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New and Exotic Self-Organized Patternsfor Modulated Nanoscale Systems

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 charac-

terized 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 block-

copolymer systems,18,19 type I superconductors,20 steady-state

reaction-diffusion (Turing) patterns,21 ferrofluids,22 Swift-

Hohenberg 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 usedto 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,

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 topologi-

cal constraints can arise from a variety of physical situa-

tions: 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

(r,) at position r as a function of time . The phenom-enological free energy functional F(suitably adimension-

alized) is expressed in terms of a Ginzburg-

Landau expan-sion based on (r) and its gradient. It consists of both a local

and a nonlocal term:

In the first term, the gradient-squared term represents the

lowest-order approximation to the cost of creating a domain* To whom correspondence should be addressed. E-mail: sagui@

unity.ncsu.edu.

F[(r)] ) d2r[1

2()

2+ f() - H] +

R

2 d2rd2r(r) g(|r - r|)(r) (1)

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LETTERS

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wall or interface; f () is the local free energy; and -H 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 as-

sociated 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 () ) -(1/2) 2

+ (1/4) ,4 so that the minima of the resulting double-well

structure identify each of two coexisting phases. These phasesare represented by either positive or negative values of -

(r): (r) > 0 corresponds to regions where the spins point

in the up direction, while (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 sketchedin 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 Tbut 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 low-

density 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 re-

gime) with a long period. Thus, quenches for high temper-

atures (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

The time evolution of the system is obtained from the

corresponding Langevin equation

with (r,) representing the dimensionless thermal noise of

strength , which obeys the standard fluctuation-dissipationrelation (r,)(r,) ) (r - r)(- ). This equationwas 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 initialrandom noise 0. For quenches starting in the initial hard-wall bubble configurations this initial random noise did not

seem to make a difference (here 0 as high as 10% of the

amplitude of (r,) in the initial patterns gave the sameresults as the 0 ) 0 case). On the other hand, for quenchesfrom the stripe phase, such noise was found to be essential:

without 0 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 2-5% of the amplitude of (r,), noise

does not effect the final configurations for the low-R and

high-R regimes. It is important to point out that a lack ofnoise does not necessarily imply zero temperature, and that

this situation is similar in spirit to the experimental realiza-

tions for ferrimagnetic films. For lamellar patterns, experi-

ments5,6,8,9 and previous theoretical40,41 considerations show

that, outside of the small critical region, temperature fluctua-

tions 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 Experi-

mentally, 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

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 homo-geneous phases. The bubble phase is a low-density triangular lattice.Typical profiles of the order parameter are also sketched.

(r,)

) -

F[]

+ (r,)

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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 isaccomplished 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 con-

straints 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 Hand 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 thenumber 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: im-

mediately 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

dislocations are on the same stripe line. The longitudinal

component of this interaction force is attractive for disloca-

tions with opposite Burgers vectors, and repulsive if these

are parallel.39 This adds to the Peach-Koehler force facilitat-

ing 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 theirwidth 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 ac-

cumulates 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 thenucleation 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 Rs, 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 Ifd+(-) is thestripe width with magnetization parallel (antiparallel) to H,

then the surface pressure due to the curvature of the

dislocation tip is pL ) 2/d+(-) (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

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Figure 2. (a) Period adaptation of symmetric stripes under compressive strain through dislocation nucleation and climb and stripe ejectionfor a direct quench R ) 0.34 f 0.06 (H ) 0.0, ) 500, 1700, 2900, 9500). (b) Final frozen configurations at different values of dilativestrain 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 initialasymmetric stripes under compressive strain after a direct quench R ) 0.34 f 0.08 at constant field H ) 0.08 ( ) 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 () 0, 2500, 5500, 37000). (e) Peristaltic modes and necking instability in a stripe-bubble transition (R ) 0.34, H) 0.0f 0.25, ) 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 0.15f 0.0f -0.15f -0.22 (initial triangular lattice not shown). For visualization purposes the domains in (e) and (f) havebeen enlarged four times (i.e., one-fourth of the system is shown).

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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

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

Figure 3. Temperature-induced dilative strain on initial triangular lattices at different values of the field H. R is increased from left toright, 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.

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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 ofR. 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 thick-ness, 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-to-

stripe 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, hard-

wall 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 temper-

ature-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 hon-

eycomb lattice of white bubbles.

Temperature-Induced Strain on Bubbles. The most exotic

patterns are formed when triangular lattices are placed under

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

ofR. 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 experiencea 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 trajec-

tories 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

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; ) 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; ) 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; ) 50, 200, 600, 1900, 2500-15000.

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stripe-bubble coexistence, and stepwise or direct trajectories

determine the stripe or bubble nature of the final configu-

ration.

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 fragmenta-

tion, 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 presentedelsewhere.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.

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