Simulation study on the assembly of rod-coil diblock copolymers within coil-selective nanoslits

1 Introduction

The self-assembly of diblock copolymer (DBC) is an important issue in science and technology. Significant efforts both experimentally and theoretically have been devoted to understanding the physics behind the self-assembly and further controlling it (1, 2). When DBCs are confined in thin film, the polymer-surface interaction and the film thickness play significant roles in the self-assembly of DBCs. Therefore, the self-assembly of DBCs in film becomes more complex (2–5). For example, when smectic rod-coil (RC) DBC melts in bulk were confined in film, a lot of new structures, such as ordered honeycomb structures with hexagonal, close-packed air-holes, islands and holes, zigzag morphology, were observed in experiments (6–11).

The new structures could be generally explained as that smectic RC DBC melts in film were strongly constrained (12), thus the difference in chain conformational entropy between rod and coil blocks influences the assembly of DBCs (6, 13–15). Islands and holes in RC DBC films were attributed to the incommensurability between the initial film thickness and the natural domain spacing of DBCs (6). However, the mechanism for the self-assembly of RC DBCs in film is not clear. In experiments, it was found to be very difficult to obtain equilibrium assembly structures of RC DBCs in thin films as they could form aggregates or micelles and the structures were thus often kinetically trapped (6, 11). Therefore, theoretical studies and computer simulations are important tools to understand the equilibrium assembly structures of RC DBCs confined in thin films.

By using extended scaling methods, Nowak and Vilgis (16) concluded that when RC DBCs were adsorbed on the surface, the rod blocks tended to arrange parallel to surface to gain entropy and to therefore lower their confinement energy. By self-consistent field theory (SCFT) calculation, Yang et al. (11) found that the rod block showed a strong tendency to segregate near surfaces in all structures due to its less conformational entropy loss than the coil block. The conclusion is correct for DBCs confined with neutral or rod-selective surface. In our previous paper, we have investigated the self-assembly of lamella-forming RC DBCs within a rod-selective slit based on dissipative particle dynamics (DPD) simulations (17). The self-assembled structure was found to be sensitively dependent on the rigidity and fraction of the rod block and the slit thickness. Several ordered structures, such as hexagonally packed cylinders perpendicular to surfaces, lamellae perpendicular and parallel to surfaces, are assembled. And RC DBCs with longer and more rigid rod block favor forming perpendicular lamellae (17).

However, it was pointed out that different surface preferences of the two blocks could directly affect the orientation of rod blocks near the surfaces as well as the equilibrium structure of RC DBCs (7, 18). Therefore, RC DBCs in the coil-selective slit could show different structures from that in rod-selective one. Moreover, it was found that the assembly structure was dependent on many factors. For example, Radzilowski (9) observed that the morphology of RC DBCs in film changed from alternating strips to hexagonal superlattice of rod aggregates with a decrease in the fraction of the rod block. Therefore, rod length is also an important factor in the assembly of DBCs.

In the present work, the self-assembly of RC DBCs confined in a slit with two coil-selective surfaces was studied by DPD simulations. The effect of rod length and slit thickness on the assembly structure was systematically investigated. In the simulations we used RC DBCs with small volume fraction of rod block which form separated aggregates in bulk (19). Less attention has been paid to the aggregate-forming RC DBC systems either from experiments or from simulations because it is more difficult to imagine the structures in detail when they are confined in slits (5). We found several ordered structures, such as perpendicular cylinders, parallel cylinders, and puck-shaped structure, and we presented a morphological phase diagram as a function of slit thickness and rod length. In the assembly structures, long-range rod-rod orientational order was observed when the rod length exceeded a critical rod length.

2 Model and simulation method

The DPD method was developed by Hoogerbrugge and Koelman (20) and cast in the present form by Español (21). In DPD simulations, fluid elements including polymers and solvents are coarse-grained into DPD particles and they interact with each other via pairwise forces that locally conserve momentum leading to a correct hydrodynamic description (22). The pairwise forces contain the conservative force Fij(C)=aijw(rij)r^ij, dissipative force Fij(D)=-γw2(rij)(r^ij⋅vij)r^ij, and random force Fij(R)=σw(rij)θijr^ij. Here r_i=|r_ij|=|r_i-r_j|, r^ij=rij/rij,v_ij=v_i-v_j, θ_ij 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=2γk_BT. Here k_B is the Boltzmann constant and T is the temperature. The softness of the interaction is determined by the weight function w(r_ij) with a commonly used choice: w(rij)=1-rij/rc for r_ij≤r_c and w(r_ij)=0 for r_ij>r_c, where r_c is the cutoff radius. Interaction parameter a_ij is the strength of the repulsive interaction dependent on the species of particles i and j. Values of a_ij will be presented subsequently.

DPD simulations are carried out in a slit consisting of two identical impenetrable parallel walls along z direction. Each wall with a thickness of 1r_c is constructed by four layers of DPD particles that are arranged in a face-centered cubic lattice. The centers of the wall DPD particles along the normal direction are placed at 1/8, 3/8, 5/8, and 7/8 of the wall. The high density of particles in the wall prevents other DPD particles from penetrating the solid wall, although the interactions between coarse-grained particles are soft. The high density of wall introduces an asymmetry in the system and results in an extra force on DPD particles in the vicinity of the wall, giving rise to a density fluctuation near the wall. However, the fluctuation does not obviously influence the equilibrium structure. For simplicity, the wall DPD particles are motionless in simulations. The slit thickness H is defined as the distance between the top of the lower wall and the bottom of the upper wall. Periodic boundary conditions (PBCs) are applied in the x and y directions.

RC DBC is modeled as a coarse-grained linear chain RNRCNC with N_R DPD particles in the rod (R) block and N_C DPD particles in the coil (C) block. Finitely extensible non-linear elastic (FENE) interaction is adopted for the chemically bonded monomers. The bonded FENE interaction U_FENE is represented by (23)

where the equilibrium bond length r_eq=0.8, the maximum bond length r_max=1.3, and the elastic coefficient k_F=200. For the rod block, an additional bending energy between consecutive bonds is introduced as (24)

with k_θ and θ₀=π are the bending modulus and the equilibrium angle between two consecutive bonds, respectively. Specially, k_θ=0 corresponds to a flexible block like the coil block in the present model. The rigidity of the rod block increases with an increase in k_θ. In this work, we choose a relatively large value k_θ=50. We find that the average bond angle <θ> in the rod block is about 170°, indicating that the rod block is very rigid.

We have checked the bond length distribution in R₅C₁₃ diblock copolymers in bulk. We find that they are Gaussian distribution and roughly the same for the rod and coil blocks, indicating that the distribution is determined by the FENE potential itself. The distribution peak and the average bond length are close to r_eq=0.8. Moreover, we find roughly the same bond length distribution for copolymers in bulk with different system sizes or even confined within slit.

The overall particle density ρ is set as 3 throughout this study. In our system, there are three species of DPD particles: R and C monomers in the copolymer chain, and wall particles of slit (W). The repulsive interaction parameter between identical species is set to be 25, i.e. a_RR=a_CC=a_WW=25. The slit surface is assumed to be preference for coil block with a_WC<25. While we keep a_WR=25 that means the slit surface is repulsive for the rod block.

In our DPD simulations, all DPD particles are of the same mass with m=1. We set the cutoff distance r_c=1 as the unit of length and k_BT=1 as the scale of energy. Thus, the unit of force is k_BT/r_c, and that of time is τ=mrc2/kBT. In this work, the amplitude of random noise is set as σ=3.

DPD particles move according to Newton’s equation mid2ridt2=fi, with f_i is the force experienced by particle i. The position and velocity of particles are solved using a modified velocity-Verlet algorithm proposed by Groot and Warren (25). In the simulations, a time step Δt=0.01τ is adopted for the integration of Newton’s equation. We have monitored the structure of system at every 1000τ. It usually takes less than 80,000τ to form an ordered structure. However, the time increases with the increase in the length of rod block. For N_R=10, the dynamic process usually lasts for 100,000τ. After ordered structure is formed, another 10,000τ is spent to ensure the structure is stable.

3 Results and discussion

In the present work, the volume fraction of R block in the RC chain, f_R=N_R/(N_R+N_C), is approximately fixed at 0.28. The interaction parameter between R and C monomers is chosen as a_RC=32.9 to ensure the formation of aggregate phase in bulk for R₅C₁₃ DBCs (19). Our DPD simulation shows that R₅C₁₃ DBCs form separated rod aggregates in bulk as shown in Figure 1. The simulation is performed in a 30×30×30 box with PBCs in all three directions. It is clear to see that R-rich domains form separated aggregates embathed inside the C-rich matrix, in agreement with theoretical prediction based on SCF lattice model (19).

Figure 1:

The equilibrium structure assembled by R₅C₁₃ DBCs in bulk. R blocks are represented by red and C blocks by blue, respectively. Same colors are used in the remaining figures. System size L_x=L_y=L_z=30, and interaction parameter a_RC=32.9.

When R₅C₁₃ DBCs are confined within coil-selective nanoslits, different structures are assembled depending on the interaction between the slit surface and coil block a_WC and slit thickness H. Previous work showed that rod length was a major factor which led to the disorder-order transition of rod orientation in rod-coil copolymer systems and the coil length had a secondary effect on the orientational order of rod blocks (26, 27). Thus, we study the effect of rod length by varying N_R from 5 to 9 with f_R kept at 0.28. For some typical structures, N_R=10 is also studied. We find that the assembly structure of RC DBCs is sensitively dependent on the rod length N_R.

Simulations for the confined RC DBCs are performed in systems with lateral size L_x=L_y. We find that the assembly structures are almost independent of the lateral size when it reaches 40 for N_R≤8. While L_x=L_y=50 is sufficiently large for N_R=9 and 10. The effect of lateral size on the assembly structure is negligible in this work.

The first finding of our simulation is that only cylinders perpendicular to surfaces (C_⊥) are formed when RC DBCs are confined within a weak coil-selective slit with 15≤a_WC<25. And the structure is independent of slit thickness H. Figure 2A shows a typical equilibrium structure formed by R₅C₁₃ in a slit with a_WC=20 and H=8. To quantitatively analyze the C_⊥ structure, we have calculated two-dimensional (2D) structure factor for the rod blocks. The structure factor is defined as (17)

where r→j=(xi,yi) is the position vector (parallel to surface) of the jth R monomer, the wave vector k→ is also a 2D vector parallel to the slit surface, and N′R is the total number of R monomers used in the calculation of S(k→). The 2D structure factors correspond to patterns obtained in X-ray scattering experiments. Bright spot in S(k→) plot indicates large value of its structure factor. There are six bright spots at the first six reciprocal vectors in the structure factor in Figure 2B, indicating that rod blocks in C_⊥ are packed in a hexagonal lattice perpendicular to surface. Such a hexagonal arrangement in thin film has been proved experimentally and theoretically (5, 28).

Figure 2:

The equilibrium structure of R₅C₁₃ DBCs formed in a weak coil-selective slit with a_WC=20 and H=8. (A) Side view (top) and top view (bottom), and (B) plot of structure factor.

However, the assembly structure is dependent on rod length N_R and slit thickness H when RC DBCs are confined within the strong coil-selective slits with a_WC=5. The simulation time for obtaining assembly structures increases dramatically with the increase in N_R or H as the relaxation time increases with N_R and the number of particles increases with H. Therefore, we restrict our simulations for N_R varied from 5 to 9 and H from 4 to 20 in the present study.

The assembly structure of DBCs is dependent on the surface property of slit. At a_WC=5, R blocks of R₅C₁₃ DBCs assemble into parallel cylinders in the interior of slit with H=8. We have calculated the density distribution along the Z-axis for the assembly structures of R₅C₁₃ DBCs at a_WC=5 and 20. The results are presented in Figure 3. Here we set z=0 at the center of two surfaces. Particles are distributed symmetrically in the slit from z=-4 to z=4, indicating that no copolymer particle can penetrate into the wall. The result shows that the high density of slit surface can prevent polymer particles from penetrating the surface. However, it introduces the disruption of homogeneity in the system and results in a density fluctuation near the surface (29). We find that there are density fluctuations near surface for both R and C blocks for the weak selective slit with a_WC=20. Whereas for the strong selective one with a_WC=5, coil blocks are attractive to surfaces that drives rod blocks to interior space in the slit. In this case, the density fluctuation near surface is found only for C blocks. However, the density fluctuations near surface in these systems are quite weak comparing with the whole density distribution. On the other hand, a significant difference in density distributions is found for these two systems. Thus, our result indicates that the different assembly structures of these two systems are resulted from the different properties of surface.

Figure 3:

Density distributions for R₅C₁₃ DBCs within different slits. (A) a_WC=5 and (B) a_WC=20. Slit thickness is H=8, and the center of the system is set as z=0.

Figure 4 presents the assembly structures of RC DBCs against N_R and H. It shows that the assembly structure is dependent on H. We observe C_⊥ and C_||,ν (ν represents the layers of parallel cylinders) structures dependent on H for N_R=5. Within narrow slits with H<10, R blocks assemble into C_||,1 structure. The radius of cylinders increases with an increase in the slit thickness. Interestingly, degenerate structures of C_||,1/C_⊥ and C_||,2/C_⊥ are observed at H=10 and 12, respectively. When the slit thickness is in the region of 12<H<20, two-layer cylinders (C_||,2) of R blocks are formed. And three-layer cylinders (C_||,3) are formed at H=20. From the side view, we can see that the cylinders are hexagonally packed. The series of structures C_||,1-C_||/C_⊥-C_||,2-C_||,3 of R₅C₁₃ DBCs are different from C_⊥-C_||,1-PL₁-C_||,1-C_||,2-PL₂ phases of fully flexible DBCs with the same composition (30), implying the important role of the chain rigidity playing in the confined assembly. While for the DBCs with N_R=6, 7, and 8, no degenerate structure is observed. Only C_⊥ is formed before C_||,1 is converted into C_||,2. For R₉C₂₃ DBCs, C_||,1 is formed for H varied from 6 to 18, and C_||,2 is observed at H=20.

Figure 4:

Dependence of the morphologies of RNRCNC DBCs on the rod length N_R and slit thickness H within coil-selective slits with a_WC=5. Shown structures are: perpendicular cylinders (∇), one-layer parallel cylinders (▼), two-layer parallel cylinders (□), three-layer parallel cylinders (★), puck-shaped structure (◊). Typical snapshots of four structures (▼, □, ★, and ◊) are presented.

We also notice in Figure 4 that the structure is dependent on N_R, and different structures are observed for big N_R at the same H. For example, at H=4, C_||,1 is assembled in the interior of slit for N_R=5, 6, and 7. However, a puck-shaped structure is formed at N_R=8 and 9. The inset of Figure 4 shows the puck-shaped structure formed at N_R=9. Similar puck-shaped phase has been reported in previous papers (11, 31). The authors thought that in the puck-shaped phase the majority component coil blocks can fan out into large region of space, hence stretching less while inducing the interfacial energy penalty (stretching energy is proportional to the volume fraction of coils) (11, 31). They proposed that the arrangement of RC DBCs in the puck-shaped phase was a combination of smectic-C bilayers and monolayer structure.

Our results above show that some phases of RC DBCs confined in slit are different from that of fully flexible DBCs. It was pointed out that the effect of rod block became more evident for longer rod blocks and there was a critical rod length ≈8–9 inducing the disorder-order transition in the alignment of rod blocks (31). To quantify the alignment of rod blocks within the C_||,1 structure, a rod-rod orientational order parameter s(r) is calculated. It is calculated for each single cylinder and then averaged over all cylinders. Similar to that defined in Ref. [26], s(r) is defined as

in cylinder. Here θ_ij is the included angle between rod i and rod j, r_ij is the distance between mass centers of rod i and rod j. A sketch for θ_ij and r_ij is presented in Figure 5. The delta function is defined as

Figure 5:

A sketch of two RC DBCs, rod i and rod j, near surface. Here θ_ij is the included angle between two rods, r_ij is the distance between mass centers of rod i and rod j, r→i is the position vector of the mass center of rod i, and ϕ_i is the angle between rod i and the surface normal direction.

By this definition, s(r) represents the space correlation between rod blocks. We have s=0 for a completely random and isotropic sample and s=1 for a perfectly aligned sample. For a typical liquid crystal sample, s is in the order of 0.3 to 0.8 (26).

We find C_||,1 is formed at H=6 for N_R varied from 5 to 9. The corresponding s(r) in C_||,1 is calculated and shown in Figure 6A. For N_R≤7, s(r) is small and decays quickly with distance, showing that rod blocks are roughly randomly oriented in long length scale. However, for N_R≥8, s(r) is large and decays slowly with distance, indicating a long-range orientation order for rod blocks. In Figure 6B, we present the mean orientational order parameter <s> for different N_R. The value <s> is averaged over all rod-rod pairs in the C_||,1. We find that <s> increases obviously from N_R=8, indicating a disorder-order transition at a critical rod length N_R*=8, which is close to N_R*=9 for RC copolymers in solution (26). Our results indicate that the coil-selective surface does not obviously affect the critical rod length N_R* because the cylinders do not locate near the surface.

Figure 6:

The rod-rod orientational order parameter for different DBCs in C_||,1 structure within a coil-selective slit with a_WC=5 and H=6. (A) Dependence of the rod-rod orientational order parameter s(r) on rod-rod distance r, and (B) Dependence of the mean rod-rod orientational order parameter <s> on rod length N_R.

The orientation of rod blocks in the C_||,1 structure is further analyzed by the angle probability distribution P(ϕ) as well as the mean angle <ϕ>. We have calculated the angle ϕ_i between the rod i and surface normal direction, i.e. cosϕi=R→i⋅e→zRi with e→z the unit vector normal to surface. Here ϕ_i represents the orientation of rod i as shown in Figure 5. Thus <ϕ>=0° indicates a perpendicular orientation of rods and 90° indicates a parallel orientation of rods. Figure 7 presents the angle distribution P(ϕ) of rod blocks for R₅C₁₃, R₇C₁₈, and R₉C₂₃ DBCs, respectively. We find P(ϕ) of R₅C₁₃ DBCs is close to random probability distribution P(ϕ)=sinϕ, indicating that rods are almost randomly oriented. And <ϕ>=62° (Inset in Figure 7) of R₅C₁₃ DBCs is also close to the average angle 57° for random orientation. Such a random direction of the rod blocks produces an orientational freedom for the system. Whereas more rod blocks are oriented parallel to surface for R₇C₁₈ and R₉C₂₃ DBCs as the probability becomes big at high ϕ and increases with N_R. It is clearly observed that the fraction of rod blocks parallel to the surface increases with N_R, and thus the mean angle <ϕ> increases with N_R, see the inset in Figure 7. Thus, although the coil-selective surface does not obviously affect the critical rod length N_R*, rod blocks tend to orient parallel to the surface for rod length above N_R*.

Figure 7:

Probability distribution of angle P(ϕ) of rod blocks for different RNRCNC DBCs in C_||,1 structure within a coil-selective slit with a_WC=5 and H=6. The black solid line represents the probability distribution of sinϕ. The inset plots the mean angle <ϕ> versus N_R.

4 Conclusion

In this work, the self-assembly of rod-coil RNRCNC DBCs confined in coil-selective slit is studied. We find that RC copolymers with f_R≈0.28 form separated aggregates in bulk but they form different assembly structures in slit. The assembled structures of confined RC DBCs are dependent on the property of slit. In weak coil-selective slits, only C_⊥ structure is observed independent of the slit thickness. Whereas in strong attractive slit, the structure depends on rod length N_R and slit thickness H. A morphological phase diagram of assembly structures against H and N_R is presented for the strong coil-attractive surface.

The rod-rod orientational order is studied for the typical C_||,1 structure in strong coil-selective slit at H=6. For N_R≤7, the order parameter is small and decays quickly with distance. And a critical rod length N_R*=8 is observed for a disorder-order transition of rod orientation. Rod blocks orientate parallel to surface for long rod block larger than N_R*. Our results show that the coil-selective slit influences the assembly structure as well as the rod orientation of RC copolymers.