Monte Carlo Study of Degenerate Behavior of AB Diblock Copolymer

Jul 26, 2016 - Degenerate behavior (i.e., forming different self-assembled structures for a given block copolymer (BCP) under the same confinement) ...
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Monte Carlo Study of Degenerate Behavior of AB Diblock Copolymer/Nanoparticle under Cylindrical Confinement Yingying Wang,†,‡,§ Yuanyuan Han,*,† Jie Cui,*,† Wei Jiang,† and Yingchun Sun‡ †

State Key Laboratory of Polymer Physics and Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun 130022, Jilin, China ‡ School of Physics, Northeast Normal University, Changchun 130024, Jilin, China § University of Chinese Academy of Sciences, Beijing 100049, People’s Republic of China S Supporting Information *

ABSTRACT: Degenerate behavior (i.e., forming different self-assembled structures for a given block copolymer (BCP) under the same confinement) commonly exists in various confined systems. Understanding degenerate behavior is crucial for precise control over the structures formed by selfassembly systems under confinement. In this study, the degenerate behavior of a self-assembled AB diblock copolymer/nanoparticle (NP) mixture in a cylindrical pore is studied using Monte Carlo simulation. We find that the degenerate behavior of such a mixture depends on the introduction of the NP. Under different pore sizes, four typical degenerate structures [i.e., single helices (S-helices), double helices (Dhelices), parallel cylinders, and stacked toroids] can be obtained if the NP content is zero. However, when the NP content in the mixture is increased, it is found that the number of degenerate structures decreases, that is, only blocky structures can be obtained in the case of high NP content. Moreover, the probability of forming S-helices decreases, whereas the probability of forming Dhelices increases with increase in the NP content. Analysis of the interactive enthalpy densities and the chain conformation of the systems indicates that entropy plays an important role in the degenerate structure formation. This study provides some new insights into the degenerate behavior of a BCP/NP mixture under confinement, which can offer a theoretical reference for further experiments.

1. INTRODUCTION It is known that under a confined environment, pure block copolymer (BCP) exhibits rich self-assembly behaviors that are quite different from those in bulk, forming various novel and controllable self-assembled structures with long-range order.1−5 On the other hand, highly ordered nanoparticle (NP) distributions are required to achieve additional interesting properties, such as surface-enhanced plasmon resonance and fluorescence resonance energy transfer.6−9 Materials fabricated using BCP/NP mixtures possess both the controllability of the structures endowed by BCPs and the unique properties endowed by NPs. Therefore, using BCPs as structure-directing agents to induce highly ordered distributions of NPs under various confined conditions has become an effective way to fabricate materials with novel properties and functionalities. In recent years, a lot of efforts have been devoted to the study of the self-assembly behavior of the BCP/NP mixture in a thin film,10−20 in a cylindrical nanopore,21−26 and in other types of confinements.27,28 On the one hand, researchers expect to take advantage of the self-assembly of BCPs under confinement to induce the ordered spacial distribution of the NPs. On the other hand, it has been shown in previous works that with an increase in the NP content, the introduction of NPs can in turn © XXXX American Chemical Society

induce changes in the spacial orientation or even in the nanostructures of the self-assembled microdomains,16−18,29,30 meaning that NPs can have considerable effects on the selfassembly behavior of BCPs. The NPs can play an active role in the self-assembly process of the BCP/NP mixture, rather than being passively introduced into the microdomains selfassembled by BCP. Therefore, one of the purposes of previous studies was to find the relationships between the various controlling parameters (including the size, content, and surface chemistry of NPs) and the novel nanostructures, so that at a given condition, desired and specific nanostructures can be obtained through a rational design of the constituent of the BCP/NP mixture and the confining condition. However, during the investigations on the self-assembly behavior of pure BCPs under confinement, it has been found that degenerate structures (i.e., different self-assembled structures for a given BCP obtained under the same confinement) were frequently observed under some confined conditions.31−36 For instance, a set of degenerate structures Received: March 20, 2016 Revised: July 22, 2016

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DOI: 10.1021/acs.langmuir.6b01090 Langmuir XXXX, XXX, XXX−XXX

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will decrease with the increase in the NP size, which will lead to the macrophase separation between the diblock copolymers and the NPs with larger size at a relatively lower NP content. To incorporate as many NPs as possible into the microdomains formed by AB diblock copolymers, the cubic particle with the smallest size (i.e., a cube with a side length of 1 lattice space) was chosen in the current work. In addition, the concentration (lattice sites occupied by all the BCPs and cubic NPs) of the system is 70%, and the remaining 30% lattice sites are vacancies. The evolution of the composite system is achieved through the exchange move between a monomer (or NP) and a vacancy. Each step of the motion is generated through a micro-relaxation model, which has proven to be highly efficient in relaxing a local chain conformation on the lattice.40−42 The micro-relaxation modes are defined as follows: A monomer (or NP) is randomly chosen to exchange with one of its 18 nearest-neighbors (one of the 6 nearest-neighbor sites for the NP). If the neighbor is a vacancy, an exchange with the monomer (or NP) is attempted. If the exchange does not violate the bond length restriction implemented on the polymer chains, the exchange is allowed. This process constitutes a single movement. If the exchange would break two chain connections, it is disallowed. If the exchange creates a single break in the chain, the vacancy will continue to exchange with the subsequent monomers along the chain until reconnection of the links occurs. The acceptance or rejection of the attempted move is further governed by the Metropolis rule:43 If the energy change ΔE is negative, the exchange is accepted; otherwise, the exchange is accepted with a probability of p = exp[−ΔE/(kBT)], where ΔE = ∑ijΔNijεij is the energy change caused by the attempted move; ΔNij is the number difference of the nearest-neighbor pairs between the components i and j before and after the movement, where i, j = A, B, P (the NP), or W (the pore wall); εij is the interaction intensity between i and j; and kB is the Boltzmann constant and assumed to be 1 in the whole simulation. The parameter T is the temperature. In the current work, the incompatibility between the blocks A and B is represented by setting εAB = 1. The interactions between different blocks and NPs are respectively set to be εAP = 0 and εBP = 1, which means that the NPs are compatible with the block A but incompatible with the block B. To obtain structures in equilibrium state, the simulated annealing MC strategy, which has been widely used by Li’s group to obtain the lowest-energy “ground state” and has been successfully applied in the study of the self-assembly of BCP/NP composite system, is employed.12,32,44 The initial states of all the composite systems are homogeneous caused by running the systems at a very high temperature. Then the usual annealing schedule, Tj+1 = f Tj, is used to equilibrate homogeneous disordered systems. Tj is the temperature used in the jth annealing step, and f is the scaling factor. The annealing procedure begins from T0 to a given lower temperature TF. We set f = 0.99, T0 = 10, and the temperature reaches the final TF after 400 annealing steps. At each annealing step, 10 000 Monte Carlo steps (MCSs) are performed. Here, 1 MCS means that on average all the monomers and NPs have one trial move.

[single helices (S-helices), stacked toroids (S-toroids), and double helices (D-helices)] have been found by Yu et al. at a certain pore size under cylindrical confinement.31 Later, it was found that the high symmetry of a spherical pore can enable more complicated multiple-degenerate structures.36 The probabilities of occurrence for these degenerate structures are different. The existence of the degenerate phenomenon implies that in experiments, we cannot repeatedly obtain the same results at a given experimental condition. As for the BCP/NP mixture system, the effect of NPs on the degenerate behavior under confinement has seldom been reported up to now.21 Some problems still remain unclear. For example, what is the effect of the NP content on the degenerate structures formed by BCP/NP mixture under confinement? What is the effect of the NPs on the chain conformation and orientation? So far, the effect of the introduction of NPs on the degenerate behavior of the BCP/NP mixture under confinement has still been far from being well understood, which hampers a deep understanding of the degenerate behavior and limits the full use of the novel materials fabricated by the BCP/NP mixture. Herein, we performed systematic Monte Carlo (MC) simulations under cylindrical confinement to investigate the effect of NP content on the degenerate behavior of the AB diblock copolymer/NP mixture. Four degenerate nanostructures formed by AB diblock copolymer/NP mixtures were observed under different cylindrical pore sizes. A morphological phase diagram showing the variations of the degenerate structures of the AB diblock copolymer/NP mixture with the NP content under different pore sizes has been provided. The interactive enthalpy density, the chain conformation (measured by the mean-square end-to-end distance of the polymer chains), and the spacial orientation of the polymer chains in different structures were also studied under different pore sizes and different NP contents to deeply understand the physics behind the degenerate behavior existing in the AB diblock copolymer/NP mixture system under cylindrical confinement.

2. MODEL AND METHOD The lattice Monte Carlo simulation method, as proposed by Larson et al., is used in this study.37−39 The system is embedded in a simple cubic lattice with volume V = Lx × Ly × Lz. For bulk cases, periodic boundary conditions are imposed in all three directions. For cylindrical confinement, a periodic boundary condition is imposed only along the z direction (i.e., the long axis of the cylindrical pore is along the z direction). Lx and Ly are set equal and are chosen to be at least 2 lattice spaces larger than the pore diameter D. The pore axis is located at xa = Lx/2 and ya = Ly/2. The cylindrical pore is constructed by all the lattice sites within a distance of D/2 to the pore axis. Lattice sites outside the pore constitute the pore wall, which cannot be occupied by polymers and NPs. The ANfABN(1−fA) diblock copolymer/NP composite with varying NP content [measured by the volume fraction of NP ( f P) in the composite] is loaded in the cylindrical pore, where N is the total chain length and fA is the chain length ratio of block A. As for the polymer chains, each monomer occupies only one lattice site, and the monomers are self-avoiding and mutually avoiding, which ensures that only one monomer occupies a single lattice site. The bond length permitted in our simulation is confined between 1 and 2 . Thus, each lattice site has 18 nearest-neighbor sites in a three-dimensional space. As for the NPs, each NP occupies 8 lattices, forming a small cube with a side length of 1 lattice space.25 Each NP has 6 nearest-neighbor sites because each NP is allowed to move only 1 lattice space. Similar to the chain monomer, the NPs are mutually avoiding. Note that according to the work of Balazs’s group,25 the loading capacity of the microdomains formed by the AB diblock copolymers for particles

3. RESULTS AND DISCUSSION In this paper, bulk cylinder-forming diblock copolymers are selected to mix with NPs for obtaining a diblock copolymer/ NP mixture. According to the literature, hexagonally packed cylinders can be formed by ANfABN(1−fA) diblock copolymers in bulk, in which fA ranges from 1/4 to 1/6.31,32 Hence, the diblock copolymer A5B15 (i.e., fA = 1/4 and N = 20) is selected. The purpose of selecting relatively larger fA and longer chain length N is to incorporate as many NPs as possible into the microdomains formed by the minority block A. 3.1. Self-Assembly Behavior of Pure AB Diblock Copolymers in a Cylindrical Pore. First of all, the selfassembly and degenerate behaviors of the pure A5B15 diblock copolymers in a cylindrical pore are studied at the NP content f P = 0 in this subsection. The hexagonally packed cylinders B

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that have equal total free energies (including contributions from entropy and enthalpy). The free energy for a polymer system can be easily investigated using field-based simulation methods.45−48 However, for MC simulations, it is difficult to calculate the entropy contribution to the total free energy of the systems. Therefore, we considered only the interactive enthalpy of each structure in this work. Taking the case of D/L0 = 1.33 and f P = 0 as an example (Figure 1d), three structures, that is, S-helices, D-helices, and S-toroids, are obtained. Their interactive enthalpy densities are respectively 1.035, 1.032, and 1.035, meaning that these structures have almost equal interactive enthalpies. Therefore, these structures are considered to be energetically degenerate.49 From the top and side views of the pores shown in Figure 1, we can see that all the nanostructures formed by block A (blue) distribute around the domains formed by block B (green), and only one A-monolayer is formed under all pore sizes. Four typical structures of A-monolayer are observed in Figure 1, that is, S-helices, parallel cylinders (P-cylinders), D-helices, and Storoids. Among the various structures shown in Figure 1, Shelices can be obtained under all pore sizes used in the current paper, whereas D-helices can be seen under most of the pore sizes, except for the smallest one (i.e., D/L0 = 0.95). In addition to the single- and double-helix structures, P-cylinders can be seen under smaller pore sizes (D/L0 = 0.95 and 1.20), whereas S-toroids can be observed under larger pore sizes (D/L0 = 1.33 and 1.46). Interestingly, from Figure 1, we find that in our simulations, the degenerate phenomenon occurs under all pore sizes, which is different from the results reported by Yu et al.31,32 This is probably because the chain length used in the current work (N = 20) is longer than that used in their work. Moreover, because calculating the entropic contribution to the total free energy of the system is difficult from the MC simulation, we cannot predict whether these structures are stable or metastable from the angle of total free energy. However, we can ensure that the structures are all at their equilibrium states because the structures obtained after annealing do not change with time any more. 3.2. Effect of the NP Content on the Degenerate Behavior. In this section, we incorporate NPs into each system shown in Figure 1. The volume fraction of the NP ( f P) in the BCP/NP mixture is the main factor that we focus on, which varies from 1% to 15% in each system (the whole volume fraction of the BCP/NP mixture in each cylindrical pore remains unchanged, i.e., 70%). In this section, the pore size is also denoted by D/L0, where L0 is the equilibrium period of the hexagonal cylinders formed by pure diblock copolymers. In fact, the period for a specific bulk phase (e.g., cylinders), even the bulk phase itself, can be modified by the NP content. This means that different nanoparticle contents may correspond to different bulk structures that have individual equilibrium periods L0. If we do not adopt a unified L0, a pore with larger diameter D will probably have a smaller D/L0, which will cause confusion in expressing the real pore size. Therefore, the pore size is usually denoted by the reduced form of D/L0, in which L0 is the equilibrium period of pure BCP in bulk.12,22 3.2.1. Morphologies under Confinement. In this section, the effect of f P on the degenerate behavior under different pore sizes is elucidated from the angle of structural changes. A morphological phase diagram as a function of the NP content ( f P) and the pore size (D/L0) is given in Figure 2. The phase diagram can be roughly divided into two regions, which are separated by the dashed line shown in Figure 2. One is the

formed by the A5B15 diblock copolymers in bulk condition are obtained (Figure S1a). Then, an important parameter, the equilibrium repeating period L0 of the cylinders in bulk (Figure S1b), is calculated. Note that L0 is averaged by many independent simulations with different initial states (or simulation box sizes), and the value of L0 is approximately equal to 15.78 lattice spacing. In addition, it is reported that the pore length Lz can affect the structures formed by the AB diblock copolymers, as well as the period and stability of these structures in a cylindrical pore.31,45,46 In this paper, if not specified, the pore lengths Lz are set as 80, and other pore lengths have also been used for illustrating the effect of pore length on the self-assembly behavior of the BCP/NP mixture (section 3.2.1). Various novel nanostructures (e.g., S-helices, D-helices, and S-toroids) have been observed in the past decade. According to the literature, it can be seen that with the increase in the pore size (measured by the ratio D/L0, where D is the pore diameter), the confinement-induced nanostructure becomes more and more complicated. These complicated morphologies can be classified in terms of the number of A-monolayers,34 that is, the larger the pore size is, the more A-monolayers will be formed and more computing times will be needed. Because our attention is focused on the degenerate behavior, it is appropriate to choose a model system that is as simple as possible. Therefore, in this paper, our studies are restricted to systems with a small pore size, in which only one A-monolayer is formed around the pore axis. On the other hand, the boundary−polymer interaction can also affect the confinementinduced nanostructures. However, Yu et al. has pointed out that the trends or sequences of the morphological transition with the pore size are generally similar under different boundary conditions such as selective boundary and neutral boundary. According to their results, to obtain similar structures, pore sizes with neutral boundary are usually smaller than those with selective boundaries.31,32 Therefore, in the current paper, for the purpose of improving the computational efficiency, only neutral boundary condition (i.e., εiW = 0, i = A, B, or P) is considered, and the pore size with the neutral boundary ranges from 0.95 to 1.46 to obtain one A-monolayer. Simulation results of the pure A5B15 diblock copolymers (without NPs) in the cylindrical neutral pores with different sizes are given in Figure 1. It should be noted that the most

Figure 1. Nanostructures self-assembled by pure A5B15 diblock copolymers in cylindrical pores as a function of the pore size (D/ L0) under neutral boundary condition. The minority block A and majority block B are represented by blue and green, respectively.

effective way of searching degenerate structures is to increase the number of samples. Therefore, more than 100 independent simulations with different initial states under each pore size are performed to search for as many degenerate states as possible. To determine whether the structures obtained under the same condition are degenerate, the interactive enthalpies of different structures under each pore size are calculated. According to the angle of thermodynamics, degenerate structures are structures C

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Figure 2. Morphological phase diagram showing the effect of the NP content ( f P) and the pore size (D/L0) on the degenerate behavior of the BCP/ NP mixture under cylindrical confinement. The dashed line is a guide for eyes to roughly distinguish the transition region between the top-left degenerate region and the bottom-right nondegenerate region. The minority block A, majority block B, and NPs are represented in blue, green, and yellow, respectively.

degenerate behavior disappears with further increase in the NP content, leaving only the blocky structure in the pore (Figure 2c7−c9). More interestingly, in larger pores (D/L0 = 1.33 and 1.46), the increase in the NP content not only decreases the amount of the degenerate structures but also changes the degenerate structures. Taking the case of D/L0 = 1.33 as an example, when the NP content is increased from 3% to 5%, the S-helices and the S-toroids (Figure 2d3) are replaced by the Pcylinders (Figure 2d4). When the NP content is as high as 10%, only D-helices (Figure 2d6) can be obtained in the simulation. In addition, in the case of larger pores (D/L0 = 1.33 and 1.46), disordered structures with broken twisted cylinders appear with further increase in the NP content to f P ≥ 13%. Typical disordered states are shown in Figure S2, and it can be found that those structures are quite similar to the D-helices. The reason why a disordered state appears instead of D-helices is that the amount of BCPs is rather low when the NP content is rather high, and it is difficult for such a small amount of BCPs to form continuous ordered structures such as D-helices in a relatively large space. Figure 2 shows that in the degenerate region, the NP contents have a significant effect on the formation of degenerate structures. Actually, we can see that adding a small amount of NPs (e.g., 1%) does not change the degenerate behavior of the system under each pore size, indicating that each degenerate structure obtained at f P = 0 has, to some extent, the ability to accommodate NPs. Obviously, the ability to accommodate NPs is limited. For instance, when the NP content is increased to 5% at D/L0 = 1.20, the S-helices and the P-cylinders disappear (Figure 2c4), leaving only the Dhelices in the system. This is because, under this circumstance, the S-helices and the P-cylinders cannot accommodate more NPs. On the other hand, it can be seen that when D/L0 = 1.33, a new degenerate structure (i.e., P-cylinders) emerges, and the S-helices and the S-toroids disappear with the increase in the NP content (Figure 2d4). This implies that the ability of Pcylinders to accommodate NPs is greater than that of the S-

degenerate region (the top-left region), in which the degenerate behavior of the BCP/NP mixture is strongly influenced by the NPs. The other is the nondegenerate region (the bottom-right region), in which the systems are either in a single ordered state (i.e., an array of discrete blocky structures arranged in a zigzag pattern) or in disordered states. From Figure 2, it can be seen that the blocky structures will ultimately be obtained with the increase in the NP content f P for most pore sizes, forming the nondegenerate region. This is because with the increase in f P, the amount of AB diblock copolymers in the pore will decrease to a certain level, below which the amount of the minority block A is no longer enough to form a continuous structure, and hence only isolated blocky structures can be formed. Unlike in the nondegenerate region, the phase behavior of the BCP/NP mixture in the degenerate region is much more diverse with varying NP content. From the degenerate region shown in Figure 2, it can be found that the influence of the NPs on the degenerate structures is very strong in the case of small pore sizes. For instance, in the smallest pore (D/L0 = 0.95), the degenerate behavior of the system can be easily influenced by incorporating only a small amount of NPs, that is, adding 1% of NPs results in the emergence of the transition structure (the blocky structure shown in Figure 2a2). Continuous increase in the NP content will lead to the disappearance of the S-helices and the P-cylinders, leaving only the blocky structure in the pore (Figure 2a3−a8). A similar phenomenon can also be observed when D/L0 = 1.08; only the NP content for the emergence of the blocky structure is larger than that in the case of D/L0 = 0.95. When the pore size is further increased to D/L0 = 1.20, before the blocky structure emerges, the degenerate structures (i.e., the S-helices and the P-cylinders) first disappear with the increase in the NP content to f P = 5%, leaving only Dhelices in the pore and thus leading to the disappearance of the degenerate behavior (Figure 2c4). However, with further increase in the NP content up to f P = 8%, the transition structure (Figure 2c6) emerges again. And after that, the D

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Figure 3. Variations in the occurrence probabilities of S-helices (a) and D-helices (b) with the NP content (f P) under different pore sizes (D/L0), (c) phase diagram as a function of f P and D/L0 showing the regions in which the probabilities of obtaining S-helices and D-helices are higher than 60%.

Figure 4. Variations of the x-, y-, and z-components of the mean-square end-to-end distances ⟨R2⟩ of AB diblock copolymer chains with NP content f P in the cases of forming S-helices (a) and D-helices (b).

easier to obtain at every region where they emerge? For this question, statistical studies are performed to give the probabilities of obtaining S-helices and D-helices among the degenerate structures formed at each set of NP content and pore size shown in Figure 2. It is seen from Figure 3a that in the smallest pore (D/L0 = 0.95), although an increase (from 0% to 1%) in the NP content can slightly increase the probability of obtaining S-helices, further increase in the NP content leads to an abrupt decrease in the probability of obtaining S-helices. In other larger pore sizes, it can be seen that increasing the NP content either decreases the probability of obtaining S-helices (D/L0 = 1.08, 1.33, and 1.46) or have almost no effect on obtaining S-helices (D/L0 = 1.20, in which the probability of obtaining S-helices is kept very low). In general, Figure 3a indicates that increasing the NP content is not favorable for obtaining S-helices. However, it can be seen from Figure 3b that under all pore sizes, the probabilities of obtaining D-helices first increase and then decrease abruptly with a continuous increase in the NP content. In smaller pores (D/L0 = 1.08 and 1.20), because the probabilities of obtaining D-helices are already high when f P = 0 (i.e., larger than 80%), the increase in the probabilities of obtaining D-helices is relatively small when the NP content is increased. On the contrary, in larger pores (D/L0 = 1.33), the probability of obtaining D-helices is much lower than that in smaller pores when the NP content is very low. A considerable increase in the probability of obtaining Dhelices is observed when the NP content is increased. Figure 3b indicates that there exists a threshold in the NP content. Before the threshold, increasing the NP content will be beneficial for obtaining D-helices. After the threshold, although D-helices containing large amounts of NPs can still be obtained, the probability of obtaining such D-helices will remarkably decrease. In fact, Figure 3a,b indicates that the probabilities of obtaining S-helices and D-helices are different at different

helices and the S-toroids. The difference in the ability of different structures to accommodate NPs is the main reason why the NPs have a significant effect on the formation of degenerate structures. In addition, we also performed simulations under different pore lengths to test the effect of pore length Lz on the selfassembly behaviors. Four typical systems in the phase diagram shown in Figure 2 were selected (i.e., systems with D/L0 = 1.33 and f P = 0, D/L0 = 1.33 and f P = 0.05, D/L0 = 1.20 and f P = 0, and D/L0 = 1.20 and f P = 0.05). The pore lengths of these four systems were varied from 48 to 76 with an increment of 4 (i.e., Lz/L0 varies from 3.04 to 4.82). For each system, 30 parallel simulations with different initial states were run. The simulation results are provided in Figure S3. In general, Figure S3 shows that the structures obtained under different pore lengths are almost the same as those obtained in the case of Lz = 80. No new structures are formed by varying the pore length. The simulation results indicate that the pore length has less effect on the confined structure. Furthermore, the interactive enthalpy densities of the structures obtained under different pore lengths (Figure S3) were calculated. The interactive enthalpy densities of the D-helices shown in Figure S3 are given in Tables S1 and S2 as an example. It can be seen that the interactive enthalpy densities of the D-helices obtained under different pore lengths are quite similar. Therefore, the simulation results shown in Figure S3 and Tables S1 and S2 suggest that changing the pore length cannot significantly change the confined structures and their interactive enthalpy densities. From Figure 2 it can also be found that among the four typical structures shown in Figure 1, S-helices and D-helices (which are believed to have more potential in applications such as optical and electronic devices5) can be obtained in a large area relative to the other structures in the degenerate region. Does this phenomenon mean that S-helices and D-helices are E

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Langmuir regions in the phase diagram shown in Figure 2. To show the regions in which S-helices and D-helices are easier to obtain, we provide a phase diagram as a function of the NP content and the pore size showing the region in which the probabilities of obtaining S-helices and D-helices are higher than 60% in Figure 3c, respectively. Figure 3c indicates that S-helices are easier to obtain in the case of lower NP content and larger pore sizes, whereas it can be seen that in the case of larger NP content and larger pore size, D-helices are easier to obtain. Moreover, it is seen that the area for easily obtaining D-helices in the phase diagram is apparently larger than that for S-helices, implying that D-helices may be the structure that can be obtained more frequently in experiments. To deeply understand the phenomenon observed in Figure 3c at the micro-scale level, we calculate the three components of the mean-square end-to-end distance ⟨R2⟩ of the diblock copolymer chains forming S-helices and D-helices under different pore sizes and NP contents. The results obtained at D/L0 = 1.33 are shown in Figure 4. It can be seen that the zcomponent of the mean-square end-to-end distance (⟨Rz2⟩) is always apparently larger than the x- and y-components (⟨Rx2⟩ and ⟨Ry2⟩) in the case of S-helices at each NP content (Figure 4a). However, for D-helices, the values of the three components (⟨Rx2⟩, ⟨Ry2⟩, and ⟨Rz2⟩) are very close to each other (Figure 4b). This phenomenon indicates that the chain conformation in the D-helices is roughly isotropic.31 Interestingly, an increase in the NP content does not destroy this isotropy of the polymer chains, instead making the polymer chains even more isotropic, that is, the values of ⟨Rx2⟩, ⟨Ry2⟩, and ⟨Rz2⟩ are almost the same when the NP content is high. Figure 4 reveals that at a micro-scale, the chain conformation in the D-helices is more isotropic than that in S-helices. On the other hand, from the angle of self-assembly structure, D-helices have twofold rotational symmetry with respect to the pore axis, whereas S-helices have no such symmetry. Therefore, we think that the isotropy in the micro-scale chain conformation and the rotational symmetry of the self-assembly structure increases the ability of the D-helices to accommodate NPs; hence, D-helices can be obtained in a larger area shown in Figure 3c. 3.2.2. Interactive Enthalpy Density and Chain Conformation. From a morphology angle, the simulation results in subsection 3.2.1 show the effect of the NP content on the degenerate self-assembly behavior of the BCP/NP mixture under different pore sizes. On the other hand, we know that incorporation of NPs into the polymer system is generally associated with changes in entropy and enthalpy.50 To gain deeper insights into the effect of the NP content on the degenerate behavior, we first computed the interactive enthalpy densities for the systems in the degenerate region shown in Figure 2. In the present simulation, the interactive enthalpy comes from two sources: (1) the repulsive interaction between monomer A and monomer B and (2) the repulsive interaction between monomer B and the NP. Thus, the interactive enthalpy can be calculated by F = (NABεAB + NBPεBP)/Nch, where Nij are the interactive pairs between components i and j and εij is the interactive intensity between components i and j. Nch is the chain number in each system. Under each pore size and NP content in the degenerate region shown in Figure 2, we computed the interactive enthalpy densities (averaged by all the ordered structures) for each system on the basis of a large number of independent simulations. The statistical results are given in Table 1. We can see that the values of the interactive enthalpy density shown in Table 1 are quite similar, that is,

Table 1. Average Interactive Enthalpy Densities under Different Pore Sizes (D/L0) and NP Contents (f P) NP content ( f P) pore size (D/L0)

0.00

0.01

0.03

0.05

0.07

0.10

1.46 1.33 1.20 1.08 0.95

1.032 1.031 1.029 1.035 1.027

1.031 1.032 1.048 1.032 1.029

1.030 1.028 1.029 1.032

1.026 1.025 1.026 1.022

1.022 1.025 1.027

1.016 1.023

almost all the values shown in Table 1 are around 1.029, and the relative deviations of these values to 1.029 are within the range of about 0.7%. This phenomenon indicates that the interactive enthalpy densities almost remain unchanged when the NP content and the pore size are changed, which reveals that the interactive enthalpy density of the system basically does not depend on the NP content and the pore size. Hence, it can be deduced that entropy may result in the variations of the degenerate behavior of the BCP/NP mixture. It is known that the chain conformation is related to the chain conformational entropy. Therefore, we investigated the effects of the pore size and the NP content on the chain conformation for the purpose of qualitatively analyzing the conformational entropy in each system. To this end, we computed the mean-square end-to-end distance (⟨R2⟩) of the AB diblock copolymer chains under each pore size and NP content shown in the degenerate region in Figure 2 (the topleft region). For comparison with the chain conformation in bulk, the mean-square end-to-end distance is defined as the ratio r2 = ⟨R2⟩/⟨RB2⟩, where ⟨RB2⟩ is the mean-square end-toend distance of the AB diblock copolymer chains in bulk cylindrical phase. According to the definition of r2, it is apparent that r2 ≈ 1 means that the conformation of the polymer chains under confinement is similar to that in bulk, whereas r2 > 1 means that the polymer chains under confinement are more stretched than those in bulk. In addition, it is noticed that the degenerate structures obtained under the same NP content and pore size have similar values of the mean-square end-to-end distance. Therefore, we did not distinguish the degenerate structures at each point shown in the degenerate region (Figure 2), that is, the average values of the mean-square end-to-end distances of all degenerate structures at each set of f P and D/L0 shown in Figure 2 are calculated and shown in Figure 5. In general, Figure 5 shows that the effect of the NP content on r2 depends on the pore size. When the pore size is D/L0 ≤ 1.20 (close to integer 1), it can be seen that the values of r2 remain almost unchanged with the increase in the NP content. Meanwhile, it is found that the values of r2 are close to 1. This is because the value of D/L0 is close to integer 1, which means that the pore size is commensurate with the bulk equilibrium repeating period L0. In this case, the mean-square end-to-end distance (i.e., chain conformation) of the polymer chains in a cylindrical confinement is quite similar to that in bulk, which results in r2 ≈ 1. The variations in r2 with f P under the pore size of D/L0 ≤ 1.20 indicate that the incorporation of the NPs has almost no effect on the chain conformation when the pore size is commensurate with the bulk equilibrium repeating period L0. However, when the pore size is increased to D/L0 ≥ 1.33, the values of r2 with different f P are all apparently larger than 1. This is because the values of D/L0 are close to half-integer 1.5 in this case, which means that the pore size is the most incommensurate with the bulk equilibrium repeating period L0. F

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conformation in each system shown in the degenerate region in Figure 2 has apparent changes with the change in the NP content or the pore size. Therefore, it can be concluded that entropy plays an important role in the variations of the degenerate behavior shown in Figure 2. 3.2.3. Average Chain Orientation. In this subsection, to further understand the effect of the NP content and the pore size on the degenerate behavior shown in Figure 2 at the microscale level, we studied the effect of the NP content and the pore size on the orientation of the polymer chains in different degenerate structures because the chain orientations are usually in close relationships with the properties of materials (e.g., the mechanical, optical, and transmission properties, etc.).51,52 Therefore, it is important to reveal the effects of various factors (such as the self-assembled structure, pore size, and NP content) on the chain orientation. To reveal the orientation of the AB diblock copolymer chains in different structures, we studied the distribution of θ (i.e., P(θ)) under each pore size and NP content shown in Figure 2, where θ is defined as the angle of orientation of the end-to-end distance vector of the entire chain (i.e., the angle between the end-to-end distance vector and the central axis of the pore as shown in Figure 6a). First, under different NP contents, we studied the effect of the pore size on the distributions of the angle of orientation for all the degenerate structures shown in Figure 2. Taking f P = 0 as an example, Figure 6b−e shows the distribution curves of θ for different degenerate structures obtained under different pore sizes. Typical morphological structures corresponding to each distribution curve are also given in the corresponding figures. Figure 6b shows the distribution curves of θ for the S-helices obtained under different pore sizes. It is seen, in general, that the polymer chains have a wide distribution of θ in the Shelices. The angles corresponding to the peaks on the

Figure 5. Variations in the mean-square end-to-end distances r2 for the AB diblock copolymer chains with NP content (f P) under different pore sizes (D/L0). The NP contents above which the degenerate behavior disappears under different pore sizes are denoted by the vertical dashed lines.

The value of r2 > 1 means that the polymer chains in the case of D/L0 ≥ 1.33 are more stretched (which leads to a decrease in the conformational entropy) than those in bulk. More importantly, Figure 5 shows that when the pore size is large (D/L0 ≥ 1.33) and close to half-integer 1.5, the value of r2 decreases with the increase in the NP content f P, which results in the value of r2 being close to 1. This means that the meansquare end-to-end distance of the polymer chains tends to be the same as that in bulk, indicating that the incorporation of the NPs is favorable for the chain relaxation when the pore size is incommensurate with the bulk equilibrium repeating period L0. From the simulation results shown in Table 1 and Figure 5, it can be seen that the interactive enthalpy densities of each system are almost equal. On the other hand, the chain

Figure 6. (a) Schematic of the chain orientation angle θ, (b)−(e) distribution curves of the orientation angle θ for different degenerate structures obtained under different pore sizes (D/L0) when the NP content is f P = 0. G

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Figure 7. Distribution curves of the orientation angle θ for different degenerate structures obtained under different NP contents ( f P) when the pore size is D/L0 = 1.33.

distribution curves are all less than 45°, which means that most polymer chains in the S-helices tend to orient along the pore axis. However, in the D-helices (Figure 6c), although the polymer chains have a wide distribution of θ, the angles corresponding to the peaks on the distribution curves are around 60°, meaning that most polymer chains in the D-helices tend to orient along the direction perpendicular to the pore axis. As for S-toroids (Figure 6d), the distribution curves of θ are quite similar to those obtained in the S-helices (Figure 6b). In P-cylinders (Figure 6e), the polymer chains have relatively narrower distributions of θ compared with the other cases shown in Figure 6b−d. It can also be seen that in P-cylinders, the peak values of the distribution curves of θ are a little larger than those in the D-helices (Figure 6c), which means that the polymer chains in the P-cylinders are more likely to orient along the direction perpendicular to the pore axis. From the distributions of θ for different degenerate structures shown in Figure 6, it can be found that each degenerate structure has an individual chain orientation. Furthermore, by comparing the distribution curves of θ corresponding to a certain structure obtained under different pore sizes, we can find another interesting phenomenon: that is, the pore size has almost no effect on the distribution of θ for each degenerate structure. Taking the case of S-helices shown in Figure 6b as an example, the distribution curves of θ for the polymer chains in the Shelices do not change significantly (remain almost unchanged) with the change in the pore size. The same phenomenon can also be found in the cases of other typical structures as shown in Figure 6c−e. It should be noted that this phenomenon can also be observed under other NP contents (as shown in the Figures S4−S8). Therefore, we can conclude that the distribution of the chain orientation in a specific structure is basically independent of the pore size.

Subsequently, we investigated the effect of the NP content on the distribution of θ for the polymer chains in different degenerate structures. Taking the case of D/L0 = 1.33 as an example, we can see from Figure 7 that each degenerate structure has a specific chain orientation. From Figure 7a, it can be seen that the distribution curve of θ for the polymer chains in the S-helices remains almost unchanged with the change in the NP content. In other cases shown in Figure 7b−d, it can also be seen that the distribution curve corresponding to a certain structure basically remains unchanged with the change in the NP content. Likewise, we also studied the effect of the NP content on the orientation of the polymer chains in different degenerate structures under other pore sizes. The statistical results show that the distribution curves of θ for the same structure are almost independent of the NP content. Thus, combining the statistical results shown in Figures 6 and 7, we find that in the current systems, the orientation of the polymer chains is mainly determined by the structures assembled by the AB diblock copolymer/NP mixture system, whereas it is independent of the pore size and the NP content. This result would be valuable for analyzing material properties that are related to chain orientation in experiments.

4. CONCLUSIONS Using MC simulations, the effect of the NP content on the degenerate behavior of the AB diblock copolymer/NP mixture was studied in a cylindrical pore with different pore sizes. Under all pore sizes, the degenerate structure formed by the AB diblock copolymer/NP mixture decreases in amount with the increase in the NP content, ultimately forming either a blocky structure (in smaller pore) or a disorder state (in larger pore). Moreover, increasing the NP content is not favorable for obtaining the S-helices but was favorable to some extent for H

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(7) Kao, J.; Thorkelsson, K.; Bai, P.; Rancatore, B. J.; Xu, T. Toward Functional Nanocomposites: Taking the Best of Nanoparticles, Polymers, and Small Molecules. Chem. Soc. Rev. 2013, 42, 2654−2678. (8) Haryono, A.; Binder, W. H. Controlled Arrangement of Nanoparticle Arrays in Block-Copolymer Domains. Small 2006, 2, 600−611. (9) Wang, H.; Levin, C. S.; Halas, N. J. Nanosphere Arrays with Controlled Sub-10 nm Gaps as Surface-Enhanced Raman Spectroscopy Substrates. J. Am. Chem. Soc. 2005, 127, 14992−14993. (10) Lee, J. Y.; Shou, Z.; Balazs, A. C. Modeling the Self-Assembly of Copolymer-Nanoparticle Mixtures Confined between Solid Surfaces. Phys. Rev. Lett. 2003, 91, 136103. (11) Lee, J. Y.; Shou, Z.; Balazs, A. C. Predicting the Morphologies of Confined Copolymer/Nanoparticle Mixtures. Macromolecules 2003, 36, 7730−7739. (12) Wu, X.; Chen, P.; Feng, X.; Xia, R.; Qian, J. Effect of Selective Nanoparticles on Phase Separation of Copolymer-Nanoparticle Composites Confined between Two Neutral Surfaces. Soft Matter 2013, 9, 5909−5915. (13) Chiu, J. J.; Kim, B. J.; Kramer, E. J.; Pine, D. J. Control of Nanoparticle Location in Block Copolymers. J. Am. Chem. Soc. 2005, 127, 5036−5037. (14) Kim, B. J.; Bang, J.; Hawker, C. J.; Kramer, E. J. Effect of Areal Chain Density on the Location of Polymer-Modified Gold Nanoparticles in a Block Copolymer Template. Macromolecules 2006, 39, 4108−4114. (15) Kim, B. J.; Fredrickson, G. H.; Kramer, E. J. Effect of Polymer Ligand Molecular Weight on Polymer-Coated Nanoparticle Location in Block Copolymers. Macromolecules 2008, 41, 436−447. (16) Yoo, M.; Kim, S.; Jang, S. G.; Choi, S.-H.; Yang, H.; Kramer, E. J.; Lee, W. B.; Kim, B. J.; Bang, J. Controlling the Orientation of Block Copolymer Thin Films using Thermally-Stable Gold Nanoparticles with Tuned Surface Chemistry. Macromolecules 2011, 44, 9356−9365. (17) Kim, B. J.; Chiu, J. J.; Yi, G.-R.; Pine, D. J.; Kramer, E. J. Nanoparticle-Induced Phase Transitions in Diblock-Copolymer Films. Adv. Mater. 2005, 17, 2618−2622. (18) Shagolsem, L. S.; Sommer, J.-U. Order and Phase Behavior of Thin Film of Diblock Copolymer-Selective Nanoparticle Mixtures: A Molecular Dynamics Simulation Study. Macromolecules 2014, 47, 830−839. (19) Kao, J.; Bai, P.; Chuang, V. P.; Jiang, Z.; Ercius, P.; Xu, T. Nanoparticle Assemblies in Thin Films of Supramolecular Nanocomposites. Nano Lett. 2012, 12, 2610−2618. (20) Kao, J.; Xu, T. Nanoparticle Assemblies in Supramolecular Nanocomposite Thin Films: Concentration Dependence. J. Am. Chem. Soc. 2015, 137, 6356−6365. (21) Huang, J.-H.; Li, X.-Z. Self-assembly of Double Hydrophilic Block Copolymer-Nanoparticle Mixtures within Nanotubes. Soft Matter 2012, 8, 5881−5887. (22) Park, J. H.; Yin, J.; Kalra, V.; Joo, Y. L. Role of Nanoparticle Selectivity in the Symmetry Breaking of Cylindrically Confined Block Copolymers. J. Phys. Chem. C 2014, 118, 7653−7668. (23) Yang, Q.; Li, M.; Tong, C.; Zhu, Y. Phase Behaviors of Diblock Copolymer-Nanoparticle Films under Nanopore Confinement. J. Chem. Phys. 2009, 130, 094903. (24) Park, J. H.; Kalra, V.; Joo, Y. L. Cylindrically Confined Assembly of Asymmetrical Block Copolymers with and without Nanoparticles. Soft Matter 2012, 8, 1845−1857. (25) Huh, J.; Ginzburg, V. V.; Balazs, A. C. Thermodynamic Behavior of Particle/Diblock Copolymer Mixtures: Simulation and Theory. Macromolecules 2000, 33, 8085−8096. (26) Kalra, V.; Lee, J.; Lee, J. H.; Lee, S. G.; Marquez, M.; Wiesner, U.; Joo, Y. L. Controlling Nanoparticle Location via Confined Assembly in Electrospun Block Copolymer Nanofibers. Small 2008, 4, 2067−2073. (27) Pan, Q.; Tong, C.; Zhu, Y. Self-Consistent-Field and Hybrid Particle-Field Theory Simulation of Confined Copolymer and Nanoparticle Mixtures. ACS Nano 2011, 5, 123−128.

obtaining the D-helices. The investigations about the chain conformation and the interactive enthalpy of the systems indicate that entropy plays an important role in the formation of the degenerate structures. Furthermore, it is found at the micro-scale level that the distribution of the chain orientation in a cylindrical pore is mainly determined by the aggregated structures formed by the AB diblock copolymer/NP mixture, but not by the NP content and the pore size. Our simulation results offer deeper insights into the self-assembly behavior of the BCP/NP mixture under confinement and can provide a theoretical reference for nanostructure control in experiments.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.6b01090. Typical hexagonally packed cylindrical phase formed by A5B15 diblock copolymers in bulk; typical disordered structures formed by A5B15 diblock copolymer/NP mixture in cylindrical pores; nanostructures selfassembled by the BCP/NP mixture in cylindrical pores with various pore lengths; average interactive enthalpy densities of double helices under different pore lengths; the distribution curves of the orientation angle θ for different degenerate structures (PDF)

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (Y.H.). *E-mail: [email protected] (J.C.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was financially supported by the National Natural Science Foundation of China for General Program (21474107), (51173056), and (21274145) and major Program (51433009). The resource provided by Computing Center of Jilin Province is gratefully acknowledged.

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REFERENCES

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