Cylindrically confined assembly of diblock copolymer under oscillatory shear flow

Manipulating the self-assembly nanostructures with combined different control measures is emerging as a promising route for numerous applications to generate templates and scaffolds for nanostructured materials. Here, the two different control measures are a cylindrical confinement and an oscillatory shear flow. We study the phase behavior of diblock copolymer confined in nanopore under oscillatory shear by considering different $D/L_0$ ($D$ is the diameter of the cylindrical nanopore, $L_0$ is the domain spacing) and different shears via Cell Dynamics Simulation. Under different $D/L_0$, in the system occurs different morphology evolution and phase transition with the changing of amplitude and frequency. Meanwhile, it forms a series of novel morphologies. For each $D/L_0$, we construct a phase diagram of different forms and analyze the reason why the phase transition occurs. We find that although the morphologies are different under different $D/L_0$, the reason of the phase transition with the changing of amplitude and frequency is roughly the same, all caused by the interplay of the field effect and confinement effect. These results can guide an experimentalist to an easy method of creating the ordered, defect-free nanostructured materials using a combination of the cylindrical confinement and oscillatory shear flow.


Introduction
Block copolymer is a class of soft material capable of self-assembling to form ordered structures at nanometer scales. These structures possess a potential for various applications to nanotechnologies such as lithographic templates for nanowires, photonic crystals, high-density magnetic storage media, drug delivery and biomineralization. So, the self-assembly of block copolymer has attracted much interest as an efficient and effective means to create structures at nanometer scales. Obviously, how to manipulate these self-assembled nanostructures have become the key of the research, so understanding the controlling factors is an intriguing field of research that holds promise for further expanding the nanofabrication. In general, the control measures mainly include a confined environment [1,2], imposed external field [3,4], substrate induction [5,6], doping [7,8], and branched molecular architectures [9] etc. Thereinto, confinement effect makes the polymer system form a series of novel morphologies that are not accessible in the bulk, and the external effect makes the structures more ordered. They both provide an effective route to manipulate the self-assembled nanostructures.
During the past years, the effects of a confined environment have been extensively explored based on a large number of experimental and theoretical studies. The internal branching is one of the confinements, i.e., one imposes some restrictions on the conformational freedom. For example, Ilnytskyi et al. analyzed the peculiarities of the equilibrium morphologies observed for the star and comb diblock polymers with equal molecular mass but with the differences in both intramolecular architecture and composition fraction [9]. In addition, the geometrical confinement can be classified as one-, two-, and three-dimensionally confined systems. The one-dimensionally confined system [10,11] is the simplest case, and its research is very mature. For a more complex three-dimensionally confined system, the studies are relatively few so far [12,13]. The research area of two-dimensional confinement is developing rapidly. When placing the block copolymer in a cylindrical pore, a series of novel morphologies that were not accessible in the bulk or in the one-dimensionally confined systems were observed at different degrees of confinement D/L 0 . Russell and co-workers discovered that the lamellae-, cylinder-, and sphere-forming block copolymer of polystyrene-b-polybutadiene (PS-b-PBD) were confined in the pores of anodized aluminumoxide (AAO) membranes experimentally [14][15][16][17][18]. They find that under cylindrical confinement, block copolymer formed various kinds of novel structures that were not accessible in the bulk, such as stacked disks, concentric cylinders, torus-like structure, core-shell structure, single row and zigzag arrangement of spherical microdomains, single-, double-, triple-helical structures etc. Sun et al. also observed the diameter-dependence of morphology confined within the ordered porous alumina templates, but the block copolymer that they used was polystyrene-b-poly (methyl methacrylate) (PS-b-PMMA) [19] and polyethylene-co-butylene-b-polyethylene oxide (PHB-b-PEO) experimentally [20]. For symmetric lamellae-forming diblock copolymer in cylindrical pores, many novel morphologies were predicted based on dynamic density functional theory (DDFT) [21][22][23], Monte Carlo (MC) [24,25], selfconsistent field theory (SCFT) [26], and simulated annealing technique (SAT) [27]. For the asymmetric cylinder-forming diblock copolymer, the structures in the cylindrical pores severely deviated from the bulk. The novel structures of spontaneous formation, such as stacked toroids, single helix, double helix and perforated tubes, were observed by means of a SAT [28,29] and SCFT [30][31][32]. For the asymmetric sphere-forming diblock copolymer, the study of it was rare. Pinna et al. predicted tremendous rich morphologies of sphere-forming diblock copolymer in cylindrical nanopores by using the cell dynamics simulation (CDS) [33]. Recently, the phase behavior of sphere-forming triblock copolymer confined in nanopores was investigated by Hao et al. by means of a DDFT [34]. They observed that typical structures were different from the bulk morphologies, which were consistent with the available experiments.
As we all know, the polymer system can form all kinds of novel structures that are different from the bulk morphologies under cylindrical confinement and form ordered structures under shear flow. So, we wonder what the phase behavior of polymer system is under both these two control measures, and it should be interesting when these two control measures affect the system. Previously, Pinna et al. [60] combined the two control measures of the one-dimension confinement and steady shear flow. They demonstrated the shear alignment and the shear-induced transitions in sphere-forming diblock copolymer single layer and bilayer films by cell dynamics simulation that was observed experimentally by Hong et al. [61], and for the first time presented a nontrivial alignment mechanism of a single layer of spherical domains in shear. On this basis, we dedicate ourselves to investigate the phase behavior of diblock copolymer within cylindrical pores under oscillatory shear flow by means of the CDS of time dependent Ginzbrug-Landau (TDGL) theory proposed by Oono and co-workers [62][63][64][65][66]. It will further provide some guiding function for experimentalists. The other parts of this paper are organized as follows: section 2 is devoted to the description of the model and simulation method; section 3 is the numerical results and discussions; and finally, section 4 gives a brief conclusion in this work.

Models and simulation methods
Our simulations are performed in a neutral cylindrical nanopore with a diameter D and a length L z .
We employ a TDGL approach in the form of the CDS, which is a very fast computational technique. For AB diblock copolymer, the structure can be described by an order parameter φ(r, t ) = φ A −φ B +(1−2 f ), where φ A and φ B are the local volume fractions of A and B monomers, respectively. The fraction of A monomers in a diblock copolymer chain is denoted by f , f = N A /(N A +N B ). The kinetics and morphological evolution are described, in the spirit of linear irreversible thermodynamics, by the TDGL equation for a diffusive field coupled with an external velocity field, which can be written as [67] ∂φ(r, t ) ∂t where M is a phenomenological mobility constant and is set to 1. r is an external velocity field.
For simplicity, we set the shear rate as ν(r, t ) = 0, 0, γωy cos(ωt ) , where γ is amplitude, and ω is frequency. We set z-axis (the direction of the arrows) is the shear direction.
Here, we use a two-order-parameter model in [68,69]. The long-range part F L and the short-range part F S are given by and respectively. The long-range part is relatively simple, in which G(r, r ) is the Green's function defined by the equation −∇ 2 G(r, r ) = δ(r − r ), while α is a parameter that introduces a chain-length dependence to the free-energy, φ 0 is the spatial averages of φ, we should set φ 0 = 0 in the case of symmetric copolymers. As for the short-range part, D is a positive constant that plays the role of a diffusion coefficient, and f (φ) = −A tanh(φ)+φ. We carry out computer simulations of the model system by using the CDS proposed by Oono and Puri [62][63][64][65][66]. In three-dimensional CDS, the system is discretized on a L x × L y × L z cubic lattice, and the order parameter for each cell is defined as φ(n, t ), where n = (n x , n y , n z ) is the lattice position and n x , n y , and n z are integers between 1 and L. The Laplacian in CDS is approximated by (2.5) where 〈〈φ(n)〉〉 represents the following summation of φ(n) for the nearest neighbors (n.), the nextnearest neighbors (n.n.), and the next-next-nearest neighbors (n.n.n.) [66] 〈〈φ(n)〉〉 = 6 80 n=n. φ(r) + 3 80 n=n.n. φ(r) + 1 80 n=n.n.n. φ(r). (2.6) In our simulations, we set ∆x, ∆y, ∆z are 1, and ∆t = 1.0. Then, equation (2.1) can be transformed into the following difference equation: We have chosen z-axis as the flow direction, y-axis as the velocity gradient direction and x-axis as the vorticity axis. In addition to Dirichlet boundary condition [33], a shear periodic boundary condition proposed by Ohta et al. [70][71][72] has been applied to z direction along the pore. With the shear strain Υ, this boundary condition is written as φ(n x , n y , n z , t ) = φ n x + N x L, n y + N y L, n z + N y L + Υ(t )N y L , (2.9) where N x , N y , and N z are arbitrary integers. All parameters in this paper are scaled, and all of them are dimensionless [62].

Numerical results and discussion
In order to simulate the confined self-assembly of AB diblock copolymer under oscillatory shear flow, we construct the cylindrical nanopore of diameter D in cubic lattice. The volume is V = L x × L y × L z with L x = L y = D + m and L pore = L z , where m = 2 when D is odd and m = 3 when D is even. Polymers cannot occupy the wall sites which are the lattice sites outside the cylinder with diameter D and the wall is impenetrable. The extra m in L x and L y ensures that each site inside the cylindrical pore can find its nearest neighbor, the next-nearest neighbor, and the next-next-nearest neighbor cells. These sites are either inside the pore or in the wall. Thus, the polymers are confined in the cylindrical nanopore of diameter D. In this paper, we investigate the phase behavior under oscillatory shear of diblock copolymer in the cylindrical pore with different D/L 0 , which L 0 is the domain spacing, refer to [32]. Firstly, we investigate the effect of oscillatory shear flow in cylindrical nanopore of D/L 0 = 1.01, L pore /L 0 = 8.1. It is noted that the oscillatory shear is imposed on the disordered structures rather than the final self-assembled structures. With the simulation on the continuous variation of shear amplitude and shear frequency, we roughly construct a structure diagram in figure 1. In figure 1, symbol • represents single-helical structure that is abbreviated as SH, represents half-cylinder structure that is abbreviated as HC. The half-cylinder refers to A phase occupying half of the cylinder along the pipe axis.
In figure 1, the amplitude, γ, varies from 0 to 1.0, and the frequency, ω, varies from 0 to 0.1. We can clearly see that in small amplitudes, such as γ 0.15, the field effect is too weak to have a sufficient capability to affect the morphology, therefore, no phase transition is observed at the given frequency, and the single-helical structure is always preserved. Typical snapshots of this case are shown in figures 2 (a)-(d).
Moreover, in the small frequencies ω 0.02, the structure is single-helical at all amplitudes. At amplitudes 0.2 < γ < 0.35, with an increasing frequency, the single-helical structure transforms to the halfcylinder, and then reverses to the single-helical structure. At the larger amplitude, when we increase the frequency, the phase transition occurs more than once. In addition to the phase transition from the singlehelical structure to half-cylinder, and then to the single-helical structure, the phase morphology turns to half-cylinder finally, as shown in figures 2 (e)-(h). Furthermore, we also observe that the morphology turns directly to the half-cylinder from the single-helical structure with an increasing amplitude at the frequency 0.09. At some frequencies, such as 0.07 ω 0.08, while increasing the amplitude, the phase transition occurs from single-helical structure to half-cylinder, then reverses to a single-helical structure, and then transforms to a half-cylinder structure. On the whole, the single-helical structure mainly concentrates on the region of the weak oscillatory shear flow, which can be seen in the bottom left-hand corner in figure 1; the half-cylinder structure mainly concentrates on the region of the strong oscillatory shear flow, which can be seen in the top right-hand corner in figure 1. Moreover, the critical shear frequency of the phase transition becomes smaller and smaller with an increasing amplitude.
As seen in figure 2 mentioned before, it shows the morphology evolution with an increasing frequency at two typical amplitudes, γ = 0.01 and γ = 0.41. Phase A is represented by black regions, phase B by gray regions. In figures 2 (a)-(d), no phase transition occurs during the frequency and in figures 2 (e)-(h), the phase transition occurs more than once. These results are mainly due to the interplay of two factors: the field effect caused by the oscillatory shear flow and the confinement effect produced by the confinement boundary. Meanwhile, the field effect is the combined effect of both amplitude and frequency. For the single-helical structure in the bottom left-hand corner, it is the same as the structure that is confined in the cylinder with no shear of the same diameter, which is in agreement with the previous studies by Morita [73] and Pinna [33]. They found that the helical structure was usually observed at incommensurate conditions. In our simulation, the length and diameter are slightly incommensurable with the domain spacing; the amplitude and frequency are relatively low, the field effect is weaker while the confinement effect is dominant. Thus, similarly, the single-helical structure is due to the extensional forces when the length, diameter and the lamellae spacing are not commensurate. For the half-cylinder structure in the top right-hand corner, the amplitude and frequency are higher. Relatively speaking, the field effect plays an important role, so we get the half-cylinder structure along the direction of oscillatory shear flow. For the repeated phase transition with an increasing frequency at a certain amplitude in figures 2 (e)-(h), we consider that the coupling of amplitude and frequency plays a certain role. Compared with the field effect, the confinement effect has an absolute superiority when the frequency is weaker. It is easier to form a single-helical structure [figure 2 (e)]. With increasing the frequency, the movement of segments A and B strengthens along the oscillatory shear direction, the coarse graining process is also intensified along the field direction and then the ordered half-cylinder structure is formed [figure 2 (f)]. With a further increase of the frequency, the movement is faster, more and more A and B segments accumulate, they push each other and then it formes a single-helical structure again [figure 2 (g)], and the nature of this structure is different from the helical under a weaker shear. When the frequency is high enough, the field effect plays a leading role, it forms the ordered half-cylinder structure along the pipe axis [ figure 2 (h)].
Then, we concentrate on the phase behavior under oscillatory shear flow in cylindrical nanopore of D/L 0 = 0.76 and L pore /L 0 = 8.1. The simulation snapshots under oscillatory shear flow at different amplitudes γ and frequencies ω are shown in figure 3. As we can see in figure 3, we control the frequency between 0 and 1, and consider the amplitude of various stages. Overall, it preferentially adopts a piledpancakes structure when the shear is low; while it adopts a half-cylinder structure that is parallel to the pipe axis when the shear is high. When γ = 0.005 and γ = 0.01, the domain structure remains the piledpancakes in the entire frequency range. However, the number of pancakes decreases with an increase of frequency when the amplitude γ is 0.01, at the same time, the layer spacing becomes wider. * represents the morphology and has no change, and it is used to distinguish these two different cases. When γ = 0.05 and γ = 0.1, with an increasing frequency, the piled-pancakes structure evolves into half-cylinder structure due to the aggravation of the coarse graining process along the pipe axis. At the further increase of frequency, at the coupling of amplitude and frequency, the system transforms to the piled-pancakes structure and the number of pancakes decreases. Undoubtedly, then it turns to the half-cylinder structure when the shear is strong. It is certain that the critical shear frequency of phase transition and frequency range of each structure are different from each other. This phenomenon also exists in γ = 0.5 and γ = 1.0.
The higher the amplitude is, the lower the critical shear frequency is. The frequency range in the same structure is smaller when the amplitude is higher, just like the piled-pancaked in 0 < ω < 0.0014, γ = 0.5 and 0 < ω 0.0006, γ = 1.0; and the half-cylinder in 0.0014 ω < 0.052, γ = 0.5 and 0.0006 < ω < 0.04, γ = 1.0. Also, the number of pancakes that formed again are less than before, as well, the layer spacing is wider than before in γ = 0.5. However, it forms a mixed structure between piled-pancakes and halfcylinder just in a small range (i.e., 0.04 ω < 0.06) under γ = 1.0. This is chiefly because at γ = 1.0, the vibration amplitude along the pipe axis is large, it is difficult to stack and easy to stretch along the field direction. Certainly, in the end of the frequency range, the system turns to a stable half-cylinder structure of paralleling to the pipe axis.
Similar to the reason of the phenomenon in nanopore of D/L 0 = 1.01, it is the result of the combination of the confinement effect and field effect. At a smaller amplitude, no phase transition mainly depends on the confinement effect. However, the phase transition at a larger amplitude is mainly decided by the size of the frequency. The slower the frequency is, the weaker the field effect is. Thus, it is easier to form piled-pancakes structure, as well as the structure with no shear, which is in agreement with the study by Pinna [33], as a result of the competition between the effect of the neutral surface and the effect of incommensurability. When the shear is weaker, the surface effect prevails at a small diameter, while above a certain diameter, the bulk effect takes over. The first phase transition to the half-cylinder with an increase of the frequency is due to the intensified graining process along the pipe axis caused by the strong oscillation of A and B segments. In case of a further increase of the frequency, these segments would be more piled up in z-axis and jostle each other to form the piled-pancakes structure due to the too fast vibration. The last phase transition is caused by a strong field effect, and the half-cylinder structure is obtained so that A phase and B phase are completely separated. Furthermore, the decrease of the critical frequency with an increase of amplitude also indicates that the phase transition occurs only if the amplitude and frequency intercouple with each other.

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occurs from the concentric ring to the parallel lamellae. Certainly, the intermediate state (i.e., the "Swiss roll" and "horseshoe" structures) also exist. However, when the amplitude decreases to 0.6, we can find that all the structures are a concentric ring in the given frequency range. Due to the amplitude being relatively small, even if the frequency increases to 0.008, the field effect is still weaker than the confinement effect.
As seen in figure 6, it shows the morphologies of full view that corresponds to γ = 1.0 in figure 5.
The three-dimensional morphology evolution with an increase of the frequency in figure 6 gives us a better visual perception than that in figure 6. As for the reason why this phase transition occurs, we can also interpret it as the interplay of the field effect caused by the oscillatory shear and the confinement effect produced by the confinement boundary. The concentric ring structure is formed under this large nanopore with no shear, and the same as the structure of the weaker shears, because the field effect is almost neglected. When the shear is strong, the field effect prevails, then it forms the parallel lamellae along the shear axis. The phase transition in the given frequency range is mainly induced by the amplitude and frequency intercoupling with each other as before.

Conclusions
In the present work, we have combined two different control measures of the cylindrical confinement and oscillatory shear flow to manipulate the self-assembly nanostructures and obtained novel morphologies. Specifically, we have predicted the phase behavior of diblock copolymer confined in nanopore under oscillatory shear by considering the different D/L 0 and different shears. We have found that in nanopore of D/L 0 = 1.01, when the amplitude is small, there is no phase transition observed at a given frequency; while at a certain amplitude, the system undergoes a phase transition from single-helical structure to half-cylinder, and then reverses to a single-helical structure, transforms to half-cylinder structure with an increase of shear frequency, finally. When we put the diblock copolymer in nanopore of D/L 0 = 0.76, the situation is similar, except that the structure of single-helical structure is replaced by piled-pancakes.