Block copolymer templated self-assembly of disk-shaped molecules. Abstract

Transcription

1 Block copolymer templated self-assembly of disk-shaped molecules J.L. Aragones a and A. Alexander-Katz b Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA. (Dated: May 29, 2017) Abstract Stacking of disk-shaped organic molecules is a promising strategy to develop electronic and photovoltaic devices. Here, we investigate the capability of a soft block copolymer matrix that microphase separates into a cylindrical phase to direct the self-assembly of disk-shaped molecules by means of molecular simulations. We show that two disk molecules confined in the cylinder domain experience a depletion force, induced by the polymer chains, which results in the formation of stacks of disks. This entropic interaction, and the soft confinement provided by the matrix, are both responsible for the structures that can be self-assembled, which include slanted or columnar stacks. In addition, we evidence the transmission of stresses between the different minority domains of the microphase, which results in the establishment of a long-ranged interaction between disk molecules embedded in different domains; this interaction is of the order of the microphase periodicity and may be exploited to direct assembly of disks at larger scales. a aragones@mit.edu b aalexand@mit.edu 1

2 Introduction Disk-shaped organic molecules have interesting applications in different fields such as organic photovoltaics [1] and organic electronics [2] due to their unique electronic properties. Therefore, the study of this type of molecules has been an intensive area of research during the past decades. Due to the alternance between single and double bonds in these type of molecules the p-orbitals are connected, which result in a delocalization of the electrons. Thus, these molecules exhibit π-π interactions, which are relatively strong and orientation dependent interactions. Hence, π-stacking interactions play a central role in the self-assembly of diskshaped heteroaromatic molecules and it is particularly suited to produce electron transfer between these conjugated molecules, being excellent candidates to develop organic electronic and photovoltaic devices [3, 4]. However, the strong and orientation sensitive nature of π- stacking interactions may result in the frustration of the assembled structures, making hard to control their self-assembly. Several approaches are possible to direct the self-assembly to ordered structures by removing metastable states. A common practice is to functionalize the disk molecules with alkyl chains on the disk periphery, which direct the self-assembly of ordered structures such as lamellar, perforated lamellar and crystal phases [3, 5, 6]. Other possible approach is to confine these molecules to reduce the degrees of freedom, and in this way reduce the number of accessible metastable states. This route is exploited by some bacteria such as the Chlorobiaceae to template by means of endosomic compartments the self-assembly of chlorophyll molecules in an optimum configuration forming the chlorosome, a very efficient photosynthetic antenna [7]. The soft confinement provided by the lipid sac templates and directs the assembly of the chlorophyll molecules into such a efficient structure. Synthetic approaches have tried to mimic nature and how it uses templates to direct the self-assembly of dyes. The most typical for assembling chromophores is self-assembled monolayers [8], and more recently using DNA as a template [9]. Motivated by this we decided to replace the lipid scaffold of the chlorosome by a block copolymer (BCP) matrix to direct the self-assembly of disk-shaped molecules like the chlorophill. BCPs have been extensively used as matrixes to incorporate nanoparticles, and thus, control their self-assembly [10 14]. Several theoretical and computational approaches have studied the fractionation and localization of nanoparticles in BCP matrixes [15 20]. In addition, the effect of the size 2

3 and shape of nanoparticles on their organization in polymer brushes has been theoretically studied [21]. In this work, we explore the block copolymer templated self-assembly of disk-shaped molecules by means of molecular simulations. Specifically, we study the behavior of diskshaped molecules confined within the cylinders formed by the minority block of a microphase separated BCP matrix. To understand the effects of soft confinement only, and discard specific chemical interactions, we use a coarse grained model. In particular, we only consider excluded volume interactions and shoulder-repulsive interactions between non-chemically compatible sites. With this minimalistic model we study the structures that can be selfassembled as a function of the disk size and number under the soft confinement imposed by the BCP matrix. We show that only under BCPs soft confinement, and in the absence of any other interaction, it is possible to obtain different self-assembled structures of the disk such as slanted stacks or columnar stacks. The paper is organized as follows: First, we describe the model and the simulation details. After that we present and discuss the results obtained, and finally we present our conclusions. Methodology and simulation details To describe the block copolymer matrix, we implement a molecular model in which each block copolymer chain is made of 20 monomers (m p ); these are described by fully flexible chains of tangent hard spheres of diameter σ. Thus, the bond length between monomers is set at L = σ. To account for the amphiphilic character of the block copolymer molecules, we consider two types of monomers per polymer chain, types A and B. The interactions between like-type monomers are hard sphere-like, while the interactions between unlike monomer types are described by a square-shoulder potential of shoulder height ɛ between σ and 2σ [22]. We consider both inter and intramolecular interactions between monomers of different types. In this model, the energy of the system is set by the number of contacts between unlike monomers. To describe the disk-shaped molecules, we use 19 tangent hard spheres (m d ) of diameter σ in a hexagonal packing arrangement, as shown in Fig. 1A. Thus, the radius of the solid of revolution of these disk-shaped molecules is R disk = 2σ. We also study the case where the radius of the disk-shaped molecules is R disk = σ. The disk 3

4 monomers are labeled as type B to make them compatible with one of the blocks of the polymer molecules. The initial configurations of the BCP-disks mixtures are prepared by adding disk molecules to either of the cylinder blocks of a pre-formed cylindrical phase. A" B" C" +" FIG. 1. Schematic representation of the system. The disk-shaped molecules (A) that are confined within a BCP matrix (B and C), which is microphase separated into a cylinder phase where the majority domaine is form by the red monomers and the minority domaine by the blue monomers. The simulations are carried out, using the Monte Carlo (MC) method, in the NpT ensemble with an anisotropic scaling of the simulation box sides length; thus, the shape of the simulation box remains orthorhombic. To improve the sampling efficiency we apply configurational-bias moves to the polymer chains [23], in which a part of the polymer chain is deleted and randomly regrown taking into account the Boltzmann weights associated with its non bonded interactions. The trial configuration regrowth is accepted or rejected based on the ratio of Rosenbluth factors of the new and old configurations. We use N p = 206 polymer molecules and a number of disk molecules ranging from = 1 to 80. We perform five independent simulation runs per number of disks, each with cycles for equilibration, followed by cycles for production. A MC cycle includes a trial volume change and a trial move per molecule. For polymer molecules, this can be a translation with a probability of 45 %, a rotation with probability of 45 % or a configurational-bias movement with a probability of 10 %. For disk molecules, only translation and rotation are possible, each one with a probability of 50 %. Results We determine the conditions at which our BCP model undergoes microphase separation into a cylindrical phase [22]. We find that such conditions correspond to polymer chain volume 4

5 fractions of f = and packing fractions of about ν = 0.23, which corresponds to a reduce pressure, p, of 0.2. Since the interactions between like monomers are hard interactions, the temperature is only important at the interface where the interactions between unlike monomers are described by a square-shoulder potential. Thus, we do not modify systematically the reduced temperature of the system, T, instead it is set to T = 0.2. This model has the advantage that the chains do not experience thermal expansion. Under these conditions, we observe that a BCP mixture of N = 1728 molecules with m p = 20 microphase separate into a cylindrical phase of periodicity d = 15.0 σ and that the radius of the cylinders is of about 3.5(2) σ, which is bigger than the radius of the disk, R disk = 2σ, as shown in Fig. 1C. To study the self-assembly of disks confined in the minority domain of the microphase separated BCP, we use the minimal repeating unit of the cylindrical phase, which contains two cylinders, as shown in Fig. 1B. The initial configurations of the composites of BCP and disk molecules are built by growing pairs of disks within the BCP cylinders, one disk in each cylinder at each time, in the NVT ensemble. Since the number of disks in both cylinders is always the same, the statistics of our measurements are better. Although in principle one could simulate different number of disks in each cylinder, this may bias their self-organization as the length of both cylinders is coupled by the periodic boundary conditions, which may result in artifacts due to the different deformation at the interfaces of the cylinders. First, we investigate a single disk confined in a cylinder formed by the minority domain of the BCP, i.e. sites type B (blue beads in Fig. 1). To compare with previous self-consistent field theory (SCFT) studies [24], we characterize the relative orientation of the disk with respect to the cylinder by defining the angle, θ, between the normal vector on the disk, n, and the cylinder axis, z, as depicted in the inset of the Fig 2. Since the disk is big enough to deform the interface between the two blocks, its preferential orientation is that which minimizes such deformation. For example, if the normal vector on the disk lays parallel to the cylinder axis, θ = 0 deg, it would deform the cylinder radially, which corresponds to the maximum deformation of the cylinder interface. On the contrary, if n is perpendicular to the cylinder axis, θ = 90 deg, the disk only disrupts the interface between the two blocks longitudinally, i.e. in two points. Therefore, we expect a preferential orientation of the disk of 90 degrees with respect to the cylinder axis. We compute the θ distribution function, f(θ), 5

6 FIG. 2. Orientational potentials of a disk molecule of radius 2σ (black line) and σ (red line) within a cylinder. Inset: Schematic representation that shows the definition of the angle form between the normal vector on the disk (n) and the cylinder axis (z). and corroborate our hypothesis. Moreover, from this orientation distribution function, we compute the orientational potential of one disk within a cylinder, U(θ), by assuming that the orientation distribution follows a Boltzmann distribution law, f (θ) = exp( U (θ)/kb T ), as shown in Fig. 2. The disk orientational potential shows that the most favorable orientation of the disk in the cylinder is of about θ = 90 deg. This result is in agreement with previous simulations using an hybrid method that combines SCFT with disk-shaped cavities [24]. Our new methodology, referred hereafter as particle-based, overcomes the computational limitations of the previous field-based approach. The agreement between both methodologies thus serves as a validation of our model and of this particle-based methodology. We also study the case of disk molecules of smaller size. If their size is half of that studied previously, the disruption of the interface between both blocks is smaller and only the parallel orientation of the disk with the cylinder is energetically penalized, as shown in Fig. 2. If more than one disk are present in the cylinder block of the BCP, we observe that they tend to form ordered structures such as stacks, slanted-stacks or close packed columnar aggregates. To gain insight into the driving force of the aggregation of disks under the 6

7 A g(r) B φ(r ij ) / degrees R disk = 2σ R disk = σ r ij / σ r ij / σ FIG. 3. A) Radial distribution function between the centers of the disks in a system of two disks. B) Average angle between the normal vectors (φ) of two disk of radius R disk = 2σ (black) and R disk = σ (red) as a function of the distance between the disk centers. soft confinement provided by the BCP matrix, we investigate the interactions between two disk molecules embedded in a cylinder block. We calculate the radial distribution function (RDF) between the center of mass of the disks, as shown in Fig. 3A. We find that there is a high probability of finding the center of the two disks at a distance of about 1.1σ. This very small distance between both disk centers is only compatible with a stacking of the two disks, one on the top of each other. To confirm this formation of stacks between two disks, we calculate the average angle between the normal vectors of both disk molecules, φ, as a function of the center-to-center distance, as shown in Fig. 3B. As the two disks become closer, they tend to align their normal vectors in parallel, from angles of about φ 50 deg at a distance of about 4σ to φ = 0 deg when they form the stack. From the RDF of the two disk molecules, we compute the reversible work of moving both disks at a distance r, so-called potential of mean force (PMF), w (2) (r), g(r) = exp( w (2) (r)/k B T ) (1) We observe that the two disk molecules experience a short range attraction, as shown in Fig. 4. The minority block of the polymer chains induces a depletion force between the disks, which results in stacking of the disk molecules. Moreover, the depletion force between disk molecules strongly depends on the conformational phase space of the polymer chains. We also perform simulations of two disk molecules embedded in a polymeric matrix composed by an homopolymer of the same chain length than the minority block in the cylindrical 7

8 w (2) (r) / k B T 5 0 Homopolymer Lamellar Cylindrical (R disk = 2σ) Cylindrical (R disk = σ) r ij / σ FIG. 4. Potential of mean force (PMF) between two disk molecules of radius 2σ embedded in an homopolymer matrix (black line), one block of a lamellar phase (red line) and in the minority block of a cylindrical phase (blue line). The green line corresponds to the PMF between two disks of radius σ within the cylinder block. phase system. We observe that the strength of the depletion interaction between the disk molecules is significantly reduced with respect to the disks confined with in the cylinder blocks of the block copolymer matrix, as shown in Fig. 4. In addition, we computed the potential of mean force between two disk molecules within one block of a lamellar phase of the same chain length than in the cylindrical block. The depletion force experienced by the two disk molecules in the lamellar phase is slightly smaller, about a 6% than in the cylindrical phase. Therefore, this depletion interaction between disks is more intense in matrices in which polymer chains have a tethered-end due to the conformational entropy reduction of the polymers. Additionally, the cylindrical confinement impacts further. We observe that the strength of the depletion interaction correlates with the polymer radius of gyrations, which increases from the homopolymer to the block copolymer in its cylindrical 8

9 phase. This trend is in agreement with the standard depletion interaction model between two parallel plates induced by ideal polymer chains [25]. We do not observe the formation of stacks by smaller disks, Rdisk = σ. Therefore, we also consider the effect of the disk size on this entropically driven interaction in polymeric matrices. As shown in Fig. 4 (green line), the depth of the PMF well is half that of the double size disks in the cylindrical phase. Furthermore, as a consequence of the gained rotational degrees of freedom, smaller disks exhibit this minimum in the PMF at 2.5σ. We also observe this shift for disk with Rdisk =σ in an homopolymer matrix and lamellar phase, although the depth of the potential is even shallower. The orientation of disk molecules is biased by the presence of neighboring disks at distances shorter than 2 Rdisk ; thus, the orientation interaction potential between two disks shows a steeper descent for smaller disks, as shown in Fig. 3B. Nd = 1 Nd = 5 E D C B A Nd = 10 Nd = 15 Nd = 18 FIG. 5. Representative configurations of the system for different number of disks: A) Ndisk = 1, B) Ndisk = 5, C) Ndisk = 10, D) Ndisk = 15, E) Ndisk = 18. The polymer chains induce a depletion force between the disks molecules, which results in the formation of stacks of disks. Thus, at a higher number of disks molecules, the structures we observe are determined by the self-assembly of these stacks. Attending to the structures we observe as a function of the number of disk molecules in each of the cylinders we define four types of assemblies: i) Small stacks of up to 3 disks, referred as unit stacks that orient perpendicular to the cylinder axis, as shown in the inset of Fig. 3A), ii) Slanted stacks that appear tilted a certain angle with respect to the perpendicular orientation of the unit stacks, 9

10 α = π/2 θ (Fig. 5B and D), iii) close packed columnar stacks, where all disks within a cylinder block form a single stack oriented with the cylinder axis (Fig. 5E), and iv) columnar stacks with interfacial disks at higher number of disks. The disk molecules added to the BCP localize within the cylinder block. Thus, the volume of the cylinder increases with the number of disks, which modifies the volume fraction of the BCP matrix. We define the effective volume fraction, f eff, as the volume ratio between the minority and the majority blocks, the minority block containing the disks (i.e. f eff = V cylinders (V system V cylinders ). As it can be seen in Fig. 6, the effective volume fraction ) increases from f eff = 0.2 in the absence of disks up to f eff = 0.25 with 8 disks embedded in each cylinder block, and it remains almost constant between 8 and 14 disks. Then, f eff increases up to f eff = 0.3 with the presence of 25 disks. For this coarse-grained model at these thermodynamics conditions (i.e. packing fraction), it has been shown that at volume fractions between f = 0.2 and 0.3 the stable phase is the cylindrical [22]. In this model, a perforated lamellar microstructure dominates at about f = 0.4 and we reach the maximum effective volume fraction at f eff = 0.36 with 40 disks embedded in each cylinder block. Therefore, at these number of disks, we do not expect any morphological transition of the block copolymer matrix by their presence. However, we carry out annealing simulations of the final configurations obtained for 10, 15, 20, 25 and 35 disks within each cylinder domain. We apply two different annealing protocols, consisting on: i) Gradually reduce the system temperature up to T* = 0.02, and ii) reduce the system pressure at p* = 0.06 (where the perforated lamellar structure is the stable phase in the absence of disk molecules) and then gradually increase it up to p* = 0.2. We do not observe any morphological transition using these annealing protocols. However, we do observe that the order of the disk molecules inside the cylinders increases as the structure is annealed. For instance, in the case of isk = 15, we observe that the disk molecules tend to align with the cylinder axis as the temperature decreases, resulting into the formation of perfect columnar stacks. Similar trend is observed using the annealing protocol ii). At p* = 0.08 we observe the formation of unit stacks randomly arranged in the cylinder blocks, and as the pressure increases they assemble up to the formation of columnar stacks. At small number of disks, up to isk = 3, all disks in each cylinder tend to form a single stack. We refer to these structures of stacks of 2 and 3 disk molecules as unit stacks. These 10

11 0.4 effective volume fraction isks FIG. 6. System effective volume fraction as a function of the number of disk molecules. The effective volume fraction, f eff, is evaluated as the volume ratio between the minority block (i.e. cylinder block, V cylinder = V disks + V B ) and the volume of the majority block (V A = V total - V cylinder ). unit stacks orient perpendicular to the cylinder axis, θ = 90 deg, as shown in Fig. 7 by the orientation distribution of the disk molecules, f(θ). For these unit stacks, the perpendicular orientation with respect to the cylinder axis is still more favorable than the radial deformation of the cylinder interface. The deformation of the cylinder interface produced by stacks of 2 or 3 disks is higher than in the case of a single disk molecule, as shown by the increase in the cylinder radius in Fig. 8A. Thus, the orientation distribution broadens around θ = 90 deg. At slightly higher number of disks, between 5 and 9 disk molecules, we observe that all disks in each cylinder arrange in a single slanted stack as shown in Fig. 5B. Stacks bigger that 3 disks are unstable in a perpendicular orientation with respect to the cylinder axis because of the high deformation of the interface between the two blocks. Therefore, to minimize the 11

12 deformation of the cylinder interface the stack of disk orient parallel to the cylinder axis but tilted a certain angle with respect to the perpendicular orientation, α = π/2 θ. The tilt angle, α, depends on the size of the stack; the bigger the stack, the higher the value of the tilt angle, as shown in Fig. 7. At these number of disks, the disk molecules do not expand the entire cylinder length; instead, they aggregate on one side of the cylinder, which allows the system to reduce the length of the cylinder, as shown in Fig. 8B. Although this produces an asymmetric deformation of the cylinders (Fig. 5B), the cylinder radius increase is compensated by the cylinder length reduction, which result in the interfacial area between the two polymer blocks remaining almost constant as the number of disks increases, as shown in Fig. 8C. As it can be seen in Fig. 5B, the stack of one cylinder is offset with respect to the stack in the other cylinder; we attribute this fact to the long-range interactions occurring between disk molecules located in different cylinders through the polymeric matrix. f(θ) θ / degrees = 1 = 2 = 3 = 5 = 10 = 11 = 12 = 13 = 14 = 15 = 16 = 18 FIG. 7. Distribution function of the average angle between the normal vector of the disks and the cylinder axis (θ) for the different number of disks: isk = 1 (black line), isk = 2 (red line), isk = 3 (orange line), isk = 5 (neon green), isk = 10 (blue line), isk = 11 (brown line), isk = 12 (green line), isk = 13 (purple line), isk = 14 (cyan line), isk = 15 (magenta line), isk = 16 (orange line), isk = 18 (violet line). At intermediate number of disks, between isk = 10 and 13, unit stacks coexist with slated 12

13 stacks. All disk molecules within a cylinder block expand the entire cylinder length, as shown by the configuration presented in Fig. 5C. In fact, the coexistence between both structures, unit and slanted stacks, is required for the disks to expand the entire cylinder block domain length. If all disk formed unit stacks, which align perpendicular to the cylinder axis, they would produce an increase in the length of the cylinder, which would be penalized by an increase in the interfacial area between both blocks of the BCP. The interfacial area produces an increase in the number of contacts between incompatible sites, and thus in the energy of the system. On the contrary, if all disk molecules arranged in a single slanted stack, they would allow the system to reduce the length of the cylinder block, but this would increase the radius of the cylinders in a larger extent, which would result in a constant interfacial area as the disk molecules number increases only if both effects compensate each other. Therefore, unit stacks that produce smaller distortions of the cylinders coexist with slanted stacks, which their more packed configuration allows shorter cylinder lengths, thereby compensating the interfacial area increase produced by unit stacks, as shown in Fig. 8C. As it can be seen in Fig. 7, the probability of finding unit stacks decreases with the number of disks (intensity of the peak around 90 deg), while the intensity of the slanted stacks peaks increases. Moreover, the tilt angle of the slanted stacks, α, shifts towards smaller angles as the number of disks increases. At high number of disks, between 14 and 20 disks per cylinder, we observe a single slanted stack that expands the entire cylinder block, as the one presented in Fig. 5D. As it can be seen in Fig. 7, in this case the orientation distributions of the disks do not show any pronounced maxima around θ = 90 deg, fingerprint of unit stacks. As the number of disks increases, the disk molecules tend to align with the cylinder axis. Thus, the tilt angle of the slanted stack, α, progressively increases up to the formation of a single close packed columnar stack of disks aligned with the cylinder axis, α 90 deg, at isk = 18. In this fashion, adding a new disk to the cylinder produces a small disturbance in the system, it only represents an small angle change of the assembled disk structure. In fact, during this gradual transition from the slanted stack towards the close packed columnar stack, the density of the system is hardly affected by the increasing number of disk-shaped molecules. In addition, the increase in the number of disks does not produce any increase in the cylinder block radius (Fig. 8A). However, the length of the cylinder blocks increases between isk 13

14 A" B" C" D" FIG. 8. Cylinder measurements as a function of the number of disks: A) Radii of the cylinders, B) length of the cylinders, C) area of the cylinders and D) density inside the cylinders (N atoms /V cylinder ). = 14 and isk = 18 (Fig. 8B), which results in an increase of the interfacial area between the polymer blocks, (Fig. 8C). As the number of disks increases between isk = 0 and 5, the presence of the disk molecules within the cylinder block produces an increase of the cylinder radius (Fig. 8A), which is not fully compensated by a decrease of the cylinder length (Fig. 8B) resulting in an increase of the interfacial area between the two polymer blocks. Therefore, at these numbers of disks the energy of the system is slightly increased, as shown by the black diamonds in Fig. 9; although this energy increment is within the systems energy fluctuations (error bars). The interfacial area between the minority and majority blocks remains almost constant between isk = 6 and 14, where the cylinder radius increase is compensated by a cylinder length 14

15 U * inter,intra U * total 8 Intermolecular Intramolecular Total isk 22 FIG. 9. Energy of the system as a function of the number of disks: Intramolecular contribution (red line), Intermoleculer contribution (blue line) and total energy (black line). contraction. However, the energy of the system increases, as shown in Fig. 9. To understand this behavior, we split the system energy into intermolecular (U inter) and intramolecular (U intra) contributions. The energy coming from the contacts between the disk molecules and the majority block is depreciable at these numbers of disks, i.e. U disk < As it can be seen in Fig 9, the intermolecular energy is almost constant between isk = 11 and 14. However, the intramolecular energy is significantly increased with the number of disks. Therefore, the increase in the system energy is due to the increase in the number of contacts between beads of the same chain. Although the interfacial area is constant between isk = 6 and 14, the increase in the number of disk molecules produces a depletion of the polymer chains from the center of the cylinder, which not only produces the coiling of the minority block chain, but also of the majority block, as shown in Fig. 10. This results in an increases in the number of contacts between the beads of the same chain, increasing U intra, and thereby the total energy of the system. Remember that U inter is almost constant at these numbers of disks. The minority block of the BCP, i.e. type B, is completely depleted from the center of the cylinder and compressed at the interface between the two blocks. Between isk = 14 and 20, the radius of the cylinders remains almost constant while the 15

16 A B FIG. 10. End-to-end distance of the polymer chain block as a function of the number of disks. The red squares correspond to the majority block, while the blue circles correspond to the minority block. length of the cylinder increases with the number of disks, which translates into an increase of the cylinder interfacial area. Contra-intuitively, as the interfacial area between the two blocks increases, the energy of the system decreases, see Fig 9. Again, at these number of disks the intermolecular part of the energy remains almost constant, but the intramolecular energy decreases up to a minimum at which the close packed columnar stack is observed, isk = At these number of disks, the end-to-end distance of both parts of the polymer chain, the majority and the minority, have reached a plateau, as shown in Fig. 10. This indicates that the contraction of the polymer chain is maximum at this number of disks due to the presence of the disk molecules. Therefore, the increase in the cylinder length allows the chains to rearrange along the cylinder axis reducing the number of contacts within the same polymer chain, which reduces the intramolecular energy; this results in a minimum in the system energy at isk =

17 f(θ) = 19 = 20 = 25 = 30 = 35 = θ / degrees FIG. 11. Distribution function of the average angle between the normal vector of the disks and the cylinder axis (θ) for the different number of disks: isk = 19 (black line), isk = 20 (red line), isk = 25 (green line), isk = 30 (blue line), isk = 35 (orange line), and isk = 40 (magenta line). At number of disks slightly above isk = 19, between 20 and 25 disks, the close packed columnar stack still appears, but the extra disks do not fit within the columnar stack and then locate at the interface between the stack and the polymers chains. Therefore, they adopt a perpendicular orientation with respect to the cylinder axis, as shown by the small peak around θ = 90 degrees in Fig. 11. At even higher number of disks, from 30 to 40, they aggregate forming big stacks that are almost evenly distributed between parallel and perpendicular orientations with respect to the cylinder axis. As it can be seen in Fig. 11, the orientation distribution of disks at these numbers of disks is almost uniform. The radius of the cylinders increases with the number of disks, and thus the simulation box reduces its length along the cylinder axis to compensate the increment in the interfacial area between the two blocks. Therefore, the minority domain is completely depleted from the cylinder center and highly compressed at the interface between the majority domain and the disks. The number of contacts between the majority block and the disks increases, but it is still smaller than one in average. Thus, even at these high numbers of disks the minority block 17

18 is able to cover the entire surface area between the disk molecules and the majority block. However, this compression of the minority block at the interface increases the energy of the system due to intramolecular contacts, as it can be seen in Fig. 9. Remarkably, at these number of disks the density within the cylinder block remains almost constant, as shown in Fig. 8D. As presented above, we observe the formation of a single slanted stack at relatively small number of disk molecules, between isk = 4 and 9. In fact, the slanted stack configuration at these number of disks is not observed in the case in which we only include disks in one of the cylinders, leaving the other one free of disks. In Fig. 12A and B, configurations of 5 and 8 disks are presented for both cases, systems where disks are confined only in one of the cylinders and systems where disks are present in both cylinders. In the former case, disks tend to form unit stacks which expand the entire cylinder length. On the contrary, when the two cylinders of the simulation box contain the same number of disks we observe the formation of a single slanted stack per cylinder, and that the stack of one cylinder is offset with respect to the other. The increase in the cylinder radii produced by the presence of the disks results in the compression of the majority block, which compresses the neighboring cylinder. Therefore, the stack formed in one cylinder feels the stack assembled in the neighboring cylinder. This transmission of the stresses from one cylinder to the other through the polymeric matrix effectively confine the stacks along the cylinder axis, which increases the tilt angle of sleanted the stacks, α. In Fig. 12C the orientation distribution of the disks, f(θ), in each case are presented. As it can be seen, when the disks are confined in only one of the cylinders, solid lines in Fig. 12C, the preferred orientation for the disks is θ = 90 deg. However, as the number of disks increases, the disks tend to tilt a certain angle as described before. Differently, in the case where the disks fill both cylinders, we observe that the disks are significantly tilted with respect to the 90 deg configuration due to the confinement along the cylinder-axis direction induced by the stack in the neighboring cylinder. 18

19 A" B" C" FIG. 12. Representative configurations of systems with asymmetric disk distributions with Ndisk = 5 (A) and Ndisk = 8. The left panel of A and B corresponds to systems in which one of the cylinders contains disk and the other is empty of disks. The right panels of A and B contains the same number of disks in each cylinder. C) Distribution function of the average angle between the normal vector of the disks and the cylinder axis (θ) for the different disk and distribution symmetries: the black solid line corresponds to Ndisk = 5 in one of the cylinders, the red solid line corresponds to Ndisk = 8 located in one of the cylinders, the green dashed line corresponds to Ndisk = 5 in each cylinder and the blue dashed line to Ndisk = 8 in each cylinder. Conclusions In this work we have studied the self-assembly of disk-shaped molecules under the soft confinement of a cylinder block of a microphase separated BCP. We have confirmed that 19

20 disk molecules of about half of the size of the cylinder block, orient forming an angle of θ = 90 deg between the normal vector of the disk and the cylinder axis. This orientation produces the minimal disruption of the interface between the two polymer blocks. Therefore, this bias in the orientation of the disk molecules is significantly reduced with the disk size. We have also observed that two disk molecules confined in a cylinder block experience a depletion force induced by the polymer chains, which results in the formation of stacks of disks. However, we have not observed the formation of stack of small disk molecules because the depletion force is no longer effective. The presence of disks within the cylinder block produces an increase of the cylinder radius. To compensate the increment in interfacial area that this causes, the system reduces the length of the cylinders. These two competing effects determine the interfacial area between the two blocks, which ultimately controls the self-assembly of disks at higher number of disks. As the size of the disk stacks increases, the disruption of the cylinder interface also increases, and the perpendicular orientation of the disk is no longer the preferred one. Instead, the disks tilt a certain angle with respect to the perpendicular orientation towards a parallel orientation with the cylinder axis, thereby forming slanted stacks. As the number of disk molecules is increased, the tilt angle of the slanted stacks is increased up to the assembly of close packed columnar stacks around isk = 18, where all disks align with the cylinder axis. At higher number of disks, between isk = 20 and 25, we still observe the formation of close packed columnar stacks, but some disks localize at the interface between the columnar stack and the majority block of the BCP. At even higher numbers of disks, we observe the formation of big stacks of 6-7 disk molecules that orient randomly, compressing the minority block at the interface. At relatively small number of disks, between isk = 4 and 9, we observe a long range interaction between the stacks of neighboring cylinders, which results in the confinement of the stacks along the cylinder axis shifting the formation of slanted stack towards smaller number of disks. This long range interaction between disks in neighboring cylinders may be exploited to control the disk assembly at larger scales. Moreover, this may serve as a tool to direct the self-assembly of interacting molecules that exhibit different structures depending on their relative orientation. For example, this long range interaction could be used to obtain slanted stacks at small number of disk molecules that interact if oriented in parallel. 20

21 Acknowledgments This work was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Science and Engineering under award No. #ER [1] J. L. Delgado, P.-A. Bouit, S. Filippone, M. A. Herranz, and N. Martin, Chem. Commun. 46, 4853 (2010). [2] C. W. Tang and S. A. VanSlyke, Appl. Phys. Lett. 51, 913 (1987). [3] L. Schmidt-Mende, A. Fechtenkötter, K. Müllen, E. Moons, R. H. Friend, and J. D. MacKenzie, Science 293, 1119 (2001). [4] J. Nelson, Science 293, 1059 (2001). [5] Y. Kim, E. Ha, and A. Alexander-Katz, Macromolecules 44, 7016 (2011). [6] Y. Kim and A. Alexander-Katz, The Journal of Chemical Physics 135, (2011). [7] G. T. Oostergetel, H. van Amerongen, and E. J. Boekema, Photosynthesis Research 104, 245 (2010). [8] M. Halik and A. Hirsch, Advanced Materials 23, 2689 (2011). [9] Z. J. Gartner, B. N. Tse, R. Grubina, J. B. Doyon, T. M. Snyder, and D. R. Liu, Science 305, 1601 (2004). [10] Y. Lin, A. Boker, J. He, K. Sill, H. Xiang, C. Abetz, X. Li, J. Wang, T. Emrick, S. Long, Q. Wang, A. Balazs, and T. P. Russell, Nature 434, 55 (2005). [11] A. C. Edrington, A. M. Urbas, P. DeRege, and C. X. Chen, Advanced... 13, 421 (2001). [12] M. R. Bockstaller, R. A. Mickiewicz, and E. L. Thomas, Advanced Materials 17, 1331 (2005). [13] M. A. Horsch, Z. Zhang, and S. C. Glotzer, Soft Matter 6, 945 (2010). [14] B. J. Kim, J. J. Chiu, G. R. Yi, D. J. Pine, and E. J. Kramer, Advanced Materials 17, 2618 (2005). [15] H. Chen and E. Ruckenstein, Journal of Colloid and Interface Science 363, 573 (2011). [16] S. W. Sides, B. J. Kim, E. J. Kramer, and G. H. Fredrickson, Physical Review Letters 96, (2006). [17] J. Y. Lee, Z. Shou, Balazs, and A. C, Macromolecules 36, 7730 (2003). 21

22 [18] F. A. Detcheverry, D. Q. Pike, U. Nagpal, P. F. Nealey, and J. J. de Pablo, Soft Matter 5, 4858 (2009). [19] S. Goy-López, E. Castro, P. Taboada, and V. Mosquera, Langmuir 24, (2008). [20] A. J. Schultz, C. K. Hall, and J. Genzer, Macromolecules 38, 3007 (2005). [21] J. U. Kim and B. O Shaughnessy, Macromolecules 39, 413 (2006). [22] A. J. Schultz, C. K. Hall, and J. Genzer, The Journal of Chemical Physics 117, (2002). [23] J. I. Siepmann and D. Frenkel, Molecular Physics 75, 59 (1992). [24] Y. Kim and A. Alexander-Katz, Submitted (2016). [25] R. Tuinier, G. A. Vliegenthart, and H. N. W. Lekkerkerker, The Journal of Chemical Physics 113, (2000). 22

23 A" B" C" +"

24

25 A g(r) B φ(r ij ) / degrees R disk = 2σ R disk = σ r ij / σ r ij / σ

26 w (2) (r) / k B T 5 0 Homopolymer Lamellar Cylindrical (R disk = 2σ) Cylindrical (R disk = σ) r ij / σ

27 Nd = 1 Nd = 5 E D C B A Nd = 10 Nd = 15 Nd = 18

28 0.4 effective volume fraction isks

29 f(θ) θ / degrees = 1 = 2 = 3 = 5 = 10 = 11 = 12 = 13 = 14 = 15 = 16 = 18

30 A" B" C" D"

31 U * inter,intra U * total 8 Intermolecular Intramolecular Total isk

32 A B

33 θ / degrees f(θ) = 19 = 20 = 25 = 30 = 35 = 40 1

34 A" C" B"