Shear-induced alignment of Janus particle lamellar ... - ACS Publications

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Shear-Induced Alignment of Janus Particle Lamellar Structures Ronal A. DeLaCruz-Araujo,† Daniel J. Beltran-Villegas,‡ Ronald G. Larson,‡ and Ubaldo M. Córdova-Figueroa*,† †

Department of Chemical Engineering, University of Puerto RicoMayagüez, Mayagüez, Puerto Rico 00681, United States Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48109, United States

‡

S Supporting Information *

ABSTRACT: Control over the alignment of colloidal structures plays a crucial role in advanced reconfigurable materials. In this work, we study the alignment of Janus particle lamellar structures under shear flow via Brownian dynamics simulations. Lamellar alignment (orientation relative to flow direction) is measured as a function of the Péclet number (Pe)the ratio of the viscous shear to the Brownian forcesthe particle volume fraction, and the strength of the anisotropic interaction potential made dimensionless with thermal energy. Under conditions where lamellar structures are formed, three orientation regimes are observed: (1) random orientation for very small Pe, (2) parallel orientationlamellae with their normals parallel to the direction of the velocity gradientfor intermediate values of Pe, and (3) perpendicular orientationlamellae with their normals parallel to the vorticity directionfor large Pe. To understand the alignment mechanism, we carry out a scaling analysis of competing torques between a pair of particles in the lamellar structure. Our results suggest that the change of parallel to perpendicular orientation is independent of the particle volume fraction and is caused by the hydrodynamic and Brownian torques on the particles overcoming the torques resulting from the interparticle interactions. This initial study of shear-induced alignment on lamellar structures formed by Janus colloidal particles also opens the door for future applications where a reversible actuator for structure orientation is required.

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lamellae.14,15 The combination of the structure versatility of Janus particles and actuation is a promising route to reconfigurable metamaterials. The actuation of Janus particle systems has been achieved with electric and magnetic fields,17,18 shear flow,16,19−22 convective assembly at a liquid−liquid interface,23 and switching amphiphilicity by controlled wetting of one of the Janus caps,24 among other ways. In a previous publication, we proposed the use of shear flow as an easy-to-implement, reversible actuator alternative for suspensions of Janus colloids. 16 We studied via Brownian dynamics (BD) simulations the effect of steady shear on exemplary Janus particle systems showcasing the canonical set of structures, shedding light on a rich variety of structural transitions, ranging from spherical micelle growth at intermediate Péclet number (Pe)the ratio of viscous shear and Brownian forcesto the subsequent staged micelle breakage. We noticed that the systems that formed lamellar micelles under quiescent conditions showed changing preferential alignment with increasing Pe. Although significant work has been devoted to

INTRODUCTION Materials whose function depends on their structure rather than their composition, also known as metamaterials,1,2 have found use in emerging technologies such as antennae,3 superlenses,4 cloaking,5 seismic protection,6 and sound filtering.7 Soft metamaterials, in which macro- or supramolecular driving forces are of the same order as thermal energy,8 thus being thermodynamically close to structure transitions, are prone to external manipulation or actuation, enabling structural reconfigurability. Among soft matter systems with promise as effective and efficient ways to realize reconfigurable metamaterials, Janus particles,9 which combine the properties of its two distinct faces, serve as an outstanding model surfactant system in interfacial science in which the effect of internal degrees of freedom of traditional molecular and macromolecular amphiphiles can be decoupled from the amphiphilic character. They stand out as versatile building blocks that can realize a wide variety of structures beyond those formed by isotropic colloidal spheres. The equilibrium assembly of Janus particles at high concentrations produces close-packed structures with different degrees of orientational order.10,11 At low concentrations, Janus particles can form micellar structures akin to those formed by molecular and macromolecular amphiphiles, yielding the canonical12 set of micelle structures formed by molecular and macromolecular amphiphiles (i.e., surfactants and block copolymers): spheres, cylinders,13−16 and © 2017 American Chemical Society

Special Issue: Early Career Authors in Fundamental Colloid and Interface Science Received: August 17, 2017 Revised: October 25, 2017 Published: October 27, 2017 1051

DOI: 10.1021/acs.langmuir.7b02921 Langmuir 2018, 34, 1051−1060

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Langmuir the shear alignment of molecular and macromolecular amphiphiles at high concentrations, both experimentally25,26 and in simulation27−30 and theory,31,32 showing the transition from a parallel orientation of lamellae characterized by layers being parallel to the velocity−vorticity plane to perpendicular, with layers parallel to the velocity−velocity gradient plane, and traverse, parallel to the velocity gradient−vorticity plane, studies of the alignment of Janus spheres in the lamellar state under shearing flow are as yet unknown. However, Janus spheres are perhaps the simplest system capable of forming the lamellar state, so their response to shearing flow should help to assess the basic mechanism of shear alignments in such systems. Here we report the shear-induced alignment of Janus lamellar states as a function of the Péclet number, the attractive strength of the Janus particles, and the particle concentration under the conditions where lamellar micelles form, as reported in our previous work.16 We characterize the different alignment phases in terms of the relative lamellar orientation with respect to the flow axes and map the observed behavior. The reported alignment transitions are then compared with a simple dimensional analysis of the competing forces and torques that drive the observed transitions. Finally, we explore the reversibility of the alignment transitions. Our results report an easy-to-implement structure change that can be harnessed in novel applications as well as a dimensional analysis that provides helpful design parameters for similar lamellar-forming systems. This article is organized as follows. In the Theory section, we introduce our model system and discuss the interaction potential between Janus particles. In the Methods section, we discuss the BD simulation method and the analysis techniques used to identify the different lamellar alignments. The Results section shows our results, including the observed aligned structures from simulation, the scaling analysis to explain these transitions, and our exploration of the structural reversibility of transitions. We summarize our findings in the Conclusions section.

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THEORY Model System for Janus Particles under Steady Shear. Our system is composed of N colloidal Janus particles of radius a subject to a simple shear flow with rate γ̇. The model corresponds to the one used in our previous work on shearinduced effects on Janus particles.16 Figures 1a and 1b shows the relevant parameters in the model, including the attractive cap size, i.e., the Janus balance α and imposed shear flow, with velocity direction ex, vorticity direction ey, and velocity gradient direction ez. Janus Particle Interaction Potential. The anisotropic pairwise additive interparticle potential that we use follows our previous publication16 and is compliant with similar implementation of Janus particle potentials described in the literature.11,14 The total interaction potential energy between Janus particles i and j, Φ, is given by the sum of repulsive (Weeks−Chandler−Andersen, WCA33), ΦWCA, and attractive (Morse), ΦM, potentials as Φ(rij , θi , θj) = ΦWCA (rij) + ΦM (rij) f (θi) f (θj)

Figure 1. Panel (a) shows a schematic representation of relevant interaction parameters for Janus particles, with attractive caps shown in blue, under simple shear flow. Over the attractive caps, the interparticle potential binding region (red) is shown. Panel (b) shows a simulation rendering of an exemplary case with lamellar structures oriented in different directions with respect to the shear axes (velocity (ex), vorticity (ey), and velocity gradient (ez) directions). Panel (c) shows schematically two ideal cases of lamellar orientations (parallel orientationlamellae with their normals (n) parallel to the velocity gradientand perpendicular orientationlamellae with their normals (n) parallel to the vorticity direction) along with the respective ideal values of average orientations relative to shear flow axes.

vector between particles, rij, and f (θ) is an orientation modulation factor. The repulsive WCA potential is given by ⎧ ⎡⎛ ⎞12 ⎛ ⎞6 ⎤ ⎪ ⎢⎜ 2a ⎟ 2a ⎟ 1⎥ 1/6 ⎜ 4 ε − + ⎪ ⎢⎜ ⎟ ⎜r ⎟ ⎥ , if rij ≤ 2 2a 4 r ⎝ ij ⎠ ΦWCA (rij) = ⎨ ⎣⎝ ij ⎠ ⎦ ⎪ ⎪ 0, if rij > 21/62a ⎩

(1)

where rij is the center-to-center distance between Janus particles, θi and θj are the relative orientation angles between Janus particle directors, ni and nj, and the center-to-center

(2) 1052


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Langmuir where ε/kT is the potential strength in units of thermal energy, k is Boltzmann’s constant, and T is the temperature. The attractive Morse potential is given by 2 ⎞ ⎛⎡ ⎛ rij − req ⎞⎤ ⎥ − 1⎟ ΦM (rij) = Mdε⎜⎢1 − exp⎜ − ⎟ ⎟ ⎜⎢⎣ Mr ⎠⎥⎦ ⎝ ⎠ ⎝

with r̂ij being the unit center-to-center vector between particles i and j.38 Figure 1a schematically shows two particles that satisfy the criterion for clustering, where the interactive regions of both particles, shown as red regions, overlap. We identify all clusters in the system and calculate relevant metrics, including the number of clusters, nc, the number of particles belonging to each cluster, Nc, the weight-averaged mean cluster size,39 ⟨Nc⟩, and the cluster size distribution, P(Nc). With the different clusters identified, we proceed to analyze their structure. Structure Analysis and Alignment Quantification. In our previous work, we identified different types of Janus particle clusters by measuring the relative orientation of neighboring particles and the relative orientation of particles with respect to the center of mass of a cluster to distinguish between fluid-like configurations and spherical, cylindrical, or lamellar clusters. Under the conditions studied in the present work, the only phases formed are lamellar clusters and fluidlike configurations, which can be distinguished by the cluster size distribution, P(Nc). To distinguish between the different ways lamellar clusters can align with respect to the shear flow axes, we calculate the relative orientation of clusters with respect to the main axes of the imposed shear flow as ⟨n·ei⟩, defined as

(3)

where Md is the Morse potential depth factor, Mr is the Morse potential range parameter, and req is the Morse potential equilibrium position. The orientational modulation factor, f (θ), is given by f (θ ) =

1 1 + exp{−ω[cos(θ ) − cos(α)]}

(4)

where ω is a parameter that determines the steepness of the angular transition from repulsive to attractive potential along the edge of attractive caps.

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METHODS

Brownian Dynamics Simulations. The evolution of a suspension of colloidal Janus particles subject to shear flow is evaluated using BD simulations.34,35 A detailed description of the equations of motion for particle positions, ri, and orientations, ni, used can be found in our previous work.16 Shear flow is characterized by the shear rate, shown in Figure 1a, and made nondimensional by means of the Péclet number, Pe, defined as Pe = γ̇a2/D0, where D0 is the translational diffusivity of an isolated colloidal particle. The diffusivity D0 is defined as D0 = kT/ 6πηa, where η is the viscosity of the fluid. The length is nondimensionalized with the particle radius, a, and the time is nondimensionalized with the Brownian diffusion time, τD (τD = a2/ D0). Particles are simulated with random initial positions and orientations in a periodic box of volume fractions ranging from ϕ = 0.05 to 0.30 with Lees−Edwards boundary conditions. The choice of interaction range, Janus balance, and interaction strength ensures the formation of lamellar clusters of Janus particles, as seen in Figure 1b. As a side note, the range of interactions is similar to the square-welltype Kern−Frenkel potentials used in the work by Sciortino and coworkers36,37 to study the phase behavior of Janus particles. In this work, we use a Janus balance of α = 90°. System size effects are studied by changing the total number of Janus particles in the simulation box from 250 to 1000. Simulations are run until steady state is reached, as explained below. A statistical analysis of clusters is made for at least 1200 statistically independent configurations at steady state and is replicated at least four times for each condition studied. Table 1 shows a summary of the different interaction potential parameters used. Cluster Identification. We identify lamellar clusters by grouping particles following the procedure used in our previous publication.16 Two particles i and j are part of the same cluster if the center-to-center distance between them is below a coordination distance, i.e., rij/2a < rd, where rd = 2.0 and each particle points its attractive cap toward the center of the other particle, i.e., ni·r̂ij > cos(α) and −nj·r̂ij > cos(α),

⟨n · ei⟩ ≡

value

α Mdb reqc Mrd ωe

90° 2.294 1.878a 1.0a 42.05

a

N

∑

nj· ei (5)

j=1

where ei represents the unit vectors in the velocity (i = x), velocity gradient (i = z), and vorticity (i = y) directions. Equation 5 is calculated over time, and when the values reach steady-state behavior, we evaluate the steady-state values using 1200 independent configurations and various simulation repetitions. Additional, the distribution of the relative lamellar orientations, P(⟨n·ei⟩), are studied at steady state. We say that a lamellar cluster phase is aligned with the velocity gradient direction if ⟨n·ez⟩ > 0.8 and small values in the other directions, hereafter called a parallel orientation, as shown in Figure 1c. A lamellar cluster phase is aligned with the vorticity direction if ⟨n·ey⟩ > 0.8 with small values in the other directions, hereafter called a perpendicular orientation, as also shown in Figure 1c. Lamellar structures that are neither parallel nor perpendicular are identified as random lamellae. We do not find lamellae oriented predominantly in the velocity direction ex. Table 2 summarizes our criteria for structure identification.

Table 2. Summary of Structure and Alignment Identification Criteria phase fluid-like parallel alignment perpendicular alignment random alignment

cluster size ⟨Nc⟩ ⟨Nc⟩ ⟨Nc⟩ ⟨Nc⟩

< > > >

5 5 5 5

alignment criteria ⟨n·ex,y,z⟩ ≈ 0.5 ⟨n·ez⟩ > 0.8 and ⟨n·ex,y⟩ ≤ 0.5 ⟨n·ey⟩ > 0.8 and ⟨n·ex,z⟩ ≤ 0.5 any other conditions

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RESULTS AND DISCUSSION Shear-Induced Lamellar Alignment under Steady Shear. We begin our discussion by quantifying the different structures we observe from BD simulations of Janus particles under steady shear. The main structures observed at steady state are shown in Figure 2, corresponding to N = 500 at different values of Pe at a volume fraction of ϕ = 0.1 and dimensionless interaction strength ε/kT = 5. Here, we present cluster size distributions (Figure 2a−d), P(Nc), relative orientation to shear axis distributions (Figure 2e−h), P(⟨n· ei⟩), and simulation renderings (Figure 2i−l). Under quiescent conditions, Pe = 0, the cluster size distribution (Figure 2a) shows two small peaks at Nc = 1 (singlets) and 60, and a large peak is observed for Nc > 400, which indicates a high

Table 1. Relevant Parameters Used for the Interaction Potential parameter

1 N

a Janus balance. bMorse potential depth factor. cMorse potential equilibrium position. dMorse potential range. eOrientation potential hardness value.

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Figure 2. Effect of steady shear on lamellar-forming Janus particle systems. Panels (a)−(d) show cluster size distributions. Panels (e)−(h) show relative orientation distributions. Corresponding simulation renderings showing the structural alignment are shown in panels (i)−(l). Panels (m)− (p) show schematic representations of the different structures with the criteria used to define randomly aligned, parallel, or perpendicular lamellar alignment as well as fluid-like structure, respectively. Random lamellar alignment is shown at Pe = 0 (panels (a), (e), (i), and (m)). Parallel lamellar alignment is shown at Pe = 0.1 (panels (b), (f), (j), and (n)). Perpendicular lamellar alignment is shown at Pe = 20 (panels (c), (g), (k), and (o)). Fluid-like structure is shown at Pe = 200 (panels (d), (h), (l), and (p)).

probability of finding large clusters that comprise most of the particles in the simulation box. This is consistent with the rendering observed in Figure 2i. At Pe = 0.1 (Figure 2b), the number of small clusters is reduced (small peak at Nc = 1) and the shear flow contributes to the aggregation of clusters of intermediate size into large clusters of Nc > 400. At Pe = 20 (Figure 2c), the shear flow causes cluster breakage, which is shown by a peak in the range of 40 ≤ Nc ≤ 400 shifted to the left with respect to the peak observed at Pe = 0.1 and by an increased number of singlets (large peak at Nc = 1). The structures of large clusters at Pe = 0.1 and Pe = 20 are lamellae as observed in Figures 2j and 2k, respectively. Finally, at Pe = 200 (Figure 2d) the population distribution shows a large peak at Nc = 1 with a number of clusters in the size range of 1 ≤ Nc ≤ 10 indicating the predominance of singlets in the suspension and with only a few small aggregates of Nc ≤ 10. At this large

Pe, the hydrodynamic force is expected to dominate the Brownian and interparticle attractive forces; the formation of lamellar clusters is hindered, and a fluid-like system results (Figure 2l). The above-mentioned results are consistent with our previous findings.16 We now shift our focus to the way the lamellar structures are aligned with respect to the shear flow axes, shown in Figure 2e−h. At Pe = 0 (Figure 2e), the distributions of orientation relative to the velocity, ⟨n·ex⟩, velocity gradient, ⟨n·ez⟩, and vorticity, ⟨n·ey⟩, directions span the range between the minimum and the maximum values of ⟨n·ei⟩, meaning that there is no preferential alignment of lamellar structures. At Pe = 0.1 (Figure 2f), we see a predominant alignment in the velocity gradient direction, shown as a peak in P(⟨n·ez⟩) at ⟨n·ez⟩ > 0.8, with low values of alignment along the other axes, i.e., ⟨n·ex,y⟩ < 0.5. At Pe = 20 1054


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As stated in the Methods section, several volume fractions, ranging from ϕ = 0.05 to 0.3, were explored. For ϕ > 0.1, the transition between parallel and perpendicular alignments occurs at essentially the same conditions as for ϕ = 0.1, as shown in Figure 3. A thorough discussion of alignment at low volume fractions, ϕ < 0.1, is presented below. Additionally, we perform a system size dependency check for system sizes ranging from N = 250 to 1000 to corroborate that the observed structures and transitions are not a product of a small simulation box. In a previous publication,16 we showed that, although lamellar structures lead to cluster sizes that are system-size-dependent, the transitions between structures are not. The lamellar orientations and transitions between them are system-sizeindependent for the range of system sizes explored in this work. Parallel-to-Perpendicular Alignment Transition Scaling Analysis. The dependence on interaction strength, ε/kT, of the Péclet number required to transition between parallel and perpendicular lamellar alignments in the orientation diagram (Figure 3) suggests a competitive effect among Brownian, hydrodynamic shear, and interparticle potential forces and torques. The interplay between the hydrodynamic and interparticle potential forces causes a deformation and breakage of aggregates into small clusters as reported in a previous work.40 The Brownian motion is relevant at small Pe because it is responsible for restoring a system back to equilibrium41,42 or for breaking aggregates when their cohesive energy is on the order of the thermal energy, kT. In Janus particle assemblies, the torques resulting from interparticle interactions are responsible for stabilizing the relative orientation between neighboring particles in a given cluster structure, as pointed out by Walther and collaborators.43 These torques, in the absence or presence of other fields, contribute to the formation of a large number of structures as reported for amphiphilic,14,44 electric,45 or magnetic46,47 Janus particles. Given that parallel- and perpendicularly aligned lamellar structures are composed of large clusters of Janus particles, as seen in Figures 2b and 2c, shear forces are not expected to break interparticle bonds completely where these structures are formed. On the other hand, at strong shear forces, Pe ≫ 1, the breakage of bonds is expected (Figure 2l). Therefore, we perform a simple scaling analysis of the competing Brownian, interparticle potential, and hydrodynamic (derived from the imposed shear flow) torques on a pair of colloids in a lamellar structure, bonded by the Janus particle potential and subject to simple shear flow. Figure 4 shows schematic diagrams of competing torques on a pair of neighboring particles in parallel- or perpendicularly aligned lamellar structures. Our analysis is focused on small Pe, where the parallel orientation is more probable. The torques acting on a pair of particles include Brownian, T B, hydrodynamic shear, T s, and interaction potential, T p. Hydrodynamic shear and Brownian torques contribute to bond breakage, and the interaction-potential torque contributes to restoring particle orientations. The ratio of breaking and restoring torques, RT, is formulated as

(Figure 2g), the predominant lamellar orientation is in the vorticity direction, seen as a peak in P(⟨n·ey⟩) at ⟨n·ey⟩ > 0.8, with values of alignment in the other directions being low, i.e. ⟨n·ex,z⟩ < 0.5. Finally, at Pe = 200 (Figure 2h) the distributions of lamellar orientation relative to the shear flow axes are in the range of 0.4 < ⟨n·ex,y,z⟩ < 0.6. The randomness in orientation along with the small size of clusters for high Pe helps us distinguish these as fluid-like configurations. Thus, the combination of cluster size distributions and relative alignment to shear axes, as shown in Table 2, provides us a quantitative way to distinguish the four different structures discussed in the rest of this article. Lamellar Orientation Diagram at Volume Fraction ϕ = 0.1. In this section, we study the effect of Pe on lamellar alignment under steady shear for various dimensionless interaction strengths, ε/kT, for a volume fraction of ϕ = 0.1. To identify the different ways that lamellar structures align in the presence of a steady shear flow, we analyze cluster size distributions and distributions of relative orientation, applying the criteria shown in Table 2 in a similar fashion to the treatment of the results shown in Figure 2. We organize the data into a map of the different lamellar alignment regimes in ε/kT vs Pe space as shown in Figure 3. Random lamellar

Figure 3. Orientation diagram of Janus particle lamellar structures under steady shear at ϕ = 0.1. Random lamellar alignment is shown with crosses, parallel lamellar alignment is shown with red squares, and perpendicular lamellar alignment is shown with blue circles. Fluid-like configurations are shown with black diamonds. The solid line represents RT = 1 (eq 8).

orientation, shown in Figure 3 as crosses, is expected for very small Pe, i.e., Pe ≪ 1, because Brownian forces are larger than shear forces. We see other cases where random lamellar orientation is seen, especially near the transition between parallel (red squares) and perpendicular (blue circles) lamellar alignment, where it can be between the velocity gradient and vorticity directions. Consistent with the observations shown in Figure 2, we see a clear division between regions of parallel and perpendicular lamellar orientations, with the Péclet number at the boundary dependent on the interparticle strength, ε/kT. As ε/kT increases, the Pe required to shift the lamellar alignment increases. Figure 3 also shows, in black diamonds, fluid-like configurations where no lamellar structure can form as a result of the strength of the shear flow.

RT ≡

breaking torque TB + Ts ≡ restoring torque Tp

(6)

The Brownian torque is on the order of the thermal energy, T B ∼ kT, the interaction potential torque scales with the interaction strength, T p ∼ ε, and the hydrodynamic torque 1055


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and suggesting that the transition is independent of volume fraction, at least in the range of volume fractions tested. In aggregating suspensions, hydrodynamic interactions (HI) play a key role in nonequilibrium phenomena, especially at moderate to high particle volume fractions.48,49 Although recent research efforts have dealt with the hydrodynamics of Janus particle dispersions under shear,19,21 to our knowledge no such work has been devoted to lamellar-forming Janus particle systems. Our expectation on the parallel-to-perpendicular transition is that Pe* will decrease once HI is taken into account because of an increase in the drag coefficient resulting from the local high particle concentration,50 which leads to an increase in T s and thus RT (eq 6). Previous experimental work on lamellar-forming block copolymer solutions,51 which serve as a polymeric analog to the simple Janus particle surfactants, reports a parallel-toperpendicular transition of lamellar phases under conditions consistent with the ones we report here and that the transition occurs at shear rates comparable to the inverse single-chain relaxation time, consistent with our picture of breaking and restoring torques driving the structure transition. Lamellar Alignment at Small Volume Fractions. Figure 5 shows the orientation diagram for a small volume fraction of ϕ = 0.05. The main difference in the observed structures (Figure 5a), compared to those observed at higher volume fraction, is that on the left side of RT = 1 only a few cases are identified as parallel structures following the criteria established in Table 2. We focus on one particular randomly aligned structure close to the RT = 1 line, i.e., Pe = 0.5 and ε/kT = 3, to elucidate the reason that parallel orientations are scarce at the smaller volume fraction. The renderings in Figure 5b reveal tumbling, or rotating, lamellae with a rotation axis parallel to the vorticity direction. The calculated relative orientations as functions of simulation time, Figure 5c, show sinusoidal fluctuations of relative orientations with a predominance in the velocity gradient direction (ez), but the tumbling of lamellae prevent numerical values of relative orientations from reaching the prescribed thresholds to meet the definitions of either parallel or perpendicular orientation identification. The tumbling lamellae seen in Figure 5b provide a clue to the reason parallel orientations are not seen. The renderings in Figure 5b show that the tumbling lamellae do not interact much as they tumble. If more tumbling lamellae were to be present, close enough to interact with the present lamellae, then the fusion of some clusters would result in larger lamella that could interact with other structures nearby, extending the lamella to produce a large lamellar cluster in the parallel orientation. At low volume fractions, the space between the solitary tumblers and other structures is evidently too large to allow the formation of more extended structures. It is important to note that the scaling analysis (i.e., RT = 1) is able to predict the onset of perpendicular lamellar structure formation, as seen in Figure 5a. Realignment of Lamellar Structures under Shear. As mentioned in the Methods section, we start our simulations with random initial particle positions and orientations. The structure formed is dictated by the balance between the Janus interaction forces, Brownian motion, and the imposed shear flow, as shown in Figures 3 and 5. In the interest of achieving structure reconfigurability, we address whether changing conditions, i.e., changing the imposed steady shear flow, results in structural change, specifically for parallel and perpendicular lamellar structures. To this end, we carry out simulations

Figure 4. Schematic representation of the competition among Brownian (TB), hydrodynamic shear (Ts), and interpaticle interaction potential (Tp) torques. Panel (a) shows the competition between torques in a parallel-oriented lamellar structure. Panel (b) shows competition between torques in a perpendicularly oriented lamellar structure. Both panels show the torque competition, only in the direction of the hydrodynamic shear torque, for a pair of neighboring Janus particles.

depends on the shear rate as T s = 8πηa3(γ̇/2) . Substituting these scalings into eq 6, the ratio of torques is expressed as 2

RT =

1 + 3 Pe ε kT

(7)

The ratio of torques, RT, represents the relative balance between breaking and restoring torques. At RT ≪ 1, the restoring torque dominates the net breaking torque and the lamellar structure in parallel alignment is expected to be maintained, whereas at RT ≫ 1, the breaking torque dominates the restoring torque and the lamellar structure in parallel alignment is expected to fall apart because of bond breakage. The competition of torques for perpendicular lamellar structures is shown schematically in Figure 4b. In this configuration, the shear torque does not contribute to bond breaking and results only in Janus particle rotation around its director vector, ni (Figure 1a). Therefore, as Pe increases and RT reaches unity, the parallel lamellar structure breaks apart and the system can instead form a perpendicular lamellar structure that is initially impervious to the effects of shear flow. Taking RT = 1 as the onset of parallel structure breakage, the critical Péclet number, Pe*, required to transition between parallel and perpendicular lamellar orientations is Pe* =

⎞ 3 ⎛⎜ ε − 1⎟ ⎠ 2 ⎝ kT

(8)

We plot this simple relation between interaction strength and Péclet number as a solid line in Figure 3, where it coincides with the observed transition between parallel and perpendicular lamellar structures. We test the scaling analysis for other volume fractions in the range of 0.05 ≤ ϕ ≤ 0.3 (Supporting Information Figures S1 and S2), proving it be a good predictor of the parallel-to-perpendicular lamellar orientation transition 1056


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seen in Figure 6a at t/τD = 25, at the beginning of the simulation is evident from the high value of the relative orientation parallel to the velocity gradient direction, ez. After the increase in Pe, we observe a progressive loss of orientational order as shown by renderings at t/τD = 52 and t/τD = 60 in Figure 6a as well as relative orientations with respect to all shear flow axes tending to values of ≈0.5 as shown in Figure 6b. As the simulation time progresses, the system progressively recovers its orientational order by forming a lamellar structure in the perpendicular orientation (t/τD = 70 and t/τD = 90 in Figure 6a), producing a high relative orientation in the vorticity direction, ey, as shown in Figure 6b. The results shown in Figure 6a,b show that parallel-to-perpendicular lamellar realignment is possible for Janus particles using simple shear as a structure actuator, where the structure change first requires a breakdown of the previous parallel-aligned structure before the perpendicularly aligned structure can form. Figures 6c and 6d shows results for a simulation at a volume fraction of ϕ = 0.1 and an interaction strength of ε/kT = 3 starting at Pe = 10 and shifting to Pe = 0.1 at t/τD = 50, which is the reverse of the flow history shown in Figure 6a,b. Both simulation renderings and orientations relative to shear flow axes show that the initial perpendicular lamellar alignment does not change to parallel with the decrease in Pe. The reason the perpendicular orientation does not change with a decrease in shear rate stems from the fact that the hydrodynamic shear torque does not drive a change in particle orientation, as we schematically show in Figure 4b. A way to bypass the apparent barrier in lamellar realignment would be to increase Pe to large values (e.g., Pe > 200, see Figure 3), where the lamellar structure is taken completely apart, and then change it, for example, to Pe = 0.1. Our inability to observe the perpendicular-to-parallel transition raises an issue: the perpendicular orientation being, apparently, stable at low Pe contradicts the observation of parallel phases at Pe < Pe*. In other words, why is the parallel orientation seen at all? Past work on sheared colloidal suspensions reports a propensity for particles to align in the shear (velocity) direction.40 We argue that the initial lamellar structures formed from random configurations will have preferentially parallel alignment, which would then permeate to the larger lamellae formed at later times. In Supporting Information video V2, we see such an occurrence. Computational work on block copolymer melts under shear reports a similar occurrence of parallel lamellae with initial random configurations. Another issue to take into account is that of the time required for parallel-to-perpendicular and perpendicularto-parallel transitions, which has been reported in past work by Mykhaylyk et al. on block copolymer solutions under shear.51 Mykhaylyk et al. report that the perpendicular-to-parallel transition occurs after an induction time that the parallel-toperpendicular transition does not have. Such an induction time could be beyond the longest time scales simulated in this work and our current computational capabilities. This study of lamellar realignment reinforces the idea that the interplay between the directional Janus torque and imposed shear flow explains the change from parallel to perpendicular Janus particle lamellar alignment under steady shear flow. The analysis can be extended to include the effect of hydrodynamic interactions (HIs), as noted in the development of the simple scaling analysis, or other possible directional interactions, such as magnetic fields, electric fields, and active particle rotation.52

Figure 5. Orientation diagram of Janus particle lamellar structures under steady shear at ϕ = 0.05. Panel (a) shows the orientation diagram with different regions showing random (crosses), parallel (filled red squares), and perpendicular (filled blue circles) lamellar alignments and a region showing a fluid-like structure (filled black diamonds). The solid line represents the function RT = 1 (eq 8). Panel (b) shows simulation renderings for ε/kT = 3 and Pe = 0.5 for different times during the simulation whose results are shown in panel (c). Panel (c) shows the relative orientations with respect to the shear flow axes at ε/kT = 3 and Pe = 0.5 as functions of time.

starting with a steady state parallel lamellar orientation, and after a certain time, we change Pe to a condition where a perpendicular lamellar orientation is expected. We then perform the reverse process, i.e., start with a perpendicular lamellar orientation and change Pe to a condition where a parallel lamellar orientation is expected. Figure 6 shows the simulation results of reconfiguration simulations. Figures 6a and 6b show results for a simulation at a volume fraction of ϕ = 0.1 and an interaction strength of ε/kT = 3 starting at Pe = 0.1, where a parallel orientation is expected (Figure 3), and then at t/τD = 50, shift to Pe = 10, where a perpendicular orientation is expected. The parallel alignment, 1057


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Figure 6. Reconfiguration simulations of Janus particle lamellar structures at ε/kT = 3 and ϕ = 0.1. Panels (a) and (b) show, respectively, simulation renderings and relative orientations relative to shear flow axes as functions of time for Pe = 0.1 (t/τD < 50, zone A) and Pe = 10 (t/τD > 50, zone B). Panels (c) and (d) show, respectively, simulation renderings and relative orientations relative to shear flow axes as functions of time for Pe = 10 (t/τD < 50, zone B) and Pe = 0.1 (t/τD > 50, zone A). The insets show schematically an orientation diagram where reorientation from parallel to perpendicular (A → B) lamellar orientation is possible (inset in panel (b)) and where a lamellar reorientation from perpendicular to parallel (B → A) is not possible (inset in panel (d)).

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CONCLUSIONS Various microstructures assemble under equilibrium conditions for Janus particles depending on the Janus patch size and interaction strength and range. In this study, we have conducted Brownian dynamics simulations to elucidate the effect of shear flow in the orientation of Janus particle lamellar structures, which are typically obtained for Janus balance α ≥ 90. We found that the alignment of the lamellar structure with respect to the direction of the shear flow depends on both the strength of the shear flow and the interaction strength of the directional Janus particle potential. To elucidate the main physics dictating lamellar alignment, a simple scaling argument was derived resulting from a torque balance on a pair of Janus particles in the lamellar structure. This analysis predicted the interface between parallel and

perpendicular alignment regions. This torque balance produced a control parameter RT, which is the ratio of structure-breaking torques from Brownian motion and imposed shear to the aligning torque from the attractive interactions of the Janus caps. For RT ≪ 1, the orientation of the lamellar structures is completely random whereas alignment parallel to the shear flow occurs as RT is increased. Near RT = 1, the interparticle torque is overcome by the sum of the Brownian and hydrodynamic torques; therefore, the parallel lamellar structure disassembles and reconfigures into a perpendicular lamellar structure. The theoretical assumption, in the scaling analysis, that the structure stability is fully dictated by a pair-level description is corroborated by the simulation results that show that the lamellar alignment is independent of the particle concentration. This observation was corroborated with simulations of particle 1058


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support from the U.S. Army Research Office through a MURI grant (award number W911NF10-1-0518). R.G.L. also acknowledges NSF support under grant CBET-1602183.

volume fractions of up to 30%. This study also carries implications for the effect of fluid flows on lamellar structures composed of block copolymers and surfactants, where the bulk of the work has been devoted to large concentrations (i.e., melts).25−27 Previous work indicates transitions between parallel and perpendicular alignment that vary from one system to the next and may depend on molecular relaxation and the contrast in viscosities between lamellar layers in the case of block copolymers. Goulian and Milner32 found from a field theoretical analysis that the parallel state should be unstable to the formation of the perpendicular state, a result we find consistent with our results, but only at shear rate large enough to rotate Janus spheres against interparticle interaction torque. Our particularly simple results indicate the instability of parallel alignment under this condition but that the perpendicular state is stable unless it is broken down by very high shear rates. These results for our system that lacks the complications and many degrees of freedom of block copolymers and other molecular systems should provide a baseline against which those more complicated systems can be compared. The precise alignment of lamellar structures could enable applications in photonics, microfluidics, and sensors. As shown previously, the alignment in lamellae can be tuned either by the flow or the energy of interaction between particles; however, the latter is hard to accomplish experimentally. A hypothetical flow-operated device could convert parallel to perpendicular lamellae by a simple increase in the shear flow, whereas the perpendicular to lamellar realignment could be accomplished by increasing the flow until full breakage of the structures occurs and then reducing the flow to a value where RT < 1; therefore, the alignment is fully reconfigurable. Future work should include the effect of hydrodynamic interactions and large particle concentrations, where a direct comparison with molecular and macromolecular amphiphiles can be made and should lead to additional physical insights on the phenomenon of shear-induced alignment.

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.7b02921. Orientation diagrams for ϕ = 0.15 and 0.20 and four videos showing random alignment (V1), parallel alignment (V2), perpendicular alignment (V3), and fluid-like structure (V4) (ZIP)

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REFERENCES

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Daniel J. Beltran-Villegas: 0000-0003-1667-3181 Ronald G. Larson: 0000-0001-7465-1963 Ubaldo M. Córdova-Figueroa: 0000-0003-4891-5325 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS R.A.D.-A. and U.M.C.-F. acknowledge support from the National Science Foundation (NSF) in the form of a CAREER award (CBET-1055284). D.J.B.-V. and R.G.L. acknowledge 1059


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