Coarsening versus pattern formation

0.26 MB·English

by  A. A. Nepomnyashchy

Comptes
Rendus
Physique
ComptesRendusPhysique00(2015)1–14
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Coarsening versus pattern formation
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2

r
a A.A.Nepomnyashchy
M
DepartmentofMathematics,Technion-IsraelInstituteofTechnology,Haifa32000,Israel

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]
h
c
e Abstract
m
It is known that similar physical systems can reveal two quite different ways of behavior, either coarsening, which creates a
- uniformstateoralarge-scalestructure,orformationoforderedordisorderedpatterns,whichareneverhomogenized. Wepresent
t
a adescriptionofcoarseningusingsimplebasicmodels,theAllen-CahnequationandtheCahn-Hilliardequation,anddiscussthe
st factorsthatmayslowdownandarresttheprocessofcoarsening. Amongthemarepinningofdomainwallsoninhomogeneities,
. oscillatorytailsofdomainwalls,nonlocalinteractions,andothers.Coarseningofpatterndomainsisalsodiscussed.
t
a
m
c 2015PublishedbyElsevierLtd.
(cid:13)
-
d Keywords:
n coarsening,patternformation,domainwalls
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c
[ 1. Introduction

2
Formanydecades, thenonlineardevelopmentofinstabilitiesin physicalsystemswas anobjectof extensivein-
v
vestigations. Themostspectacularconsequencesofinstabilitiesaretheappearanceoforderedspatiallynon-uniform
4
1 structures(patternformation,see [1]- [3])orirregularmotions(spatio-temporalchaos[4]) underuniformexternal
5 conditions. However,thereisonemorescenarioofaninstabilitydevelopment: thatisagradualgrowthofthechar-
5 acteristicscalewithtime(coarsening)[5],[6]. Differentevolutionscenarioscantakeplaceinrathersimilarphysical
0
systems.
.
1 As an example, let us consider the phase separation in binary alloys that consist of two kinds of atoms, A and
0 B, with volumefractionsφ (x,t) and φ (x,t), respectively. There exists a temperatureT such thatfor T > T the
A B c c
5 componentsaremixed,i.e.,theorderparameterφ(x,t) = φ (x,t) φ (x,t)vanisheseverywhere,whileforT < T
1 A − B c
they are separated, i.e., there exist two thermodynamicallystable phases, one with φ > 0 (“A-rich phase”) and the
:
v other with φ < 0 (“B-rich phase”). A mathematical model of that phenomenon has been suggested by Cahn and
i
X Hilliard[7]. Underthesimplestassumptionofa constantmobility,thekineticsofthephaseseparationisdescribed
bythefollowingnon-dimensionalequation(theCahn-Hilliardequation),
r
a
φ = 2( φ+φ3 2φ). (1)
t
∇ − −∇
Theuniformphaseφ = 0isunstablewhiletheuniformphasesφ = 1arestable. Theinstabilityofthephaseφ = 0
±
createsamosaicofislandsofbothstablephases. Thesizeoftheseislands(domains)growsduetocoarsening,which
eventuallyleadstoacompleteseparationofstablephases[5],[6].
Adiblockcopolymer,whichconsistsofmonomersAandBwithreducedequallocaldensitiesφ andφ ,isquite
A B
similar to a binary alloy. The basic difference is the existence of a long-range interaction of monomers [8] - [10]
1
/ComptesRendusPhysique00(2015)1–14 2
whichprovidesanadditionaltermintheevolutionequationfortheorderparameter:
φ = 2( φ+φ3 2φ) Γφ, Γ>0, φ =0. (2)
t
∇ − −∇ − h i
Therearenootherspatiallyuniformstationarysolutionsexceptφ=0. Therefore,whenthelattersolutionisunstable
(atΓ < 1/4),atransitiontoanon-uniformstateisunavoidable[11]. AtsmallΓ,theinitialevolutionofdisturbances
issimilartothatintheCahn-Hilliardequation,butitisstoppedwhenstripeswitha definitepatternwavelengthare
created[12].
Theexistenceoflong-wavelinearinstabilityandmultiplehomogeneousstatesdoesnotguaranteethecreationof
spatiallyuniformdomainsthroughcoarsening.Asanexample,letusdiscussthenonlineardynamicsgovernedbythe
one-dimensionalKuramoto-Sivashinskyequation,
φ = φ φ (φ2) , φ =0, (3)
t xx xxxx x
− − − h i
whichisusedforthedescriptionofinstabilitiesinreaction-diffusionsystems[13], [14], instabilitiesofflamefronts
[15],andfilmflowinstabilities[16].Inthatcase,thelinearpartoftheequationisidenticaltothatoftheCahn-Hilliard
equation,andanyconstantsolution,φ = φ ,satisfiesequation(3). However,allsolutionscorrespondingtouniform
0
statesareunstable. TheKuramoto-Sivashinskyequationisaparadigmaticmodelofthespatio-temporalchaos[14];
stableperiodicpatternsarealsopossible[16],[17],buttheattractiondomainofthatregimeissmall. Generally,the
wayoftheinstabilitydevelopmentdependssignificantlyonthedetailsofthesystemnonlinearityandsymmetry[18],
[19].
Notethatpatternformationandcoarseningarenotincompatiblephenomena. Letusreturntomodel(2)thatde-
scribesformationofstripes. Becauseoftherotationalinvarianceoftheproblem,theorientationofstripesisarbitrary.
Initially,adisorderedsystemofstripesisdevelopedfromrandominitialconditions,andthenthemeansizeofordered
domainsgrowswith time, i.e., domaincoarseningtakesplace for differentlyorientedstripe patternsratherthan for
differentuniformphases[11],[20].
Onecanseethattheinterplaybetweencoarseningandpatternformationisnontrivial,anditisthesubjectofthe
present chapter. Let us emphasize that here we discuss only dynamic models of coarsening and pattern formation,
whichdonotincludeanykindofnoise.Thephenomenacausedbythermalfluctuationsareconsideredinotherpapers
ofthepresentissueandinthecomprehensivebook[21],wherethereadercanfindmanyadditionalreferencesonthat
subject.
2. Coarseninginonedimension: dynamicsofdomainwalls
Whenconsideringthekineticsofcoarsening,onehastotakeintoaccountthefollowingbasicfactors.
1. The existence of a Lyapunov functional. If the temporal evolution of an n-component order parameter φ,
i
i=1,...,nisdescribedbyagradientevolutionequation
n δF
φ = D , i=1,...,n, (4)
i,t − ikδφ
Xk=1 k
whereF = L(φ,φ ,...)dxistheLyapunovfunctionalofthesystem,andD isapositivedefinitematrix,then
x ij
R
δF δF δF
F = φ˙ dx= D dx 0. (5)
t Xi Z iδφi −Xi,k Z ikδφiδφk ≤
In the case of an equilibriumphase transition, the existence of the Lyapunovfunctional(free energy)is the conse-
quenceofthethermodynamics. Thenonlineardevelopmentofinstabilitiesinsystemsfarfromequilibriuminsome
casesisalsogoverned,atleastapproximately,bypotentialsystemsofequationsthatpossessLyapunovfunctionals.
2. Theexistenceofaconservationlaw. Inthecasewheretheorderparameterisadensityofaconservedquantity
(e.g.,thenumberofmolecules),theevolutionequationshaveadivergenceform,
φ = J, i=1,...,N, (6)
i,t i
−∇·
wherethefluxJ isafunctionoftheorderparameteranditsderivatives.
i
Inordertounderstandhowbothfactorsinfluencethecoarseningkinetics,letusconsideranumberofexamples.
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/ComptesRendusPhysique00(2015)1–14 3
2.1. Non-conservedorderparameter
2.1.1. Allen-Cahnequation
WestartwiththeAllen-Cahnequation[22]
φ =φ +φ φ3, (7)
t xx
−
which describes a phase transition in the absence of a conservation law for the order parameter φ. The physical
interpretationofthatmodelcanbeasfollows[6]: φisthespontaneousmagnetizationdirectedalongthedefiniteaxis
(duetothecrystalanisotropy).Also,theAllen-Cahnequationisthesimplestexampleofanorderparameterequation
for a pattern forming system far from the thermodynamicequilibrium. Let us consider the onset of convection in
ahorizontalcylinderheatedfrombelow. Abovethe instabilitythreshold,therotationof theliquidinthe transverse
sectionofthecylindercanbeeithercounterclockwise(φ > 0)orclockwise(φ < 0). Thetemporalevolutionofthe
order parameter φ(x,t) (x is the coordinate along the axis of the cylinder) is governedby the Allen-Cahn equation
[23].
Equation(7)hasaLyapunovfunctional,
1 1
F(t)= L(x,t)dx, L(x,t)= φ2+ (φ2 1)2 0. (8)
Z 2 x 4 − ≥
Itsderivative
∂L ∂L
F˙ = φ + φ dx= ( φ+φ3 φ )φdx= dxφ2 0 (9)
Z ∂φ t ∂φx xt! Z − − xx t −Z t ≤
isnon-positive,hencetheLyapunovfunctionaldecreasesmonotonicallywithtimeuntilastationarystateisreached.
It is obvious that equation (7) has three fixed points corresponding to uniform phases, φ = 0 (unstable paramag-
neticphase)andφ = 1(ferromagneticphaseswithoppositeorientationsofthemagnetization);thelattersolutions
correspondtotheabso±luteminimaoftheLyapunovfunctional,F =0. Theseparatrices
x ξ
φ = tanh − , ξ =const (10)
± ± √2
(kinkandantikink)describedomainwallsseparatingsemi-infinitedomainswithdifferentsignsoftheorderparameter.
Notethat
φ 1 2exp[ (x ξ)√2] asx . (11)
± ∼±{ − ∓ − } →±∞
ThecontributionofadomainwalltotheLyapunovfunctionalisF =2√2/3>0.
0
Otherstationarysolutionsof(7)describespatiallyperiodicpatternswhichcanbeconsideredasperiodicarrays
ofdomainswithalternatingsignsofφ. TheyareexpressedthroughtheellipticJacobifunction.
Letusimposeaninfinitesimaldisturbanceφˆ(x,t)onthestationarysolutionφ(x)andconsideritstemporalevolution
governedbythelinearizedproblem,
φˆ =φˆ +(1 3φ2(x))φˆ, φˆ < asx . (12)
t xx
− | | ∞ →±∞
Fornormaldisturbancesφ˜(x)exp(σt)weobtainaneigenvalueproblem,
σφ˜ =φ˜ +(1 3φ2)φ˜, φ˜( < . (13)
′′
− | ±∞| ∞
It is obviousthat solution φ = 0 is unstable: σ(k˜) = 1 k˜2 for φ˜ = eik˜x, and the solutions φ = 1 are stable:
σ(k˜) = 2 k˜2 for the same φ˜. The domain wall solution−s (10) are neutrally stable. One can sho±w that all the
− −
spatially-periodicsolutionsmentionedaboveareunstable.
Letusdiscussnowthetemporalevolutionofthesystem, whentheinitialstate istheunstablephaseφ = 0with
a certain initial, spatialdisordered,perturbation. Accordingto the stability analysis presentedabove,the final state
shouldbeuniformorcontainasingledefect,adomainwall. However,itisclearthatthedecompositionofthephase
φ = 0 can produce numerous domains with alternating signs of φ. Such a state can be characterized by a certain
density of defects decreasing with time, or by a mean domain length which grows with time. At late stages of the
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/ComptesRendusPhysique00(2015)1–14 4
evolution,whenthetypicaldistancebetweendomainwallsislargetheanalysiscanbedonebymeansofasymptotic
methods[24],[25]. Oneconsidersasetofdomainwalls(10)ofalternatingsignscenteredatξ = ξ(t),i = 1,2,...,
i i
andslowlymovingbecauseoftheirinteraction. Bymeansofasymptoticexpansions,theoriginalnonlinearproblem
istransformedintoaninfinitesystemofinhomogeneouslinearequations. Theirsolvabilityconditionsdeterminethe
followingequationsofmotionforthecentersofdomainwalls[26]:
2√2 ∂U
ξ˙ = , (14)
3 i −∂ξ
i
where
U = W(ξ ξ ), W(ξ ξ )= 8√2exp √2(ξ ξ ) . (15)
i i 1 i i 1 i i 1
Xi − − − − − h− − − i
Thus, the domain walls attract to each other according to the exponential interaction law (14) which reflects the
exponentialdomainwallasymptotics(11).
System(14)hasafamilyofstationarysolutions
ξ =a+ jl, j=0, 1, 2,... (16)
j
± ±
corresponding to periodic patterns with the spatial period 2l. The attractive interaction makes all these solutions
unstable.
Considertheinteractionofapairofdomainwalls. Settingl(t) = ξ (t) ξ (t),wefindthatthedistancebetween
2 1
−
domainwallsisgovernedbytheequation
2√2
l(t)= 32e l√2. (17)
′ −
3 −
Ifl(0)=l 1,thesolutionis
0
≫ 1
l=l0+ ln 1 48e−√2l0t . (18)
√2 (cid:18) − (cid:19)
ThedistancebetweentwodomainwallsbecomesofO(1)at
1
t t0 = e√2l0 +O(1). (19)
∼ 48
Finally,thedomainwallsreachthedistanceoforderO(1)andannihilate. Duringthetimeintervalt,onlythedomain
wallswiththeoriginalseparationgreaterthan ln(48t)/√2cansurvive. Thus,alogarithmiccoarseningtakesplace.
∼
InalargebutfinitesystemwiththelengthL,thenumberofdomainwallsN(t) L/l(t).
≤
Asanexampleofthesituationwhentwolocallystablephasesareenergeticallynon-equivalent,letusconsidera
systemwiththeLyapunovfunctional
1 1
F[φ(x)]= φ2+ (1 φ2)2 hφ dx (20)
Z "2 x 4 − − #
whichcorrespondstothedynamicequation
φ =φ +φ φ3+h. (21)
t xx
−
Inthecaseofamagneticsystem,thelasttermintheexpression(20)describestheinfluenceofanexternalmagnetic
field,whichmakestheorientationofthemagnetizationinthedirectionofthefieldpreferable.Theuniformstationary
statessatisfytheequation
φ φ3+h=0. (22)
−
In the interval h < h < h , equation (22) has three solutions: stable solution φ = φ+ > 1/√3, another stable
solutionφ=φ −<∗ 1/√3,an∗danintermediateunstablesolutionφ , 1/√3<φ <1/√3. Forh>0,thephasewith
0 0
φ=φ+ >0iss−tabl−e,andthephasewithφ=φ ismetastable. −
−
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Ifhissmall,domainwallsaredescribedbyformulas(10)attheleadingorder.Bymeansofanasymptoticanalysis,
onecanfindthatthemotionofeachdomainwallisgovernedbytheequation[26]
2√2dξ
= 2h (23)
3 dt ∓
(theuppersignisforakinkandthelowersignisforanantikink). Thismotioncreatesacoarseningprocess,which
issignificantlyfasterthanthatinthecaseofenergeticallyequivalentphases. Thatprocessleadstotheeliminationof
themetastablephase.
2.1.2. FractionalAllen-Cahnequation
Thelogarithmiclawoftheone-dimensionalcoarseningiscausedbytheexponentiallyweakinteraction,duetothe
exponentialasymptotics(11)characteristicforsolutionsofpartialdifferentialequations.However,thebasicequations
governingthenaturalphenomenaareoftenintegro-differentialequationsratherthanpartialdifferentialequations.For
instance,thefreeenergydensityofafluiddependsonthefluiddensityinanonlocalway[27]. Theconventionalvan
derWaals’expressionforthefluidfreeenergywhichcontainsasquareddensitygradient[28]correspondstoacertain
asymptoticlimitofthe basicnonlocalexpression. The asymptoticsofa domainwall (i.e., thegas-liquidboundary)
in the framework of local and nonlocal models are strongly different: while a local model predicts an exponential
decay,anonlocalmodelsuggestsapower-lawdecay[27]. Frontsthatcannotbegovernedbylocalpartialdifferential
equations have been found also in studies of transitions with long range interactions [29], vacancy diffusion and
domaingrowthinbinaryalloys[30], andorderingkineticonfractalstructures[31](forthelattersubject,see[32]).
Asanexampleletusdiscussthefrontpropagationinsystemswithsuperdiffusion[33]. Whilethenormaldiffusionis
associatedwithaGaussiannon-correlatedrandomwalkofparticles,thesuperdiffusionisobservedinnon-equilibrium
systemswithanalgebraicallydecayingjumplengthdistribution,wherethecentrallimittheoremisnovalid. Among
theexamplesarewaveturbulence[34],transportinporousmedia[35],andforagetrajectoriesofanimals[36]. One
canuseasuperdiffusivegeneralizationoftheAllen-Cahnequation,
φ = Dγ φ+φ φ3, 1<γ<2. (24)
t x −
| |
HereDγ denotesthefractionalRieszderivative,whichcanbedefinedbyitsactionintheFourierspace:
x
| |
F Dγ φ(x) (k)= kγF(φ(x))(k), (25)
x −
(cid:16) | | (cid:17)
whereFisthesymboloftheFouriertransform.Equation(24)hasaLyapunovfunctional[37].Inacontradistinctionto
theexponentialasymptotics(11)ofthedomainwallsinthelocalPDE(7),thedomainwallsintheintegro-differential
equation(24)hasanalgebraictail,
sec(πγ/2)
φ+ ∼1− 2Γ(2 γ)x−γ, x≫1. (26)
−
Thatleadstoapowerlawforthekink-antikinkattraction,
l(t)= Cl γ (27)
′ −
−
(cf.(17)),andforthetemporaldecayofthenumberofdomainwallsonafinitespatialinterval,N t 1/γ [37].
−
∼
2.1.3. Non-potentialsystems
For systems far from equilibrium, e.g., in the case of longwave instabilities of flows, the Lyapunov functional
generallydoes not exist. Nevertheless, coarseningmay take place if the domain walls are described by monotonic
functions.Asanexample,letusmentiontheamplitudeequationthatgovernsthefixed-fluxconvectioninatiltedslot
[38]:
φ =φ +φ φ3+2αφφ , α>0. (28)
t xx x
−
Becausethesymmetryx xisviolated,thekinkandantikinkdomainwallshavedifferentwidths:
→−
1
φ (x)=tanhβ (x ξ), β = (α √α2+2). (29)
± ± − ± 2 ±
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/ComptesRendusPhysique00(2015)1–14 6
Theinteractionofdomainwallsisattractivebutasymmetric.Forapairwhichconsistsofakinkwiththecenterinthe
pointξ (t)andanantikinkwiththecenterinthepointξ (t),theequationsofmotionlookas
1 2
ξ1′ = F+exp(−2|β−|(ξ2−ξ1)), ξ2′ = F−exp(−2β+(ξ2−ξ1)), (30)
where F+ , F . Alltheperiodicstationarysolutions(patterns)areunstable,andalogarithmicallyslowcoarsening
−
takesplace,similarlytothecaseoftheAllen-Cahndynamics.
2.2. Conservedorderparameter
2.2.1. Potentialsystems
LetusreturntotheCahn-Hilliardequation(1),whichcanbewrittenas
φ + j =0, j=(φ +φ φ3) ; l x l. (31)
t x xx x
− − ≤ ≤
Beingagenericnonlinearequationgoverninglongwaveinstabilitiesinthepresenceoftheconservationlaw[18],that
equationhasbeenrevealedinnumerousproblemsofdifferentphysicalnature,includingsecondaryflowsproducedby
theinstabilityoftheKolmogorovflow[39],Marangoniinstabilityofatwo-layersystemwithadeformableinterface
[40],nonlineardevelopmentofzigzaginstabilityofconvectionrolls[41],andevencoarseningofordereddomainsin
oscillatorypatternsgovernedbythe complexSwift-Hohenbergequation[42], which describesoscillationsin lasers
[43],[44]andopticalparametricoscillations[45]-[47].
Forsakeofsimplicity,applytheboundaryconditionsφ =φ =0atx= l;then
x xxx
±
d l
j( l)=0, φdx=0 (32)
± dtZ
l
−
(thetotallengthofdomainsofeachphaseisconserved).TheLyapunovfunctional(8)decreaseswithtime:
F = (φ +φ φ3)2dx 0. (33)
t xx
−Z − ≤
Usingtheboundaryconditions,onecanpresentequation(31)intheform[25]
1 l
∂ 2φ +φ +φ φ3+h(t)=0, ∂ 2φ(x,t)= x yφ(y,t)dy. (34)
−x t xx − −x t 2Z | − | t
l
−
Note that despite the energetical equivalence of phases φ = 1, an efficient field h(t) appears, which must to be
±
determinedself-consistently.Themotionofakinkandanantikinktowardseachotherwouldchangethetotallengths
ofdomainsofdifferentphases,andhenceitisnotpossible. Twokinkscanmovesimultaneouslytowardsanantikink
placedbetweenthem,orkink-antikinkpairscanmoveasawhole. Thecorrelatedmotionofndomainwallswiththe
centersatξ,i=1,...,n,sufficientlyfarfromeachother,isgovernedbythesystem[25]
i
n
− 2(−1)i−j|ξi−ξj|ξ′j =16 e−|ξi−ξj|√2sign(ξj−ξi)+2(−1)ih(t), i=1,...,n, (35)
Xj=1 Xj,i
supplementedbytheconservationlaw
n
( 1)iξ =0. (36)
− i′
Xi=1
Forakink-antikinkpair,theattractioniscompensatedbythefieldh = 8exp[ (ξ ξ)√2],hencethedomainwalls
2 i
aremotionless. Fora symmetrickink-antikink-kinktripletwithcoordinateso−fdom−ainwallsξ = l(t), ξ = 0and
1 2
ξ =l(t),oneobtainsh=0,l = 8exp( l√2),hencetheannihilationtimedependsonl =l(0)as −
3 ′ 0
− −
1
t0 l0el0√2
∼ 8√2
(cf.(19)).
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2.2.2. Non-potentialsystems
As an example of a non-potentialsystem with a conservation law let us consider the convective Cahn-Hilliard
equation,
D
φ +(φ +φ φ3) (φ2) =0, < x< , (37)
t xx xx x
− − 2 −∞ ∞
whichhasbeensuggestedtodescribeseveralphysicalprocesses,namelyspinodaldecompositionofphaseseparating
systems in an external field [53] - [55], step instability on a crystal surface [56], faceting of growing, thermody-
namically unstable surfaces [57] - [61], evolving nanofoams[62] as well as dewetting of a thin film flowing down
an inclined plane [63]. Thatequationprovides“a bridge”between the Cahn-Hilliard equation(1) (D = 0) and the
Kuramoto-Sivashinskyequation(3)(φ 2φ/D,D 1).
Stationarypatternsφ=φ(x)aredes→crib−edbythep≫roblem
Dφ2 DA
φ +(φ φ3) = , < x< ; A>0; x , φ < . (38)
′′′ ′
− − 2 − 2 −∞ ∞ →±∞ | | ∞
ForanyD,0,thesetofsolutionsofequation(38)isincomparablymorecomplexthanthatoftheusualCahn-Hilliard
equation.Onecaneasilyfindsomeexactsolutionsoftheproblem.Theconstantsolutions,
φ=φ = √A, (39)
± ±
correspondto two stable phases. For domainwalls, there existexact solutions[53], one for a kinkwith A = A+ =
1+D/√2,
φ0
φ=φ+(x)=φ0+tanh + (x ξ), φ0+ = 1+D/√2, ξ =const, (40)
√2 − q
andtheotherforanantikinkwithA=A =1 D/√2,D< √2,
− −
φ0
φ=φ (x)= φ0tanh − (x ξ), φ0 = 1 D/√2, ξ =const. (41)
− − − √2 − − q −
However,thesetofstationarysolutionsismuchmorerich. Specifically,solution(40)isjustonerepresentativeofa
familyofkinksφ+(x;A). Inadditiontothemonotonicantikink(41),thereexistsalsoadiscretesetofnon-monotonic
antikinksolutions[64](thatphenomenonistypicalformodelscontaininghigher-orderspatialderivatives,see [65]-
[67]). Thekink-antikinkpairisformedbyantikink(41)andarepresentativeofthefamilyofkinkswithA= A .
If0< D< D = √2/3.Inthatregion,thecoarseningisobserved[55],[58],[59],[61]. Becauseoftheasym−metry
0
betweenkinksand antikinks, a kink-antikinkpair movesspontaneouslywith a definite velocityv (D,L). The most
2
typicalprocessobservedbycoarseningistheannihilationofdomainwalltriplets,whentwokinksofthesamesign
approachwithvelocities v (D,L)thekinkoftheoppositesignsituatedbetweenthem.Exactexpressionsforv (D,L)
3 2
±
andv (D,L)canbefoundin[68]. InthelimitofsmallD[61],
3
v (D,L) v (D,L) (D2√2/4)exp( DL/2).
2 3
∼ ∼− −
Therefore,thecoarseninglawislogarithmic.
ForD> D ,thedomainwallshaveoscillatorytails. Thatcasewillbediscussedinthenextsection.
0
3. Factorshinderingcoarsening
Inthepresentsection,wediscusssometypicalsituationswherethesystemcannotreachauniformstateoranother
energeticallypreferredstatebycoarsening.
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3.1. Externalinhomogeneities
Themotionofdomainwallsleadingtoannihilationcanbestoppedbyinhomogeneityofthemedium. Recallthat
weconsiderthephenomenaintheabsenceofnoise.Coarseningininhomogeneoussystemsinthepresenceofthermal
fluctuationsisconsideredin[69].
Inapotentialsystem,thedomainwallwould“prefer”thelocationwhereitsenergywillbesmallerthaninother
locations.Forexample,letusconsiderthefollowingmodificationoftheone-dimensionalAllen-Cahnequation[23]:
φ =φ +[1+ǫf(x)]φ φ3. (42)
t xx
−
Attheleadingorderinǫ,theequationofmotionforadomainwallofanykindis
2√2dξ d 1
= V (ξ), whereV (ξ)= ∞ f(ξ+y)cosh 2(y)dy. (43)
3 dt −dξ ih ih 2Z −
−∞
Specifically,iftheinhomogeneityhasaδ-likeshape,
f(x)= 2V δ(x x ),
0
− − ∗
theinteractionpotentialis
x x
Vih(x0)= V0cosh−2 0− ∗.
− √2
Generally,theshapeofthepotentialisalineartransformationoftheinhomogeneityshape,accordingto(43).
If there are many domain walls and many inhomogeneities, the motion of domain walls is determined by the
systemofequations(14)withthepotential
U = W(ξ ξ )+ V (x),
i i 1 ih i
Xi − − Xi
where W(ξ ξ ) is determined by equation (15), and V (x) corresponds to (43). Thus, the problem of finding
i i 1 ih i
− −
stationarysolutionsof(42) isequivalenttofindingequilibriumconfigurationsof achainofparticlesin theexternal
potential(43),interactingaccordingtothelaw(15). Thismodelresemblesthewell-knownFrenkel–Kontorovamodel
(seee.g. [70]).
Asanexampleletusconsidertwodomainwallswithcoordinatesξ andξ whicharenearthedistantδ-shaped
1 2
attractinginhomogeneities:
f(x)= 2V δ(x x ) 2V δ(x x ), V >0, x > x .
0 1 0 2 0 2 1
− − ∗ − − ∗ ∗ ∗
x x =l 1, ξ x =O(1), ξ x =O(1). Theequationofmotionfortheleftdomainwallis:
2 1 1 1 2 2
∗− ∗ ∗ ≫ | − ∗| | − ∗|
2√2dξ ξ ξ ξ x
1 =16e−(ξ2−ξ1)√2 √2V0sinh 1− 1∗ cosh−3 1− 1∗. (44)
3 dt − √2 √2
Thefirsttermintheright-handsideofequationcanbeestimatedas16exp( l √2).Theminimumofthesecondterm
intheright-handsideoftheequationisequalto 2√2V /3√3. Thus,wec−om∗etotheconclusionthatif
0
−
2√2
V >16e l √2, (45)
0 −∗
3√3
thedomainwallwillnotbeabletoescapefromthepotentialwellcreatedbytheinhomogeneity.Hence,thecoarsening
willbestoppedwhenthedistancesbetweentheneighbordomainwallssatisfytheinequality(45).
Similarly, in the case of domain walls pushed by the asymmetry of phases (see (23), we find the criterion of
pinning:
3√3
V > h.
0
√2
If f(x)isaperiodicfunction,thesequenceofpinningsites(minimaofthepotential)filledbypinneddomainwalls
canberegular(“commensuratepatterns”)[71]orirregular(“spatialchaos”)[72].
8
/ComptesRendusPhysique00(2015)1–14 9
3.2. Oscillatorytailsofdomainwallsandstabilityofstationarypatterns
IntheexamplesconsideredinSec.2,thedomainwallsaredescribedbymonotonicfunctionslike(10).Themono-
tonicityoftheasymptoticbehaviorofthedomainwallsolutionontheinfinityleadstoasign-preserving(attracting)
interaction between domain walls. Oscillatory tails of domain walls create a sign-alternating interaction potential.
Thedomainwallscanbecapturednearthepotentialminima,thereforestablepatternsareformedduetopinningofa
domainwallbyaninhomogeneitycreatedbyanotherdomainwall.
As an example, let usconsiderthe stability of periodicsolutionsof the convectiveCahn-Hilliardequation(37),
whichsatisfytheconditionφ(x+l) = φ(x). Atlargel,thesesolutionsresembleperiodicsequencesofdomainwalls.
DefinethepatternwavenumberK =2π/l.ThenormaldisturbancesofaperiodicsolutionhavetheshapeofaFloquet-
Blochfunction,φˆ(x,t)=φ˜(x)exp(ikx+σt),whereφ˜(x+l)=φ˜(x),andkisaquasi-wavenumber, k < K/2.Aperiodic
solutionis alwaysneutrallystable (σ = 0) withrespectto a spatialshift, φ˜(x) = φ (x), k = 0.|T|herefore,a special
x
attentionhastobepayedtopotentiallyunstablelongwavedisturbanceswithsmallk. Theirgrowthrateσ(k;K)can
bepresentedas
σ(k;K)=σ (K)k+σ (K)k2+...
1 2
Onecanshowthatthesignofσ2(K)dependsonthedependenceofthesquaredpatternamplitude
1
1 l
A= φ2(x)dx
l Z
0
(see (38)) on the pattern wavenumber K = 2π/l [16], [73], [64]. If dA/dK < 0 for any K, which takes place for
D < D = √2/3, thenσ2(K) > 0 forany K, thereforeallperiodicsolutionsare unstable. Thatis compatiblewith
0 1
theattractiveinteractionbetweendomainwalls. ForD> D ,thefunctionA(K)isnotmonotonic,andtheextremaof
0
A(K)separatetheregionsofamonotonicgrowthoflongwavedisturbances,σ2(K) > 0,andthoseoftheoscillatory
1
responseofpatternstodilationsandcompressions,σ2(K) < 0[74]. Theregionofoscillatoryresponsecan contain
1
a subinterval of stable patterns (where σ (K) < 0) [60], [64]. That is possible because of the alternating sign of
2
theinteractionbetweendomainwalls. Ontheboundariesofthestabilityinterval,thepatternsbecomeunstablewith
respecttoeitherlongwave(phase)disturbances(see[75],[76])orshortwavedisturbanceswithk = K/2,leadingtoa
spatialperioddoubling.
Becausethepotentialoftheinteractionbetweendomainwallshasmultipleminima,thedistancebetweendomain
wallsisnotselectedina uniqueway. Thepatternincludeselementswith “short”and“long”distancesbetweenthe
maximathatalternateinaratherirregularway[60].
3.3. Nonlocalinteraction
Aspecifickindofanarrestedcoarseningprocesshasbeenfoundforequation(2)[12],whichcanbewrittenalso
as
∂ 2(φ +Γφ)+φ +φ φ3+h(t)=0. (46)
−x t xx −
ForsmallΓthelowestdensityoftheLyapunovfunctionalisachievedforpatternswithwavelengthλ = O(Γ 1/3).
opt −
Accordingtothelinearstabilitytheory,thedisturbancewithlargestgrowthratehasawavelengthλ = O(1). There-
c
fore, one could expectthat the energeticallypreferable, long-wavepattern will be developedfrom the initial short-
wavepatternbycoarsening.Thecoarseningprocesstakesplaceindeed,butitisstoppedwhenthewavelengthreaches
a much smaller value, λ = O(ln(1/Γ)). The criterion of the pattern stabilization is similar to (45), but now the
min
stabilizingfactoristhenonlocalinteractionwhichisproportionaltoΓ.
3.4. Pattern-inducedpinningofadomainwall.
Thestabilityregionsofaperiodicpatternandauniformstatemayoverlap.Inthatcase,thebehaviorofadomain
wall between the pattern and the uniform state is crucial. Near the threshold of the instability creating short-wave
patterns,wherethewidthofadomainwallislargecomparedtothecharacteristicpatternwavelength,onecandescribe
thedynamicsofadomainwallusingtheenvelopefunctionapproach[77],[78].Intheframeworkofthatapproachone
comestotheconclusionthatadomainwallbetweenthepatternandtheuniformstatemoveswithaconstantvelocity,
whichisproportionalto thedifferencebetweenthe Lyapunovfunctionaldensitiesofthephases[41]. However,the
9
/ComptesRendusPhysique00(2015)1–14 10
influence of the underlyingperiodic pattern leads to some qualitative changes of the domain wall dynamics. First,
themotionofthedomainwallisanoscillatoryprocess;duringoneperiod,onestripeiscreatedormelted[79],[80].
Secondly, because of the pinning effect, there is a finite interval of the parameter value where the domain wall is
motionless, i.e., a patternand a uniformstate coexist. Near the thresholdof the patternappearance, thatintervalis
transcendentallysmall[81],[41].
Asanexample,letusconsiderthecompetitionandcoexistencebetweenpatternsanduniformstatesforasystem
governedbytheSwift-Hohenbergequation,
∂2 2
φ = ǫ +1 φ φ3 (47)
t  − ∂x2 !  −
 
whichcorrespondstotheLyapunovfunctional
ǫ 1 1 ∂2 2
F φ = φ2+ φ4+ +1 φ dx. (48)
{ } Z −2 4 2" ∂x2 ! # 
 
Thatmodelwas suggested for studyinghydrodynamicsfluctuationsnear theinstability threshold [82] and used for
modellingBe´nardconvection[83]. Atǫ >0,periodicpatternsexistwithwavenumberskintheinterval1 √ǫ <k2 <
1+ √ǫ,andtheyarestableinasubintervalk (ǫ) < k < k+(ǫ). Atǫ > 1,constantnonzerosolutionsφ −= √ǫ 1
− ± ± −
appear. Atǫ > 3/2theybecomestable withrespectto smalldisturbances;kinkswith oscillatorytailsconnectboth
stableuniformphasesφ [84].
±
ThevalueoftheLyapunovfunctionaldensityfortheregularpatternwithanoptimalwavenumberislowerthan
thatoftheuniformstatewhenǫ <ǫ 6.3[84],[79].Nevertheless,thedomainwallbetweenbothstatesisimmobile
m
≈
for much smaller values of ǫ, ǫ > ǫ 1.7574. The reason is the self-induced pinning caused by the oscillatory
c
asymptoticperturbationof the uniform≈state. Similarly, the pinningeffect preventsthe replacementof a pattern by
a uniformstate at ǫ > ǫ . The stability intervalfor a finite fragmentof patterns sandwichedbetween semi-infinite
m
regions of a uniform state slightly depends on the length of that fragment [79]. Note that the noise activates the
transitionforametastablestatetoatrulystable,energeticallypreferred,state[79].
Thecoexistenceofpatternsanduniformstateshasbeenrevealedformanypattern-formingsystems(forareview,
see[85]).
4. Coarseningintwoandthreedimensions: curvatureeffects
4.1. Phaseseparation
First,letusconsiderdomaincoarseningforspatiallyuniformstatesinasystemwithouttheconservationlaw.
In a two-dimensional(three-dimensional)potentialsystem, the Lyapunovfunctionalcan be diminishedwithout
annihilationofa domainwall, justby diminishingits length(area). Letus considera two-dimensionalAllen-Cahn
equation,
φ =φ +φ +φ φ3+h, (49)
t xx yy
−
and present the isoline φ(x,y,t) = 0 (“a front”), which describes the center of a curved domain walls between the
stable uniformphases, in the formy = H(x,t): φ(x,y,t) < 0 as y < H(x,t), φ(x,y,t) > 0 as y > H(x,t). One can
showthatthemotionofthedomainwallisdetermined,inthelimitofsmallhandasmallcurvatureofthefront,by
theequation[86],[26],
H H 3√2 3√2
t = xx h, orv=κ h, (50)
1+Hx2 (1+Hx2)3/2 − 2 − 2
wherevisthenormalvelocity,anpdκisthecurvatureofthefront.Specifically,inthecaseh=0(bothphaseshavethe
samefreeenergy),wegetjusttherelationv=κ,whichiscalledcurvatureflow.
Asanexample,letusconsiderarounddropletofthephaseφ intheinfiniteseaofthephaseφ+. Becauseκ=1/R,
inthecaseh=0thedropletradiusischangedaccordingtothel−aw
R2(t)=R(0)2 2t.
−
10