Homopolymer

Dec 9, 2011 - The phase behavior of the blends is examined for four typical cases, .... The idea of blending in a new species of polymer (or solvent) ...
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Phase Behavior of Binary Blends of Diblock Copolymer/Homopolymer Confined in Spherical Nanopores Rongqiao Yang and Baohui Li* School of Physics and Key Laboratory of Functional Polymer Materials of Ministry of Education, Nankai University, Tianjin 300071, China

An-Chang Shi* Department of Physics and Astronomy, McMaster University, Hamilton, Ontario L8S 4M1, Canada

ABSTRACT: Binary blends of a diblock copolymer (AB) and an incompatible homopolymer (C) confined in spherical cavities are studied using a simulated annealing technique. The phase behavior of the blends is examined for four typical cases, representing the different selectivity of the pore surface to the A, B, and C species. The internal morphology of the spherical polymeric particles is controlled by the homopolymer volume fraction, the degree of confinement, and the composition of the copolymer. Inside a particle, the homopolymers segregate to form one or, under some conditions, two domains; thus, the homopolymers may act as an additional controlling parameter of the shape and symmetry of the copolymer domain. A rich array of confinement-induced novel diblock copolymer morphologies is predicted. In particular, core−shell particles with the copolymers as the shell wrapping around a homopolymer core or a copolymer−homopolymer combined core and Janus-like particles with the copolymers and the homopolymers on different sides are obtained.

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INTRODUCTION Block copolymers have attracted tremendous scientific interests for many years due to their ability to self-assemble into rich nanostructures and their potential applications.1−5 For the simplest block copolymer system, the AB-diblock copolymer melts, their bulk phase behavior has been well understood experimentally6 and theoretically.7,8 The formation of various microphase-separated structures, including lamellae, hexagonally packed cylinders, body centered-cubic array of spheres, and a bicontinuous gyroid structure, can be controlled by the degree of segregation χN (χ is the Flory−Huggins interaction parameter and N is the total degree of polymerization) and the volume fractions (fA or f B) of the blocks. Besides synthesizing more complex block copolymers, various novel morphologies can be obtained by placing block copolymers under geometrical confinements. In confined spaces, the size, shape, and wall selectivity introduce new parameters to tune the morphologies of block copolymers, resulting in structures which are not available in bulk systems.9−24 Although previous study has been mostly on one-dimensionally (thin films)25,26 and two-dimensionally (cylindrical pores)10−24 confined systems, block copolymers under three-dimensional confinement (spherical and elliptical pores) have attracted considerable attention recently.27−44 Experimentally, block copolymer © 2011 American Chemical Society

nanoparticles with rich internal morphologies (such as onionlike concentric lamellae, one directionally stacked lamellae, and Janus-like, tennis-ball-, mushroom-, wheel-, or screwlike structures) were prepared by placing block copolymers in nanoscale inverse silica colloidal crystals27,28 or by using an aerosol approach,29 emulsion,30−33 and precipitation34−37 methods. Moreover, block copolymer particles with similar and even richer internal morphologies were predicted from theoretical and simulation studies.38−44 For block copolymer droplets, it was also found that the shape of a block copolymer particle can be nonspherical.33,44 An excellent review of our current knowledge on the interplay between symmetries of block copolymer phases and confining geometries is given by Stewart-Sloan and Thomas.45 Besides the copolymer volume fraction, it is known the addition of homopolymers to block copolymer melts can be used to control the self-assembled morphology because blends of block copolymers and homopolymers also exhibit complex phase behavior.46,47 It is expected that the combination of confinement and blending with homopolymers will offer new Received: November 12, 2011 Revised: December 2, 2011 Published: December 9, 2011 1569

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concentration controls the shape and size of the confining boundary of the copolymer domain. The idea of blending in a new species of polymer (or solvent) to control the shape of the confining boundary for block copolymer systems should be very useful in manipulating their phase behavior. In the present paper, we examine the application and validity of using blending to control the shape of the confining boundary and to produce new block copolymer morphologies. As an example of the blending system, the phase behavior of binary mixtures of a diblock copolymer AB and an incompatible homopolymer C confined in spherical nanopores is studied using computer simulations. In our model system, the homopolymers segregate into one or two domains which may further control the size and shape of the confining boundary of block copolymers. Four typical cases are investigated, where the pore surface has different selectivity for the three species of A, B, and C. Spherical core−shell and Janus-like particles with rich internal morphologies are predicted.

and exciting opportunities to manipulate the morphologies of block copolymers. Experimentally, nanoparticles consisting of a binary blend of a diblock copolymer and a homopolymer (or a nonsolvent) have been prepared from small droplets or aqueous suspension through the solvent evaporation. In particular, Thomas and co-workers examined the stuctures of nanoparticles consisting of blends of diblock copolymers and homopolymers.29 For the bulk lamella-forming, cylinderforming, and gyroid-forming systems, they observed concentric lamellar structures, curved concentric arrays, and a honeycomblike structure, respectively, in the droplet. Using emulsion and subsequent solvent evaporation techniques, Jeon et al. prepared nanoparticles of different morphologies by blending nearly symmetric poly(styrene)-block-poly(butadiene) (PS-b-PB) and polystyrene homopolymer (hPS).48 They found that the microphase structures of the PS-b-PB block copolymer particles are controlled by the weight fraction of hPS (ϕ), the particle size (characterized by the ratio of the diameter, D, of the emulsion drop to the feature spacing, L0, of the phase-separated periodic domains), and the ratio r of the molecular weight of hPS to that of the PS block in the PS-b-PB block copolymer. In particular, they observed that the effect of spherically confining geometry was pronounced for D/L0 < 15, in which directionally oriented structures such as helices and hoops were observed. For r < 1, the PB domains exhibit transitions among morphological microphases of concentric spherical lamellae, concentric spherical perforated lamellae, circular helices or stacked hoops, and spheres as ϕ is increased. In this case, the hPS swells the PS blocks, and hence increasing ϕ is equivalent to increasing the volume fraction of the PS block in the PS-bPB block copolymer. This effective volume argument can be used to understand why this series of particle morphologies with increasing ϕ in the blending case is similar to that predicted with increasing the volume fraction of one block for the neat diblock copolymers under spherical confinement.38,42 Jeon et al. also found that for r > 1 macrophase separation occurs when ϕ is relatively larger.48 In this case the hPS segregates to the outermost region rather than staying inside the blend particle or swelling the PS blocks. For r ∼ 1, complex structures consisting of spherical lamellae, domelike lamellae, perforated lamellae, and hoops were produced due to the combined effects of micro- and macrophase separations. They also showed that the internal morphology of the nanoparticles can be controlled by using one or two surfactants.33 More recently, phase behavior of binary blends of block copolymer poly(styrene)-block-poly(methyl methacrylate) (PS-b-PMMA) and hexadecane in miniemulsion droplets was investigated by Staff et al.49 In their experiments, nanocapsules consisting of PS-b-PMMA copolymer shell and with the hexadecane as liquid core material, and Janus-like nanoparticles of the copolymer and hexadecane were observed. They found that the particle diameter was controlled by varying the concentration of surfactant dodecyl sulfate (SDS) in the miniemulsion, while the shell thickness was controlled by the ratio of hexadecane/ copolymer. The dependence of the internal morphology of block copolymer assemblies on the nanoconfinement was also investigated in their work, and an onion-like or concentric spherical morphology was identified in the shell of a nanocapsule and also at the center of a Janus-like particle, whereas the outside of a Janus-like particle consisted of bent lamellae (or truncated concentric spherical morphology). In their capsules and Janus-like particles, the liquid hexadecane can be regarded as a new controlling parameter whose

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MODEL AND METHOD In the current study, equilibrium structures are explored using a simulated annealing method50 applied to the “single-site bond fluctuation” model of polymers.51,52 Our previous studies have established that the method and model are appropriate for studying the self-assembly of block copolymers in a confined environment,22,23,38,42 and details of the model and algorithm can be found therein. In principle, the phase behavior of block copolymers can be investigated by a number of methods, including the simulated annealing technique and the widely used self-consistent field theory (SCFT). It has been demonstrated previously that consistent results were obtained from these two methods for diblock copolymers confined in cylindrical pores.18,21 One advantage of the simulated annealing technique is that it does not require a priori assumptions of the underlying structure symmetries. The ground state of the system can be obtained spontaneously by executing a series of Monte Carlo simulations at decreasing temperatures. The simulations are performed on a model system that is embedded in a simple cubic lattice of volume V = L × L × L. For each blend system, a spherical pore with diameter D is constructed from a cubic box with length L = D + 3. The pore includes those lattice sites whose distance to the pore center is less than D/2. Each monomer occupies one lattice site inside the pore, and the polymers are self- and mutual-avoiding. The lattice sites outside the pore constitute the pore surface which cannot be occupied by the polymers. The bond length is set to be 1 and √2 lattice spacing, so that each site has 18 nearest neighbors. The total monomer concentration in the system is kept at 90%. One model diblock copolymer chain used in the simulations is of the type AnBN−n, where N = 12 is the total number of monomers and n is the number of A-monomers. In our previous studies, the phase diagram of the diblock copolymers AnB12−n has been constructed. Specifically, lamellae and hexagonally packed cylinders are formed for n = 6, 5 and n = 2, 3, respectively, and gyroids are formed for n = 4.22,23,38,42 In the current study, we focus on the lamella-forming and cylinder-forming diblock copolymers A6B6 with fA = 1/2, A2B10 with fA = 1/6, and A3B9 with fA = 1/4. The bulk periods of these diblock copolymer ordered structures are L0 = 9.33, 9.93, and 10.67, respectively.22,23,42 One model homopolymer chain contains Nh C 1570

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monomers, where Nh = 12 is chosen throughout the paper. Our further studies show that the chain length of the homopolymer has little effects on the results. In each blend, the numbers of the copolymer and homopolymer chains are n1 and n2, respectively. Thus, the volume fraction of the homopolymer in the blend is defined by ϕh = Nhn2/(Nn1 + Nhn2). Only nearest-neighbor interactions are considered. The repulsive interaction between any two different species is modeled by εAB = εAC = εBC = 1.0kBTref, where kB is the Boltzmann constant and Tref is a reference temperature. The A−A, B−B, and C−C interactions are set to be zero. Interactions between pore surface and monomers A, B, and C are characterized by εSA, εSB, and εSC, respectively. These three parameters have different values in the different cases examined in this study. The degree of confinement is represented by the parameter D/L0. The initial temperature, the annealing schedule, and the number Monte Carlo steps performed at each temperature are all the same as those used before.38

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RESULTS AND DISCUSSION For blends of AB-diblock copolymers and C-homopolymers, a systematic study of the spherical confinement-induced morphologies is carried out. In this section, results are presented for four typical cases, in which the pore surface is equally selective to A, B but repulsive C (Case 1), selective to B but repulsive to A and C (Case 2), neutral to all the three species (Case 3), and selective to B and C but repulsive to A (Case 4). In each case, one series of blends including symmetric diblock copolymers ( fA = 0.5) and homopolymers is studied, and in Case 4, two more series of blends with asymmetric diblock copolymers ( fA = 1/6 and 1/4) and homopolymers are also investigated. In each series of blends, particles with phaseseparated internal morphologies are obtained as functions of the degree of confinement (D/L0) and the volume fraction of the homopolymer (ϕh) in the blend. For a given pair of values of D/L0 and ϕh, 30−70 simulations with different random number generator seeds have usually been performed to ensure the reproducibility of the observed morphologies.

Figure 1. Typical particle morphologies (a, c) and phase diagram (b) obtained from the series of blends of symmetric AB-diblock copolymers A6B6 and C homopolymers as functions of D/L0 and ϕh in Case 1. Morphologies of the blends (with the B-domain in transparent) and those of the A- and B-domains are shown. In (a), the first six morphologies are obtained in relative smaller pores with D/L0 = 2.7−3.1; other morphologies are in relatively larger pores with D/L0= 3.3−4.4. In (b), the morphologies in parentheses have a low occurrence probability. In (c), the morphologies are obtained in relatively larger pores with D/L0 = 4.2−4.4. Color scheme: A-block (green), B-block (red), and homopolymer C (blue).

εSA = εSB = 0, εSC = εAB In this case, the pore surface is equally selective to the two blocks (A and B) of the copolymers but repulsive to the homopolymers (C). Typical particle morphologies obtained from the series of blends of symmetric AB-diblock copolymers (fA = 1/2) and C-homopolymers are shown in Figure 1. As seen in Figure 1, the overall morphology of the particles is of the core−shell type, in which the homopolymers form the core to avoid contacting with the pore surface, while the copolymers segregate to well phase-separated structures in the shell wrapping around the C-core. The detailed morphology of the diblock copolymers in the shell depends on the pore diameter D/L0 and the homopolymer concentration ϕh. We observe 12 classes of shell morphologies in the range of D/L0 =1.5−4.4 and ϕh = 0.05−0.8, as shown in Figure 1. In these morphologies, the A (or B)-blocks themselves form one or several bands packed in various ways, resulting in one of the following basic structures: stacked toroids (ST), a single helix (H1), double helices with all ends free (H2) or in one side with the two ends connected (OH2) or in both sides with the twoends connected (TH2), two rings (R2) with each in a saddle shape, a composite of one or two saddle-shaped rings and one or two arcs (RiAj, where the subscript i is the number of rings and the subscript j the number of arcs in the morphology), a randomly coiled-band (RB1, RB2, and RB3) where one curved Case 1:

band coils in the shell in a self-avoiding way, a handshake structure from helices (HH), helices from a ring (HR), and stacked arcs (SA) where the C-core is partially surrounded by the copolymer shell. Each of the 12 classes shell morphologies plotted in Figure 1 is a combination of the two basic structures mentioned above, with one from the A-domain and the other from the B-domain. We use the abbreviated notations of the basic structures to name the core−shell morphologies, and a sign “−” is added between notations for the A- and B-domains when they do not belong to the same class of basic structures, while only one notation is given when they belong to the same class. It is noted that inside each particle the two basic structures from the A- and B-domains usually belong to the same class or similar classes. The phase diagram shown in Figure 1b is organized as functions of D/L0 and ϕh in a table form, where three main 1571

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groups of morphologies can be identified depending on the ϕh value. In the range of ϕh = 0.05−0.1, only the ST morphology occurs when D/L0 = 1.5−2.5, while degenerated morphologies including the H1, OH2, and H2-TH2, besides the ST also occur when D/L0 = 2.7−4.4. However, the OH2 and H2-TH2 are found with low occurring probabilities. In the range of ϕh = 0.2− 0.6, also only the ST occurs when D/L0 = 1.5−2.5, and the number of degenerated morphologies increases with the increase of the pore diameter. When D/L0 = 2.7−3.1, the first six morphologies shown in Figure 1a (ST, H1, OH2, H2-TH2, RB1, R1A1) all occur. When D/L0 = 3.3−4.0, nine degenerated morphologies occur, including the first six morphologies shown in Figure 1a, and the RB2, RB3, and R1A2-R2. When D/L0 = 4.2− 4.4, 10 degenerated morphologies occur, including the first six morphologies shown in Figure 1a, and the RB2, RB3, HH-HR, and R2A1. In the range ϕh = 0.7−0.8, the copolymer shell cannot completely cover the C-core so that only the SA morphology occurs, and it holds in the whole range of D/L0 due to the small volume fraction of the copolymers in the blends. In Figure 1b, it is noted that morphologies having the same type of microstructures may occur at different pore sizes. An examination of these morphologies shows that the D/L0 value does influence the details of the resulting morphology. However, it does not change the essential feature of the structure. Figure 1c shows the six morphologies, corresponding to the first six morphologies shown in Figure 1a, observed at relatively larger pores. When compared the morphologies shown in Figure 1a with those shown in Figure 1c, it is noted that the number of periods in morphology with a given type is increasing with increasing D/L0. It is noted that for the helical or coiled-band morphologies shown in Figure 1, both lefthanded and right-handed structures have been observed, and inside each particle the two basic structures from the A- and Bdomains have identical chirality. Figure 2a shows that in the core−shell particles the C-core increases gradually in size and accordingly the thickness of the

Li et al. investigated the morphologies of AB-diblock copolymer and ABC-triblock copolymer systems confined on the surface of a sphere.53 For symmetric AB-diblock copolymer system, they observed three classes of striped patterns, including ringform, spiral-form, and cage-form ribbon patterns, with changing the ratio of the perimeter of the sphere to the average domain size. The copolymer morphologies in the shell of the core−shell particles obtained in our 3D model are similar to those predicted by Li et al. The ST, R1A1, R1A2-R2, R2A1, and SA morphologies obtained in our simulations are similar to their ringform ribbon patterns, while the other morphologies observed in our simulations are similar to their spiral-form ribbon patterns. Also using SCFT calculation with the idealized 2D model of infinitely thin films, Chantawansri et al. investigated the selfassembly of thin diblock copolymer film confined to the surface of a sphere.54 For symmetric AB-diblock copolymer system, they obtained the spiral, hedgehog, and quasi-baseball phases and found the hedgehog configuration was almost always lowest in energy for sphere with small radii. For sphere with larger radii, there was competition between the hedgehog and spiral configurations. Quasi-baseball configurations were found metastable but close in energy to the spiral. The H1, ST, and R1A1 morphologies found in our simulations are similar to the spiral, hedgehog and quasi-baseball phases of Chantawansri et al., respectively. Using cell dynamic simulations, Pinna et al. performed a systematic study of thin diblock copolymer films around a nanoparticle (between two spherical interfaces).55 Lamella-, cylinder-, and sphere-forming diblock copolymers were investigated with respect to different film thicknesses, particle radii, and boundary conditions at the film interfaces. They also performed some experiments on the model system. In their case when both the inner and the outer spherical surfaces are neutral to the two species of the symmetric AB-diblock copolymers, they found that the lamellae of the A- and B-domains stand up on the spherical surfaces. No such morphology is observed in our simulations, which may be due to the difference in the polymer− surface interactions between our case and theirs. On the other hand, some of the basic structures formed by the A- or B-blocks in our simulations are similar to the nanoshells in their case formed by the minority blocks of the cylinder-forming asymmetric diblock copolymers.55 For the block copolymer morphology in the AB-shell, another related system is pure diblock copolymers under 3D confinement of a spherical pore. Using simulated annealing method, Yu et al. investigated the morphologies of the symmetric38 and asymmetric42 AB-diblock copolymers confined in spherical pores. For the neat symmetric AB-diblock copolymers, single-helix and/or double helices with two ends connected or free were obtained only in the range of D/L0 = 2.7−3.3 and when the pore surface is neutral or very weakly preferential to one block of the diblocks.38 The helical morphologies observed there are similar to the H1, OH2, and H2-TH2 obtained in the present simulations in the range of ϕh = 0.05−0.1. On the other hand, for the asymmetric cylinder-forming diblock copolymers with fA = 1/6 confined in spherical pores with surface attractive to the majority B-blocks, Yu et al. found a rich array of confinement-induced morphologies formed by the minority A-block, including stacked toroids, single helix, double helices with all ends free or in one side with the two ends connected or both sides with the two ends connected, a composite of one or two saddle-related structures and one or two curved cylinders or a composite of two saddles, and curved A-cylinders which is randomly coiled in the A-shell in a self-avoiding way, etc.42 Moreover, similar morphologies were

Figure 2. (a) Variation of the core size with ϕh in the core−shell particles having the ST morphology at D/L0 = 3.1 in Case 1. (b) Locations of the homopolymer core in particles having the R1A1 morphology at D/L0 = 3.1 and ϕh = 0.2. The color code is the same as that in Figure 1.

shell decreases with the increase of ϕh. Finally, the shell is partially broken when ϕh ≥ 0.7. Moreover, it is noted that the location of the C-core may deviate somewhat from the particle center. This deviation usually does not affect the resulting morphology in the copolymer shell, as shown in Figure 2b. However, the thickness of the copolymer shell becomes not uniform when the location of the C-core deviates from the particle center. Our above simulation results can be compared with some related experiments and theories. For the diblock copolymer morphology in the shell, the most relevant system is the thin diblock copolymer films around a sphere. Using SCFT calculation with an idealized 2D model of infinitely thin films, 1572

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the copolymer shell occur as the unique observed morphology. It is also noted that the ϕh values needed to form the Janus-like and the unique core−shell particles decrease with the decrease of D/L0, and in small pores with D/L0 < 2.5, these two kinds of particles are the only observed morphologies in the whole ϕh range. In relatively larger pores, other type of particles also occurs in the limit of a smaller ϕh. In pores with 2.5 < D/L0 ≤ 3.1, core− shell particles where the core consists of a combination of the Cdomain and the innermost B-domain wrapped around by the copolymer shell are the unique observed morphologies at ϕh ≤ 0.1. In such a particle, the C-domain is usually in a partialspherical shape. In larger pores with D/L0 > 3.1, particles with a C-domain locating in the peripheral region also occur besides the core−shell particles with a spherical C-core or a copolymer− homopolymer combined core, and in this latter case, the core consists of a combination of the C-domain and the innermost B (and/or A)-domain. The morphologies shown in Figure 3 can be compared with the recently experimental results of Staff et al.49 In their work, phase behavior of binary mixtures of a nearly symmetric diblock copolymer PS-b-PMMA and a nonsolvent liquid hexadecane in miniemulsion droplets was investigated. Their results showed that at very low concentrations of hexadecane (5.26 wt %) the hexadecane was not completely but partially engulfed, whereas upon an increase in hexadecane (10 wt %) nanocapsules consisting of the PS-b-PMMA copolymer shell and with the hexadecane as liquid core were obtained, and a further increase in hexadecane (60−80 wt %) induced that the capsules became unstable and the shell became thinner, leading to collapse of the capsules as the hexadecane is a liquid.49 Furthermore, onionlike morphology was identified in the nanocapsules, and bent lamellae were identified in the outside of a Janus-like particle. The sequence of particle morphologies observed by Staff et al. with increasing the hexadecane concentrations49 (the partially engulfed capsules, the capsules, and the collapsed capsules)49 is similar to that obtained in our simulations with increasing ϕh in relatively larger pores (the particles with the C-domain locating in the peripheral region, the core−shell particles, and the Januslike particles).

also predicted when asymmetric bulk cylinder-forming diblock copolymers are under the soft confinement of poor solvent.41,44 All of these morphologies formed by the minority block of the cylinder-forming diblock copolymers are similar to the basic structures formed by the A- or B-blocks in our simulations shown in Figure 1. Using real-space SCFT calculation, Li et al.43 investigated the morphologies and phase diagram of diblock copolymers under spherical confinement of neutral surface, and a rich variety of geometric frustration phases with specific symmetries were obtained. The morphologies in the outermost A-layer of distorted cylinder structures formed by the minority block of the cylinder-forming asymmetric diblock copolymers in their case are also similar to the basic structures formed by the Aor B-blocks in the present simulations. However, in the abovementioned studies,41−44 the morphologies are formed from the asymmetric diblock copolymers, whereas in our present study, they are formed from symmetric diblock copolymers. Another difference is that, in the above-mentioned studies, only the minority-block forms the complex morphologies which embed in the matrix of the majority-block domain, whereas in the present study both the A- and B-blocks form complex structures.

εSA = −εSB = εAB, εSC = 0.5εAB In this case, the pore surface is selective to one block but repulsive to the other block of the copolymer and the homopolymer. Typical particle morphologies obtained from the series of blends of symmetric AB-diblock copolymers and C homopolymers as functions of ϕh and D/L0 are plotted in Figure 3. In this case, the copolymers always separate into Case 2:

Case 3:

εSA = εSB = εSC = 0

In this case, the pore surface is neutral to all the three species from the copolymers and the homopolymers. Typical particle morphologies obtained from the series of blends of symmetric AB-diblock copolymers (fA = 1/2) and C homopolymers as functions of ϕh and D/L0 are plotted in Figure 4. In these particles, the homopolymers usually form one partial-spherical cap shaped domain locating in the peripheral region, while the copolymers segregate into well-phase-separated lateral structures occupying the rest space. It is also noted that the homopolymers may also form two domains occupying opposite peripheral regions of a particle at ϕh = 0.2−0.3 when D/L0 ≥ 3.8. For a particle with only one C-domain, its overall outside morphology is similar to a Janus particle in which the C-domain is in one side while the copolymer domain with well-phase-separated lateral structures occupying the other side of the particle. It is also noted that the interface between the C-domain and the copolymer domain is not always flat but gradually changes from convex to concave in the C-domain side with increasing ϕh. In Figure 4, perpendicular lamellae occur frequently in the copolymer domain; they are similar to those obtained when the corresponding neat symmetric diblock copolymers are confined in spherical nanopores with a neutral surface38 but truncated by

Figure 3. Typical particle morphologies obtained from the series of blends of symmetric AB-diblock copolymers A6B6 and C homopolymers as functions of D/L0 and ϕh in Case 2, where cross-section views are shown. The color code is the same as that in Figure 1.

onion-like morphologies, whereas the homopolymers separate into one domain locating in the central or peripheral regions of a particle depending on the ϕh and D/L0 values. It is noted that particle morphologies shown in Figure 3 are relatively simple at the larger ϕh limit, where Janus-like particles are observed. Each Janus-like particle consists of a nearly spherical C-domain which is partially wrapped by a copolymer shell. At the smaller ϕh value side before the occurrence of the Janus-like particles, there is a narrow ϕh region where core−shell particles with each having a spherical C-core completely wrapped around by 1573

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Figure 4. Typical particle morphologies obtained from the series of blends of symmetric AB-diblock copolymers A6B6 and C homopolymers confined in spherical pores with neutral surfaces (Case 3) as functions of ϕh and D/L0. Morphologies of the blends and those of the C and copolymer domains are shown. The color code is the same as that in Figure 1.

Figure 5. Typical particle morphologies obtained from the series of blends of symmetric AB-diblock copolymers A6B6 and C homopolymers confined in spherical nanopores as functions of ϕh and D/L0 in Case 4. Morphologies of the blends (side views and/or cross-section views) and those of the copolymer domains are shown. The color code is the same as that in Figure 1.

A-block of the copolymers. Particles with different morphologies are obtained. In a particle, the homopolymers usually just form one partial-spherical cap shaped domain occupying a peripheral region of the particle, while the copolymers segregate into a well-phase-separated structure with an outermost B-layer occupying the rest space. The overall outside morphology of a particle is similar to a Janus particle in which the C-domain is in one side while the copolymer domain with an outermost Blayer is in the other side of the particle. In some larger pores, however, the homopolymers may form two partial-spherical cap shaped domains occupying opposite peripheral regions of a particle. Morphologies for f A = 1/2. Typical particle morphologies obtained from the series of blends of symmetric AB-diblock copolymers and C homopolymers are plotted in Figure 5. It is noted that in the copolymer domain truncated concentric spherical structures are observed. In these structures, the

the homopolymer domain. It is interesting to notice that at a given D/L0 value the number of the A (or B)-layers in the perpendicular lamellae is independent of ϕh. It is also noted that when D/L0 ≈ 1.5−2.0, embedded structures, where one block forms a dumbbell-like core while the other block forms a band wrapping around the core at the central part, are observed in the copolymer domain in the range of ϕh = 0.05−0.3. In pores with D/L0 = 2.5−3.5, a few helical structures, besides perpendicular lamellae, also occur as degenerated structures in the range of ϕh = 0.05−0.1. The embedded structures and helical structures are also the truncated counterparts of those obtained when the corresponding neat symmetric diblock copolymers are confined in smaller spherical pores with a neutral surface.38

εSA = −εSB = −εSC = εAB In this case, the pore surface is selective to the B-block of the copolymers and the C-homopolymers but repulsive to the Case 4:

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Figure 6. Dimensionless thicknesses of the outermost B-rich layers, dB1, in the truncated concentric spherical structures. (a) dB1 as a function of D/L0 at different ϕh values. (b) dB1 as a function of ϕh at different D/L0 values.

number of A−B interfaces increases with increasing D/L0, just like that found when the corresponding neat diblock copolymers are confined in a spherical pore.38 On the other hand, it is interesting to find that the number of A−B interfaces does not depend on the ϕh value for a given pore diameter. Another interesting finding is that in a truncated concentric spherical structure the innermost A (or B)-sphere is always in contact with the C-domain, instead of in the particle center. Therefore, the symmetry of these truncated structures is totally different from their nontruncated (ideal) counterparts. It is also noted that when ϕh = 0.2−0.3 and the pore diameters in the ranges of D/L0 ≈ 4.0−4.6, the homopolymers can segregate into either one or two domains. In particles with two Cdomains, the sizes of the two C-domains are usually not equal, and thus the symmetry of these particles is also different from their ideal counterparts. Our further computations show that the dimensionless thickness of the outermost B-rich layer, dB1, in the truncated concentric spheres increases with increasing D/L0 at a given ϕh value (Figure 6a); this trend is consistent with that found for the neat diblock copolymers under the same confinement.38 Furthermore, dB1 also increases with increasing ϕh at a given D/L0 value (Figure 6b), indicating that the layer thicknesses in these truncated concentric spheres are different from those in the ideal concentric spheres.38 This may be due to the difference in their symmetries. Our above simulation results can be compared with the experiments of Staff et al.49 In their work, they controlled the interfacial property and the size of the nanoparticles and nanocapsules by varying the concentration of SDS in the miniemulsion process, and they observed Janus-like particles of polymer and hexadecane in samples with relatively higher SDS concentration. Inside such a Janus-like particle, onion-like morphology and bent lamellae were identified at the center and the outside of the half-spherical polymer particle, respectively. The Janus-like particles with truncated concentric spherical structure predicted in our simulations are similar to the Januslike particles of Staff et al.49 both in shape and in the internal morphologies. Morphologies for f A = 1/6. Typical particle morphologies obtained from the series of blends of asymmetric AB-diblock copolymers (with fA = 1/6) and C-homopolymers are plotted in Figure 7. It is noted that in the copolymer domain the minority A-blocks form shell structures located inside the Bmatrix. Figure 7 shows that in pores with D/L0 = 3.3 the A blocks form two A shells, and the inner A-shell is composed of

Figure 7. Typical particle morphologies obtained from the series of blends of asymmetric AB-diblock copolymers A2B10 (fA = 1/6) and C-homopolymers confined in spherical pores with D/L0 = 3.3 in Case 4, where morphologies of the blends (with the B-domain in transparent) and those of the A- and C-domains are shown. (a) ϕh = 0.05 and (b) morphologies as a function of ϕh. The color code is the same as that in Figure 1.

one A-sphere whereas the morphology of the outer A-shell depends on ϕh. It is noted that when ϕh = 0.05, the outer Ashell can be multiple degenerated structures (Figure 7a), similar to those obtained for the corresponding neat diblock copolymers under the same confinement.42 When ϕh ≥ 0.1, the outer A-shell always consists of stacked toroids or tacked toroids with vertex spheres (Figure 7b). This result reflects that pores with a truncation (or irregularity) of ∼10% may stabilize the stacked toroids and destabilize the helical morphologies. Similar morphologies and tendency are found in pores with different diameters. It is noted that in larger pores the inner Ashell may also consist of stacked toroids, and the number of Atoroids in the A-shells increases with increasing D/L0 or with decreasing ϕh. Morphologies for f A = 1/4. Typical particle morphologies obtained from the series of blends of asymmetric AB-diblock copolymers (with fA = 1/4) and C-homopolymers are plotted in Figure 8. It is noted that in the copolymer domain the minority A-blocks also form truncated shell structures located inside the B-matrix. Figure 8a shows that in pores with D/L0 = 3.3 the A-blocks form two A-shells, where the inner A-shell is composed of one A-sphere, whereas the morphology of the outer A-shell is always a truncated cage, insensitive to the ϕh value. These truncated cages are just the truncated version of cages obtained for the corresponding neat diblock copolymers 1575

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D/L0, indicating that in smaller pores a relatively larger percent of C-monomers are in contact with the pore surface.

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CONCLUSIONS We have performed simulated annealing simulations on a lattice model of homopolymer/diblock copolymer blends confined in spherical cavities. The investigation focuses on the study of the phase behavior of blends of an AB-diblock copolymer and an incompatible C-homopolymer under three-dimensional confinement. Four typical cases are chosen to represent systems in which the selectivity of the pore surface is different. In each case, the homopolymers segregate into one or two domains which may act as an additional controlling parameter changing the shape and symmetry of the resulting copolymer domain and therefore affects the internal morphologies of the copolymer. Rich and complex confinement-induced copolymer morphologies, depending on the volume fraction of the homopolymer in the blend, the degree of confinement, and the composition of the copolymer, characterized by ϕh, D/L0, and fA, respectively, are predicted. When the pore surface is equally selective to the two species of the copolymer but repulsive to the homopolymer, core−shell particles with the homopolymers as the core and the copolymers as the shell are observed for blends of symmetric diblock copolymers/homopolymers. In this case, the homopolymer domain acts as an internal confining boundary and therefore changes the spherical space into a spherical shell for the copolymer domain. The copolymer morphologies in the shell are rich and multiple degenerated. Inside each particle, the A- and Bdomains form the same class or similar classes of structures, such as stacked toroids and helical structures. When the pore surface is selective to one block of the copolymer, but repulsive to the other block of the copolymer and the homopolymer, particles with a Cdomain locating either in the central or peripheral regions, core− shell particles, and Janus-like particles occur in sequence with increasing the homopolymer concentration. When the pore surface is neutral to all the three species, or when the pore surface is selective to the homopolymer and one block of the copolymer but repulsive to the other block of the copolymer, Janus-like particles with the homopolymers in one side while the copolymers in the other side are observed for blends of diblock copolymers/homopolymers. In these cases, the homopolymer domain truncates the spherical confining space into a partial spherical one for the copolymer domain. In these partial spheres, the copolymers segregate into well

Figure 8. Typical particle morphologies obtained from the series of blends of asymmetric AB-diblock copolymers A3B9 (fA = 1/4) and Chomopolymers confined in spherical pores as a function of ϕh in Case 4, where morphologies of the blends (with the B-domain in transparent) and those of the A- and C-domains are shown. The color code is the same as that in Figure 1. (a) D/L0 = 3.3 and morphologies for the A-domains are plotted in two perpendicular directions. (b) D/L0 = 3.84.

under the same confinement.42 Our further simulations show that in larger pores the inner A-shell may also consist of a cage, and it is truncated when ϕh is large enough. It is also noted that in some large pores structures with two C-domains, truncating each A-cage on two opposite sides (see Figure 8b), may also occur as a degenerated morphology for that with only one Cdomain. The number of holes in the A-cage(s) increases with increasing D/L0 or with decreasing ϕh. In the Janus-like particles formed in Case 4, the surface coverage of C-monomers, defined as the ratio of the number of C-monomers in contacting with the pore surface to the total number of C-monomers in a particle, is plotted as a function of ϕh for different pore sizes in Figure 9. It is noted that each surface coverage curve gradually decreases with increasing ϕh, indicating that a relatively larger percent of C-monomers are in contact with the pore surface when ϕh is smaller. This result may be due to the entropic effect. It is also noted that the surface coverage of C-monomers decreases with increasing

Figure 9. Surface coverage of C-monomers as a function of ϕh at different D/L0 values in Case 4. The composition of the copolymer is (a) fA = 1/2 and (b) fA = 1/6. 1576

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phase-separated lateral or longitudinal structures. For blends of an asymmetric diblock copolymer AB (with fA = 1/6) and homopolymer C, it is found that pores with a truncation (or irregularity) equal to or higher than 10% may stabilize the stacked toroids and destabilize the helical morphologies. When comparing our present study with those of the corresponding neat copolymers under the same confinement,38−44 our results clearly demonstrate that the geometry and size of the confining space have a large effect on the structure and symmetry of the self-assembled morphologies. On the other hand, Higuchi et al. have prepared Janus-type particles by mixing PS and polyisoprene (PI) homopolymers using a simple self-organized precipitation method, where diblock copolymers PS−PI are added in the PS/PI homopolymers blends as compatibilizers changing the Janus-type structures.36 Furthermore, Saito et al. have prepared particles with dimple, acorn, or spherical shapes by mixing PS and PMMA homopolymers.56 If one of the two homopolymers used in their experiments is replaced by a diblock copolymer, the situation would be similar to ours. It is expected that the Janus-like and core−shell particles predicted in the present study can be observed experimentally in the near future.

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AUTHOR INFORMATION

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ACKNOWLEDGMENTS

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REFERENCES

Corresponding Author *E-mail: [email protected] (B.L.); [email protected] (A.-C.S.).

This research is supported by the National Natural Science Foundation of China (20774052 and 20990234), by the National Science Fund for Distinguished Young Scholars of China (20925414), and by Nankai University ISC. A.-C.S. gratefully acknowledges the supports from the Natural Sciences and Engineering Research Council (NSERC) of Canada.

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