Dynamic, Directed Self-Assembly of Nanoparticles via Toggled

Dynamic, Directed Self-Assembly of Nanoparticles via Toggled Interactions Zachary M. Sherman and James W. Swan* Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States S Supporting Information *

ABSTRACT: Crystals self-assembled from nanoparticles have useful properties such as optical activity and sensing capability. During fabrication, however, gelation and glassification often leave these materials arrested in defective or disordered metastable states. This is a key difficulty preventing adoption of self-assembled nanoparticle materials at scale. Processes which suppress kinetic arrest and defect formation while accelerating growth of ordered materials are essential for bottom-up approaches to creating nanomaterials. Dynamic, directed self-assembly processes in which the interactions between self-assembling components are actuated temporally offer one promising methodology for accelerating and controlling bottom-up growth of nanostructures. In this article, we show through simulation and theory how time-dependent, periodically toggled interparticle attractions can avoid kinetic barriers and yield well-ordered crystalline domains for a dispersion of nanoparticles interacting via a short-ranged, isotropic potential. The growth mechanism and terminal structure of the dispersion are controlled by parameters of the toggling protocol. This control allows for selection of processes that yield rapid self-assembled, low defect crystals. Although self-assembly via periodically toggled attractions is inherently unsteady and out-of-equilibrium, its outcome is predicted by a first-principles theory of nonequilibrium thermodynamics. The theory necessitates equality of the time average of pressure and chemical potential in coexisting phases of the dispersion. These quantities are evaluated using well known equations of state. The phase behavior predicted by this theory agrees well with measurements made in Brownian dynamics simulations of sedimentation equilibrium and homogeneous nucleation. The theory can easily be extended to model dynamic self-assembly directed by other toggled conservative force fields. KEYWORDS: nonequilibrium self-assembly, nanoparticle crystallization, Brownian dynamics, nonequilibrium thermodynamics

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serious barrier to the self-assembly of functional materials at a commercial scale. In fact, for many colloidal systems, the envelope of experimentally achievable ordered states is quite small.23 For each self-assembled material, interactions must be carefully tuned to find that envelope, and within it, the rate of self-assembly is necessarily slow. Stronger interactions lead to faster aggregation because of a stronger thermodynamic driving force but also produce a significantly higher rate of defect formation. However, several experiments with paramagnetic nanoparticles have shown that rapidly toggled magnetic fields can be used to avoid kinetically arrested states even when magnetic interactions are hundreds of times stronger than thermal forces in the dispersion.24,25 The time-dependent magnetic field modulates the interparticle potential in time, opening kinetic pathways that are not available to paramagnetic dispersions exposed to a steady magnetic field. Swan and co-workers investigated self-assembly in toggled paramagnetic suspensions further.26,27 They found an optimal

rdered colloidal and nanoparticle structures are useful materials with applications as photonic crystals,1−3 microwires,4 templates for inverse opals and macroporous membranes,5 lithographic masks,6 and thermal and chemical sensors.7,8 However, it can be difficult to fabricate nanoparticle crystals quickly and inexpensively, taking days in many cases to grow small crystallites from supercritical dispersions.9,10 To speed up self-assembly, interparticle attractions can be engineered to encourage aggregation. For example, functionalizing particles with DNA,11 adding depletants,12 applying electric4,13,14 or magnetic fields,15,16 introducing hydrodynamic flow fields,17−19 and surface tension20−22 have all been used to promote controlled crystallization of nanoparticles. In most of these cases, the particle interactions are fixed temporally. Although the driving force for crystallization may vary in time, the self-assembly pathway is constrained by relaxation to thermodynamic equilibrium. However, relaxation may be slow because these same attractive interactions also cause the nanoparticles to become trapped in defective or disordered, kinetically arrested states, i.e., glasses or gels. Metastable states persist for great lengths of time, presenting a © XXXX American Chemical Society

Received: February 10, 2016 Accepted: April 20, 2016

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DOI: 10.1021/acsnano.6b01050 ACS Nano XXXX, XXX, XXX−XXX

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effective attraction of colloidal spheres in the presence of a nonadsorbing depletant,37,38

toggling frequency at which crystallization occurred the fastest. Similar enhancement in the rate of self-assembly was observed in kinetic Monte Carlo simulations of self-replicating dispersions with toggled interparticle interactions.28 Swan and co-workers also noticed that the terminal structure of the paramagnetic suspension was sensitive to the toggling frequency. At low frequencies, the suspension separated into dense and dilute fluid phases. At moderate frequencies, large well-ordered crystalline domains formed. At large frequencies, gelation occurred and the dynamics of the suspension slowed dramatically. Crystalline phases are predicted by thermodynamic calculations of dipolar hard spheres with steady interactions, but fluid/fluid coexistence is not.29 Thus, the toggling protocol can stabilize nonequilibrium phases that are not observed in equilibrium. Nonequilibrium structures have also been seen in Brownian dynamics simulations of colloidal suspensions with time-oscillatory, short-range interactions.30,31 The ability to control both the rate of assembly and the microstructure is a powerful tool for the fabrication of nanostructured materials. Because the interparticle potentials are time-dependent, these type of processes belong to a class of nonequilibrium selfassembly processes termed dynamic self-assembly.32 Despite numerous advantages illustrated by the toggled magnetic field experiments, there are currently no methods to predict the phase behavior of materials processed this way. To gain insight into dynamic self-assembly, we have used simulations to investigate nanoparticle dispersions driven to self-assemble by toggled interparticle attractions. While it would be useful to replicate self-assembly via toggled magnetic interactions in silico, to directly compare to the experiments, there are a few challenges. First, magnetic dipole/dipole interactions are longranged. To properly incorporate these in simulation, sophisticated summation algorithms must be implemented.33 The computational expense of computing long-range interactions limits both the system size and simulation duration. Additionally, the equilibrium phase diagram of paramagnetic suspensions is complex, and there are no complete analytical expressions describing the pressure and chemical potential in such dispersions.29 It is likely that the out-of-equilibrium toggled analogue will be even more complicated. For these reasons, we choose a simpler model: a monodisperse suspension of spherical nanoparticles that interact with isotropic, short-ranged attractions that are toggled on and off periodically in time. Such systems have already been realized experimentally by decorating nanoparticles with photoswitchable ligands to induce short-ranged attractions that can be reversibly actuated by exposure to UV irradiation.34,35 Shortranged interactions can be handled rapidly in simulations with efficient neighbor list algorithms, and the equilibrium phase behavior is well understood.36,37 Toggled short-ranged attractions provide a useful prototype dynamic, directed selfassembly that can be investigated experimentally, computationally, and via first-principles theory. A fundamental understanding of this simple model will lead to methods for describing more complex processes, such as assembly via longranged, toggled electric or magnetic suspensions. As we shall show, the assembly process modeled by toggled short-ranged attractions shares many features in common with experiments having more complicated interparticle interactions. For the functional form of the interparticle potential, we choose to use the Asakura−Oosawa form, which represents the

U (r ) = −

3 ε(a + δ)3 ⎛ 3 r 1 ⎛ r ⎞⎞ ⎜ ⎟ ⎟ ⎜ 1 − + 4a+δ 16 ⎝ a + δ ⎠ ⎠ δ 2(3/2 + δ) ⎝

(1)

when 2a ≤ r ≤ 2(a + δ). The particles also interact via a hardcore repulsion at contact (r = 2a). Here, r is the center-tocenter distance between two particles, a is their radius, ε is the interaction strength at contact, and δ is the interaction range. We make the particular choice of the depletion potential as a model for short-ranged attractions in general because it is continuous and differentiable (beyond contact), making it straightforward to incorporate into simulation, and there are analytic expressions for the phase behavior of particles interacting via this potential.37 While ε and δ certainly affect phase behavior, the specific shape of the short-ranged attractive interaction is irrelevant, as the Noro-Frenkel principle of corresponding states demonstrates that the phase behavior of all dispersions with short-ranged attractions is generic.36 For this work, we take the interaction to be strong compared to the thermal energy, ε = 5 − 15kBT, and short in range compared to the particle size, δ = 0.1a. Steady interparticle attraction with this strength results in rapid, irreversible aggregation and kinetically arrested structures with a large number of defects. When the attraction is toggled cyclically, on for a time ton and off for a time toff, the arrested structures relax in the off halfcycle and defects are annealed, as sketched in Figure 1. This is a

Figure 1. Schematic showing how toggled interactions can anneal defects and avoid kinetic arrest in a nanoparticle suspension. The particles diffuse as hard spheres for a time toff in the “off” half-cycle and experience strong, short-ranged attractions for a time ton in the “on” half-cycle. The left column shows a kinetically arrested configuration with a vacancy defect. If toff is too short (top row), the particles cannot diffuse sufficiently far from their original positions in the off half-cycle, and the defect persists when the attractions are turned on. If toff is too long (bottom row), the particles diffuse away farther than the range of the interaction in the off half-cycle and do not reaggregate when the attractions are turned on. If toff = δ2/D (middle row), the particles diffuse far enough to relax the defected structure but still reaggregate when the attractions are turned on and find their thermodynamically favorable crystalline configuration. B


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ACS Nano physical analogue to the flashing Brownian rachet, in which thermal motion drives a net particle current across an asymmetric, time-periodic energy landscape, but not in a time-independent one.39 “Toggling” implies immediate step changes between the “on” state in which particles are mutually attractive and the “off” state in which particles behave as hardspheres. There are innumerable other forms of time-varying control that could be used to switch between the on and off states. However, toggling is simple to implement in experiments and simulations, and straightforward to analyze in theory. An interesting coupling between the symmetry of the potential energy landscape and the symmetry of the time signal is discussed elsewhere.39 Additionally, some dynamic selfassembly simulations have shown that other periodic signals lead to essentially the same phase behavior.30 For simplicity and more direct comparison to the toggled magnetic field experiments, we choose the square wave, toggled signal. The signal is characterized by only two parameters, ton and toff. Equivalently, the signal can be described by its period ; ≡ ton + toff , frequency ω ≡ 1/; , duty η ≡ ton/toff, or duty fraction ≡ ton/(ton + toff) = η/(η + 1). Risbud and Swan recently investigated the same model in two-dimensional Brownian dynamics simulations.40 They demonstrated that both the self-assembly rate and crystal quality improved when the interparticle attractions were toggled. They argued that an optimal choice for toff exists and is comparable to the time it takes for a particle to diffuse the range of the interaction, toff = δ2/D, where D is the particle diffusivity. This choice for toff allows for the largest amount of structural rearrangement without dissolving aggregated structures (Figure 1). This hypothesis was supported by their simulations as well as by the toggled magnetic field experiments of Swan and co-workers.26,27 In the magnetic field experiments, the magnetic capture radius took the place of the range of the interaction to yield a prediction for the optimal toff. Figure 2 demonstrates that large crystalline domains indeed form when the interactions are toggled with toff = δ2/D, but not with steady interactions of the same overall or time-averaged strength. Here, we develop a formal description of the phase behavior of dispersions self-assembled via periodically toggled attractions. The article is organized as follows. First, we measure the volume fractions of coexisting phases with a fixed toff = δ2/D while varying ton and the volume fraction ϕ. Three different simulation techniques are used to make this measurement: crystal nucleation from a homogeneous fluid, fluid nucleation from a homogeneous crystal, and sedimentation equilibrium. We also measure the coexisting phases while fixing the volume fraction ϕ and varying ton and toff independently. Finally, we develop a first-principles theory to predict the concentrations of the coexisting phases which shows that the phase behavior is accurately predicted by appropriate time-averages of analytical equilibrium equations of state for the dispersion in the attractive mode and the purely repulsive mode. We conclude with a discussion of the results and their implication for dynamic nanoparticle self-assembly.

Figure 2. Snapshots of Brownian dynamics simulations of 64 000 hard spheres at different volume fractions ϕ with steady and toggled depletion interactions of strength ε, duty η, and off duration toff = δ2/D. With steady interactions, the dispersions become kinetically arrested and never crystallize. With toggled interactions, large crystalline domains emerge, and the dispersions reach a periodic-steady-state of fluid/crystal coexistence. The timevariation is crucial for enhanced self-assembly as neither steady case with equal overall strength nor equal time-averaged strength shows large ordered domains.

been shown to lead to enhanced crystal growth,40 and consider only variations in ton, or η ≡ ton/toff equivalently, and ϕ. We will consider variations in toff and ε in later sections. Here, τD = a2γ/kBT is the characteristic diffusion time of a particle of radius a and drag γ. If η is large, there is sufficient time in the on half-cycle for the particles to aggregate. The crystal structure will relax and heal in the off portion, and so we expect to see large crystalline domains emerge after many toggle cycles. If η is too small, the particles do not have enough time to aggregate sufficiently. Any small aggregated clusters dissolve in the off phase, and the suspension will remain as a homogeneous fluid. The phase boundary η*f (ϕ) separates these two regions. Below η*f (ϕ), crystal nucleation from the homogeneous fluid is not observed. Similarly, we can determine the phase boundary ηc*(ϕ) below which fluid nucleation from a homogeneous crystal is not observed. In the on half-cycle, the particles try to move away from their lattice positions and collapse into closest packings. However, because the hard sphere crystal is stable at large densities, the particles tend to move back to their lattice positions in the off half-cycle. If η is small, the particles do not deviate sufficiently far in the on half-cycle and always return back to their lattice positions in the off half-cycle. If η is large enough, there is enough time in the on half-cycle for density fluctuations to drive portions of the crystal to collapse into closest packings and open up a pocket in which the crystal melts. We performed Brownian dynamics simulations of toggled suspensions initially in a homogeneous fluid state with 64 000 particles and a homogeneous face-centered-cubic (fcc) crystal state with 62 500 particles (Figure 3). The interaction strength

SIMULATION RESULTS Homogeneous Nucleation. To determine what range of parameters lead to self-assembly, there are four parameters that can vary: the on duration ton, the off duration toff, the strength of the attraction at contact ε, and the volume fraction ϕ. We do not consider the interaction range δ as an adjustable parameter. We fix ε = 10kBT and toff = δ2/D = 0.01τD, as these values have C


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visually, as seen in Figure 3. Both phase boundaries are shown in Figure 4. The fluid phase boundary η*f decreases with increasing ϕ, while the crystal phase boundary η*c generally increases with increasing ϕ.

Figure 4. Phase boundaries determined from homogeneous nucleation simulations at fixed ε = 10kBT and toff = δ2/D = 0.01τD. The points correspond to the lowest duty at which crystal nucleation in a homogeneous fluid (green) and fluid nucleation in a homogeneous fcc crystal (purple) was observed after 105 toggle cycles. Below these curves, the suspensions remains homogeneous while above them phase separation occurs.

Sedimentation Equilibrium. Under a weak, constant gravitational force Fg = −Fgêz, the competing processes of sedimentation and diffusion in a suspension result in a nonhomogeneous equilibrium number density profile n(z) as a function of height z that satisfies, ∂P(z) = −n(z)Fg ∂z

(2)

where P is the (osmotic) pressure of the colloid particles. Note that the number density and volume fraction are related by v0n = ϕ, where v0 = 4πa3/3 is the volume of a single particle. If the vessel containing the suspension is sufficiently tall, i.e., a tower, the suspension is so dilute at large z that the pressure at the top vanishes. Integrating from the top of the tower z = L to any height z,

Figure 3. Mechanisms for phase separation in pulsed suspensions (color online). Left: Sedimentation under a weak gravitational force downward crystallizes lower particles under the weight of those above. Right top: One-step crystal nucleation in a homogeneous metastable fluid. Right middle: Two-step nucleation in a homogeneous metastable fluid. A dense fluid drop nucleates first and grows before crystal nucleation inside the droplet occurs. Crystalline (light blue) and liquid-like (dark-blue) particles are distinguished. Right bottom: Fluid nucleation in a homogeneous metastable crystal. In this 2-D projection, white space indicates that the particles are aligned in the crystal phase.

P(z) = −Fg

∫L

z

dz′n(z)

(3)

Therefore, if we measure the number density profile n(z), we can obtain the pressure profile P(z) using eq 3, from which we can infer the equation of state P(n). If there is sufficient pressure, the bottom of the sample will crystallize under the weight of the particles above. Thus, the equation of state will capture phase coexistence as well. In particular, the densities on either side of the phase boundary are the coexisting densities of the two phases. This is a common technique used to obtain equations of state and phase diagrams of nanoparticle suspensions.41−43 Equations 2 and 3 only hold for equilibrium, timeindependent systems. With time-periodic interparticle interactions, these equations will never hold, and the system is always instantaneously out of equilibrium. Instead, with a

and off duration were fixed, ε = 10kBT and toff = 0.01τD, while the toggling duty and volume fraction were varied. Each simulation was run for 105 toggle cycles. The lowest duty at which crystal nucleation was observed in the fluid for different volume fractions was designated as η*f (ϕ) and the lowest duty at which fluid nucleation was observed in the crystal was designated as ηc*(ϕ). The nucleating phase was easily observed D


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ACS Nano ⎛ ∂μ ⎞ + Fg ⎟ Jz = Mn⎜ ⎝ ∂z ⎠

periodic driving force, the suspension evolves until it reaches a periodic-steady-state (PSS) where it does not change from cycle to cycle, n(x , t ) = n(x , t + ;) for all future t. The number density evolves according to the conservation of mass law, ∂n = −∇·J ∂t

Using the Gibbs−Duhem relation, n dμ = dP, ⎛ ∂P ⎞ Jz = M ⎜ + nFg ⎟ ⎝ ∂z ⎠

(4)

∫t

⎛ ∂P ̅ ⎞ + nF Jz̅ = M ⎜ ̅ g ⎟⎠ ⎝ ∂z

t+;

dt J

∂z

=0

(6)

where the overbar indicates a quantity time-averaged over a pulse cycle, X̅ (t ) ≡

1 ;

∫t

t+;

dt X

∂P ̅ = − nF ̅ g ∂z

(7)

Therefore, the time-averaged flux is constant. Because particles cannot escape, there is a no-flux boundary condition at the bottom of the tower. Thus, the time-averaged flux must vanish everywhere at periodic-steady-state, J ̅ = 0. To make further progress, a constitutive relation is needed for the time-averaged particle flux. In the linear regime of nonequilibrium thermodynamics, the particle flux is, at constant T, proportional to the gradient in chemical potential,

J = −L∇μ′

P ̅(z) = −Fg

(8)

(9)

As discussed in the Supporting Information, colloidal suspensions are overdamped, and so the velocity is linearly related to the force with the constant of proportionality being the mobility M. −∇μ′ has the units of force, so we identify L = Mn and thus, J = −Mn∇μ′

(10)

The chemical potential is

μ′ =

∂f ′ ∂n

(11)

where f ′ is the free energy density, which can be written as f ′(n , z) = f (n) + nzFg

(12)

with f(n) as the free energy density of a homogeneous system at density n in the absence of the gravitational field. Therefore, μ′(n , z) = μ(n) + zFg

∫L

z

d z′ n ̅ (z )

(18)

and eqs 2 and 3 hold for time-averaged quantities in the pulsed suspension. In particular, the validity of eq 18 implies that there exists an equation of state P̅ = P̅(n)̅ that relates the time averaged pressure and density. It is not immediately obvious that such an equation should be valid. Additionally, if phase separation occurs in the suspension, we can use eq 18 to extract coexistence points and determine the phase diagram of the pulsed suspension. Because the average in eq 18 is over a single toggle cycle, the equation of state and the phase behavior are independent of frequency. Only the duty, or duty fraction, should affect the phase behavior. We performed Brownian dynamics simulations of 32 076 particles at an overall volume fraction of ϕ = 0.50 sedimenting under a weak, constant gravitational force Fg = 0.1kBT/a (Figure 3). The depletion interaction strength and off half-cycle were set to ε = 10kBT and toff = δ2/D = 0.01τD, and the duty was varied. To test the result that the phase behavior is independent of frequency, we performed simulations at two other off durations toff = 0.005τD and toff = 0.02τD and varying duty. We also performed simulations with toff = 0.01τD but different interaction strengths ε = 5kBT and ε = 15kBT. The suspension was supported below by a wall that repelled particles with a strong, short-ranged Lennard-Jones potential truncated at its minimum for purely repulsive interactions. The aspect ratio of the simulation cell was 16, oriented in the direction of Fg, to ensure that the pressure near the bottom of the suspension was large enough to sustain a crystal. While traditional sedimentation experiments start with a homogeneous fluid and then allow the particles to sediment downward, this setup is quite slow in simulation. We choose to implement the reverse scheme where all particles are initially in a closest packed fcc crystal at the base of the tower and allowed to melt from the interface until periodic-steady-state is achieved. While the terminal density profile is identical in either the downward

where L is a positive phenomenological coefficient. The particle flux can also be written as J = nu, where u is the velocity field, so we can rearrange eq 8 to

L ( −∇μ′) n

(17)

and

44

u=

(16)

In this step, we have assumed that M is constant within a pulse cycle. The mobility is a function of only the particle configuration. The only effect the interparticle potential has on the mobility is through its effect on the particle configuration. For short toggle periods, the configuration of the particles cannot change appreciably, and so the mobility is nearly constant within a cycle. Note that this is not true for the pressure, which is highly sensitive to the interparticle potential. Because the time-averaged particle flux vanishes at periodicsteady-state,

(5)

At periodic-steady-state, the left side vanishes. The system only varies in the z direction so, eq 5 simplifies to ∂Jz̅

(15)

Taking the time-average of this equation,

where J is the particle flux. Integrating eq 4 over a pulse cycle and dividing by the pulse period ; , 1 1 (n(x , t + ;) − n(x , t )) = −∇· ; ;

(14)

(13)

where μ is the chemical potential of a homogeneous system at density n in the absence of the gravitational field. Substituting eq 13 into the flux expression of eq 10, E


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ACS Nano or upward sedimentation schemes, the equilibration time in the upward setup is much shorter. Once periodic-steady-state was achieved, the density profile ϕ(z) was determined by dividing the tower into thin, rectangular volume slices and measuring the fraction of slice volume occupied by particles. This profile was then time-averaged over many pulse cycles and smoothed over space using Gaussian convolution. The equation of state P(ϕ) was then obtained from eq 18. The coexistence points are the densities on either side of the crystal/fluid interface. These points appear as cusps in the equation of state and can be extracted from a plot of P̅ versus ϕ̅ . Because the cusp was not always sharp, error bars were also computed as a reasonable range of volume fractions around the cusp. The results are plotted in Figure 5.

Figure 5. Fluid/crystal (F/C) phase diagram for the toggled depletion suspension (color online). The phase behavior is completely determined by the volume fraction ϕ and time-averaged strength εη/(1 + η), made dimensionless by the thermal energy kBT. The solid lines are the coexisting volume fractions predicted by the time-average of the equilibrium equations of state, and the points are those extracted from Brownian dynamics sedimentation equilibrium simulations with a variety of strengths ε, duties η, and off durations toff, where t* = δ2/D. For clarity, error bars are rendered only for a single set of data. Also shown in the inset are the data from Figure 4, the phase boundaries determined from homogeneous nucleation in a metastable fluid (green squares) and in a metastable crystal (purple squares).

Figure 6. Phase diagram of the toggled suspension obtained from varying ton and toff independently at fixed volume fraction ϕ = 0.20 and strength ε = 10kBT (color online). Points indicate simulation results, while lines are approximate guides to delineate different regions. There are four main regions: no self-assembly (crosses), percolated (squares), fluid/crystal coexistence (circles), and fluid/ fluid coxistence (triangles). The crystalline region is further separated into crystals formed by one-step (filled) or two-step (hollow) nucleation mechanisms.

Varying ton and toff Independently. Until now, all simulations have operated around the “optimal” toff = δ2/D = 0.01τD based off the work of Risbud and Swan.40 However, their analysis was limited to order of magnitude changes in toff and a fixed duty η. Here, we more rigorously analyze the effects of varying ton and toff independently. We performed Brownian dynamics simulations of pulsed suspensions with N = 64 000 particles at fixed volume fraction ϕ = 0.20 and strength ε = 10kBT and varying on and off durations. All suspensions began in a homogeneous fluid state and ran for around 10 000τD. The terminal structures after 10 000τD fall into 4 classes, shown in Figure 6. For small ton (small duty fraction η/η+1), 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 phase. The phase boundary is nonmonotonic with respect to toff; initially,

the duty fraction required for self-assembly increases with increasing toff but eventually peaks and decreases with increasing toff. At small toff and large ton (duty fraction close to unity), the system initially forms a disordered, percolated gel. In each off half-cycle, the structure is able to relax and heal, so the gel slowly coarsens and becomes more and more ordered. However, the coarsening is slow and the system remains percolated with a large amount of surface area. Percolated structures are not observed at large toff, even for duty fractions close to unity. Between these two regions, large, high quality crystals form into surface area minimizing shapes (e.g., spheres and cylinders). Within this region, at low toff, the crystals directly nucleate in the initial homogeneous suspension. At larger toff within this region, a dense fluid phase nucleates first and grows. Then, crystal nucleates within the dense fluid and grows inside the dense drop. This two-step nucleation mechanism has been reported for short-ranged attractive F


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ACS Nano colloid and protein systems.45−47 Although the terminal states are similar, the mechanisms for one-step and two-step nucleation are fundamentally different, so these points are differentiated within the crystalline region in Figure 6. At large toff and η/(η + 1) above the phase boundary, fluid/fluid phase separation occurs, but crystal nucleation within the dense fluid does not. Fluid/fluid coexistence was also observed in the toggled magnetic field experiments at low frequencies.26,27

toggled suspension, so the suspension is always instantaneously out of equilibrium. However, the nonequilibrium periodicsteady-state phase diagram can be determined solely from the equilibrium equations of state using eqs 23. Choose t in the time average to be at the start of the on halfcycle. We split the integral over a toggle period ; into two integrals over the half-cycles, P̅ =

THEORETICAL PHASE DIAGRAM The appearance of the time-averaged equation of state P̅ = P̅(ϕ̅ ) in the periodic-steady-state condition for sedimentation equilibrium suggests that the toggled suspension may be described by equilibrium thermodynamics if instantaneous quantities are replaced by their time averages. Suppose a toggled suspension phase separates into bulk dense and dilute phases such that the interface between them is flat and reaches periodic-steady-state. This sort of phase coexistence is observed in the present simulations as well as elsewhere.40 The requirement of macroscopic phase separation restricts the following derivation to ton and toff short compared to the time to homogeneously nucleate one phase within the bulk of the other. Otherwise, if nucleation occurs, the dispersion is not well described by two homogeneous bulk phases. If we designate z as the direction normal to the interface, the analysis of the time evolution of the volume fraction profile is similar to the sedimentation analysis without the gravitational field. In particular, eqs 14 and 15 imply ϕ

∂μ ∂P ̅ = v0 =0 ∂z ∂z

P̅ =

P̅ =

dt ϕ

∂μ ∂ 1 =ϕ ∂z ∂z ;

∫;

dt μ = ϕ

P ̅ = constant

PI̅ (ϕI) = PII̅ (ϕII)

toff

dt Poff )

(24)

(25)

ηPon + Poff η+1

(26)

Pdep = PHS + εf (ϕ)

(27)

μdep = μHS + εg(ϕ)

(28)

ηε g (ϕ) η+1

(30)

Comparing to eq 27, the time-averaged EoS is equal to the equilibrium depletion EoS with the interaction strength scaled by the duty fraction η/(η + 1). The phase diagram for the pulsed system, Figure 7, obtained by solving eq 23 is therefore a scaled version of the equilibrium depletion phase diagram, which has already been calculated.37 Thus, the nonequilibrium states of the pulsed assembly can be calculated solely from equilibrium equations of state. Additionally, the out-ofequilibrium phase behavior depends only on the time-averaged interaction strength εη/(η + 1) and is independent of the pulse frequency. The functional forms of the equations of state used to generate the phase diagram are explained in detail in the Supporting Information. For the hard sphere fluid equation of state, we use an extension of an equation of state derived by Torquato that diverges at random close packing.48,49 For the hard sphere crystal EoS, the Lennard-Jones-Devonshire lattice

(21)

(22)

In particular, far away from the interface, the time-averaged quantities are given by their bulk values, so the time-averaged chemical potential and the time-averaged pressure must be equal in the two bulk phases at periodic-steady-state, μI̅ (ϕI) = μII̅ (ϕII)

tonPon + toff Poff ton + toff

μ ̅ = μHS +

Therefore, for coexistence in a periodic-steady-state, μ ̅ = constant

∫

where f(ϕ) and g(ϕ) are functions of only volume fraction.37 The time-averaged equations of state are then ηε P ̅ = PHS + f (ϕ) η+1 (29)

(20)

∂μ ̅ ∂z

dt Pon +

and similarly for the time-averaged chemical potential μ̅. Within a half-cycle, the suspension behaves as either an equilibrium hard sphere system or an equilibrium depletion system, albeit in a state far from equilibrium. Thus, the pressure and chemical potential of each phase in the off and on half-cycles are given by the equilibrium equations of state (EoS) for hard spheres (HS) and depletion (dep), respectively: Poff = PHS, μoff = μHS, Pon = Pdep, and μon = μdep. The depletion equations of state are given by free volume and scaled particle theories and are of the form

This approximation is good when the pulse period is small compared to the time scale on which the particles move. At larger time scales, this approximation may break down. With ϕ constant within a pulse cycle, the left side of eq 19 becomes

∫;

ton

or equivalently

where we have used the result that the time-averaged particle flux vanishes everywhere at periodic-steady-state. The pressure and chemical potential are very sensitive to the interparticle interaction potential. Because we are modulating the potential over time, the pressure and chemical potential vary greatly within a single pulse cycle. However, the volume fraction profile does not. At periodic-steady-state, the volume fraction satisfies ϕ(x , t ) = ϕ(x , t + ;). If ϕ is to return to its profile at the start of a cycle, it cannot deviate too far within the cycle. Thus, ϕ is relatively time-independent at periodic-steady-state,

1 ;

∫

where the subscripts of P indicate which half-cycle the quantity is evaluated in, and the limits of integration indicate that the integrals span entire half-cycles. In general Pon and Poff may change slowly over many cycles, but because the volume fraction profile is essentially constant within a cycle, the pressure does not change significantly within a half-cycle. Thus, we may approximate the time-average as

(19)

ϕ(x , t ) = ϕ(x , t + ;) ≈ ϕPSS(x)

1 ( ;

(23)

This is analogous to the coexistence criteria of equal pressure and chemical potential in the equilibrium case. Of course, the equilibrium coexistence conditions will never hold in the G


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only knowledge of equilibrium equations of state. This is a general result that holds for toggling between any two arbitrary interparticle potentials. Second, for the specific case of toggling between short-ranged attractions and hard sphere repulsions, the dimensionality of the parameter space describing phase behavior reduces from four (ton, toff, ε, and ϕ) to two (ε̅ and ϕ). Notably, this implies that the limiting phase behavior is independent of the pulse frequency ω. These conclusions hold when ton and toff are short compared to the time scale for homogeneous nucleation of one phase within the bulk of the other. If this is true, all the particle flux occurs at the interface between the two phases in periodicsteady-state, and the bulk phases are homogeneous. We find that ton and toff up to order τD are short enough to suppress homogeneous nucleation and lead to structures that are well described by our theory. For ton and toff between 2τD and 10τD, significant nucleation occurs within the bulk phases in coexistence. The two phases are quite heterogeneous and it is difficult to assign a single value of ϕ to them. For ton and toff above 20τD, the on and off half-cycles are so long that the suspension forms a percolated, disordered gel in every on halfcycle which completely dissolves in the subsequent off halfcycle. Thus, no sustained, macroscopic phase separation was observed. Although we did not account for hydrodynamic interactions between particles in our simulations, the way we have formulated our analysis, hydrodynamic interactions do not affect the criteria of equal time-averaged pressure and timeaveraged chemical potential in two coexisting phases at periodic-steady-state. Hydrodynamic interactions will only affect the value of the particle mobility M in eq 4, but they will not change the fact that M does not vary much over a toggle cycle, and so we will arrive at the same periodic-steadystate criteria whether or not hydrodynamics are accounted for. The self-assembly kinetics, on the other hand, will be affected if hydrodynamic interactions are included. Short-ranged lubrication interactions can effectively be scaled out of the problem with an appropriate renormalization of time. Long-range hydrodynamics tend to enhance the rate of aggregation of particle clusters.52 We believe that the effect is only a quantitative correction, and that including hydrodynamics will not qualitatively affect the conclusions drawn here. Thus, some of the boundaries in Figure 6 may shift when hydrodynamics are accounted for, but the types of terminal structures and trends of the boundaries are expected to remain unchanged. While the limiting phase behavior of the toggled suspension is not dependent on both ton and toff, the kinetics of phase separation are sensitive to the dynamic toggling parameters. For example, the inset of Figure 5 shows the phase boundary extracted from homogeneous nucleation simulations overlaid with the predicted phase boundary. Although free energy barriers delay the onset of homogeneous nucleation, in the thermodynamic infinite time limit, these boundaries should agree. However, because all simulations have finite duration, it may be exceedingly improbable to observe nucleation as the phase boundary is approached from above. In this case, the suspension will remain in its metastable homogeneous state, and coexistence is inaccessible. The inset of Figure 5 shows that the fluid phase boundary ηf* predicted from homogeneous nucleation lies above the predicted binodal, whereas the crystal phase boundary η*c coincides well with the prediction. This suggests that fluid nucleation in the crystal has a relatively small nucleation barrier, while crystal nucleation in the fluid has a

Figure 7. Fluid/crystal (F/C) and fluid/fluid (F/F) phase diagram of the toggled depletion suspension (color online). The data are Brownian dynamics simulation results at fixed strength ε = 10kBT and different off durations, made dimensionless by the diffusion time τD, while the lines are fluid/crystal and fluid/fluid coexistence curves predicted from the time-average of equilibrium equations of state. The circles are the data from Figure 5 obtained from sedimentation equilibrium. The diamonds and triangles are the densities of the coexisting fluid phases from the simulations in Figure 6. The pluses and crosses are data from Monte Carlo simulations of fluid/fluid (pluses) and fluid/crystal (crosses) coexistence in an equilibrium depletion suspension.51

model is used.50 The depletion equations of state are derived from these hard sphere EoS using free volume and scaled particle theories.37 The phase behavior predicted by the time average of these equations of state is in good agreement with the coexistence points extracted from our simulations, Figures 5 and 7.

DISCUSSION In the limit of very large toggle frequencies, particles experience a constant, effective time-average interparticle potential.31,39,40 Although kinetic arrest may make it difficult, the suspension will eventually relax to the thermodynamic equilibrium for the effective interparticle potential. For the specific case of toggling short-ranged attractions on and off in a hard sphere suspension, the effective potential leads to time-averaged equations of state that take the form of equilibrium depletion equations of state with a time-averaged strength ε̅ = ηε/(η + 1) in place of the actual strength ε. Therefore, the phase behavior is described exactly by a scaled version of the equilibrium depletion phase diagram. Our analysis suggests that, even outside the large frequency limit, the phase diagram derived from time-averaged equations of state is still descriptive of coexistence in toggled suspensions. The phase behavior is therefore controlled entirely by the parameters ε̅ and ϕ. Figures 5 and 7 show fluid/fluid and fluid/crystal coexistence points extracted from simulations. While these points correspond to a variety of different interaction strengths, duties, and frequencies, the data collapse together when plotted in terms of ε̅ and ϕ and agree well with the phase behavior predicted by the time-averaged equations of state as well as coexistence points from Monte Carlo simulations of equilibrium depletion suspensions.51 These results have two important consequences. First, the out-of-equilibrium periodic-steady-states can be predicted with H


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growing crystal provides particles to the crystal and enhances the rate of crystal growth. Two-step nucleation is also observed in the toggled suspensions. Although crystals grow by one-step nucleation at high frequencies, toff ≤ 0.02τD, crystals grow by two-step nucleation at moderate frequencies, 0.05 τD ≤ toff ≤ 0.2τD. At low frequencies, toff ≳ 0.5τD, crystallization is suppressed entirely. Two-step nucleation is only possible within a metastable fluid/fluid coexistence region. Outside of a fluid/ fluid coexistence region there is no thermodynamic driving force for the first fluid nucleation. If fluid/fluid coexistence was thermodynamically stable relative to fluid/crystal coexistence rather than metastable, there would be no thermodynamic driving force for the second crystal nucleation. The timeaveraged equations of state do predict fluid/fluid coexistence, Figure 7, suggesting that the observed two-step nucleation likely is rooted in the periodic coexistence diagram. But from Figure 6, one-step crystal nucleation, two-step nucleation, and simple fluid/fluid phase separation are all observed in the fluid/ fluid envelope at different values of toff. While ε̅ and ϕ are still descriptive of the terminal structure, the frequency too is important for the growth mechanism and to select whether the terminal coexistence is fluid/fluid or fluid/crystal. In an equilibrium short-ranged attractive suspension, fluid/ fluid coexistence is metastable to fluid/crystal coexistence for sufficiently short attractions (the δ = 0.1a used in this work satisfies this condition). Even if fluid/fluid phase separation occurs first, the suspension must eventually crystallize. In the out-of-equilibrium toggled suspension, there is nothing to preclude fluid/fluid coexistence from an acceptable periodicsteady-state. Energy is constantly added to and removed from the system, so the same thermodynamic variational principles that favor fluid/crystal over fluid/fluid coexistence in the equilibrium suspension are not valid in the toggled suspension. This is not to say that the periodic-steady-state criteria derived from a dynamic analysis, P̅I = P̅II and μ̅I = μ̅II are invalid. Only that we cannot make any assertions on the relative stabilities of two different phase coexistences that both satisfy the periodicsteady-state conditions. We cannot predict that phase separation will occur or which coexistence is favored, only that if it does, the phases must satisfy eq 23. The fact that we observe fluid/crystal and fluid/fluid coexistence at different frequencies but same ε̅ is therefore not an inconsistency within our analysis, but rather a fascinating result that the out-ofequilibrium self-assembly process can stabilize structures that are only metastable in equilibrium self-assembly. Why low pulse frequencies should favor fluid/fluid phase separation rather than fluid/crystal phase separation is presently unclear. One may hypothesize that at low pulse frequencies, the particles are more mobile in the off half-cycle and thus the crystal phase may be kinetically unstable. However, we do not observe instabilities in the crystal at large frequencies. Figure 8 shows that if we increase the pulse frequency in a fluid/fluid coexisting suspension, crystal nucleation and growth occurs within the dense fluid. However, if we then decrease the frequency to its original value, the suspension remains in fluid/ crystal coexistence rather than return to its fluid/fluid state. This hysteresis implies that the terminal state is path dependent, since both fluid/fluid and fluid/crystal coexistence are observed at the same point in parameter space.

larger nucleation barrier that limits the envelope of selfassembly. Note that this is purely a kinetic effect, and the homogeneous crystal nucleation results are still consistent with the phase behavior predicted by eq 23, as they lie within the coexistence region. Additionally, the nucleation boundaries are qualitatively similar to the predicted phase boundaries. This is partially a thermodynamic effect, as it is favorable to be close to the phase boundary, and partially a kinetic effect, as it becomes easier (i.e., requires a lower duty) to form a critical crystal nucleus in a fluid at larger densities, where the interparticle separation decreases, and vice versa for nucleating a fluid in a crystal. The sedimentation results lie much closer to the phase boundary than the homogeneous nucleation results. In the classical downward sedimentation setup, the supporting wall at the bottom of suspension offers a surface for heterogeneous crystal nucleation. In the reverse scheme that we implemented, the interface at the top of the initial crystal block allows for heterogeneous fluid nucleation, i.e., interfacial melting. In either case, the nucleation barrier for heterogeneous nucleation is lower than that for homogeneous nucleation. Because the nucleation induction time decreases with decreased barrier height, it is much easier to observe phase separation with finite duration simulations via sedimentation rather than homogeneous nucleation, and so the results are in better agreement with the theory. These kinetic effects cannot be analyzed in terms of only ε̅ and ϕ, but rather depend on both ton and toff. For example, our theory predicts a single value for the fluid phase boundary at a given volume fraction, say ϕ = 0.20. However, Figure 6 shows that as we vary toff at fixed ϕ = 0.20 and ε = 10kBT, the fluid phase boundary η*f changes. In fact, the fluid phase boundary is a nonmonotonic function of toff. At small toff, the duty η*f required for self-assembly increases with increasing toff. Here, one-step nucleation is observed and crystal nucleates directly in the homogeneous fluid phase. ηf* peaks at the first appearance of dense fluid nucleation. Beyond this point, dense fluid nucleation occurs in the homogeneous fluid rather than crystal nucleation. While η*f increased with increasing toff for crystal nucleation, ηf* now decreases with increasing toff for fluid nucleation. Because of this, it is possible to observe four different terminal structures for a single value of ε̅ = ηε/(η + 1), images of which are shown in Figure 6. Clearly both ton and toff are important for capturing the dynamics of the toggled assemblies. Several researchers have reported fluid/fluid phase separation in short-range attractive colloidal suspension in the context of two-step nucleation before.45−47 In the two-step nucleation mechanism, a dense fluid drop nucleates in a less dense homogeneous fluid suspension and grows. Within the dense fluid, a crystal then nucleates and grows. The two-step mechanism is an alternative to the classical one-step mechanism where the crystal nucleates directly in the initial homogeneous suspension. Because the dense fluid is closer in density and structure to the initial homogeneous fluid suspension than the crystal phase, the fluid/fluid interfacial tension is lower than the fluid/crystal interfacial tension. Thus, the barrier to nucleation is smaller for the dense fluid than the crystal, and the fluid nucleates first. Similarly, the interfacial tension between the dense fluid and crystal is less than that for the dilute fluid and crystal, so crystal nucleation preferentially occurs within the dense fluid. The increased density of the fluid layer around the I


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detailed view of such a model dynamically self-assembled material, and thus may be applied quite generally to a broad range of materials driven to assemble by toggling protocols.

METHOD We perform Brownian dynamics simulations of a monodisperse suspension of spheres of radius a in a solvent of viscosity μs. We summarize the method here, but a more detailed description is given in the Supporting Information. Brownian dynamics integrates the overdamped equations of motion of dispersions as

x i(t + Δt ) = x i(t ) +

1 Fi(t )Δt γi

(31)

where xi is the position of particle i, γi = 6πμsai is its Stokes drag coefficient, Δt is the integration time step, and Fi ≡ FBi + FPi + FEi is the sum of all nonhydrodynamic forces on the particle including a stochastic Brownian force FBi , an interparticle force FPi arising from interaction potentials, and an external force field FEi . FP is a sum of a hard sphere repulsion and, in the on phase of a toggle cycle, an attraction due to the depletion potential. The functional form of the depletion potential is given in eq 1. We model hard sphere interactions with a Hookean spring potential

Figure 8. Path dependence of the periodic-steady-state of the toggled suspension. A suspension toggled at ton = 1.5 τD, toff = 0.5τD reaches fluid/fluid coexistence (right bottom). If the toggle frequency is raised, crystallization occurs in the dense fluid and the suspension reaches fluid/crystal coexistence (right top). A suspension toggled at ton = 0.3τD, toff = 0.1τD reaches fluid/crystal coexistence (left top). If the toggle frequency is lowered, the suspension remains in fluid/crystal coexistence (left bottom). Crystalline (light blue) and liquid-like (dark-blue) particles are distinguished.

U HS(r ) =

1 k(2a − r )2 , 2

r < 2a

(32)

with a time step dependent spring constant of γ k≡ 2Δt

(33)

With this form of the hard sphere potential, particles that overlap due to other forces over the course of a time step are moved exactly to contact. This is a simple implementation of the Heyes-Melrose “potential-free” algorithm for hard repulsions53 that converges to the hard sphere potential as Δt → 0, is stable for all Δt, and adapts to system parameters to always bring overlapped particles exactly to contact. We perform all Brownian dynamics simulations in HOOMD-blue, an MD package optimized for graphics processing units (GPUs).54−56 Note that although we use an MD package, we are able to simulate overdamped, hard nanoparticles (see Supporting Information for more details). All simulations are performed with around 64 000 identical particles for around 10 000τD. The integration time step must be 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. We use a time step of Δt = 10−4 τD when both ton and toff are longer than 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 to 0.01 τD or below, we use a smaller time step Δt = 10−5 τD to resolve the dynamics within each individual half-cycle. All lengths are presented in units of particle radius a, energies in units of thermal energy kBT, and time in units of diffusion time τD. A variety of quantities were recorded at the end of an on half-cycle to aid in structural analysis, track self-assembly progress, and distinguish crystalline and fluid-like particles. Each particle’s contact number was determined by finding the number of neighboring particles within a threshold distance of 2.1a. The local density around each particle was determined by looking at a sphere of radius 6a centered around a particle and calculating the fraction of that 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 a local density 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. Bond order parameters based on spherical harmonics were also calculated to describe the local orientational order around each

CONCLUSION In this work, we analyzed suspensions of nanoparticles interacting with short-ranged attractions that were toggled on and off in time using Brownian dynamics simulations and theory. We found that the limiting out-of-equilibrium phase behavior was described with only two parameters, the timeaveraged strength of interaction ε̅ and the volume fraction ϕ. Additionally, we showed that the out-of-equilibrium structures could be predicted in terms of only equilibrium equations of state. This revealed that structures that are metastable in equilibrium can be stabilized during dynamic self-assembly. Finally, we showed that the phase separation kinetics of toggled nanoparticle suspensions are sensitive to the dynamic toggling parameters toff and ton. Because of this, the mode of growth can be controlled simply by varying the toggling protocol. Fabricating nanoscale materials on a commercial scale requires well-characterized, efficient processing techniques. Dynamic self-assembly methods hold much promise in this area over their equilibrium counterparts in three critical areas: speed of assembly, quality of product, and ease of control. In fact, nearly all materials in nature are self-assembled using such out-of-equilibrium pathways. Out-of-equilibrium methods are also capable of forming novel structures that equilibrium methods cannot. As the nature of materials becomes more complex, dynamic self-assembly will play a bigger role in nanomaterial fabrication. The main obstacle to implementing dynamic self-assembly processing techniques in practice so far has been a lack of understanding of their governing principles and little exploration into the large parameter spaces involved. Through our research, we have answered some key questions about dynamic self-assembly. We believe that the present results for the toggled isotropic potential represent the first J


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ACS Nano particle.57 Finally, the static structure factor of the suspension was calculated. Because we consider only strong, short-ranged attractions, any crystal phase that forms is always nearly closest packed ϕ ≈ 0.74. In three-dimensions, the largest packing density that a disordered packing can achieve is ϕ ≈ 0.64. Because disordered configurations are excluded from the densities at which we observe crystal, we can distinguish phases based on density alone. In fact, the density difference between coexisting phases is large and the phase domains are typically expansive, so the two phases can be clearly distinguished visually, as seen in Figure 6.

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ASSOCIATED CONTENT S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsnano.6b01050. Additional details of the simulation method and equations of state used in this work (PDF)

AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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Dynamic, Directed Self-Assembly of Nanoparticles via Toggled

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