Percolating Behavior of Nanoparticles in Block Copolymer Host

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Article Cite This: J. Phys. Chem. C 2017, 121, 23705-23715

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Percolating Behavior of Nanoparticles in Block Copolymer Host: Hybrid Particle-Field Simulations Qian Zhang, Liangshun Zhang,* and Jiaping Lin* Shanghai Key Laboratory of Advanced Polymeric Materials, State Key Laboratory of Bioreactor Engineering, Key Laboratory for Ultrafine Materials of Ministry of Education, School of Materials Science and Engineering, East China University of Science and Technology, Shanghai 200237, China S Supporting Information *

ABSTRACT: The hybrid particle−field method is extended to investigate selfassembly and percolating behavior of nanocomposites containing block copolymers and nanoparticles. The self-assembled nanostructures serve as templates to guide organization and distribution of nanoparticles. The percolation threshold of nanoparticles in the host of block copolymers is discussed in terms of composition of block copolymers, radius, and aspect ratio of nanoparticles. The simulated results demonstrate that the block copolymers have a synergistic or antagonistic effect on emergence of percolating network of nanoparticles, depending on the copolymer composition. There exists an optimal value of copolymer composition for lowering the percolation threshold of nanoparticles. In addition, regulating the geometrical shape of nanoparticles further lowers the percolation threshold of nanoparticles dispersed in the block copolymers. These simulated results provide useful guidelines for designing high-performance composite materials with light weight.

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INTRODUCTION Combining the desirable functionality of nanoparticle fillers with the easy processability of polymer host holds great promise for designing novel materials with enhanced mechanical, optical, and electrical properties, which make them attractive for large-scale industrial applications in the fields of photovoltaic and optoelectronic devices.1−7 In particular, if the loading of nanoparticles reaches a critical value termed as percolation threshold, a continuous network of nanoparticles spans the polymer host, typically leading to insulator-to-conductor transition in composites of conductive fillers and insulating host.8−10 Recently, theoretical and experimental studies show that the percolation threshold of nanoparticles depends on their geometrical shape, interaction between them, as well as properties of polymers.11−14 However, creating the functional materials that simultaneously possess the mechanical reinforcement, electrical conductivity, as well as lightweight feature still remains challenging, especially for the case of nonspherical nanoparticles. In order to further lower the loading of nanoparticles, a comprehensive understanding about the effect of their physiochemical properties on the percolating behavior becomes important. An effective strategy to reduce the percolation threshold of nanoparticles is introduction of inhomogeneity into the polymer host.15,16 Because block copolymers self-assemble into a variety of periodically ordered nanostructures depending on their composition, these materials offer a means to achieve both the structural continuity and the dimensional stability.17−24 This allows them to serve as templates for controlling © 2017 American Chemical Society

organization and distribution of nanoparticles. As a typical example, Balsara and co-workers investigated the effect of adding nanoparticles on ionic conductivity of block copolymer electrolyte.23,24 The (ethylene oxide)-rich domains provide the conductive pathway, while the nonconductive domains impart the desired mechanical properties. Remarkably, it is demonstrated that the addition of nanoparticles leads to the formation of a three-dimensional percolating network and the increase of ionic conductivity. Despite the significant success in experiments, there remain great challenges to characterize the relationship between the self-assembled nanostructures and the percolation probability as well as to achieve a fundamental understanding about the experimental observations. Computer modeling and simulations are playing an everincreasing role in exploring the percolating behavior of polymer/nanoparticle mixtures. There are numbers of computational studies focusing on the systems of nanoparticles dispersed in simple fluids or homopolymers.11,25−30 However, only a few reports involve the percolating network of nanoparticles in the host of block copolymers, where the selfassembled nanostructures offer a confined environment to guide the location and orientation of nanoparticles.31−33 In earlier computational studies, Balazs and co-workers developed a phenomenological model to probe into the microphaseseparated structures and the percolating behavior of nanoReceived: July 25, 2017 Revised: October 1, 2017 Published: October 9, 2017 23705

DOI: 10.1021/acs.jpcc.7b07337 J. Phys. Chem. C 2017, 121, 23705−23715

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The Journal of Physical Chemistry C particles in the host of block copolymers.31 It is revealed that the percolation threshold of nanoparticles dispersed in the block copolymers is lower than that in the host of homopolymers. Recently, Zhao et al. used molecular dynamics simulations to investigate the self-assembled nanostructures and the percolation probability of nanofillers in the block copolymer template.32 It is demonstrated that the block copolymer/nanotube composites have a lower value of percolation threshold. Although the above examples illustrate the potential utility of harnessing the self-assembled nanostructures to yield the percolating network of nanoparticles, the effects of composition of block copolymers and geometry of nanoparticles have not been comprehensively evaluated. In addition, there remains a need to further lower the percolation threshold of nanoparticles through designing the nanocomposites (e.g., composition of block copolymers, size, and shape of nanoparticles), which facilitates experimentalists to fabricate lightweight composite materials encompassing characteristics of mechanical reinforcement and electrical conductivity. Motivated by the above issues, we herein utilize the mescoscopic model to probe into the self-assembled nanostructures and the percolating behavior of block copolymer/ nanoparticle mixtures. Our model is based on the selfconsistent field theory (SCFT) of inhomogeneous polymeric fluids,34 which is widely applied to tackle issues of complex selfassembly behavior of block copolymers. In our previous works, a theoretical approach originally proposed by the Balazs group is extended to address the polymer/particle nanocomposites by coupling the SCFT for polymers and density functional theory for nanoparticles.35−38 One limitation of this approach is that the particle coordinates are imposed as an approximate energy functional. Thus, this approach fails to capture the percolation phenomenon of nanocomposites. Recently, the hybrid particlefield (HPF) model, in which the particle coordinates are explicitly retained as degrees of freedom, is developed to predict the hierarchically self-assembled nanostructures of block copolymers with embedded spherical nanoparticles.39−45 It should be emphasized that the contribution from the geometrical shape of nanoparticles is not incorporated into the field-theoretic framework. In this contribution, the HPF model is extended to build up the computational model for the mixtures of block copolymers and nanoparticles with nonspherical shape. We herein focus on the percolating behavior of nanoparticles dispersed in the block copolymers. The effects of composition of block copolymers, radius, and shape of nanoparticles on the percolation threshold of nanoparticles are examined in detail. More importantly, we introduce synergistic strength to describe the percolation probability of nanoparticles. The lower loading of nanoparticles to form the percolating network is obtained via evaluating the synergistic strength in terms of the physicochemical parameters of nanocomposites. We expect that the present study may provide meaningful guidance for experimentalists to design and fabricate the nanocomposites with light weight and electrical conductivity.

copolymer chains and nP nanoparticles. The block copolymers are modeled as the Gaussian chains with a length of N and gyration radius of R g = b N/6 , where b is statistical Kuhn length. The composition of block copolymers is indicated by the volume fraction fA of A blocks per chain. The i-th nanoparticle with center Ri and orientation θi is described by a continuous cavity function hi(r) given by39 hi(r) = 1 − tanh( x*2/αRP 2 + αy*2 /RP 2 /λ)

(1)

where r* ≡ (x*, y*) = A(θi)·(r − Ri) is a rotated version of the relative coordination r − Ri, and A(θi) is a rotation matrix. The effective radius RP and aspect ratio α are used to characterize the size and shape of nanoparticles, respectively. λ defines the width of the particle−fluid interface. The schematic illustrations of nanoparticles with various aspect ratios in the two-dimensional space are depicted in Figure S1 of Supporting Information (SI). The local volume fraction of nanoparticles is n represented by φP(r) = ∑i =P 1 hi(r). The volume fraction of nanoparticles in the mixtures is denoted by cP = VP/V, where VP = ∫ drφP(r) is the total volume of nanoparticles. The free energy functional (in units of kBT) for the block copolymer/nanoparticle mixtures is written as T

F = UPP + UAB − TS

(2)

The first term represents the contribution from the interparticle interactions given by nP − 1 nP

UPP =

∑ ∑ ∫ dr′hi(r′)∫ drhj(r)u(|r′ − r|) (3)

i=1 j>i

where u(r) denotes the repulsive part of the Lennard−Jones potential. The second and third terms of eq 2 constitute the energy from the contributions of the block copolymers and the copolymer−nanoparticle interactions UAB − TS = n(1 − c P)ln +

n V

V (1 − c P) Q

⎧ ⎪

⎫ ⎪

∫ dr⎨ 12 ∑ χIJ NφIφJ + 2κ (∑ φI − 1)2 − ∑ ωK φK ⎬ ⎪

⎩

I,J

I

K

⎪

⎭ (4)

where φI (I, J = A, B, and P) is the local volume fraction of Itype component and ωK (K = A and B) denotes the potential field of the K-type component. χIJ represents the Flory− Huggins interaction parameter between distinct components. The Helfand-type parameter κ controls the local compressibility of polymer fluids. Q is the normalized single-chain partition function of block copolymers. Minimization of the free energy functional with respect to the variables ωK and φK yields the self-consistent field equations

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COMPUTATIONAL MODEL Below, we provide a brief description of the extended HPF model (i.e., incorporation of geometrical shape of nanoparticles) to study the self-assembly and the percolating behavior of block copolymer/anisotropic nanoparticle mixtures. We consider a system with volume V, containing n AB block

δF =0 δωK

(5)

δF =0 δφK

(6)

The translation and rotation of nanoparticles obey the Newton’s motion equations. The force Fi and torque Ni exerted on the i-th nanoparticle are written as40,46,47 Fi = 23706

∫ drfi(r)

(7) DOI: 10.1021/acs.jpcc.7b07337 J. Phys. Chem. C 2017, 121, 23705−23715

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The Journal of Physical Chemistry C Ni =

∫ dr(r − ri) × fi(r)

= 0.125Rg. The edge length of square simulation cells is set as 32.0Rg, and the corresponding cells are discretized by 256 × 256 grids. To check the effect of finite size of simulation cells on the self-assembly behavior of block copolymer/nanoparticle mixtures, we also perform a series of simulations with various cell sizes. From Figure S2 of the SI, it is verified that the percolating networks of nanocomposites are maintained in spite of further extension of cell size, and the values of percolation thresholds have a high accuracy under the parameter setting of cell size of 32.0Rg. 1. Percolating Behavior of Block Copolymer/Nanoparticle Mixtures. Figure 1 shows the self-assembled

(8)

Here, fi is the body force that acts throughout the volume of the i-th nanoparticle fi(r) = hi(r)( −∇μi (r) − ζ vi(r) − ∇η)

(9)

where μi(r) ≡ δF/δhi(r) is the chemical potential of the i-th nanoparticle. ζ is the Stokes friction coefficient, and vi is the velocity of the i-th nanoparticle. The last term of eq 9 is a Gaussian white noise satisfying the fluctuation−dissipation theorem. All the simulations are implemented in the two-dimensional boxes with periodic boundary conditions. Initially, a number of nanoparticles are randomly placed in the simulation boxes, and the potential fields ωA and ωB for the A and B blocks are guessed. Subsequently, the algorithm for solving the HPF equations is performed as follows: (i) For a given configuration of nanoparticles, the self-consistent field equations (eqs 5 and 6) for the polymer fluids are numerically solved in the real space:36−38 (ii) the force Fi and torque Ni exerted on the i-th nanoparticle are calculated through eqs 7−9; (iii) the position and orientation of nanoparticles are updated by integrating the Newton’s motion equations; (iv) step (i) is repeated until the mean-square displacement of nanoparticles is less than 10−6Rg2, and the condition of |1 − φA − φB − φP| < 10−4 is satisfied. Under these conditions, the self-assembled nanostructures of block copolymer/nanoparticle mixtures are “kinetically” arrested configurations with lots of defects.42 While the system will tend toward the thermodynamic limit of a perfect lamella/ cylinder phase, the computational time for reaching this state is prohibitively long due to high energy barrier of structural rearrangement under the condition of large-cell simulations. It should be mentioned that such “kinetically” arrested structures are also observed in the experiments and simulations.48−50 For each choice of parameter settings, ten independent runs with different random seeds and initial configurations are carried out to ensure the reproducibility of simulated results. The configurations obtained from the HPF simulations are used to analyze the percolating behavior of block copolymer/ nanoparticle mixtures. Initially, a site of the grid is assumed to be occupied by one component as the local volume fraction of this component at this site is larger than those of other components. Based on this criterion, three types of sites (i.e., A, B, and P sites) are identified for given configurations. Subsequently, the cluster multiple labeling technique proposed by Hoshen and Kopelman is used to search the clusters of distinct components.51 The areas of various clusters in the twodimensional space are obtained by counting the number of corresponding sites. Finally, normalized cluster size SI is defined as the area of cluster divided by the total area of the corresponding I-type component. When SI is close to one, almost all the I-type component belongs to one cluster, and the percolation of this component takes place. As SI approaches zero, the I-type component forms plenty of small clusters.

Figure 1. (a and c) Self-assembled structures of nanocomposites and (b and d) normalized size distributions of A-rich, B-rich, and nanoparticle clusters for various volume fractions cP of nanoparticles. (a and b) cP = 0.10 and (c and d) cP = 0.21. The green, red, and blue colors denote the A-rich domains, B-rich domains, and nanoparticles, respectively. The normalized size S of I-type (I = A, B, and P) cluster is defined as the area of cluster divided by the total area of I-type component. Insets of images (a) and (c) illustrate the results of cluster analysis of nanoparticle. The arrows in image (c) highlight the isolated B-rich domains. The composition of copolymers is fixed at fA = 0.50, and the radius of nanoparticles is set as RP = 0.50Rg.

structures of block copolymer/nanoparticle mixtures at the volume fraction of nanoparticles cP = 0.10 and 0.21. The composition of block copolymers is fixed at fA = 0.50, and the radius of nanoparticles is set as RP = 0.50Rg. At low loading of nanoparticles (i.e., cP = 0.10), the symmetric block copolymers self-assemble into lamellae with short-range ordering, and the A-selective nanoparticles are sequestered in the A-rich domains (Figure 1a). As the volume fraction of nanoparticles is increased to cP = 0.21, a small portion of continuous B-rich domains are isolated due to the swelling of A-rich domains from the strong interparticle repulsions, as highlighted by the arrows in Figure 1c. It should be pointed out that the structural transition from the lamellae to the cylinders is triggered by an increase of nanoparticle loading in our simulations, which is verified by the existing results of simulations and experiments.39,52−56 The cluster searching technique is employed to recognize the clusters of A blocks, B blocks, and nanoparticles. As a typical example, insets of Figure 1 illustrate the results of cluster analysis of nanoparticles, where each isolated cluster is labeled by a distinct color to provide visual information on their connectivity. It should be mentioned that the nanoparticles

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RESULTS AND DISCUSSION In our simulations, the Flory−Huggins interaction parameter between A and B blocks is chosen as χABN = 16.0. The values for the interactions between nanoparticles and blocks are set as χAPN = 0.0 and χBPN = 16.0, implying that the nanoparticles are compatible with the A blocks and incompatible with the B blocks. The width of the particle−fluid interface has a value of λ 23707

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Figure 2a shows the normalized size Smax A of the largest A-rich cluster as a function of the volume fraction cP of nanoparticles

belong to the same cluster if the minimum distance between their lattice sites is less than one lattice unit. When the volume fraction of nanoparticles has a value of cP = 0.10, the clusters are marked by multiple colors (inset of Figure 1a), indicating that the nanoparticles are organized into a large number of isolated clusters. At higher loading of nanoparticles, a single-colorized cluster of nanoparticles is identified instead of multiplecolorized clusters (inset of Figure 1c). This phenomenon signals the fact that the organization of nanoparticles produces a percolating network spanning the entire sample. To quantitatively characterize the percolating behavior of block copolymer/nanoparticle mixtures, the distributions of normalized cluster size of various components at the nanoparticle loading cP = 0.10 and 0.21 are calculated and plotted in Figures 1b and 1d, respectively. As a reference, the corresponding distributions of A and B components in the system of pure block copolymers are shown in Figure S3 of SI. The normalized sizes of A- and B-rich clusters are both smaller than 0.65. As the nanoparticles are incorporated into the system of block copolymers, a slight difference emerges. In the case of block copolymer/nanoparticle mixtures with cP = 0.10, the normalized size of the largest A-rich cluster (denoted by Smax A ) is 0.85, but those of B-rich clusters are smaller than 0.40 (Figure 1b). In particular, most of the nanoparticle clusters have a normalized size of 0.05, suggesting that the nanoparticles form isolated clusters in the host. As the volume fraction of nanoparticles is increased to cP = 0.21, the block copolymer/ nanoparticle mixtures produce the doubly percolating structure, where both the clusters of A blocks and nanoparticles have the normalized size of 1.0 (Figure 1d). In comparison with the case of nanoparticles dispersed in the host of homopolymers or homopolymer blend, the percolating behavior of block copolymer/nanoparticle mixtures has some salient features. In the homopolymer/nanoparticle system, the species uniformly fill the simulation boxes, and the percolation formation of nanoparticles originates from strong interparticle repulsions. As other incompatible homopolymers are also incorporated into the system, the nanoparticles are almost homogeneously located in the domains of compatible homopolymers due to the macrophase separation of the polymer blend. However, for the block copolymer/nanoparticle mixtures, the cooperative effects (arising from the microphase separation of block copolymers, the preferential wetting interactions and the interparticle repulsions) drive the isolated domains to form a continuous network spanning the entire sample. Meanwhile, the network structures of block copolymers serve as templates to guide the nanoparticles to form the continuous pathway. It is worthy to point out that the percolating behavior of nanoparticles in the host of diblock copolymers not only depends on their geometrical properties but also relies on the composition of block copolymers. Below, we explore the effects of composition of block copolymers, radius, and anisotropy of nanoparticles on the percolating behavior of nanocomposites. 2. Effect of Composition of Block Copolymers. To elucidate the effect of copolymer composition on the formation of the percolating network of nanocomposites, a series of simulations are performed for the block copolymer/nanoparticle mixtures with the copolymer compositions ranging from 0.30 to 0.70. The radius of nanoparticles is fixed at an intermediate value of RP = 0.50Rg. Since the nanoparticles are sequestered in the A-rich domains, we below concentrate on the percolating behaviors of A blocks and nanoparticles.

Figure 2. (a) Normalized size of the largest A-rich domain Smax A as a function of the volume fraction cP of nanoparticles at the copolymer compositions fA = 0.30, 0.50, and 0.70. (b) Normalized size of the largest nanoparticle cluster Smax P as a function of the volume fraction cP of nanoparticles. As a control group, the cases of nanoparticles in the homopolymer host are plotted by dashed line in image (b). The percolation thresholds (PAC and PPC) of A-rich domains and nanoparticles are labeled by the arrows in images (a) and (b), respectively. The radius of nanoparticles is fixed at RP = 0.50Rg.

under various compositions fA of block copolymers. Each point in the plot is averaged over ten independent simulations, whereas the statistical deviations are not displayed for the sake of clarity. As indicated by the arrow, the percolation threshold PAC of A blocks is defined as the lowest loading of nanoparticles when Smax approaches one. In the case of fA = 0.30, a higher A percolation threshold of A blocks is attributed to the structural transition from the isolated A-rich domains to the percolating network. When the block copolymers become symmetric, the percolation threshold of A blocks is decreased to PAC = 0.12, owing to the existence of elongated A-rich domains. As the composition of block copolymers has a value of fA = 0.70, a percolating network of A blocks is achieved without extra addition of nanoparticles. Figure 2b illustrates the normalized size Smax of the largest P nanoparticle cluster in terms of the volume fraction cP of nanoparticles at the copolymer compositions fA = 0.30, 0.50, and 0.70. For the sake of comparison, the data for the homopolymer/nanoparticle mixtures are also presented in Figure 2b. The percolation thresholds PPC of nanoparticles are highlighted by the arrows. One can identify a general tendency for the percolation formation of nanoparticles dispersed in the block copolymers (i.e., the percolating network of nanoparticles takes place with an increase of their loading). However, the percolation threshold PPC of nanoparticles is strongly dependent upon the composition of block copolymers. In the case of fA = 0.30, the percolation threshold of nanoparticles is higher than that in the homopolymer system, implying that the percolation 23708

DOI: 10.1021/acs.jpcc.7b07337 J. Phys. Chem. C 2017, 121, 23705−23715

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nanoparticles with radius of RP = 0.50Rg, the block copolymers with the composition fA = 0.56 (highlighted by dashed lines) provide the strongest synergistic effect. A further increase of copolymer composition does not have a beneficial effect and is even detrimental with respect to the increase of Δ. The percolating behavior of nanoparticles in the host of block copolymers with different compositions can be understood as follows: For the system of A-minority block copolymers, the A-selective nanoparticles are strongly confined in the isolated A-rich domains. Introduction of nanoparticles stretches the isolated domains into elongated domains. Further increase of particle loading drives the elongated domains to form a percolating network. This phenomenon leads to a decrease of PPC as the copolymer composition is increased. On the other extreme, an increase of composition of A-majority copolymers increases the spatial freedom of nanoparticles, leading to a decrease of contact probability of nanoparticles. The behavior gives rise to the appearance of a number of relatively large nanoparticle clusters. Therefore, increasing the copolymer composition results in an increase of PPC. Between the two extremes, there is an optimal value of copolymer composition, where the block copolymers form the percolating network and their self-assembled nanostructures have a suitable space to accommodate the nanoparticles. It is worthwhile to compare the percolation thresholds (PPC)′ of nanoparticles in the A-rich domains with these PPC in the host of block copolymers. The volume fraction of nanoparticles in A-rich domains is defined as cP′ = VP/(VP + VA), where VI (I = A and P) represents the volume of I-type components. The corresponding definition of (PPC)′ in the A-rich domains is illustrated in Figure S4a of SI. Figure S4b of SI shows the percolation threshold PPC in the block copolymers and (PPC)′ in the A-rich domains as a function of the copolymer composition. In the range of copolymer compositions studied here, the values of (PPC)′ are larger than these of PPC. Nevertheless, there still exists a synergistic region, where the values of (PPC)′ are lower than those in the homopolymer system. Through the simulations, we can estimate the nanoparticle loading that is needed to form a percolating network in the systems of the homopolymers and the block copolymers. For the host of homopolymers, the percolation threshold of nanoparticles has a value of 0.32. What is striking is that the block copolymers with the composition fA = 0.56 possess a lower percolation threshold PPC = 0.19. Thus, one can reduce the percolation threshold of nanoparticles through utilizing the near-symmetric block copolymers instead of the homopolymers. 3. Effect of Radius of Nanoparticles. It is also important to comprehend the effects of geometrical parameters of nanoparticles on the self-assembly and percolating behavior of block copolymer/nanoparticle mixtures. To clarify these issues, we perform a series of simulations for the systems containing nanoparticles with various radii RP. It should be pointed out that the radius of nanoparticles is not in excess of 0.60Rg, ensuring that the local interfacial curvature of microphase-separated structures is not strongly distorted by the nanoparticles. Figures 4a and 4b show the typical self-assembled structures of nanocomposites at the radii RP = 0.50Rg and 0.25Rg of nanoparticles, respectively. The volume fraction of nanoparticles is fixed at cP = 0.31. The copolymer composition is chosen as fA = 0.30, where the pure block copolymers selfassemble into the isolated A-rich domains. As the A-selective

formation of nanoparticles is suppressed. For the block copolymers with fA = 0.50 and fA = 0.70, the nanoparticles yield a percolating network at lower loading of nanoparticles, indicating that the percolation formation of nanoparticles is promoted. Especially, since the larger A-rich domains accommodate more nanoparticles, the percolation threshold of nanoparticles dispersed in the A-majority copolymers (case of fA = 0.70) is higher than that in the symmetric block copolymers (fA = 0.50). These findings in Figure 2 manifest the fact that the percolation thresholds of distinct components have a tight relation with both the composition of copolymers and the loading of nanoparticles. To evaluate the generality of percolating behavior of block copolymer/nanoparticle mixtures, the relation between the percolation threshold and the copolymer composition is depicted in Figure 3. The dotted line highlights the percolation

Figure 3. Percolation thresholds (PAC and PPC) of A-rich domains and nanoparticles as a function of the copolymer composition fA. The percolation threshold P0C of nanoparticles in the host of homopolymers is highlighted by the dotted line. The color band shown on the top axis represents the synergistic strength Δ = P0C − PPC as a function of the copolymer composition fA. The solid line divides the color band into two regions: the synergistic region with Δ > 0 and the antagonistic region with Δ < 0. The dashed line indicates the optimal composition of block copolymers, where the synergistic strength Δ has the largest value. The color bar indicates the value of Δ.

threshold P0C of nanoparticles in the host of homopolymers. The percolation threshold PAC of A blocks as a function of the copolymer composition fA displays a monotonic behavior. Particularly, PAC reaches a plateau and maintains zero in the region of fA > 0.56. Conversely, the percolation threshold PPC of nanoparticles exhibits a nonmonotonic behavior as the composition of copolymers is changed. In other words, there exists an optimal value of copolymer composition for lowering the percolation threshold of nanoparticles. Furthermore, in the region of 0.30 < fA < 0.40, the percolation formation of nanoparticles is suppressed (i.e., PPC ≥ P0C). Beyond the region, the block copolymers and nanoparticles provide a synergy for the formation of a percolating network. In order to quantitatively describe the synergy between the nanoparticles and block copolymers, we introduce a synergistic strength Δ = P0C − PPC defined as the difference between the percolation thresholds of nanoparticles in the hosts of homopolymers and block copolymers. The top panel of Figure 3 displays the one-dimensional color band of synergistic strength Δ in terms of the copolymer composition fA. The areas with positive and negative values of Δ separated by a solid line correspond to the synergistic and antagonistic regions of the percolating network of nanoparticles, respectively. In addition, a larger value of Δ indicates the stronger synergistic effect on lowering the percolation threshold of nanoparticles. For the 23709

DOI: 10.1021/acs.jpcc.7b07337 J. Phys. Chem. C 2017, 121, 23705−23715

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Figure 5. Percolation threshold PPC of nanoparticles as a function of the radius RP of nanoparticles at the copolymer compositions fA = 0.30, 0.50, and 0.70. The dashed line indicates the percolation threshold of nanoparticles in the host of homopolymers.

sketch some common features for the PPC versus RP relation. On the one hand, a decrease of RP at fixed loading of nanoparticles leads to an increase of the number of nanoparticles, which boosts the contact probability of nanoparticles and promotes their percolation (corresponding to a decrease of PPC). On the other hand, the strength of this promoting effect depends on the composition of block copolymers. Specifically, in the case of fA = 0.30, the nanoparticles are confined in the isolated A-rich domains, and a structural transition from the isolated domains to a continuous network is required for the emergence of percolation of nanoparticles. Under this circumstance, PPC has a weak relation with the radius of nanoparticles and is almost higher than the value of P0C, corresponding to the antagonistic region. For the block copolymers with fA = 0.50 and 0.70, decreasing RP promotes the formation of the percolating network of nanoparticles due to the increase of interparticle repulsions. For the purpose of investigating the effects of the copolymer composition fA and the nanoparticle radius RP on the synergy between the copolymers and the nanoparticles, we systematically calculate the synergistic strength Δ as functions of fA and RP, allowing us to construct the landscape of synergistic strength presented by the contour map in the fA−RP plane, which is depicted in Figure 6. The solid line separates the plane into the synergistic (Δ > 0) and antagonistic (Δ < 0) regions. The dashed line highlights the optimal composition of block copolymers for lowering the percolation threshold of nanoparticles. An important outcome of our simulations is that the nanocomposites with large nanoparticles have a wide synergistic region and a strong synergistic strength. Especially,

Figure 4. Self-assembled structures of block copolymer/nanoparticle mixtures for various radii of nanoparticles: (a) RP = 0.50Rg and (b) RP = 0.25Rg. Insets of images (a) and (b) illustrate the corresponding results of cluster analysis of nanoparticles. The representations of colors are the same as those of Figure 1. The composition of copolymers is set as fA = 0.30, and the volume fraction of nanoparticles is fixed at cP = 0.31.

nanoparticles with radius of RP = 0.50Rg are incorporated into the polymer systems, the isolated domains are elongated due to the interparticle repulsions (Figure 4a). The cluster analysis confirms that the nanoparticles form the separated clusters (inset of Figure 4a). As the radius of nanoparticles is decreased at fixed loading of nanoparticles, the number of particles primarily located in the A-rich domains is increased. This leads to strong repulsions of nanoparticles in the confined environment. To alleviate the contribution of repulsions, the isolated domains are connected to form a network, which simultaneously directs the nanoparticles to yield a percolated network (Figure 4b). Such observations are further verified by the normalized size distributions of clusters (Figure S5 of SI). We below focus on the percolating network of nanoparticles. Figure 5 plots the percolation threshold PPC of nanoparticles as a function of the radius RP of nanoparticles. As a reference, the percolation threshold P0C of nanoparticles in the host of homopolymers is also presented in Figure 5. These curves

Figure 6. Contour map of synergistic strength Δ = P0C − PPC in the plane of the effective radius RP of nanoparticles and the composition fA of copolymers. The solid line divides the contour map into two regions: the synergistic region with Δ > 0 and the antagonistic region with Δ < 0. The dashed line represents the optimal value of copolymer composition. The color bar indicates the value of Δ. 23710

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The Journal of Physical Chemistry C at the particle radius RP = 0.60Rg, the block copolymers offer a maximum synergistic strength of 0.15. Another important outcome is that the boundary between the synergistic and antagonistic regions is regulated by the composition of block copolymers and the radius of nanoparticles. However, the optimal composition of copolymers for the percolation of nanoparticles has a weak relation with the radius of nanoparticles. 4. Effect of Geometric Anisotropy of Nanoparticles. We now turn our attention to the percolating behavior of nonspherical nanoparticles. Currently, there exist only a few investigations on the percolating network of nonspherical nanoparticles dispersed in the block copolymers.11,31,32 To achieve understanding about the percolating behavior of such nanoparticles, the model of nanoparticles with various aspect ratios α are constructed, and the influence of geometric anisotropy of nanoparticles on the percolating network of nanoparticles is investigated. The nanoparticles with α > 1 are designated as nanorods. The effective radius of nanoparticles is set as RP = 0.50Rg (i.e., the area of the nanoparticle is fixed as the aspect ratio is changed). Figure 7 shows the self-assembled structures of nanocomposites containing nanoparticles with various aspect ratios α. The copolymer composition has a value of fA = 0.30. The volume fraction of nanoparticles is set as cP = 0.28, where the spherical nanoparticles cannot form the percolating network (Figure 7a). As the aspect ratio of nanoparticles is increased to α = 3.0, the addition of nanorods is in favor of the elongation of isolated A-rich domains, leading to the production of larger clusters (Figure 7b). When the aspect ratio of nanoparticles is further increased to α = 5.0, the long axis of nanorods induces the structural transition of A-rich domains (Figure 7c). The phenomenon results in the achievement of doubly percolating networks of A blocks and nanoparticles, which are further confirmed by the normalized size distributions of clusters (Figure S6 of SI). For the block copolymer/nanorod mixtures, an important issue is how the geometrical anisotropy of nanoparticles influences their percolation threshold. Figure 8 shows the percolation threshold PPC of nanoparticles as a function of the aspect ratio α of nanoparticles at the copolymer compositions fA = 0.30, 0.50, and 0.70. The dashed line represents the percolation threshold of nanoparticles in the host of homopolymers. In both the hosts of homopolymers and block copolymers, increasing their aspect ratio results in the lower percolation threshold of nanoparticles. What is striking is that fewer repelling nanorods are required to form a percolating network if the nanofillers are dispersed in the block copolymers. It is also revealed that the percolation threshold has a strong relation with the copolymer composition. Unlike the effect of nanoparticle radius (Figure 5), increasing the aspect ratio of nanoparticles efficiently reduces the percolation threshold of nanoparticles even at fA = 0.30. Figure 9 shows the landscape of synergistic strength Δ presented by the contour map in the fA−α plane. On the basis of the value of synergistic strength, the image is divided into two characteristic zones separated by the solid line: the synergistic (Δ > 0) and antagonistic (Δ < 0) regions. For the nanoparticles with the same effective radius, the synergistic region of nanorods is wider than that of spherical nanoparticles. The dashed line highlights the optimal value of copolymer composition for lowering the percolation threshold of nanoparticles. As the aspect ratio of nanoparticles is increased, the

Figure 7. Self-assembled structures of block copolymer/nanoparticle mixtures for various aspect ratios of nanoparticles: (a) α = 1.0, (b) α = 3.0, and (c) α = 5.0. Insets illustrate the results of cluster analysis of nanoparticles. The composition of copolymers is set as fA = 0.30, and the volume fraction of nanoparticles is fixed at cP = 0.28.

Figure 8. Percolation threshold PPC of nanoparticles as a function of the aspect ratio α of nanoparticles at the copolymer composition fA = 0.30, 0.50, and 0.70. The dashed line represents the percolation threshold of nanoparticles in the host of homopolymers. The effective radius of nanoparticles is set as RP = 0.50Rg.

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from the HPF simulations. It should be pointed out that the phenomenological model proposed by Balazs’s group ignores the contribution of conformation entropy of polymer chains as well as the effect of excluded volume between the nanoparticles and polymer components, which play critical roles in controlling the self-assembled nanostructures of hybrid systems. In our model, the conformation entropy of polymer chains is microscopically described by the self-consistent field theory, and the effect of excluded volume is incorporated into the hybrid system via the cavity function. The previously simulated works focus on the case of symmetric block copolymers (i.e., the effect of copolymer composition is not involved).31−33 In the present investigation, a series of simulations are performed for the nanocomposites with various copolymer compositions. It is found that the block copolymers have a synergistic or antagonistic effect on emergence of percolating network of nanoparticles, strongly depending on the copolymer composition (Figures 3, 6, and 9). The synergistic effect of symmetric block copolymers is also demonstrated in the work of Balazs’s group.31 However, as the block copolymers become extremely asymmetric, the antagonistic effect emerges in the nanocomposites due to the transition of self-assembled nanostructures. Thus, these computational results clarify the previously experimental and theoretical observations (i.e., the composition of block copolymers has a strong effect on the percolation threshold of nanoparticles).31 To the best of our knowledge, this is the first example for clearly demonstrating the antagonistic effect of block copolymers on the percolating behavior of nanoparticles. Such a finding provides a useful guideline for experimentalists to design the composition of block copolymers for the achievement of percolating network with lower loading of nanoparticles. More importantly, our HPF simulations predict that there is an optimal value of copolymer composition, where the percolation threshold of nanoparticles reaches a minimum value. In addition, such percolation threshold has a tight relation with the geometrical shape of nanoparticles. Specifically, the block copolymers with the composition fA = 0.56 possess a lower percolation threshold PPC = 0.19 of spherical nanoparticles (Figure 3). Regulating the radius and aspect ratio of nanoparticles further lowers the percolation threshold of nanoparticles dispersed in the near-symmetric block copolymers (Figures 5 and 8). Such findings shed light on the material design and application of nanocomposites consisting of the block copolymers and functional nanoparticles. For example, the mixtures of symmetric block copolymers and anisotropic nanoparticles offer the potentials to fabricate the highperformance materials encompassing the characteristics of high electrical conductivity and light weight, because of lower percolation threshold of nanoparticles. Although the HPF simulations presented here capture the fundamental features of percolating behavior of nanoparticles dispersed in the block copolymers, there are still some limitations for the HPF model and predicted results. Here, a few comments on the model and results are listed in order. First of all, the value of percolation threshold of nanoparticles in our simulations is higher than that obtained from other methods. For instance, the value of percolation threshold of nanoparticles in the work of Balazs’s group is 0.09,31 which is lower than the value of 0.19 in our predictions. Such numerical discrepancy originates from the following aspects: The definition of the percolating network of nanoparticles is

Figure 9. Contour map of synergistic strength Δ = P0C − PPC in the plane of the aspect ratio α of nanoparticles and the composition fA of copolymers. The representations of lines and color bar are the same as those of Figure 6. The effective radius of nanoparticles is set as RP = 0.50Rg.

synergistic strength between nanoparticles and copolymers changes slightly. However, the percolation threshold of nanoparticles has a further decrease. For instance, the spherical nanoparticles have a percolation threshold PPC = 0.19. As the aspect ratio of nanoparticles has a value of α = 7.0, the value is decreased to PPC = 0.12. These findings provide a useful guideline to lower the loading of nanoparticles for the achievement of high-performance composite materials with light weight (i.e., an increase of aspect ratio of nanoparticles is able to further reduce their percolation threshold in the host of diblock copolymers). It should be pointed out that the interparticle potential plays an important role in the self-assembly of block copolymer/ nanorod mixtures. Figure S7 of the SI shows the effect of interparticle potential (including Lennard−Jones, exponential decay and angle-dependent potentials) on the arrangement and percolation threshold of nanorods. The Lennard-Jones and exponential decay potentials of nanorods favor the side-by-side arrangement, leading to higher percolation thresholds. As the interaction of nanorods is modeled by the angle-dependent potentials,57 the nanorods mostly adopt the end-to-end arrangement in the host of polymers, which promotes the formation of percolating network and results in a lower percolation threshold. Detailed investigation based on the HPF method will be devoted to thoroughly elucidating the generality of nanoparticle arrangement in terms of the interparticle potential. 5. Discussion. In our simulations, it is confirmed that the host of block copolymers can be employed to enhance the formation of percolating network due to the introduction of inhomogeneity from the microphase separation (Figure 2). Our computational consideration of anisotropic nanoparticles also predicts that the nanoparticles with higher aspect ratio have the capability to further lower their percolation threshold (Figure 8). Over the past decade, several groups independently developed the modeling of block copolymer/nanoparticle mixtures to probe into the percolating behavior of nanoparticles.31−33 For instance, Balazs’s group combined the cell dynamics simulations with Langevin dynamics to investigate the clustering of nanoparticles in the host of block copolymers.31 It is found that the percolation threshold of nanoparticles is reduced by a factor of 2 if the diblock copolymers are used as a host instead of the homopolymers. Their simulations also revealed that the anisotropy of nanoparticles plays a significant role in engineering the doubly percolating networks of block copolymers and nanoparticles. These observations are in general agreement with the results 23712

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considerably strict in our work. An infinite network of nanoparticles is regarded as the percolating structures, instead of the formation of a pathway from one side to the opposite side. The other aspect is that the criterion for the connection of different nanoparticles is rigorous in our work.58,59 As the minimum distance between lattice sites of nanoparticles is less than one lattice unit, the nanoparticles are regarded as the same cluster. The above two aspects contribute to a high value of the percolation threshold in our simulations. Second, the volume fraction of nanoparticles in our simulations is much larger than the typical amount of nanoparticles in experiments, where the aggregation of nanoparticles occurs.60−64 As a representative example, when the volume fraction of nanoparticles exceeds 0.2, the nanoparticles form large clusters due to their high interfacial energy, and macrophase separation between nanoparticle clusters and block copolymers is identified.60 However, because of strong repulsions of nanoparticles sequestered in the favored domains, the nanoparticles have the capability to disperse in the microphase-separated nanostructures of block copolymers. From the experimentalist’s point of view, this computational observation provides useful information to improve the dispersion of nanoparticles in the polymer matrix via surface functionalization. For example, under the case of high amount of additions, the nanorods grafted by homopolymers are well dispersed in the polymer matrix via rationally designing the relative length of polymer chains.65 Third, it is demonstrated that the percolation threshold has a tight relation with the dimension of simulation cells.66 However, it should be mentioned that the computational intensity of the hybrid particle-field method in three-dimensional (3D) space is huge, owing to incorporation of particle’s motion into the field-theoretic framework of inhomogeneous fluids. In consideration of computational power, it is currently difficult to perform the 3D simulations with large cell size. Yet the computational findings in the two-dimensional space provide useful guidelines for rationally designing the nanocomposites with light weight (i.e., there exists an optimal composition of block copolymers, which yields maximum difference of percolation thresholds of nanoparticles embedded in block copolymers and homopolymers).

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Article

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.7b07337. Model of nanoparticles and additional images of selfassembled nanostructures and cluster distributions (PDF)

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (L. Zhang). *E-mail: [email protected] (J. Lin). ORCID

Liangshun Zhang: 0000-0002-0182-7486 Jiaping Lin: 0000-0001-9633-4483 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (21574040 and 21234002) and the Key Laboratory of Advanced Polymer Materials of Shanghai (ZD20170201).

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REFERENCES

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CONCLUSIONS

In summary, we combine the hybrid particle-field method and the cluster multiple labeling technique to probe into the selfassembled nanostructures and the percolating behavior of diblock copolymer/nanoparticle composites. Our simulated results demonstrate that the nanoparticles are guided to locate in the selective domains of self-assembled nanostructures of diblock copolymers. Depending on the copolymer composition, the diblock copolymers have the capability to lower the percolation threshold of nanoparticles due to the incorporation of confined environment. More importantly, there exists an optimal value of the copolymer composition for the lowest loading of nanoparticles to form the percolating network. In addition, the percolation threshold of nanoparticles in the host of diblock copolymers is further reduced by modulating the geometrical parameters of nanoparticles, such as their radius and aspect ratio. 23713

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