New and Exotic Self-Organized Patterns for Modulated Nanoscale

In this work we have generated a variety of new and exotic patterns, which represent either metastable or glassy states. These patterns arise as a com...
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NANO LETTERS

New and Exotic Self-Organized Patterns for Modulated Nanoscale Systems

2005 Vol. 5, No. 2 389-395

Celeste Sagui,* Eliana Asciutto, and Christopher Roland Center for High Performance Simulation and Department of Physics, The North Carolina State UniVersity, Raleigh, North Carolina 27695-8202 Received October 27, 2004; Revised Manuscript Received December 16, 2004

ABSTRACT The self-assembled domain patterns of modulated systems are the result of competing short-range attractive and long-range repulsive interactions found in diverse physical and chemical systems. From an application point of view, there is considerable interest in these domain patterns, as they form templates suitable for the fabrication of nanostructures. In this work we have generated a variety of new and exotic patterns, which represent either metastable or glassy states. These patterns arise as a compromise between the required equilibrium modulation period and the strain resulting from topologically constrained trajectories in phase space that effectively preclude the equilibrium configuration.

Introduction. A large variety of quasi-two-dimensional physical, chemical, and even biological systems are characterized by a high degree of universality, all displaying the same kinds of structural motifs and dynamical mechanisms, albeit on very different length and time scales.1,2 This is irrespective of the physical origins of the underlying microscopic interactions, which may indeed be very different. Universal features are particularly striking in modulated systems, which are characterized by short-range attractive and real or effective long-range repulsive interactions (LRRI). Here, the interactions conspire to produce patterns based on lamellar “stripe” and “bubble” motifs. Prototypical examples of modulated systems include such diverse examples as magnetic garnet films,1-11 Langmuir monolayers,12-17 blockcopolymer systems,18,19 type I superconductors,20 steady-state reaction-diffusion (Turing) patterns,21 ferrofluids,22 SwiftHohenberg fluid systems,23-25 liquid-crystal systems,26,27 surface science,28,29 and the primate visual cortex.2 Exploring the genesis of different configurations in these modulated systems is a problem of fundamental importance, which recently has been given new urgency with the advent of nanotechnologies for molecular electronic, biomedical, and photonic applications. Modulated systems have been used to produce nanolithographic templates for self-assembly applications with unprecedented characteristics,30,31 relying on the spectacular long-range ordering and the selective placements of defects achievable in these systems. In particular, soft-condensed matter systems such as block copolymers and related surface systems have proven to be particularly versatile, because of the tunability of the size, * To whom correspondence should be addressed. E-mail: sagui@ unity.ncsu.edu. 10.1021/nl048224t CCC: $30.25 Published on Web 01/22/2005

© 2005 American Chemical Society

shape, and periodicity of the resulting patterns. Current patterns are, for the most part, based on the self-assembly of stripes (lamellae) and bubbles. Here, we present results based on a successful phase field model,1,32-37 that reveal a much larger set of unexplored patterns, so that the types of templates that can be produced for applications is actually more varied than what has been considered to date. These new and exotic patterns are formed by successively taking the system through different trajectories inside the phase diagram. The trajectories are chosen such that the topological constraints in the system create strained patterns that need not evolve to the global free energy minimum. The topological constraints can arise from a variety of physical situations: high energy barriers for the nucleation of stripes, the bending stiffness of the stripes, packing constraints in the initial highly geometrically ordered configurations, etc. Model and Simulations. It is convenient to discuss the modulated patterns in the language of two-dimensional, uniaxial ferromagnetic thin films.1,3-11 The standard model for this system gives a description of the order parameter φ(r,τ) at position r as a function of time τ. The phenomenological free energy functional F (suitably adimensionalized) is expressed in terms of a Ginzburg-Landau expansion based on φ(r) and its gradient. It consists of both a local and a nonlocal term: F[φ(r)] )

∫ d2r[21(∇φ)2 + f(φ) - Hφ] + R ∫ ∫ d2r d2r′φ(r′) g(|r - r′|)φ(r) 2

(1)

In the first term, the gradient-squared term represents the lowest-order approximation to the cost of creating a domain

The time evolution of the system is obtained from the corresponding Langevin equation ∂φ(r,τ) δF [φ] )+ xµζ(r,τ) ∂τ δφ Figure 1. Sketch of a phase diagram for a ferrimagnetic thin film. The phase diagram is symmetric with respect to the magnetic field H. First-order transition lines separate stripe, bubble, and homogeneous phases. The bubble phase is a low-density triangular lattice. Typical profiles of the order parameter are also sketched.

wall or interface; f (φ) is the local free energy; and -Hφ is the coupling between the external magnetic field H (oriented perpendicular to the film) and the order parameter. The local free energy has the standard temperature dependence associated with phase transitions:1,32-37 for temperatures T greater than the critical temperature Tc, the local free energy has a single-well structure that represents the uniform phase, for T < Tc (the case of our simulations), f (φ) ) -(1/2)φ 2 + (1/4)φ ,4 so that the minima of the resulting double-well structure identify each of two coexisting phases. These phases are represented by either positive or negative values of φ(r): φ(r) > 0 corresponds to regions where the spins point in the “up” direction, while φ(r) < 0 corresponds to spins pointing in the “down” direction. H > 0 favors “up” spins, which in our graphics are represented by the white phase. The double integral represents the LRRI, with the long-range repulsive kernel given by g(|r - r′|) ) |r - r′|-1 - [|r r′|2 + L2]-1/2; L is the film thickness. In the limit of very thin films (L f 0), the kernel g(|r - r′|) f L2/(2|r - r′|3) becomes a purely repulsive, dipolar interaction. The relative strength of the LRRI is given by the temperature-dependent parameter R.32,34 The phase diagram as a function of R and H is sketched in Figure 1. Here, we use R and T, somewhat loosely, as being interchangeable because they play similar roles in regulating the characteristic length scale of the modulated phases (R depends on T but in a nontrivial, system-dependent way3,4,8). The phase diagram is symmetric with respect to H, with first-order transition lines separating the stripe, bubble, and homogeneous phases. Symmetric stripe patterns at H ) 0 (zero net magnetization) become asymmetric as H is increased, where the stripe domains with magnetization parallel (antiparallel) to the field become wider (thinner). Above a critical value of H, there is a transition to a bubble phase consisting of cylindrical domains arranged on a lowdensity triangular lattice. A crucial characteristic of the system is that at high R or high T, where the LRRI predominate, the order parameter profile is a small-amplitude sinusoidal (the “soft-wall” regime) with a short period, while at low R or low T, where the LRRI are very weak, it has a large-amplitude “square”-well profile (the “hard-wall” regime) with a long period. Thus, quenches for high temperatures (shallow quenches) are mimicked in our simulations by values of R close to Rc ≈ 0.385, while quenches for low temperatures (deep quenches) are mimicked by values of R that are much smaller than Rc.33-37 390

with ζ(r,τ) representing the dimensionless thermal noise of strength µ, which obeys the standard fluctuation-dissipation relation 〈ζ(r,τ)ζ(r′,τ′)〉 ) δ(r - r′)δ(τ - τ′). This equation was discretized on grids with sizes ranging from 2562 to 5122 and numerically integrated using standard pseudospectral methods with periodic boundary conditions.32,34 The initial patterns for the simulations consisted of highly ordered stripe or bubble arrays that were constructed with their proper, equilibrium wavelength characteristic of the given point of the phase diagram. These structures were then further equilibrated, without noise, to produce the equilibrium patterns. These were then used as initial conditions for the exploration of the patterns presented in this work, which were produced by means of subsequent quenches in R and H. To initiate the time evolution from the equilibrium patterns, some of the configurations required the addition of initial random noise µ0. For quenches starting in the initial hardwall bubble configurations this initial random noise did not seem to make a difference (here µ0 as high as 10% of the amplitude of φ(r,τ) in the initial patterns gave the same results as the µ0 ) 0 case). On the other hand, for quenches from the stripe phase, such noise was found to be essential: without µ0 the perfect lamellar patterns were too stable. Most of the simulations were conducted without noise, but we explicitly checked, for a number of cases, that the patterns remained robust in the presence of noise. In general, we observed for µ ∼ 2-5% of the amplitude of φ(r,τ), noise does not effect the final configurations for the “low-R” and “high-R” regimes. It is important to point out that a lack of noise does not necessarily imply zero temperature, and that this situation is similar in spirit to the experimental realizations for ferrimagnetic films. For lamellar patterns, experiments5,6,8,9 and previous theoretical40,41 considerations show that, outside of the small critical region, temperature fluctuations are irrelevant and the only role of temperature is to modulate the characteristic period (analogously to the parameter R). The same is true for the bubble patterns, where the coercive friction associated with microscopic roughness suppresses the effects of any thermal fluctuations.11 Experimentally, when fluctuations are needed to initiate the time evolution, these are “simulated” by adding a small ac H-field to the system. One of the main goals of this work is to provide a comprehensive understanding of the evolution of highly ordered equilibrium patterns under temperature-induced or field-induced strain. For a given film thickness, the patterns depend not only on R and H but also on the initial configuration as given by the shape and size of the domains, along with their geometrical arrangement. In addition, modulated systems are strongly history dependent, so that how a specific point in the phase diagram is reached is very important. Many trajectories do not give the same patterns Nano Lett., Vol. 5, No. 2, 2005

when the quenches are reversed, and changes in R and H often do not commute. Initial and final values of R in the system can be linked through a “direct” quench (R0 f Rf) or through a “stepwise” quench (R0 f R1 f R2 ... f Rf), with intermediate equilibration (similarly for the field). Again, this can lead to radically different configurations because of the way the strain is accommodated. Stepwise trajectories tend to produce smaller strain, leading to affine shape transformations. Direct trajectories can lead to a considerable accumulation of strain, whose fast release is accomplished by the quick fragmentation of the domains, or by nucleation of “domains within domains”. There are innumerable ways of straining the system; in the simple cases reported here, the patterns are either under compressive strain (too many domains when fewer are required for equilibrium) or under dilative strain (too few domains when more are required). Strain generally is a result of topological constraints on the system arising from a variety of physical situations: high energy barriers for the nucleation of stripes, the bending stiffness of the stripes, packing constraints in the initial highly geometrically ordered configurations. Results and Discussion. To understand the origin of strain in the quenched patterns, assume a configuration of stripes at H ) 0. Let Llat be the lateral dimensions of the film, such that all the stripes are perpendicular to the side of the film. Let do represent the equilibrium stripe period and No the number of lamellae in the initial equilibrium pattern, and let dR and NR be the corresponding stripe period and number for the quenched system. The dependence of d on the parameters H and R may be found numerically,3,4,7 although in terms of temperature, do ∼ |T - Tc|1/4. Clearly, Nlat ) Nodo ) NRdR in equilibrium. Immediately after the quench, when the number of stripes has not changed, the strain produced by the quench in R is  ) (dR - do)/do. When R is decreased, the equilibrium stripe period is larger and the number of stripes is correspondingly smaller. Immediately after such a quench, there is an excess number of stripes which, therefore, are under a compressive strain ( > 0). The reverse situation occurs when R is increased: immediately after the quench, the number of stripes is lower than what is required by the equilibrium condition and the initial system is under dilative or extensional strain ( < 0). To investigate the patterns produced, we have exhaustively explored different quench trajectories. Here, we present only the main results. Temperature-Induced Strain on Stripes. Consider an initial pattern of symmetric stripes (H ) 0) at high R (soft-wall regime, small period) quenched to a low R (hard-wall regime, large period). The system is therefore under compressive strain. In this case, strain release takes place by means of dislocation nucleation and climb (the topological process by which a dislocation gradually shortens its length until it disappears: the original stripe is “ejected”), as illustrated in Figure 2a. The process is facilitated by the Peach-Koehler force,38 which results from the strain-induced curvature of the stripes surrounding the dislocation core. In addition, there is a force due to the elastic interactions between the dislocations: the force is approximately zero when the Nano Lett., Vol. 5, No. 2, 2005

dislocations are on the same stripe line. The longitudinal component of this interaction force is attractive for dislocations with opposite Burger’s vectors, and repulsive if these are parallel.39 This adds to the Peach-Koehler force facilitating dislocation climb, while the perpendicular component provides a mechanism for the clustering of dislocations to form a domain wall or grain boundary. The stripe ejection allows the pattern to accommodate the increase in the stripe period induced by the lowering of R while preserving the stripe pattern (stripes do not disappear by reducing their width to zero, but by shortening their length). Dislocation interaction forces play a role when more dislocations are nucleated. The large change in R forced onto the system by the quench allows for the nucleation of several dislocations in both phases. Eventually, the tips of these dislocations separate incommensurate regions of different periods. This is clearly seen in the last panels of Figure 2a, where two regions of shorter period alternate with two regions of larger period. The reverse quench, increasing R on an initial pattern of ordered stripes at low R, subjects the stripes to dilative strain. Nucleation of additional stripes should release the strain, but this is precluded by the large energetic barriers to the nucleation of Bloch wall pairs. Rather, the excess dilative strain is reduced by an undulation or buckling instability as shown in Figure 2b. The free energy for the stripe phase may be recast as an effective Hamiltonian for a lyotropic liquid crystal:40 the undulation instability arises from the competition between the elastic extensional energy and the opposing elastic bending energy. As dilative strain accumulates with increasing R, a collective buckling of the lamellae on macroscopic scales results in stable undulation patterns, and in stable chevron or zigzag patterns at higher R characterized by sharp cusps. Further increasing the dilative strain, leads to a “melting” of the chevron pattern via the nucleation of disclination dipoles that have their origin in the sharp tips of the zigzags. These new tethers are oriented at 120° with respect to the original chevron walls. This process of line branching (also known as pincement in liquid crystals) relieves the strain by adding lamellae. For even higher R’s, the disclinations unbind completely, and drive the system to a glassy stripe phase. These results are in agreement with previous experimental observations.7-9 New and unexplored patterns emerge for initial asymmetric (H * 0) stripes undergoing temperature quenches (Figure 2c,d). There is an additional force coming from the action of the magnetic field on the dislocation core, which qualitatively may be understood as follows.41 If d+(-) is the stripe width with magnetization parallel (antiparallel) to H, then the surface pressure due to the curvature of the dislocation tip is pL ) 2σ/d+(-) (σ is the wall surface tension and 2/d+(-) the dislocation curvature). This force decreases for + dislocations pulling them in, while the ejection of (-) dislocations is facilitated. When the lamellae are under compression, the process of dislocation climb and ejection is similar to that for symmetric stripes when H is small. However, at larger (constant) fields, the process of strain release takes place by the rupturing of stripes and subsequent 391

Figure 2. (a) Period adaptation of symmetric stripes under compressive strain through dislocation nucleation and climb and stripe ejection for a direct quench R ) 0.34 f 0.06 (H ) 0.0, τ ) 500, 1700, 2900, 9500). (b) Final frozen configurations at different values of dilative strain for a stepwise trajectory R ) 0.08 f 0.10 f 0.16 f 0.18 f 0.28 (initial symmetric stripes not shown). (c) Time evolution of initial asymmetric stripes under compressive strain after a direct quench R ) 0.34 f 0.08 at constant field H ) 0.08 (τ ) 1800, 2000, 2200, 4000). (d) Time evolution for asymmetric stripes under dilative strain after a step quench R ) 0.14 f 0.16 at constant field H ) 0.10 (τ ) 0, 2500, 5500, 37000). (e) Peristaltic modes and necking instability in a stripe-bubble transition (R ) 0.34, H ) 0.0 f 0.25, τ ) 0, 600, 700, 800). The system ends in a perfect triangular lattice. (f) Final frozen configurations at different values of field for a stepwise trajectory H ) 0.25 f 0.15 f 0.0 f -0.15 f -0.22 (initial triangular lattice not shown). For visualization purposes the domains in (e) and (f) have been enlarged four times (i.e., one-fourth of the system is shown). 392

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Figure 3. Temperature-induced dilative strain on initial triangular lattices at different values of the field H. R is increased from left to right, either in stepwise [S] trajectories (where each configuration on the left acts as the initial configuration for the next configuration on the right) or direct [D] trajectories (where each configuration is obtained through a single quench starting from the configurations at R ) 0.08). All the configurations are effectively frozen except for those in gray, that are still evolving slowly.

coarsening of segments. This process is triggered by the formation of two dislocation pairs separated by a single stripe, which thickens in the region surrounded by the gaps left by the dislocations. This is a highly correlated process, with the thickened region of stripes inducing the pinching of neighboring stripes. Eventually, incommensurate domains of thick and thin stripes appear. Thinner domains disappear by shortening their length, ultimately forming regions of parallel stripes of the right thickness separated from each Nano Lett., Vol. 5, No. 2, 2005

other by grain boundaries consisting of either bubbles or arrays of segment tips. New features also emerge when asymmetric lamellae are under dilative strain. The undulation patterns arise in a fashion similar to the symmetric case. However, the asymmetric stripes do not form a chevron pattern. Rather, some undulation grooves in the minority black phase increase their amplitude and become more square, while others decrease their amplitude and become more triangular. The square black profiles eventually frame 393

Figure 4. Time evolution for the following: (a) Direct quench R ) 0.08 f 0.34, H ) 0.25, on initial configuration (R, H) ) (0.08, 0.25) in Figure 3; τ ) 10, 70, 100, 900, 1400-15000. (b) Direct quench R ) 0.08 f 0.30, H ) 0.0, on initial configuration (R, H) ) (0.08, 0.0) in Figure 3; τ ) 10, 30, 200, 500, 3500-15000. (c) Step quench R ) 0.26 f 0.34, H ) 0.0, on initial configuration (R, H) ) (0.26, 0.0) in Figure 3; τ ) 50, 200, 600, 1900, 2500-15000.

short, narrow areas of the majority white phase, which effectively become disclination dipoles; unlike the symmetric case, the dipoles do not grow as tethers out of sharp chevron tips. Field-Induced Strain on Stripes and Bubbles. The effect of varying the field at constant R strongly depends on the value of R. In the high-R, soft-wall regime, a change in field brings about the well-known reversible stripe-to-bubble transition shown in Figure 2e. Above a threshold field, the stripes experience inhomogeneous variations of their thickness, known as peristaltic modes in the lyotropic liquid crystal effective Hamiltonian.41 Pinching of the stripes follows, ending in a bubble lattice. The inverse bubble-tostripe transition takes place via the “stripe-out” instability, where bubbles elongate along a given direction and touch each other, melting into stripes. This process, entirely reversible, is like reading Figure 2e from right to left. More interesting and unexpected results appear in the low-R, hardwall regime. If a field is imposed on a lamellar pattern, the stripes favored by the field just grow in thickness. If, however, a dislocation is nucleated, period adjustment proceeds by irreversible dislocation climb and ejection. Reducing back the field creates the undulation patterns, and the entire dynamics is similar to that produced by temperature-induced strain. Now consider a triangular lattice. Figure 2f shows resulting patterns under successive field quenches H ) 0.25 f 0.15 f 0.0 f -0.15, whose only effect is to increase the area of the black bubbles, which eventually become the majority phase. Finally, a larger field (H ) -0.22) produces morphology changes, resulting in a honeycomb lattice of white bubbles. Temperature-Induced Strain on Bubbles. The most exotic patterns are formed when triangular lattices are placed under 394

dilative strain, i.e., by taking the configurations that are formed when an initial hard-wall, low-R bubble lattice is subject to a field quench (the patterns shown in Figure 2f), and then further subjecting these to a quench to higher values of R. Roughly speaking, these patterns, illustrated in Figure 3, fall into four regimes based on the final value of R: (i) Low temperature 0.08 j R j 0.165 regime. Domains of the minority phase experience an elliptical instability and end up as ordered lattices of either dumb bells or rounded segments. Domains in the equal or majority phase experience a higher-harmonic shape transition, and end up as “Y” shapes with trigonal symmetry. As R increases, the center of the “Y” becomes thinner and the tips more rounded. In all cases, the final patterns are independent of whether the quenches are stepwise or direct. (ii) Lower intermediate temperature 0.165 j R < 0.22 regime. The final configurations depend very much on whether the quenches are stepwise or direct. Domains of the minority phase are wavy stripe segments if the trajectory is stepwise, or form a bubble lattice if a direct quench is involved. Equal or majority phase domains (under larger dilative strain) acquire a “Y” shape in the stepwise trajectories or form rings under direct quenches. (iii) Higher intermediate temperature 0.22 j R j 0.31 regime. Configurations in this regime also depend strongly on whether the quenches are stepwise or direct. Except for the minority phase at high field, most configurations are glassy states of melted stripes/segments and bubbles in various proportions. (iv) High temperature 0.31 j R j 0.36 regime. In the high-R, soft-wall regime, domains have high mobilities and reach their equilibrium configurations. Interestingly, the points given by (R, H) ) (0.34, (0.15) correspond to the Nano Lett., Vol. 5, No. 2, 2005

stripe-bubble coexistence, and stepwise or direct trajectories determine the stripe or bubble nature of the final configuration. Finally, the time-evolution of sample systems is shown in Figure 4. Nonlinear instabilities in these systems trigger nontrivial temporal patterns, including the nucleation of opposite-phase bubbles inside domains, domain fragmentation, coexistence of serpentine stripes and bubbles, etc. In the course of our investigations, we have obtained many more such exotic patterns, most of which will be presented elsewhere.42 It may well be that these kinds of transient patterns will prove to be useful experimentally. The key issue here revolves around the long-time stabilization of these patterns, which presumably may be achieved by means of kinetic freezing. Summary. We have investigated the pattern formation of modulated systems by means of a reliable phase field model. This model faithfully reproduces experimental and theoretical results in previously explored regions of phase space. However, by tuning the strain that arises through the inherent dependence of the patterns on the temperature and the field, and through the topological contraints of the system, we have shown that new metastable or glassy patterns are formed that, to date, have been largely unexplored. These patterns are the result of a complex mix of ordering mechanisms and instabilities that will require considerable more theoretical and experimental study before a detailed understanding is achieved. Ultimately, it is hoped that these new patterns will find their application as templates for nanotechnology applications, complementing the patterns that are in current use. Acknowledgment. We gratefully acknowledge financial support from NSF ITR-0312105, NSF CAREER-0348039, and DOE DE-FG-02-98ER45t85 grants. References (1) Seul, M.; Andelman, D. Science 1995, 267, 476. (2) Bowman, C.; Newell, A. C. ReV. Mod. Phys. 1998, 70, 289. (3) Kooy, C.; Enz, U. Philips Res. Rep. 1960, 15, 7.

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