Directed Copolymer Assembly on Chemical Substrate Patterns: A

1284

Langmuir 2008, 24, 1284-1295

Directed Copolymer Assembly on Chemical Substrate Patterns: A Phenomenological and Single-Chain-in-Mean-Field Simulations Study of the Influence of Roughness in the Substrate Pattern† Kostas Ch. Daoulas* and Marcus Mu¨ller Institut fu¨r Theoretische Physik, Georg-August UniVersita¨t, 37077 Go¨ttingen, Germany

Mark P. Stoykovich,‡ Huiman Kang, Juan J. de Pablo, and Paul F. Nealey Department of Chemical and Biological Engineering, UniVersity of Wisconsin-Madison, Madison, Wisconsin 53706-1691 ReceiVed August 10, 2007. In Final Form: October 13, 2007 The directed assembly of lamella-forming copolymer systems on substrates chemically patterned with rough stripes has been studied using a Helfrich-type, phenomenological theory and Single-Chain-in-Mean-Field (SCMF) simulations.The stripe period matches that of the lamellar spacing in the bulk. The effect of the line edge roughness (LER) of the substrate pattern on the microphase-separated morphology was investigated considering two generic types of substrate LER with a single characteristic wavelength imposed on the edges of the stripes: undulation and peristaltic LER. In both cases, the domain interfaces are pinned to the rough stripe boundary at the substrate and, thus, are deformed. We study how this deformation decays as a function of the distance from the substrate. The simple theory and the SCMF simulations demonstrate that one of the basic factors determining the decay of the roughness transferred into the self-assembled morphology is the characteristic LER wavelength of the substrate pattern; i.e., the distance over which the roughness propagates away from the substrate increases with wavelength. However, both approaches reveal that, for a quantitative understanding of the consequences of substrate LER, it is important to consider the interplay of the pattern wavelength with the other characteristic length scales of the system, such as the film thickness and the bulk lamellar spacing. For instance, in thin films, the induced deformation of the lamellar interface decays slower with distance from the patterned surface than in thicker films. It is shown that the phenomenological theory can capture many of the same qualitative results as the SCMF simulations for copolymer assembly on substrate patterns with LER, but, at the same time, is limited by an incomplete description of the constraints on the polymer chain conformations imposed by the substrate.

I. Introduction The natural tendency of block copolymers to self-organize on the molecular scale into dense, regular arrays of structures makes them attractive for various nanotechnology-related applications.1 However, the structures formed during spontaneous self-assembly commonly contain many defects that limit their technological application. Thus, various strategies such as chemically patterned surfaces,2-4a,4b graphoepitaxy,5,6 solvent evaporation,7,8 shear,9,10 external electrical fields,11-13 and directional solidification14,15a have been proposed to control orientation and long-range ordering of the domains. Directed assembly of symmetric copolymers in †

Part of the Molecular and Surface Forces special issue. * Corresponding author. Electronic address: daoulas@ theorie.physik.uni-goettingen.de. ‡ Present address: Department of Materials Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801. (1) Xia, Y.; Rogers, J. A.; Paul, K. E.; Whitesides, G. M. Chem. ReV. 1999, 99, 1823. (2) Kim, S. O.; Solak, H. H.; Stoykovich, M. P.; Ferrier, N. J.; de Pablo, J. J.; Nealey, P. F. Nature 2003, 424, 411. (3) Edwards, E. W.; Montague, M. F.; Solak, H. H.; Hawker, C. J.; Nealey, P. F. AdV. Mater. 2004, 16, 1315. (4) (a) Stoykovich, M. P.; Mu¨ller, M.; Kim, S. O.; Solak, H. H.; Edwards, E. W.; de Pablo, J. J.; Nealey, P. F. Science 2005, 308, 1442. (b) Stoykovich, M. P.; Kang, H.; Daoulas, K. Ch.; Liu, G.; Liu, C.; de Pablo, J. J.; Mu¨ller, M.; Nealey, P. F. ACS Nano 2007, 1, 168. (5) Segalman, R. A.; Yokoyama, H.; Kramer, E. J. AdV. Mater. 2001, 13, 1152. (6) Sundrani, D.; Darling, S. B.; Sibener, S. J. Nano Lett. 2004, 4, 273. (7) Kim, S. H.; Misner, M. J.; Russell, T. P. AdV. Mater. 2004, 16, 2119. (8) Kim, S. H.; Misner, M. J.; Xu, T.; Kimura, M.; Russell, T. AdV. Mater. 2004, 16, 226. (9) Angelescu, D. E.; Waller, J. H.; Adamson, D. H.; Deshpande, P.; Chou, S. Y.; Register, R. A.; Chaikin, P. M. AdV. Mater. 2004, 16, 1736. (10) Angelescu, D. E.; Waller, J. H.; Register, R. A.; Chaikin, P. M. AdV. Mater. 2005, 17, 1878.

thin films on chemically striped substrates was recently demonstrated to form defect-free arrays of aligned lamellae over arbitrarily large areas and to precisely position or register the copolymer domains with respect to the substrate pattern.2-4a This approach also can fabricate device-oriented structures at the nanoscale, including those required for applications such as magnetic storage media, flash memory devices, capacitors, and complete integrated circuit layouts.4b Local properties of the self-assembled morphology, however, can be equally important as the long-range order for the suitability of copolymer nanostructures in many applications. For instance, the line edge roughness (LER) of features produced by conventional lithographic processes utilizing chemically amplified photoresists is an important issue. When patterning a stripe serving as a transistor gate,16,17 the LER on the left and right edges leads to variations in the gate length. The variations on small length scales generated by short-wavelength LER can seriously impair the functionality of individual transistors, while larger lengthscale fluctuations, due to long wavelength LER, cause variations in the performance of entire transistor groups (see, e.g., Figure 1 of ref 17). (11) Morkved, T. L.; Lu, M.; Urbas, A. M.; Ehrichs, E. E.; Jaeger, H. M.; Mansky, P.; Russell, T. P. Science 1996, 273, 931. (12) Li, H. W.; Huck, W. T. S. Nano Lett. 2004, 4, 1633. (13) Thurn-Albrecht, T.; Schotter, J.; Ka¨stle, G. A.; Emley, N.; Shibauchi, T.; Krusin-Elbaum, L.; Guarini, K.; Black, C. T.; Tuominen, M. T.; Russell, T. P. Science 2000, 290, 2126. (14) Bodycomb, J.; Funaki, Y.; Kimishima, K.; Hashimoto, T. Macromolecules 1999, 32, 2075. (15) Rosa, C. D.; Park, C.; Thomas, E. L.; Lotz, B. Nature 2000, 405, 433. (16) Yamaguchi, A.; Komuro, O. Jpn. J. Appl. Phys. 2003, 42, 3763. (17) Yamaguchi, A.; Fukuda, H.; Arai, T.; Yamamoto, J.; Hirayama, T.; Shiono, D.; Hada, H.; Onodera, J. J. Vac. Sci. Technol. B 2005, 23, 2711.

10.1021/la702482z CCC: $40.75 © 2008 American Chemical Society Published on Web 12/08/2007

Directed Copolymer Assembly on Patterned Substrates

In the case of directed copolymer assembly on chemically patterned substrates, the boundaries between the chemically distinct regions of the substrate pattern also have some LER (denoted as “substrate LER” to distinguish it from the distortion of the internal interfaces of the soft copolymer material). Thus, it is of interest to understand to what extent the roughness of the substrate pattern propagates into the copolymer morphology. Since the equilibrium properties of the interfaces in a copolymer morphology are dictated by thermodynamics, the interfacial tension and rigidity favors smooth interfaces, and the effects of small pattern imperfections can be mitigated; i.e., the morphology is expected to exhibit a degree of self-healing. Copolymers should smooth-out the local LER of the substrate pattern, and the substrate LER is expected not to affect the copolymer morphology farther away from the substrate. Thus, block copolymer lithography, in principle, could offer better means for controlling the dimensions of the patterned device features than conventional photoresist-based methods. On the other hand, the free-energy costs of deforming the morphology are only on the order of kBT per molecule in these soft matter systems. Therefore, the strong interactions with the substrate pattern have a pronounced influence on the reconstruction of the morphology. In fact, for the film thicknesses considered here and previously,2-4a,18 the substrate interactions can influence the morphology over the entire film thickness. This softness of the morphologies enables us to fabricate irregular (e.g., nonperiodic) patterns in films that have no analog in the bulk. In view of the subtle balance between the strong forces at the chemically patterned substrate, the softness of the copolymer morphology, and the spatial confinement, a quantitative approach is warranted. The small free-energy costs associated with the deformation of the soft, self-assembled morphology inevitably brings forth the role of thermal fluctuations. Even if assembled on substrates with no substrate LER, the instantaneous shape of the interfaces exhibits a certain degree of roughness induced by thermal fluctuations. Recent developments in in-situ scanning force microscopy19-22 allow for monitoring of the shape changes of the soft domains in time and exploring to what extent the interfacial roughness generated by the thermal fluctuations can influence the morphology. In the past, the interface between two simple liquids preferentially wetting the two different regions of a patterned substrate separated by a wavy boundary has been considered by theory.23 At the substrate, the interface is pinned to the rough boundary and deforms. This deformation increases the area of the interface and, like capillary waves, it raises the free energy by an amount that is proportional to the interface tension and the excess area. The deformation exponentially decays farther away from the substrate, and the characteristic decay length perpendicular to the substrate is proportional to the characteristic wavelength of the wavy boundary along the substrate. Thus, we expect that a roughness with a small wavelength propagates less into the liquid than one with a large wavelength. Although, this limiting case of a simple liquid offers an important insight into the self-healing properties of liquid interfaces, in the case of (18) Daoulas, K. Ch.; Mu¨ller, M.; Stoykovich, M. P.; Park, S. M.; Papakonstantopoulos, Y. J.; de Pablo, J. J.; Nealey, P. F.; Solak, H. H. Phys. ReV. Lett. 2006, 96, 36104. (19) Knoll, A.; Magerle, R.; Krausch, G. Macromolecules 2001, 34, 4159. (20) Tsarkova, L.; Knoll, A.; Magerle, R. Nano Lett. 2006, 6, 1574. (21) Knoll, A.; Lyakhova, K. S.; Horvat, A.; Krausch, G.; Sevink, G. J. A.; Zvelindovsky, A. V.; Magerle, R. Nat. Mater. 2004, 3, 886. (22) Horvat, A.; Knoll, A.; Krausch, G.; Lyakhova, K. S.; Sevink, G. J. A.; Zvelindovsky, A. V.; Magerle, R. Macromolecules 2007, 40, 6930. (23) de Gennes, P. G.; Brochard-Wyart, F.; Quere, D. Capillarity and Wetting Phenomena; Springer: New York, 2004.

Langmuir, Vol. 24, No. 4, 2008 1285

copolymer morphologies, additional factors such as the coupling of neighboring interfaces, interfacial bending rigidity, and the geometrical constraints imposed on the extended macromolecules must be accounted for to obtain a quantitative description. Various aspects of the influence of substrate LER in the copolymer systems could be addressed through a wealth of theoretical methods, including Landau-Ginzburg theory,24-26 self-consistent field (SCF) theory,27-31 dynamic self-consistent field theory,22,32-35 and analytical approaches.36-40 A rather simple approach consists in considering the elastic properties of the copolymer interfaces within Helfrich-type Hamiltonians.41 In this framework, one describes the interface as an infinitely thin sheet that is characterized by a few material parameters: tension, spontaneous curvature, and bending rigidities. These coarse-grained parameters can be estimated from strong segregation theory (SST)42-44 or SCF theory45,46 for elastic properties of a diblock copolymer monolayer. Approaches of this kind have been used to describe instabilities (undulations) in strained lamellar phases, where the strain was generated either by uniaxial stress44 or by the assembly of lamellae on a striped surface with a mismatch between stripe periodicity and bulk lamellar spacing.47 Within a similar phenomenological description48 and experiments/Single-Chain-in-Mean-Field (SCMF) simulations,49 the effect of substrate patterns comprising asymmetric stripe widths on the copolymer morphology has been considered. More recently, a phenomenological approach was used in combination with MC simulations to study the morphology of copolymer melts confined in nanopores.50 In all the above cases, the edges of the substrate pattern were perfectly straight, which corresponds to the limit where the wavelength of the substrate LER diverges. In this work we present a combined phenomenological and simulation study of the effects of substrate LER on the average shape of copolymer interfaces for the case of lamella-forming copolymer melts assembled on substrates chemically patterned with stripes. The substrate pattern with no LER is a sequence of alternating A- and B-attracting stripes with period L. If L matches within approximately 10% the lamellar spacing Lo in the bulk, the polymer can register perfectly with the pattern.2 Substrate LER consists of sinusoidal fluctuations of the stripe (24) Tsory, Y.; Andelman, D. J. Chem. Phys. 2001, 115, 1970; Europhys. Lett. 2001, 53, 722; Macromolecules 2001, 34, 2719; Interface Sci. 2003, 11, 259. (25) Kielhorn, L.; Muthukumar, M. J. Chem. Phys. 1999, 111, 2259. (26) Wu, X. F.; Dzenis, Y. A. J. Chem. Phys. 2006, 125, 174707. (27) Petera, D.; Muthukumar, M. J. Chem. Phys. 1998, 109, 5101. (28) Matsen, M. W. J. Chem. Phys. 1999, 110, 4658. (29) Chen, H. Y.; Fredrickson, G. H. J. Chem. Phys. 2002, 116, 1137. (30) Miao, B.; Yan, D.; Wickham, R. A.; Shi, A. C. Polymer 2007, 48, 4278. (31) Huang, K.; Balazs, A. Phys. ReV. Lett. 1991, 66, 620. (32) Katz, A. A.; Fredrickson, G. H. Macromolecules 2007, 40, 4075. (33) Lyakhova, K. S.; Horvat, A.; Zvelindovsky, A. V.; Sevink, G. J. A. Langmuir 2006, 22, 5848. (34) Morita, H.; Kawakatsu, T.; Doi, M. Macromolecules 2001, 34, 8777. (35) Hasegawa, R.; Doi, M. Macromolecules 1997, 30, 3086. (36) Stepanow, S.; Fedorenko, A. A. Europhys. Lett 2002, 58, 368. (37) Fredrickson, G. H. Macromolecules 1987, 20, 2535. (38) Halperin, A.; Sommer, J. U.; Daoud, M. Europhys. Lett. 1995, 29, 297. (39) Angerman, H. J.; Johner, A.; Semenov, A. N. Macromolecules 2006, 39, 6210. (40) Turner, M. S.; Joanny, J. F. Macromolecules 1992, 25, 6681. (41) Helfrich, W. Z. Naturforsch., C 1973, 28, 693. (42) Wang, Z. G.; Safran, S. A. J. Phys. (Paris) 1990, 51, 185. (43) Wang, Z. G.; Safran, S. A. J. Chem. Phys. 1991, 94, 679. (44) Wang, Z. G. J. Chem. Phys. 1994, 100, 2298. (45) Matsen, M. W. J. Chem. Phys. 1999, 110, 4658. (46) Mu¨ller, M.; Gompper, G. Phys. ReV. E 2002, 66, 041805. (47) Perreira, G. G.; Williams, D. R. M. Europhys. Lett. 1998, 44, 302. (48) Wang, Q.; Nath, S. K.; Graham, M. D.; Nealey, P. F.; de Pablo, J. J. J. Chem. Phys. 2000, 112, 9996. (49) Edwards, E. W.; Mu¨ller, M.; Stoykovich, M. P.; Solak, H. H.; de Pablo, J. J.; Nealey, P. F. Macromolecules 2007, 40, 90. (50) Wang, Q. J. Chem. Phys. 2007, 126, 2007.

1286 Langmuir, Vol. 24, No. 4, 2008

Daoulas et al.

Figure 1. Sketch of an AB copolymer morphology assembled on patterns with undulation (left) and peristaltic roughness (right). The left and the right insets show the substrate pattern for the cases of undulation and peristaltic roughness, respectively. The A and the B monomerattracting regions are shown in yellow and blue, respectively. The two main panels present the detailed morphologies assembled on patterns with undulation (left) and peristaltic roughness (right). For clarity, only the A-rich phase is shown in yellow. The wavy lines are examples of the polymer chain trajectories, which are assumed to be parallel to the local normal of the AB interface, while the straight lines clarify the geometric construction used for the phenomenological description of chain stretching (see main text). For the peristaltic roughness, the two shaded surfaces mark the vertical midplanes of an A-rich and a B-rich slab.

edges with a specific wavelength, LLER. Two model cases ofsubstrate LERsperistaltic and undulational (see inset of Figure 1)sare considered, and, for these examples, a phenomenological theory of the deformation of the copolymer interfaces generated by the rough stripe boundary is formulated. Phenomenological approaches are attractive because of their simplicity and ability to highlight the basic, qualitative behavior of complex interface assemblies in polymeric systems. On the other hand, they invoke significant simplifications and important factors determining the behavior of the studied systems might be ignored. Additionally, a coarse-grained, particle-based model for studying the effects of substrate LER is investigated, and a highly accurate, yet approximate solution of this model is obtained via SCMF simulations.51,52 Basic aspects of the influence of the substrate LER on the assembled polymer morphologies derived by both approaches are presented. Particular emphasis is placed upon the role of substrate LER wavelength and its interplay with other characteristic length scales of the system, including the film thickness and the dimensions of the natural periodicity of the bulk morphology. Our manuscript is organized as follows: section II presents a brief description of the coarse-grained polymer model and the SCMF simulation technique, and section III presents a phenomenological approach to describe the effects of the LER of the substrate pattern. Section IV discusses the results of the phenomenological approach and SCMF simulations. Finally, section V summarizes the basic conclusions of our study.

II. Coarse-Grained Modeling of Substrate LER Effects on Copolymer Assembly A. Molecular Model and LER Representation. The effect of substrate LER on the assembled polymeric structures is studied for the case of two copolymer systems. First, symmetric diblock AB copolymer melts will be considered allowing a direct comparison with a phenomenological theory presented in the following section. Second, in the spirit of recent experiments,2-4a,4b,18 symmetric ternary AB copolymer blends with their respective A and B homopolymers will be briefly investigated. The above (51) Mu¨ller, M.; Smith, G. D. J. Polym. Sci., Part B: Polym. Phys. 2005, 43, 934. (52) Daoulas, K. Ch.; Mu¨ller, M. J. Chem. Phys. 2006, 125, 184904.

lamella-forming systems are described through a minimal coarsegrained model aimed at reproducing universal features of polymeric behavior on a mesoscopic scale. The chain connectivity is modeled through a discretized Edwards Hamiltonian, Ho:

Ho(m)[r(s)] kBT

N(m) - 1

)

∑ s)1

3(N(m) - 1) [ri(s + 1) - ri(s)]2 (1) 2 2Re(m)

where ri(s) is the coordinate of sth bead of polymer i, and m stands for the chemical species of the bead and the type of the molecule, respectively. Re(m)2denotes the mean squared end-toend distance of the unperturbed polymer coil, while kB and T are the Boltzmann constant and temperature, respectively. The endto-end distance, Re(m) is the only parameter defining the chain architecture, through which the length scale of the coarse-grained representation and an experimental system can be related. In the following, all lengths are referred to in units of the end-to-end distance of the diblock copolymer, Re. Although the numbers of beads, N(m), per molecule explicitly appear in eq 1, they do not have a specific physical meaning; i.e., different contour discretizations describe the same physical realization. Here, in both pure diblock melts and ternary blends, the diblock chains are discretized with N ) 32 beads. In pure diblock melts, the molecules are perfectly symmetric, and the asymmetry parameter of the diblock, f, is set to f ) 0.5. In the case of ternary blends, we select a system that has previously been studied by experiments and simulation.4a,18 A value of f ) 0.468 for the asymmetry parameter is chosen, while the number of beads in the A and B blocks is set to 15 and 17, respectively. The ternary blends contain 60% AB diblock copolymers, 20% A and 20% B homopolymers, composed of NA ) 12 and NB ) 13 beads, respectively. Excluded volume effects are described via Helfand’s potential,53 and the incompatibility between the A and B species of the diblock is characterized by a Flory-Huggins approach. The polymer-substrate interactions are modeled as external fields. The Hamiltonian of the nonbonded polymer-polymer and polymer-substrate interactions is given by (53) Helfand, E.; Tagami, Y. J. Chem. Phys. 1971, 56, 3592.


Hnb Fo ) kBT N

Langmuir, Vol. 24, No. 4, 2008 1287

χN (φˆ + φˆ B - 1)2 - (φˆ A - φˆ B)2] + ∫ d3r[κN 2 A 4 Fo N

∫ d3rNUW(r)(φˆ A - φˆ B)

(2)

In the above equation, Fo denotes the average monomer number density, while χ is the Flory-Huggins parameter. The local A and B monomer volume fractions are denoted as φˆ A(r) and φˆ B(r). Like the Re for the chain connectivity, only coarse-grained parameter combinations will be important for defining the nonbonded interactions: the “invariants” of the representation. Since they have been discussed elsewhere,53-56 it is only mentioned that χN sets the incompatibility between the A and the B blocks, while κN controls the amplitude of the local density fluctuations, being proportional to the inverse compressibility. Here, we set χN ) 37.6, which is typical for polystyrene-blockpoly(methyl methacrylate) (PS-b-PMMA) melts that have been used in templated block copolymer assembly experiments.2-4a In a dense polymer melt, local density fluctuations are negligible; therefore, κN should be chosen high enough to mimic the incompressible behavior on the mesoscopic length scale.52 Here we choose κN ) 50. This value is sufficient to suppress total density fluctuations on the scale of a small fraction of the molecular size and simultaneously allows for computational efficiency. The role of fluctuations in determining the physical properties of a polymer system is controlled by the invariant degree of polymerization, N h ≡ (F0Re3/N)2, characterizing the degree of interdigitation of polymer chains. It constitutes an additional invariant of the coarse-grained model and, in the current work, is set to N h ) 14884, which corresponds to the typical molecular weight in experiments.2-4a For these systems, the equilibrium bulk lamellar spacing is Lo ) 1.72Re for the pure copolymer melt and Lo ) 2.27Re for the ternary blend. The last term in eq 2 accounts for the polymer-substrate interactions. Since the coarse-grained model does not incorporate local fluid structure or details of chain architecture, it cannot address properties on length scales smaller than a small fraction of the molecular size. Thus, it is consistent to adopt the same level of description with respect to the patterned substrate, ignoring the fine details of the interactions with the polymer on the length scale of angstroms. We rather try to elucidate the universal effects of LER of the substrate pattern, considering wavelengths on the order of or above the Re scale. Specifically, the substrate is placed at x ) 0, and its interactions with the A and B monomers are described through the potential UW(r):

UW(r) )

[ ]

Λf(y,z) x2 exp - 2 /Re 2

(3)

It can be seen that, in the x direction perpendicular to the substrate, the details of monomer-substrate interactions are neglected, and they are represented, in a coarse-grained way, through a generic function. This function is short-ranged, and its decay on the sub-Re scale is defined by a single characteristic length, . Here, the value ) 0.15Re is used. The parameter Λ describes the strength of the polymer-substrate interactions, and the chemical pattern is encoded through the function f(y,z), taking values of R or -β depending on whether the lateral y and z (54) Helfand, E.; Sapse, A. M. J. Chem. Phys. 1975, 62, 1327. (55) Mu¨ller, M. In Soft Matter; Gompper, G., Schick, M., Eds.; Wiley VCH: Weinheim, Germany, 2005; Vol. 1. (56) Daoulas, K. Ch.; Mu¨ller, M.; Stoykovich, M. P.; Papakonstantopoulos, Y. J.; de Pablo, J. J; Nealey, P. F.; Park, S. M.; Solak, H. H. J Polym. Sci., Part B: Polym. Phys. 2006, 43, 3444.

coordinates of the bead correspond to an A- or B-preferential region of the pattern. The substrate LER is introduced as a perturbation to the edges of the stripes. The simplest case of LER with a single characteristic wavelength will be considered, and the perturbation imposed on a stripe edge takes the form of a simple periodic function:

Ao cos(ωoy + φ), with ωo )

2π LLER

(4)

where y is the coordinate along the edge, Ao is the LER amplitude, ωo is the inverse wavelength, and LLER is the corresponding wavelength. The cases of peristaltic and undulational LER are realized through the phase shift factor φ (see inset of Figure 1). In particular, for the case of undulational LER, for all edges, φ ) 0. For modeling peristaltic LER, the edges of the stripes are numbered such that φ ) 0 for even edges, and φ ) π for odd edges. The amplitude, Ao, of the pattern LER is Ao ) 0.2Re. Generally, the registration of the polymer with a pattern that differs from the natural symmetry and geometry of the bulk morphology is controlled by the strength of the interactions. If the polymer-substrate interactions are strong enough, a surface reconstruction of the soft morphology will take place.18 The assembly of a lamella-forming copolymer melt on stripes with LER can be also conceived as a “local” surface reconstruction because substrate LER deforms the AB interface from its planar geometry, which is thermodynamically favored in the bulk. In the case of weak polymer-substrate interactions and/or pronounced substrate LER, one expects an interface depinning to occur; i.e., the polymer morphology does not closely follow the rough pattern. Substitution of eq 3 into eq 2 shows that ΛN is the parameter setting the energy scale of the polymer-substrate interactions. Earlier studies4a,18,56 have shown that an order of magnitude ΛN ∼ 1 is a realistic choice for achieving good agreement with the experimentally observed morphologies in PS-b-PMMA melts assembled on chemically patterned substrates. Here we choose ΛN ) 3, in accord with a recent study18 of surface-induced reconstruction, yielding excellent agreement with experiments. In pure diblock melts, the interactions of the A beads with the A-preferential regions of the substrate and the interactions of the B beads with the B-preferential regions are the same; i.e., R ) -1 and -β ) 1. This symmetric choice allows for a direct comparison with the phenomenological theory. In the case of ternary blends, the interactions of the A beads4a,18 are chosen to be stronger, and the values R ) -1 and -β ) 0.6 are used. Finally, the upper layer of the film is considered to be exposed to air. Assuming that there is no preferential A or B segregation, it is modeled as a hard wall, positioned at a distance H above the substrate. We consider two film thicknesses, H ) 1.43Re and H ) 2.86Re, the former being a typical value for the thickness of spin-coated films used in previous studies.56 With these ingredients, the coarse-grained model incorporates (a) the characteristic free energy scales that influence the thermodynamics of the directed assembly of the soft matter on a chemically patterned substrate set by χN and ΛN, (b) the strength of fluctuations, characterized by N, and (c) the relevant length scales, LLER, L, Lo, and H. The first, LLER, characterizes the length of the roughness of the substrate pattern. In principle, patterns combining many characteristic wavelengths could be considered, in the same spirit. Second, the model accounts for the equilibrium periodicity, Lo, of the bulk lamellar structure and, third, the periodicity of the pattern L. As was discussed above, their mismatch must not be excessive for achieving perfect longrange order. However, even a small mismatch creates a stress

1288 Langmuir, Vol. 24, No. 4, 2008

Daoulas et al.

in the structure, which could affect the propagation of substrate LER. Finally, the finite film thickness, H, which will turn out to be an important length scale when considering substrate LER effects is also accounted for. B. SCMF Simulations. Within the above coarse-grained model, the equilibrium properties of the system can be obtained via SCMF simulations. This is an approximate, particle-based simulation technique inspired by the standard SCF theory. It retains the computational efficiency of SCF theory, including, at the same time, fluctuation effects. A more detailed description of this technique can be found in ref 52. We mention that, for the implementation of SCMF simulations, the simulation box is discretized with a grid of mesh size ∆L, such that the integrals in eq 2 are approximated by a summation over grid points. To each grid point m, volume fractions φˆ R,m are assigned by counting the beads of R species within the corresponding grid cell. The spatial discretization introduces an additional parameter, the grid mesh size ∆L, into the coarse-grained model. The mesh size plays the role of an interaction range. Here, we use a grid size of ∆L ∼ Re/6, which is comparable to the amplitude of the substrate LER and much smaller than its wavelength. Periodically, during the SCMF simulations, the polymer chains are considered independent, being placed in fluctuating external fields, wR,m, defined by

wR,m ∂Hnb N N δHnb ) ≡ kBT k TF ∆L3 ∂φR,m kBTFo δφR B o

(5)

These external fields approximate the instantaneous interactions of an R-type segment with its surroundings, and they are frequently recalculated from the instantaneous spatial distribution of the components. Between field updates, the chains evolve in these background fields via a short Monte Carlo (MC) simulation. To keep the SCMF simulations accurate,52 one typically uses local MC moves such as the random segmental displacement or slithering-snake moves. The moves are accepted according to a Metropolis criterion, pacc ) min [1,e-∆Hb - ∆W], where ∆Hb is the difference in bonded energy between the proposed and the old configuration, while ∆W is the change in the interactions with the external fields (including the polymer-substrate interactions). After a small number of MC steps, the densities are recalculated from the monomer positions, and the fields are updated according to eq 5. This short MC simulation and the density/field update constitutes one SCMF simulation cycle. It is pointed out that the recalculation of the fields from the density distribution of the ensemble recovers the correlations between the polymer chains, and, in the limit that the external fields follow the polymer spatial distribution instantaneously, the SCMF simulations become exact. In this case they are equivalent to a standard MC simulation57,58 of Hamiltonians 1 and 2. However, a short decoupling of chains is advantageous because it allows for an efficient implementation on parallel computers. This short decoupling of interactions constitutes the quasi-instantaneous field approximation52 which is the basis of the SCMF simulations. This approximation is controlled by a small parameter ) (V/nN2∆L3) ) (1/N2xN h) (Reo/∆L)3, which plays a role similar to that of the Ginzburg parameter in mean field theory. The small value of ) 0.002 guarantees that the simulations are accurate. In this work, defect-free lamellar morphologies assembled on a pattern with no roughness were utilized as an initial configuration (57) Laradji, M.; Guo, H.; Zuckermann, M. J. J. Phys. ReV. E 1994, 49, 3199. Soga, K. G.; Zuckermann, M. J.; Guo, H. Macromolecules 1996, 29, 1998. Miao, L.; Guo, H.; Zuckermann, M. J. Macromolecules 1996, 29, 2289. (58) Detcheverry, F. A.; Daoulas, K. Ch.; Mu¨ller, M.; de Pablo, J. J., in press.

for SCMF simulations. The assembly was facilitated by an external field, acting in the bulk of the film, in combination with nonlocal MC moves such as chain translation. The purpose was to achieve the ideal limit of nearly equal distribution of molecules among the stripes. If at this stage, one used SCMF simulations (which only utilize local moves), one would mimic the dynamics of the spatial organization of the real polymer system. During the process of phase separation, a local surplus of polymer chains in a stripe area can be created. Then, once the structure is formed, it is difficult to equalize the number of chains among different lamellae through diffusion because of the chemical potential barrier; e.g., the A monomers are slow to “tunnel” through the B-rich domains and vice versa. In a rather small system, this could affect the shape of the lamellae and would create difficulties in elucidating the effect of substrate LER. The assembled lamellae were subjected to SCMF simulations invoking local displacement and slithering snake moves. For local monomer displacements, the external fields are updated after 100% of the monomers were attempted on average to be moved. For slithering-snake moves, we updated after attempting to reptate 10% of all chains. To calculate the average shape of the assembled polymer morphologies on the rough pattern, we typically employed 500 configurations, saved every 5000 SCMF simulation cycles after the initial morphology is equilibrated. The equilibration stage lasts typically 5 × 105 SCMF cycles. Because of periodic boundary conditions in lateral directions, the dimensions of the simulation cell are a multiple of the stripe spacing of the pattern and the characteristic wavelength of the LER. The system geometry is 1.43(2.86)Re × 5.16Re × 10.32Re for pure diblock melts and 1.43(2.86)Re × 6.81Re × 13.62Re for ternary blends. The corresponding number of grid nodes was 9(18) × 36 × 64 and 9(18) × 48 × 96, respectively. The number in parentheses denotes the value employed for thicker films. To obtain the desired value of the invariant polymerization degree, N, the simulation cell contains 9290(18580) AB copolymer chains in the case of pure diblock melts, while 11217(22434) copolymers, 9962(19924) A homopolymers, and 9202(18404) B homopolymers are used in ternary blends. These numbers illustrate the computational efficiency of the SCMF simulations technique. For example, the generation of 2.5 × 106 MC steps for the large ternary blend system, typically takes 6 days on 10 IBM p690, Power4+, 1.7 GHz processors. Additionally, we also used larger system sizes to explore finite size effects (cf. Figure 6). Although the present work focuses on the effect of LER on the substrate pattern on the aVerage shape of the copolymer interfaces, we emphasize that SCMF simulations do incorporate interface fluctuations (see discussion of Figure 3). The interplay between substrate LER, the mismatch between the substrate pattern and bulk lamellar spacing, and thermal interfacial fluctuations will be addressed in a future study.

III. Phenomenological Theory of LER Propagation A. Free Energy of Modulated Interfaces. The starting point of the phenomenological theory on how the LER on the substrate pattern propagates into the polymer film is the assumption that, at the substrate, the AB copolymer interface is pinned to the rough boundary of the two chemically different regions. Thus, the rough pattern edge acts as a boundary condition for the AB interface at the substrate, and the problem is reduced to describing how this boundary condition affects the shape of the internal interface farther from the substrate. The free energy, F, per chain in a monolayer of a modulated lamella can be expressed as a sum50 of chain stretching (Fel), interfacial (FAB), bending (Fbend), and Gaussian curvature (FGauss) terms:


F ) Fel + FAB + Fbend + FGauss

Langmuir, Vol. 24, No. 4, 2008 1289

(6)

The last three terms are equivalent to the Helfrich Hamiltonian41 frequently used to describe the free energy of a fluid membrane:

H)

(

)

κ

∫ d2A γ + 2b(c1 + c2)2 + κjc1c2

(7)

where the integration is carried out over the membrane surface, c1 and c2 are the local principal curvatures of the membrane, γ denotes the lateral membrane tension, and κb and κj characterize the bending rigidity and saddle-splay modulus, respectively. In eq 6, the Gaussian curvature term appears in the free energy because it refers to one vertical monolayer sheet, i.e., an “open” surface. Therefore, the Gauss-Bonnet theorem cannot be invoked to a priori disregard FGauss, as is common practice when applying Helfrich-type Hamiltonians to biological membranes.59 For a lamella of period L, the elastic free energy per chain, within SST, is given by

Fel ) RL

2

(8)

where R is a constant that, as was shown by Semenov,60 depends on the details of the distribution of the diblock ends in the lamellae. When the end-to-end distances, DA and DB, of the A and B blocks are different, eq 8 becomes48

Fel ) 8R(DA2 + DB2)

(9)

The interfacial energy per chain in the lamella is61

FAB )

2γAB FL

(10)

where γAB is the surface tension between the A and B blocks, while F is the polymer chain number density. Within SST, eqs 8 and 10 are combined, expressing the free energy of the bulk lamellar phase. Minimization yields the bulk lamellar spacing Lo as a function of γAB and R. This expression can be inverted, giving

R)

γAB FLo3

(11)

This expression will be used in the following to derive a dimensionless free energy. The bending and the Gaussian curvature free energies per chain are given by42,44

Fbend )

3 4 RL (c1 + c2)2 64

FGauss ) -

1 4 RL c1c2 40

(12)

where c1and c2 are the local principal curvatures of the monolayer. The expressions for the various components of the free energy per chain can be utilized to describe the free energy of a modulated copolymer interface, after the symmetries of the considered system are taken into account. The two cases of undulation and peristaltic roughness are sketched in Figure 1. In both cases, all monolayers are equivalent because of symmetry; thus it is sufficient to consider one single AB interface. Assuming that the deformations are (59) Mu¨ller, M.; Katsov, K.; Schick, M. Phys. Rep. 2006, 434, 114. (60) Semenov, A. N. SoV. Phys. JETP 1985, 61, 773. [Zh. Eksp. Teor. Fiz. 1985, 88, 1242]. (61) Ohta, T.; Kawasaki, K. Macromolecules 1986, 19, 2621.

rather small, interface overhangs along the axis normal to the lamellae (i.e., the z-axis in Figure 1) can be neglected, and the deviation of the AB interface from the flat, non-deformed, interface of the no-LER case, can be described via a function z ) h(x,y). In this Monge representation, the x-axis is chosen normal to the patterned substrate, and the y-axis is parallel to the symmetry plane of the lamellae (see Figure 1). From eqs 10, 11, and 12, using the standard expressions for mean (c1 + c2)/2 and Gaussian curvatures c1c2 within the Monge representation,59 the FAB, Fbend, and FGauss parts of the free energy for one monolayer are written as

FAB ) γAB Fbend )

∫0H dx ∫-∞∞ dyx1 + hx2 + hy2

3γABL5 3

128Lo

FGauss ) -

γABL5 3

80Lo

∫0H dx ∫-∞∞ dy [hxx + hyy]2 ∫0H dx ∫-∞∞ dy [hxxhyy - hxy2]

(13)

where H denotes the film thickness. Total incompressibility is assumed such that the number of chains of one monolayer nmono, per dxdy area, is nmono ) FLdxdy/2. These equations are a special case of the Helfrich Hamiltonian applied to a diblock copolymer monolayer, and it is instructive to compare them to eq 7. It can be seen that, in the case of the diblock monolayer, one gets for the bending rigidity and the saddle-splay modulus κb ) 3γABL5/ (64Lo3) and κj ) - γABL5/(80Lo3), respectively. The Fel term is specific to undulation and peristaltic roughness. To derive it, it is crucial to assume that the copolymer chains (shown as wavy lines in Figure 1) follow the local interface normal on average. The unit vector along the local normal is62 n ) [ - ∂xh, - ∂yh,1]/x1+(∇h)2 thus, if θ is the angle between the local normal n and the z-axis (BAC angle in Figure 1), then,

cos(θ) )

1

x1 + (∇h)

1 ≈ 1 - (hx2 + hy2) 2 2

(14)

In the case of undulations, the end-to-end distance of each A or B block is (L/4) cos(θ) ) (L/4)(1 - (1/2)(∇h)2) as sketched by the AC and AC′ segments on the left panel of Figure 1. (Note that AD ) L/2, where L is the lamellar thickness.) This expression invokes the additional assumption that the normal n is a common normal to the opposite copolymer interfaces (AE and AE′ segments in Figure 1) such that the chains that are parallel to AE and AE′ reach the midpoints, C and C′. Substituting this expression into eq 9, using eq 11, and assuming that nmono ) FLdxdy/2, one gets for the elastic energy of an interface deformed by undulation LER of the substrate pattern

Fel )

γABL3 2Lo

3

∫0H dx ∫-∞∞ dy [1 - 21(hx2 + hy2)]

2

(15)

which is equivalent to the expression of ref 47. To derive Fel for peristaltic LER, we assume that two A or B blocks emanating from points of two opposite copolymer interfaces (cf. points A and A′ in Figure 1) follow the local normals, n and n′, and meet at the vertical midplane of the A- or the B-rich slab. The distance spanned by the A and B blocks emanating from a point A will be [L/4 + h(x,y)]/cos(θ) and [L/4 - h(x,y)]/ (62) Kamien, R. D. ReV. Mod. Phys. 2002, 74, 953.

1290 Langmuir, Vol. 24, No. 4, 2008

Daoulas et al.

cos(θ), respectively. This is illustrated by the AC and AC′ segments in the right panel of Figure 1. Combining eqs 14, 9, and 11 and using nmono ) FLdxdy/2, we obtain the elastic energy for peristaltic substrate LER: Fel )

γABL3 2Lo3

∫

H

0

dx

∫

[

∞

dy 1 + -∞

][

16h2 L2

]

1 1 + (hx2 + hy2) 2

2

(16)

At this point, all components of the phenomenological free energy of the modulated interface, eq 6, for the cases of peristaltic and undulational LER, are available, and it is possible to proceed to quantify the roughness effect on assembled polymer structures. B. Estimating the Modulation of Copolymer Interfaces Induced by Substrate LER. The deformation of the AB domain interface due to the LER of the surface pattern was characterized following a procedure similar to the one of ref 47. It is assumed that no other wavelengths are present in the interface modulation apart from the one that corresponds to the substrate LER. Then the ansatz23,44,47,63,64

h(x,y) ) f(x) cos(ωoy)

(17)

is made, where f(x) is an unknown function describing the decay of amplitude of modulation in the film. It will be determined from the free energy, eq 6. In particular, all the terms in eqs 15, 16, and 13 are expanded up to second-order in h(x,y) and its derivatives. The ansatz for h(x,y) is substituted into the equations, and an integration over one LER wavelength, LLER, is performed in the y direction. Expressing all lengths in units of LLER, we introduce the dimensionless quantities: x˜ ) x/LLER, L˜ ) L/LLER, L˜ o ) Lo/LLER, ω ˜ o ) ωoLLER ) 2π, ˜f ) f/LLER, and γ˜ AB ) γABLLER2 and obtain the following expressions for the contributions to the free energy:

FAB )

Fbend )

γ˜ ABπ 2ω õ

3γ˜ ABπL˜ 5 128L˜ o ω õ 3

FGauss )

∫0H˜ dx˜ [f˜ ′2 + ˜f 2ω˜ o2 + 4]

∫0H˜ dx˜ [f˜ ′′2 - 2f˜ ′′f˜ω˜ o2 + ˜f 2ω˜ o4]

γ˜ ABπL˜ 5ω õ

(18)

∫0H˜ dx˜ [2 - ˜f 2ω˜ o2 - ˜f ′2]

(19)

80L˜ o

and

Fuel )

Fpel )

γ˜ ABπL˜ 2L˜ o ω õ 3

γ˜ ABπL˜ 3 2L˜ o ω õ 3

∫0H˜ dx˜ [f˜ ′2L˜ 2 + 16f˜2 + ω˜ o2˜f 2L˜ 2 + 2L˜ 2]

[

]

3 3 d4˜f 64 (L˜ - L˜ o ) d2˜f + - 2ω ˜ o2 4 5 3 dx˜ L˜ dx˜ 2 3 3 128 (L˜ - L˜ o ) -ω ˜ o2 ω ˜ o2˜f ) 0 (21) 3 L˜ 5

[

for peristaltic, and

[

]

]

3 3 64 (L˜ + L˜ o ) d2˜f d4˜f + 2ω ˜ o2 + 4 5 3 dx˜ L˜ dx˜ 2 3 3 ˜ o2 1024 1 64 (L˜ + L˜ o )ω + +ω ˜ o4 ˜f ) 0 (22) 3 3 L˜ 4 L˜ 5

[

]

for undulational LER. In the most general case, the solutions of the above homogeneous, linear, fourth-order equations will be linear combinations of the terms x˜ m exp(ν1x˜ ) cos(ν2x˜ ) and x˜ m exp(ν1x˜ ) sin(ν2x˜ ). ν ) ν1 + iν2 is the solution of the characteristic equation of eq 21 or 22. If k is its multiplicity, then m ) 0,...,k - 1. The coefficients of the linear combination are determined from the boundary conditions.

IV. Results

∫0H˜ dx˜ [f˜ ′′f˜ + ˜f ′2]

3

to B-preferential stripes are equal, it can be assumed that the contact angle is 90° such that ˜f′(0) ) 0. When discussing the SCMF simulations, it will be demonstrated that this boundary condition can have a qualitative influence on the predictions regarding the shape of the modulated interface giving rise to a rather non-trivial behavior of ˜f(x˜ ). Assuming that the top film surface has no preference for the A or B component, we set ˜f ′(H ˜) ) 0. The above three boundary conditions constitute the fixed48 boundary conditions for the problem. The fourth, natural, boundary condition, ˜f ′′′(H ˜ ) ) 0, sets the boundary term ˜f ′′′δf˜ |H0˜ , appearing during the variation, to zero.47,48 Interestingly, with the above boundary conditions, the variation of the Gaussian curvature term in eq 18 is zero; i.e., it has no effect on interface deformation behavior. The condition δF/δf˜ ) 0 yields for ˜f(x) a differential equation of the form

(20)

where Fuel and Fpel are the elastic energy for undulation and peristaltic roughness, respectively. We utilize these expressions in the free-energy functional, eq 6, and vary it with respect to the unknown function ˜f(x). The boundary conditions for this variation are obtained as follows: First, it should be ˜f(0) ) Ao/ LLER ) A ˜ o because the interface is pinned to the rough pattern boundary. Then, if the strengths of attraction of A beads to the A-preferential stripes and the strength of attraction of the B beads (63) de Gennes, P. G. The Physics of Liquid Crystals; Oxford University Press/ Clarendon Press: NewYork/Oxford, 1974. (64) Delrieu, J. M. J. Chem. Phys. 1974, 60, 1081.

A. Predictions of the Phenomenological Approach. Equations 21 and 22 show that a mismatch between the period of the stripe pattern, L˜ , and the bulk lamellar spacing, L˜ o, affects the propagation of the substrate LER into the assembled morphologies. In fact, a sufficiently large mismatch between L˜ o and L˜ will give rise to tilting or buckling of the assembled lamellae, even in the absence of substrate LER since it acts as a mechanism of stress relief. Such buckling in stressed, lamella-forming, block copolymers has been discussed theoretically for bulk systems44 and thin films,47,48 and it has been observed experimentally.65 If one increases the mismatch between pattern period and bulk lamellar spacing further, the self-assembled morphology no longer registers with the substrate pattern throughout the film thickness, and defects form.3 In order to achieve perfect registration, no mismatch larger than 15% is tolerable.3 Thus, we restrict our study to the case where the lamellar thickness equals the bulk lamellar spacing, L ) Lo. In this case, both eqs 21 and 22 greatly simplify. For undulation LER, the solution of eq 21 takes the form ˜f ) A1 exp(ω ˜ ox˜ ) + A2x˜ exp(ω ˜ ox˜ ) + A3 exp(-ω ˜ ox˜ ) + A4x˜ exp(-ω ˜ ox˜ ). The determination of the coefficients from the boundary conditions is straightforward, and, for the case of a thick film, H ˜ f∞, the solution takes a particularly compact form: (65) Kim, S. O.; Kim, B. H.; Kim, K.; Koo, C. M.; Stoykovich, M. P.; Nealey, P. F.; Solak, H. H. Macromolecules 2006, 39, 5466.


f(x) ) Ao(1 + ωox) exp(-ωox)

Langmuir, Vol. 24, No. 4, 2008 1291

(23)

where normal length units have been recovered. This equation demonstrates that, in a thick film, the amplitude of modulation of AB interfaces generated by the substrate LER decays exponentially with distance from the substrate. Most interestingly, the length scale of this decay in copolymer systems is determined only by the characteristic wavelength of the substrate LER, similarly to the case of an interface between two simple liquids preferentially wetting two different regions of a patterned substrate separated by a rough boundary.23 This is a rather remarkable result, taking into account the more complex nature of copolymer interfaces and their coupling. Other characteristics, like the surface tension γAB of the A and B blocks, are irrelevant to the propagation of undulation LER. For the case of a thin film, apart from the LER wavelength, the film thickness H also has a significant effect on the roughness decay. To demonstrate this, the main panel of Figure 2 shows the decay of f(x) obtained for Lo ) 1.72Re and LLER ) 2Lo from eq 21 at different values of film thickness. All lengths are scaled by LLER. Comparing the results for the thicknesses H ˜ ) 0.415 (H ) 1.43Re), H ˜ ) 0.831 (H ) 2.86Re), and H ˜ f∞, it can be clearly seen that reducing H amplifies the roughness effect: at the same distance from the substrate, the modulation of AB copolymer interfaces is more prominent in the thin films for patterns with the same LLER. For the peristaltic roughness, the solution, even in the L ) Lo case, is more complicated. In particular, it has the form ˜f ) A1 exp(t1ω ˜ ox˜ ) + A2 exp(-t1ω ˜ ox˜ ) + A3 exp(t2ω ˜ ox˜ ) + A4 exp(-t2ω ˜ ox˜ ), where t1 ) x32/(3ω ˜ o2L˜ 2)+1 and t2 ) x32/(ω ˜ o2L˜ 2)+1. For an infinitely thick film, H ˜ f∞, the solution (in normal length units) is

f(x) )

Ao [t exp(-t2ωox) - t2 exp(-t1ωox)] (24) t1 - t 2 1

Contrary to the case of undulation roughness, the decay of the substrate pattern roughness into the morphology is dictated by two parameters: LLER and ωoL. The latter parameter, ωoL, quantifies the difference between the LER wavelength and the natural lamellar thickness, Lo. The inset of Figure 2 presents the profile f(x) for Hf∞, at a fixed LER wavelength, and various values of bulk lamellar spacing, Lo (in units of length scaled by LLER). It can be seen that the distances from the substrate up to which the AB interface remains modulated become larger as Lo becomes smaller than LLER (as L˜ o decreases). Taking into account the tendency to miniaturize the features of the assembled copolymer morphologies by selecting systems with smaller Lo, this result indicates a possible limitation: copolymer morphologies with small Lo might be effected more by the substrate LER. Apart from the ωoL parameter, eq 24 demonstrates, as in the case of undulation roughness, that the length scale of the exponential decay of the modulations of the interface is set by the characteristic wavelength, LLER. In peristaltic roughness, just like in the undulation case, reducing H amplifies the roughness effect. This can also be seen in the main panel of Figure 2, where the f(x) profiles for LLER ) 2Lo obtained from eq 22 at H ˜ ) 0.415 (H ) 1.43Re), H ˜ ) 0.831 (H ) 2.86Re), and H ˜ f∞ are shown (circles and solid line). B. Predictions of SCMF Simulations. Within SCMF simulations, the effect of undulation and peristaltic substrate LER was studied considering patterns with period L equal to the natural lamellar spacing, Lo. In both diblock melts and ternary blends, two LER wavelengths were modeled: LLER ) Lo and LLER ) 2Lo. A value of Ao ) 0.2Re (see eq 4) was chosen for the LER

Figure 2. The main panel shows the predictions of the phenomenological theory for the decay of the amplitude, f(x), of the AB interface deformation in a diblock melt, as a function of distance, x, from the substrate. For all graphs, LLER ) 2Lo, with Lo ) 1.72Re, and the lengths are scaled with the LER wavelength, LLER; i.e., x˜ ) x/LLER. For two film thicknesses, H ˜ ) 0.415 (H ) 1.43Re) and H ˜ ) 0.831 (H ) 2.86Re), both cases of undulation and peristaltic pattern LER (period LLER ) 2Lo) are shown with squares and circles, respectively. The dotted and solid lines denote the corresponding limiting cases for an infinitely thick film, given by eq 23 and eq 24, respectively. The inset shows the decay of the amplitude of the interface deformation (derived from eq 24) for copolymer melts with different bulk lamellar spacings, Lo, assembled on a pattern with peristaltic LER of fixed wavelength.

amplitude. In both systems, two film thicknesses, H ) 1.43Re and H ) 2.86Re, were considered. The left and the right bottom panels of Figure 3 show an averaged morphology of the copolymer melt in an H ) 2.86Re thick film assembled on a pattern with undulation and peristaltic roughness (left and right image, respectively) and period LLER ) 2Lo. For clarity, only the A-rich phase is shown in yellow, while the blue wire mesh marks the AB interface. To present the morphology close to the substrate, a cut-through is shown for one A-rich slab. It can be seen that, near the substrate, the AB interface is modulated by being pinned to the wavy, “rough” stripe boundary. Farther away from the substrate, the amplitude of the deformation continuously decreases toward the free surface of the film. Unlike the phenomenological approach, SCMF simulations incorporate thermal fluctuations. These capillary waves are observable in the instantaneous configuration snapshots presented in the two top panels of Figure 3. The wire mesh marks the AB interface, which, in the instantaneous snapshots, is rough. To quantitatively compare the simulation results to the prediction of the phenomenological approach, we have averaged the fluctuating morphologies in the simulation over a long run. These aVeraged morphologies are depicted in the two lower panels of Figure 3. This procedure averages out the thermal fluctuations, and the average shape of the AB interface is much smoother. To quantify the behavior of this average shape of the AB interface, its position is calculated along the stripe edge at various distances from the substrate. This is achieved66 by dividing the system in the plane vertical to the substrate (i.e., the xy plane) into parallelepipeds with a base of size ∆L, i.e., commensurate with the grid used in the SCMF simulations. This procedure is sketched in the top panels of Figure 3. The z coordinate of the (66) Werner, A.; Schmid, F.; Mu¨ller, M.; Binder, K. Phys. ReV. E 1999, 59, 728.

1292 Langmuir, Vol. 24, No. 4, 2008

Daoulas et al.

Figure 3. The left and right panels depict two instantaneous three-dimensional snapshots of the copolymer morphology assembled on stripes with undulation (left) and peristaltic (right) LER, obtained from SCMF simulations. In both cases, LLER ) 2Lo. For clarity, only the A-rich phase is shown in yellow, while the blue mesh marks the AB interface. To illustrate the shape of the lamella near the rough substrate, a cut-through is shown for one A-rich slab. The sketched parallelepiped explains the construction of parallelepipeds utilized for calculating the AB interface in SCMF simulations as a Gibbs dividing surface. The left and right bottom panels present the averaged structures of the same systems obtained by averaging over thermal fluctuations. These averaged morphologies are used to calculate the averaged shape of the interface and to compare with the phenomenological approach.

Figure 4. The average shape of the copolymer interface, h(x,y), at various distances, x, from the substrate is shown on the left (undulation LER) and right (peristaltic LER) panels, for a film with thickness H ) 1.43Re. The pattern LER wavelength in both cases is LLER ) 2Lo, and the rough edge of the stripe is presented on both graphs for reference (solid line). The line-shaded area, to the left of the pattern edge, is B-preferential.

center of the parallelepipeds, zo, corresponds to the position of the stripe edge in the straight pattern in the absence of LER, while its length is d ) Lo/2. Then the number nA of A beads and the number nB of B beads in the parallelepipeds are counted, and, for the considered morphology, the position h of the interface with respect to zo - d/2 is defined via an integral criterion: h ) nAd/(nA + nB). The average shapes of h at various distances from the substrate are shown in the left (undulation LER) and right (peristaltic LER) panels of Figure 4, together with the rough edge of the substrate pattern stripe, which is presented for reference. The data refers to a diblock copolymer melt in a film of thickness H ) 1.43Re assembled on a pattern with undulation

and peristaltic roughness (left and right image, respectively) with period LLER ) 2Lo. The line-shaded area, left of the pattern edge, denotes the B-preferential area of the substrate. The shape of the edge is due to the mapping of the smooth potential of eq 4 onto the grid used for defining the interactions in the SCMF simulations. Apart from the basic, LER inverse wavelength ωo, higher harmonics (3ωo, 5ωo,...) are contained in the pattern. These higher wavenumbers, however, have a small amplitude, and their decay length that characterizes their effect on the morphology as a function of the distance from the substrate decreases with the value of the inverse wavelength (see below). Thus, it is not surprising that, from the very first layers in Figure 4, the interface


Langmuir, Vol. 24, No. 4, 2008 1293

Figure 5. The amplitude of AB interface deformation, f(x), in a diblock copolymer melt at various distances, x, from the substrate is shown in the left and right panels. The film thickness is H ) 1.43Re and H ) 2.86Re, respectively. Two cases of pattern LER, undulation (squares) and peristaltic (circles), are considered for two wavelengths LLER ) Lo (open symbols) and LLER ) 2Lo (solid symbols). The predictions of the phenomenological theory for undulation roughness, eq 21, are also shown as solid (LLER ) Lo) and dashed (LLER ) 2Lo) lines. The inset demonstrates the importance of boundary effects. At H ) 2.86Re, for the case of peristaltic LER (LLER ) 2Lo), f(x) is calculated from eq 22, setting f'(0) ) 0 (open circles) and f'(0) ) b, with b < 0 (solid circles). With the latter boundary conditions, the phenomenological theory qualitatively captures the behavior of SCMF simulations (cf. main panel).

is well described by the single harmonic variation with an inverse wavelength ωo. An interesting observation related to Figure 4 is the rather non-trivial behavior of the interfacial deformation for peristaltic substrate LER. In the case of undulation LER, the oscillations of the AB interface at all levels are in phase with the stripe edge; i.e., edge protrusions into the A-preferential area always correspond to AB interface protrusions into the A-rich phase and vice versa. This is not the case for peristaltic roughness. Starting approximately at a distance of x ) 0.7Re from the substrate, the variations of h(x,y) are 180° out-of-phase with the underlying pattern (see the two curves shown in red and green). To quantify the interface modulation and to compare it with the phenomenological theory, the interface position h(x,y) obtained by SCMF simulations of pure diblock melts was fitted to the ansatz eq 17 at each height level, x. The fits are of excellent quality, and the fitted values for ωo always match well the characteristic inverse wavelength of the substrate LER. The f(x) extracted from these fits is shown in the left and right main panels of Figure 5 for undulation and peristaltic LER and for two wavelengths LLER ) Lo and LLER ) 2Lo. The film thicknesses H ) 1.43Re and H ) 2.86Re were utilized, respectively. The SCMF simulations confirm the phenomenological predictions that the modulation of the AB interface in systems assembled on rough patterns with smaller LLER decays faster with the distance from the substrate when compared to larger LLER. For undulational LER, the f(x) profiles predicted by eq 21 are shown for LLER ) Lo and LLER ) 2Lo in the same graph (dashed and solid lines, respectively). It can be seen that, for undulational roughness and for both film thicknesses, the phenomenological theory captures the qualitative behavior of the decay of the modulation of the copolymer interface. In the case of the thin film, one even observes quantitative agreement between SCMF simulations and the simple theory. The quality of the phenomenological approach, however, should not be overestimated. Its limitations become apparent for the case of peristaltic LER. In the SCMF simulations, the f(x) profile becomes negative; i.e., as discussed in Figure 4, the

modulation at the substrate pattern and farther away from the substrate are 180° out-of-phase. On the other hand, the calculations of the phenomenological theory, when performed with boundary conditions ˜f(0) ) Ao/LLER ) A ˜ o, ˜f ′(0) ) 0, ˜f ′(H ˜ ) ) 0, and ˜f ′′′(H ˜) ) 0, yield a monotonically decaying positive profile, ˜f(x˜ ). For the case of H ) 2.86Re and LLER ) 2Lo, the f(x) profile is presented with open circles in the inset of Figure 5. Although the simple phenomenological arguments provide an insight into the influence of substrate LER on the local shape of morphologies in assembled copolymer systems, they suffer from significant simplifications. An important point is that the phenomenological theory does not incorporate the constraints imposed on the chain configurations by the film boundaries.48 Therefore, the shapes of the AB interface for which the emanating copolymer blocks intersect the solid substrate are also allowed. In the case of peristaltic roughness, for example, the blocks are allowed to reach the midplane of the A or B-rich slab in the unphysical region “below” the substrate. The importance of boundary effects is illustrated in the inset of Figure 5, where, by setting ad hoc f ′(0) to negative values, we are able to reproduce qualitatively the “phase flip” at large x (see the curve shown in solid circles) observed in SCMF simulations. This qualitative agreement also holds for the thinner film, which is omitted in the graph for clarity. It should be pointed out,48 however, that setting f ′(0) < 0 in combination with the basic assumption that the polymer chains, on the average, emanate normal to the AB interface leads to the violation of the substrate impenetrability at x˜ ) 0. Therefore, the use of this condition is more a mathematical construction for illustrating that the phenomena near the patterned substrate could have a quite nonlocal effect on the shape of the assembled structures. Figure 6 demonstrates that the finite system size does not affect the results of f(x). The graph compares the SCMF results for film thickness, H ) 1.43Re, and all of the types of substrate LER considered in the current work as obtained from simulation cells with lateral dimensions Ly × Lz ) 5.16Re × 10.32Re and Ly × Lz ) 10.32Re × 20.64Re. The graph clearly shows that the data derived with the two different system sizes are identical within the statistical error. This absence of system

1294 Langmuir, Vol. 24, No. 4, 2008

Figure 6. Demonstration of the absence of finite size effects in the SCMF simulations. The graph compares the decay, f(x), of the interface deformation obtained from simulations with lateral dimensions Ly × Lz ) 5.16Re × 10.32Re and Ly × Lz ) 10.32Re × 20.64Re and film thickness, H ) 1.43Re. Undulation and peristaltic substrate LER with two wavelengths are considered.

size effects is expected because the system geometry is adopted to fluctuations with the wavelength of the substrate LER and all higher harmonics.

Daoulas et al.

Finally, the SCMF simulations confirm the prediction of the phenomenological approach that reducing film thickness enhances the roughness effect. The comparison of f(x) profiles in Figure 5 for undulation and peristaltic substrate pattern LER and different film thicknesses (H ) 1.46Re and H ) 2.86Re) shows that the deformation decays faster with x in the thicker films for both types of roughness. The parameters of the Helfrich Hamiltonian of the phenomenological theory are specific to the case of the symmetric, pure diblock, copolymer melt assembled on stripes with symmetric interactions. The values of these parameters will be different in more complicated copolymer systems,45 and it is expected that a change in the elastic properties of the AB interface will affect the substrate LER propagation into the assembled polymer. However, the SCMF simulations show that the polymer morphologies in ternary blends and pure diblock melts are affected by substrate LER in a similar way. In some cases, this similarity is even quantitative. To demonstrate this, each of the four panels of Figure 7 compares the f(x) profiles derived from SCMF simulations of the blend and the diblock systems at a certain film thickness (H ) 1.46Re or H ) 2.86Re) and a certain type of roughness (undulation or peristaltic). Each panel presents four f(x) profiles. Two of them correspond to pure diblock systems assembled on patterns with LLER ) Lo and LLER ) 2Lo (Lo ) 1.72Re), while the others correspond to the ternary blend assembled on patterns with LLER ) Lo and LLER ) 2Lo (Lo )

Figure 7. The four panels present the amplitude of AB interface deformation, f(x), for pure diblock melts (filled symbols) and ternary copolymer/homopolymer blends (open symbols). The systems that are presented on the same panel have the same film thickness, in normal length units, H (H ) 1.43Re or H ) 2.86Re) and the same type of roughness (undulation or peristaltic). All profiles are scaled with the corresponding LER wavelength, x˜ ) x/LLER.


2.27Re). The distance from the substrate, x, is scaled with corresponding roughness wavelength LLER: x˜ ) x/LLER. This rescaling of x is motivated by the phenomenological theory, particularly by eqs 23 and 24. Specifically, for the case of undulation roughness, the only parameter determining the deformation decay is the product ωox ) 2πx˜ in the limit Hf∞. Thus the f(x)/Ao profiles in rescaled x˜ units are expected to collapse onto a master curve, irrespective of LLER. For peristaltic roughness, the deformation decay is determined by two parameters, ωox and ωoLo, in the limit Hf∞. In this case, the f(x)/Ao profiles as a function of x˜ are also expected to follow a master curve for systems with same ωoLo value (i.e., the same Lo/LLER ratio) but different LLER. The phenomenological theory proposes a scaling for the decay of AB interface deformation in systems with different Lo and/or LLER. It can be observed in Figure 7 that the f(x) profiles for the symmetric diblock and the ternary blend systems resemble each other in this rescaled representation. The two left panels show data for undulation roughness and film thicknesses, H ) 1.46Re and H ) 2.86Re, (upper and lower left panels, respectively). For the case of undulation roughness (at a given H), the f(x) profiles exhibit a common envelope with some deviations at the outer surface. Interestingly, for peristaltic roughness and film thicknesses H ) 1.46Re and H ) 2.86Re (upper and lower right panels, respectively), the diblock melts and blends have the tendency to separate into two envelopes, depending on the ratio Lo/LLER. Part of the deviation of the simulation results from the simple scaling suggested by the phenomenological theory can be attributed to the relatively small thickness of the films and to the fact that, in the rescaled LLER units, all polymer films shown within the same panel have a different rescaled thickness, H ˜ . For example, the deviations in the scaled profiles on the top left panel of Figure 7 resemble the differences between the scaled profiles that have been calculated theoretically for undulation roughness for H ˜ ) 0.415, H ˜ ) 0.831, and H ˜ f∞ (cf. Figure 2, main panel, square symbols and dotted line). Overall, Figure 7 suggests that, within the range of compositions investigated, the blending of copolymers with the homopolymers does not change the elastic properties of AB interfaces to an extent that significantly alters the substrate LER propagation into the film.

V. Conclusions A combined phenomenological and simulation study of directed assembly of lamella-forming block copolymer systems on substrates patterned with stripes has been performed. Two prototypical model roughness schemessundulation and peristaltics were imposed on the edges of the stripes patterned on the substrate, and their effect on the local morphology of the assembled copolymer morphologies was investigated. This analysis was done through a phenomenological theory considering the elastic properties of deformed copolymer interfaces within a Helfrichtype41 Hamiltonian, and a much more detailed particle-based model investigated by SCMF simulations.51,52

Langmuir, Vol. 24, No. 4, 2008 1295

Both the simple theory and the SCMF simulations highlight the significant role of the characteristic pattern roughness wavelength in determining the local shape of the assembled copolymer structures. In this work, the LER of the substrate pattern (substrate LER) is characterized by a single characteristic wavelength, LLER, and, in all cases, the copolymer interface is strictly pinned to the rough stripe edge at the substrate. Both the simple phenomenological approach and the SCMF simulations predict that the smaller the LLER, the faster the deformation of the copolymer interface decays with the distance from the substrate. For peristaltic roughness, the phenomenological theory predicts that the propagation of roughness into the polymer morphology depends on both the wavelength LLER of the substrate pattern roughness and the ratio between LLER and the bulk lamellar spacing, Lo. In particular, it shows that the distances from the substrate up to which the copolymer interface remains deformed decrease as LLER becomes smaller than Lo. This suggests that the assembled copolymer morphologies in systems characterized by small Lo might be more affected by the substrate LER. An additional important observation supported by both the phenomenological theory and the SCMF simulations is that the influence of the LER of the substrate pattern on the assembled copolymer structures depends on film thickness. Reducing film thickness amplifies the roughness effect: when comparing patterns with the same LER characteristics, the thinner the film, the slower the copolymer interface deformation decays with distance from the substrate. A basic limitation of the simple theory is that it does not incorporate the restriction that the substrate imposes on the chain conformations.48 For instance, it cannot reproduce certain qualitative aspects of the behavior of copolymer interface deformation without additional, ad hoc assumptions regarding the structure of the copolymer morphology at the boundary. SCMF simulations do not invoke these approximations and can provide a quantitative description of the complex interplay between polymer-substrate interactions, confinement, and reconstruction of the soft morphology. They also provide an efficient way of considering the substrate LER effects in the case of more complex polymer systems, where the formulation of a phenomenological theory is formidable. In this context, ternary AB copolymer blends with their respective A and B homopolymers have been simulated and compared to pure diblock melts. Interestingly, the SCMF simulations demonstrate that the blending of copolymer with the homopolymers does not have a significant effect on the properties of substrate LER propagation into the assembled film for the blend composition considered in this work. Acknowledgment. It is a great pleasure to thank M. Deserno for helpful discussions. Financial support was provided by the Volkswagen foundation and the DFG under grant Mu 1764/4. Computer resources were provided by the NIC Ju¨lich and the HLRN Hannover. LA702482Z

Directed Copolymer Assembly on Chemical Substrate Patterns: A

Recommend Documents