen-bonding 35

electric field magnetic field

dielectric contrast magnetic contrast

dipolar, 36,37 dielectrophoretic 38 dipolar, 39 magnetophoretic 40

shear flow

any

entropic, 41 viscoelastic, 42 hydrodynamic 43

Results and discussion

the suspension will phase separate into dense crystal and dilute fluid. If ε is close to the fluid phase boundary, phase separation occurs via nucleation and growth. Because the thermodynamic driving force is small near phase boundaries, the dynamics are necessarily slow and tend to create compact, low-defect crystals. The extent of crystallization is quantified with the crystal fraction Xc ≡ Nc /N , defined as the ratio of the number of particles incorporated into the crystal, Nc , to the number of total particles, N (see Experimental Section for how we determine Nc ). Figure 2 shows the crystallization kinetics in simulations of steady attractive suspensions with different attraction strengths. Near the phase boundary, a large fraction of the suspension crystallizes, but only after a significant induction period followed by slow growth. If ε increases quickly and quenches the suspension deep into the coexistence region, the thermodynamic driving force for self-assembly is significantly enhanced. Initially, the aggregation speed is high, but the suspension quickly forms a disordered, percolated phase. Conversion from this metastable gel to the thermodynamic equilibrium crystalline state is incredibly slow, and the terminal state is treated as arrested over the time scales of interest. These kinetics are quantified in our simulations in Figure 2, where only a small fraction of the suspension crystallizes at large ε. Therefore, for passive self-assembly, the kinetics and terminal structures are coupled; we can either form ordered crystals slowly, or disordered states

Steady versus Toggled Interactions In typical nanoparticle self-assembly processes, the particle interactions driving structure formation are constant in time. This type of self-assembly has a few equivalent names including equilibrium self-assembly, passive selfassembly, and static self-assembly. 44 For passively self-assembling materials, the phase transition kinetics are intimately related to the equilibrium phase diagram. Figure 1 sketches an equilibrium phase diagram for a suspension of monodisperse hard spherical colloidal particles with an effective short-ranged isotropic attraction in the presence of nonadsorbing depletant. 45 While the functional form of Figure 1 is specific to this particular interparticle potential (see Experimental Section), the equilibrium phase diagram for any short-ranged isotropic attractive potential is similar, 46 and Figure 1 serves as a generic phase diagram for any typical short-range attractive colloidal suspension. To navigate the equilibrium phase diagram, we can tune two parameters: the particle volume fraction φ and the attraction strength ε relative to the thermal energy kB T . The range of the attraction δ also weakly affects the phase behavior, but is kept fixed at δ = 0.1a, where a is the particle radius, in this work. We consider the case of an initially homogeneous, disordered suspension at constant volume fraction, say φ = 0.20, and ε below the phase boundary. As ε increases across the phase boundary,

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Figure 1: Phase diagram from Ref. 25 for suspensions of particles interacting with steady or toggled depletion attractions of fixed range δ = 0.1a, where a is the particle radius. The points are the volume fractions φ at periodicsteady-state of coexisting fluid (F) and crystal (C) phases (circles) or coexisting dilute and dense fluid phases (squares) in simulations of toggled suspensions for various time-averaged strengths εξ, made dimensionless by the thermal energy kB T . Each color represents a data set of varying ton at a particular ε and toff , all of which collapse together when plotted in terms of εξ. The solid lines are the coexisting volume fractions predicted by the time-average of equilibrium equations of state, which corresponds exactly with the phase diagram for steady depletion attractions with εξ substituted for ε.

previously shown that two phases at periodicsteady-state may only coexist if their timeaveraged pressure and time-averaged chemical potentials are equal. 25 These time-averages can be constructed from known equations of state valid for equilibrium, steady interactions. For the case here of toggling short-ranged isotropic attractions of hard spherical particles, the timeaveraged equations of state take a particularly simple form, and the phase behavior is completely described by only two parameters, the volume fraction φ and the time-averaged attraction strength εξ, rather than four, (φ, ε, ξ, toff ). The phase diagram describing periodic-steadystates in toggled attractive dispersions, Figure 1, is identical to that for steady attractive dispersions with εξ in place of ε. Note that in the toggled suspensions we refer to ε as the absolute attraction strength and εξ as the time-averaged attraction strength. Unlike in the steady attractive suspension, the self-assembly kinetics in the toggled attractive suspension are not constrained by the location in the phase diagram. Because only two parameters (φ, εξ) are needed to specify a location in the phase diagram, the other two (ξ, toff )

quickly. In the case of toggled particle interactions, two extra parameters, ton and toff , in addition to φ and ε influence the suspension selfassembly. It is convenient to normalize ton by the toggle period T ≡ ton + toff and treat the duty fraction ξ ≡ ton /(ton + toff ) as an independent variable rather than ton . Though we gain additional control over the self-assembly, we lose the predictive power of equilibrium thermodynamics, which is no longer valid with a time-dependent interaction potential. We have

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Phase Separation Kinetic Mechanisms

can be optimized to yield desired self-assembly kinetics. Figure 3 shows that for any particular value of εξ, the toggled attractions produce more crystal after 10000τD than steady attractions of equivalent absolute or time-averaged strength, where τD ≡ 6πµa3 /kB T is the characteristic time for a particle to diffuse its own radius a through a fluid of viscosity µ. Equivalently, the rate of crystallization is always faster with toggled attractions than with steady attractions of equivalent absolute or time-averaged strength. Generally, the crystallization rate increases as the toggle frequency decreases. Toggled attractions also produce, larger and higher quality crystals than steady attractions. The quality of crystal is quantified by Lc ≡ Nc /Ni , the ratio of the number of particles in the crystal interior to the number of particles at the crystal interface Ni (see Experimental Section for how we determine Ni ). Nc scales with the crystal volume, while Ni scales with the crystal surface area, so Lc estimates a quantity proportional to the length scale of bulk crystal. Lc tends to increase with crystal size, but fluid/crystal interface, grain boundaries, and defects all contribute to Ni and penalize Lc . At a given Xc , dispersions with a few large, high quality crystalline domains have a larger Lc than those with many small, defective or polycrystalline domains. Thus, Lc serves as one measure for overall crystal quality. Figure 3 shows that for any crystal fraction, Lc is always larger for toggled attractions than steady attractions. The crystal quality generally increases as the toggle frequency decreases. A broad range of ξ and toff at constant ε for toggled attractions lead to fast kinetics and high quality structures whereas only a narrow range of ε for steady attractions produces acceptable crystallization rates and qualities. Finding this narrow crystallization slot for steady attractions and then designing a control scheme to ensure the dispersion stays within it can be challenging. With toggled attractions, control over ε is not important, and only the toggle parameters, which are easily controlled, need to be adjusted.

Figure 4 shows the crystal fraction in many simulations after 10000τD as a color density plot in toff − ξ space overlayed with regions where different kinetic mechanisms are observed in the toggled attractive suspensions. For small duty fractions, there is insufficient driving force for self-assembly, and the suspension remains a homogeneous fluid. While some aggregation occurs in the on half-cycle, toff is too long and the aggregated structures all dissolve in the off half-cycle. The self-assembly boundary ξ ∗ (toff ), below which no phase separation is observed, is nonmonotonic with respect to toff . Initially, ξ ∗ increases with with increasing toff but eventually peaks and decreases with increasing toff . Above ξ ∗ are regions of fluid/crystal and fluid/fluid coexistence. At low toff ≤ 0.02τD , the suspension phase separates into fluid and crystal phases by one-step nucleation, where crystal nucleates directly in the initial fluid (third row of Figure 5). At moderate toff ≥ 0.05τD and ξ ≥ ξc2 (toff ), the suspension reaches fluid/crystal coexistence by two-step nucleation, where droplets of dense fluid nucleate first and grow before crystal nucleates and grows within the dense fluid (fourth row of Figure 5). ξc2 (toff ) is the boundary above which the second crystal nucleation occurs. The transition between crystal nucleation and fluid nucleation occurs between 0.02τD ≤ toff ≤ 0.05τD for all ξ. The maximum in the phase boundary ξ ∗ (toff ) occurs precisely at this transition between crystal and fluid nucleation. For sufficiently large toff and ξ < ξc2 , the second crystal nucleation is suppressed entirely and only fluid/fluid phase separation occurs (fifth row of Figure 5). The boundary ξc2 sharply increases with toff . In fact, at toff ≥ τD , crystal nucleation is suppressed entirely for all ξ we simulated. At small toff and large ξ, the suspension undergoes a spinodal-like decomposition and initially forms a disordered, percolated gel. The toggling protocol allows the structure to relax defects, and the gel slowly coarsens and becomes more crystalline (first row of Figure 5). The coarsening is sufficiently slow that

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Figure 3: Left: The crystal fraction Xc as a function of steady attraction strength ε or toggled timeaveraged attraction strength εξ, made dimensionless by the thermal energy kB T , for various values of toff . Right: The ratio of number of crystalline particles Nc to number of interfacial particles Ni as a function of crystal fraction. For the steady attractions, ε varies, but for the toggled attractions, ε = 10kB T is fixed and ξ varies. At any particular steady ε, the crystallinity is higher in toggled suspensions of equivalent εξ. At any particular crystallinity, Nc /Ni is larger in toggled suspensions. For clarity, not all of the data collected in Figure 4 is shown here. the structure remains percolated at the end of 10000τD . The gel line ξgel (toff ), above which the global structure is percolated, is a monotonically increasing function of toff . Below ξgel , phase separation may still proceed by gel coarsening, but the rate of coarsening is sufficiently high that the structure depercolates and condenses into more compact domains within the duration of observation. Thus, ξgel depends on the finite observation time. It should be noted that while the space around ξ ∗ was sampled with high resolution, ξgel was not, so the shape of ξgel is less certain than the shape of ξ ∗ . In the one-step fluid/crystal, two-step fluid/crystal, and fluid/fluid coexistence regions, points on or very close to the boundary ξ ∗ exhibit nucleation and growth of a few, noninteracting nuclei, shown in the bottom three rows of Figure 5. Farther from the phase boundary, many nuclei form and significant coalescence and Ostwald ripening occur, as in the second row of Figure 5. As we approach the gel line ξgel from below, spinodal-like decomposition starts to occur along with nucleation and growth, coalescence, and ripening, and there is a complicated combination of many different

kinetic mechanisms. Necessarily, the crystal fraction is zero in the fluid/fluid and homogeneous fluid regions. In the one-step fluid/crystal and gel regions, the crystal fraction is generally a monotonically decreasing function of ξ moving away from the ξ ∗ boundary at constant toff . In the two-step fluid/crystal region, the maximum in Xc at constant toff shifts away from the phase boundary. Both the one-step and two-step fluid/crystal regions achieve similar crystallinities, but for any particular Xc , the size and quality of the crystals are significantly higher in the two-step region than in the one-step region, as shown in Figure 3.

Gel Coarsening Colloidal gels formed by steady, strong, and short-ranged attractions coarsen slowly over time according to a power law, L = L0 + ktn ,

(1)

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Figure 5: Kinetic mechanisms observed in simulations of toggled attractive suspensions. Each row shows snapshots of the self-assembling suspension over time, with the snapshots in the first column taken at time t = 0 and in the last column at around t = 10000τD . Snapshots in the other columns do not correspond to the same time points. Crystalline particles are colored light blue in rows two through five and dense fluid is colored pink in rows four and five. ACS Paragon Plus Environment

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tial network formed, k is a rate constant dependent on the attraction strength ε and particle volume fraction φ, and n is the power law exponent. The power law exponent is set by the kinetic mechanism that dominates coarsening. n = 1/3 when the kinetics are limited by bulk diffusion, n = 1/4 when the kinetics are limited by surface diffusion, 47 and n = 1/6 is observed at early times in spinodal decomposition. 48 Although the strong attractions favor a compact, crystalline conformation in thermodynamic equilibrium, the local microstructure is arrested in a disordered configuration, with an average contact number around Nnb = 6 – 7 49 rather than a fully occupied nearest neighbor shell of Nnb = 12. To coarsen, thermal fluctuations must break the interparticle “bonds" in the gel to allow the particles to rearrange. Because the bond strength is O(ε) and many times larger than the thermal energy in the gel-forming regime, bondbreaking is incredibly slow and the coarsening process takes on the order of days or weeks. 48 In our simulations at ε = 10kB T , steady attractions form an arrested gel that only reaches a crystal fraction of Xc = 0.005 in 10000τD (see Figure 2). If this average crystallization rate held constant, it would take around 106 τD (or around 10 days for a typical τD = 1 s) to crystallize half of the particles, consistent with experimental time scales. Toggling the particle attractions greatly enhances the rate of coarsening by accelerating the rate of bond-breaking. In the off halfcycle, all particle bonds are immediately broken, so there is no kinetic barrier preventing microstructural rearrangement. Figure 6 shows that the coarsening rate of the toggled attractive gel is orders of magnitude faster than the coarsening rate of the steady attractive gel at ε = 10kB T . Unlike the steady attractive gel, the local microstructure of the toggled attractive gel is highly crystalline with Nnb = 12. Although the global structure is disordered and percolated, the toggling protocol anneals local defects and allows the particles to find low energy, closest-packed crystalline configurations. If the time scale for local crystallization is short compared to the time scale for global coars-

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ening, the gel strands are nearly entirely crystalline and slowly grow larger over time, as in the top row of Figure 5. We observe this to hold in nearly the entire range of the gel coarsening regime, with the local crystallization and coarsening rates becoming comparable only at the large toff ≥ 0.05τD edge of the region. In this case, Lc = Nc /Ni serves as a good estimate for a quantity proportional to the characteristic length scale L of the gel strands. We performed Brownian dynamics simulations of toggled attractive gels at φ = 0.20, ε = 10kB T , and a variety of points (toff , ξ) in the gel coarsening region of Figure 4. Figure 6 shows the evolution of the gel length scale Lc over time. We observe that coarsening of toggled attractive gels is also described by the power law in equation (1). (1) is relevant on time scales long compared to the toggle period, and is not intended to describe the fast dynamics within a single toggle period, which are of little interest compared to the slow dynamics of phase separation. The rate constant k for the steady attractive gel is a function of ε and φ. For the toggled attractive gel, k is also a function of ton and toff , even with ε and φ fixed. The dependence of k on the toggle parameters reflects the competing processes of aggregation in the on half-cycles and structural rearrangement in the off half-cycles. Because the attraction is strong compared to the thermal noise, the suspension microstructure quickly freezes in place at the start of each new on half-cycle. For the remaining duration of the on half-cycle, essentially no more structural changes occur because the strong bonds are difficult to break. Time is “wasted" sitting in the frozen configuration until the bonds are broken at the end of the on half-cycle. The value of ton has no effect on the rate of coarsening other than delaying the next off half-cycle. The larger ton , the longer it takes between structural rearrangements in the off half-cycles, and the slower the coarsening rate. We can remove the effect of ton by tracking the coarsening kinetics in terms of the number of toggle cycles that have occurred, Ncycle ≡ t/T , rather than the absolute time t.

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Recasting (1) in terms of Ncycle ,  n t 0 L = L0 + k , T

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for all simulations. A good choice for ` is the capture radius rc , the separation distance at which the strength of the interparticle attraction |U (r)| is equal to the thermal energy kB T . With this choice, particles that diffuse r < rc in the off half-cycle have |U | > kB T and are recaptured when the attractions are turned on while particles that diffuse r > rc have |U | < kB T , with enough thermal energy to avoid capture. For the particular form of the interparticle attraction we have chosen, rc = 0.14a (see Experimental Section). We estimate f by integrating the probability distribution P (x, t = toff ) for ideal Gaussian diffusion, Z ∞ 1 2 e−|x| /4toff (4) dx √ f= 4πtoff rc rc −rc2 /4toff rc =√ e + erfc √ . (5) πtoff 4toff

(2)

where the new rate constant k 0 (toff ) ≡ kT n is a function of toff only. ton only affects the rate constant k through its effect on the period T . Figure 6 shows that when Lc is plotted versus Ncycle , all kinetic curves with the same toff , but different ton , collapse together. Equation (2) holds when ton  toff , but may break down for ton ≈ toff , where the amount of rearrangement in the on and off half-cycles become comparable. The extent of structural rearrangement in the off half-cycle depends on the amount of time the particles diffuse. k 0 (toff ) increases with increasing toff , indicating that particles with more time to rearrange tend to promote enhanced coarsening. If toff is short, particles will not diffuse very far from their initial positions in the off half-cycle, and the amount of structural rearrangement will be small. It is difficult for the suspension to sample new configurations, and coarsening proceeds slowly. Presumably then, it is easier to sample new configurations the longer the particles diffuse, and coarsening occurs faster at longer toff . One way to quantify the extent of structural “arrangement" is to track the fraction of particles f that diffuse a sufficient distance ` in the off half-cycle. We assume that the 1 − f fraction of particles that failed to diffuse sufficiently far in the off halfcycle return to their same configuration when the attractions are turned back on and cannot contribute to coarsening. The f fraction of particles that succeeded in diffusing a distance ` find a new configuration and can contribute to coarsening. At every cycle then, the coarsening rate dL/dNcycle is a factor of f smaller than we might expect if all particles could contribute to coarsening. Thus, we can recast (2) again as  n t 00 , (3) L = L0 + k f T

In reality, the particles have hard repulsions that modify their diffusion. Some particles will diffuse more slowly than ideal diffusion because they are caged while others will diffuse more quickly due to stronger osmotic pressure gradients, so the true f is different than the one here. Figure 6 shows that when Lc /f is plotted versus t/T , all the kinetic data collapse to a single master curve. The best fit values of the power law exponent and the rate constant in (3) are n = 0.24 ± 0.05 and k 00 = 0.057 ± 0.021 (see Experimental Section for details on the fit). We ignore the effect of L0 , which only affects the short time kinetics, by assuming L0 = 0 for the fits. The average value of n = 0.24 supports that surface diffusion (n = 1/4) likely plays the dominant role in setting the coarsening rate, though it is difficult to distinguish between the other possible power laws (e.g. n = 1/3 or n = 1/6). Maximizing the Coarsening Rate As toff increases, the fraction of particles f able to diffuse rc in toff increases, enhancing coarsening. However, increasing toff also increases the toggle period T = ton + toff , which reduces the number of toggle cycles and hinders coarsening. There is an optimal toff = t∗off given by ∂k/∂toff = 0, where k ≡ k 00 f /T n , at which the

where now the rate constant k 00 ≡ kT n /f is independent of both ton and toff and is constant

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Figure 6: Evolution of the characteristic length scale Lc = Nc /Ni of toggled dispersions in the gel coarsening region of Figure 4 for fixed ε = 10kB T and varying ton and toff , made dimensionless by the diffusion time τD . The top row shows data in the regime where our model applies. A: The length scale increases as a power law in time t, dimensionless on τD . B: In terms of number of cycles, Ncycle ≡ t/(ton + toff ), curves with the same toff collapse together. C: Dividing Lc by the fraction of particles f diffusing the capture radius rc from equation (4), the curves all collapse to a single universal curve (dashed line). The bottom row shows data outside of where our model applies. In moving from panel D to panel E, the curves at toff = 0.001 do not collapse, while the curves at toff = 0.05 show a different kinetic mechanism, indicating that our model does not describe these regimes. Only the toff = 0.01 curves behave like the model predicts.

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Langmuir

hancement (a factor of ≈ 2) in the crystallization rate for the toggled gel. The approach to an effectively steady attraction may prevent collapsing the coarsening data in terms of Ncycle . At larger toff ≥ 0.05τD (with ξ ≥ ξgel ), the time scale for local crystallization is slow compared to the time scale for global coarsening. As a consequence, fluid/fluid phase separation occurs rather than fluid/crystal phase separation and the local microstructure of the percolated network is fluid-like initially. After some time, crystal nucleates within the dense fluid network and rapidly grows to crystallize the entire network, at which point the structure slowly coarsens like usual in the gel coarsening regime. From the way we estimate Lc = Nc /Ni , the gel length scale rapidly increases after crystal nucleation, even though the true length scale set by the dense fluid network is already large here. Interestingly, the data still collapse when plotted in terms of Ncycle , but equation (3) does not account for two-step nucleation and cannot describe the full suite of coarsening kinetics.

rate constant k (and therefore the coarsening rate dL/dt) is maximized. At constant ton , the value of t∗off depends on ton . At constant ξ, t∗off is independent of ξ because T = toff /(1 − ξ) and so ξ contributes only to the coefficient in k = k 00 (1−ξ)n f /tnoff . Maximizing k with respect to toff at constant ξ yields ∂k = 0, ⇒ t∗off = 0.011τD . (6) ∂toff ξ This is very close to the time it takes for a particle to diffuse the range of the interaction, δ 2 /D = 0.01τD , allowing for the most structural rearrangement in the off half-cycle while still reaggregating in the on half-cycle. A toff = 0.01τD had been argued to be optimal in previous work on these toggled attractive suspensions. 23,25 The local maximum toff ≈ 0.01τD at constant ξ can be seen in the crystal fraction Xc in Figure 4 at large ξ > 0.9 in the gel region, where our kinetic model works best. For any particular toff , k also increases with decreasing ξ. This agrees with the monotonic trend of the crystal fraction at constant toff in the left half of Figure 4 with toff ≤ 0.02τD .

Nucleation and Growth In the fluid/crystal and fluid/fluid coexistence regions of Figure 4, distinct clusters nucleate in the toggled suspensions rather than many nuclei forming a network as in the gel region. The clusters grow, coalesce, and Ostwald ripen as the suspension heads toward a periodic-steadystate of two-phase coexistence. As the selfassembly boundary ξ ∗ is approached from above by lowering the duty fraction, the driving force for self-assembly decreases and few nuclei form. Very close to the phase boundary, only a single nucleation event occurs. Here, coalescence and Ostwald ripening are entirely suppressed, and phase separation occurs purely by nucleation and growth. This is the simplest place to understand the self-assembly kinetics in the coexistence regions, so we focus on a kinetic model for the nucleation and growth of a single, spherical nuclei close to the phase boundary, as in the bottom three rows of Figure 5. Suppose a dense spherical nucleus of radius R and volume fraction φn has already formed in the surrounding dilute fluid at volume fraction

Relevant Domain of the Model Equation (3) describes the coarsening data well in the decade 0.002τD ≤ toff ≤ 0.02τD when ton  toff , but breaks down elsewhere in the gel coarsening regime. For toff ≤ 0.001τD (with ξ ≥ ξgel ), the coarsening data does not collapse together when plotted in terms of Ncycle , shown in the bottom row of Figure 6. Although the coarsening rate at constant toff increases with decreasing ton (as in (3)), evidently the value of ton influences k more than just its effect on the toggle period. One possibility is that the time scale for toggling (T ) is so fast relative to time scale for particle diffusive motion (τD ) that the particles see a steady attraction of timeaveraged strength εξ. Figure 3 shows the crystallinity of the toggled attractive gel approaching that of the steady attractive gel of equivalent time-averaged strength as toff decreases, although even at the shortest times investigated here, toff = 0.001τD , there is still decent en-

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φf . The nucleus can be composed of crystal, dense fluid, or both. When the attractions are on, particles prefer the dense phase and the nucleus grows. Particles within δ of the nucleus are quickly driven towards the nucleus at the beginning of the on half-cycle by the strong interparticle attractions. The number of particles in this δ layer is Nlayer =

δφlayer A , v0

of a single particle in a homogeneous phase, the collective diffusivity describes motion of a collection of particles moving along concentration gradients. The two diffusivities are related as Dc = Ds φ(∂(µ/kB T )/∂φ), so Dc contains additional thermodynamic information capturing interactions between particles. Here, Dnc /a estimates the enhanced speed (over Dns /a) of particles out of the nucleus when the attractions are off due to the high osmotic pressure of the dense nucleus. At constant volume and number of particles, the background fluid becomes depleted in particles as the nucleus grows because φn > φf ,

(7)

where A = 4πR2 is the surface area of the nucleus, v0 = 4πa3 /3 is the volume of a single particle, and φlayer is the volume fraction of the layer, which is unknown but between the densities of the nucleus and the surrounding fluid, φf ≤ φlayer ≤ φn . The time it takes for a particle δ away from the nucleus to reach the nucleus directed by the interparticle force of ε/δ is tlayer ≈ δ 2 kB T τD /a2 ε. For ε = 10kB T and δ = 0.1a, as in the all the simulations presented, tlayer = 10−3 τD . If ton ≥ tlayer , all of the particles in the thin δ layer will be incorporated into the nucleus in the on half-cycle, independent of ton . Because the attractions are strong and shortranged, the remaining particle flux into the nucleus in the on half-cycle is diffusion-limited, Jon

Dfs φf , =− av0

φ0 − φn 4πR3 nnuc /3 (10) 1 − 4πR3 nnuc /3 4πnnuc 3 = φ0 + (φ0 − φn ) R + O(φ2nuc ), (11) 3

φf =

where φ0 is the overall particle volume fraction, nnuc is the number density of nuclei, and φnuc ≡ 4πR3 nnuc /3 is the overall volume of the growing nuclei relative to the system volume (and not the volume fraction within the nuclei itself, φn ). The largest φnuc can become is φ0 , when all particles are incorporated into the nucleus, so we can drop the O(φ2nuc ) terms and higher. Here, φ0 = 0.20, so neglecting higher order terms suffers only a 4% error in the worst-case scenario and only near the end of the phase separation. While φf decreases as the nucleus grows, we assume φn is constant and given as the known coexisting volume fraction at periodic-steady-state from Figure 1. 25 If ton and toff are not too long compared to 2 a /Dfs and a2 /Dnc , the nucleus cannot change size appreciably within a single half-cycle, and the change in the number of particles in the nucleus, ∆Nn , over one toggle cycle may be approximated as,

(8)

where the self diffusivity in the fluid Dfs is used to estimate a characteristic diffusive speed Dfs /a of the fluid particles. Notice that the flux is negative, indicating that particles are moving into the nucleus. When the attractions are off, particles are pushed out of the dense phase and the nucleus shrinks. The particles interact only through hard repulsions, so the particle flux out of the nucleus is also diffusive, Joff

Dc φn = n , av0

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1 ∆Nn = (Nlayer − Jon Aton − Joff Atoff ) , T T

(9)

(12)

On time scales long compared to the toggle period T , ∆Nn /T ≈ dNn /dt, and we have a first-order ordinary differential equation for the

but the appropriate diffusivity is the collective diffusivity Dnc of the particles in the nucleus. While the self-diffusivity describes the motion

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˜ at t˜, we obtain an expression implicit in R ˜ R √ √ ˜+R ˜2 ˜ 1 + R 3 3R 1 −1 + tan . (19) t˜ = ln ˜ 2 ˜ 6 3 (1 − R) 2+R

nucleus radius,  Dfs φ0 dR 1 D c φn = δφlayer + ton − n toff dt T φn a a  s Df 4πnnuc 3 (13) − ton (φn − φ0 ) R a 3 ≡ k1 − k2 R3 , (14)

˜= We could have begun the integration from R ˜ ∗ at t˜ = t˜i , where R ˜ ∗ is the initial size of the R nucleus at the nucleation event occurring after an induction time ti , which adds a constant factor to the right side of (19). We observe that ˜ ∗ . 0.1 for nucleation events that occur in the R simulations, so we ignore this term, with the understanding that t˜ refers to time after nucle˜ → 1 with the asymptotic ation. As t˜ → ∞, R form √ √ ˜ ≈ 1 − Ce−3t˜, C ≡ 3e 3π/6 . (20) R

where δφlayer Dfs φ0 Dnc k1 = + ξ− (1 − ξ), T φn aφn a Dfs 4πnnuc k2 = ξ (φn − φ0 ) , aφn 3

(15) (16)

and we have used Nn = φn (R/a)3 to write dNn /dt = (3φn R2 /a3 ) dR/dt. k2 > 0, so the second term, arising from the depletion of the background particle concentration, acts to shrink the nucleus. k1 must be positive for any growth to occur, so the condition k1 > 0 sets a critical duty fraction ξ ∗ necessary for growth. Whether this critical ξ ∗ coincides with the predicted phase boundary in Figure 1 from 25 is unclear. From (14), we immediately extract the limiting behavior. Initially, the growing nucleus is small, and so the second term in (14) is negligible. The nucleus grows linearly in time at a constant rate k1 . As the nucleus grows, so does the magnitude of the O(R3 ) term. Eventually, growth ceases when the two terms in equation (14) exactly cancel and the nucleus reaches a terminal size,  3 k1 , (17) Rf ≡ k2

Figure 7 shows the growth of nuclei in Brownian dynamics simulations of toggled attractive dispersions at different points close to the phase boundary ξ ∗ in Figure 4. Each curve corresponds to a different point along ξ ∗ and a single curve represents the average growth for 5-10 replicates of the same point. The terminal radius Rf of each curve was easily extracted from the late time data, and the rate constant k1 was determined from a least squares linear fit of (19) to the data. The values of k1 and Rf along the phase boundary ξ ∗ are shown in Figure 8. All of the data collapses to a single curve ˜ = R/Rf versus t˜ = k1 t and when plotted as R agrees well with (19) and (20). Estimating k1 and Rf Parameters The terminal nucleus size Rf is set by the number of particles leaving the nucleus in the off half-cycle balancing the number of particle condensing to the nucleus in the on half-cycle. One way to estimate Rf is to estimate the kinetic parameters k1 and k2 and then use equation (17), which reflects the balance between particle fluxes in the on and off half-cycles. At periodicsteady-state, the dense phase in the nucleus is in coexistence with the surrounding dilute fluid phase. Therefore, another way to estimate Rf is to use the phase diagram in Figure 1 to compute the nucleus size that yields nucleus and background fluid volume fractions equal to their

where the suspension is at periodic-steady˜ ≡ R/Rf state. Recasting (14) in terms of R and t˜ = k1 t, ˜ dR ˜3. =1−R (18) ˜ dt While R(t) in (14) depends on the values of ton ˜ t˜) in (18) does not, and the kinetics and toff , R( have universal behavior. Separating variables ˜ = 0 at t˜ = 0 to in (18) and integrating from R

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Figure 7: Left: Growing radius R, made dimensionless by the particle radius a, of a single, spherical nucleus in the toggled suspensions as a function of time t after nucleation, made dimensionless by the diffusion time τD , for different points along the nucleation boundary ξ ∗ in Figure 4. Right: The data collapse to a single master curve when the radius is scaled by the terminal radius Rf and time multiplied by the fit rate constant k1 . The curve agrees well with the solid line from equation (19). The inset shows that the asymptotic behavior of the data as R/Rf → 1 also agrees with the dashed line from equation (20). coexisting volume fractions. For each duty fraction ξ, we extract the coexisting volume fractions φn and φf . With a known overall volume fraction φ0 , the lever rule sets the total fraction (by volume) of the dense phase to be (φ0 −φf )/(φn −φf ). From the known nucleation density nnuc = 1/V , where V is the total dispersion volume in our simulations, we can then compute the terminal volume and radius of the dense phase. The predicted Rf is shown in Figure 8 along with the Rf from the simulations. Generally, Rf decreases in the simulations as toff increases. The predicted Rf overestimates the true Rf and remains fairly constant with toff until it sharply decreases at large toff , where the coexisting volume fractions of the dense and dilute phases approach each other. The predictions assume a sharp interface between the nucleus at φn and the surrounding fluid at φf , to compute a precise nucleus radius. However, the real interface is diffuse, with the volume fraction decaying continuously from φn to φf moving away from the nucleus. The discrepancy

between the simulation data and the prediction could be due to the diffuse interface region growing larger as toff increases. In the worse case scenario, the theory only overpredicts the true Rf by ≈ 20% and can still provide a fair estimate of its magnitude over a wide parameter range. The initial growth rate k1 is more difficult to estimate. From equation (15), k1 is a sum of three terms. The first term represents the particles in the thin δ layer that rapidly condense to the nucleus at the beginning of each on halfcycle. This happens once per cycle, so the contribution to k1 decreases as 1/T as the cycles become longer. The second and third terms in equation (15) are from the diffusive contributions in the on and off half-cycles. Assuming that all the parameters in (15) do not change significantly compared to toff as we move along the nucleation boundary ξ ∗ (toff ), k1 follows the form C1 + C2 , (21) k1 ≈ T

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where C1 and C2 are constant. Figure 8 shows that the k1 observed in the simulations agrees with (21) with the linear least squares fit values of C1 = 10−5 a and C2 = 4 × 10−4 a/τD . We can estimate C1 and C2 by estimating the terms in (15). φ0 , φlayer , φn , and ξ are all O(0.1 – 1). If we assume that the diffusivities Dfs and Dnc are on the same order of magnitude as the diffusivity of an isolated particle, O(a2 /τD ), equation (15) predicts that C1 ≈ O(0.01a) and C2 ≈ O(0.1a/τD ) about 103 times larger than observed in the simulations. There is likely a multiplicative prefactor missing in (15) that slows the growth time scales by several orders of magnitude. Though we cannot quantitatively predict the C1 and C2 parameters, the variation of k1 with toff agrees with our model.

The first term is the surface energy of the nucleus and contains the interfacial tension γ between the two phases. This term is positive and always contributes an energy penalty to phase separation. The second term is the bulk Gibbs energy difference between the nucleus and the equivalent number of particles in the surrounding phase and contains the difference in chemical potential ∆µ ≡ µn (φn )−µf (φf ) between the two phases. This term is negative in the phase coexistence regime. Whether the nucleus grows or shrinks is dictated by the sign of d∆G/dNn . Because the surface and bulk terms in (22) scale differently with R, there is a critical nucleus size R∗ =

40

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(23)

at which ∂∆G/∂Nn = 0 and ∆G has a maximum. Clusters smaller than R∗ reduce ∆G by shrinking, while clusters larger than R∗ reduce ∆G by growing. Because only clusters with R > R∗ can grow, there is a finite induction time ti for fluctuations in the dispersion to produce a critical cluster before growth can occur. The induction time grows exponentially with the nucleation barrier height, ∆G(R∗ ), 50

44

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1

off

ti ∼ e∆G(R

Figure 8: Left: Growth rate constant k1 in equation (14), made dimensionless by a/τD , as a function of toff , made dimensionless by τD , along the nucleation bounday ξ ∗ from Figure 4. The points are data extracted from simulations while the dashed line is a fit of (21). Right: The terminal nucleus radius Rf , made dimensionless by the particle radius a, extracted from simulations (filled points) and predicted (open points) from the phase diagram in Figure 1.

∗ )/k

BT

= e4πγR

∗2 /3k T B

.

(24)

∗ a nuEquation (23) gives the critical size Rstdy cleus must reach to grow for steady attractions. For toggled attractions, all nuclei will shrink in the off half-cycle and only those larger than ∗ Rstdy will grow in the on half-cycle. The critical radius R∗ for toggled attractions must be at ∗ ∗ least as large as Rstdy . We can estimate Rstdy using the equations of state for short-ranged attractions, 45 which gives ∆µ = −22kB T between the equilibrium crystal phase φc = 0.730 and metastable fluid phase φf = 0.20 at ε = 10kB T . The interfacial tension of colloidal dispersions with short-ranged attractions is small and on the order γ = O(0.1 – 1kB T /a2 ). 51–53 Thus, ∗ Rstdy ≈ a and individual particles are already the size of the critical nucleus. Because the in∗ 2 duction time goes as ti ∼ e(R /a) , there is essentially no barrier to nucleation and nucleation should occur immediately. However, we do ob-

Critical Radius In classical nucleation theory, the change in Gibbs free energy ∆G of forming a spherical nucleus of radius R at volume fraction φn in a homogeneous phase at volume fraction φf is given as, 50  3 R 2 ∆G = 4πR γ + ∆µφn . (22) a

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serve a finite induction time in our simulations. This implies that there must be another mechanism that is setting a second critical radius ∗ R∗ > Rstdy . One possibility is that R∗ must be the size of the smallest cluster that survives a single off half-cycle. If R < R∗ , the cluster will completely dissolve in the off half-cycle and cannot grow in the subsequent on half-cycle. If R > R∗ , the cluster does not dissolve completely in the off half-cycle and continues to grow in the on half-cycle. This seems plausible for longer toggle periods, where toff is long enough for significant dissolution, but does not necessarily explain the finite induction time for short toggle periods. For example, a cluster of any size is unlikely to dissolve completely in toff = 0.001τD , yet we still observe a significant induction time in this case. Whatever the mechanism is to set R∗ , it is certain that this critical cluster must form in the on half-cycle (or over several consecutive on half-cycles), as all clusters shrink in the off halfcycle. The longer ton , or equivalently the larger ξ, the more likely it is to form a cluster of the critical size. When ξ becomes large enough that nucleation of a critical cluster becomes likely in the simulation time, a nucleation event will occur. This sets the boundary ξ ∗ for a given finite observation time. This ξ ∗ must be at least the ξ that guarantees k1 > 0 in equation (15), because k1 > 0 is required for any cluster to grow (according to our model).

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dense fluid droplets nucleate, they rapidly coalesce and ripen to form a single bulk dense fluid domain. Crystal nucleates preferentially in the dense fluid, so all crystal domains are localized rather than dispersed throughout the suspension volume. It is easy for the crystals to coalesce and ripen into a single crystalline domain with very little interface and high quality. In contrast, crystals nucleate throughout the entire suspension volume in the one-step region. Coalescence is limited by the time it takes for the large crystal clusters to diffuse toward one another, which is long because of the reduced diffusivity of the cluster compared to individual particles. Ripening is limited by the number of particles in the crystalline domains that can diffuse sufficiently far in the off-cycle to escape the crystal and enter the bulk fluid. Because toff is short in the one-step region, very few particles detach from the crystals and ripening is also slow. Reducing interface is therefore a very slow process in the one-step fluid/crystal region and so Lc is smaller here. Effect of Excluding Hydrodynamics Our Brownian dynamics simulations neglect interparticle hydrodynamic interactions (HI) and instead assume that each particle experiences the hydrodynamic resistance of an isolated particle in a solvent. This neglects both the long-ranged, many-bodied far-field hydrodynamic couplings as well as the pairwise nearfield lubrication interactions. 54 Without capturing the correct fluid mechanics of the solvent in which the particles are immersed, our simulation model is not a completely accurate picture of real suspensions. We should be careful to consider how choosing not to incorporate these physics impacts our results. The lubrication interactions are only important for particles that are very close together. As two particles approach each other, an increasingly large force is needed to squeeze out the thin fluid layer in the gap between them, with the force increasing as the reciprocal of gap width. This tends to slow down the dynamics of nearly touching particles and prevents them from coming into contact. This is likely

Discussion Self-Assembling the Largest Crystals In Figure 4, there are two regions of high crystallinity. The first is at small toff and very close to the lower self-assembly boundary ξ ∗ in the one-step fluid/crystal region, and the second is away from the self-assembly boundary in the two-step nucleation region. The crystal quality as measured by Lc = Nc /Ni in Figure 3 however, is much higher for the latter. Because fluid particles are more mobile than crystalline particles, the initial fluid/fluid phase separation in the two-step region is very fast. Even if many

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Conclusion

not important for condensation in our suspension with strong attractions, because the attractions allow the particles to get “close enough" to condense before lubrication plays a significant role. It is possible that lubrication makes it more difficult for a particle to diffuse away from dense clusters in the off half-cycle, but is likely a small effect because of the strong osmotic driving force for dissolution of the dense phases with hard repulsions. Simulations of colloidal gelation have found that including lubrication does not modify the general self-assembly behavior of suspensions. 49,55 The long-ranged far-field hydrodynamic couplings on the other hand are crucial for capturing the correct kinetic behavior of suspensions. Without the far-field hydrodynamic couplings, each particle in a cluster contributes to the diffusivity of the entire cluster, and so the cluster diffusivity decreases as R−3 , where R is the cluster radius. With the far-field hydrodynamics, only the overall cluster size affects the diffusivity so the cluster diffusivity obeys the correct scaling of R−1 . 49 Within dense phases, hydrodynamic interactions are screened and do not contribute to local rearrangements of individual particles. The enhanced cluster aggregation relative to local crystallization tends to promote gelation when far-field hydrodynamics are included 49 and may shift some of the boundaries in Figure 4. Within the gel coarsening region, the microstructure evolves from surface diffusion of individual particles. While the diffusivity of these particles is modified by HI, the mechanism is not. The progression of equation (1) (L vs. t) to equation (2) (L vs. Ncycle ) to equation (3) (L/f vs. Ncycle ) is still valid with inclusion of HI, though the value of k 00 in (3) may change quantitatively. Similarly, our model of nucleation and growth only accounts for a single growing nucleus, whose cluster diffusivity is not relevant. HI will change the diffusivities Dfs and Dnc in (13), but not the growth mechanism.

Dynamically self-assembling processes are ubiquitous in biological systems, from the microscopic scale of DNA replication to the macroscopic scale of fish schooling. 44 These dynamic biological systems have inspired an emerging type of material, active matter, in which selfassembling components are engineered to have mechanical activity, allowing them to, for example, self-propel, 56,57 self-rotate, 58,59 or grow. 60 Because dynamic self-assembly processes are not thermodynamically constrained, they can avoid kinetic arrest and rapidly form ordered structures 13 making them promising methodologies for scalable manufacturing of hierarchical nanomaterials. However, the structures formed by dynamic self-assembly are inherently out-of-equilibrium and cannot be described by the theories of equilibrium thermodynamics. There is currently no general theoretical framework to predict the outcome of dynamic selfassembling processes. New theories must be developed before dynamic self-assembly can be effectively harnessed to design novel nanomaterial fabrication techniques. In this work, we examined the phase separation kinetics of one such dynamically selfassembling system, a nanoparticle suspensions with interparticle attractions toggled on and off cyclically in time. Because dynamic selfassembly with toggled attractions is only a “first-order" perturbation away from equilibrium, passive self-assembly, we were able to develop simple models to describe the complex kinetics of the toggled suspensions. These models complement a theory we previously developed for predicting the terminal structures of toggled self-assembly. 25 Knowledge of the final out-ofequilibrium structures as well as their rates of formation, will aid in the design of scalable dynamic self-assembly processes to synthesize useful nanomaterials.

Experimental Section We perform dynamic simulations of a monodisperse suspension of N = 64000 spherical colACS Paragon Plus Environment

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loidal particles of radius a in a solvent of viscosity µ. The equations of motion are overdamped at the colloidal scale and can be numerically integrated as xi (t + ∆t) = xi (t) +

N X

of a time step due to other forces are moved exactly to contact. 62 For simulations with toggled attractions, the hard sphere potential U HS (r) is always present, but the depletion potential U dep is turned off for a time toff and back on for a time ton cyclically in time (equivalently, ε in (28) is switched between a finite value and zero). All lengths are made dimensionless by the particle radius a, all energies are made dimensionless by the thermal energy kB T , and all times are made dimensionless on the characteristic particle diffusion time τD ≡ 6πµa3 /kB T . In this work, the particle volume fraction is fixed to φ = 0.20 and the range of the depletion attraction is fixed to δ = 0.1a. For steady attractions, the attraction strength ε varies for different simulations, but for toggled attractions, the attraction strength is fixed to ε = 10kB T , with only the duty fraction ξ ≡ ton /(ton + toff ) and toff changing. In the kinetic data in Figure 3, the data is plotted in terms of εξ on the x-axis, but only ξ is changing with fixed ε = 10kB T . Note that the εξ y-axis of the phase diagram in Figure 1 from Ref. 25 holds for any combination of ε and ξ. All simulations began from a homogeneous, disordered configuration and ran for around 10000τD . The integration time step was chosen sufficiently small to resolve the fastest time scale in the suspension. For long ton and toff , the diffusive motion of a single particle is the fastest time scale, and we use a time step of ∆t = 10−4 τD when both ton and toff are longer than or equal to 0.01τD . This time step is sufficiently short to accurately capture the dynamics of colloidal particles with our simulation method. Where either ton or toff fall below 0.01τD , we use a smaller time step ∆t = 10−5 τD to resolve the dynamics within each individual half-cycle. At φ = 0.20, Figure 1 shows that the crystals that form are nearly closest-packed φ ≈ 0.74 for almost any ε/kB T , with each particle contacting Nnb = 12 neighboring particles. The densest disordered packing is much lower, φ ≈ 0.64, so disordered particles contact fewer than Nnb < 12 particles. We can distinguish between crystalline and fluid-like particles using the contact number. We consider particles separated

Mij · Fj (t)∆t, (25)

j=1

where xi is the position of particle i, Mij is the hydrodynamic mobility tensor that couples a force on particle j to the velocity of particle i, ∆t is the integration time step, and Fi is the P total nonhydrodynamic force Fi ≡ FB i + Fi including a normally-distributed stochastic BrowP nian force FB i and an interparticle force Fi arising from interaction potentials. In Brownian dynamics (BD) simulations, interparticle hydrodynamic interactions are neglected, so the mobility tensor is Mii =

1 I, 6πµa

Mij = 0,

(26)

where 6πµa is the Stokes drag coefficient and I is the identity tensor. In this case, the Brownian force satisfies, hFi i = 0,

hFi Fj i =

12πµakB T δij . ∆t

(27)

The interparticle force is a sum of a short-range attraction given by the Asakura-Oosawa depletion potential, 61 U

dep

ε(a + δ)3 3 rij (r) = − 2 1− δ (3/2 + δ) 4a+δ  3 ! 1 rij + , (28) 16 a + δ

when 2a ≤ r ≤ 2(a + δ) and a hard sphere repulsion given by the Hookean spring potential 1 U HS (r) = k(2a − r)2 , 2

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(29)

when r < 2a, with a time step dependent spring constant of k ≡ 3πµa/∆t. This form for the hard sphere repulsion is functionally identical to the Heyes-Melrose “potential-free" algorithm, in which particles that overlap over the course

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by a distance less than r ≤ 2.1a as “contacting". A particle is designated as “crystalline" if it has a contact number of Nnb = 12. We denote the number of crystalline particles as Nc and the crystal fraction as Xc ≡ Nc /N . A particle is designated as “interfacial" if it is contacting a crystalline particle, but doesn’t have 12 contacts itself. These particles are not necessarily disordered, because they sit on a crystal lattice site, but do not have the same local structure as the bulk crystal particles due the fluid phase on one side. We denote the number of interfacial particles as Ni . A particle that is neither crystalline nor interfacial (i.e. has a contact number less than 12 and is not contacting a crystalline particle) is designated “fluid-like". In the case of fluid/fluid phase separation, we can distinguish between particles belonging to the dense and dilute fluid phases using the particles’ local volume fraction φ` . The local volume fraction was determined by looking at a sphere of radius 6a centered around a particle and calculating the fraction of this search sphere’s volume that was occupied by particles. The central particle’s volume was included as well as partial intersections of particle volume with the search sphere surface. Strictly, this represents density only conditionally around a central particle and so overestimates the true local density. However, this allows us to assign a local density unambiguously to a particle rather than to a position in space, and so is more useful as a particle-based order parameter. The error in using our approach is small for all but the most dilute regions. The power law fit for the gel coarsening data in Figure 6 was computed by first performing a two parameter linear least squares fit of log(Lc /f ) versus log Ncycle data to find n and k 00 in equation (3) for each data set in Figure 6. The values of n for each data set were averaged together to give the average n, with an uncertainty given by the standard deviation. Then, an additional one parameter linear least squares fit was performed to find k 00 again for each data set, assuming the average value of n = 0.24. The values of k 00 for each data set were averaged together, and the uncertainty is given by the standard deviation. This procedure ensures

that each data set contributes equally to the fit parameters rather than biasing the fit toward the data sets containing more data points. To determine the size of the nucleus in Figures 7 and 8, we used R = a((Nc + Ni + Nd )/φn )1/3 , where Nd is the the number of dense fluid particles and φn is the average volume fraction of the nucleus. This allows us to track one-step crystal growth, two-step crystal growth, and dense fluid growth. Particles were designated as dense fluid if they were neither crystalline nor interfacial and have a local volume fraction larger than some threshold volume fraction. We select φ = 0.35 as the threshold for one-step crystal nucleation and fluid nucleation, and φ = 0.40 for two-step nucleation. φn was measured when the nucleus reached its terminal size and assumed constant throughout the growth process. In the case where two-step nucleation occurs, both crystal and dense fluid are present in the nucleus, so we average the densities of both phases (φc and φd ), weighted by the fraction of particles in each phase, φn = (Nc + Ni )φc /(Nc + Ni + Nd ) + Nd φd /(Nc + Ni + Nd ).

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Graphical TOC Entry gel coarsening

ton = 0.1 toff = 0.01 one-step crystal nucleation

ton = 0.046

toff = 0.02 two-step crystal nucleation ton = 0.24 toff = 0.1

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