Dynamics of Terrace Formation in a Nanostructured Thin Block

May 20, 2006 - Effect of the supporting pattern on the orientation of hexagonal morphology in thin films of diblock copolymers. Yu. A. Kriksin , P. G...
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Langmuir 2006, 22, 5848-5855

Dynamics of Terrace Formation in a Nanostructured Thin Block Copolymer Film K. S. Lyakhova,† A. Horvat,‡ A. V. Zvelindovsky,*,§ and G. J. A. Sevink*,| Polymer Physics, EindhoVen UniVersity of Technology, P.O. Box 513, 5600 MB EindhoVen, The Netherlands, Physicalische Chemie II, UniVersitat Bayreuth, D-95440 Bayreuth, Germany, Centre for Materials Science, Department of Physics, Astronomy and Mathematics, UniVersity of Central Lancashire, Preston PR1 2HE, United Kingdom, and Leiden Institute of Chemistry, Leiden UniVersity, P.O. Box 9502, 2300 RA Leiden, The Netherlands ReceiVed January 27, 2006. In Final Form: April 3, 2006 We have used dynamic self-consistent field (DSCF) theory to investigate the structural evolution of an ABA block copolymer thin film placed between a solid substrate and a free surface. In line with the few existing theoretical studies for pure homopolymers and mixtures, the free interface is introduced by a void component. In our calculations, the free surface experiences surface roughening and eventually the formation of terraces, as in the experiments. The kinetic pathway of the microstructures was compared to findings of an existing detailed experimental study (Knoll, A.; Lyakhova, K. S.; Horvat, A.; Krausch, G.; Sevink, G. J. A.; Zvelindovsky, A. V.; Magerle, R. Nat. Mater. 2004, 3, 886) and was found to be equivalent in detail. This corroborates our assumption in this earlier work that the pathway due to changing film thickness is similar to a pathway due to changing surface energetics. Moreover, our calculations show for the first time that microstructural transitions are a driving force of polymer/air interface curving and the formation of terraces.

I. Introduction Thin films of polymers play an important role in various industrial applications, in coating solid surfaces, and in controlling adhesion properties and gas separation. In particular, thin films made of block copolymers are gaining interest as the enhancement of functionality (one block versus many and micro versus macrostructure) increases the possibility of material optimization in a rational design strategy. In gel electrophoresis, for instance, properties of AB diblock copolymer thin film coatings can be tuned such that the A block is chemically linked to the solid surface (coating), and the B block can be chosen to have a preferential interaction with specific throughput material components (selectivity). The existence of a “natural” microdomain spacing on the mesoscale (1-1000 nm), originating from the block copolymer tendency to microphase separate into patterns under certain conditions, is also highly desired from an application point of view, for instance, in permeable membranes for gas separation. However, the introduction of interfaces that are the result of confining block copolymers in thin to very thin films may give rise to issues of compatibility, in particular when the thickness of a film is incompatible with this inherent length scale. For films confined between two solid substrates, the microstructures have to adapt to a fixed film thickness. When one or two of the surfaces of the film are exposed to the air or a liquid, these interfaces should be regarded as a free surface, and the film can adjust its thickness locally. In this case, thin films often separate into thinner and thicker regions. In supported films (one free surface), this process is called terrace formation. Terrace formation in supported block copolymer systems was studied experimentally for AB block copolymer systems forming lamellar1-4 and cylindrical5-8 phases. In general, it was found * Corresponding authors. E-mail: A.V.Z., uclan.ac.uk; G.J.A.S., [email protected]. † Eindhoven University of Technology. ‡ Universitat Bayreuth. § University of Central Lancashire. | Leiden University.

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that the behavior of these systems depends on the interaction of the blocks with the two interfaces (the surface fields) and the mismatch of the natural domain distance with the particular film thickness (incommensurability). For lamellar systems of initially incommensurable film thickness, islands of different thickness form. Here, incommensurablity depends on the nature of the two surface fields. For asymmetric surface fields (different blocks have lower interfacial energies at the interfaces), it exists if the initial thickness deviates from nc0. For a symmetric surface field (one block has a lower interfacial energy at both interfaces), this situation occurs if (n + 1/2)c0. In both cases, c0 is the (lamellar) period and n is an integer. In the case in which islands are formed, initially many small islands exist. With time, these islands coalesce, a process that is driven by a decrease in the total surface area. Within these islands, the microstructure is lamellar. At the slope between islands of different thicknesses, other (hybrid) structures can be found. Terrace formation in cylinder-forming systems was first found and studied in ref 5. In recent detailed experiments, the phase behavior of a confined concentrated cylinder-forming polystyreneblock-polybutadiene-block-polystyrene triblock copolymer solution has been studied for varying film thickness and solvent concentration.8 It was found that terrace formation occurs following the same principle as in the lamellar case, where c0 is now the repeat distance of the hexagonal lattice. However, the (1) Coulon, C.; Russell, T. P.; Deline V. R.; Green, P. F. Macromolecules 1989, 22, 2581. (2) Russell, T. P.; Coulon, C.; Deline V. R.; Miller, D. C. Macromolecules 1989, 22, 4600. (3) Russell, T. P.; Anastasiadis, S. H.; Menelle, A.; Felcher, G. P.; Satija, S. K. Macromolecules 1991, 24, 1575. (4) Anastasiadis, S. H.; Russell, T. P.; Satija, S. K.; Majkrzak, C. F. J. Chem. Phys. 1990, 92, 5677. (5) Van Dijk, M. A.; Van den Berg, R. Macromolecules 1995, 28, 6773. (6) Radzilowski, L. H.; Carvalho, B. L.; Thomas, E. L. J. Polym. Sci., Part B 1996, 34, 3081. (7) Harrison, Ch.; Park, M.; Chaikin, P. M.; Register, R. A.; Adamson, D. H.; Yao, N. Macromolecules 1998, 31, 2185. Harrison, Ch.; Park, M.; Chaikin, P. M.; Register, R. A.; Yao, N.; Adamson, D. H. Polymer 1998, 39, 2733. (8) Knoll, A.; Magerle, R.; Krausch, G. J. J. Chem. Phys. 2004, 120, 1105.

10.1021/la060265c CCC: $33.50 © 2006 American Chemical Society Published on Web 05/20/2006

Terrace Formation in a Block Copolymer Film

experiments also showed that confinement can give rise to nonbulk structures such as perforated lamellae and lamellae.8 In a comparative theoretical study,9 the detailed phase behavior for a varying surface field was considered and shown to be equivalent to the phase behavior for varying solvent concentration in the experiments. It was found that a strong surface field typically gives rise to nonbulk structures or surface reconstructions such as a wetting layer, perforated lamellae, and lamellae in the vicinity of the interface. Different from lamellar systems, the structure on the terraces may therefore deviate from the bulk cylindrical structure. For symmetric and weak surface fields, parallel cylinders were found to be stable, but for an increased surface field, perforated lamellae (or even lamellae) can be found. At the slopes of the terraces (for incommensurable film thickness), perpendicular cylinders (weak surface fields) or surface reconstructions (stronger surface fields) can be found.9,10 For asymmetric surface fields, the situation is even more complex, and combinations of (sometimes interconnected) parallel, perpendicular cylinders and surface reconstructions can exist.11 Sphere-forming systems can also form terraces.12 In these experiments, islands and holes consisting of layers of spherical domains of asymmetric diblock copolymers were observed similar to the islands and holes in lamellar block copolymer systems. As the thickness of the film increases, the shape of the island becomes diffuse. With further increases in the thickness, no islands or holes are formed, but the layered structure in the film is distorted when the original film thickness is not the natural thickness of the diblock copolymer. Theoretical studies of terrace formation in thin films are sparse and focus on simulating/calculating the properties of the polymer/ void interface for pure homopolymers and mixtures13-16 or polymer brushes.17 Recently, the hydrodynamic properties of block copolymer film spreading in the regime of complete wetting were investigated theoretically and experimentally,18 and the time dependence of the flow at later stages was found to be diffusion-like (t-1/2). Nevertheless, the fundamentals of terrace formation, that is, how a developing and rearranging microstructure determines (or is determined by) the shape of the free interface, remain to be uncovered. In the present study, we show the results of a dynamic self-consistent field theory calculation19-21 for terrace formation in an ABA triblock copolymer thin film with a free surface. We focus on the dynamics of terrace formation and the relation between film thickness and microstructure. Moreover, we compare one of the transitions in the calculation, the parallel cylinder-to-perforated lamellae transition on the lower terrace, to the detailed experimental observations in ref 22. (9) Knoll, A.; Horvat, A.; Lyakhova, K. S.; Krausch, G.; Sevink, G. J. A.; Zvelindovsky, A. V.; Magerle, R. Phys. ReV. Lett. 2004, 89, 035501. (10) Horvat, A.; Lyakhova, K. S.; Sevink, G. J. A.; Zvelindovsky, A. V.; Magerle, R. J. Chem. Phys. 2004, 120, 1117. (11) Lyakhova, K. S.; Horvat, A.; Magerle, R.; Sevink, G. J. A.; Zvelindovsky, A. V. J. Chem. Phys. 2004, 120, 1127. (12) Yokoyama, H.; Mates, T. E.; Kramer, E. J. Macromolecules 2000, 33, 1888. (13) Cifra, P.; Nies, E.; Karasz, F. E. Macromolecules 1994, 27, 1166. (14) Theodorou, D. N. Macromolecules 1989, 22, 4578. (15) Hariharan, A.; Kumar, S. K.; Russell, T. P. J. Chem. Phys. 1993, 98, 6516. (16) Morita, H.; Kawakatsu, T.; Doi, M. Macromolecules 2001, 34, 8777. (17) Fredrickson, G. H.; Ajdali, A.; Leibler, L.; Carton, J.-P. Macromolecules 1992, 25, 2882. (18) Belyi, V. A.; Witten T. A. J. Chem. Phys. 2004, 120, 5476. (19) Fraaije, J. G. E. M. J. Chem. Phys. 1993, 99, 9202. (20) Fraaije, J. G. E. M.; Van Vlimmeren, B. A. C.; Maurits, N. M.; Postma, M.; Evers, O. A.; Hoffmann, C.; Altevogt, P.; Goldbeck-Wood, G. J. Chem. Phys. 1997, 106, 4260. (21) Sevink, G. J. A.; Zvelindovsky, A. V.; Van Vlimmeren, B. A. C.; Maurits, N. M.; Fraaije, J. G. E. M. J. Chem. Phys. 1999, 110, 2250.

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II. Method We give a short outline of the theory used in the calculations; for more details, see refs 19-21 and references therein. We model the pattern formation that occurs when a block copolymer melt or solution is brought into a state where the chemically different blocks phase separate on a mesoscopic level (1-1000 nm). In our model, a block copolymer molecule is represented by a Gaussian chain consisting of N beads. (We use the same A3B12A3 chain as in ref 9; NA ) 6, NB ) 12, and N ) 18.) Each bead (A or B) typically represents a number of chemical monomers. Differences in monomers give rise to different bead species. The 3D volume used in the calculations (periodic boundary conditions apply) is denoted by Vsystem and contains n Gaussian chains. The presence of a free surface can be taken into account by explicitly considering a void component (referred to as “air” in the remainder of the article) in the system.14-16 It is treated as a solvent in the standard path-integral calculation for a polymer solution and incorporated as a single bead V.23 Its volume is assumed to be the same as the volume of the polymer beads. The interchain interactions are incorporated via a mean field with the interaction strength controlled by the Flory-Huggins (FH) parameters χIJ. In line with our earlier work,20,21 the interactions are specified by the parameters 0IJ (in kJ/mol),23 which are directly related to the dimensionless FH parameters by χIJ ) 0IJ/NAkBT (where NA is Avogadro’s number, kB is the Boltzmann constant, and T ) 300 is the temperature in Kelvin). We note that the choice for 0IJ in kJ/mol is for our convenience only; to relate given values to the more general FH parameters, 0IJ should be recalculated in J/mol in order to obtain the correct dimensionality. The A-B interactions are given by 0AB ) 6.5, in correspondence with earlier work.9 Both interactions of 0 and 0BV, are taken to be large the polymer blocks with the void, AV to avoid the diffusion of polymers into the air. In practice, we cannot completely avoid polymers diffusion into the air (void) phase, but we choose the parameters such that the concentration of polymer in the void phase is less than 1%. The microstructure patterns are described by coarse-grained variables, which are the density fields FI(r) of the different species I. Given these density fields, a free-energy functional F[F] can be defined as follows:19-21 F[F] ) -kT ln

Ψn n!

-

∑∫ I

Vsystem

UI(r) FI(r) dr + Fnid[F]

(1)

Here, Ψ is the partition function for the ideal Gaussian chain in the external fields UI, and Fnid is the contribution due to the nonideal mean field interactions. The external potentials UI and the density fields FI are bijectively related in a self-consistent way via a density functional for Gaussian chains. In the case that a solid wall is incorporated into Vsystem, there is an extra contribution to Fnid due to the interaction of the wall with the polymer/void, and relevant boundary conditions at the solid wall apply. For details, see ref 21. The notation used here here is in accordance with this work. Several methods can be employed to find the minimum of freeenergy (1) and equilibrium density fields FI(r). They can roughly be divided into static and dynamic methods, although a number of hybrids exist that are generally referred to as quasi-dynamic methods. (See ref 24.) A rather complete recent review is given in ref 25. In this article, we use a dynamic scheme that has been developed within our group. An advantage of this scheme is that it intrinsically considers dynamic pathways towards a free-energy minimum, including visits to long-living metastable states. In this sense, the model can be seen to mimic experimental reality when compared to static schemes, which are optimizations based upon mathematical arguments. The thermodynamic forces driving the pattern formation in time are the gradients of the chemical potential µI(r) ) δF/δFI.19,23 (22) Knoll, A.; Lyakhova, K. S.; Horvat, A.; Krausch, G.; Sevink, G. J. A.; Zvelindovsky, A. V.; Magerle, R. Nat. Mater. 2004, 3, 886. (23) Van Vlimmeren, B. A. C.; Maurits, N. M.; Zvelindovsky, A. V.; Sevink, G. J. A.; Fraaije, J. G. E. M. Macromolecules 1999, 32, 646. (24) Drolet, F.; Fredrickson, G. H. Phys. ReV. Lett. 1999, 83, 4317.

5850 Langmuir, Vol. 22, No. 13, 2006 ∂FI ) MI∇‚FI∇µI + ηI ∂t

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where MI is a constant mobility for bead I and ηI(r) is a noise field distributed according to the fluctuation-dissipation theorem. The starting configuration is shown in Figure 1. A solid substrate M1 and wall M2 confine the 3D calculation volume from both sides. At the unconstraint boundaries, periodicity applies; reflective boundary conditions are considered at the solid objects present in the melt. The calculation volume is taken to be Lx × Ly × Lz ) 128 × 64 × 20. Typically, one lateral dimension is taken to be larger in order to reduce the influence of the constraint of periodicity on the air/polymer interface while keeping the calculation time within reasonable limits. The solid walls are located at z ) 0 and 19. We find it convenient to introduce dimensionless concentration fields θ(r) ) νF(r), with ν being the molecular volume (0 e θ(r) e 1). The volume percentage of the polymer is θ h P(r) ) 0.5 (where θP ) θA + θB, and j‚ denotes the spatial avarage of a field over the whole volume except the part of the volume occupied by the solid walls); the polymer-rich region is supported by a hard substrate M1 (with 0 0 0 energetic interactions AM , BM , and VM ).21 1 1 1 Instead of the density fields, the numerical schemes explicitly consider the external potential fields UI, which are bijectively related to the densities by the density functional FI[U].20 Therefore, the input input input fields Finput V (r), FA (r), and FB (r) cannot be directly imported but are generated by iterative optimization of UI(r), with initial value UI(r) ) 0 (and consequently FI[UI ) 0](r) ) FjI) prior to the application of the dynamic scheme of eq 2. An update of the external potentials is accepted when the difference between the input densities Finput and FI[UI] is smaller than some preset value δ , 1. I For arbitrary input fields (for instance, density fields with an unphysical step profile in the z direction), the difference will never vanish completely. In this way, a very “natural” polymer/air interface is automatically generated. For our purposes, we found it advantageous to choose the input void density field to be a step function: θV(x, y, z) ) 0.03 for z < (Ly/2) - 1, and θV(x, y, z) ) 0.97 for z g (Ly/2) - 1. The initial void concentration (3%) in the polymer is slightly higher than the upper boundary of 1% mentioned above. We do this on purpose to check whether the demixing of polymers and voids actually takes place. The input polymer densities θI(r) () NI/N‚θP(r), I ) A or B) are related to the input void density by θP(r) ) 1 - θV(r), ensuring incompressibility. The local thickness of the polymer film h(x, y) depends on the definition of the dividing interface. We define h(x, y) as the integer for which the dimensionless concentration θV(x, y, h(x, y) - 1) < 0.5 whereas θV(x, y, h(x, y)) g 0.5). In the initial stages, the system is not microphase separated. The upper half of the calculation volume (next to M2) is occupied by the void component. We assume gravity to play no significant role; its contribution to the system is neglected. As a reference for our study, we use the experimental system from refs 8 and 9. Here, the behavior of thin polystyrene-blockpolybutadiene-block-polystyrene (SBS) films on a silicon substrate was studied with tapping mode scanning force microscopy (SFM). In refs 9 and 10, this experimental system was parametrized in detail for our method. Although the experimental system is a solution, the solvent is neutral for both blocks. Its presence can be rescaled in the energetic parameters for a melt. Previously, we have calculated the interdomain or repeat distance c0 for the cylindrical morphology to be approximately 6 grid cells.10 We will disregard the presence of a wetting layer in the experiments on the supported side. We have shown earlier11 that the presence of a wetting layer can be taken into account by a reduced film thickness and effective interaction. The initial film thickness is chosen to be completely incompatible with the microdomain distance (h(x, y) ) 3/2c0 ) 9 grid points), consistent with the experiments of Knoll et al.8 The incompatibility of domain distances was shown to lead to the formation of thicker and thinner regions with film thickness corresponding to nc0 (n ) 1, 2) in the experiments; the structures at the terraces are either one or two layers of cylinders. On the lower terrace, cylinders eventually transform into perforated lamellae at later stages, a transition that is accompanied by an additional (tiny) decrease in the film thickness.

Figure 1. Schematic representation of the calculation volume. The block copolymer film is confined between a solid substrate (M1) and void (modeled as solvent). The boundary between the void phase and the polymer film corresponds to a free surface. The number of “free” parameters in the system is rather large (eight): six interaction parameters of bead species I (A, B, and V) 0 with walls Mi (IM ) and two interaction parameters of polymer bead i 0 and 0BV). In the general case, species A and B with void bead V (AV there will be no polymer close to M2 and no void near M1, so we can rewrite the six bead-wall interaction parameters into two 0 0 0 0 effective interactions M ) AM - BM and VM , reducing the 1 1 1 2 number of important parameters to four.

III. Results and Discussion A. Notation Issues. For the assignment of the calculated 3D structures in confinement (i.e., in the presence of one or two solid interfaces), we use the notation in our previous work.9-11 In the case in which the interface energetics is symmetric (similar interfaces), these structures can be surface structures of parallel (C|) and perpendicular (C⊥) oriented cylinders or surface reconstructions such as perforated lamellae (PL), lamellae (L), and a wetting layer (W). Because the symmetry in these systems is broken only by the presence of the interface, these surface reconstructions appear only in layers parallel to the interface. For asymmetric surface energetics (dissimilar interfaces), a number of so-called hybrid structures can also be found. We use the following classification scheme:11 (a) Two structured layers separated by a structureless layer are denoted by a hyphen (-) between the particular structures in the structured layers, for instance, PL-C|. When these structures are equal, we use a shorthand notation, for instance, C|,2 instead of C|-C|. For consistency purposes, we will use the more specific notation C|,1 when the presence of a single layer of parallel cylinders is relevant. (b) The lateral coexistence of different structures in a single layer is denoted by a slash (/), for example, PL/C|, denoting a parallel structure that is a mixture of parallel cylinders and perforated lamellae. (c) No separating symbol is used for structures that are not separated by an unstructured layer, for instance, C|C⊥, denoting cylinders with necks. In ref 11, we found that the wetting layer can be structured as well. Moreover, it can be classified as a half structure of thickness c0/2. Because of this, the wetting layer is alternatively denoted by S1/2, where S ) L, PL, C|, or C⊥. In SFM experiments, one obtains only structural information for the top part of the film. These experimental surface structures can be directly compared to surface structures derived from DSCF calculations by volumetric averaging of the calculated density fields over a thin 3D top slice of the calculation volume (with an avaraging thickness that is taken to be c0/2 ) 3 in the remainder). In general, the 2D surface structures can be identified by a 2D analogue of the 3D classification scheme mentioned above. The issue of determining the actual 3D morphology from

Terrace Formation in a Block Copolymer Film

experimental 2D surface structures is not straightforward and was one of the accomplishments of our previous work.9 For instance, SFM measurements for different 3D structures (C⊥ perpendicular cylinders, C|C⊥ - cylinders with necks, and PL-C⊥ - perpendicular cylinders separated by a structureless layer from a parallel perforated lamellar structure) give visually comparable surface structures (dots in this case). Only a detailed analysis of the surface structure spacing (for surface structures with a low defect density) enables the experimentalist to relate the 2D surface structure to the underlying 3D structure. B. Parametrization: General Considerations. In this article, we focus on the interplay of microphase separation and transitions in the bulk of the free film on one hand and the polymer/air interface curving (the formation of terraces) on the other hand. These phenomena are clearly connected but take place on completely different length and time scales. For instance, the cylindrical microdomain spacing in the experiments9 is on the order of c0 ) 30 nm () 6 gridpoints in our calculation), whereas the lateral dimension of a terrace is on the order of a few micrometers. (For one sample, the distance between two consecutive terraces, measured from the center of the terrace, is approximately ∆l ) 2 µm.) The final image of the terrace structures in ref 9 was taken after more than 7 h, whereas the very slow microphase transition at the lower terrace, the C|-to-PL transition, takes approximately 5 h.22 The microphase separation and formation of the initial terraces with parallel cylinders is much faster. As to the terrace formation itself, one may pose the philosophical chicken-and-the-egg question: which is the leading factor, the curving of the free surface or the microphase transitions? For the free surface to curve, the underlying microstructure has to redistribute itself over the accessible and no longer laterally symmetric polymer volume. Although the correlation lengths are on the molecular level, one should keep in mind that a local transition of the microstructure also gives rise to a redistribution of the domain distances (and packing) naturally assigned to the former and new microstructures as well as changes in the nature of the grain boundaries between the laterally connected microstructures. Structure rearrangement and redistribution by movement of defects is known, but the details of this phenomenon in multiple laterally coexisting structures of changing and adaptive thickness have not been studied much and are far from understood. Assuming a different order, the redistribution of volume through microstructure transitions may be a cause of free surface curving. Structural transitions may result from small local density variations that lower the free energy in a coordinated manner. In principle, parametrization of the actual experimental free film system is possible. The most important new parameter is the energetic interaction between the void (representing the air) and the different polymer blocks, which was used previously for a different polymer in an aqueous environment.26 Because the mixing of the void and polymer is extremely unfavorable, the Flory-Huggins parameters will be very high, 0AV ≈ 0BV ≈ 20-30. For such high interactions, our calculations show no terrace formation within the calculation time and volume dimensions considered. This is no surprise because terrace formation in experimental systems is a very slow process, and terraces of very low aspect ratio (∆v/∆l≈30/2000, with ∆v being the height of the terrace) are formed, which would translate in a lateral dimension of the calculation volume of at least 400 (25) Fredrickson, G. H.; Ganesan, V.; Drolet, F. Macromolecules 2002, 35, 16. (26) Linse, P.; Hatton, T. A. Langmuir 1997, 13, 4066.

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gridpoints. Even for such large calculation volumes, the periodic boundary conditions would have an effect on the terraces. One can even add to the previous considerations that modeling of the terrace formation in block copolymer systems was long believed to be intractable because of the complexety of the interface phenomena and the many different length and time scales involved. Although one question is how we can model the actual terrace formation, the truly interesting question is whether we are able to model such complex phenomena within the SCF framework, which is based on several assumptions. The fact that our model is successful in this respect (as one can see from the remainder) raises both questions and propects. One wonders if it would be possible to simplify the model further (e.g., by replacing our computational-intensive SCF free energy with more computational-efficient phenomenological descriptions, such as a Landau-Ginzburg Hamiltonian in a cell dynamics method). A benefit of such a reduced model would not only be its efficiency but also that it could serve as a starting point for theoretical investigations. Because our DSCF approach models all details for only one set of parameters (i.e., without tweaking our model), we conclude that the underlying physics is governed by/contained in the physical parameters in our model, and we can enter the next stage in computer-aided research: the use of our method for finding and understanding new phenomena. A practical and sufficient approach to deal with the issues raised above is to determine system parameters that give rise to the same quasi-equilibrium sequence of top structures for varying film thickness as the reference, the experimentally observed sequence of ref 9. We rely on top structures because the underlying structure is experimentally mostly unresolved. (Imaging techniques such as SFM can image only top layers.) This set was 0 0 determined to be M ) 12 (for completeness, VM ) 8), 0AV ) 1 1 0 16, and BV ) 10. The choice for the interaction of the void and M2 is less important but should be taken negative to avoid 0 repulsion (VM ) -4). In search of this set, the four parameters 2 were varied over a wide range. Our choice to present only data for this set in this article is further rationalized by the following reasoning. In the remainder, we corrobrate a finding from earlier work9 that identified a strong relation between particular microstructures and the (changing) local thickness of the film. Moreover, calculated and experimental diagrams9-11 show a structure-height dependence that is very gradual in both parameters (for fixed energetic surface interactions). Small perturbations to the parameters of the chosen set will therefore lead to only slightly different pathways. The largest deviation will be for thickness/energetic interaction pairs close to phase boundaries. In this respect, the generality of this study lies in the transition mechanism itself, that is, how a laterally coexisting structure transforms from one combination to the other and how this affects the local film thickness. Moreover, the calculation result allows us to formulate an answer to the philosophical question at the beginning: what is/are the determining factor(s)? The details of the parameter study (including a comparison of the experimental and calculated terrace formation kinetics for different parameter sets) will be the subject of a much more elaborate publication. C. Comparison of Surface Structures after Terrace Formation. In Figure 2, we compare the experimental image of ref 9 (h(x, y) ∈ [c0, 2c0]) to the top layer of the result at later stages for the chosen set of parameters. Experimentally, the following surface structures were identified with increasing film thickness: C|, PL, C|/C⊥, C⊥, C|. We note that the 2D surface structure notation C|/C⊥ denotes a mixture of dots and stripes.

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Figure 2. (a, top) Detail of a tapping mode scanning force microscopy phase image of a thin SBS film on a silicon substrate after annealing in chloroform vapor (reproduced from ref 9 with the permission of the American Physical Society). Bright (dark) corresponds to PS (PB) microdomains. (a, bottom) Schematic height profile of the phase image shown on top. (b, top) Calculated surface structure (θsurf(x, y) ) θA(x, y, h - 2)/3 + θA(x, y, h - 1)/3 + θA(x, y, h)/3 with h ) h(x, y)) of an A3B12A3 block copolymer film confined between a solid substrate and free surface (56 200 time steps). The calculation volume is 128 × 64 × 20; only the part of the structure between the two extremes in film thickness is shown. (b, bottom) Calculated height profile of the phase image shown on top. The relevant interaction parameters used in the calculation are 0AB ) 6.5, 0 0 0 ) 12, AV ) 16, 0BV ) 10, and VM ) -4 (all in kJ/mol). M 1 2

Earlier, experimental analysis27 (as well as 3D calculations9) showed that in three dimensions this phase can be identified as parallel cylinders with necks. (See also the assignment in Figure 2.) From Figure 2, we note that the structures (PL, C|,2) on the terraces (for h(x, y) ≈ nc0, with n being an integer) are well reproduced by the calculations. At the slope, we observe a small region of mixed structures (parallel cylinders with a single neck, C|C⊥, see also Figure 3e). The experimental C⊥ structure is not found for this intermediate film thickness. Conclusions about this mismatch between experiments and calculations are complicated by the fact that the thickness gradient in the calculated system is much higher than in the experiments. This is most likely an effect of our choice of parameters. (See also the discussion in the parametrization paragraph above.) The reduced (27) Konrad, M.; Knoll, A.; Krausch, G.; Magerle, R. Macromolecules 2000, 33, 5518.

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lateral extent of the slope region represents a geometrical constraint. For a predefined thickness profile (a wedge-shaped calculation volume) with a much lower thickness gradient,9 all experimental structures, including the C⊥ structure at intermediate thickness, were reproduced in the correct order and for the correct film thickness. Most likely, in the current system the steep slope suppresses the formation of two different adjacent (C|/C⊥ and C⊥) structures. D. Dynamics of Terrace Formation. The dynamics of terrace formation can be seen from Figures 3 and 4. In Figure 3, we show isodensity surfaces for the A block, leftmost column (θA ) 0.5), and isodensity surfaces of the total polymer field, rightmost column (θA + θB ) θP ) 0.5), at corresponding times. Because the void component is almost absent in the polymer-rich phase, this phase can be considered to be a melt. Moreover, in this melt 1 ) θA + θB + θV ≈ θA + θB ) θP because of the incompressibility of the system. For this reason, the left column shows the microstructure for the shortest (A) block in the polymerrich phase, and the right column shows the shape of the polymer/ air interface (in line with our definition of the free surface, see Model section). Isodensity surfaces for the B component, θB ) θP - θA, are not shown. They can be easily deduced from the left and right columns in Figure 3. We complement the 3D structures by calculated 2D surface structures. The calculated surface structures (middle column) are the result of averaging θA(r) over three layers away from the polymer/air interface (θsurf(x, y) ) θA(x, y, h - 2)/3 + θA(x, y, h - 1)/3 + θA(x, y, h)/3 with h ) h(x, y)). For clarity, we added isothickness lines of the polymer/air interface itself. At small time scales (time step 200), one clearly observes the early stages of microphase separation. This microphase separation is surface-induced; the deviation of the density value from the average (homogeneous) density value is largest in the layer next to the substrate. In the layer closest to the polymer/air interface, there is some moderate microphase separation causing small variations in h(x, y) (Figure 3a). The structures are not yet well evolved, but the overall microstructure resembles PL-C1/2 ⊥ (denoting a structured wetting layer consisting of C⊥ or half spheres) with a large number of defects. Because the interface is still flat, we are tempted to compare this system to a system confined between two solid walls M1, being the same as in this free film with M1 ) 12, and M2, representing the polymer/air interface with M2 ≈ AV - BV ) 6. In Figure 6a in ref 11, showing the most stable phases within this dynamic approach for confinement in a slit with asymmetric surface fields, we see that this is indeed the most stable structure for this combination of film thickness (3/2c0) and surface fields. However, in the latter situation the structure has to adapt to confinement by the second, solid interface, whereas the polymer/air interface can adapt its shape to the energetically and entropically most preferred structures. However, at the next stage (time step 12 200) the polymer/air interface is still flat. Only at the boundary of the two different microphases are three moderate thickness deviations present (Figure 3b). On the left-hand side of the film, the initial PL structure with a large number of defects at the solid substrate has evolved into an almost perfect PL region. A wetting layer C1/2 ⊥ is still present on top of this PL, but the maximum value of the minority component θA in this microphase-separated structure is much smaller than the one in the PL layer. As a result, the spheres are almost invisible for this isodensity value. On the left-hand side, the initial PL-C1/2 ⊥ has evolved into cylinders with necks (C|C⊥). Close to the region of PL-C1/2 ⊥ , these necks can be classified as undulations of the underlying cylinders; further away, these undulations have developed into

Terrace Formation in a Block Copolymer Film

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Figure 3. Dynamics of terrace formation in an A3B12A3 block copolymer film. The calculation volume is 128 × 64 × 20. (Left) Isodensity surface θA ) 0.5 at time steps of (a) 200, (b) 12 200, (c) 32 200, (d) 50 200, (e) 56 200, and (f) 86 200. (Middle) Calculated surface structures at corresponding time steps combined with isoheight lines of the polymer/air interface. (Right) Polymer/solvent interface (isodensity surface of θpol ) θA + θB ) 0.5).

short perpendicular cylinders C⊥ connected to parallel cylinders C| (Figure 4). It is important to notice that the maximum value of the minority component is highest in C|C⊥ and lowest in the structured wetting layer C1/2 ⊥ . Apparently, the polymers tend to mix in this layer. Moreover, different phases coexist while the polymer/air interface remains mostly flat. At later stages (time step 32 200), terraces have been formed. The film has developed into thicker and thinner regions. Somewhat elongated parallel cylinders (C|) on the lower terrace coexist with mostly C|C⊥ on the higher terrace. In the thickest domains (Figure 3c), the onset of a transition of C|C⊥ to two layers of cylinders C|,2 can be seen. At later times (time steps 50 200 and 56 200), the C| structure at the lower terrace slowly transforms into PL, initially at the same film thickness as the perfect C| structure (h(x, y) ) c0 ) 6), but later, when the percentage of PL increases, at a reduced thickness (h(x, y) ) 5). On the higher terrace, the initial abundance of C|C⊥ decreases in favor of parallel cylinders (C|,2) while the film thickness increases. The C|,2 microphase takes over completely at later stages. At the slopes between the two terraces,

a C|/C|C⊥ mixture survives. We stopped our calculations after time step 86 200, where we observe a transition from PL to an unstructured wetting layer W at the lower terrace, which is a perfectly sensible structure for this thickness h(x, y) ) 3 ) 1/2c0. E. Phase Transition Details: The Lower Terrace. Having clarified the nontrivial details of terrace formation, we now focus on the C| f PL transition studied in ref 22. In the experimental and calculation diagrams,9 the regions where PL is found to be the most stable structure are nested within regions of stable C|; the transition from one to the other can therefore be induced by a variation of either of the two parameters of the diagram, the film thickness or the interaction with the interface. In the experiments, this transition is accompanied by a small decrease in the film thickness. The resulting slightly lower terrace with a PL microstructure is found to be stable for the experimental measurement times. In ref 22, we compared the detailed experimental dynamics of this transition with calculation results obtained for a constant film thickness and a change in surface interaction. We found a good match. This is an indication that

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Figure 4. Details of the microstructure in an A3B12A3 block copolymer film prior to curving of the polymer/air interface (time step 12 200).

the dynamic pathways are rather similar. Here, we want to validate this indication by considering the details of the C| f PL transition on the lower terrace in Figure 3 (time steps 32 20056 000). In Figure 5, the details of this transition are highlighted. We note that the area in which this transition takes place in this free film calculation is rather small compared to the work in ref 22. Nevertheless, we clearly observe a nucleation and growth mechanism. We see a decrease in the film thickness in two consecutive steps. At time step 32 200, there is a single defect in a perfect C| structure, which is elongated in the z direction. Between time steps 32 200 and 40 200, this defect migrates while the elongated cylinders flatten. At some stage, a 3-fold connection is formed, exactly as in ref 22. The 3-fold connection serves as a nucleus for the first ring, being the initial stage of the PL phase. Consequently, the PL phase grows. The migration of the nucleus of the PL phase to other parts of the film is new and differs from the pathway in ref 22. The cause of this effect is the packingsthe rearrangement of the structure on a larger scale that is necessary to satisfy the natural domain distances of the new phase. We conclude this from a more detailed analysis of the air/polymer interface that shows that the film thickness reduction (shown here by isolines for integer values of the film thickness) takes place simultaneously in a much larger area than the size of the PL cluster. We also observe the next stage of ref 22: annihilation of C| islands in the PL matrix and rearrangement of defects. The rate at which this transition takes place cannot easily be quantitatively compared to earlier work. First, the transition takes place in a rather small part of the calculation volume giving rise to only a single nucleus of the PL phase. Moreover, the thickness gradient of the terrace is much larger than in the experiment, resulting in a higher pressure at the boundaries of this transition region. This will not influence the dynamic pathway of this transition but may well lead to an increase in the rate at which this transition takes place. The experimental results indicate that the driving force for the formation of a terrace with surface reconstruction (PL) is a decrease in the total free energy compared to a situation where C|,1 is the lower terrace structure (for a higher thickness). The calculation results strengthen this conclusions because they are obtained by a method that minimizes the free energy based on diffusive dynamics of the concentration fields. For this transition, it is not easy to separate the change in structure from the reduction in film thickness. At best it is a cooperative effect, both the

Figure 5. Details of the C|-PL transition on the lower terrace. (Left) Isodensity surface θA ) 0.5 at time steps of (a) 32 200, (b) 40 200, (c) 45 000, (d) 50 600, (e) 54 200, and (f) 58 200. (Right) Calculated surface structures at corresponding time steps combined with isoheight lines of the polymer/air interface.

thickness and the microstructure change in order to get to this situation. The free-energy difference between these two should be rather small because in reality the transition occurs on relatively long time scales. The defects in the structure mediate the domain

Terrace Formation in a Block Copolymer Film

rearrangement, the film thickness change, and phase transition as a whole but are not the cause of them.

IV. Conclusions In the present study, we have for the first time calculated the detailed real space dynamics of terrace formation in a microstructured thin film. The thin film system is chosen to model an experimental ABA block copolymer. It is placed between a solid substrate and void phase such that this polymer/void interface can adapt its shape. The calculations are based on dynamic selfconsistent field theory. We show that under certain conditions the film forms terraces. We have investigated in detail a series of order-to-order transitions in the thin film. We also show that in early stages different microstructures laterally coexist under an almost flat polymer/air interface. Prior to the formation of a lower terrace, the microstructure next to the polymer/air interface at that position “melts” by polymer mixing. Our calculation is somewhat biased by the reduced lateral extent of the simulation volume with respect to the experimentally observed terraces and the condition that the amount of volume occupied by the polymer and void is fixed within a chosen calculation volume. This does not affect our conclusion, however, that the formation of terraces is dictated by the decrease in free energy that is acquired by separating into regions with different underlying microstructure, as already suggested in ref 28. Moreover, curving of the polymer/air interface depends heavily on microstructural reorganization and transitions, a process that is mediated by defect movement. Because of the strong relation between film thickness

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and microstructure, the observed behavior is rather general although the pathway as a whole (the stages at which certain transitions take place) may deviate from the experimental system. Because in reality (and in our simulations) block copolymer systems can be trapped in long-living metastable states, a consequence of this finding is that the shape of the polymer/air interface may also be trapped and may not necessarily reflect the shape assigned to the equilibrium structure. In our free surface calculations, all parameters have been optimized with respect to a particular experimental system22 and remain fixed during the calculation. Subsequently, all order-to-order transitions are observed only because of changing film thickness, just as in this experiment. We compared our findings to the earlier published experimental results and obtained the same quasi-equilibrium behaviorsmicrostructure dependence on thicknesssas in the experiments. We also compared our results to an earlier studied C| f PL transition. We show that this transition is accompanied by a small change in film thickness and find a sequence of transient states (nucleation, annihilation, rearrangement of defects) similar to the one observed in these experiments. Acknowledgment. Supercomputer time was provided by the Stichting Nationale Computer Faciliteiten (NCF). LA060265C (28) Huinink, H. P.; Brokken-Zijp, J. C. M.; van Dijk, M. A.; Sevink, G. J. A. J. Chem. Phys. 2000, 112, 2452.