Comptes Rendus Physique ComptesRendusPhysique00(2015)1–14 5 1 Coarsening versus pattern formation 0 2 r a A.A.Nepomnyashchy M DepartmentofMathematics,Technion-IsraelInstituteofTechnology,Haifa32000,Israel 3 1 ] 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 o 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. 2 /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 3 /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. − − 4 /ComptesRendusPhysique00(2015)1–14 5 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 ± 5 /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)). 6 /ComptesRendusPhysique00(2015)1–14 7 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. 7 /ComptesRendusPhysique00(2015)1–14 8 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 modellingBeńardconvection[83]. Atǫ >0,periodicpatternsexistwithwavenumberskintheinterval1 √ǫ 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

Coarsening versus pattern formation

Similar Coarsening versus pattern formation