Growth of Order in An Anisotropic Swift-Hohenberg

arXiv:cond-mat/0506703v2[cond-mat.soft]6Jan2006GrowthofOrderinAnAnisotropiSwift-HohenbergModelHaiQianandGeneF.MazenkoTheJamesFrankInstituteandDepartmentofPhysis,TheUniversityofChiago,Chiago,Illinois60637(Dated:Sep.10,2005)AbstratWehavestudiedtheorderingkinetisofatwo-dimensionalanisotropiSwift-Hohenberg(SH)modelnumerially.ThedefetstrutureforthismodelissimplerthanfortheisotropiSHmodel.Onendsonlydisloationsinthealignedorderingstripedsystem.Themotionofthesepointdefetsisstronglyinuenedbytheanisotropinatureofthesystem.Wedevelopedauratenumerialmethodsforfollowingthetrajetoriesofdisloations.Thisallowsustoarryoutadetailedstatistialanalysisofthedynamisofthedisloations.Theaveragespeedsforthemotionofthedisloationsinthetwoorthogonaldiretionsobeypowerlawsintimewithdierentamplitudesbutthesameexponents.Thepositionandveloitydistributionfuntionsareonlyweaklyanisotropi.PACSnumbers:05.70.Ln,64.60.Cn,64.60.My,64.75.+g1I.INTRODUCTIONThereisongoinginterestinthegrowthkinetisofstripeformingsystems.Therehasbeenprogressviaexperimental[1,2℄andnumerial[3,4℄studiesofgrowthafteraquenhfromanisotropiinitialstate.Howeverthetheoretialunderstandingofsuhsystemsremainslimited.Thisismostlyduetotheomplexityofthedefetstruturesgeneratedduringorderinginsuhsystems.Forexample,intheSwift-Hohenbergmodel,therearegrainboundaries,dislinationsanddisloationsgeneratedintheorderingproess.Theo-existeneofallthesedierentdefetstrutureshashinderedthetheoretialanalysisofthestripedphaseorderingsystems.Inthispaper,westudyananisotropiSwift-Hohenberg(SH)model,whereonlydisloationsareproduedintheorderingproess.Ourgoalistounderstandthestatistialpropertiesofthesedefetsmuhaswenowunderstandthosepropertiesforsimplevortexproduingmodels.Thereareformalarguments[5℄thatifwebreakthesymmetryoftheisotropiSHmodelbyapplying,forexampleaneletrield,thenthesystemanbemappedontoananisotropiTDGLmodel.ThissuggestsaL≈t1/2growthlawomparedtomuhslowergrowthintheisotropiSHmodel.Wendsupportforthishypothesis.Somepreviousstudieshavefousedontheevolutionofafewdisloations[6,7,8,9,10,11,12℄.TesauroandCross[6℄studiedthesteadystatelimbingmotion(movealongthediretionofstripes)ofisolateddisloationsboththeoretiallyandnumeriallyinseveraltwo-dimensionalmodelsystemsinludingtheSHmodel.Theyfoundthatthewavenumberseletedbydisloationlimbismarginallystableonlyforpotentialmodels.Bodenshatzetal.[11℄studiedthelimbingmotionofdisloationswithamplitudeequationsappropriateforsystemswithanaxialanisotropy.ThePeah-Kohler(PK)fore(theeetivewavenumbermismath)drivesthedisloationmotion,justasinRef.[6℄.TheyalsoonsidertheinterationbetweentwodisloationstogetherwiththePKfore.Gorenetal.[7,8,9℄studiedtheonvetioninathinlayerofanematimaterialexperimentally.Theyintroduedagauge-eldtheoretialtreatmenttostudythelimbingofdisloationsinastressedbakgroundeldwherethePKforeplaysarole.Thetheory[12℄preditsthatlimbingandglidingmotionsofasingledisloationareequivalent(afterthepropersalingfortheanisotropisystem)andduetothePKmehanism.BraunandSteinberg[10℄studiedthesameexperimentalsystem.TheymeasuredtheglidingmotionofdisloationsduetoapureinterationbetweenthemembersofthepairwithoutthePKmehanism.Theyfoundthatthelimbandglidingmotionhavedierentharaters.Boyer[13℄simulatedananisotropistripeformingmodel[14℄basedontheSwift-Hohenbergmodel.Hismodelismoreompliatedthanours.Inhismodelthestripeshavetwopreferreddiretionsandazig-zagpatternisformed,andthedisloationstendtostaytogethertoformlarge2domainwalls.Theauthorfoundthatforsmallquenhestheenergy,thedisloationenergyandtheharateristilengthnormaltothestripesallsaleast±1/2(+fortheharateristilength).Healsofoundthatfordeepquenhesthesystemwasfrozen.Thepinningeetbeomesimportantasthequenhdepthinreases.Thezig-zagpatternwasexperimentallyrealizedinRef.[15℄.Herewestudyanensembleofwellseparateddisloationsintheontextofdomaingrowth.Themotionofthedisloationsinthismodelishighlyanisotropi.Theytendtomovearossthestripes.Theaveragespeedsarossandalongthestripesobeysimplepowerlawsintimewithdierentamplitudesbutapproximatelythesameexponent.Thedistributionsofthedefetveloitiesalongthetwoorthogonaldiretionshavesameformandlargeveloitypower-lawtailswithapproximatelythesameexponents.Twobulkmeasurementsoftheordering,thedeayoftheeetiveenergyandthenumberofdisloations,obeyasimplepowerlawintimewithalogarithmiorretion,asfortheXY-model[18℄.ThetwodimensionalisotropiSwift-Hohenberg(SH)model[19℄isdenedbyaLangevinequa-tion∂ψ(x,t)∂t=−δH[ψ]δψ(x,t)+ξ(x,t),(1)whereψistheorderingeld,andtheeetiveHamiltonianisgivenbyH[ψ]=Zd2r−ǫ2ψ2+12(∇2+1)ψ2+14ψ4,(2)whereǫisapositiveonstant.Allthequantitiesinthispaperhavebeenputindimensionlessform.Thenoiseξsatiseshξ(x,t)ξ(x′,t′)i=2Tδ(x−x′)δ(t−t′),whereTisthetemperatureafterthequenh.Inthefollowing,wesetT=0whiheliminatesthenoisetermfromtheanalysis.Startingfromarandominitialonditionwithoutlongdistaneorrelations,theSHequation(1)generatesstripeswithperiod2π.InthesimulationsfortheisotropiSHmodel,wefound[4℄thatthegrainboundaries’motiondominatetheorderingdynamisofthesystem,whihisdierentfromwhatisseeninsomeex-periments[1,2℄,wher