Kinetics of Phase Ordering in a Two-Component Flui

arXiv:cond-mat/9603027v15Mar1996KineticsofPhaseOrderinginaTwo-ComponentFluidMembraneP.B.SunilKumarandMadanRaoInstituteofMathematicalSciences,Taramani,Madras600113,India(February6,2008)AbstractWepresentanovelMonteCarlosimulationofthephaseseparationdynamicsofamodelfluidmembrane.Suchaphasesegregationinducesshapechangesofthemembraneandresultsinlocal‘budding’underfavourableconditions.Wepresentapreliminaryinvestigationofthenucleationregimeandcomputeavarietyofgrowthlaws.Ourstudyindicatesthatmembraneswithhighviscosity,buckleonaccountofphasesegregation.Wediscusstherelationbetweenthisdynamicalbucklingandthephenomenonof‘capping’seeninbiologicalmembranes.I.INTRODUCTIONOwingtostronghydrophobicinteractions,phospholipidmoleculeslikeDMPCorSOPC,spontaneouslyaggregateinanaqueousmediumtoformabilayeredfluidmembrane.Overlengthscaleslargerthanitsbilayerthickness(∼50˚A),themembranecanberepresentedbyatwo-dimensionalclosedsurface(vesicle)ofdimension10−20μm,embeddedina3-dimensionalaqueousmedium.Inequilibrium,thevesicleexhibitsawell-definedshapeandtopologywhensubjecttoconstraintsoffixedsurfaceareaandvolume.Unlikesoapfilms,theseequilibriumconformationsareprimarilydeterminedbybendingelasticity[1].Shapeandtopologychangescanbeinducedbychangingparameterslikethetemperatureoros-moticconditions.Experimentslikephasecontrastvideomicrographyhaveestablishedan1entiretaxonomyofequilibriumshapesandshapetransitionsastheseparametersarevaried,themostdramaticbeingthe‘budding’transitionatwhichalarge‘parent’vesicleformsasphericalappendageattachedtoitbyanarrowumbilical[2].Anextensivestudyofcer-taintheoreticalmodels[3]haveresultedincomplicatedshape-phasediagrams,withshapechangesoccuringatdiscontinuousorcontinuousphaseboundaries.Thesequenceofshapechangespredictedbythesetheoriesarelargelyinconformitywiththeexperimentsmentionedabove.Althoughtheaboveshapechangeshavebeenobservedinartificialmembranes,similarandmorecomplexshapechangeshavebeenseenin‘real’biologicalmembranes(e.g.,plasmamembraneofanimalcells[4]).Thesechanges,oftentriggeredbyachangeinthechemicalenvironment,areassociatedwithdistinctbiologicalfunctions(e.g.,endocytosis,exocyto-sis).Unlikethesimplesingle-componentartificialmembranesdiscussedabove,biologicalmembranespossessarounda1000differentspeciesoflipids.Theselipidsmaydifferintheirheadgroups,lengthofchains,ornumberandpositionoftheunsaturatedbondswithinthechain.Suchmulti-componentlipidmembranescanalsoexhibitshapeandtopologychanges,astheareafractionsoftheconstituentlipidsarealtered.Indeed,recenttheoretical[5]andexperimental[6]studiesontheshapesoftwo-componentartificialmembranes,haveshownthat‘budding’andeven‘fission’[6]maybeinduced,followingthephasesegregationofthelipidspeciesalongthemembrane.Inthispaperwearguethatthetimescaleoverwhichtheshapeofthemembranechanges,isofthesameorderasthetimescaleoverwhichphasesegregationoccurs.Thisimpliesthatthestudyofshapechangesinducedbyphaseseparationisanonequilibriumproblem.Inthelastfewyearstherehavebeensomestudiesonthedynamicalbehaviouroffluidmembranes(seeRef.[7]andreferencestherein).Mostoftheefforthasfocussedonevaluatingdynamicalcorrelationfunctions(andrenormalizedtransportcoefficients)ofmembranesslightlyawayfromequilibrium.Ontheotherhand,weareconcernedwiththeshapedynamicsofmem-branesfarfromequilibrium.Inthispaper,wepresentpreliminaryresultsofaMonteCarlostudyofthedynamicsofphaseseparationofatwo-componentlipidfluidmembrane,that2incorporatesthemomentumdensityofthelipids.Asweshallseebelow,therelativelipidconcentrationiscoupledtothelocalcurvature—thustheshapeofthemembranechangesasphaseseparationproceeds.Ourstudyisrestrictedtothenucleationregime,wherethegenericlipidconcentrationprofileshowspatchesoftheminorityspeciesinthebackgroundoftheother.Followingatemperaturequench,themembranesproutsbudslocally,withtheminorityspeciesformingthebud.Attheseearlytimes,coalescencedoesnotoccur.Fromthedomaininterfacialenergydensityandthetotalmeancurvaturedensity,weextracttwolengthscalesRφandRH,whichweshowarerelatedtotheaverage‘neck’radiusandtheaveragecurvatureofthebudrespectively.ThelengthRH,increaseswithtimeast1/4.Sub-sequentcoarseninghappenswiththebudscoalescingormovingtowardstheedge(thelatterisanartifactofour‘free’boundaryconditions).Wenextinvestigatethetimeevolutionofasingleisolatedpatchoftheminorityspeciesstartingfromaflatconfiguration.Wefindthatthelatetimemorphologydependscruciallyonthein-planeviscosityofthelipids.Wefindthatanincreaseintheviscositycausesthefluidmembranetobuckleonphasesegregation.Thisbuckledshapeisthusaneffectofdynamics;itisnotanequilibriumconfigurationofthefluidmembrane.Thisbucklingcomesaboutbecauseofthepresenceofpositivedisclinationslocalisedatthedomaininterface[8].Thisclusteringinducedbucklingofafluidmembrane,whosein-planeviscosityishigh,mightbeofrelevanceinthephenomenonof‘capping’[4].II.DYNAMICSOFPHASESEPARATIONOFTWOCOMPONENTFLUIDSLetusrecountthatwhentwoimmisciblefluids,likewater+toluene,arecooledbelowtheircoexistencecurve,theyphasesegregateintowater-richandtoluene-richregions,separatedbysharpinterfaces.Forequalvolumefractionsofwaterandtoluen