Vertical Domain Orientation in Cylinder-Forming Diblock Copolymer

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Vertical Domain Orientation in Cylinder-Forming Diblock Copolymer Films upon Solvent Vapor Annealing Anatoly V. Berezkin,† Christine M. Papadakis,† and Igor I. Potemkin*,‡ †

Physik-Department, Physik weicher Materie, Technische Universität München, James-Franck-Str. 1, 85748 Garching, Germany Physics Department, Lomonosov Moscow State University, Moscow 119991, Russian Federation

‡

ABSTRACT: Dissipative particle dynamics was applied to simulate solvent vapor annealing in films of cylinder-forming diblock copolymers. Our simulations reveal a wide range of conditions, under which the vertical domain orientation is promoted by the fast solvent evaporation. First, the order-to-disorder transition must be induced upon swelling of the film. This condition ensures the independence of the structure formation during drying on the history, i.e., the preparation conditions. Second, the weak segregation regime has to be achievable during the evaporation process. This condition provides more “pathways” for structural transformations which are not accompanied by a strong penalty in the interfacial energy. The third condition for vertical orientation of the cylindrical nanodomains is a weak selectivity of the solvent. In the swollen state, a stronger swelling of the majority domains results in the formation of spherical micelles (rather than cylindrical). Upon drying of the film, these spherical micelles join to form cylinders, which align in parallel to the (vertical) solvent flow. Possible mechanisms of the alignment are discussed. In addition, the effects of varying film thickness, the degree of swelling, segregation regime, and the selectivity of the free surface of the film are studied. Computer simulations of the solvent vapor annealing and thermal annealing of equivalent dry films reveal considerable differences in the final film structures.

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morphology of a swollen film can be significantly different from the dry one. It depends on the concentration and selectivity of the solvent, the molar volume of the solvent molecules, the maximal degree of film swelling, and the interfacial tensions affected by the solvent content. Equilibrium morphologies of solvent-free block copolymer films confined between two solid walls have been intensively studied within the past two decades.26−35 These simple systems demonstrate an enormous morphological diversity35 with the morphology depending on the copolymer composition, block segregation strength, film thickness, and wetting of the walls by the different blocks. The nanodomain orientation between nonselective (neutral) walls is mainly controlled by the (in)commensurability of the film thickness, h, and the copolymer chain size, Rg,36−38 preferentially stabilizing vertical or parallel orientations of nanodomains, respectively. A specific feature of asymmetric copolymer films is the entropy-driven adsorption of the shorter blocks to neutral surfaces,26,39,40 which is not clearly understood yet. Vertical cylinders in relatively thick films are stable only when this entropic attraction is compensated by a weak selective physisorption of the longer blocks,27,39−41 and the range of appropriate adsorption energies in this case is very narrow.27,42

INTRODUCTION Nanostructured block copolymer (BC) films formed due to microphase separation of chemically different blocks are promising materials for nanolithography,1−5 as functional templates for photovoltaics6−8 and functional arrays,9−13 for the fabrication of nanoporous films,14,15 photonics, and polarizers,16 in molecular biology, biomineralization, colloid science, and supramolecular chemistry.17 The copolymer films are usually prepared via the fast spin-coating process. As a result, a nonequilibrium, poorly ordered nanostructure may be formed. To implement long-range order and to control the orientation of the nanodomains, different post-treatment techniques have been proposed: thermal annealing, directed self-assembly on prepatterned substrates, application of mechanical stress, electric field, etc.18,19 Among others, solvent vapor annealing (SVA) is known20 as a simple and fast method, where a copolymer film is exposed to the vapor of a solvent and swells up to a certain controlled degree, before being dried at a controlled rate. As initially demonstrated by Albalak et al.,21 Kim and Libera,22 Kim et al.,23 Fukunaga et al.,24 and Knoll et al.,25 SVA can be highly efficient, but a comprehensive understanding of this method has not been established yet.20 The main effect of SVA is related to the reduction of the glass transition temperature of the polymer which accelerates the molecular mobility and subsequently facilitates the relaxation of the nanodomain structure toward the equilibrium morphology.21 The complexity of SVA comes from the fact that the equilibrium © 2015 American Chemical Society

Received: August 10, 2015 Revised: December 8, 2015 Published: December 22, 2015 415

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transformations in the film upon drying different from that observed in recent Dynamical Self-Consistent Field Theory (DSCFT) simulations by Paradiso et al.60

More realistic thin copolymer films, which are confined by one solid wall (substrate) and a second free surface, which is in contact with the air, are much less studied theoretically. Some understanding of the nanodomain orientation was achieved for the simplest case of symmetric lamella-forming diblock copolymers.45−47 In particular, it has been demonstrated that the parallel domain orientation prevails over the vertical one if the blocks have different spreading coefficients.45 The presence of solvent as well as a gradient of solvent concentration within the film changes the ratio of nanodomain volumes, the interaction between the immiscible blocks of the copolymer, and their surface tension coefficients. As a result, it can lead to structural transitions including the order-to-disorder transition (ODT), variation of the size of nanodomains, and their orientation.47 Thus, the knowledge of the equilibrium structure of a dry block copolymer film is insufficient for the prediction of SVA results. The solvent effect has only recently been simulated in a complete swelling-drying cycle of SVA50−53 as well as at its different stages47,49,54−60 using molecular dynamics,54,56 dynamic Monte Carlo,50,51,53 the Flory−Huggins Cahn−Hilliard model,55 self-consistent field theory (SCFT),52,60 dissipative particle dynamics (DPD),47−49 analytical theory,57 and a theoretically informed coarse-grained (TICG) model.53 The mechanism of domain orientation and ordering during SVA is a subject of intensive discussions.53,55−57,60 It is wellknown that a nonselective solvent screens unfavorable contacts between immiscible polymeric components in the films and reduces the surface tension between the polymer nanodomains. The decrease of the solvent volume fraction (ϕS) near the air− film interface due to evaporation of the solvent initiates demixing of components which is followed by orientation of nanodomains along the gradient of solvent concentration.55,56 Buxton and Clarke55 and Morita et al.56 have found this orientation at fast evaporation of nonselective solvent even in a blend of two immiscible homopolymers. It shows that the domain orientation is a quite general phenomenon that appears not only due to the solvent selectivity or the specific nanodomain structure of the copolymer film. In simple systems,55,56 this can be related to the interfacial flows, driven by the gradient of the surface tension, also known as the Gibbs−Marangoni effect.55,58 Most of the theoretical studies of SVA deal with symmetric copolymers, and there are only few of them51,52,60 which are focused on asymmetric cylinder-forming copolymers. In the equilibrium dry diblock copolymer film,30,35,40 there is only a narrow range of copolymer compositions and film thicknesses, where vertically oriented and hexagonally ordered nanostructures are formed, which are demanded in nanolithography and membrane applications. The formation of the vertical cylindrical structure during SVA also requires a careful choice of conditions. In the present paper, we demonstrate that SVA can change the orientation of cylinders from the equilibrium parallel to the vertical one. We have discovered a wide range of parameters, where the latter morphology reproducibly appear, provided the films are sufficiently thick. Our simulations were performed with the DPD technique which is believed to be more accurate in simulations of initial stage of copolymer film drying than the field-theoretical methods. The latter usually neglect non-Gaussian chain conformations in rapidly thinning films and underestimate thermal fluctuations or nonlocal correlations in the dynamics of chemically bonded monomers, which lead to overestimated rates of spinodal decomposition or nanodomain ordering,61 as discussed below. As a result, we find a character of structural

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SIMULATION TECHNIQUE A. Dissipative Particle Dynamics. DPD is a coarse-grained molecular dynamics technique which is frequently used for simulations of simple fluids, suspensions, and polymers. The method was proposed by Hoogerbrugge and Koelman62,63 for the simulation of liquid suspensions and extended to polymer systems by Espanol, Groot, and Warren.64,65 Because of its coarse-grained character, the DPD method has well-known limitations: the polymer chains are Gaussian-like, and neither their entanglements nor the glassy state can be reproduced. However, the thermal fluctuations and polymer-specific dynamics naturally appear in DPD, and interactions of particles can be directly derived from the Flory−Huggins theory. This makes DPD convenient for simulations of solvent vapor annealing.47,49 Pagonabarraga and Frenkel66 proposed multibody dissipative particle dynamics (MDPD) for explicit simulations of gas−liquid equilibria and capillary effects in porous materials. MDPD was subsequently revised by several authors67,68 and recently mapped onto the Flory−Huggins theory.69 However, in our work, we use the simpler classical DPD because the most significant effects and interactions in our system appear in the condensed phase, while the gas phase mainly plays a role of a solvent sink and can be treated in a simple way. Additionally, the classical DPD operates in a smaller parameter space that significantly facilitates modeling of our multicomponent system. The DPD particles, each representing a group of small molecules or extensive molecular fragments, interact by conservative, dissipative, and random forces, which are pairwise additive. The net force fi = ∑j≠i(FCij + FDij + FRij ) exerted on a given ith particle is calculated by summation over all other particles within a certain cutoff radius, rc. Let rc, m, and kBT be the unit distance, the particle mass, and the thermal energy, respectively, thus defining the unit time as τ = rc(m/kBT)1/2. The conservative force represents the excluded volume interactions as well as the interactions of chemically bonded particles i and j in the dimensionless form FCij = aij(1 − rij)ȓij − ksrij, where rij = ri − rj, rij = |rij|, ȓij = rij/rij, aij is the maximum repulsion between particles attained at ri = rj, and ks is a spring constant taken to be nonzero for neighboring particles in a polymer chain and equal to zero for nonbonded particles. The dissipative and random forces, FDij = −γω(rij)2(ȓij·vij)ȓij and FijR = σω(rij)ȓijζ/(δt)1/2, respectively, constitute the Groot−Warren thermostat,65 where γ is a friction coefficient which is related to the thermal noise amplitude σ via the fluctuation−dissipation theorem, σ2 = 2γ, ω(r) is a weight function, ζ is a normally distributed random variable with zero mean and unit variance that is uncorrelated for different particle pairs, δt is the time step of an integration scheme, and vij = vi − vj is the relative velocity of the ith and the jth particles. Following ref 65, we chose σ = 3.0, ω(r) = 1 − r, ks = 4, the average density of particles ρ0 = 3, and the repulsion parameter between chemically identical particles aii = 25, while immiscible particles are more repulsive, i.e., aij ≥ 25. The latter parameter can be derived from the Flory−Huggins interaction parameter, as discussed below. The equations of particle motion, dri/dt = vi and dvi/dt = Fi, are solved using the so-called DPD-VV integration scheme70 (modified velocity-Verlet algorithm) with a time step δt = 0.04. B. Simulation Box. In the simulation box, the polymer film (bottom layer) and the vapor above the film are confined by two 416

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transfer in this part is expected to be much faster than near the surface. Thus, the concentration of the solvent at the distance δ above the film is equal to its concentration in the bulk of the gas phase. Since we consider the case of drying with pure inert gas, the bulk solvent concentration has to be zero which is mimicked as follows. Those solvent particles, which are located above the film at a distance exceeding a certain value δ, immediately and irreversibly disappear by transformation into “gas”. We found that choosing δ values between 2rc and 10rc does not have a notable impact on the solvent evaporation rate. Thus, the range of acceptable δ values is quite wide. The described transformation of the solvent particles into gas leads to a gradual solvent evaporation, i.e., formation of the dry film. The case of a fixed partial pressure of the vapor above the film, i.e., fixed fractions of solvent and gas beads, and control of the film swelling will be considered in our forthcoming publication. In the classical DPD technique, one can relate the repulsion between particles to the Flory−Huggins parameter, χij, and to the mutual solubility of components known from experiments. For molecules consisting of single DPD particles, χij reads65

solid walls which are impenetrable in vertical (z) direction (substrate and upper wall in Figure 1). In the x and y directions,

Figure 1. Schematic representation of the simulation box. The swollen diblock copolymer film (left image) reveals a spatially disordered structure. The space above the film is filled by vapor, namely a homogeneous mixture of solvent and gas beads. Drying of the copolymer film (right image) is accompanied by the formation of a cylindrical structure.

χij = (0.286 ± 0.002)(aij − 25)

(1a)

and for polymer chains of length N > 2 periodic boundary conditions are imposed. Four types of particles are considered: solvent particles (S), inert gas particles (G), and monomer units of type A and B which constitute the linear diblock copolymer chains (Figure 1). Each chain has N = 20 beads and the composition fA = 0.25; i.e., it consists of NA = 5 and NB = 15 beads of type A and B, respectively. The gas phase in classical DPD is simulated as a dense fluid of Gparticles.47−49,56,58 The wall-forming particles have infinite mass; i.e., they cannot move. To avoid temperature drops near the walls and the wall slippage effect, they are formally assigned nonzero velocities obeying the Maxwell−Boltzmann distribution. At the initial moment of the simulations, the bottom part of the box is occupied by a random mixture of copolymers and solvent particles of a certain initial volume fraction, ϕS0, which corresponds to a swollen film. In different simulation runs, the value of ϕS0 ranges from 0.2 to 0.5; it controls the rate of solvent evaporation. In most cases, the swollen film has an initial thickness h = lz − 20rc. The vapor occupies the remaining space, which has a thickness 20rc and consists of a mixture of solvent and gas particles. To prevent a too fast initial solvent evaporation, the initial solvent fraction in this layer was always chosen at 0.5. We used a periodic box of the size lx × ly × lz = 43 × 43 × 80 rc3 and also repeated some simulations in boxes of the size lx × ly × lz = 50 × 50 × 70 rc3 (giving similar results) to eliminate the effect of the periodic box dimensions on the stability of block copolymer morphologies.26,40 The typical thickness of the dry film in our simulations h0 = 30rc can accommodate up to four layers of parallel hexagonally packed cylinders. The solvent evaporation is a complex process involving convective mass transfer in the gas phase. To keep things as simple as possible, we built the model of this process on the basis of the simple theory of Lewis and Whitman.71 In this theory, the gas phase can be divided into two unequal parts. One is the thin diffusion boundary layer of constant thickness (δ), adjacent to the film surface, where only molecular diffusion of solvent proceeds. This layer determines the mass transfer coefficient in the gas phase, βg = Dg/δ, where Dg is the molecular diffusion coefficient of the solvent in the gas. The remaining part of the gas phase is assumed to be homogeneous, while turbulent mass

χij = (0.306 ± 0.003)(aij − 25)

(1b)

72

Further simulations have shown that the mapping of eqs 1 has to be corrected for short chains, giving χij =

(0.306 ± 0.003)(aij − 25) 1 + 3.9N −0.51

(2)

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Moreover, recent work has demonstrated that accurate mapping of DPD onto the Flory−Huggins theory is not straightforward, which should be kept in mind when applying DPD results to experimental systems. To avoid ambiguities, we used the absolute values of repulsion parameters aij. According to the SCFT,74 the ODT in the pure copolymer melt of composition fA = 0.25 occurs at χABN ≈ 17.3. At N = 20, fluctuation corrections (eq 2) shift the critical value to χABN ≈ 31.9 and the critical repulsion parameter to aAB = 30.2. This conforms with DPD simulations of Gavrilov et al.,75 who obtained the critical value χABN ≈ 42 at N = 16. In our simulations, the values of aAB range from 35 to 50, which covers the weak and strong segregation regimes, χABN ≈ 33−83, respectively, as calculated from eq 2. Polymer/solvent interactions are governed by the parameters aAS and aBS. They are varied between 25 and 50, but in most of the simulations, smaller values were used to avoid macrophase separation of polymer/solvent mixture, namely 25 ≤ aAS ≤ 35 and 25 ≤ aBS ≤ 28. This choice of the parameters ensures a good solvent quality for the longer (B) block and a good (aAS = 25) to slightly worse (aAS = 35) quality for the shorter (A) block. The surface tension of the block copolymer at the free surface is characterized by the parameters aAG and aBG. In most of the simulations, they were fixed at aAG = aBG = 100. This choice leads to the complete immiscibility of gas and polymer phases, preventing the “evaporation” of polymers. The case of selective adsorption of the B-block at the film/gas interface (aAG > aBG) was also simulated. In all cases, the solvent/gas repulsion was chosen to be minimal, aSG = 25, to ensure their homogeneous mixing and to maximize the solvent evaporation rate. Polymer/ substrate interactions are expected to be nonselective in all simulations with aWA = aWB = 25 and aWS = aAS, where the 417

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Figure 2. Evolution of the distributions of A (red), B (green), and S (blue) particles in the film along the film normal under “standard” conditions (aAB = 35, aAS = 35, aBS = 28, δ = 2, ϕP0 = 0.5); t is the dimensionless time.

Figure 3. Internal structure of the film and topography of the free surface at different moments of time (aAB = 35, aAS = 35, aBS = 28, δ = 2, ϕP0 = 0.5). Isosurfaces of A-monomer density are shown for the density value ρ = 1.5. Minor (A) and major (B) copolymer blocks are shown in blue and yellow, respectively.

subscript “W” means “wall”. The simulations lasted 4 × 106 steps or 1.6 × 105 dimensionless time units, which is usually sufficient for a complete solvent removal from the block copolymer film.

= 28. The initial volume fraction of the solvent in the film was ϕS0 = 0.5, and the thickness of the diffusion boundary layer in the gas phase, δ = 2, was small enough to ensure a high solvent evaporation rate. Figure 2 shows that at all times the solvent forms an adsorption layer atop the film which is evident as a local maximum of the solvent distribution function. The reason is that the repulsive interactions of the polymer blocks and the solvent are not as strong as the ones between the polymer blocks and the gas. In other words, the solvent shields unfavorable contacts between the polymer chains and the gas, which reduces the surface energy of the free surface of the film. This effect was previously observed in molecular simulations.47,49,56,58,76 The same effect is

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RESULTS AND DISCUSSION A. Film Structure upon Drying and Mechanism of Domain Reorientation. Figure 2 shows the density distribution of particles A, B, and S along the film normal, i.e., the zcoordinate in the film during the solvent evaporation under typical conditions of SVA, namely a weak block segregation regime at aAB = 35 and a selective solvent. The solvent was chosen to be poor for the cylinder-forming A blocks with an interaction parameter aAS = 35 and good for the B blocks with aBS 418

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Figure 4. Film structures after simulated thermal annealing via a gradual increase of the repulsion parameter aAB from 25 to 35 (upper row) and after SVA (lower rows, aAS = 35, aBS = 28, δ = 2, and ϕP0 = 0.5) at different values of final film thickness (h0).

the whole volume of the film in the swollen state (Figure 3, t = 0). Because of the thermal motion and weak selectivity, they continuously arise, move, and disintegrate, in contrast to the spherical micelles of well-defined shape in strongly selective solvents.77,78 A drop of the solvent concentration near the free film surface in the initial stage of the solvent evaporation is accompanied by merging of the spherical micelles within this region into elongated ones (Figure 3, t = 2400). The elongated micelles serve as initiators of growth of the cylindrical domains upon film drying. During solvent evaporation, propagation of the randomly oriented cylindrical micelles down to the substrate occurs via joining of the spherical micelles (Figure 3, 2400 < t < 7200). The late stages of drying (Figure 3, t ≥ 9600) are characterized by cooperative vertical alignment of the domains which is more pronounced near the free surface of the film and propagates through the whole film thickness. The hexagonally ordered structure is also clearly seen from the topography of the free surface in the late stages of drying (upper images in Figure 3 at t > 9600). The above-described evolution of the film structure is different from the results of recent DSCFT simulations by Paradiso et al.60 while the sequence of arising nanodomain shapes is the same. Such difference comes from the fact that the field theories simplify the structure and dynamics of the system even as compared with DPD. First of all, DSCFT assumes Gaussian chain conformations which is valid on time scales exceeding Rouse relaxation time tR = Rg2/DN, where DN is the polymer diffusivity. On the other hand, the characteristic time of drying of the film to the depth of one interdomain spacing due to the solvent diffusion, tD = Rg2/Dp, where Dp is the solvent diffusivity, is approximately N times shorter since DN ∝ Dp/N. Thus, DSCFT results can hardly be valid at the very initial stages of film drying, when the drying depth does not exceed 4 or 5 interdomain distances. Moreover, DSCFT in the local-coupling approximation used in ref 60 neglects nonlocal coupling and interdependent motion of bonded monomers, producing wellknown artifacts: overestimated growth rates of density fluctuations during spinodal decomposition79 and also overestimated sizes of the ordered region in the phase diagram of homopolymer/copolymer mixture,61 which is quite similar to the

responsible for the preferential localization of a nonselective solvent at the AB interfaces of nanostructured BC films.47 Figure 2 also illustrates the formation of a solvent-poor region near the free surface of the film, i.e., a minimum of the solvent distribution function or a maximum of the total polymer volume fraction (A + B). This region gradually grows with time, becoming both wider and denser. In all our simulations, the evaporation rate is limited by the solvent diffusion through the polymer film, since the relative intensity of mass transfer in the gas phase is much higher. This conclusion is supported by the fact that the maximal gradient of solvent concentration is observed within the film. Quantitatively, a relative intensity of the mass transfer in gaseous and polymeric phases can be characterized by the Biot number Bi = mg/pβgh/Dp,57 where mg/p is a distribution coefficient of the solvent between vapor and polymer, βg ≈ Dg/δ is a mass transfer coefficient in the gas phase, h is the film thickness, and Dg and Dp are the solvent diffusivities in gas and polymer phases, respectively. Bi exceeds unity in all simulations. For example, at mg/p ≈ 0.1, βg = Dg/δ = 0.303/2 = 0.151, h = 60, and Dp ≈ 0.22, which were found during preliminary simulations, the Biot number Bi ≈ 4. It is close to Bi = 5, considered by Paradiso et al.60 In the later stages of the solvent evaporation, the solvent density profile in the film reveals a rather gradual decay with z instead of a step-like shape, which is characteristic for the initial stage of the drying. The polymer density profiles also have different features in different time intervals. In the beginning of the drying (t < 2400), oscillations on small length scales are characteristic for the densities of both A and B segments. Then, the length scale of the oscillations increases (t > 2400). More detailed information about the evolution of the film structure is accessible via visualization of the film morphology at different stages of the drying (Figure 3). Thermal fluctuations in DPD simulations are at the origin of the formation of dynamic “nuclei”spatial regions with an excess of A and B monomer units. Furthermore, the asymmetry of the BCs ( fA = 0.25) and the solvent selectivity to the longer block are responsible for the formation of disordered spherical micelles: The short A blocks and the long B blocks constitute the cores and the coronae of the micelles, respectively. Such spherical micelles appear throughout 419

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worms are subjected to the flow of evaporating solvent for longer time. One can conclude that only the nonequilibrium character of the fast drying process is responsible for the formation of the metastable vertical orientation of the cylinders in our simulations. The simulations reveal a wide range of conditions under which the perpendicular orientation of cylindrical microdomains occurs. For the preparation of well-ordered structures after SVA, it is a prerequisite that the swollen film is disordered and becomes ordered during solvent evaporation. To fulfill this condition, a high initial solvent concentration in the film (i.e., a high swelling coefficient) or a relatively weak segregation between the two blocks is needed. Also, a certain solvent selectivity for the longer matrix blocks is necessary because it facilitates the vertical domain orientation in equilibrium26,39,40 and also can amplify this effect under SVA conditions due to different permeabilities of nanodomains and matrix. We can assume that reorientation of domains is driven by the vertical flows of the components, arising upon the solvent evaporation. Inside the domains, these flows appear due to molecular diffusion. In the case of a B-selective solvent, cylindrical nanodomains and matrix have slightly different viscosities and permeabilities. Less swollen and permeable cylindrical nanodomains orient along the flow to minimize viscous friction (drag force). At the interdomain interfaces, additional flows may appear due to the gradient of the surface tension originated by the nonuniform vertical solvent distribution and followed by a contraction or an expansion of the interfacial regions (per chain) with high and low surface tensions, respectively. These interfacial flows are known as the Gibbs− Marangoni effect.55,58 Different degrees of orientation of the worms near the free surface (vertical) and the solid wall (random) can be seen in Figure 3, t > 12 000. This difference is due to the fact that the upper layers of the film are subjected to an intensive solvent flow for longer time whereas the bottom layers are less affected. We note that the mechanism of vertical orientation of the cylindrical nanodomains identified in the present work is different from the one proposed in ref 60, which is related to morphologydependent density correlations present at the ordering front. B. Effect of Polymer−Solvent Interactions. In the course of more extensive DPD simulations, we confirmed the wellknown experimental fact that the orientation of nanodomains and their ordering after SVA drastically depend on the solvent, namely its quality and selectivity.1,4,20,52 To investigate the effect of the solvent in more detail, we have considered only thick films with h0 = 30 where the parallel cylindrical structure in the dry state is thermodynamically stable (Figure 4) and where the formation of other morphologies is evidently related to the SVA conditions. The diagram of states in dependence on the solvent quality for A (aAS) and for B (aBS) blocks is shown in Figure 5. The initial swollen structure of the film (δ = 2, ϕP0 = 0.5) is disordered for a weak incompatibility of A and B blocks (aAB = 35). In the case of a nonselective solvent, aAS = aBS (dashed line in Figure 5), only a randomly oriented cylindrical structure emerges after SVA, similarly to recent experimental studies.57 Formation of regular morphologies requires a certain selectivity of the solvent. Filled green and red circles in Figure 5 indicate highly ordered vertical and parallel cylinders, respectively. The corresponding open circles depict less ordered morphologies, in which defect-free domain alignment appears near the free surface, whereas the bottom part of the film features randomly

solvent/copolymer mixture. In other words, this method tends to overestimate nanodomain ordering rate, which can give rise to unrealistically fast formation of ordered morphologies. DPD, which is free from these artifacts, properly captures thermal fluctuations which postpone domain ordering to the late stages of the film drying. In contrast to ref 60, we do not observe any evident boundary between ordered and disordered regions (cf. Figure 3). Without such boundary, a correlation hole effect proposed in ref 60 to explain the instability of the vertical domain orientation at high evaporation rate is not strong enough to influence the morphology in DPD. The following questions arise: what is the reason for the vertical orientation of the cylinders in our simulations, and is this the equilibrium one or a metastable one and does it possibly originate from nonequilibrium character of SVA? To answer these questions, we simulated also the thermal annealing of dry films differing in film thickness and compared the ultimate equilibrated structures with those obtained after SVA. Nanodomain structures of films from asymmetric block copolymers confined between two walls were theoretically studied in detail.25−27,30,35,39,40 The orientation of the cylinders was shown to be controlled by the (in)commensurability of the film thickness (h0) with the natural spacing of the nanodomains l0 (this parameter is equal to 9.35rc for the copolymer at aAB = 35) or with the distance between parallel domain planes, lp = √3l0/2 ≈ 8.1rc. Parallel cylinders between neutral surfaces appear when the ratio h0/lp is close to an integer number.30,40 However, the films on the basis of cylinder-forming copolymers with a free surface were not considered in such detail, and we present results of our simulations in Figure 4. The initial spatially homogeneous melt of the dry diblock copolymer films was obtained from randomly placed copolymer molecules having completely miscible blocks at aAB = 25. This mimics the interactions and the film structure above the ODT temperature. The space above the film is filled by the gas molecules which are strongly immiscible with the polymer, aAG = aBG = 100. Because of this strong repulsion, providing a high surface tension at the free surface, and the finite size of the simulation box, we do not observe formation of terraces in Figure 4, which are usually observed in experiments in the case of the parallel domain orientation.25,26,80−82 Cooling down the film to “room” temperature (i.e., below the ODT) is simulated via a gradual increase of the block segregation parameter aAB from 25 to 35 (weak segregation regime) within 2 × 106 simulation steps, followed by relaxation within the same time interval at a fixed repulsion parameter aAB = 35. As seen in Figure 4 (upper row), in the most of the cases the values of the ratio h0/lp are close to integer numbers which are in favor of stabilization of parallel worms. Only a very thin film (h0 = 5) demonstrates an in-plane hexagonal ordering of nearly spherical micelles, and at h0 = 20, the parallel morphology is perturbed by an incommensurability between the optimum interdomain distance and the film thickness. The thermal annealing does not lead to any vertical structures for the considered film thicknesses. On the other hand, SVA of the polymer films of the same thicknesses under the conditions shown in Figure 3 leads mostly to vertical cylinders as demonstrated in the bottom row of Figure 4. Only at h0 = 5, the resulting morphology is similar to that after the thermal annealing. The intermediate mixed structure of parallel and vertical cylinders is found at h0 = 10, and thicker films exhibit the vertical orientation (bottom row of Figure 4). The mixed structure at h0 = 10 is a result of the short-term action of the solvent flow as compared to the thicker films where the 420

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unfavorable, and the ordering is hindered. This is why for example in the case of aAS = 35 and aBS = 25, only randomly oriented cylinders are formed. Another argument in favor of vertical domain orientation in weakly selective solvents can be drawn from the conditions for the equilibrium vertical orientation of the cylinders when the confining surfaces are slightly selective to the longer blocks.28,39−41 In the SVA, this selectivity of the free film interface is attained because of inhomogeneous surface distribution of the solvent (as seen in Figure 2), providing a slightly higher affinity of the free surface providing a slightly higher affinity of the free surfacefor the B blocks. The opposite solvent selectivity, aAS < aBS, produces parallel cylinders marked by red circles in Figure 5. In this case, the matrix is less soluble and permeable than the cylindrical nanodomains. Evaporation of the solvent slows down in comparison with the case of selectivity to the longer blocks because the matrix occupies an approximately 3 times larger volume of the film ( fA = 0.25). This argument is supported by longer evaporation time, shown in Figure 9 (see below). Therefore, we can assume that the orienting action of the evaporating solvent weakens, and a presumably equilibrium parallel morphology is stabilized. If the solvent is equally poor for both blocks, polymer- and solvent-rich phases are formed in the swollen state of the film (an analogue of macrophase separation in concentrated solutions, black circles in Figure 5). This regime is probably less efficient for SVA because the solvent makes the film inhomogeneous at large length scales before its evaporation. C. Effects of Solvent Evaporation Rate. Initial SVA experiments of Kim and Libera22 have shown that a too fast evaporation of the solvent can freeze a disordered nanodomain structure if the time of drying is shorter than the characteristic time of defect relaxation via macromolecular diffusion. On the other hand, the evaporation rate has to be high enough to maintain a large gradient of the solvent concentration in the film that is prerequisite of nanodomain reorientation. There are a number of ways to control the evaporation rate which include the choice of the initial degree of swelling of the film, the volatility and selectivity of the solvent, the control of the partial pressure of the vapor, etc. In the present section, we study the effects of the degree of swelling and the selectivity of the solvent on the evaporation rate and the duration of the drying. Figure 6a shows that the solvent evaporation rate ∂h/∂t (the slope) notably depends on the initial solvent concentration: the higher the concentration, the higher the rate at each moment of

Figure 5. Diagram of states of the film after SVA at different values of polymer/solvent interaction parameters aAS and aBS. Filled green and red circles correspond to well-ordered vertical and parallel cylinders, respectively. The corresponding open circles depict less ordered morphologies in which defect-free domain alignment appears near the free surface, whereas the bottom part of the film consists of randomly oriented cylinders. The randomly oriented cylindrical structure is shown by gray circles. Black circles depict the region of separation into polymer- and solvent-rich phases in the film. The dashed line corresponds to the case of nonselective solvent. Parameters of the simulations are aAB = 35, δ = 2, ϕS0 = 0.5, and h0 = 30.

oriented cylinders. The randomly oriented cylindrical structure is shown by gray circles. The best conditions for the vertical domain orientation correspond to a weak selectivity of the solvent for the long blocks, aAS > aBS, when the shorter blocks are less soluble than the longer ones. Under these conditions, the minor A-rich nanodomains are less permeable than the swollen major B-rich matrix that can give a rise to the domain reorientation by the solvent flow. The weak selectivity of the solvent together with the weak segregation strength of the blocks (aAB = 35) ensure a relatively small surface tension coefficient at the AB interfaces. As a result, any transformations of the cylinders during SVA, which are accompanied by the increase of the area of AB interfaces (as in the case of rupture), do not increase significantly the interfacial energy of the system. Therefore, the potential barriers for the structure evolution are not so high, and there are many “pathways” for relaxation of the defects in the initial stage of film drying. On the contrary, a strong selectivity of the solvent and a high interfacial energy of the nanodomains lead to the formation of thicker cylinders (with higher polymer concentration), whose rupture and re-formation are energetically

Figure 6. Degree of swelling of the film, h/h0, as a function of the simulation time t for different values of the initial fraction of the solvent in the film, ϕS0, and aAS = 35, δ = 2 (a) and at different values of the solubility parameter for the A block, aAS, at ϕS0 = 0.5 (b). 421

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Macromolecules time. The rate decays with time, approaching zero at h/h0 = 1. The overall drying time is proportional to the initial degree of swelling of the film, h(t = 0)/h0 = 1/(1 − ϕS0). Figure 6b demonstrates that the thickness of the diffusion-boundary layer in the gas phase, δ, has no effect on the drying kinetics (blue lines in Figure 6b). Thus, evaporation is mainly limited by the solvent diffusion within the film. Figure 6b also shows that the decrease of solvent solubility in the minor domains (increase of aAS) slows down the solvent evaporation (h/h0 decays faster at aAS = 28 than at aAS = 35) because the minor domains start to play the role of impenetrable or weakly penetrable obstacles. Figure 7 shows structures of dry films after SVA formed at different initial values of polymer volume fraction ϕP0 = 1 − ϕS0.

small repulsion parameters evaporate faster than poor solvent, and a poorer solvent solubility in the minor phase also slows down the evaporation. The time t1/2 in Figure 8 can differ by more than a factor of 3, depending on the solvent quality and the resulting morphology. Potentially, the difference can be even larger.49 In principle, the longer drying time should facilitate relaxation of the defects in the nanodomain structure. However, in our simulations, it is not the only condition of proper domain alignment. On the one hand, the time needed to create wellordered vertical cylinders (aBS = 27) is much shorter than the one needed to create parallel cylinders (aBS = 30). On the other hand, at a comparable time t1/2 the domain orientation notably depends on the solvent choice, in particular, on its selectivity (Figure 8). D. Effect of Block Segregation. To understand the effect of block segregation on the final structure of the film, we have varied the repulsion parameter aAB in the range from 30 to 50, which correspond to the weak and the strong segregation regimes, respectively. The other parameters were chosen at aAS = 35, aBS = 28, δ = 2, and ϕP0 = 0.5. Figure 9 demonstrates the emergence of cylinders of segregated blocks. These are not ordered when the dry film is in the weak segregation regime (aAB = 30). The best vertical ordering of the cylinders is observed when the dry film has a slightly higher interaction parameter (aAB = 35). However, higher values of the incompatibility of the two blocks (up to aAB = 50) degrade the ordering again. We attribute this behavior to the fact that the strong segregation of the blocks increases the time of defect relaxation.53 E. Effect of Surface Tension at the Free Surface of the Film. Finally, we would like to stress that neutrality/nonselectivity of the free surface is important for the orientation of the cylindrical nanodomains because the selective adsorption destabilizes the vertical domain orientation. In particular, this is the case when the blocks have different surface tensions. The structures of the films after SVA with different values of the interaction parameter of the longer B-block with the gas phase, aBG, and a fixed value of aAG = 100 are presented in Figure 10. Lower values of aBG (i.e., lower surface energies of the B block) result in more favorable contacts of the B-blocks with the gas molecules, which induces the parallel orientation of the cylinders despite the orienting action of the SVA. A gradual increase of aBG toward a neutral free surface first leads to a certain frustration (aBG = 70), when an equilibrium parallel domain orientation competes with the evaporationdriven vertical alignment. Finally, a weak asymmetry of the interaction parameters (aBG = 80−100 vs aAG = 100) cannot suppress the orienting action of the solvent flow.

Figure 7. Final morphology of the film after drying starting from different values of the initial polymer volume fraction, ϕP0. The parameters of the system correspond to those in Figure 3.

In all cases, the solvent is slightly selective to long block (aAS = 35 and aBS = 28), the block segregation aAB = 35 is weak, δ = 2, and h0 = 30. The initial structure of the film is disordered (i.e., a micellar solution, as at t = 0 in Figure 3), and the final structures of the vertical cylinders after SVA turn out to be very similar. In this particular case, the final structure is only weakly dependent on the initial solvent concentration, and in order to obtain vertical cylinders from parallel ones in the equilibrium dry film (Figure 4), it is sufficient to swell the film in a weakly selective solvent until the initial domain structure (obtained after spincoating) is dissolved. The subsequent drying results in growth of the vertical cylinders. As Figure 6b demonstrates, the drying kinetics is notably affected by solvent quality and selectivity. In order to get more detailed information and to relate the kinetics to the morphology of the film, we construct a diagram (Figure 8) using Figure 5 and mapping the time t1/2 needed to evaporate half of the initial amount of the solvent. Figure 8 shows that good solvents with

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CONCLUSIONS Using dissipative particle dynamics, we investigated solvent vapor annealing in films of cylinder-forming diblock copolymers under different conditions, when the film is dried from the spatially disordered swollen state. According to our simulations, gradual removal of the solvent increases the number of contacts between chemically different copolymer blocks and causes the microphase separation in the film. It starts near the free surface of the film and propagates down to the substrate upon the solvent evaporation. Disordered micelles appear everywhere in the film, and, due to the decrease of the solvent concentration, merge into worm-like cylindrical nanodomains. We found that formation of well-ordered cylinders by SVA requires an initially disordered swollen state. A vertical orientation of the cylinders is observed under a high evaporation

Figure 8. Effect of polymer/solvent interaction parameters on the characteristic time t1/2 of the solvent evaporation. The simulation conditions are the same as in Figure 5. 422

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Figure 9. Effect of block segregation on final structure of the film after SVA (top views and internal structure of the minor component) for aAB = 30, 35, 40, 45, and 50.

Figure 10. Film structure after SVA at different values of interaction parameter aBG of the major B block with the gas phase and a fixed value of aAG = 100. Other parameters are aAS = 35, aBS = 28, δ = 2, and ϕP0 = 0.5.

lations were performed on the multiteraflop supercomputers Lomonosov and Chebyshev at Moscow State University.

rate, which is necessary to create a pronounced gradient of the solvent concentration within the film. Also, a certain solvent selectivity for the longer matrix blocks is necessary. We suppose that the vertical orientation of nanodomains is governed by the diffusion-driven solvent flow through the system of nanodomains having different viscosities and permeabilities in a selective solvent as well as by the interfacial flows of components along interdomain interfaces, triggered by the gradient of surface tension (so-called Gibbs−Marangoni effect). Some of our findings are in agreement with recent mean-field type calculations53,60 which also predict that crossing of the ODT upon film drying and a weak segregation regime are needed for an efficient vertical alignment of nanodomains. However, ref 60 predicts that, in the case of the vertical cylindrical structure, the local microphase separation near the free film surface is followed by immediate ordering of the nanodomains while the remaining part of the film stays spatially homogeneous.60 On the contrary, we have detected that the ordering of nanodomains in that region is limited by macromolecular diffusion which is a much slower process than the evaporation of small solvent molecules. The latter is controlled by fast diffusion (evaporation) of small solvent molecules. As a result, the initial structure of the nanodomains in the beginning of solvent evaporation is less ordered than predicted within the DSCFT approach.60

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AUTHOR INFORMATION

Corresponding Author

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

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The financial support from the Deutsche Forschungsgemeinschaft within project Pa771/10-1 (A.V.B. and C.M.P.) and from the Ministry of Education and Science (Russian Federation) within agreement 14.613.21.0027, identifier RFMEFI61314X0027 (I.I.P.) is gratefully acknowledged. The simu423

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