Article pubs.acs.org/Langmuir
Self-Assembly of Rod−Coil Diblock Copolymers within a RodSelective Slit: A Dissipative Particle Dynamics Simulation Study Jian-Hua Huang,*,† Ze-Xin Ma,† and Meng-Bo Luo‡ †
Department of Chemistry, Zhejiang Sci-Tech University, Hangzhou 310018, China Department of Physics, Zhejiang University, Hangzhou 310027, China
‡
ABSTRACT: Dissipative particle dynamics simulations are performed to investigate the self-assembly of rod−coil diblock copolymers RNRCN−NR within a rod-selective slit. The self-assembled structure of the confined system is sensitively dependent on the rigidity kθ and the fraction f R of the rod block and the slit height H. From the phase diagram of structures with respect to kθ and f R for N = 12 and H = 6, we observe four main structures including disordered cylinder (DC) structure, hexagonally packed cylinders (HPC) perpendicular to the slit surfaces, and lamellar structures parallel (L∥) and perpendicular (L⊥) to surfaces. And structure transitions can be achieved by tuning kθ. The effect of the slit height on the self-assembled structure is also studied for R6C6 and R7C5 copolymers with large kθ. For R6C6, different structures near surfaces and in the interior of slit are observed in relatively wide slits. Whereas for R7C5, L⊥ structure, whose lamellar domain spacing decays exponentially with H, is generally generated. Our results suggest an effective way to control the ordering of rod−coil diblock copolymers under nanoscale confinement.
I. INTRODUCTION The self-assembly of confined diblock copolymers (DBCs) has attracted considerable attention in recent years because of its wide applications in a variety of fields.1 In thin films, the volume fraction of the blocks, the molecular weight, the interaction between the block species and confining surfaces, and the film thickness relative to the natural periodicity of DBC lamellae would lead to a many-dimensional phase diagram which could be used to tune the order and morphology of polymer films.2−5 For instance, perpendicular orientation was observed for DBCs confined by nonselective surfaces,6,7 whereas coil−coil DBC lamellae generally oriented parallel to selective surfaces.8,9 On the other hand, tilted or deformed lamellar structure, or even coexistence of lamellae with different orientations, was observed when the film thickness was not compatible with the natural thickness of the lamellae.10 Compared with the classical coil−coil DBCs, the difference in chain conformational entropy between the rod and coil blocks and anisotropic interactions between rod blocks offer intriguing possibilities for the self-assembly of rod−coil (RC) DBC film.11,12 It was observed that RC DBCs could form highly ordered honeycomb structures with hexagonal, closepacked air-holes.13 However, it was difficult to obtain equilibrium structures of RC DBCs assembled in thin films in experiments since it was easy for the DBCs to form aggregates or micelles in solution and the structures were often kinetically trapped and deposited into the film.1,11 Therefore, theoretical studies and computer simulations are needed to understand the self-assembled structures of RC DBCs confined in thin films. Hybrid density function theory (DFT) study showed that RC © 2014 American Chemical Society
DBCs in slit could form ordered structures beyond a critical polymer concentration.14 The rigidity of the rod block in RC DBCs is an important factor in determining the structure of RC DBC films. Pereira and Williams reported that smectic rod−coil melts confined between two flat surfaces formed either parallel lamellae (L∥) or perpendicular lamellae (L⊥) for relatively thick films through the change in the rod tilt angle adjacent to the surfaces.15 By using extended scaling methods, Nowak and Vilgis pointed out that the rod blocks would prefer to lie parallel to the surfaces to gain entropy and to therefore lower their confinement energy when the aggregates of RC DBC adsorbed on surfaces.16 Again, a self-consistent field theory (SCFT) study found that the rod block had a strong tendency to segregate near the surfaces in all structures, due to its less conformational entropy loss than the coil block.1 Monte Carlo (MC) simulation on the aggregation of rod−coil−rod (RCR) triblock copolymers between two impenetrable nonselective surfaces showed that the RCR copolymers were likely to form L⊥ structure with increasing rigidity of rod block.17 Experiments and theoretical studies found that the equilibrium phase of RC DBCs in solution was strongly dependent on the fraction f R of rod block.18−20 The structure of RC DBCs in films was dependent not only on f R but also on the property of surfaces. Surfaces could directly affect the orientation of the rod block near the surfaces as well as the Received: March 16, 2014 Revised: April 28, 2014 Published: May 6, 2014 6267
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equilibrium structure of RC DBCs.21,22 The property of surface is represented by the interaction between surface and polymer, so the surface is often qualitatively named as rod-selective (Rselective) surface, coil-selective surface, or nonselective surface. RC DBCs with f R = 0.6, which formed a lamellar structure in bulk, could form stable L⊥ structure within nonselective surfaces.1 Whereas for the R-selective surfaces, the phase transition between L⊥ and L∥ was strongly dependent on the competition between the degree of the surface selectivity and the film thickness. When the surface attraction to rod blocks was relatively weak, the L⊥ phase was stable and independent of the film thickness.1 Experiment found that the morphology of RC DBCs in film changed from alternating strips to hexagonal superlattice of rod aggregates with the decrease in f R from 0.36 to 0.25.23 SCFT study on 2D system reported the transition from lamellar structure to rod puck-shaped structure with the decrease in f R from 0.65 to 0.4.1 However, the effect of the rigidity and fraction of the rod block on the structure of RC DBC film is still not clear and deserves careful and systematic studies.1,14,17,24,25 In the present work, the self-assembly of RC DBCs confined within a Rselective slit with two identical flat surfaces is investigated by using dissipative particle dynamics (DPD) method. The effects of the rigidity and fraction of the rod block in the RC DBCs on the self-assembled structures are systematically simulated.
determined by the weight function w(rij) with a commonly used choice w(rij) = 1−rij/rc for rij ≤ rc and w(rij) = 0 for rij > rc. Here rc is the cutoff radius for all pairwise forces. Reduced units are adopted in the DPD simulation. All DPD particles in our simulations are of the same mass with m = 1. We set the cutoff distance rc = 1 as the unit of length, and energy scale kBT = 1 as the unit of energy. Thus, the unit of force is kBT/rc. In this work, the amplitude of thermal noise is set as σ = 3 and thus the friction coefficient is γ = 9/2. The interaction amplitude aij in F(C) ij is the strength of the repulsive interaction between particles i and j, and the value is dependent on particle types. In our system, there are four types of DPD particles: rod monomer (R) and coil monomer (C) in the copolymer chain, surface (w), and solvent (s). The repulsive interaction parameter between the two identical particles is set to be 25.29 We assume that the rod block R is solvophilic with aRs = 20, whereas the coil block C is solvophobic with aCs = 30. And blocks R and C repel each other with aRC = 35. We consider the slit surfaces to be selective to R but repulsive to C with interaction values awR = 5 and awC = 25, respectively. And surfaces are nonselective to solvents with aws = 25. The motion of every DPD particle obeys Newton’s equation, i.e. mi (d2ri)/(dt2) = fi with fi being the force experienced by particle i. The position and velocity of DPD particle are solved using a modified velocity−Verlet algorithm proposed by Groot and Warren.29 The unit of time is τ = ((mr2c )/(kBT))1/2 as all DPD particles are of the same mass m. In the simulation, a time step Δt = 0.01τ is adopted for integration of Newton’s equation. B. Models for Slit and RC DBC Chain. Simulations are carried out in a slit constructed by two impenetrable parallel surfaces. The two surfaces are identical and each surface is constructed by four layers of DPD particles that are arranged in a face-centered cubic lattice. The centers of the slit DPD particles along the normal direction are placed at 1/8, 3/8, 5/8, and 7/8 of the surface. For simplicity, the slit DPD particles are motionless in the simulations. The transverse lengths parallel to surfaces are Lx = Ly = 40 in the x and y directions with periodic boundary conditions (PBC). The z direction is normal to surfaces. The RC DBC is modeled as a coarse-grained linear chain RNRCN−NR with NR DPD particles in the rod block and N−NR DPD particles in the coil block. Finitely extensible nonlinear elastic (FENE) interaction is adopted for the chemically bonded monomers. The bonded FENE interaction UFENE is represented by30
II. MODEL AND SIMULATION METHOD A. DPD Algorithm. The DPD method was developed by Hoogerbrugge and Koelman26 and cast in the present form by Español.27 In DPD simulation, all elements, including solvent, polymer, and surface, are coarse-grained into DPD particles. These DPD particles interact with each other via pairwise forces that locally conserve momentum leading to a correct hydrodynamic description.28 The pairwise forces contain the conservative force F(C) = aijw(rij)r̂ij, ij = −γw2(rij)(r̂ij·vij)r̂ij, and random thermal force dissipative force F(D) ij F(R) ij = σw(rij)θijr̂ij. Here rij = |rij| = |ri − rj|, r̂ij = rij/rij, vij = vi − vj, θij is an uncorrelated symmetric random noise with zero mean and unit variance, σ is the amplitude of the thermal noise, and γ is a friction coefficient. The combined effect of the dissipative and random forces is that of a thermostat, leading to σ2 = 2γkBT. Here kB is Boltzmann constant and T is the temperature. The softness of the interaction is
UiFENE ,i+1
⎧ 2⎤ ⎡ ⎛r − req ⎞ ⎥ ⎪ ⎪− kF (r − r )2 ln⎢1 − ⎜ i , i + 1 eq ⎜ r − r ⎟⎟ ⎥ for 2req − rmax < ri , i + 1 < rmax ⎢ = ⎨ 2 max ⎝ max eq ⎠ ⎦ ⎣ ⎪ ⎪ ⎩∞ otherwise
where the equilibrium bond length req = 0.8, the maximum bond length rmax = 1.3, and the elastic coefficient kF = 200. For the rod block, an additional bond-bending force between consecutive bonds is introduced as31 Ubend =
1 k θ(θ − θ0)2 2
(1)
enough concentration.14 We find that the self-assembled morphology of DBCs does not change with the concentration when Cp reaches a certain high value, say Cp > 0.8 in the present case. In the present work, the chain length of RC DBC is fixed at N = 12.
III. RESULTS AND DISCUSSION At the beginning of the simulation, all polymers and solvents are randomly put into the system. Then we simulate the evolution of the system for a sufficient long time. During the simulation, the system energy is calculated at every 100τ. After the system energy presents only a small fluctuation with time within a period of 1000τ, we assume the system reaches an equilibrium state. Then, we run additional 5000τ for statistics. In all simulations, the slit surfaces are assumed to be Rselective. Within narrow slits, the equilibrium structures are complicated. It is believed that the structure is mainly
(2)
with kθ the bending modulus and θ0 = π the equilibrium angle between consecutive bonds, respectively. kθ = 0 corresponds to a flexible block, such as the coil block in the present model. In this work, the rigidity of the rod block R is represented by the value kθ. Thus, “rod” represents semiflexible block with finite value of kθ instead of infinitely rigid with kθ = ∞. The overall particle density ρ is set as 3 throughout this study. The concentration of RNRCN−NR chains Cp = NNp/ρV is fixed at 0.9; here V is the volume of slit, and N and Np are the length and the number of chains, respectively. It is known that ordered structures appear at high 6268
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determined by the balance among the immiscibility between rod and coil blocks, the surface selectivity, and the elastic free energy penalty for deformation of chain at the rod−coil interfaces and near the film surfaces, and the rod−coil interfacial energy.1 In this work, we have studied the effects of the rigidity of rod block, the fraction of rod block, and the slit height on the self-assembled structure of RC DBCs in the Rselective slit. A. Effect of the Rigidity of Rod Block. The equilibrium conformational dimensions of R7C5 DBCs in bulk solution and in a slit with height H = 6 are studied at the beginning. For the bulk simulation we use Lz = 40 with PBC in the z direction instead of two solid surfaces. The polymer concentration is the same in these two cases. The mean square end-to-end distance and the mean square radius of gyration are calculated for the R7C5 DBCs. Here represents the average over all Np DBCs and independent samples. The dependence of and on the rigidity kθ of the rod block is plotted in Figure 1. The behavior of polymer in the slit is similar to that
We then gradually increase the rigidity of the rod block by increasing the bending modulus kθ. At kθ = 5, the system selfassembles into cylinders perpendicular to surfaces but arranged randomly as shown in Figure 2a. Such kind of structure is
Figure 2. Top views of the equilibrium structures assembled by R7C5 DBCs within R-selective slit at different rigidity kθ of the rod block. From panel (a) to (c), kθ is 5, 10, and 50, respectively. Rod block is represented by red and coil block is represented by blue. The same symbols are used in the remaining figures. For clarity, slit and solvent DPD particles are not shown in all snapshots.
named as disordered cylinders (DC). When kθ is increased to 10, ordered cylinders arranged in a hexagonal structure appear as shown in Figure 2b. Such kind of ordered structure is named as hexagonally packing cylinders (HPC). Finally, L⊥ structure is formed at large kθ from about 30 up to the maximum 100 used in simulations. Figure 2c shows the L⊥ structure obtained at kθ = 50 as an example. Similar aggregation morphologies were observed in experiment,23 and L⊥ was also observed for RC DBCs with f R = 0.6 within neutral and R-selective slits by SCF theory.1 MC simulation showed that RCR triblock copolymers liked to form L⊥ structure in neutral slit with increasing rigidity of rod block.17 And Li and Gersappe found that, even in bulk solution, lamellar structure could replace part of the cylindrical domain in the phase diagram when the flexible block of the RC copolymer became more rigid.33 These results show that the rigidity of the rod block plays an important role in the formation of ordered structures. However, little attention was paid on the ordered cylinder structure in previous reports. Lattice MC simulation reported a morphology change from disordered structure to L⊥ structure with increasing the rigidity of R8C14R8 triblock copolymers in a nonselective slit. But they did not observe such an ordered cylinder structure.17 The possible reason is that the ordered cylinder structure exists in a small parameter region for the lattice model. In the present DPD model, ordered cylinder structure can be observed for moderate rigid RC DBCs. To quantitatively analyze the ordered cylinder structure formed at kθ = 10, we have calculated two-dimensional (2D) structure factor for the rod block. The structure factor is defined as34,35
Figure 1. Variation of the mean square end-to-end distance of R7C5 DBCs in bulk solution and in slit with the rigidity kθ of the rod block. The slit height H = 6. The inset presents the variation of the mean square radius of gyration with kθ.
of polymer in the bulk solution. With the increase in kθ, both and increase rapidly at first and then roughly reach a saturated value when kθ is around 30. However, the effect of slit on the dimension of DBCs is obvious. The equilibrium conformational dimension of R7C5 DBCs in the slit is significantly smaller than that in the bulk solution. That indicates that DBCs are compressed by the surfaces and thus DBCs are in an entropy penalty state. We have also simulated the equilibrium structures of R7C5 DBCs in bulk solution and within R-selective slit with height H = 6, respectively. In the bulk solution, the equilibrium structure of R7C5 DBCs is always a lamellar structure and is almost independent of the rigidity kθ of the rod block (data not shown). However, the equilibrium structure of R7C5 DBCs within the R-selective slit is strongly dependent on kθ. With the increase in kθ we observe three kinds of ordered structures: L∥ at small kθ, cylinders perpendicular to the slit surface at moderate kθ, and L⊥ at large kθ. For the fully flexible R7C5 copolymers (kθ = 0 for the block R), L∥ is formed with the blocks R locating near the slit surfaces due to the attraction between the block R and surfaces, whereas the blocks C form the interior layer in the slit (data not shown). This result is consistent with the previous DPD simulations for symmetric diblock copolymers confined between planar surfaces.32
S(k ⃗ ) =
1 N ′R
2
N ′R
∑ exp(ik ⃗· rj⃗) j=1
(3)
where rj⃗ = (xi, yi) is the position vector (parallel to surface) of the jth rod monomer, N′R is the total number of rod monomers used in the calculation, and the wave vector k⃗ is also a 2D vector parallel to the slit surface. By this definition, S(k)⃗ can be used to analyze perpendicular structures. To avoid the influence of the thin rod layer near surfaces, which is caused by the attraction between the slit surfaces and rod block, rod monomers near surfaces with a distance less than rc are ⃗ Figure 3a plots S(k)⃗ for the excluded in the calculation of S(k). 6269
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clearly observed that the fraction of rod blocks parallel to the surface increases with the rigidity of the rod block. But the coil blocks are always randomly oriented in these three cases (data not shown). Our simulation results clearly show that the surface parallel packing tendency of the rod blocks is a driving force to drive the system into anisotropic structures. The underlying reason is that an aggregate of copolymers adsorbed with the rods parallel to the surface could allow the polymers to gain entropy.16 Therefore, the self-assembly of RC copolymers arises not only from the general thermodynamic incompatibility of the blocks but also from the surface parallel packing tendency of the rod blocks. B. Effect of the Fraction of Rod Block. We have also studied the effect of the fraction of rod block on the selfassembled structures. The relative length of the rod block is represented by the fraction f R = NR/N in RNRCN−NR DBC. In the present simulation, we set the chain length N to 12, the interaction parameter awR to 5, the rigidity kθ of the rod block to 100, and the slit height H = 6. Figure 5 shows the self-
Figure 3. Plots of structure factor of HPC structure at kθ = 10 (a) and that of L⊥ structure at kθ = 50 (b). The unit of wave vector k is 2π/rc (rc = 1). The scale bars shown in plots are 0.1 (2π/rc).
cylinder structure at kθ = 10 shown in Figure 2b. A bright spot in S(k)⃗ plot means its structure factor S(k)⃗ is large. The 2D structure factors correspond to patterns obtained in small-angle X-ray scattering experiments. There are six bright spots at the first six reciprocal vectors in the structure factor shown in Figure 3a, indicating that it is an HPC structure at kθ = 10. The average value of the structure factor at the six reciprocal vectors, S(Q), is about 0.18 for the HPC structure at kθ = 10, whereas it is about 0.09 for the DC structure at kθ = 5. Besides the small value of S(Q) for the DC structure, a more significant difference is that there are more than six bright points in the S(k)⃗ for the DC structure. In addition, the L⊥ structure formed at kθ = 50 can be characterized by two bright spots in the 2D structure factor as shown in Figure 3b. The two bright spots correspond to onedimensional reciprocal vector of the L⊥ structure. The mean value of the vector length of the first reciprocal vectors is about 0.13 (2π/rc) for HPC and 0.11(2π/rc) for L⊥. We then estimate the lattice spacing (the periodic distance of the ordered lattice) between cylinders is about 8.9 and that of lamellae is about 9.1. The rod block of aggregated copolymers could have two possible configurations: parallel or perpendicular to surfaces.16 We have calculated the orientation of the rod blocks in the L∥, HPC, and L⊥ structures at kθ = 0, 10, and 50, respectively. The orientation is characterized by the cosine of the angle θ between the surface normal and the end-to-end vector of the rod block, i.e., cos θ = (R⃗ ·ez⃗ )/(R) with R⃗ the end-to-end vector and ez⃗ a unit vector normal to surface. Thus, cos θ = 0 (i.e., θ = 90°) indicates a parallel orientation, whereas cos θ = 1 (i.e., θ = 0°) indicates a perpendicular orientation. The distribution functions f(cos θ) at various kθs are presented in Figure 4. We find that rod blocks are randomly oriented with a roughly uniform distribution in the L∥ structure, whereas more rod blocks are oriented parallel to surface in the L⊥ structure. It is
Figure 5. Top views of the equilibrium structures assembled by RNRCN−NR DBCs within R-selective slit for three values of NR = 4 (a), 6 (b), and 8 (c). Other parameters are awR = 5, kθ = 100, and H = 6.
assembled equilibrium structures for different RC DBCs within the slit. For short rod block with f R ≤ 1/3, such as NR = 4, disordered structure is observed (Figure 5a). Cylinders are assembled for NR = 5 and 6. Figure 5b shows the equilibrium HPC structure formed by R6C6 copolymers. It is found that the lateral size of cylinders becomes bigger with increasing NR. When NR is increased to 7, 8, and 9, ordered L⊥ are observed, as shown in Figure 5c for NR = 8. However, with further increase of NR to 10, a disordered structure appears again (data not shown). It can be briefly summarized that RNRCN−NR DBCs with N = 12 can form HPC structure when f R is about 1/2 and form well ordered L⊥ when f R is in the region of 7/12−3/4. Therefore, the dependence of the self-assembled morphology on the length and the rigidity of the rod block in a slit can be summarized in the phase diagram shown in Figure 6. The phase diagram is obtained by checking structure at different NR with increment step 1 and different kθ with increment step 1 near phase boundary. There are mainly five structures: disordered structure (D), L∥, L⊥, DC, and HPC. In the phase diagram, small coexisting phase regions, such as coexistence of DC and L∥ near the boundary of DC and L∥, are not shown, but a large coexisting region of DC and L⊥ (DCL⊥) is presented. Specially, we have D at NR ≤ 4 and NR ≥ 8, and L∥ at 4 < NR < 8 for coil−coil copolymers with kθ = 0, respectively. The result of the symmetric phases at kθ = 0 is consistent with the mean field result of copolymer in bulk solution.36 However, we do not observe HPC at kθ = 0. The possible reason is that χN is small in our case, where χ is the Flory−Huggins rod/coil interaction parameter. For long copolymers with large N, one may observe HPC between DC and L∥, whereas asymmetric phases are
Figure 4. Probability distribution functions f(cos θ) for the orientation of rod block in R7C5 DBCs at various bending modulus kθ of the rod block. 6270
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Here we focus our simulations on two confined systems R6C6 and R7C5 DBCs with a high rod rigidity kθ = 100. These two systems self-assemble into different structures at H = 6 as shown in Figure 6: R6C6 forms an HPC structure while R7C5 forms an L⊥ structure. We have simulated the self-assembled structures of R6C6 and R7C5 DBCs in slits with H changing from 4 to 30, and we have checked the structures at different heights z in the slit as in experiments.21 For each equilibrium structure, we check the top view of the structure at different z by removing all DPD particles above z. Here z is varied from 0 to H. The structures are symmetric to z = H/2 layer since two surfaces are identical. Figure 7 shows the dependence of equilibrium structures on the slit height H for R6C6 copolymers. For narrow slits with H
Figure 6. Phase diagram of the self-assembled structures by RNRCN−NR DBCs within an R-selective slit with respect to the rod length (NR) and the rigidity (kθ) of the rod block. The slit height H = 6 is used. Notations in the phase diagram are D, disordered structure; L∥, parallel lamellar structure; DC, disordered cylinder structure; L⊥, perpendicular lamellar structure; HPC, hexagonally packed cylinder structure; and DCL⊥, coexistence of DC and L⊥.
observed for the RC copolymers at kθ > 0 due to the confinement of slit and the asymmetry of copolymers themselves. For short RC DBCs with NR ≤ 4, the interaction between rod blocks is too weak to set up an ordered structure. Therefore, we observe disordered structure below NR = 4 for all kθs. Other structures for RC DBCs with NR > 4 are determined by the competition between the chain-deformation energy and the attraction energy between rod blocks and surfaces. The L⊥ structure, which can freely relax to that of the bulk spacing, can reduce the chain-deformation energy in thin films,1 whereas the L∥ structure can reduce the surface energy because of the attraction between surface and rod blocks but increase the chain-deformation energy. In contrast to the coil−coil DBC thin films, the chain-deformation energy of RC DBC may dominate because of the intense restriction of copolymers under confinement, thus L⊥ occupies predominantly a wide region in the phase diagram at high f R and kθ. However, the physics of forming HPC is not yet clear. Similar hexagonalstructured film was successfully fabricated by drop casting coil− rod−coil triblock copolymers containing di(styryl)-anthracene unit as the rod block and oxadiazole homopolymers as the flexible block.37 Jenehke et al.38 reported the HPC structure of poly(phenylquinoline)-block-polystyrene RC DBC film. Similar ordered microporous honeycomblike structure was also observed in poly(p-phenylenevinylene)-b-polystyrene RC DBC film.39 It is considered that such supramolecular organization arises not only from the general thermodynamic incompatibility of the blocks but also by the tendency of the rod blocks to order into anisotropic structures. C. Effect of the Slit Height. It is known that film surfaces have a profound effect on the structure of coil−coil block copolymers by inducing surface reconstructions.40 Our results in the above two sections show that slit surfaces have a more profound effect on the structure of RC DBCs. In this section, the effect of the slit height is investigated. Segalman’s group discovered that the orientation of lamellae in thin films was dependent on the depth for poly(alkoxyphenylenevinylene-bisoprene) (PPV-b-PI) RC block copolymers.11,21 Especially, an exponential decay of domain spacing of L⊥ with increasing film thickness was reported.21 SCFT study also showed that the structure of RC DBCs in the interior might be different from that near the surfaces.1
Figure 7. Dependence of self-assembled structures at different layer z of R6C6 DBCs on the slit height H. Two surfaces are at z = 0 and z = H. Inset: panel (a) shows gyroid-like (G) structure formed at H = 12; panels (b) and (c) show HPC and L⊥ structure assembled in the slit with H = 18, respectively.
< 8, we find HPC structure throughout the whole slit. The copolymer is strongly confined in the narrow slits, and thus the configuration of copolymer is distorted. The result implies that the formation of the HPC structure is due to the confinement of surfaces as well as the rigidity of rod block. The confinement of slit weakens with the increase of H. However, a mixture structure of HPC near surfaces and L⊥ in the interior region is observed at H = 10 and 11. This indicates that the confinement effect of surfaces on the RC DBCs exists near surfaces but dies away in the interior away from the surfaces at large H. Thus, an L⊥ structure forms in interior region of slit. In thin films, L⊥ can relax to that of the bulk spacing.1 Meanwhile, the cylinder structures near the surfaces still exist. Interestingly, gyroid-like (G) structure is observed in the interior region of slit with H = 12 and 13, although the cylinder structure near the surfaces remains. Panel (a) in Figure 7 shows the G structure seen on the sixth layer (z = 6) in the slit with H = 12. When H goes up to 14−20, the G structure disappears and L⊥ occurs again in the interior of film. As an example, panels (b) and (c) in Figure 7 show the top view taken on the surface layer (z = 0) and the central layer (z = 9) at H = 18, respectively. With further increase of H to 21 and 22, the G structure appears again. Afterward, L⊥ or L∥ (represented by the symbol B in Figure 7) is observed in the interior region of slit. In wide slits, the structure in the interior region shows the bulk property. The results clearly imply that the confinement of surfaces on the RC DBCs is important in the formation of the HPC structure. The 6271
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coexistence of different structures at high H is due to the competition between surface and bulk. Surface tends to establish HPC structure, whereas the interior region presenting bulk property tends to establish lamellar structure. However, these two structures are stable and will not transform from one to another. On the other hand, for R7C5 copolymers, except the G structure formed at H = 12 and 13, L⊥ structure is always formed throughout the film when H changes from 4 to 30. We have measured the lamellar domain spacing DL, i.e., the period of the L⊥ structure. The dependence of DL on H is presented in Figure 8 for H varying from 4 to 18. The variation of DL with H
copolymers with high rigidity in the rod block. For R6C6 copolymers, different structures near surfaces and in the interior region are observed in a relatively wide slit. A phase diagram of the self-assembled structures with respect to the slit height is obtained. However, for R7C5 copolymers, L⊥ structure is the most likely structure, and the lamellar domain spacing decays exponentially with the slit height.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China under Grants 21171145 and 11374255.
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REFERENCES
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Figure 8. Domain spacing DL of R7C5 perpendicular lamellae as a function of slit height H. Here G represents the gyroid-like structure formed at H = 12 and 13. The solid line, a guide to eyes, shows an exponential decay.
can be expressed as an exponential decay, in agreement with previous results.1,21 Such an exponential decay is valid for large H > 18 though the values of DL are not presented in Figure 8. For wide slits with H > 28, the system gradually shows bulk property as we find that the lamellar structure can be either perpendicular or parallel to surface.
IV. CONCLUSION In this work, dissipative particle dynamics simulation is performed to investigate the effect of the rigidity kθ and the fraction f R of rod block on the self-assembly of RNRCN−NR rod− coil diblock copolymers in R-selective slit. We find that both the rigidity and the fraction of the rod block play important roles in the self-assembly of rod−coil diblock copolymers. The results are summarized in a phase diagram presenting the selfassembled structures with respect to the rigidity and the fraction of the rod block for N = 12. There are four main structures in the phase diagram including disordered cylinder (DC) structure, lamellar structures parallel (L ∥ ) and perpendicular (L⊥) to surface, and hexagonally packed cylinders (HPC) perpendicular to the slit surfaces. HPC structure is observed in a large region of kθ for the copolymers with f R close to 1/2. Copolymers with longer rod block favor to form L⊥ with the increase in kθ. We also observe phase transitions between different phases. For example, the transition from L∥ to HPC and ultimately to L⊥ with the increase in the rigidity of the rod block for the confined R7C5 rod−coil diblock copolymers. Our results suggest an effective way to control the ordering of rod−coil diblock copolymers within nanoscale confinement. We have also studied the effect of slit height on the selfassembled structure of R6C6 and R7C5 rod−coil diblock 6272
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