Coassembly of Janus Nanoparticles in Asymmetric Diblock Copolymer

Jun 23, 2014 - Tailoring Interfacial Nanoparticle Organization through Entropy ... Theoretical Studies and Tailored Computer Simulations in Self-Assem...
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Coassembly of Janus Nanoparticles in Asymmetric Diblock Copolymer Scaffolds: Unconventional Entropy Effect and Role of Interfacial Topology Bojun Dong, Ruohai Guo, and Li-Tang Yan* Key Laboratory of Advanced Materials (MOE), Department of Chemical Engineering, Tsinghua University, Beijing 100084, P. R. China S Supporting Information *

ABSTRACT: The coassembly of Janus nanoparticles and block copolymers offers a unique approach to control the spatial organization of nanoparticles. Herein, using computer simulations and theoretical analysis, we explore the hierarchical structures and underlying mechanisms of the coassembly of symmetric Janus nanoparticles in asymmetric block copolymers. Our simulations constitute the first study clarifying that Janus nanoparticles with two symmetric surface moieties do not take symmetric distribution in the interfaces of asymmetric block copolymers. Rather, they take various but controllable off-center arrangements from the interfaces upon tailoring the molecular architectures of block copolymers and thereby controlling their resulted mesostructural topology. We examine the detailed mechanism of this mesostructural topology-mediated hierarchical assembly and find that the structural asymmetry of the block segments causes unconventional entropy effect at the molecular scale, and the curved interfaces can lead to topology mismatching between Janus nanoparticles and polymer interfaces at the mesoscale. Furthermore, we employ a micromechanical model to demonstrate that the deviation of the Janus nanoparticles from the interface can significantly influence the mechanical properties of the nanocomposites. These findings enrich our understanding on the thermodynamic nature of polymer nanocomposites and may suggest a novel design approach to precisely program the spatial organization of nanoparticles in polymer matrix. phase with flat interface since it can make the problem significantly tractable. For the lamellar phase formed by the symmetric block copolymers, Janus nanoparticles with two symmetric moieties have been demonstrated to take symmetric arrangement in the interfaces of this scaffold.9−11 For the nonlamellar phase, the phase interface however becomes tortuous where structure asymmetry can be induced between two sides of the interface.16 This leads to the question: do symmetric Janus nanoparticles still take symmetric interfacial distribution in asymmetric diblock copolymer scaffolds? Understanding the detailed position of symmetric Janus nanoparticles in asymmetric block copolymers provides fundamental information on the control of the hierarchical structures of these BNs and is related to the determination for their structure−property relationships. Thereby, it is a critical issue to be addressed before further application of Janus nanoparticles in the construction of novel BNs with ordered hierarchical structures and unique properties. For this purpose, we herein conduct an extensive investigation of the hierarchical structures and formation mechanisms of the coassembly of symmetric Janus nano-

1. INTRODUCTION Controlling the morphology of nanoscale additives in a polymer matrix is critical for tuning the macroscopic properties of the resulting polymer nanocomposite.1−6 Of particular interest is controlling the assembly of nanoparticles at the interface between different domains of block copolymers.7,8 Janus nanoparticles with two chemically different compartments are ideal building blocks for realizing this purpose due to their high interfacial activity and amphiphilicity.9−12 While the structure formation mechanism of block copolymer-based nanocomposites (BNs) has been understood to a certain extent,5,6 establishing more detailed knowledge regarding the hierarchical structure control for the BNs containing Janus nanoparticles is indubitably required. A great challenge in this aspect is clarifying the thermodynamic nature underlying various phenomena of the nanocomposites, specially, the entropy effect. In fact, entropy keeps springing nonintuitive findings in the manipulation of the self-assembly of nanoparticles and the structural formation of some other soft matter systems and thereby is key to the fundamental understanding of various phenomena in BNs containing Janus nanoparticles.5,13,14 On the other hand, despite the repertoire of available morphologies of block copolymers,15,16 most theoretical studies of BNs containing Janus nanoparticles focus on the lamellar © 2014 American Chemical Society

Received: January 21, 2014 Revised: April 28, 2014 Published: June 23, 2014 4369

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Table 1. Equilibrium Values of Various Structural Parameters of Different Diblock Copolymersa ⟨RgA⟩ ⟨RgB⟩ ⟨Rg⟩ ⟨lbA⟩ ⟨lbB⟩ ⟨lb⟩

A6B6

A6B7

A6B8

A6B9

A6B10

A6B11

A6B12

A6B13

A6B14

0.93 0.94 1.60 0.93 0.93 0.95

0.95 1.12 1.75 0.94 0.94 0.96

0.95 1.20 1.78 0.93 0.94 0.95

0.95 1.20 1.87 0.93 0.94 0.95

0.96 1.27 1.91 0.93 0.94 0.95

0.93 1.36 1.99 0.93 0.95 0.95

0.96 1.39 2.02 0.94 0.94 0.95

0.96 1.48 2.11 0.94 0.94 0.95

0.95 1.53 2.13 0.93 0.94 0.95

a RgA, RgB, and Rg are the radii of gyration of the A segment, B segment, and the whole chain. lbA, lbB, and lb are the averaged bond lengths of the A segment, B segment, and the whole chain. ⟨ ⟩ denotes the ensemble-averaged results (unit: rc).

zero mean and unit variance. The noise amplitude, σ, is fixed at σ = 3 in the present simulations. The bonds between beads in the polymer chain are represented by FSij = Crij with a stiffness constant C = −4. The radius of Janus nanoparticles is fixed at rn = 1.11rc, and a low volume fraction of nanoparticles, i.e., VP = 0.02, is chosen so that the effect of the particle−particle interaction is trivial, except where noted otherwise. The radius of gyration, Rg, and the averaged bond length, lb, of diblocks are listed in Table 1. We use a modified velocity-Verlet algorithm to solve the motion equation.17 The simulation box is (20rc)3 in size and with periodic boundary condition in all directions, which is large enough to avoid the finite size effect. Each structure is determined over three independent runs, and each run is equilibrated with more than 5 million time steps (for a time step, Δ t = 0.04τ where τ is the time unit of DPD).

particles in asymmetric diblock copolymer scaffolds by combining systematic computer simulations and theoretical analysis. To the best of our knowledge, this is the first study demonstrating that Janus nanoparticles with two symmetric surface moieties do not take symmetric distribution in the interfaces of asymmetric block copolymers. In particular, our results show that the symmetry Janus nanoparticles can take different off-center arrangements upon tailoring the molecular architectures of block copolymers and thereby controlling their resulted mesostructural topology. The mechanism of the mesostructural topology-mediated hierarchical assembly is examined by evaluating various energetic contributions as well as the multiscale structures of the systems. We also employ a micromechanical model to demonstrate that the deviation of the Janus nanoparticles from the interface can significantly influence the mechanical behaviors, e.g., deformation and fracture, of the nanocomposites.

3. RESULTS AND DISCUSSION 3.1. Kinetics Pathway. We begin by examining the detailed kinetic pathway of the coassembly of symmetric Janus nanoparticles in asymmetric diblock copolymers. Figure 1 is an example showing the time evolution of the morphology for the A6B7 block copolymer system where the top and bottom snapshots at each time display the phase structure of the nanocomposite and its interfacial structure, respectively. It demonstrates that, at the earlier stage, the block copolymers undergo spontaneous phase separation that seems to be faster than the transition and rotation of Janus nanoparticles. Thereby the nanoparticles are not positioned at the interfaces at this stage, as marked by the green circles. With the increasing time, the strong interactions between polymer blocks and nanoparticle surface make the Janus nanoparticles to attach to the interfaces. The Janus nanoparticles anchoring at the interface will depress the evolution of nanocomposites and lead to sluggish kinetics due to their slower diffusion ability.18 Moreover, Janus nanoparticles can induce large fluctuation or defects between two neighboring interfaces, as denoted by the pink circles. Overcoming these additional defects requires a longer time. Thereby the kinetics of Janus nanoparticles turns out to be very slow at the later stage. Figure 1 clearly reveals how the diffusion of Janus nanoparticles and the evolution of block copolymers driven by the chemical potential interplay in the kinetic pathway of this coassembly process. To examine the effect of nanoparticles on the kinetic behaviors, the structures are quantified by calculating the temporal evolution of the order parameter, S, defined by Saupe tensor, that is, Qαβ = −1.5r̂αr̂β − 0.5δαβ, where α and β are Cartesian indices, δ is the Kronecker symbol, and r̂ is a unit vector along the chain axis from the center of block A to the center of block B. The largest eigenvalue of the volume average Qαβ is the order parameter S.19 S is zero for block copolymer in the completely disordered state and unity for block copolymer

2. METHODOLOGY In our coarse-grained molecular dynamics (CGMD) simulations, a cluster of atoms is represented as a single bead located at the center of mass. The simulations are based on the method of dissipative particle dynamics (DPD).17 We consider the mixtures of AB diblock copolymers filled with spherical Janus nanoparticles. Each Janus nanoparticle consists of two sites with equal surface areas.11 In DPD, a bead i at position ri surrounded by beads j ≠ i at rj (distance vector rij = ri − rj and unit vector eij = rij/rij with rij = |rij|) experiences a force with the components of conservative interaction force FC, dissipative force FD, random force FR, and bond force FS; i.e., f i = ∑j≠i(FCij + FDij + FRij + FSij) where the sum runs over all beads j. The conservative force is given by FCij = αijω(rij)eij, where αij is the maximum repulsion between beads i and j, and has a linear relationship with Flory−Huggins χ parameter: χij ≈ (αij − αii)/ 3.27.17 αii = 25 is the interaction between like species (i.e., χii = 0.0). Here χAB = 6.0, χPQ = 7.6, χAQ = 1.1, and χAp = χBQ = 0.0 are set for various interactions where A, B indicate the beads of two blocks in diblock copolymers and P, Q indicate the beads of two compartments of Janus nanoparticles.11 Clearly, the A and B blocks have strong affinities to the P and Q sites of Janus nanoparticles, respectively. The repulsion between B and P, χBP = 1.1, is set except where noted otherwise. For the purpose of comparison, we also alter the surface property of Janus nanoparticle by “coating” them with homogeneous surface chemistry that has preferential interaction with the B segment, i.e., χBQ = χBP = 0.0 and χAP = χAQ = 1.1. The weight function ω(rij) is chosen as ω(rij) = 1 − rij/rc for rij < rc and ω(rij) = 0 for rij ≥ rc, where rc is the truncate distance. The random force FRij and the dissipative force FDij are given by FRij = σω(rij)ξijΔt−1/2eij and FRij = −1/2σ2ω(rij)(vij·eij)eij, where vij = vi − vj and vi denotes the velocity of bead i. ξij is a random number which has 4370

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behaviors because the locally structural coarsening dominates the kinetic process, as shown in Figure 1. With the increasing time, the nanoparticles tend to play an important role in the kinetic process because their attraction to neighboring polymer chains can significantly modify the local diffusion of these chains.4,20 This leads to the slower kinetics for the nanocomposites than pure block copolymers in the later stage, as marked by the pattern in Figure 2. Moreover, the two moieties of Janus nanoparticles present stronger attraction to their preferential phases and thereby have a more significant effect on the kinetics of structural evolution than the homogeneous surface of neutral particles. Indeed, as demonstrated in Figure 2, there is evident difference in their kinetic behaviors at the later stage. In fact, it has also been demonstrated that, due to the higher adsorption energy of Janus particles, the emulsions made of Janus particles are more stable against the coalescence of the droplets since energy barrier to break a droplet is primarily determined by the energy of the interface.21 3.2. Hierarchical Structures and Interface-Topology Dependence. In this section we turn to quantify the interfacial topologies and examine the precise positions of Janus nanoparticles in block copolymer scaffolds. Here the interfacial topologies of AB diblocks are controlled by gradually increasing the bead number of the B block segment, lB, from 6 to 14 while the bead number of the A block segment is maintained at 6. Figure 3a displays some typical snapshots showing the morphology transition of the BNs with Janus nanoparticles in response to various lB. From lB = 6 to 8, the block copolymers present lamellar phase with flat phase interfaces. The phase interfaces deform for more asymmetry block copolymers with larger lB, leading to the stable perforated lamellar phase. At A6B13, the morphology of the system is clearly cylindrical. The simulated morphologies are in agreement with the results of theoretical calculation22 and experiments.23 To gain more detailed insight into the dependence of the interfacial topology on lB, the mean curvature, 2H, of each system is calculated by employing Minkowski functionals describing the topology information contained in an image by numbers that are proportional to geometrical quantities (see Figure S1 and its description in the Supporting Information for details).24 The circles in Figure 3b show how the interface curvature 2H varies as a function of lB. Indeed, as lB is no larger than 8, the interface curvature is close to zero. Further increasing lB results in more curved phase interfaces with larger 2H. The snapshots in Figure 3a demonstrate that Janus nanoparticles are basically restricted to their energetically favorable interfaces, being the typical interfacially directed selfassembly.10,11 To determine the precise position of the Janus nanoparticles with respect to interfaces, the fraction of Janusnanoparticle surface wrapped by the beads of A segment, f w, is calculated. f w = 0.5 indicates that the center of a Janus nanoparticle is exactly located at the interface while a smaller f w denotes that its center deviates from the interface and turns to located in the B phase, and vice versa. It is worth noting that some particles are lightly misoriented with respect to the interface in Figure 3a. This is due to the fact that the motion, including transition and rotation, of nanoparticles will present a low degree of random fluctuation resulting from the thermoset consisting of the dissipative and random forces. However, the general trend of their orientation cannot be affected by this fluctuation. In the present work, each structure is determined over three independent runs, and its f w is obtained through averaging the values from these different runs. Indeed, the

Figure 1. Kinetic pathway of A6B7 block copolymer−Janus nanoparticle system. The evolution time t = 0 (a), 400τ (b), 800τ (c), 2000τ (d), 4000τ (e), 8000τ (f), 20000τ (g), and 40000τ (h). At each time the top snapshot is the isosurface phase structure of the nanocomposite where the interface between phases A and B is colored yellow and phase A is colored blue. Phase B is fully transparent. The bottom snapshot highlights the interfacial structure of the nanocomposite. The green and pink circles mark the spatial distribution of Janus nanoparticles.

in the ordered alignment state. In Figure 2, we provide the temporal evolution of S for the systems of pure block copolymers, the composite containing homogeneous particles, and the composite containing Janus nanoparticles, where the same block copolymers, A6B6, are used in these three systems. At the initial stage, these systems present very similar kinetic

Figure 2. Order parameter, S, as a function of simulation time for various systems: (green square) diblock copolymers with Janus nanoparticles; (red circle) diblock copolymers with homogeneous nanoparticles; (blue triangle) pure diblock copolymers. The rectangle marks the plots at the later stage. The block copolymer of A6B6 is used in all systems, and volume fraction of nanoparticles is fixed at 0.02. 4371

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Figure 3. (a) Typical simulated morphologies formed by the self-assembly of Janus nanoparticles in various diblock copolymers. (b) Fraction of the Janus-nanoparticle surface wrapped by the A block segment, f w, and the mean curvature of the mesostructure, 2H, as a function of the bead number of the B block segment, lB. (c) f w as a function of the interaction parameter between the beads of the B block and the A site of the Janus nanoparticle, χBP, at various lB. The lines are to guide the eye.

calculation results present a reasonable change in respond to the change of the block-segment length. We illustrate f w in Figure 3b as a function of lB. As indicated by the dashed vertical line, it is interesting to note that a crossover occurs at lB = 8 in the f w−lB plot. Before the crossover, f w increases with the increasing of lB, revealing that the Janus nanoparticles tend to move to A domains for the longer B segments. To evaluate the generality of this rule, we vary the interaction parameter between B segments and P site of Janus nanoparticles, χBP, and Figure 3c shows the plots of f w against χBP at various lB. Clearly, despite the tendency to the A segments for a larger χBP, the rule identified in Figure 3b (from lB = 6 to 8) is always obeyed within the whole scale of χBP considered here. Subsequent to the crossover, f w is however reduced for a larger lB. In this case, the increase of B segments also causes the Janus nanoparticles to shift off-center from the interface and turn to the B domains, being asymmetric distribution at the interfaces. What is more interesting is that the crossover of the f w−lB plot is exactly the same as that of the 2H−lB plot, indicating the hierarchical structures critically depend on the interfacial topologies (see Figure S2 in the Supporting Information for more results). In Figure 3, the gyration radius of the B block segment, RgB (see Table 1), is comparable to the radius of the Janus nanoparticle, i.e., rn = 1.11rc. In order to further evaluate the generality of the dependence of f w on lB, we vary the length of the A block segment, lA, and lA = 3 and 12 are selected to consider various situations with RgB smaller and larger than the Janus particle size. Indeed, at lA = 3, the RgB increases from 0.85rc to 0.88rc when lB is increased from 3 to 8. The values are smaller than the radius of the Janus particle. At lA = 12, increasing lB from 12 to 18 leads to the change of RgB from 2.10rc to 3.10 rc, evidently larger than the particle size. To avoid the finite size effect, the simulation box size is enlarged to (40rc)3 for each system at lA = 12. Figures 4a and 4b show the

Figure 4. Fraction of the Janus-nanoparticle surface wrapped by the A block segment, f w, as a function of the bead number of the B block segment, lB, for various lengths of A block segment, lA. (a) lA = 3; (b) lA = 12. Here the size and number of Janus nanoparticles are the same with those in Figure 3. The left and right snapshots in (a) present the simulated morphologies of A3B3 and A3B6 while the left and right snapshots in (b) show present the simulated morphologies of A12B12 and A12B16. Note that the displaying ratio of the snapshots in (a) and (b) are different for the efficient organization of the figure.

f w−lB plots at lA = 3 and 12, respectively, where the size and number of the Janus nanoparticles are the same with those in 4372

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interfaces, and the nanoparticles take a narrow and predominant distribution for an equilibrium system with uniform interface curvature. Furthermore, the peaks in Figure 5 imply that the Janus nanoparticles may tend to reside in a preferential interfacial distribution determined by many factors like interfacial curvature, interfacial topology, and so on. 3.3. Structural Formation Mechanism before the Crossover. We now explore the detailed mechanism of this unique transition of the precise spatial organization of Janus nanoparticles with respect to block polymer interfaces. The case where the morphology is lamellar (i.e., before the crossover in Figure 3b) is considered first. The lB dependence of the off-center arrangement of Janus nanoparticles is to explain from examining the conformational states of A and B segments, respectively. The technique of molecular simulation used here allows us to achieve this purpose conveniently. In this paper, the conformational state of block segments is quantified by the Flory exponent, v.25,26 In the bulk of a polymer melt, the polymer chains are ideal with Gaussian state where v = 0.5.25 A higher value of v indicates a larger deviation of the chain conformation from Gaussian state. For a linear polymer chain, v can be calculated as ⟨Rg2⟩ = (1/6)N2v⟨lb2⟩, where ⟨Rg2⟩ and ⟨lb2⟩ are the mean-square radius of gyration and the mean-square bond length, respectively.27 ⟨Rg2⟩ and ⟨lb2⟩ of the block copolymers concerned in the present work are summarized in Table 1. Based on the data, v can be calculated, and Figure 6a shows what happens for the v of A and B segments when lB is increased from 6 to 14 while the bead number of the A segment is maintained at 6. Examining the points in Figure 6a, we note that there are very similar values of v for these both segments when their bead numbers are equal. In this case, v approximates to 0.51, the conformational states of these both segments being nearly Gaussian. The small deviation from 0.5 is primarily due to the slight contraction of the segments at interfaces and is similar to the results of the previous simulations.28−31 Interestingly, significant variation develops in the plots of A and B segments upon further increasing lB. As shown in Figure 6a, the points of A segments do not depend on lB and only fluctuate around a certain value while v of B segments increases largely. This implies that increasing the length of B segments results in a larger deviation from Gaussian state for the conformation of the B segments. As the state of the bulk segments is not likely to alter,26 we thereby anticipate that the change of the B-segment conformation could be attributable to their large deformation at interfaces.

Figure 3b. The crossovers also form in these both plots with the increase of lB. We find that, as demonstrated by the snapshots in Figure 4a,b, the systems present lamellar phase with flat phase interfaces whereas the phase interfaces deform for more asymmetry block copolymers with larger lB, in agreement with Figure 3a. The same changing rule of the f w−lB plots at lA = 3, 6, and 12 reveals that the fundamental mechanism behind the phenomena has no relation with the domain size of the block copolymers. In addition to the averaged interface curvature stated above, the local interface curvature is the other important factor affecting the distribution of Janus nanoparticles. To clarify this effect, we calculate the probability distribution of f w, F( f w), with the temporal evolution of the nanocomposites containing Janus nanoparticles. Figure 5 shows F(f w) for the Janus nanoparticle/

Figure 5. Probability distributions of f w, F( f w), as a function of f w at different times. Here the Janus nanoparticle/A6B6 system is selected as an example. The dashed curves are used to guide the eyes.

A6B6 system at different times, where the dashed curves clarify the statistic distributions of the points. At the initial stage, the morphology of the nanocomposite undergoes structural coarsening where the interface curvature is highly irregular (Figure 1). This leads to a wide distribution of Janus nanoparticles with respect to the interface, as demonstrated by the points at 1 × 104τ in Figure 5. With the increasing time, the system gradually evolves into the lamellar structure with uniform interface curvature. Consequently, a peak forms in the probability distribution of f w, indicating that the Janus nanoparticles tend to take a predominant interfacial distribution. The peak at 3.64 × 106τ is narrower and higher than that at 1.75 × 106τ due to the more uniform interfacial curvature at the later stage. Figure 5 shows that the distribution of Janus nanoparticles significantly depends on the local curvature of

Figure 6. (a) Flory exponent, v, of A and B segments as a function of the bead number of the B block segment, lB. (b) Average interaction energy ⟨E(n)⟩ at different monomer positions n for various block copolymers: (red) A6B6, (green) A6B7, and (blue) A6B8. The patterned rectangle marks the junction section of the block copolymers. 4373

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Copolymers consist of A and B block segments bound together, and thus their behaviors differs from that of homopolymers in blends of A and B homopolymers: In homogeneous blends the homopolymer chains contract isotropically while the block copolymer chains form the anisotropic (dumbbell) shape.28 Since the shape of the polymers is tightly related to the intra- and interchain forces, the distribution of interaction energies along the chain is a viable tool in understanding the conformational behaviors of single polymer chains. We thereby calculate the average energy (enthalpic interactions) distribution ⟨E(n)⟩ along the chain, where n counts separately the monomers in each block copolymer chain. Figure 6b shows ⟨E(n)⟩ in units of kBT as a function of n for block copolymer chains with various architectures. Remarkably, ⟨E(n)⟩ varies a lot with n, and the average energy of the monomers close to the junction point (marked by the patterned rectangle) is much higher than that of monomers located well inside the blocks, in agreement with the previous calculation based on the Monte Carlo methods.28,29 It is interestingly to find from Figure 6b that the ⟨E(n)⟩ almost does not change for the A block segments with unchangeable length while it is significantly reduced for the B block segments with increasing length. This reveals that the B segment tends to expand with the increasing of lB, leading to a more stretched deformation of B segment at interfaces. This can also be confirmed by examining the distance between the two segments, db, defined as illustrated by the inset of Figure 7. Indeed, Figure 7 displays that db increases when lB is

Figure 8. (a) Radial distribution function between the interfacial and the bulk B-segment beads, g(z), at various lB. The inset shows the lB dependence of the excess entropy per bead Snp estimated from the pair correlation function between the B-segment beads and the B-site beads of Janus nanoparticles. (b) Schematic diagram showing the unconventional entropy effect within the system of Janus nanoparticle and block copolymers.

bulk polymer chains,25,26 the structural change is largely attributable to the interfacial B segments, providing the other evidence for the enhanced expansion of the longer B segments at interfaces. The radial distribution function (RDF) between the interfacial and the bulk beads of B segments, g(z), is also calculated, which measures correlations between B beads in these both sections. Figure 8a plots g(z) for lB = 6, 7, and 8. There are two characteristic peaks occurring at the same positions in these three plots: the first is at z ≈ 0.82rc, which is the equilibrium distance for two interacting DPD beads; the second is at z ≈ 1.62rc and is the diffraction peak. As indicated by the dashed arrows, the heights of these both peaks decrease in magnitude as lB increases. This reveals that the correlation between the interfacial and the bulk B beads is reduced as response to the increase of the B segments due to the structural change of any component. In view of the invariably state of the bulk polymer chains,25,26 the structural change is largely attributable to the interfacial B segments, providing the other evidence for the enhanced expansion of the longer B segments at interfaces. Physically, the interaction between a nanoparticle and its neighboring polymer chains will incur a loss of conformational entropy of the polymer chains since they must stretch to get around the particles.5,6 The interfacial B segments can lose less conformational entropy in an expanded conformation in comparison with the interfacial A segment of the Gaussian state. One consequence is that expanded conformation makes it difficult for the Janus nanoparticles to be positioned in B segments. The deviation is enhanced for more asymmetry diblocks with longer lB, rationalizing the observations for Figure 3b (before the crossover) and Figure 3c. To gain more detailed insight into the entropic contribution from B segments to the position deviation of the Janus nanoparticles, we turn to the excess entropy defined as the difference between the thermodynamic entropy and the entropy of the ideal gas under the same temperature and density conditions.32−34 The two-body approximation on the basis of RDF has been demonstrated to provide a reasonable estimation of the total excess entropy.32 We herein calculate excess entropy per bead, Snp, from the pair correlation function

Figure 7. Distance between two blocks, db, as a function of the length of B block segment, lB. The inserting image shows the schematic diagram of the definition of db, where the red and green lines indicate the A and B block segments, respectively.

increased from 6 to 8, implying a more asymmetric dumbbell conformation of the block copolymer chains with a more stretched state of B segment near interfaces (note from Table 1 that the average size of B segments almost does not change). The radial distribution function (RDF) between the interfacial and the bulk beads of B segments, g(z), is also calculated, which measures correlations between B beads in these both sections. Figure 8a plots g(z) for lB = 6, 7, and 8. There are two characteristic peaks occurring at the same positions in these three plots: the first is at z ≈ 0.82rc, which is the equilibrium distance for two interacting DPD beads; the second is at z ≈ 1.62rc and is the diffraction peak. As indicated by the dashed arrows, the heights of these both peaks decrease in magnitude as lB increases. This reveals that the correlation between the interfacial and the bulk B beads is reduced as response to the increase of the B segments due to the structural change of any component. In view of the invariably state of the 4374

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between the B-segment beads and the beads of Janus nanoparticles as follows:32,33 Snp = −2π

∫ {gnp(z) ln gnp(z) − [gnp(z) − 1]}z 2 dz

(1)

Here n and p represent the two types of beads, respectively. The inset of Figure 8a is the histogram showing the values of Snp at lB = 6, 7, and 8. Clearly, increasing lB leads to a higher Snp, indicating that the position deviation of Janus nanoparticles from B domains to A domains is thermodynamically favored by reducing the entropic loss of the B segments, as illustrated by the schematic diagram in Figure 8b. This result is particularly interesting if one considers that, for nanoparticles with homogeneous surface chemistry, entropy favors the formation of uniformly dispersed nanoparticle within their preferential segments when the ratio of segment length to nanoparticle size is increased.5,6 To examine this case in more detail, we alter the surface property of Janus nanoparticle by “coating” them with homogeneous surface chemistry that has preferential interaction with the B segment, i.e., χBQ = χBP = 0.0 and χAP = χAQ = 1.1. Figure 9 presents the averaged distance

Figure 10. (a) Snapshot of A6B6 system showing an example of the self-assembly of homogeneous nanoparticles at the interfaces of diblock copolymers. (left) The isosurface structure where the interface between phases A and B is colored yellow and phase A is colored blue. Phase B is fully transparent. (right) Only homogeneous nanoparticles and the interfaces are shown by removing phase A in the left snapshot. (b) Fraction of the homogeneous nanoparticle surface wrapped by the A block segment f w as a function of the bead number in the B segment, l B.

(Figure 3b), in contrast to the homogeneous nanoparticles with the surface chemistry preferential to one block segment.5,6 Figure 10 indicates that the homogeneous neutral particles behave very like Janus particles. However, it should be emphasized that there are many advantages of Janus particles in the construction of novel nanocomposites. In contrast to the homogeneous surface of a neutral particle, the amphiphilic surface can offer a Janus particle higher adsorption energy with the two phases at interface, which may significantly stabilize the bulk structure and arrest the kinetics of the composites.35 On the other hand, the amphiphilic surfaces can make Janus particles form novel and tunable superstructures at the interfaces of block copolymers. For example, by controlling the surface chemistry and geometry of Janus particles, these amphiphilic building blocks can self-assemble into “standing” and “lying” interfacial superstructures in the symmetric block copolymers.11 Therefore, the coassembly of Janus particles in block copolymers could lead to viable approaches to fabricate and tune novel structures or superstructures on the nanometer scale. Although the homogeneous particles seem to be easier to experimentally obtain, their surfaces still need very precise control so that they can be positioned at interface. A typical example in this aspect is the bijels where nanoparticles with equal affinities to the two components of a blend are driven to the interface.36 Experimentally, it does also be a challenge to obtain nanoparticles with this characteristic. 3.4. Structural Formation Mechanism after the Crossover. We now move to the case where the interface topology is nonlamellar (i.e., subsequent to the crossover in Figure 3b). Figure 3b shows that f w turns to decrease in response to the increasing of lB, where the morphology undergoes a transition from lamella to cylinder with more curved interfaces. To thoroughly identify the influence of the curved interfaces on the hierarchical structures, let us first consider an ideal case where a

Figure 9. Averaged distance between the centers of homogeneous nanoparticles and the interface of block copolymers, df, as a function of the length of B block lB where the length of A block is fixed at 6. The inset schematic diagram illustrates the definition of df and shows the transition of the homogeneous nanoparticle from lB = 6 to 7. The snapshots display the morphologies of the systems at lB = 6 and 7.

between the centers of homogeneous nanoparticles and the interface of block copolymers, df, as a function of the length of B block lB, demonstrating that the nanoparticle spontaneously tends to the B domain in response to the increase of lB from 6 to 7. However, it turns to the A domain with further increasing lB. As illustrated by the insetting schematic diagram, the stretched B block at the interface prompts the transition of the nanoparticle to the B domain to eliminate the entropy cost induced by the interaction between the stretched part of the B block and the nanoparticle. The crossover with further increase of lB may be explained by the entropy effect revealed by the previous simulation where the nanoparticle with smaller size tends to the interface to maximize its transition entropy.5 These opposing entropic effects are largely attributable to the role of interface in the formation of hierarchical structures of BNs with Janus nanoparticles. To confirm this hypothesis, we alter the surface property of Janus nanoparticle by “coating” them with homogeneous surface chemistry that has equal interactions with A and B segments, i.e., χAP = χBQ = χAQ = χBP = 0.0. In this case, the homogeneous nanoparticles are also anchored at interfaces in the equilibrium BNs (see Figure 10a). As shown in Figure 10b, indeed, the f w−lB plot of this system is similar to those of the BNs containing Janus nanoparticles 4375

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at curved interfaces, we calculate the area of the defect section, Ad, for a symmetry Janus particle at a cylindrical domain with various radii. The calculation is based on the ideal model as illustrated in the inset of Figure 11. The mathematical deduction gives the following form of Ad

spherical Janus nanoparticle with two symmetry sites is placed on an interface. As illustrated in the inset of Figure 11, the

Ad =

∫0

π

⎛ r sin(θ ) ⎞ 2rn 2 sin(θ ) arcsin⎜ n ⎟ dθ ⎝ 2R ⎠

(2)

where rn and R are the radii of the particle and the cylindrical domain, respectively. Figure 11 shows the defect fraction, fd, defining as the ratio of Ad to the total area of the Janus particle (4πrn2), as a function of R, where rn is set as 1.11rc and is the same with that used in simulations. Clearly, the defect section is significantly reduced for a cylindrical domain with larger radius (smaller curvature), in qualitative agreement with the observation in Figure 3b (subsequent to the crossover). More precisely, we determine the simulated results of fd−R relations for A6B13 and A6B14 since the phase domains of these two systems approximate to ideal cylinders (for example, see Figure 3a), and thereby their averaged radius can be determined easily. As shown by the points in Figure 11, the simulation results agree well with the calculation based on the model of eq 2, providing a quantitative support for the fact that the topology mismatching has a great effect on the hierarchical structures of BNs with Janus nanoparticles, whereas the enthalpic interaction between “defect” and nonpreferential domain can also influence the arrangement of Janus nanoparticles. In the above simulations, a low volume fraction of Janus particles, i.e., VP = 0.02, is chosen so that the effect of the particle−particle interaction is trivial. To consider the influence of particle concentration, we examine the f w−lB relationships for a series of systems with higher particle concentrations, and

Figure 11. Fraction of the defect section induced by the topological mismatching between one site of a Janus nanoparticle and a cylindrical domain, fd, as a function of the radius of the cylindrical domain, R. Here the radius of the particle is selected as rn = 1.11rc. The dots are the simulated results for A6B13 (left) and A6B14 (right). The inset provides schematic diagrams showing a Janus nanoparticle inserted into (a) lamellar and (b) cylindrical phase domains. The dashed circle in (b) marks the defect section induced by the topological mismatching.

equator of the particle can perfectly match a flat interface. However, a topology mismatching will always arise when the Janus nanoparticle is placed at a curved interface. As marked by the dashed green circle, a “defect” section always occurs in one site of the Janus particle which cannot be wrapped by its preferential domain. This leads to the reduced value of f w and therefore may account for the behavior of the f w−lB plots subsequent to the crossover in Figure 3b. To provide a definitive confirmation of the role of topology mismatching in the hierarchical assembly of Janus nanoparticles

Figure 12. (A) Schematic diagram showing the model used for the micromechanical simulation of a Janus nanoparticle in the matrix of block copolymers where the middle cross section with dotted lines indicates the interface between phases A and B. (B) Strain fields for systems with various distances of the particle center from the interface, di. (a) di = −1.0, (b) di = 0.0, and (c) di = 1.0. The yellow dashed line is used to mark the interface between two phases. The A phase possesses a Young’s modulus 3 times greater than that of the B phase whereas the Janus particle possesses a Young’s modulus 20 times greater than that of the A phase. The green arrows in (A) and (B) denote the tensile direction. 4376

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to each of the domains, the complex elastic deformation fields associated with the nanocomposites can be examined. Furthermore, the fracturing of the polymeric material at regions of large deformation can also be investigated. In the present work, the spring constant of the pure A phase is taken to be 3 times stiffer than the spring constant of the pure B phase; thus, the Young’s modulus of the A polymer is 3 times greater than that of the B polymer.38 The Young’s modulus of the particle is taken to be 20 times greater than that of the A polymer by assigning it an appropriate force constant. Using the dynamical LSM, we deform the systems through the application of a constant strain, at the system boundaries, along the direction normal to the phase interface (as denoted by the arrows in Figure 12). We initially consider the elastic fields of the systems with various positions of Janus nanoparticles after the systems have been extended up to a global strain of approximately 1.5%. In Figure 12B, we present the strain fields as contour plots that slice through the center of the nanoparticle, along the zaxis, and parallel to the x−z plane, where di is changed from −1 to 1, corresponding to three distinct positions of the particle. Figure 12B(b) depicts the strain field for a system where the two moieties of the Janus nanoparticle are equally distributed between the two sides of the interface (di = 0). As indicated by the color bar, the stiffer particle tends to inhibit the deformation of the surrounding softer matrix but induces strain concentrations at the poles of the droplets. By comparing the deformation degrees of the polymer phases at the two poles of the particle, we find that the deformation degree near the pole in the compliant phase is more evident. However, it can be identified that the particle presents a more significant inhibition for the deformation of its surrounding matrix in the softer polymer phase than that in the stiffer polymer phase. The reinforced region is almost exclusively within the softer polymer phase. This indicates that the reinforcement efficiency of the particle for the softer polymer phase is higher than that for the stiffer polymer phase under present conditions. Figure 12B(a) shows the strain field for the system at di = −1.0. In this case, the particle center shifts to the A phase with more surface area of the particle being in this stiffer phase. Consequently, the reinforced region in the B phase becomes smaller. Figure 12B(c) shows the strain field in a system at di = 1.0 in which the particle takes a position predominantly in the B phase. This significantly increases the area of the reinforced region. The comparison for the above images allows one to readily visualize the extent to which the local mechanical response is affected by altering the position of Janus nanoparticle with respect to the phase interface. Having established the deformation distribution in the material, we now apply our fracture criteria to the different samples to characterize the effect of the particle position on the mechanical failure of the material. Here we utilize an energybased fracture criterion (see Supporting Information for more details).41 The minimum fracture elastic energy of the A phase is taken to be 2 times lower than that of the B phase while this value for the Janus nanoparticle is 50 times greater than that of the A phase. The resultant stress−strain curves are shown in Figure 13, and the data are averaged over three independent runs. The maximum in these stress−strain curves corresponds to the maximum stress that the system can sustain before the occurrence of one dominant crack, i.e., catastrophic failure. The slope of the curves before material failure clarifies that the particle with a more negative di leads to a low modulus of the

the simulation results are presented in Figure.S3. One can find that the relationships remain even when VP is increased to 0.10. This definitely indicates that the entropy effect and the interfacial topology predominately influence the position of Janus particles with respect to the interface. In a previous theoretical analysis, Hirose et al.21 discussed the dependence of the equilibrium position of a Janus nanoparticle at a liquid−liquid interface on the surface chemistry and wettability of the nanoparticle as well as the interfacial curvature. One can find that there are some fundamental differences between this work and our present simulations. One of the important differences is that the theoretical analysis cannot consider the effect from the detailed molecular architectures of the block copolymers. It argued that the flat interface is stable for a Janus particle with two equal moieties. Actually, both our simulation in Figure 3 and theoretical phase diagram15 demonstrate that the flat interface can occur in the block copolymers with various molecular architectures, e.g., A6B6, A6B7, and A6B8. Our simulations reveal that the molecular architectures significantly influence the interfacial position of Janus nanoparticles although the interface keeps flat in all of these systems, as demonstrated in, for instance, Figure 3b. The other important difference lies at that the two-dimensional (2D) analysis they used actually corresponds to the system where a spherical Janus particle interacts with a “spherical” interface instead of the cylindrical interface considered in our simulations. In this case, the interfacial topology effect will not take place because a circular contacting line will always form when a small sphere contacts with a big sphere, compared to the topology defect defined in Figure 11. 3.5. Mechanical Properties: Deformation and Fracture. As the interfacial structure of the composites will be changed with the deviation of Janus nanoparticles from interfaces, one may anticipate that the mechanical properties of these systems can be extremely modified.37 We thereby present detailed micromechanical simulations to demonstrate how the deviation of a Janus nanoparticle from the interface influences the deformation and fracture of this nanocomposite. For this purpose, a micromechanical model, dynamical lattice spring model (LSM), is adopted.38−41 The details of this model have been systematically introduced in the Supporting Information. The model can accurately recover continuum elastic behavior through appropriately choosing the constants of springs consisting of a lattice network. In particular, the LSM has been shown to accurately capture the elastic fields corresponding to Eshelby’s well-known theoretical solutions for the elastic behavior of inhomogeneous materials38 and has been used to simulate the micromechanical behaviors of polymer-based materials.38−41 To assess the influence of the position of Janus nanoparticle with respect to the phase interface, we adopt a local but typical structure which consists of a small simulation box with equal sizes of the A and B domains (Figure 12A). A Janus nanoparticle is incorporated in the box where the x and y coordinates of the particle center are set at the same with those of the box. However, the center of the particle is changeable along the z-axis (normal to the interface), and its distance from the box center is measured as di. di = 0 indicates that the particle center coincides with the box center and the equator of the particle superposes the phase interface. A negative value of di denotes that the particle center moves to the A phase and vice versa. We can now take these structures and feed them directly into a LSM simulation. By assigning a different stiffness 4377

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implications lies at that the dependence of nanoparticle distribution on the topology of mesoscopic phase is also particularly useful for the directed self-assembly of Janus nanoparticles in the block copolymer-based supramolecules where the mesostructures can be simultaneously modified in response to the change of their environments.7,44

4. CONCLUSIONS In summary, our work constitutes the first study exploring the interfacial topology dependence of the directed self-assembly of Janus nanoparticles in block copolymer scaffolds. Particularly, our simulations reveal that symmetric Janus nanoparticles do not take symmetric distribution in the interfaces of asymmetric block copolymers. Rather, they take various but controllable off-center arrangements from the interfaces upon tailoring the mesostructural topologies of these polymer matrices. For the lamellar phase formed by the asymmetric block copolymers, the deviation of these Janus nanoparticles from interfaces can be attributable to the asymmetric enthalpic interaction from the block segments and, interestingly, the unconventional entropic effect at the molecular scale. When the interfacial topology deviates from lamella, the topology mismatching between Janus nanoparticles and curved polymer interfaces at the mesoscale however accounts for the position transition of the Janus nanoparticles. Fundamentally, the interface plays a key role in this unconventional entropy effect. By employing a micromechanical model, we demonstrate that the deviation of the Janus nanoparticles from the interface can significantly influence the mechanical properties, including deformation distribution and mechanical failure, of the nanocomposites. This study could motivate synthetic efforts, including the experimental and theoretical studies, to be directed toward obtaining controlled mesostructural topologies of block copolymers as a design tool to precisely program the spatial organization of nanoparticles in polymer matrix.

Figure 13. Stress−strain plots for systems with various distances of the particle center from the interface, di. The A phase possesses a Young’s modulus 3 times greater than that of the B phase while the Janus particle possesses a Young’s modulus 20 times greater than that of the A phase. The minimum fracture elastic energy of the A phase is taken to be 2 times lower than that of the B phase while this value for the Janus nanoparticle is 50 times greater than that of the A phase.

nanocomposite because the reinforced region in the B phase becomes smaller as shown by Figure 12B(a−c). Surprisingly, the material fracture for a positive di occurs at a smaller strain. This is presumably due to the greater particle-phase interfacial area in the compliant phase and, therefore, a greater number of regions where significant elastic disparity occurs. At these regions, elastic field concentrations are significant enhanced, inducing elevated probability of failure. The simulation results definitely demonstrate that the position of Janus particle can evidently modify the mechanical failure of the nanocomposites. In this work, our aim for employing a micromechanical model is only to demonstrate that the deviation of Janus nanoparticles from interfaces can significantly influence the mechanical properties of the nanocomposites. Nevertheless, we also perform the simulation of micromechanical property for homogeneous particles (see Figure S4). One can find that the micromechanical property for homogeneous particles is not very different from that for Janus particles when the homogeneous particles are positioned at the interface because the enhancement of modulus mainly comes from the particle. However, Janus particles possess many advantages, e.g., higher adsorption energy, superstructural construction and control, and so on. 3.6. More Implications of the Findings. The results presented here have far more reaching consequence in more aspects. For example, the rich morphologies of block copolymers with various molecular architectures provides repertoire of interfacial structures.16 The mechanism demonstrated in this article greatly facilitates us to understand the thermodynamic nature of polymer nanocomposites and to control the hierarchical structures formed by the interfacial selfassemble of Janus nanoparticles in these scaffolds. For example, the stiffness of block segments can be modified, providing various conformational entropy contributions to the structural formation of the interfacial hierarchical structures.42 The position transition of Janus nanoparticles in the polymer matrix can significantly influence the other properties or functions of the nanocomposites, determining their structure−property relationships. For example, it has been shown that the width of the band gap could be increased through the selective sequestering of inorganic nanocrystals within one of the phases of the microphase-separated diblock copolymers.43 Thereby, the transition of Janus nanoparticles will change the optical properties of the filled block copolymers. Another important

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ASSOCIATED CONTENT

* Supporting Information S

Details of methods and additional simulation results. 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] (L.-T.Y.). Author Contributions

B.D. and R.G. contributed equally. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The financial support from the National Natural Science Foundation of China (nos. 21174080 and 51273105) is gratefully appreciated.

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REFERENCES

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