计算材料学14-6-1

1陈效双中科院上海技术物理研究所、上海科技大学计算材料学计算材料学第六章介观和宏观模拟方法简介2本章提纲1.耗散粒子动力学2.相场动力学3.有限元方法36-1.耗散粒子动力学计算材料学第六章4Introductiontomesoscalemodeling•最准确的方法ofcalculatingthedynamicalbehaviorofanatomisticsystemistointegrate所有原子的运动方程inthesystem.ThisisthebasisoftheMolecularDynamicssimulationtechnique.•分子动力学的缺点isthatitoftenprovidesfarmoredetailof小尺度的涨落运动ofatomsthanisnecessaryforanunderstandingofmanyphysicalprocesses.•计算机处理器速度和内存容量对原子的数量有限制.Thisisinadequateformanychemicalprocessesthatoccuron微秒（更长）时间尺度.5Introductiontomesoscalemodeling6DissipativeParticleDynamics(DPD)•一个新的模拟技术,耗散粒子动力学(DPD),fromMolecularDynamicssimulationsandLatticeGasAutomata,openingupthe介观长度和时间范围incomplexfluidstosimulation.•DPD，保持对系统积分运动方程的原理，但是，首先积分出最小的空间自由度。在一个系统内原子的快速运动是被平均化，andtheremainingstructureisrepresentedbyasetof“beads”,of给定质量和尺寸,thatinteractviasoftpotentialswithotherbeads.一个“珠子”代表小区域的流体物质anditsmotionisassumedtobegovernedbyNewton’slaws.7•在DPD范围内,有三种形式的力:Anharmonicconservativeinteraction,Adissipativeforcerepresentingtheviscousdrag(阻尼)betweenmovingbeads,Arandomforcetomaintainenergyinputintothesysteminoppositiontothedissipation.Allforcesareshort-rangedwithafixedcut-offradius.•对“珠子”运动方程的积分generatesatrajectorythroughthesystem’sphasespace(e.g.densityfields,orderparameters,correlationfunctions,stresstensor,etc.).•相对传统的MD和布朗动力学模拟的一个巨大的优势是thatallforcesare“soft”allowingtheuseofamuchlargertimestepandcorrespondinglyshortersimulationtimes.BasicideasofDPD8BasicideasofDPD9Equationsofmotion牛顿定律控制的动力学演化ri,vi和fi分别是在第I“珠子”上的位置矢量，速度和总的力.所有“珠子”质量，mi,被假设相等，为简单起见，被视作为单位。每个“珠子”，受到近邻的三个非成键力:aconservativeinteraction,linearinthebead-beadseparation;adissipativeforce,proportionaltotherelativevelocityoftwobeads;arandomforcebetweenabeadandeachofitsneighbors.10•Hence,thetotalforceonabeadis:wherethesumisoverallbeadswithinadistancercoftheithbead.rcissettounitysothatalllengthsaremeasuredrelativetothebeadradius.•TheconservativeforceFCisarepulsivecentralforcewithamaximummagnitudeaij,asfollows:whererijisthemagnitudeofthebead-beadvectorrij,andistheunitvectorjoiningbeadsiandj.Forceonthebead11•ThedissipativeforceFDisproportionaltotherelativevelocityoftwobeadsandactssoastoreducetheirrelativemomentum:whereωD(rij)isashortrangeweightfunction.Forceonthebead12Forceonthebead•TherandomforceFRalsoactsbetweenallpairsofbeadssubjecttoasimilarshortrangecutoff,butwithadifferentfunctionωR(rij),andactssoastopumpenergyintothesystem:whereζij(t)isadelta-correlatedstochasticvariablewithzeromean,i.e.:13•Inaddition,beadsconnectedinamoleculeexperienceaspringforceduetobondedneighbors.AsimpleharmonicforcelawisusedforthespringforcefSonbeadiwherethesumrunsoverallbeadstowhichbeadiisconnected.Forceonthebead14Scalingoflength,velocityandtime•ThebeadsinDPDdonotcorrespondtorealatomsormolecules.Theyrepresentsmallregionsoffluidmaterialinteractingviaphenomenologicalforces.Quantitingwithunitsofmassandlengtharerendereddimensionless.•Thetimeunitremainstobespecified.Thetheoremofequipartitionofenergyprovidesthenecessaryrelationshipbetweenmass,length,timeandtemperature.Thisisequivalenttomeasuringtimeinunitsof(mrc2/kBT)1/2.15Integrationscheme•Amodifiedvelocity-Verletalgorithmisusedtointegratetheequationsofmotion.Thenewpositionandvelocityarethenusedtocalculatethenewforce,andthisthencorrectsthevelocity.•Thefactorλissetto0.65intheDPDsimulationcode.ItwasfoundthatthisvalueoptimizesnumericalstabilityunderthedefaultconditionsofdensityandnoiseusedinDPD.16Equationsofstate•AnexpressionforthepressurepinaDPDsystemcanbeobtainedfromthevirial(维里)theoremas:risthedensity,dFijCistheconservedpartoftheforce.Theaboveequationcanalsobeexpressedintermsoftheradialdistributionfunctiong(r):17两体相互作用力与距离的关系比较18两体相互作用力与距离的关系比较19两体相互作用力与距离的关系比较20两体相互作用力与距离的关系比较21两体相互作用力与距离的关系比较22两体相互作用力与距离的关系比较2324两体相互作用力与距离的关系比较25两体相互作用力与距离的关系比较26两体相互作用力与距离的关系比较27DPDsimulationofdrugdiffusion28DPDsimulationofphaseseparationBlendoffluororesin（氟树脂的混合）/TATB1000timesteps2000timesteps4000timesteps20000timestepsInitialstate296-2.相场动力学计算材料学第六章30有关材料相变的知识，如：材料的微观结构、拓扑结构、形态、以及微结构化学特性等方面的信息是材料性能预测的基础。在微观结构的尺度上，研究关于平衡和非平衡相变现象、特别是研究液体-固体和固体-固体相变，已经成为现代材料科学研究中的最具挑战性课题。相场动力学简介31相场动力学简介相场模型是以热力学和动力学基本原理为基础而建立起来的一个用于预测固态相变过程中微结构演化的有力工具。在相场模型中，相变的本质由一组连续的序参量场所描述。微结构演化则通过求解控制空间上不均匀的序参量场的时间关联的相场动力学方程而获得。相场模型对相变过程中可能出现的瞬时形貌和微结构不做任何事先的假设。32•有两种形式的场变量，守恒与非守恒的.•在扩散界面的描述，一个非均匀微结构系统的总自由能可以由一系列的守恒和非守恒的场变量描述:wherefisthelocalfree-energydensitythatisafunctionoffieldvariablesciandhi,andαiandβjarethegradientenergycoefficients.Phase-fieldmethod33Phase-fieldmethod•第一行积分表示thelocalcontributiontothefreeenergyfromshort-rangechemicalinteractions.•第二行积分表示anonlocaltermthatcontainsthecontributionstothetotalfreeenergyfromanyoneormoreofthelong-rangeinteractions,suchaselasticinteractions,electricdipole-dipoleinteractions,electrostaticinteractions,etc.34Free-energyfunction•Inaphase-fieldmodel，thelocalfree-energydensityfunction.Manyofthephase-fieldmodels,useadouble-wellformforthefunction,namely,.whereφisafieldvariable.•Anotherfreeenergyistheso-calleddouble-obstaclepotential,where35•Torealmaterialsprocesses,introducingmorethanonef