Coassembly of Nanorods and Photosensitive Binary Blends

Using a similar course-grained computational approach, we focused on routes for harnessing light to create periodically ordered, defect-free polymeric...
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Coassembly of Nanorods and Photosensitive Binary Blends: “Combing” with Light To Create Periodically Ordered Nanocomposites Ya Liu, Olga Kuksenok, and Anna C. Balazs* Chemical Engineering Department, University of Pittsburgh, Pittsburgh, Pennsylvania 15261, United States S Supporting Information *

ABSTRACT: Using computational modeling, we establish a means of controlling structure formation in nanocomposites that encompass nanorods and a photosensitive binary blend. The complex cooperative interactions in the system include a preferential wetting interaction between the rods and one of the phases in the blend, steric repulsion between the coated rods, and the response of the binary blend to light. Under uniform illumination, the binary mixture undergoes both phase separation and a reversible chemical reaction, leading to a morphology resembling that of a microphase-separated diblock copolymer. When a second, higher intensity light source is rastered over the sample, the binary blend and the nanorods coassemble into regular, periodically ordered structures. In particular, the system displays an essentially defect-free lamellar morphology, with the nanorods localized in the energetically favorable domains. By varying the speed at which the secondary light is rastered over the sample, we can control the directional alignment of the rods within the blend. Our approach yields an effective route for achieving morphological control of both the polymeric components and nanoparticles, providing a means of tailoring the properties and ultimate performance of the composites.

I. INTRODUCTION One of the challenges in creating high-performance polymeric nanocomposites for optoelectronic and structural applications is establishing effective and facile routes for controlling the morphology of both the polymeric components and the nanoparticles, which impart the desirable optical, electrical, and mechanical properties to the material. The ability to control morphology provides a means of tailoring the properties and ultimate performance of these hybrid materials. Achieving precise control over morphology, however, is especially challenging when the nanoparticle additives have a high aspect ratio, such as nanorods.1−5 Nanorods are remarkably useful for improving the photovoltaic properties6 and strength7,8 of the matrix materials. Hence, it is particularly important to devise routes for tailoring the structure of nanorod-filled polymers and, in this manner, optimize the desirable attributes of these composites. In previous studies,8,9 we used computational modeling to predict means of controlling the morphology of polymer-coated rods in a binary phase-separating blend. In this mixture, the steric interactions between the coated rods, the preferential wetting interactions between these rods and one of the components in the blend, and the phase separation of the blend all led to synergistic behavior, which caused the rods to form percolating structures at very low volume fractions of these fillers.9 Notably, these predictions were later confirmed experimentally.10 We also showed11 that the morphology of © 2012 American Chemical Society

such nanocomposites could be tailored by introducing an attractive interaction between the rods. As seen in the simulations, the corresponding experimental studies revealed that binary mixtures encompassing such attractive rods exhibited novel bicontinuous structures.12 Using a similar course-grained computational approach, we focused on routes for harnessing light to create periodically ordered, defect-free polymeric materials from binary and ternary mixtures of homopolymers.13−15 The latter studies were inspired by prior experimental findings on blends of transstilbene-labeled polystryene and poly(vinyl methyl ether) (PSS/PVME).16−18 Upon irradiation, the stilbene moieties on the PSS chains undergo a reversible trans−cis photoisomerization, and the mixture undergoes phase separation since the cislabeled polystyrene and PVME are immiscible.16−18 Thus, two distinct processes are occurring simultaneously within the binary blend: phase separation and a reversible chemical reaction. As a result of these simultaneous processes, the system exhibits a morphology resembling that of microphase-separated diblock copolymers.19 We also took advantage of the fact that the rate of the reversible trans−cis isomerization can be controlled by varying the light intensity,18 with the higher Received: November 6, 2012 Revised: December 11, 2012 Published: December 19, 2012 750

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∂ψ δF = M ψ ∇2 − 2Γψ ∂t δψ

intensity light also producing smaller domain sizes within the blend.19 On the basis of the above experimental findings,16−18 we simulated the behavior of a phase-separating binary blend that undergoes a reversible chemical reaction in the presence of a stationary background light (see Methodology). As noted above, this led to a structure resembling the morphology of microphase-separated diblocks. We then introduced a secondary, higher intensity light source and rastered this light over the mixture. The resulting system displayed a periodic, defectfree structure. In effect, we used the rastering light to “comb out” any defects that existed in the pattern formed by the irradiated binary blend. Thus, through the combination of the background and rastering lights, we achieved significant control over the morphology of the binary13,14 (and ternary15) blends. Thus, our previous studies provided guidelines for (1) regulating the spatial arrangement of nanorods in disordered blends and (2) creating highly ordered, regular structures from photoreactive homopolymer mixtures. The careful integration of these processes could lead to a novel approach for controlling the arrangement of nanorods within a periodically ordered, defect-free polymer matrix. Herein, we build on these previous studies8,9,13−15 to accomplish the latter goal and hence establish a procedure for coassembling and ordering the nanorods and the matrix AB binary blends. Namely, we introduce coated nanorods that have a preferential wetting interaction with one of phases of a photosensitive binary blend and then utilize both the stationary and rastering light to tailor the morphology of the system. As we show below, this approach allows us to design hybrid materials where the nanorods are localized in the energetically favorable domains and are aligned by the rastering light, and the entire structure shows regular, long-range order. Below, we first describe the computational model we developed to undertake these studies. We then discuss the effects of varying the number and characteristics of the rods as well as the speed at which the secondary light source is rastered over the sample.

where Mψ is the mobility of the order parameter, F is the free energy functional defined below, and the forward and reverse reaction rate coefficients are set to be equal: Γ+ = Γ− ≡ Γ. The nanorods are considered as one-dimensional rigid objects and characterized by the center of mass {ri} and orientation angle {θi}, referenced to a fix coordinate system.8,9 Because of the rod’s anisotropic shape, its translational motion is decomposed into components that lie parallel and perpendicular to the rod axis, each of which obeys the following Langevin equations:8,9,20,21 ⎛ δF ⎞ ∂r i = −M r ⎜u · ⎟ + ηi ∂t ⎝ δ ri ⎠ ⎛ δF ⎞ ∂r ⊥i = −M r⊥⎜v· ⎟ + ηi⊥ ∂t ⎝ δ ri ⎠

⟨ηi (t ) ·ηj (t ′)⟩ = 2kBTM r δijδ(t − t ′) ⟨ηi⊥(t ) ·ηj⊥(t ′)⟩ = 2kBTM r⊥δijδ(t − t ′)

(4)

where kB is the Boltzmann constant and T is temperature. The rotational motion of the rods is similarly governed by the following equation:8,9,20,21 ∂θi δF = − Mθ + ζi ∂t δθi

(5)

with rotational mobility Mθ and random torque ζi satisfying ⟨ζι(t)ζj(t′)⟩ = 2kBTMθδijδ(t − t′). The total free energy of the system consists of three parts: F = FGL + Fpr + Frr, which represent the contributions from the respective polymer−polymer, polymer−rod, and rod−rod interactions.8,9 The free energy for the binary blends is given by the following Ginzburg−Landau equation: FGL =

Γ+

Γ−

(3)

provided that u and v are unit vectors collinear with and perpendicular to the rod axis, respectively. M∥r (M⊥r ) is the translational mobility along the u (v) direction. η∥i (η⊥i ) is the random force acting on the ith rod along u (v) and satisfies the fluctuation−dissipation relation:20

II. METHODOLOGY Our system consists of nanorods that are immersed in an AB binary, photosensitive mixture. When the AB blend is irradiated with light, it simultaneously undergoes phase-separation between the A and B components and the following reversible chemical reaction: A⇔B

(2)

∫ [−aψ 2/2 + bψ 4/4 + c(∇ψ )2 /2] dr

(6)

The first term in eq 6 defines a local free energy and the coefficients a and b are chosen to ensure that this free energy has two equal minima, corresponding to a pure A and pure B phase.13−15 The third term in eq 6 characterizes the energy penalty for creating spatial variations of the order parameter and thus is related to the interfacial tension.14 Here, we set the respective values for a, b and c in eq 6 to be 0.9, 1.3 and 0.5. The coupling between the polymer and rods is given as8,9

(1)

where the parameters Γ+ and Γ− represent the respective forward and reverse reaction rate coefficients. As noted in the Introduction, blends of trans-stilbene-labeled polystryene and poly(vinyl methyl ether) (PSS/PVME) exemplify a system that undergoes these two processes: phase separation and a reversible chemical reaction.16−18 This incompressible AB binary blend is characterized by the continuous order parameter ψ(r), defined as the difference between the local volume fraction of the A and B components: ψ(r) = φA(r) − φB(r), with the condition φA(r) + φB(r) = 1. Thus, ψ(r) = 1 and −1 correspond to the A-rich and B-rich phases, respectively. The phase-separation dynamics is described by a modified Cahn−Hilliard equation that incorporates the contribution from the reversible chemical reaction:13−15,19

Fpr =

∫ dr ∑ ∫ V (r − si)(ψ (r) − ψw)2 dsi i

(7)

where si is the position vector on the surface of the ith rod. The constant ψw represents the preferential wetting between the coated rods and one of the components of the blend; this interaction can be tailored experimentally by altering the chains that coat the surface of the rods. For example, setting ψw = 1 (−1) specifies that it is energetically favorable for the rods to be 751

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preferentially localized in the A (B) phase. V(r − si) is a shortrange attractive potential and has the following form:8,9 V (r − si) = V0 exp( −|r − si| /r0)

It is known that in the absence of rods both the amplitude and wavelength of the order parameter decrease with an increase in the reaction rate coefficients.19,23 These values could be found analytically using the single mode approximation,23 which is an excellent approximation for higher values of Γ, or from the respective simulations in the absence of the rods.15 Namely, our simulations show that for the chosen values of Γ1 = 0.006 and Γ2 = 0.02 the bulk values of the local order parameters corresponding to the A-like (B-like) phases are ≈0.75(−0.75) and ≈0.24(−0.24), respectively. Finally, we find that the characteristic wavelengths of the lamellar patterns estimated using the single mode approximation are λ1 ≈ 5.5 and λ2 ≈ 4.1 for the patterns formed within the Γ1 and Γ2 regions, respectively. Rods with length L = 9, number N = 300, and preferential wetting ψw = 0.75 are set as the reference values in the ensuing simulations, unless specified otherwise. Note that the value of ψw that describes the coupling between the rod and the polymer is set to a value corresponding to the A-like phase within the Γ1 region. In other words, the lowest energy of the rod/polymer interaction, Fpr, corresponds to a rod surrounded by the A-like phase in the background illumination of Γ1 = 0.006 (i.e., to ψ = ψw = 0.75). In our one-dimensional representation of nanorods, the rod diameter b is not set explicitly; rather, the effective dimensionless width of the rods is set by the interaction range in the rod−mixture interaction (eq 8), which is equal to b ≈ 2. We note that the values above have been used in prior studies of the self-assembly of nanorods in binary mixtures.8,9 Independent simulations show that our results are qualitatively insensitive to the rod mobilities: similar results are observed for M⊥r = 0 (i.e., modeling rods immersed in a polymer network) and M⊥r = 0.5(i.e., corresponding to rods moving in semidilute solutions).20,24 We thus set M⊥r = M∥r = 1 as a simplification.8,9 (The insensitivity of the results to the value of M⊥r arises from the localization of the rods into long, lamellar domains. Since the length of the rods is comparable to the width of the lamellae and the width of these domains are relatively narrow, the translational motion of rods perpendicular to the rods axis is suppressed; hence, the rod motion is relatively independent of the value of M⊥r .) The dimensionless simulation parameters we chose are related to the corresponding experimental values as follows. If we relate the lattice spacing in our system15 to ζ = 100 nm, then the simulation unit of time is given as τ ≈ 0.036ζ2/Dp, where Dp is the diffusion coefficient for the polymers, provided that Mψ = 1. Taking the characteristic diffusion coefficient as Dp ≈ 10−14 m2/s, we find τ ≈ 0.036 s. Therefore, our reference dimensionless combing speed, v = 5 × 10−3 (see below), corresponds to vζ/τ ≈ 13.8 nm/s. Finally, the length of the nanorods L = 9 corresponds to a physical value of 900 nm.

(8)

where V0 is the strength and r0 is the characteristic length scale of the rod−polymer interaction, which is also tunable by varying the chemical nature and length of the chains anchored on the rods.9 We assume that the chains coating the rods are end-tethered to the rod’s surface and thus would give rise to a repulsive steric interaction as the rods come into close contact. Hence, the rod−rod interaction has the following form, which describes the steric repulsion:8,9 ⎧ χ ∑ (L − |r − r |)2 [4/3 if |r − r | < L i j i j ⎪ ⎪ ij Frr = ⎨ − cos2(θ − θ )] i j ⎪ ⎪0 if |ri − rj| ≥ L ⎩

(9)

where L is the rod length and χ is the strength of the repulsive interaction. This interaction is known to lead to an isotropic− nematic phase transition for the pure rod system.9 Notably, we recently compared the results obtained from this coarse-grained model with those obtained from dissipative particle dynamics, where the tethered chains are explicitly included, and found that the self-assembled structure of the coated rods (within immiscible blends) is remarkably similar for both approaches.8 Hence, the above equations capture the rod−rod interactions that are induced by the coating. In previous studies on photosensitive, binary blends in the absence of nanoparticles, we showed that defect-free, periodically ordered structures could be created by exposing the sample to both a uniform, background light and a moving, higher intensity light source.13−15 Within the regions illuminated by the background light, the rate reaction coefficient was Γ+ = Γ− ≡ Γ = Γ1. Since the forward and backward rate coefficients were equal, the binary mixture formed a lamellar structure. For the regions illuminated by the secondary light source, the chemical reaction rate was Γ = Γ2 where Γ2 > Γ1. Within these Γ2 regions, the domains were smaller and more intermixed than in the Γ1 domains. We observed that at each moment of time the lamellar domains within in the Γ1 region aligned perpendicular to moving interface between the Γ1 and Γ2 regions. With the proper choice of Γ2/Γ1 and velocity, v, at which the higher intensity was rastered over the sample, the mobile light source effectively combed out defects within the system, so that the mixture assembled into a regular lamellar structure.13,14 Here, we apply the same procedure to photosensitive binary mixtures that are filled with nanorods. In these simulations, we solve the governing equations on a two-dimensional 256 × 256 square lattice, with periodic boundary conditions at all the edges. The numerical discretization and integration of eq 2 are carried out using cell dynamics method.22 A Brownian dynamics simulation is utilized for the translation (eq 3) and rotation (eq 5) of the nanorods. Parameters used in these simulations are rescaled in terms of the lattice spacing units and kBT. The mobilities are fixed at Mψ = M∥r = M⊥r = 1 and Mθ = 0.1.9 The parameters for the interactions are set at V0 = 0.02, r0 = 2, and χ = 0.6. The reaction rate coefficients are chosen as Γ1 = 0.006 and Γ2 = 0.02, and the width of the rastering stripe is w = 10.

III. RESULTS AND DISCUSSION A. Mixtures with A-like Rods. Our system encompasses a 50:50 AB binary mixture that contains nanorods, which are preferentially wetted by the A phase. In the ensuing discussion, we first consider rods of length L = 9 and N = 300 of these nanorods are initially distributed randomly within the binary mixture. (At L = 9 and b ≈ 2, this represents a rod volume fraction of 8.2%.) This sample is illuminated uniformly by a background light, which initiates the chemical reaction given by eq 1. The reaction rate coefficient for this photoinduced reaction is given by Γ1. We also utilize a spatially localized light 752

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Figure 1. (a) Early time morphology of nanorods-filled binary mixture at t = 5 × 103. Mixture is represented by the value of order parameter ψ(r), as illustrated by the color bar, in which red (blue) corresponds to the A-rich (B-rich) phase. Rods are shown in black. Rastering light moves periodically along x-direction from left to right (as marked by arrow) with velocity v = 5 × 10−3. (b) Late time morphology of the system at t = 2 × 106.

source, which is indicated by the pale vertical stripe of width w = 10 lattice units in Figure 1. The localized light is taken to have a higher intensity than the background illumination and hence yields a higher reaction coefficient for the photoinduced reaction. Thus, within the vertical stripe the reaction rate coefficient is given by Γ2 > Γ1. The stripe of light is rastered over the sample, moving from left to right (along the x-direction) with a constant dimensionless velocity v = 5 × 10−3. Given that the length of our square lattice is l = 256, this means that it takes roughly 5 × 10 4 dimensionless units of time to perform one swipe over the entire sample. Under the influence of the light, the binary mixture phaseseparates into A-rich and B-rich domains (shown in red and blue, respectively). Because the reaction rate coefficients for the forward and backward reactions in eq 1 are equal, the phaseseparating system assumes a lamellar morphology,13−15,19,23 but as shown in Figure 1a, the blend does not exhibit any distinct ordering at early times. Since the nanorods are wetted by the A component (ψw = 0.75 in eq 7), they preferentially migrate into the red phase to minimize the total free energy of the system; at early times, the spatial arrangement of the rods is also disordered. Because of the higher reaction rate coefficient, Γ2, within the spatially localized stripe of light, the domains within this region are more intermixed and the average domain size is smaller;19 hence, the order parameter ψ for the blends is smaller than in the rest of the sample. This can be seen from the color bar in Figure 1, which indicates that ψ ≈ 0 within this stripe. Thus, the Γ2 region is more structurally homogeneous than the neighboring Γ1 region, which encompasses larger, distinct Alike and B-like domains. Because of this relative homogeneity within the Γ2 stripe, this stripe presents a “neutral” interface to the adjacent Γ1 region.13−15 In previous studies involving just such AB binary mixtures (no particles), we exploited the effective neutrality of the Γ2 region to drive the system into an ordered morphology and create essentially defect-free materials. To appreciate the utility of the neutral interface, it is important to recall that an A/B

interface will form a 90° contact angle with a neutral, nonselective boundary;13−15 that is, the domains orient perpendicular to a neutral boundary. Alternatively, because eq 2 can be used to describe the dynamic behavior of diblock copolymers,19 we can draw an analogy between the binary system in Figure 1 and the behavior of diblocks confined between neutral or nonselective walls. In the latter system, it is known that the lamellar domains of symmetric diblocks align perpendicular to neutral walls since this configuration lowers the free energy of the system.25 Keeping the above behavior in mind, it can be understood that the lamellar domains in the Γ1 region will reorient and align perpendicular to the Γ2 boundary. If the Γ2 region is moved across the sample, then at each moment of time, the lamellae within the Γ1 region will orient perpendicular to the Γ2 boundary. With each additional passage of the secondary light source over the material, most of remaining nonaligned domains are driven to reorient. Hence, the repeated motion of the Γ2 stripe can be said to “comb out” defects within the system and thereby cause the mixture to form ordered, defectfree structures.13−15 The above discussion helps explain the morphology seen in Figure 1b, which shows the late time structure of the system, after the Γ2 stripe was rastered over the sample multiple times. It is apparent that the A and B lamellae lie perpendicular to the Γ2 boundary. Furthermore, as the Γ2 region was repeatedly passed over the sample, it effectively “combed out” structural defects within the Γ1 region. Clearly, the binary mixture in Figure 1b appears to be highly ordered. Notably, the nanorods in Figure 1b are also highly organized. The favorable wetting interaction between the A phase and the nanoparticles not only causes the rods to be preferentially localized in the A domains but also to align along the oriented lamellar domains, i.e., along the combing direction. It is worth noting that while the rods are located within the Γ2 regime, their orientations can deviate from the x-direction to some degree, which depends on ts, the time that a region is illuminated by the stripe in one sweeping cycle. (For a rastering velocity v and stripe width w, ts = w/v.) In the example in 753

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Figure 1b, these deviations in the rods’ orientation are relatively small. (We will return to a discussion of the rods’ reorientation within the Γ2 stripe as a function of the stripe velocity in section B.) Because of the orienting effects of the A domains, such deviations are suppressed when the rods leave Γ2 and reenter the Γ1 regime. The sample in Figure 1b does exhibit some defects at t = 2 × 106; the length of the rods, L = 9, is longer than the width of the A domains, and a few of the rods are oriented perpendicular to the lamellae, causing some reorientation of the mixtures. Notably, these defects are eventually eliminated and essentially defect-free samples are created at t = 3 × 106 (not shown). To quantify the degree of order within the system, we introduce the order parameters β and S for the binary mixtures and nanorods, respectively, which are defined as 2

Figure 3. Time evolution of order parameter, S, for N = 300. From top to bottom, the curves correspond to rods length L = 3, 6, and 9. Insets show the morphology of system with L = 3 at early (t = 5 × 103, left image) and late (t = 2 × 106, right image) times.

2

β = ⟨⟨ψ (r, t )⟩x /⟨ψ (r, t )⟩x ⟩y S = ⟨2 cos2 θ − 1⟩

(10)

Here, the brackets ⟨ ⟩i in the expression for β represent the average along the i-direction within the Γ1 regime, and the brackets ⟨ ⟩ in the expression for S represent the ensemble average for all rods. θ is the orientation angle with respect to the x-direction. As illustrated in the insets in Figure 2, β = 1

As can be seen from Figures 2 and 3, the rod length does affect the time required for the mixture to reach the optimal ordering of the given system. The alignment of the rods is fastest in the case of the shortest rods. Namely, the highest possible ordering is achieved for the L = 3 case at approximately t ≈ 2 × 105 (i.e., after the higher intensity light was rastered over the sample four times). Additionally, β is higher for the L = 3 case than for the L = 6 and L = 9 examples. The inset in Figure 3 shows the structure of a binary mixture filled with rods of length L = 3 in the initially disordered phase and the defect-free, late-time state. The ordering is more efficient in the L = 3 case primarily because this rod length is smaller than the width of the lamellar domain (see inset in Figure 3). In contrast, longer rods can more easily create subdomains having an orientation that is different from the combing direction (as can be seen in the Supporting Information movies). If such subdomains are created, then typically multiple rastering cycles are required to order the mixtures. Hence, a high degree of ordering is achieved only at t ≈ 2 × 106 (i.e., after the rastering process was repeated 40 times). The volume fraction of the nanorods within the blend will also affect the overall ordering of the mixture; to illustrate this point, we increase the number of rods in our system to N = 600 and 900. With L = 9 and b ≈ 2, these numbers correspond to rod volume fractions of ∼16% and ∼25%, respectively. Figures 4a and 4b show the temporal evolution of β and S for N = 600 and N = 900, respectively. These curves show that while the rastering light does introduce long-range directional order into these particle-filled samples, increases in N decrease the magnitudes of β and S and increase the time required to reach the optimal values of these order parameters. Note that increasing N increases the crowding in the mixture and hence enhances the steric repulsion (excluded volume effects) between the neighboring rods; unable to move or navigate in these crowded systems, some of the rods become kinetically trapped and frozen into specific configurations. Figure 4c shows the late-stage morphology (at t = 5 × 106) of the mixture with N = 900 rods. Clearly, this morphology displays some degree of ordering; however, we observe the appearance of defects or subdomains with preferential orientations that are distinctly different from the combing direction. Such defects persist within the system and can no longer be eliminated by repeated combing; note that the

Figure 2. Time evolution of order parameter, β, for N = 300. From top to bottom, the curves correspond to rods length L = 3, 6, and 9, as marked in the legend. Insets show the typical morphology for order parameter for rods, S, and binary mixtures, β, within the fraction of the simulation box for the given values.

corresponds to the completely ordered phase, i.e., the lamellar domains are aligned along the combing direction (the xdirection), and β = 0 corresponds to lamellar domains that do not exhibit any degree of order along x. Similarly, S = 1 (−1) corresponds to rod alignment parallel (perpendicular) to the lamellar domains and S = 0 characterizes rods with a random distribution of orientations. The temporal evolutions of β and S for three different mixtures, which contain rods of respective lengths L = 3, 6, and 9, are shown in Figures 2 and 3. Starting from zero (corresponding to the initial intermixed states), both the order parameter curves show a rapid evolution to a plateau value that is indicative of an ordered state. The time scale for reaching a well-defined plateau is approximately the same for β and S, which indicates that the rastering light has simultaneously “combed” the polymer matrix and the nanorods into an ordered state. Small fluctuations of the order parameters are due to thermal noise that acts on the rods (see eq 4). 754

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binary mixtures requires that ts > tresp. Namely, the combing can only be effective if the rastering velocity is sufficiently slow that the domains in the Γ2 region reach the equilibrium size, orientation, and degree of intermixing that corresponds to the higher light intensity.14 In addition, we showed that the effective ordering also requires a sufficiently high ratio of Γ2/ Γ1.14 In other words, while the combing process yields a high degree of order for high ratios of Γ2/Γ1 and sufficiently slow velocities, for lower ratios of Γ2/Γ1, the ordering is significantly less efficient and depends more on the actual value of the rastering velocity.14 As we discuss below, the velocity v also has a significant effect on the behavior of the rod-filled mixtures. In the following, we fix the number of rods at N = 300. If the light is rastered at the relatively high velocity of v = 10−2 (shown in Figure 5), the system displays behavior similar to that observed for the case of v = 5 × 10−3. The rods are aligned within the red A phase throughout the sample as the light is passed over the material multiple times. Note, however, that the ordering is less efficient at v = 10−2 than at v = 5 × 10−3. Importantly, independent simulations show that binary mixtures without the rods combed with the rastering velocity v = 10−2 (with all the other simulation parameters held fixed) exhibit an even lower degree of ordering than the system shown in Figure 5. This low efficiency of the combing process at a high velocity and a relatively low Γ2/Γ1 ratio is consistent with the previous studies.14,26 Hence, our simulations indicate that the presence of rods increases the degree of ordering in the scenario in Figure 5. The situation is quite different, however, when the light is rastered at low velocities; here, we consider v = 5 × 10−4 and v = 10−4. In these scenarios, as the Γ2 stripe slowly passes over a rod, this rod remains in the higher intensity illumination for a significant time. As noted above, the average value of the order parameter characterizing the binary mixture in the Γ2 region is ψ ≈ 0, with the maximum values corresponding to the A-like and B-like phases at ψ ≈ ±0.24. With a rod in the Γ2 region for a finite time, this value of ψ is modified by the presence of the A-coating on the rod; namely, the region becomes more A-like. As discussed in the Methodology, φA(r) + φB(r) = 1 in this system, so that a buildup of A within the Γ2 region leads to a high concentration of B in the region directly adjacent to this stripe. This can be clearly seen in Figure 6, where a distinct blue band lies next to the Γ2 region. (Recall that the secondary light source is moved from left to right, and hence, the buildup of A from the coated rods is higher on the left side of the Γ2 stripe, leading to a high concentration of B adjacent to this side, as seen in the figure.) This neighboring B band exerts a repulsive force on the A-coated rods (see eq 7) that arises from the incompatibility between the A and B components. This repulsive force inhibits the A-coated rods from exiting the Γ2 stripe. As the stripe continues to be moved over the sample, the above scenario is repeated multiple times and the Γ2 region effectively collects a significant number of rods that are aligned parallel to the length of the Γ2 stripe, i.e., along the y-direction. Hence, instead of delivering the rods throughout the material, the secondary light source essentially collects the rods and clears regions to the left of the stripe of these nanoparticles. This effect of collecting rods within the Γ2 stripe at a low rastering velocity strongly depends on the number of rods in the system. To illustrate this point, we performed simulations with the same parameters as in Figure 6, including the slow

Figure 4. (a) Time evolution of order parameters, β and S, for N = 600. (b) Time evolution of order parameters, β and S, for N = 900. (c) Morphology of the system with N = 900 at t = 5 × 106.

morphology in Figure 4c was observed after the higher intensity light stripe was rastered over the sample 100 times. When the number of rods was increased even further, to N ≈ 1200, the entire sample consisted of similar subdomains with different orientations, and combing with light no longer introduced a noticeable ordering into the system. B. Varying the Rastering Speed. In prior studies involving just the photosensitive AB mixtures (no rods), we showed that the degree of ordering achieved through this “combing” procedure depended on the velocity at which the stripe of light was rastered over the sample, v.14 Specifically, in order to achieve optimal ordering, v must be smaller than a critical value. This notion can be expressed in terms of the two parameters ts and tresp, where ts = w/v is the time that a region is illuminated by the stripe in one sweeping cycle and tresp characterizes the response time required for the mixture to rearrange due to the rastering light. Effective combing of the 755

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Figure 5. (a) Time evolution of β and S, for fast rastering, v = 10−2. (b) Morphology of the system at t = 3 × 106.

Figure 6. (a) Late time morphology (t = 5 × 106) of the system with N = 300, rastered with velocity v = 5 × 10−4 . Nanorods are collected by the light stripe. (b) 3D height map for the order parameters corresponding to morphology in (a).

Figure 7. (a−d) Morphology of three rods being collected by the rastering stripe. (e−h) Morphology of single rods escaping from the rastering stripe. Images show a portion of a simulation box with N = 20 rods.

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rastering velocity v = 5 × 10−4, but with only N = 20 rods immersed within the binary mixture. Figure 7 shows a close-up of a portion of the simulation box with the top row revealing the dynamics of the Γ2 stripe passing over three rods located in close proximity to each other and the bottom row showing the Γ2 stripe moving over a single rod without any neighboring particles. In both cases, the repulsive force described above pushes the rods from the left to the right along the x-direction as the stripe passes over the rod. The interaction between the three rods (top row in Figure 7) prevents their reorientation and escape from this stripe; i.e., the block of three rods remains within this Γ2 stripe and moves together with it, similar to the effect we observed above for a larger number of rods (see Figure 6a). If, however, the rod is isolated and does not have any neighbors, it can reorient and escape from the metastabe state within the Γ2 region into the Γ1 region of the box as shown in the bottom row in Figure 7. At late times (t = 2 × 105), about 15% of the rods are able to escape from the Γ2 stripe, as measured by averaging over four independent simulation runs. We now return to the simulations with the number of rods fixed at N = 300 and focus on the effect of the rastering velocity on the dynamics in this system. The plot in Figure 8 shows the

sample and thus indicates no collecting. The typical morphology for the rods in the stripe for v = 10−2 is shown in inset (b) of Figure 8. Finally, we increased the width of the Γ2 region to w = 20, while fixing v = 5 × 10−3, N = 300, and L = 9. The time evolution of the order parameters for this case (Figure 9) shows

Figure 9. Time evolution of order parameters for w = 20 and v = 5 × 10−3. Inset shows the morphology of the system at t = 2 × 106.

that the binary mixture and nanorods are aligned into an ordered structure, yielding similar values for β and S as in the case of w = 10 (see respective curves for L = 9 in Figures 2 and 3). In other words, rastering with the higher intensity light over the sample has given rise to a nanocomposite that exhibits longrange order. These results indicate the width of the Γ2 does not have to be centered about a specific value (for the range of values considered here) for the combing technique to be effective. C. Binary Mixture of Rods. A potential advantage of this approach is that one can simultaneously order the AB mixture, the A-like rods within the A phase, and B-like rods within the B phase to create a composite that combines the desirable attributes of the A-like and the B-like nanoparticles within a spatially regular, defect-free material. For example, the A-like rods could exhibit superior optical properties, while the B-like rods could contribute advantageous mechanical properties. The defect-free composite could then be harnessed for a broad range of applications. To demonstrate this behavior, we consider an AB mixture that encompasses NA = 150 A-like rods (ψw = 0.75 and drawn in black in Figure 10) and NB = 150 B-like rods (ψw = −0.75 and drawn in white in Figure 10). All the other parameters are set at the default values defined above, with the width of the rastering stripe equal to w = 10, the rastering velocity being v = 5 × 10−3, and rod length fixed at L = 9. The rods and the binary mixture are initially randomly distributed within the simulation box. As shown in the inset of Figure 10, the binary mixture phase-separates into a disordered lamellar structure, with the black rods located in the red (A) phase and the white rods confined in blue (B) phase to minimize the total free energy of the system. As the rastering light is passed over the sample and it effectively combs out defects, both the rods and binary mixture evolve into an ordered structure, with an efficiency that is similar to that for the system encompassing just the A-like rods. The morphology at t = 2 × 106 shows a highly ordered structure, with rods partitioned into the respective, energetically favorable phases.

Figure 8. Time evolution of number of rods located within the rastering stripe for different velocities. The top two curves correspond to v = 5 × 10−4 and v = 10−4. The bottom two curves correspond v = 5 × 10−3 and v = 10−2 as marked in the legend. Inset (a) is the typical morphology around the rastering stripe for v = 5 × 10−4, in which the second layer of rods forms within the band. Inset (b) is the typical morphology for v = 10−2.

time evolution of the number of rods within the Γ2 stripe. For the low velocities of v = 5 × 10−4 and v = 10−4, the number of rods increases with time and then plateaus off at a relatively high value, indicating that the rods have been collected within the stripe (as shown in inset (a)). This “collecting” effect is the most significant for the lowest velocity considered herein. As noted above, the neighboring stripe of the B phase imposes an energy barrier that inhibits the rods from leaving the Γ2 region. Consequently, the nanorods are gathered by the stripe of high intensity light as this light moves from left to right. For the slowest velocity considered here, v = 10−4, a second rod layer develops (shown in inset (a) of Figure 8) as more rods accumulate in the Γ2 region. For the higher velocities of v = 5 × 10−3 and v = 10−2, the rod number fluctuates around 10, which is equal to the number of rods that would lie in the Γ2 area assuming a uniform distribution of rods within the entire 757

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kinetically trapped in an intermediate energy state because the rod length L = 9 is larger than the domain width. In particular, when the neutral nanorods migrate out of the stripe of light, they are preferentially oriented along the y-direction since the order parameters on the interface between the Γ1 and Γ2 regimes (along the y-direction) are close to zero. In this orientation, these neutral rods span across multiple lamellar domains and thus have a lower energy than those residing in either the A or B phase. Consequently, these long nanorods become trapped in this metastable state. This behavior is distinctively different for the shorter rods of length L = 3 (Figure 12). With the same parameters as above,

Figure 10. Time evolution of order parameters for single type (A-like) rods and binary rods (mixture of A-like and B-like rods). Insets show the early (t = 5 × 103) and late (t = 2 × 106) times configurations for binary rod system. A-like and B-like rods are shown in black and white, respectively.

It is worth noting, however, that the distribution of nanorods is more spatially uniform in the single-rod system than in this binary rod mixture, where the rods of a specific type are grouped into larger aggregates. The aggregation, for example, of the A-like rods is due to the repulsive interaction exerted by nearby B-like rods in the bordering lamellae that drives the Arods into distinct clusters. A similar effect is felt by the B-like rods from the neighboring A-like species. In other words, the distance between two different types of rods along the xdirection can be half of that in the system encompassing just one type of rod, and this can lead to stronger repulsive interactions. Therefore, rods of a given type are pushed together and driven to aggregate in order to minimize these repulsive effects. Finally, we also studied mixtures of A-like rods (ψw = 0.75, colored in black) and neutral rods, which are characterized by ψw = 0 (and colored in white in Figure 11); here L = 9 for both

Figure 12. Time evolution of order parameter S for binary rods (A-like and neutral) system with L = 3. Inset shows late time morphology at t = 3 × 106.

we observe that all neutral rods are located at the interface between the red and blue phases, and the temporal evolutions of the order parameters for the red and neutral rods collapse on the top of each other. The latter behavior is due to the fact the rod length is smaller than the domain width and the system can access the lowest free energy state (rather than being trapped in the metastable state).

IV. CONCLUSIONS In summary, we investigated the self-assembly of nanorods immersed in photosensitive binary mixtures that are irradiated by a low-intensity background light and a higher-intensity rastering light and thereby isolated conditions to create spatially regular, defect-free nanocomposites. Importantly, our findings point to an effective route for simultaneously coassembling the rods and polymers into spatially ordered structures via one processing step. Furthermore, by tailoring the coating on the nanorods, these fillers can be preferentially localized in the one of the two polymeric domains or the interfacial regions of the ordered structures. For cases involving binary mixtures of rods, our results show that the appropriately coated rods can be effectively sequestered to different regions of the material. The ability to control the spatial arrangement of two different types of rods can significantly improve the performance of the material. For example, mixtures of nanorods (in the absence of polymer) have been shown by Alivisatos and co-workers6 to provide more effective photovoltaic activity than either of the rods alone. The facile incorporation of these rods into spatially ordered polymers can yield further improvements in the

Figure 11. Time evolution of order parameter S for binary rods (A-like and neutral) system with L = 9. Inset shows late time morphology at t = 3 × 106.

types of rods. As expected, the A-like rods are aligned along the x-direction within the red A domains; however, ∼80% of the neutral rods are oriented along the y-direction (shown in the inset of Figure 11). This observation is supported by the calculated values of the order parameter S (Figure 11); for the neutral rods, S decreases to negative values. The remaining ∼20% of the neutral rods stay within the interfacial region between the red and blue domains; this location is energetically favorable for the neutral rods since the order parameter at the interface is ψ(r) ≈ 0. Most of neutral rods, however, are 758

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(2) Erb, R. M.; Libanori, R.; Rothfuchs, N.; Studart, A. R. Composites Reinforced in Three Dimensions by Using Low Magnetic Fields. Science 2012, 335, 199−204. (3) Liu, K.; Zhao, N.; Kumacheva, E. Self-assembly of Inorganic Nanorods. Chem. Soc. Rev. 2011, 40, 656−671. (4) Thorkelsson, K.; Mastroianni, A. J.; Ercius, P.; Xu, T. Direct Nanorod Assembly Using Block Copolymer-based Supramolecules. Nano Lett. 2012, 12, 498−504. (5) Ryan, K. M.; Mastroianni, A.; Stancil, K. A.; Liu, H.; Alivisatos, A. P. Electric-field-assisted Assembly of Perpendicularly Oriented Nanorod Superlattices. Nano Lett. 2006, 6, 1479−1482. (6) (a) Gur, I.; Fromer, N. A.; Geier, M. L.; Alivisatos, A. P. AirStable All-inorganic Nanocrystal Solar Cells Processed from Solution. Science 2005, 310, 462−465. (b) Milliron, D. J.; Gur, I.; Alivisatos, A. P. Hybrid Organic-nanocrystal Solar Cells. MRS Bull. 2005, 30 (1), 41−44. (7) Hull, D.; Clyne, T. W. An Introduction to Composite Materials; Clarke, D. R., Suresh, S., Ward, I. M., Eds.; Cambridge Solid State Science Series; Cambridge University Press: Cambridge, 1996. (8) Yan, L. T.; Maresov, E.; Buxton, G. A.; Balazs, A. C. Self-assembly of Mixtures of Nanorods in Binary, Phase-separating Blends. Soft Matter 2011, 7, 595−607. (9) Peng, G.; Qiu, F.; Ginzburg, V. V.; Jasnow, D.; Balazs, A. C. Forming Supramolecular Networks from Nanoscale Rods in Binary, Phase-separating Mixtures. Science 2000, 288, 1802−1804. (10) Wu, K. H.; Lu, S. Y. Preferential Partition of Nanowires in Thin Films of Immiscible Polymer Blends. Macromol. Rapid Commun. 2006, 27, 424−429. (11) Yan, L. T.; Maresov, E.; Hayward, R. C.; Emrick, T.; Russell, T. P.; Balazs, A. C. Promoting Network Formation in Nanorod-filled Binary Blends. MRS Proc. 2012, 1411. (12) Li, L.; Miesch, C.; Sudeep, P. K.; Balazs, A. C.; Emrick, T.; Russell, T. P.; Hayward, R. C. Kinetically Trapped Co-continuous Polymer Morphologies Through Intraphase Gelation of Nanoparticles. Nano Lett. 2011, 11, 1997−2003. (13) Travasso, R. D. M.; Kuksenok, O.; Balazs, A. C. Harnessing Light to Create Defect-free, Hierarchically Structured Polymeric Materials. Langmuir 2005, 21, 10912−10915. (14) Travasso, R. D. M.; Kuksenok, O.; Balazs, A. C. Exploiting Photoinduced Reactions in Polymer Blends to Create Hierarchically Ordered, Defect-free Materials. Langmuir 2006, 22, 2620−2628. (15) Kuksenok, O.; Travasso, R. D. M.; Balazs, A. C. Dynamics of Ternary Mixtures with Photosensitive Chemical Reactions: Creating Three-dimensionally Ordered Blends. Phys. Rev. E 2006, 74, 011502. (16) Tran-Cong, Q.; Kawai, J.; Endoh, K. Modes Selection in Polymer Mixtures Undergoing Phase Separation by Photochemical Reactions. Chaos 1999, 9, 298−307. (17) Nishioka, H.; Kida, K.; Yano, O.; Tran-Cong, Q. Phase Separation of a Polymer Mixture Driven by a Gradient of Light Intensity. Macromolecules 2000, 33, 4301−4303. (18) Tran-Cong, Q.; Nishigami, S.; Ito, T.; Komatsu, S.; Norisuye, T. Controlling the Morphology of Polymer Blends Using Periodic Irradiation. Nat. Mater. 2004, 3, 448−451. (19) Glotzer, S. C.; DiMarzio, E. A.; Muthukumar, M. ReactionControlled Morphology of Phase-separating Mixtures. Phys. Rev. Lett. 1995, 74, 2034−2037. (20) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Oxford University Press: Oxford, 1986. (21) Yan, L.-T.; Balazs, A. C. Self-assembly of Nanorods in Ternary Mixtures: Promoting the Percolation of the Rods and Creating Interfacially Jammed Gels. J. Mater. Chem. 2011, 21, 14178−14184. (22) Oono, Y.; Puri, S. Study of Phase-separation Dynamics by Use of Cell Dynamical Systems. I. Modeling. Phys. Rev. A 1988, 38, 434− 453. (23) Christensen, J.; Elder, K.; Fogedby, H. Phase Segregation Dynamics of a Chemically Reactive Binary Mixture. Phys. Rev. E 1996, 54, 2212−2215.

performance of photovoltaic devices. More generally, the presence of two different types of nanorods permits simultaneous control over distinct features of the ordered materials, with one of the rods, for example, providing improved mechanical behavior and the other rods enhancing the optical or electrical properties of the composite. Hence, in one processing step, one can incorporate a range of advantageous properties into the system as well as achieve significant control over the material’s long-range order. Finally, we also showed that by varying the rastering velocities, one can control the alignment of nanorods, so that these fillers lie parallel or perpendicular to the directional ordering of the polymer matrix. The 2D simulations used here are particularly appropriate for modeling the behavior of thin films. We anticipate, however, that the above behavior would also be observed in 3D bulk samples. Namely, in previous studies,15 we extended the model (in the absence of rods) to three dimensions and showed that the combination of the stationary, background, and higher intensity, rastering light could be used to create defect-free lamellar structures in 3D systems. In mixtures involving the nanorods, the preferential wetting interactions between the rods and one of the phases keeps these rods localized in the appropriate domains, and the relatively narrow width of the lamellar domains favors the orientation of the rods along the length of the lamellae. Hence, on the basis of our previous studies,15 we anticipate that the approach will yield ordered lamellar nanocomposites in three-dimensional systems. In future studies, we will harness this aspect of the “combing” process to create structures that encompass a nanogrid, with wires of nanorods that associate into rectangular structures. In future studies, we will also consider scenarios where Γ+ ≠ Γ−. In these cases, we can expand the repertoire of morphologies that can be achieved through this “combing” process.15 These efforts will allow us to design hierarchically structured nanocomposites that exhibit high degrees of ordering; these structural features can in turn lead to novel materials properties.

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ASSOCIATED CONTENT

S Supporting Information *

Two movies. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Support for this work was provided by Polymer-Based Materials for Harvesting Solar Energy, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Award DE-SC0001087. The authors gratefully acknowledge helpful conversations with Dr. Egor Maresov.

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REFERENCES

(1) Gupta, S.; Zhang, Q.; Emrick, T.; Russell, T. P. “Self-corralling” Nanorods Under an Applied Electric Field. Nano Lett. 2006, 6, 2066− 2069. 759

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(24) Fakhri, N.; MacKintosh, F. C.; Lounis, B.; Cognet, L.; Pasquali, M. Brownian Motion of Stiff Filaments in a Crowded Environment. Science 2010, 330, 1804−1807. (25) Kellog, G. J.; Walton, D. G.; Mayes, A. M.; Lambooy, P.; Russell, T. P.; Gallagher, P. D.; Satija, S. K. Observed Surface Energy Effects in Confined Diblock Copolymers. Phys. Rev. Lett. 1996, 76, 2503. (26) Generally, decreasing Γ2 decreases the degree of order in a binary mixture because for relatively low values of Γ2/Γ1, the ratio between the characteristic wavelengths in the Γ2 and Γ1 regions, λ2/λ1, is closer to unity. This results in domains attempting to match at the Γ2/Γ1 boundary, and this in turn decreases the ordering in the system (see ref 14).

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