Sequential Domain Realignment Driven by Conformational

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Sequential Domain Realignment Driven by Conformational Asymmetry in Block Copolymer Thin Films Arash Nikoubashman, Richard A. Register, and Athanassios Z. Panagiotopoulos* Department of Chemical and Biological Engineering, Princeton University, Princeton, New Jersey 08544, United Sates ABSTRACT: In this contribution, we use dissipative particle dynamics simulations to investigate the orientations of lamellaforming copolymers with two blocks of equal molecular weight confined in thin films. We employ neutral and symmetric interactions between the copolymer blocks and the walls, a situation for which contradicting observations have been made in experiments and previous theoretical studies. Our model takes into account a realistic degree of conformational asymmetry between the blocks, which stems from unequal Kuhn lengths of the constituent monomers. We perform a thorough scaling analysis to exclude finite size effects, and find, in agreement with experiments, a remarkable cascade of morphological transitions from parallel to perpendicular orientations when the film height is varied. We demonstrate that the emergence of a stable parallel configuration stems from entropic effects due to the conformational asymmetry and the confinement of the system.

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INTRODUCTION The controlled fabrication of well-aligned nanoscale patterns is one of the key challenges of nanotechnology. Such tailored materials have a very wide area of applications, including for instance data storage,1,2 the fabrication of in-plane nanowire arrays,3,4 and photovoltaic devices.5 Block copolymers are a formidable candidate for fulfilling these requirements; significant progress in the area has been made in the past few years, due to advances in synthetic chemistry coupled with improvements in theoretical modeling.6,7 These macromolecules consist of two or more chemically different blocks, tethered together through covalent bonds. Because of this bonding, the blocks cannot phase separate on a macroscopic scale as the system is quenched below the order−disorder temperature TODT but instead self-assemble into nanoscale domains that minimize the contact area between the blocks. In the bulk, the emerging morphologies essentially depend only on the segregation strength χ and the volume fractions f i of the individual blocks. In general, for symmetric diblock copolymers (f = 0.5), the lamellar phase is observed, followed by gyroid, cylindrical, and finally spherical phases as the degree of compositional asymmetry is slowly increased. The phase behavior is however more complex in confined systems, since the domain formation takes place relative to the surfaces of the film and thus the film thickness H and the polymer-surface interactions play an additional significant role.6−9 In this contribution, we focus on thin films of lamellaforming symmetric diblock copolymers, where the emerging nanodomains can be aligned either perpendicular L⊥ or parallel L∥ (or combinations of both) with respect to the substrate (see Figure 1). The phase behavior of these systems has been extensively studied during the last two decades through theory and simulations, revealing an intricate set of possible morphologies, depending on film thickness and interactions © 2014 American Chemical Society

Figure 1. Simulation snapshot of a lamella-forming diblock copolymer film, with the nanodomains oriented (a) perpendicular (L⊥) and (b) parallel (L∥) to the substrates. Particles of type A and B are colored in red and blue, respectively. Neutral walls at the top and bottom are not shown for clarity.

of the confining surfaces with the constituent blocks of the polymer.10−18 The conceptually simplest case is the one for which the substrate exhibits no preference toward either block (neutral), and both confining walls behave in the same way (symmetric). Early calculations based on strong segregation theory (SST) predicted the coexistence of the L∥ and L⊥ configuration for film thicknesses equal to half-integer domain spacings D, since the two states become energetically indistinguishable at these values of H.10 More recent self-consistent field theory (SCFT) calculations have shown, however, that a finite free energy gap exists between the L∥ and L⊥ orientation favoring the alignment of lamella perpendicular to the substrates irrespective of the film thickness.11−13 These findings have also been confirmed by lattice Monte Carlo14 and off-lattice dissipative particle dynamics (DPD) simulations.16,17 Received: December 10, 2013 Revised: January 17, 2014 Published: January 27, 2014 1193

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where kB denotes the Boltzmann constant, T the temperature, rij the distance between particles i and j, and bi the bond length of the connecting spring. Alternatively, one can also use a slightly modified version,

From an experimental point of view, it is difficult to verify these observations, as it is very challenging to fabricate systems where the confining interfaces exhibit exactly the same interactions with both blocks.6,7 In their seminal contribution,19 Kellogg et al. approached such neutral conditions by coating the substrates with a random copolymer composed of the same blocks, and they observed the L⊥ orientation at H = 2.52D and the L∥ configuration at H = 2.80D. Their observation contradicts the established theoretical predictions for films between truly neutral surfaces, leaving the question of stable orientations for lamella-forming diblock copolymers in thin films unresolved. In order to provide a conclusive answer to this question, we have conducted large-scale DPD simulations20,21 in conjunction with a theoretically informed coarse-grained model.22 Our approach differs from previous simulation studies,14,16,17 in that we consider considerably longer chains and employ a more realistic off-lattice polymer model that takes the Kuhn-length asymmetry of the block copolymers into account. Although the effect of such an asymmetry has been studied via SCFT for bulk systems,23 the impact on confined systems still remains elusive. In general, even for a symmetric diblock with f = 0.5, there are differences in conformational flexibility of the constituent blocks, resulting in unequal Kuhn lengths bi and radii of gyration. In the following, we demonstrate that this conformational asymmetry plays a significant role for alignment of diblock copolymers in thin films. In particular, we obtain a cascade of L⊥ ↔ L∥ transitions as the film thickness H is changed. The rest of this article is organized as follows. First, we present our copolymer model and briefly introduce the employed simulation technique. Then, we present our results and compare them to previous findings from experiments and simulations. Finally, we summarize the findings and draw our conclusions.

Ub 3 = (rij − bi)2 kBT 2

which yields similar results. The monomer−monomer interaction Umm between two particles i and j of either type A or B is given by:22 Umm(rij) kBT

=

N̅ [κN + χN (1 − δij)] R e3

∫V ω(r − ri)ω(r − rj) dr

(3)

Here, Re is the end-to-end distance of the copolymer in the melt, which can be estimated via the freely jointed chain model as Re2 = ∑N−1 bi2,26 and N̅ = ρRe3/N denotes the so-called i interdigitation number, which is a measure for the strength of fluctuations.27 The incompatibility of the type A and B monomers is quantified by the Flory−Huggins parameter χ, while κ sets the compressibility of the melt.22 Furthermore, δij is the Kronecker delta, which is 1 when particles i and j are of the same type, and 0 otherwise. Finally, ω(r) in eq 3 above denotes the density cloud around each particle’s center, where we chose spherical steplike functions with ω(r) = 1 for r < σ/2 and ω(r) = 0 elsewhere. Hence, the beads can be considered as soft spheres in this model. In addition to these intra- and intermolecular potentials, the interactions between the beads and the confining surfaces have to be taken into account as well. In our setup, we applied periodic boundary conditions in the x and y directions of the system and placed impenetrable walls at z = ± (H + σ)/2 in the xy-plane. Here, we chose a potential of the form:

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Uwall(z) ⎛ z ⎞−6 = εS⎜ ⎟ ⎝σ ⎠ kBT

MODEL AND SIMULATION METHOD In order to numerically study the behavior of block copolymer thin films and to make meaningful comparisons to experiments, we need to employ a theoretical model that captures the essential microscopic details of the real systems, but also allows for simulations on large length- and time-scales. To this end, we opted to employ the so-called theoretically informed coarsegrained (TICG) model22 in conjunction with the DPD simulation technique.20,21,24,25 The bead−spring nature of this model gives us access to the microscopic system properties, while the inherent softness of the interaction sites allows for rather large system sizes and time steps. In this model, each chain consists of N beads of diameter σ, where f N particles are of type A and the remaining (1 − f)N particles of type B ( f = 0.5 for a diblock copolymer). We denote the position and the velocity of the ith particle as ri and vi, respectively. The interaction between individual beads can be divided into two contributions, one arising from the chemical bonding between two beads of the same chain (Ub), and one stemming from the interactions between all beads (Umm). Here, we have chosen the following functional form for the bonded interactions: 2 Ub 3 ⎛ rij ⎞ = ⎜ ⎟ kBT 2 ⎝ bi ⎠

(2)

(4)

where z denotes here the vertical distance to each wall and εS = 3 sets the magnitude of the repulsion. In a recent contribution,9 we have successfully utilized this model for simulating thin films of cylinder-forming polystyrenepoly(n-hexyl methacrylate) (PS−PHMA) diblock copolymers, and employ the same model parameters to study lamellaforming copolymers. Table 1 shows the relevant properties of the constituent monomers, which can then be used for determining the characteristics of the diblock copolymer. We chose to model PS−PHMA 42/42, where in this notation the first number signifies the molar mass (in kg/mol) of the first block and so forth. At this point we would like to point out Table 1. Molecular Weight of the Repeat Unit M0, Molecular Weight of the Kuhn Segment MK, Mass Density ρ, and Kuhn Length b of the Constituent Styrene (S) and Hexylmethacrylate (HMA) Moleculesa monomer

M0 [kg/mol]

MK [kg/mol]

ρ [g/cm3]

b [nm]

S HMA

0.104 15 0.170 25

0.720 1.62

0.969 0.95

1.8 2.44

a

The properties of the S and HMA monomers were obtained from ref 26. and refs 28−31, respectively.

(1) 1194

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unit vector between the beads i and j, and vij = vi − vj. Furthermore, the functional form of the weight function ωD(rij) has been chosen as:

that, although we are modeling a specific copolymer system, our results are general for any lamella-forming diblock copolymer with conformation asymmetry. Therefore, we will refer to the different particle types in our computational model as A and B in lieu of S and HMA to stress the general nature of our findings. With the information on the monomeric level at hand, we can derive the properties of the experimental copolymer as a whole and find N̅ = 44.0, Re = 18.6 nm and N = 658. For the Flory−Huggins parameter χ, we used the relationship that has been established in ref 32 for PS−PHMA block copolymers in the temperature range 423 K < Tabs < 488 K (with absolute temperature Tabs): χ ≈ 0.035 + 3.93K /Tabs

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⎧ ⎪1 − rij / σ , rij < σ ωD(rij) = ⎨ ⎪ rij ≥ σ ⎩ 0,

(8)

RESULTS We studied film thicknesses in the range 6 σ ≤ H ≤ 36σ, and performed a thorough scaling analysis by varying both the number of constituent beads per chain (N = 8, 16, 24, and 32) and the size of the simulation box (40 σ ≤ L ≤ 90 σ), in order to exclude any finite-length or finite-size effects. First, we tested our simulation scheme by investigating copolymers with symmetric Kuhn lengths (bB = bA) and confirmed the L⊥ configuration for all studied values of H, independent of the coarse-graining level (N) and simulation box size (L), as expected from prior studies.11−14,16,17 We observed that the individual chains were slightly compressed at film thicknesses below H ≈ 2Rbulk leading to accordingly smaller values of D. g For thicker films, the domain spacing remained constant, and we measured D ≈ 13.5σ for bB = bA = σ, and D ≈ 17.5σ for bB = bA = 1.4σ in the case of N = 32. These data indicate that D is roughly proportional to the Kuhn length. Let us now consider the more interesting case of asymmetric Kuhn lengths. The shortest chains (N = 8) exhibited a similar behavior as the symmetric ones, namely perpendicular lamellae for all values of H. However, as soon as we increased the number of beads per chain, we observed a distinctively different behavior (see Figure 3): at film thicknesses corresponding to

(5)

We chose Tabs = 423 K and therefore used χ ≈ 0.044 in our modeling (χN = 29.1). We adjusted the degree of coarsegraining by using N = 8, 16, 24, and 32 for the chain length, while keeping the parameters f, N̅ , and χN fixed. Furthermore, we set the bond length between two A particles to bA = σ, leading to bB = 1.4 σ. We chose κN = 15 for the compressibility term in Umm and a number density of ρ = 4.5 according to the arguments brought forward in refs 22 and 33. Figure 2 shows the potential for the case of N = 32.

Figure 2. Pair potential between particle pairs of the same type UAA and dissimilar type UAB as a function of interparticle distance r for a diblock copolymer with N = 32.

Simulations were conducted using the HOOMD package.34−36 The time evolution of the particle trajectories was obtained by the standard velocity Verlet algorithm,37,38 where we have chosen a value of Δt = 0.03 for the time step. NVT conditions were established via a DPD thermostat. The DPD model consists of conservative forces (FC), random noise (FR), and dissipative forces (FD), where FC can be derived from eqs 1-4 through the relationship FC = −∇U. The random and dissipative forces on each particle are given by:20,21,24,25 FR (rij) = −ξij 3

2γkBT ωD(rij)riĵ Δt

FD(rij) = −γωD2(rij)(vij·rij)riĵ

Figure 3. Morphology diagram of a diblock copolymer thin film with f = 0.5. The diagram shows the equilibrium configuration of the system as a function of polymer length N and film thickness H. Integer multiples of the domain spacing D are indicated with horizontal bars (D ≈ 6.5σ, 10.5σ, 14.0σ, and 16.0σ for N = 8, 16, 24, and 32, respectively). Red crosses and green triangles indicate L⊥ and L∥ configurations, respectively, while blue squares correspond to the coexistence of both. Dotted lines are guides to the eye.

integer multiples of the domain spacing, the diblocks formed lamellae which were aligned parallel to the surfaces. Furthermore, we always observed for the L∥ state that the lamellae in contact with the surfaces were comprised of A particles, although the surface-polymer interactions were set to neutral. This behavior became more pronounced with

(6) (7)

where ξij denotes a uniformly distributed random number with zero mean and unit variance, γ the friction coefficient, r̂ij the 1195

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increasing N, as the thickness range increased over which the L∥ configuration was found. Our observations were independent of the employed simulation box size L and are in excellent agreement with the experimental findings presented in ref 19, since they also observed parallel lamellae already at H = 2.80 D. To better comprehend the mechanism behind this phenomenon, we need to examine the driving force behind the L⊥ configuration for symmetric Kuhn lengths at film thicknesses commensurate with the domain spacing (the cases H ≠ D are trivial): for any system with repulsive walls, a density drop perpendicular to the surfaces can be observed over some finite length, and thus particles in the vicinity of a wall have fewer interaction partners compared to particles in the center of the film. Since the L⊥ configuration has more of its unfavorable A−B contacts in this region compared to the L∥ state, it exhibits an effectively reduced free energy which makes it more favorable.12 Hence the tendency to form lamellae perpendicular to the substrate is purely due to enthalpic reasons. Calculations within the framework of SST cannot capture this effect, since they do not take into account the confinement-induced local decrease in χ near the walls. We surmise that the emergence of the L∥ state is related to the microscopic alignment of the individual diblocks in the vicinity of the confining surfaces. To show this, let us first ignore the incompatibility of the two blocks (χ = 0) and consider a homopolymer thin film. It is well-known that polymers close to a hard surface align perpendicular to it due to entropic reasons; such a configuration allows for more conformational freedom for the majority of the chain, since the system exhibits a bulk-like behavior far away from the walls. If we now extend our thought experiment to a binary homopolymer blend where the two species differ only in their rigidity, we expect that the surfaces are wet by the more flexible block, since it exhibits a shorter correlation length and can therefore restore to the bulk density within a shorter distance from the wall.39,40 Let us now tether each rigid homopolymer with its flexible counterpart, and create thereby a diblock copolymer with f = 0.5 and bA ≠ bB. Although the two blocks cannot move independently from each other anymore, we still expect a qualitatively similar wetting behavior. If we turn on the incompatibility between the A and B beads, we anticipate that the system will form lamellae aligned parallel to the walls. Here, it is important to note that our hypothesis predicts parallel configurations under the requirement that the more flexible block wets both surfaces, and hence parallel lamellae should be observed in experiments and simulations only for full integer multiples of the domain spacing D. This restriction is in line with the findings from both our simulations and previous experiments.19 We have depicted this pathway schematically in Figure 4 to illustrate the origin of the L∥ configuration. In order to verify our reasoning, we first ran simulations of homopolymer (N = 32) thin films with and without conformational asymmetry and measured the density profiles ρ(z) of the A and B blocks perpendicular to the confining walls. Results for ρ(z) are shown in Figure 5, and clear differences between the cases bA = bB and bA ≠ bB are visible. In the symmetric case, ρ(z) is indistinguishable for the A and B particles as expected, and shows a thin depletion layer in the immediate vicinity of the wall due to its purely repulsive nature. The density then exhibits a small peak followed by a minimum. At distances larger than 2σ, the density profile becomes completely flat and structureless. In the asymmetric case, we

Figure 4. Schematic showing the microscopic conformation: (a) homopolymer, (b) “homopolymer” with bA ≠ bB (χ = 0), and (c) diblock copolymer with bA ≠ bB (χ > 0).

Figure 5. Density profiles ρ(z) of a thin film (H = 16σ) comprised of homopolymers (N = 32) with (a) symmetric (bA = bB) and (b) asymmetric Kuhn (bA ≠ bB) lengths. The continuous red line corresponds to the A block, while the dashed blue line shows the B block.

find, as predicted, significantly more A particles compared to B particles next to the wall. The trend then reverses for distances larger than 2σ as we find ρB(z) > ρA(z) there. In contrast to the symmetric case, the effect of the wall is still visible at the center of the channel. Wu et al. predicted that for a binary blend of Gaussian homopolymers the polymer-wall repulsion will be effectively weakened by a term ∝ (bB − bA) for the more flexible species.39 In order to verify their prediction for our diblocks, we performed simulations of block copolymer melts with bB > bA and χ = 0 (cf. Figure 4b). In principle, it is possible to measure the surface free energies by integrating the anisotropy of the pressure tensor,38,41 but for this particular system the approach suffers from poor sampling statistics. Therefore, we followed a different approach, where we carefully adjusted the interaction between the B particles and the confining surfaces to match the density profiles ρA(z) and ρB(z). We investigated Δb = bB − bA = 0.1σ through 0.6σ in steps of 0.1σ, and quantified the difference between the profiles through the mean squared distance Δρ2, which is defined as follows: Δρ2 =

1 H

+H /2

∫−H /2

[ρA (z) − ρB (z)]2 dz

(9)

We then minimized Δρ with respect to ΔεS for each value of Δb. The inset of Figure 6 shows Δρ as a function of ΔεS for Δb = 0.4σ, and a clear minimum is discernible. The reason that the value of Δρ is nonzero at the minimum is due to the fact that the conformational asymmetry impacts the local packing in the 1196

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CONCLUSIONS We performed large scale dissipative particle dynamics simulations in combination with a theoretically informed coarse-graining protocol to study the phase behavior of lamella-forming diblock copolymers confined in thin films. Here, we focused on the case of neutral and symmetric interactions between the copolymer blocks and the walls, for which conflicting observations have been made in experiments and previous theoretical studies. We have demonstrated that the emergence of stable L∥ configurations for copolymers with conformational asymmetry is strictly due to entropic reasons. The observed L⊥ ↔ L∥ transition is in very good qualitative agreement with experiments,19 and has been overlooked in previous particle-based simulations because of overly aggressive coarse-graining.16,17 Hence, special care has to be taken during the coarse-graining procedure, since seemingly harmless simplifications on the microscopic scale can have numerous dramatic effects on macroscopic system properties, resulting in erroneous predictions. Furthermore, we quantified the magnitude of this entropically driven polymer−wall attraction and found that it increases linearly with the degree of conformational asymmetry. Our findings are highly relevant for many technological applications, since the accurate prediction of domain morphologies and their orientation with respect to the substrate is a key challenge. For instance the fabrication of nanowire arrays requires perpendicular lamellae. However, the orientation of the lamellae strongly depends on the film thickness, and thus special care is required when the templates are prepared.

Figure 6. Main plot: Symbols show simulation data of ΔεS at which Δρ was minimized as a function of (bB − bA). The solid line is a linear fit of our data. Inset: Δρ as a function of ΔεS for (bB − bA) = 0.4σ.

immediate vicinity of the walls, which cannot be suppressed by the bead−wall interaction. In contrast to our particle-based approach, computer models that treat the polymer as an (infinitely) thin thread cannot reproduce this effect. The main plot in Figure 6 shows the ΔεS at which Δρ was minimized for all investigated values of Δb, and it is well visible that the effective attraction of the flexible block is indeed proportional to Δb. Finally, we fitted our data through a line and found a slope of approximately 1.3 kBT. This value is model specific, and is somewhat higher than the factor of 2/9kBT reported for Gaussian homopolymer blends in ref 39. The suggestion that this transition is entropically driven is further corroborated by the fact that we did not observe the L∥ state for very short chains (N = 8), where the conformational entropy of the individual chains plays a lesser role. We also conducted simulations for copolymers with N = 32 where we set χN = 50, 75, and 100: as the segregation strength was increased, the thickness range over which parallel lamellae was observed became narrower, until the L∥ state entirely disappeared at χN = 100. Hence, orientation of lamella-forming copolymers with conformational asymmetry is dictated by the interplay between maximization of entropy and minimization of A−B contacts close to the confining surfaces. This effect is reminiscent of the case of lamella-forming copolymers under steady shear, where entropy leads to the L∥ state at low shear rates and enthalpy to the L⊥ configuration at high shear rates.42 At this point we want to emphasize, that although our modeling is for a specific system, the arguments we brought forward are general and independent of the interaction details; we have repeated our simulations with alternative pair and bond potentials for selected state points, and noticed no qualitative difference. Hence, the observed behavior should occur for any confined system with a sufficiently strong conformational asymmetry. SST is not able to predict this L⊥ ↔ L∥ transition, since the inclusion of conformational asymmetry only affects the effective domain spacing of the lamellae. SCFT calculations on the other hand should in principle be able to capture the preferential surface wetting due to conformational asymmetry as well.39 Here, special attention must be directed to the implementation of the walls, since reflective boundary conditions inhibit the entropically driven alignment of individual copolymers43 and therefore suppress the L∥ configuration.

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AUTHOR INFORMATION

Corresponding Author

*(A.Z.P.) E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors would like to thank Nathan A. Mahynski and Paul M. Chaikin for helpful discussions. This work has been supported by the Princeton Center for Complex Materials (PCCM), a U.S. National Science Foundation Materials Research Science and Engineering Center (Grant DMR0819860).

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