Table Of ContentGrain boundary pinning and glassy dynamics in stripe phases
Denis Boyer
School of Computational Science and Information Technology, Florida State University, Tallahassee, Florida 32306-4120.
(Present address: Instituto de F´ısica, Universidad Nacional Aut´onoma de M´exico, Apartado Postal 20-364, 01000 M´exico
D.F., M´exico )
2
0
0 Jorge Vin˜als
2 Laboratory of Computational Genomics, Donald Danforth Plant Science Center, 975 North Warson Rd, St. Louis, Missouri
63132.
n
a (February 1, 2008)
J
We study numerically and analytically the coarsening of stripe phases in two spatial dimensions,
4
and show that transient configurations do not achieve long ranged orientational order but rather
2
evolve into glassy configurations with very slow dynamics. In the absence of thermal fluctuations,
defects such as grain boundaries become pinned in an effective periodic potential that is induced
]
h by the underlyingperiodicity of the stripe pattern itself. Pinning arises without quencheddisorder
c fromthenon-adiabaticcouplingbetweentheslowlyvaryingenvelopeoftheorderparameteraround
e
adefect,anditsfastvariationoverthestripewavelength. Thecharacteristicsizeofordereddomains
m asymptotes to a finite value Rg λ0 ǫ−1/2exp(a/√ǫ), where ǫ 1 is the dimensionless distance
∼ | | ≪
- awayfromthreshold,λ0thestripewavelength,andaaconstantoforderunity. Randomfluctuations
t
a allow defect motion to resume until a new characteristic scale is reached, function of the intensity
t of the fluctuations. We finally discuss the relationship between defect pinning and the coarsening
s
. laws obtained in theintermediate time regime.
t
a
m
-
I. INTRODUCTION Farenoughfromthe bifurcationthresholdofthemod-
d
n ulatedphase,theseparationbetweenslowandfastscales
o The motion of topological defects in two dimensional no longer holds, and corrections to the amplitude equa-
c smecticphasesisstudiedatafinitedistancefromthresh- tionsappearbecauseofthecouplingbetweenbothscales.
[
old. WefocusontheSwift-HohenbergmodelofRayleigh- These corrections are generically referred to as non-
2 B´enard convection and related amplitude equations to adiabatic effects. One manifestation of non-adiabaticity
v address the role that non-adiabatic effects play in do- is that a defect that would be expected to move at con-
4 main coarsening of a modulated phase, defect pinning, stant velocity from an amplitude equation analysis may
5 and the appearance of glassy behavior. instead remain immobile or pinned [21–23]. We argue
2
Topological defects are often the longest lived modes below that non-adiabatic effects anddefect pinning have
0
1 ofanon-equilibriumsystem,withtheirmotiondetermin- important consequences for domain coarseningof modu-
1 ing the longest relaxation times of the structure. Phe- lated phases in two dimensions, and are responsible for
0 nomenological models of defect motion that are based the formation of glassy configurations.
/ on a mesoscopic description have been known for some Ourresultscomplementrecentresearchonglassyprop-
t
a time [1,2]. Such a description, valid for distances much ertiesofstripephases. Ithasbeensuggestedthatsystems
m larger than the defect core, typically involves time de- in which long ranged order is frustrated by repulsive in-
- pendent Ginzburg-Landau equations or their generaliza- teractions (the latter often leading to the formation of
d
tions. A few cases have been studied extensively, includ- stripe phases or other patterns in equilibrium) may in
n
o ing domain coarseningin O(N) models [3,4],in nematics fact exhibit the properties of structural glasses. An ex-
c [5–8],andin smectic phasesas effectively encounteredin ample are the glassy states recently observed in doped
: modelsofRayleigh-B´enardconvectionorlamellarphases semiconductors in a stripe phase [24]. Coarse grained
v
i of block copolymers [9–15]. In the case of modulated modelswithcompetinginteractionsofthetypeusedhere
X phases, the motion of a single defect has been widely (andalsousedtostudyblockcopolymermeltsinlamellar
r studied within the well known amplitude equation for- phases)havebeenreintroducedtodescribetheformation
a malism. This method describes the spatio temporal evo- of glasses in supercooled liquids [25]. Additional equi-
lution of the envelope of a base periodic or modulated librium studies of the same models in three dimensions
structure [16–20]. The amplitude equation description based on replica calculations [26] or Monte Carlo simu-
is valid only close to bifurcation points where the spa- lations [27] have been used to argue for the existence of
tialscaleofvariationoftheamplitudes islargeor“slow” an equilibrium glass transition. Structural glasses form
comparedwiththe“fast”periodofthebasepatternand, spontaneously at low temperature without the presence
in the present case, with the extent of the defect core as of any quenched disorder,and their properties remainin
well. general poorly understood. It is noteworthy that coarse
1
grainedmodels exhibitingglassybehaviorinthe absence to quantify the time dependence of the linear scale of
ofdisorderarerare,whereasexamplesofdiscretesystems the coarsening structure. The statistical self-similarity
are known(e.g. Ising models with next-nearestneighbor hypothesis asserts that after a possible transient, con-
interactions [28,29]). We present here a two dimensional secutive configurations of the coarsening structure are
study that indicates a dynamical route to the formation geometrically similar in a statistical sense. As a conse-
of glassy configurations in stripe phases. quence,anylinearscaleofthestructure(e.g.,theaverage
We first analyze the motion of a particulartype of de- size of a domain or grain of like oriented stripes) is ex-
fect, namely a grain boundary separating two domains pected to growas a powerlaw oftime l(t) t1/z, with z
∼
of differently oriented stripes. Earlier asymptotic work a characteristicexponent. Self-similarity is a well known
near onset (i.e. in the limit ǫ 0, where ǫ is the dimen- featureinsystemsthatorderinuniformphasesofbroken
→
sionlessdistanceawayfromthreshold)isextendedtothe symmetry [32,33]. However, the determination of z has
region of small but finite ǫ. In Section II, grain bound- been problematic for stripe phases. Its value appears to
ariesareshowntomoveinaneffectiveperiodicpotential depend on the quench depth (the value of ǫ), whether
of wavelength λ /2 (where λ is the periodicity of the or not fluctuations are included in the governing equa-
0 0
stripe modulation) and ofmagnitude that increases very tions, on the thermal history of the system, and on the
quickly with ǫ. Grain boundaries asymptotically pin as particular linear scale analyzed.
the driving force for grain boundary motion decreases. Recent work in the limit ǫ 0 showed that coarsen-
→
It is argued that for any finite ǫ an infinite size system ing is self-similar and that z = 3 [15]. The value z = 3
will not achieve macroscopic long range order dynami- in that limit can be justified by a dimensional analysis
cally following a quench. Rather, the characteristic size of the law of grain boundary motion. We focus here on
of a domain will not exceed typical value R that is pro- the case of finite ǫ (in practice ǫ 0.1 for the Swift-
g
≥
portional to λ ǫ 1/2exp(a/√ǫ), where a is a constant Hohenberg model), and report a slowing down of phase
0 −
| |
of order unity. ordering dynamics with increasing ǫ, in agreement with
InSectionIII,weincorporatetheeffectofrandomfluc- the literature. We attribute this behavior to partial pin-
tuations and derive the corresponding amplitude equa- ning of defects that becomes increasingly important at
tions valid for fluctuations of small amplitude. The long times as the driving force for coarsening decreases.
asymptotic motion of a grain boundary can be recast Atevenlongertimes,coarseningstopsaltogetherandthe
as an escape problem in which the effective activation system reaches a glassy state as the linear scale of the
barrier is seen to be proportional to the grain boundary structure reaches the critical value Rg(ǫ) computed in
perimeter. Section II. When random fluctuations are incorporated
Our approach must be considered only qualitative in in the model, we show that, sufficiently close to onset,
nature because of the scope of the descriptionemployed. the value of z remains independent of the intensity of
Ginzburg-Landau equations, and more generally ampli- the fluctuations, thus verifying the universality implied
tude or order parameter equations (of which the Swift- in the self-similarity hypothesis in that region. At larger
Hohenberg model described below is but one example) ǫ, we find that fluctuations accelerate ordering kinetics,
areonlyasymptotic,largelengthscaleapproximationsto also in agreement with the literature, and that, as ex-
thephysicalsystemtheymodelintheimmediatevicinity pected, defect motion is allowed beyond the scale given
ofabifurcationpoint. Thereforeanyshortscalephenom- byRg. Atevenlatertimesthesystemordersveryslowly,
ena involved in the description of non-adiabatic correc- possibly logarithmically in time.
tionsclearlyfallsbeyondtheirrangeofvalidity,atleastin
a systematically quantifiable way. It is nevertheless not
unreasonable to expect that non-adiabatic effects of the II. NON-ADIABATIC CORRECTIONS AND
sort encountered in order parameter equations will also GRAIN BOUNDARY PINNING
occurinthephysicalsystemswhichtheymodel. Further-
more,ourresultsalsoprovideinsightsintomanyexisting We consider the Swift-Hohenberg model of Rayleigh-
numericalstudiesoftheseorderparametermodels,asde- B´enardconvection[34]asaprototypicalmodelofamod-
scribed below. ulated phase. The numerical results presented below
In Section IV, we address the consequences of pinning have been obtained from a direct numerical solution of
on the domain coarsening that occurs in the intermedi- themodel. Theanalyticresults,ontheotherhand,follow
ate time regime following the quench. This subject has from the corresponding amplitude equation, and hence
been the focus of several numerical studies [10–15] and, are expected to be of somewhat wider generality. The
morerecently,ofexperimentalstudiesinblockcopolymer model equation studied here is
thin films [30]andin electro-convectionin nematics [31].
∂ψ 1
TheresultsofSectionsIIandIIIprovideapossibleinter- =ǫψ (k2+ 2)2ψ ψ3 , (1)
∂t − k4 0 ∇ −
pretationofconflictingresultsintheliterature. Previous 0
studiesofthisproblem[10–15]addressedtheexistenceof where ψ is an dimensionless order parameter related to
self-similarity during domain coarsening and attempted the verticalfluidvelocityatthe mid planeofa Rayleigh-
B´enardconvectioncell,ǫisthereducedRayleighnumber
2
(R R )/R 1 (R is the critical Rayleigh number We next derive two coupled equations for the ampli-
c c c
− ≪
for instability), and k = 2π/λ is the roll wavenumber tudes A and B that take into account the possible cou-
0 0
(in Appendix A we outline the connection between this pling between these amplitudes and the phases of the
model and other coarse-grained models with long range stripes. This coupling becomes significant at a finite
repulsive interactions [35]). value of ǫ, and hence when there is a large but finite
For 0<ǫ 1, the leading orderapproximationto the separation between the scales X ,Y and x,y .
A,B A,B
≪ { } { }
stationary solution of Eq. (1) is a sinusoidal function of We followanapproachsimilartothat usedinref.[23]to
wavenumberk . We focus in this section on a configura- study the motion of a planar front between a hexagonal
0
tion thatcontains anisolatedgrainboundary separating andauniformphase,orbetweenahexagonalandastripe
two such stationary solutions with mutually perpendic- phase. Thefirststepisamultiscaleanalysis,andisstan-
ular wavevectors (Figure 1). The reason for studying dard [16]. Equation (1) is expanded in power series of ǫ,
this perpendicular orientation is the expectation that a aswellasthesolutionψ =ǫ1/2ψ +ǫψ +ǫ3/2ψ +....
1/2 1 3/2
90◦ grain boundary is that of lowest energy, and hence The leading order solution ǫ1/2ψ1/2 is given by Eq. (2).
theprevalentboundaryangleinanextendedsystemthat At orderǫ3/2, the solvability conditions for the existence
evolvesspontaneouslyfromaninitiallydisorderedconfig-
of a nontrivial solution for ψ yield the relations that
3/2
uration (see, for example, Figure 5a,b). It is known that
A and B must satisfy,
a planar grain boundary separating two regions of uni-
f[t3or6arm]nstlhkaa0ttioisansswtlaiigtthihotnlayasrppyeer[e1tdu9r,t2bh0ea]d.tbiHsooauwnfeduvanercryt,iowunnedofeofrugtnhodeescinuarrnveaeft-. λ120 Zxx+λ0dx′Zyy+λ0dy′hL(ψ1/2)−ψ13/2ie−ik0x′ =0 (4)
ttuhreeexoftetnhseiornololsf tahheeaadsyomfpitt.otWicereasdudltrsesgsivienntihnisthsaetctrieofn- λ120 Zxx+λ0dx′Zyy+λ0dy′hL(ψ1/2)−ψ13/2ie−ik0y′ =0 (5)
erence to small but finite ǫ, and show how corrections
obNtaeianredthlreeasdhotoldb,oau9n0d◦argyrapininnbionugn.dary configuration is 2w∂ith∂the+li2n∂ea∂r op)e2r.atIonrtLhe=li1m−it∂ǫT −0k0−t4h(e∂Xf2uBnc+ti∂oYn2As +A
an approximate solution of Eq. (1) of the form anyd BYBremaixnXcoAnstant over one sp→atial period λ , and
0
therefore the only non vanishing contribution to the in-
ψ(x,y,t)= 1 A(X ,Y ,T) eik0x tegralscome fromthe terms proportionalto eik0x′ (resp.
A A
2(cid:2) eik0y′) within brackets in Eq. (4) (resp. Eq. (5)). This
+ B(X ,Y ,T) eik0y+c.c , (2)
B B standard set of coupled Ginzburg-Landau equations fol-
(cid:3) lows [19,16] . It is known, however, that additional non
where slow variables are denoted by capital letters and
perturbativecontributionsarisingfromthetermψ3 ap-
are defined as [19,20] 1/2
pear in Eqs. (4) and (5). We focus next on these con-
X =ǫ1/2x, Y =ǫ1/4y; tribution and their effect on the relaxation of a slightly
A A
perturbed grain boundary.
XB =ǫ1/4x, YB =ǫ1/2y; T =ǫt. (3) Integrals of the type x+λ0dx eimk0xAnBp in Eqs.
x ′ ′
(The coordinate x is directed along the normal to the (4) and (5) (where m, Rn and p are integers) will not
integrate to zero if the thickness of the grain bound-
reference planar boundary.)
ary profiles along the x direction is finite. (Contribu-
We recall first some known results for a planar and
tions from the direction transverse to the grain bound-
stationary grain boundary in the limit ǫ 0, a case
thatwasextensivelystudiedinrefs.[19,20].→Thestation- ary, yy+λ0dy′ eimk0y′AnBp, will be neglected. They are
ary amplitudes {A0,B0} are a function only of x. A0, typicRallyoftheorderofB2∂y2Aandhencealwayssmaller
the amplitude of the rolls parallel to the interface, van- thanthe leading analyticalterms of the amplitude equa-
ishes as exp(x√ǫ/λ0) when x , and saturates to tions.) Terms proportional to eimk0x′ will contribute to
(4ǫ/3)1/2tanh(x√ǫ/λ0) when x→ −+∞. The behavior of Eq. (4), and terms proportional to eimk0x′+ik0y′ to Eq.
→ ∞
the amplitude of the rolls perpendicular to the interface (5). If we only retain the lowest order term ǫ1/2ψ as
1/2
is slightly different: B0(x) (4ǫ/3)1/2 exp(x√ǫ/ξ0) given by Eq. (2), we find that only A3 (m = 3) con-
− ∝
when x and there exists a location x∗ such that tributes to Eq. (4), while 3A2B (m = 2), as well as
→ −∞
B0(x > x∗) 0 to a good approximation. Hence, the 3A¯2B (m= 2, with A¯ the complex conjugate of A), to
≃
grain boundary region has a thickness proportional to −
Eq. (5). Reintroducing the original unscaled variables,
λ /√ǫ. Itisimportanttonotethatatsmallǫthelocation
0 the generalized amplitude equations read
of the grain boundary decouples from the phase of the
stripes of domain A. Thus, the configuration obtained ∂A δF
gb
=
is invariant under any translation of the grain bound- ∂t − δA¯
uarnychbayngaeddi)s.tance x0 (the phase of the stripes remaining − 4λ12 Z x+λ0dx′Z y+λ0dy′ A3(x′,y′,t) ei2k0x′, (6)
0 x y
3
∂B = δFgb where κ = δx0q2 is proportional to the mean curvature
∂t − δB¯ of the stripes of domain A, φ is a constant phase, and
− 4λ32 Z x+λ0dx′Z y+λ0dy′[A2Bei2k0x′ D(ǫ)= ∞ dx [(∂xA0)2+(∂xB0)2] , (13)
0 x y Z
+A¯2Be−i2k0x′] , (7) p (ǫ)=M−a∞x 3 ∞ dx A3(x)∂ A (x)cos(2k x+θ)
θ(cid:26)4Z 0 x 0 0
where F = d~r is the standard Lyapunov func-
gb gb −∞
tionalcorrespoRndinFgto the 90◦ grainboundary. Its vari- + 3 ∞ dx [2A B2∂ A +A2B ∂ B ]cos(2k x+θ) . (14)
ational derivatives satisfy [19,20] 2Z 0 0 x 0 0 0 x 0 0 (cid:27)
−∞
Equation (12) without the oscillatory term was derived
4 i 2
δF /δA¯= ǫ A+ ∂ ∂2 A in Ref. [36] in the limit ǫ 0. The coefficient D(ǫ),
− gb k02 (cid:18) x− 2k0 y(cid:19) withdimensionsofaninverse→length,representsafriction
3 3 term that depends on the static grain boundary profile
A2A B 2A , (8)
− 4| | − 2| | A0,B0 , while the term ǫκ2 in the numerator is pro-
{ }
4 i 2 portionalto Ldy [ (x= ,y) (x= ,y)]/L,
δF /δB¯ = ǫ B+ ∂ ∂2 B 0 Fgb ∞ −Fgb −∞
− gb k02 (cid:18) y − 2k0 x(cid:19) whereFgb isRthefreeenergydensityimplicitlydefinedby
Eqs. (8)-(9)). The numerator can be understood as the
3 3
B 2B A2B . (9) leading contribution (in ǫ and κ) from an external force
− 4| | − 2| |
actingonthegrainboundary. Thisforceresultsfromthe
The last terms in the right hand sides of Eqs. (6) and difference in the free energy density gb between curved
F
(7)dependonbothfastandslowspatialscales,andthey stripesononeside,andstraightstripesontheotherside
embodytheso-callednon-adiabaticcouplingbetweenthe ofthe boundary. Note the unusualdependence ofx˙gb on
two. Analyzing the effects of these two terms on the a even power of the curvature thus indicating that the
relaxation of a perturbed grain boundary is the subject motion of the grain boundary is such that curved paral-
of remainder of this section. lel rolls of higher energy are always replaced by straight
We now introduce a small perturbation to the pla- perpendicular rolls.
nar boundary as shown schematically in Figure 1. The The last term in the right hand side of Eq. (12) is
phaseofthestripesofdomainAisdistortedbyauniform thedominantcontributionarisingfromthenon-adiabatic
perturbation of wavenumber q k (and of amplitude terms of Eqs. (6)-(7). The dimensionless quantity p(ǫ)
0
≪
δx λ ) in the direction transverse to the stripes. As plays the role of the amplitude of a periodic potential
0 0
≪
shown in ref. [36], approximate solutions to Eq. (6)-(7) of period λ0/2 within which the grain boundary moves.
are given by The major contribution to p(ǫ) comes from the integral
thatcontainsthe term∂ B inEq. (14)sincethe profile
x 0
A=A0(x−xgb(t)) eik0δx0cos(qy) , (10) B0(x) has a steeper variation than A0(x) [19]. Given
B =B0(x xgb(t)) , (11) that both amplitudes A0 and B0 are approximately of
− the form √ǫf(√ǫx/λ ), it is easy to show from Eq. (14)
0
where x (t) represents the time dependent position of that
gb
the grain boundary (averaged over y). As already dis-
p(ǫ) ǫ2 e α/√ǫ , (15)
cussed in that reference, perturbations to the phase of B ∼ −| |
areofhigherorderinǫ. Inordertoderivealawofmotion where α is a constant of order unity, corresponding
| |
forx itissimplertoneglectthelinearrelaxationofthe to the pole of the envelopes closest to the real axis in
gb
perturbed rolls, and hence assume that δx is constant. the complex plane. Hence, p behavesnon-analyticallyat
0
The amplitude δx relaxes exponentially with time but small ǫ, and increases extremely quickly with ǫ. Quali-
0
therelaxationtimeoftheperturbationisproportionalto tatively similar results were reported in [22,23] for one
q 4 andusuallymuchlongerthanthecharacteristictime dimensional fronts between conductive and convective
−
associatedwithgrainboundarymotion,λ /x˙ . Further- states, or between different convective states.
0 gb
more, as was shown in ref. [36], explicitly considering Equation (12) shows that for any finite ǫ>0 a planar
stripe relaxation does not change the law of motion for grain boundary (κ = 0) can have only two stationary
x in any quantitative way. positions per period of the stripe pattern λ . This effect
gb 0
Multiply Eq. (6) (resp. Eq. (7)) by ∂ A¯ (resp. ∂ B¯), had been observed numerically and reported in ref. [36],
t t
addtheresultsandintegratetherealpartoverthesystem with similar findings also given in ref. [20]. Equation
area. By using Eqs. (10)-(11) and integrating by parts (12) also implies that there exists a critical curvature κ
g
the non-adiabatic terms, we obtain the following law of below which the grain boundary will remain immobile.
motion for the grain boundary This critical curvature is given by,
ǫ p(ǫ) 1 3p(ǫ) 1/2
x˙ = κ2 cos(2k x +φ) , (12) κ = =k , (16)
gb 3k2D(ǫ) − D(ǫ) 0 gb g R 0(cid:18) ǫ (cid:19)
0 g
4
where R is the associated radius of curvature which di- likelythatasimilardependencebetweenthespeedofthe
g
verges non-analytically near onset defect andǫ will holdfor the motionof other topological
defects (except for dislocation climb). Hence we argue
Rg λ0 ǫ−1/2 exp |α| . (17) that a defected configuration of stripes does not macro-
∼ (cid:18)2√ǫ(cid:19) scopically order following a quench to a finite value of ǫ.
Asymptotic long time configurationsappear to exhibit a
These results have been verified by direct numerical labyrinthicandpartiallydisorderedstructure withmany
solution of the Swift-Hohenberg model with reasonably immobile defects that do not anneal away. These disor-
smallvaluesofǫ. Thenumericalalgorithmusedhasbeen dered configurations resemble those of a structural glass
described in [36,15]. Briefly, Eq. (1) is discretized on a at zero temperature which lack long range order (trans-
square grid of mesh size ∆x = 1 with 5122 nodes (2562 lational or orientational) order. They become sponta-
for ǫ = 0.5), and the wavelength is set to λ0 = 8∆x. A neously trapped in metastable configurations that are
semi-implicit spectral method is used to iterate in time. very different from the configuration of lowest free en-
The initial condition for ψ is a white and Gaussian ran- ergy (all stripes parallel to each other, or a ”crystalline”
dom field with zero averageand variance ψ2 =ǫ. Typ- state).
h i
ical long time configurations which are stationary for all Not all grain boundaries in a glassy configuration are
practical purposes are shown in Figs. 2 and 5a. These 90 boundaries. However, we expect that grain bound-
◦
figures show the field ψ in grey scale. Many topologi- aries with a different orientation would be pinned less
caldefects includingdislocations,+1/2disclinationsand efficiently (i.e. would have a higher value of α in Eq.
several 90◦ grain boundaries can be identified. Figure 2 (15)). The reason is that their stationary pl|an|ar pro-
corresponds to ǫ = 0.5 and two different times t = 104 file is smoother than that of a 90 grain boundary, and
◦
and t= 2 104, showing that the order parameter does thereforenon-adiabaticeffectsareexpectedtobeweaker.
×
not change beyond t = 104. Figure 5a corresponds to Wefinallymentionthatifbothǫandκλ arenotsmall
0
ǫ=0.4, and the configurationshownremains practically comparedtoone,bothadiabaticandnon-adiabaticterms
constant beyond t=2.3 105. will contain higher order analytic corrections which we
×
To further quantify these observations we have com- have not calculated.
puted the probability distribution function of stripe cur-
vatures P(κ,t). The stripe curvature is defined as κ =
nˆ , where nˆ is the unit normal to the lines of con- III. MOTION AT FINITE TEMPERATURE
|∇· |
stant ψ. The curvature κ is a slowly varying quantity
awayfromdefectcores,andonlytheseregionsareusedto
GiventheresultsofSectionII,itisnaturaltostudythe
compute P(κ,t) by the filtering method describedinref.
effectofrandomfluctuationsaddedtoEq. (1). Smallam-
[15] Figure 3 shows our results for ǫ = 0.4 and ǫ = 0.5.
plitude fluctuations will allow activated motion of grain
In both cases the distribution converges at long times
boundaries, and in general unpinning. We consider in
towards a limiting curve of finite width, thus indicat-
this section the stochastic Swift-Hohenberg model
ing that asymptotic configurations contain many curved
stripes and are disordered at large scales, or ”glassy”. ∂ψ 1
=ǫψ (k2+ 2)2ψ ψ3 +η(~r,t) , (18)
This behavioris to be contrastedwith that of a coarsen- ∂t − k4 0 ∇ −
ingsysteminwhichP(κ,t )wouldapproachadelta 0
→∞
function at κ = 0. We take P(κ = 0,t ) as a mea- where η is a Gaussian and white random noise of zero
→ ∞
sureofthe linearscaleofthe structureortypicaldomain mean and variance
size and compare its value with the pinning radius R
g
givenin Eq. (16). Figure 4b showsthe numericalresults η(~r,t)η(~r ′,t′) =2Fδ(~r ~r ′)δ(t t′). (19)
h i − −
togetherwith R multiplied by a (fitted) scale factorap-
g
proximately equal to 4. The pinning radius R increases The noise intensity F is proportional to the (dimension-
g
extremely quickly with decreasing ǫ, in agreement with less)temperatureaccordingtothefluctuation-dissipation
thenumericalcalculationsfortherangeofǫwhichwecan theorem. In what follows, F and ǫ are considered as in-
study (computational constraints on system sizes have dependent parameters, although they might be related
preventedusfrominvestigatingthe regionǫ<0.30). We in some particular physical systems. The stochastic
havecheckedthattheglassyconfigurationsatlongtimes Swift-Hohenberg model has been used to study hydro-
do not resultfromnumericalpinning; the results arenot dynamicfluctuationsnearonsetofRayleigh-B´enardcon-
modifiedwhenthegridspacingishalvedto∆x=λ /16. vection[37],andthermalfluctuationsofmolecularorigin
0
Although other types of defects (e.g., dislocations and in lamellar phases of diblock copolymers [38].
+1/2 disclinations) may also become pinned, and thus The stationary states of Eq. (18) in two spatial di-
contributetotheoverallstabilityofglassyconfigurations, mensions have been studied in refs. [39,11]. Above a
the predominance of grain boundaries over other defects criticalnoiseintensityFc (thatdependsonǫ),thesystem
seems to be a generic feature of the Swift-Hohenberg is disordered (lacks both translational and orientational
model (see Figures 5a,b and ref. [15]). Furthermore it is long ranged order). Below Fc a stripe phase with long
5
rangedorientationalorderbutnotranslationalorderwas x˙ = ǫ κ2 p(ǫ) cos(2k x +φ)+η˜, (23)
found. Only at F = 0 the system was seen to exhibit gb 3k2D(ǫ) − D(ǫ) 0 gb
0
both translational and orientation long ranged order. In
what follows we focus on defect dynamics in the range with η˜a (real) random white Gaussian noise satisfying,
0 < F F , so that the local stripe pattern is not very
c
distorte≪d. η˜ =0, η˜(t)η˜(t′) =2F′ δ(t t′), F′ =F/[2D(ǫ)Rgb]
h i h i −
We first derive the stochastic amplitude equations for (24)
a 90 grain boundary. Following Graham [40], we ap-
◦
proximate the effect of the noise on the amplitudes by whereR isthegrainboundaryperimeter. Asexpected,
gb
projectingitalongthetwoslowmodesofthedeterminis- the intensity of the fluctuations onthe globalcoordinate
ticequationandneglectinganycontributionarisingfrom x isproportionalto1/R . Equation(23)isastraight-
gb gb
couplingsandresonancesbetweennoiseandfastvariables forward generalization of Eq. (12), and is formally anal-
[41,42]. We start by writing the random function as, ogoustotheequationthatdescribestheonedimensional
motion of a Brownian particle in a periodic potential of
η(~x,t)= 1 eik0x η˜ (X ,Y ,T)
2 A A A amplitude 2p(ǫ)/[2D(ǫ)k0].
+(cid:2)eik0y η˜ (X ,Y ,T) + c.c. , (20) Equations (23) and (24) can be recast as
B B B
(cid:3)
where the slow variables X(Y) are given by Eq. (3), k F k F 1
A,B x˙ = 0 0 R κ2 0 0 cos(2k x +φ)
and η˜A and η˜B are two independent complex random gb (cid:18) 2D (cid:19) g −(cid:18) 2D (cid:19)Rg 0 gb
processes that satisfy the relations,
1 F 1/2
hη˜Ai=hη˜Bi=0, hη˜A2i=hη˜Aη˜Bi=hη˜Aη˜B∗i=0, +√2D (cid:18)Rgb(cid:19) ξ . (25)
η˜ η˜ = η˜ η˜ =2Fδ(~x ~x )δ(t t).
h A A∗i h B B∗i − ′ − ′ Therandomtermξ issuchthat ξ =0and ξ(t)ξ(t ) =
′
h i h i
ItisimplicitinthedecompositionthatF issmallenough 2δ(t t). We have also used Eq. (16) to eliminate p(ǫ)
′
−
so that well defined stripes exist locally. On the other from Eq. (23), and we have defined
hand,F hastobelargeenoughsothatη˜ andη˜ arenot
A B
negligible in the solvability conditions at order ǫ3/2 [43]. 2ǫ
F = . (26)
Givenboth assumptions,Equations(6) and(7)straight- 0 3k3R
0 g
forwardly generalize to
Considerthesituationwheregrainboundariesarepinned
∂A = δFgb at F =0. Since κ<κg, the first term of the right-hand-
∂t − δA¯ sideofEq. (25)isnotdominantandthepotentialbarrier
− 4λ12 Z x+λ0dx′Z y+λ0dy′ A3(x′,y′,t) ei2k0x′ tohrdateraopfiFn0n/eRdgd.eTfehcteostfoscizheasRtigcbphraosbtleomoviserncoowmeaniseosfcatphee
0 x y problem overthis potential barrier giventhe intensity of
+η˜ , (21)
A the noise term in Eq. (25). The Kramers rate of escape
is given by
∂B δF
gb
∂t =− δB¯ r exp F0Rgb . (27)
3 x+λ0dx y+λ0dy [A2Bei2k0x′ +A¯2Be i2k0x′] ∼ (cid:18)−F Rg (cid:19)
− 4λ20 Zx ′Zy ′ − Therefore a noise intensity
+η˜ . (22)
B
F
Wecannowestimatetheescaperateofagrainbound- F =Rgb R0 ∼Rgb k0−1ǫ2 e−|α|/√ǫ (28)
g
ary over the potential barrier of Eq. (12). In order to
do so, we need to estimate the projection of the noise is required to un-pin a grain boundary of length Rgb.
intensity in Eqs. (21) and (22) on the coordinate x (t)
gb
implicitly defined by Eqs. (10) and (11). A rough esti-
matethatissufficientforourpurposescanbeobtainedby IV. SLOW COARSENING DYNAMICS:
usingEqs. (10)and(11)asthetrialsolutionofEqs. (21) DEPENDENCE ON TEMPERATURE AND
and (22). Focusing on x alone ignores possible bound- QUENCH DEPTH.
gb
ary broadening because of fluctuations, or roughening.
Both phenomena will be important for grain boundary We use here the results of Sections II and III to pro-
motion above the pinning point, but their contribution vide a possible interpretation of conflicting results con-
is probably less important in the immediate vicinity of cerning domain coarsening of stripe phases. We recently
the pinning transition. By substituting Eqs. (10) and studied this issue by numerically solving the noiseless
(11) into Eqs. (21) and (22), we find Swift-Hohenberg equation (Eq. (1)) in the limit ǫ 0
→
6
[15]. Ournumericalresultssuggestedthatthe character- directly from Eq. (25):
istic scale of the structure (or the typical size of ordered
domains) increases as t1/z, with z = 3. That value of dl k0F0 Rg k0F0 1
= cos(2k l+φ)
the exponent was interpreted to follow from the domi- dt (cid:18) 2D (cid:19) l2 −(cid:18) 2D (cid:19)R 0
g
nant motion of grain boundaries through a background 1/2
1 F
ofcurvedstripes. Indisorderedconfigurations,thecurva- + ξ , (29)
tureofstripesissetbyadistributionoflargelyimmobile √2D (cid:18) l (cid:19)
+1/2disclinations. AccordingtoEq. (12),themotionof
wherewehaveassumedthat,priortopinning,thevarious
grain boundaries is driven by stripe curvature, and acts
length scales remain approximately proportionalto each
to reduce the overall curvature by replacing regions of
other. Recall from Eq. (27) that F = F is required to
curved stripes by straight ones of a different orientation. 0
unpin a configuration obtained in the absence of noise,
It also reduces the disclination density whenever their
for which l(t) R . According to Eq. (29), coarsening
core region is swept by a moving grain boundary. In the ∼ g
proceedsifF >F untilanewcharacteristicpinningsize
limit ǫ 1 we computed several measures of the lin- 0
ear scale≪, including moments of P(κ,t), moments of the is reached given by F0lF/(FRg)=1 or
structure factor of the order parameter, and the average
F eα/√ǫ
| |
distancebetweendefects. Theywereallfoundtobecome l =R k F . (30)
F g F ∼ 0 ǫ2
proportional to each other, and to grow as a power law 0
of time with an exponent 1/3. After reaching the scale l , domains are expected to
F
Grain boundary motion as described in Section II was coarsenveryslowlybythermalactivation. Whenagrain
used to provide an interpretation for the value z = 3. boundary overcomes one pinning barrier, the linear ex-
Since +1/2 disclinations generate roughly axisymmet- tent of the corresponding domain typically increases by
ric patterns of stripes around them, the characteris- an amount of order λ /2. Hence dl/dt λ r, where r is
0 0
tic stripe curvature in any given configuration is pro- givenbyEq. (27)withR replacedby∼l. Hencedomains
gb
portional to the inverse characteristic distance between areexpectedtogrowlogarithmicallyintimeaccordingto
disclinations. Under the self-similarity hypothesis, the
distance between disclinations is proportional to l(t), l(t) F ln(t/F) for l l . (31)
F
∼ ≫
hence κ 1/l(t). If grain boundaries are the class of
∼
defect the motion of which controls asymptotic coars- A numerical solution of Eq. (18) yields results qual-
ening, then the coarsening exponent can be inferred by itatively consistent with those presented above. Figure
dimensional analysis of Eq. (12). In the limit ǫ 0, the 5a shows a configuration of the order parameter field ψ
→
oscillatorytermintherighthandsideofEq. (12)canbe obtained for F = 0 and ǫ = 0.4 starting from random
neglected and we simply have dl/dt l 2 or l(t) t1/3, initial conditions. The configuration shown corresponds
−
in agreement with the numerical solu∝tion of Eq. (∼1). to very late times t = 2.3 105 at which point all de-
×
Equation (12) shows that this result changes qualita- fects are practically immobile, and domain growth has
tively further from onset. As ǫ increases the pinning po- stopped. We then set F = 0.00318, and the integration
tential energy barrier p(ǫ) increases extremely fast, and is continued. The order parameter configuration t=105
important corrections to scaling are to be expected. For timeunitslaterisshowninFig. 5b. Theaveragedomain
finite ǫ and short times many defects are present, there- size has increased substantially. Many grain boundaries
fore the characteristic curvature of the stripes is very havea90◦orientation(likeinFig. 5a),androugheningis
large, and the first term in the right hand side of Eq. limited or nonexistent. We have determined the average
(12)dominates. Ascoarseningproceeds,the characteris- domain size l from the probability distribution function
tic curvature decreases until it reaches the critical value of the quantity ζ = ψ2 + (~ψ)2/k2. Figure 6 shows
∇ 0
κ given by Eq. (16). At that point the typical velocity the probability distribution function corresponding to a
g
ofagrainboundaryvanishes,althoughthesystemisstill perfectly ordered configuration, as well as to partially
disordered. Therefore one would expect that coarsening disordered configurations. The inverse linear scale 1/l,
would stop when l(t) is of the order of R . This is pre- proportional to the defect density ρ , is extracted from
g d
ciselytheresultsshowninFig. 4withonlyoneadjustable the difference between these curves, as detailed in Ap-
parameter (a scale factor relating R given by Eq. (16) pendix B. As shown in Fig. 7 domain growth is very
g
to l(t) determined numerically from the distribution of slow, possibly logarithmic, although a precise check of
stripe curvatures). this behavior is problematic.
When random fluctuations are considered (F > 0 in Figure 8 displays the evolution of the defect density
Eqs. (18)-(19)),someofthe grainboundariesinafrozen ρ (t) as a function of time, starting from random ini-
d
configuration are expected to resume motion. We argue tial configurations. For reference we also show the case
thatthestructurewillcontinuecoarseninguntiltheaver- F =0. Increasingthevalueofǫleadstosmallereffective
agedomainsizereachesanewcharacteristicsizel >R exponents, whereas increasing F has the opposite effect.
F g
thatcanbeestimatedasfollows. Wewriteageneralphe- For sufficiently small ǫ, we find z = 3 independent of
nomenologicalevolutionequationforthedomainsizel(t) the value of F. The two bottom curves correspond to
7
systems that are close enoughto onset, and hence either state. Instead, they reach disordered metastable con-
R or l is very large compared with the linear size of figurations in which topological defects, mainly grain
g F
the system. We show our results for ǫ = 0.04 (averaged boundaries and disclinations, fail to annihilate and re-
over 40 independent runs) and for ǫ = 0.15 (averaged main with finite density. It appears that the formation
over15independentruns). Thesolidlineclosesttothese oftheseglassyconfigurationsinaquencheddisorder-free
twocurveshasaslopeof 1/3. Thedownwarddeviation system can be accounted for by the finite separation be-
−
fromlinearityatlongtimesatǫ=0.04isatypicalmani- tween “fast” length scales of the structure (associated
festationoffinitesizeeffects(thislongtimebehaviorand with stripe periodicity), and “slow” scales (associated
its dependence on the system size was studied in detail withtheextentofdefectenvelopes). Sinceatafinitedis-
in ref. [15]). tance from threshold the ratio between these two scales
With increasing ǫ and/or decreasing F, pinning be- is finite, non-adiabatic effects lead systematically to de-
comes more pronounced as evidenced the lower effective fect pinning in an infinite system. Fluctuations allow
slopes of the three upper curves in Fig. 8 (the results unpinning and a certain amount of “crystallization”, al-
areaveragesoversixindependentruns,eachcurvecorre- beit through an asymptotically slow activated motion of
sponding to the same value ǫ=0.4). The top curve cor- grain boundaries and other defects.
responds to a system without fluctuations for which ρ The present framework is far too simple to be used in
d
wascomputedwiththemethoddescribedinref.[15]. The thepredictionofaglasstransitiontemperature,ifsucha
density starts decaying roughly as an inverse power law, transitionexists. Insome respects, the situationjust de-
with an effective exponent much smaller than 1/3 (the scribed is instead very similar to that of domain growth
−
top solid line has a slope of 1/5), and after a crossover in random fields in dimension larger than two [44,45].
−
saturates at long times indicating pinning. When small There, domain walls separating magnetized domains are
amplitude noise is added (curve below denoted by dia- pinned by fixed impurities and evolve by thermal acti-
monds), the initial behavior is similar to that of F = 0, vation to other more favorable configurations. The phe-
and the decay rate also slows down considerably at long nomenologicalpinningenergyofadomainofsizeRgrows
times (where we would predict logarithmic growth, i.e. asΥRθ,wereθdependsontheproblemconsidered,yield-
ρ 1/ln(t)). The curve below, denoted by plus signs, ing an escape rate given by r exp( ΥRθ/k T), equa-
d B
∼ ∼ −
corresponds to a noise intensity three times larger than tion that is formally analogous to Eq. (27) with θ = 1.
the previous case. Its initial decay is slightly faster (it Twocrucialdifferencesarethatoursystemisglassyeven
can be fitted with an effective exponent 1/z 0.23 in two dimensions,and that defects do not need any dis-
eff
≃ −
as shown with the solid line in the figure), and the up- order to become pinned.
wardsdeviationsatlongtimes arelesspronounced. This The consequences of defect pinning on the interme-
behaviorisinqualitativeagreementwiththetheexpecta- diate time regime corresponding to domain coarsening
tion that defects overcome pinning barriersmore readily have also been investigated. A universal coarsening ex-
at higher noise intensities and pinning is postponed to ponent can be determined close to thresholdonly, where
longer times when l l . However, the effective ini- we obtain z = 3. Coarsening stops when the linear
F
tial decay is slower th∼an t 1/3 which we interpret as a size of the system is larger than the characteristic do-
−
crossovereffect resulting from non-adiabaticity. main size for pinning. In this situation, an intermediate
In summary, our results for large ǫ are in agreement crossoverregimeisanticipatedwithlowereffectivecoars-
with earlier numerical results performed at ǫ = 0.25, ening exponents, as is observedin numerical solutions of
showing that coarsening laws are very slow and depend the model. Crossover effects induced by pinning can be
on the presence of thermal fluctuations. We argue here reduced by either increasing the intensity of the fluctua-
that a coarsening exponent can be properly determined tions or approaching threshold.
onlyinthesmallǫ limit,wherethe phaseorderingkinet- We note that some of our conclusions as well as our
ics is self-similar. Our results support that the exponent interpretation of the numerical results are based on the
z = 3 is independent of F for sufficiently small ǫ, when analysis of a particular type of defect, namely a grain
pinning effects are negligible (R much larger than the boundary separating two domains with differently ori-
g
linear size of the system). ented stripes. We think it is likely that similar non-
adiabaticcorrectionstodefectmotionwillappearfordis-
location glide or disclination motion, leading to similar
V. CONCLUSIONS non perturbative corrections in ǫ to the speed of the de-
fect.
We have shown that the Swift-Hohenberg model of We believe more generally that pinning through non-
Rayleigh-B´enardconvectionexhibits glassypropertiesin adiabaticeffectsislikelytobeafeatureofawidevariety
spatially extended systems. In the absence of fluctua- ofpatternformingsystems,andisnotlimitedtothepar-
tions, and following a parameter quench across thresh- ticular model treated here. Block copolymer melts, for
old,randominitialconfigurationsdonotevolveintocom- instance, provide aninteresting case in whichthe results
pletely ordered states, a single plane-wave or crystalline obtainedcouldhavepracticalimplications(seeAppendix
A for a summary of the relevant equations and their re-
8
1 ∂ψ 1
lationship with the model studied here). We also men- = 2[ (2+ǫ)ψ+ψ3 2ψ] k2(ψ ψ ),
tion here that results qualitatively similar to ours have M ∂t ∇ − − k2∇ − 0 − ∞
0
been reported for a model with competing interactions
(A3)
(describing ferromagnetic films), that is defined by the
equations of Appendix A with a different form of the
where ψ is the boundary condition at infinity. In most
Green’s function G [46]. There, frozen polycrystalline studies,∞it is customary to set ψ = ψ , the spatial av-
configurations of stripe patterns were observed for deep erage of ψ over the sample. We∞introdhuice the amplitude
quenches as well, whereas the system could reach an or-
A of slightly modulated waves through,
dered state for shallow quenches. This same model was
alsoable to predict the formationofa frozenphase com- 1
ψ(~r,t)= [A(~r,t) eik0x+ c.c.]. (A4)
posed ofpolydisperse droplets with a near-hexagonalar- 2
rangement[47],aspreviouslyobservedinexperimentson
a Langmuir monolayer[48]. However,the pinning mech- A multi-scale analysis of Eq. (A3) in the limit ǫ 1
anisminvolvedinthislastcaseisprobablydifferentthan was conducted by Shiwa [51]. Setting M = 1/k02,≪the
the one discussedinthe presentpaper sincethe patterns resulting equation for the amplitude is
arenolongerlocallyperiodic. Nevertheless,wewouldex-
∂A 4 i 2 3
pect that our main conclusions can be readily extended =ǫA+ ∂ ∂2 A A2A , (A5)
to other systems with periodic structures such as hexag- ∂t k02 (cid:18) x− 2k0 y(cid:19) − 4| |
onal patterns [47,49].
whichisidenticaltotheamplitudeequationoftheSwift-
Hohenberg model. Note that the only effect of the con-
ACKNOWLEDGMENTS servation law on the local part of the free energy (the
Laplacian operator in front of the square bracket in Eq.
(A3))is arenormalizationofthe mobilityM. The quan-
This researchhas been supported by the U.S. Depart-
tities ǫ and k defined above play the same role as the
ment of Energy, contract No. DE-FG05-95ER14566. 0
same coefficients in the Swift-Hohenberg model (i.e. the
dimensionless distance to threshold and the dominant
wavenumber of the structure, respectively.)
APPENDIX A: MEAN-FIELD MODEL OF A
SYMMETRIC BLOCK COPOLYMER MELT
APPENDIX B: CALCULATION OF THE DEFECT
We briefly recallinthis appendix knownresultsabout
DENSITY IN THE PRESENCE OF
the relationship between the mean field description of a
FLUCTUATIONS
block copolymer melt, and the amplitude equation for
Swift-Hohenberg model(Eq. (1))atfirstorderin ǫ. The
Computation of the domain size from the probability
dynamics of micro-phase separationof block copolymers
distributionofstripecurvatureisdelicateinthepresence
is often modeled by a time-dependent Ginzburg-Landau
ofnoise. Wehaveusedadifferentmethodthanthatused
equation for a conserved order parameter [35,50],
for F = 0. We introduce an effective squared amplitude
ζ(~r,t) by
∂ψ(~r,t) δF
=M 2 , (A1)
∂t ∇ δψ(~r,t)
ζ =ψ2+(~ψ)2/k2 . (B1)
∇ 0
where
Foraperfectlyorderedsystemconsistingofaplane-wave
r u K solution of the Swift-Hohenberg equation and F = 0,
F = d~r ψ2+ ψ4+ ( ψ)2
Z (cid:18)−2 4 2 ∇ (cid:19) the probability distribution function of ζ is a delta func-
tion at ζ = 4ǫ/3. When F > 0 the probability dis-
B
+ d~rd~r ′ψ(~r,t)G(~r,~r ′)ψ(~r ′,t). (A2) tribution∞function of ζ even for a plane wave p(F)(ζ) is
2 Z Z
∞
broader because of “phonon” excitations. The function
G is the Green’s function of the Laplacian operator p(F) is plotted in Figure 6 (with dotted lines), and is
2G(~r,~r ′) = δ(~r ~r ′) and M a constant mobility th∞en used as a reference curve for a fixed F. In a par-
∇ − −
or Onsager coefficient. The scalar order parameter ψ tially disordered configuration, the presence of defects
is the local monomer concentration difference between and curved stripes further broadens the probability dis-
the two chemical species. Following Ref. [51] we set tribution function to p(F)(ζ,t) (solid lines of Figure 6).
r = 2+ ǫ,u = 1,K = 1/k2 and B = k2. Two inde- The difference between the two curves is related to the
0 0
pendent parameters k0 and (small) ǫ remain. Equation degree or disorder beyond small fluctuations away from
(A1) reduces to a perfectly ordered structure. We define the defect den-
sity by ρ (t) = Max p(F)(ζ), ζ Max p(F)(ζ,t), ζ .
d
{ ∞ }− { }
9
Since grain boundaries are seen to be the major contri- [27] M. Grousson, G. Tarjus, and P. Viot, Phys. Rev. Lett.
butiontodefectdensity,onecanintroduceacharacteris- 86, 3455 (2001).
tic length scale ρ−d1, which is further identified with the [28] J.D.Shore,M.Holzer,andJ.P.Sethna,Phys.Rev.B46,
characteristic size, or domain size l. In Figure 7, l has 11376 (1992).
been normalized so that l(t=0)=R (at the beginning [29] M.E.J. Newman and C. Moore, Phys. Rev. E 60, 5068
g
of the heating process) where R is computed from the (1999).
g
probability distribution function of stripe curvatures at [30] C. Harrison et al., Science290, 1558 (2000).
F =0. [31] L. Purvis and M. Dennin, Phys. Rev. Lett. 86, 5898
(2001).
[32] J. Gunton, M. San Miguel, and P. Sahni, in Kinetics
of first order phase transitions, Vol. 8 of Phase Transi-
tions and Critical Phenomena, edited by C. Domb and
J. Lebowitz (Academic, London,1983).
[33] A. Bray, Adv.Phys. 43, 357 (1994).
[1] K.Kawasaki, Prog. Theor. Phys. Suppl. 79, 161 (1984). [34] J. Swift and P.C. Hohenberg, Phys. Rev. A 15, 319
[2] H.Brand and K.Kawasaki, J.Phys.A17, L905 (1984). (1977).
[3] G. Mazenko, in Formation and interaction of topologi- [35] L. Leibler, Macromolecules 13, 1602 (1980).
cal defects, edited by A.-C. Davisand R. Brandenberger [36] D.BoyerandJ.Vin˜als,Phys.Rev.E63,061704(2001).
[37] G. Ahlers, M. Cross, P. Hohenberg, and S.Safran, J.
(Plenum,New York,1995).
[4] A. Bray, in Formation and interaction of topological Fluid Mech. 110, 297 (1981).
defects, edited by A.-C. Davis and R. Brandenberger [38] P.C. Hohenberg and J.B. Swift, Phys. Rev. E 52, 1828
(1995).
(Plenum,New York,1995).
[39] J. Toner and D.R.Nelson, Phys.Rev.B 23, 316 (1981).
[5] I.Chuang,B.Yurke,A.N.Pargellis,andN.Turok,Phys.
Rev.E 47, 3343 (1993). [40] R. Graham, Phys.Rev.A 10, 1762 (1974).
[6] H.Toyoki, Phys. Rev.E 47, 2558 (1993). [41] F. Drolet and J. Vin˜als, Phys.Rev.E 57, 5036 (1998).
[42] F.DroletandJ.Vin˜als,Phys.Rev.E64,026120(2001).
[7] M. Zapotocky,P.M. Goldbart, and N. Goldenfeld, Phys.
Rev.E 51, 1216 (1995). [43] F can be estimated to be of the order F = uǫ5/4, with
u 1. See also Ref. [40].
[8] N.Mason,A.N.Pargelis, andB.Yurke,Phys.Rev.Lett.
≤
70, 190 (1993). [44] T. Natterman and J. Villain, Phase Transitions 11, 5
[9] Y. Oono and M. Bahiana, Phys. Rev. Lett. 61, 1109 (1988).
[45] J.-P. Bouchaud, L. Cugliandolo, J. Kurchan, and M.
(1988).
Mezard, in Spin Glasses and Random Fields(WorldSci-
[10] K.R. Elder, J. Vin˜als, and M. Grant, Phys. Rev. Lett.
68, 3024 (1992). entific, Singapore, 1998), p. 161.
[11] K.R. Elder, J. Vin˜als, and M. Grant, Phys. Rev. A 46, [46] C. Sagui and R.C. Desai, Phys. Rev.E 49, 2225 (1994).
[47] C. Sagui and R.C. Desai, Phys. Rev.E 52, 2822 (1995).
7618 (1992).
[12] M.C. Cross and D.I. Meiron, Phys. Rev. Lett. 75, 2152 [48] M. Seul, Europhys.Lett. 28, 557 (1994).
[49] K.R. Elder, M. Katakowski, M. Haataja, and M. Grant,
(1995).
[13] Q.Hou,S.Sasa, andN.Goldenfeld, PhysicaA239,219 cond-mat/0107381.
[50] T. Ohta and K. Kawasaki, Macromolecules 19, 2621
(1997).
[14] J.J. Christensen and A.J. Bray, Phys. Rev. E 58, 5364 (1986).
[51] Y. Shiwa, Physics Letters A 228, 279 (1997).
(1998).
[15] D.BoyerandJ.Vin˜als, Phys.Rev.E64,050101 (2001).
[16] P. Manneville, Dissipative Structures and Weak Turbu-
lence (Academic, New York,1990).
[17] M. Cross and P. Hohenberg, Rev. Mod. Phys. 65, 851
(1993).
[18] E.D. Siggia and A. Zippelius, Phys. Rev. A 24, 1036
(1981).
[19] P. Manneville and Y. Pomeau, Phil. Mag. A 48, 607
(1983).
[20] G. Tesauro and M. Cross, Phil. Mag. A 56, 703 (1987).
[21] Y.Pomeau, Physica D 23, 3 (1986).
[22] D.Bensimon,B.Shraiman,andV.Croquette,Phys.Rev.
A 38, 5461 (1988).
[23] B.A. Malomed, A.A. Nepomnyashchy, and M.I. Tribel-
sky,Phys.Rev.A 42, 7244 (1990).
[24] M.-H.Julien et al.,Phys. Rev.Lett. 83, 604 (1999).
[25] D.Kivelson et al., Physica A 219, 27 (1995).
[26] J.SchmalianandP.G.Wolynes,Phys.Rev.Lett.85,836
(2000).
10
