MS516KineticProcessesinMaterialsLectureNote
3.DiffusioninGlassyStates
Byungha ShinDept.ofMSE,KAIST
1
2016SpringSemester
CourseInformationSyllabus1.Atomisticmechanismsofdiffusion (3classes)2.Macroscopicdiffusion
2.1.Diffusionunderchemicaldrivingforce (2classes)2.2.Otherdrivingforcesfordiffusion (2classes)2.3.Solvingdiffusionequations (2classes)
3.Diffusion(flow)inglassystates (2classes)4.Kineticsofsurfacesandinterfaces
4.1.Thermodynamicsofsurfacesandinterfaces (4classes)4.2.Capillary-inducedmorphologyevolution (2classes)
4.2.1.Surfaceevolution4.2.2.Coarsening
5.Phasetransformation5.1.Phenomenological theory (1class)5.2.Continuousphasetransformation (3classes)
5.2.1.Spinodal decomposition5.2.2.Order-disordertransformation
5.3.Nucleationandgrowth(Solidification) (3classes)
CourseInformationPhasechangememory
Wongetal.Proc.IEEE98,2201(2010)
Amorphoushighresistivity(RESET)
Crystallinelowresistivity(SET) Annealingat
somehightemperature
Melting+fastquenching
TypicalphasechangematerialusedinPCM:SbTe,GeTe,GST(Ge2Sb2Te5)
CourseInformationAmorphousstructure
• Crystalline:short-rangeorder+long-rangeorder• Amorphous:stillpossessesshort-rangeorder
distortioninbondangles(typicaldistortionangleina-Siisabout7degrees)
CourseInformationGlasstransition
Debenedetti andStillinger,Nature410,259(2001)
Glasstransition
temperature,T
g
• Coolingfastenoughtopreventcrystallization- AboveTg:metastableliquid- AsTg isapproached,molecularmobilitydrops- AtTg,systemfrozeninoneparticularconfiguration;glass
• Glasstransitionisakineticphenomenon(ifthemeltiscooleddownmoreslowly,Tg islower;however,inpractice,thedependenceofTg onthecoolingrateisweak)
ΔHf
~0.5ΔHf
CourseInformationGlasstransition
Heatcapacity,C
P
Entropy,S
Freeenergy,G(orµ)
T T
T
Tm TmTgTg
CourseInformationCriterionforglassformation
Tg
• Partitionless transformation:withoutchangeincomposition
• T0-line:locusofallpointsx0 fordifferenttemperaturesà partitionlesstransformationonlypossiblefortemperatures T0-line
• Ifonlypartitionless transformationcanoccur(e.g.inlaserquenching),nocrystallizationcanoccurinthecompositionrangex1 andx2 (intersectionofT0-linewithTg)à glassmustform
• Conclusion:TheT0-lineprovidesasufficient (butnotnecessary)criterionforglassformation,butonlyifjustpartitionless crystallizationcanoccur.
GL =GS
CourseInformationViscosityvstemperature
TypeA
TypeB
0 1
Tg typicallydefinedwhenη ~1013 poise
𝜂 = shearviscosity =𝜎01234𝜀0̇1234
=[𝑑𝑦𝑛𝑒/cm>][%/sec] =
𝑔cm B sec = "poise"
• Primarymanifestationofglasstransition:sharpincreaseinη(forexample,a-Se:η increasesbyfactorof3forevery1o dropinT)
• TypeA:typicalglassformer,includingpolymers,metals,oxides,ionic,chalcogenides,molecular,glasswithmodifiers
• TypeB:SiO2,GeO2,BeF2 (bestglass-formers)bondbreakingprocessà singleactivationà Arrheniusbehavior, 𝜂 = 𝜂E exp
𝑄𝑅𝑇
CourseInformationViscosityvstemperature
TypeA
TypeB
0 1
AnexampleofTypeAglassformer:glasswithmodifiers(Na2O,CaO)à Breakingupoffullyconnectednetwork,loosenetworkà Fulcher-Vogelbehavior:
largevariationinactivationenergywithtemperatureverylargeactivationenergynearTg (cooperativerearrangementofmanymolecules)
CourseInformationAtomictransportinglassFreevolumemodelbyTurnbullandCohen(Handout#7)
Originalconfiguration
• Densityfluctuationaroundatom1:localatomicvolumeincreases,coordinationincreases
• Freevolumevf =volumeofcage– volumeofatom(ion)• Atom1jumpswithvelocityu overdistanceλ;
𝐷 = 𝑔Γ𝜆> = 𝑔𝑢𝜆
FinalconfigurationIfthefreevolumeislargeenoughtoallowanewatom6topreventatom1fromreturning:• Atom1makesdiffusivejump:newneighbor8• Localshearstrain,γ0 ≈1
CourseInformationAtomictransportinglassCalculatetheprobabilitythefluctuationisbigenoughtoallowatomicrearrangement.• Perfectlyorderedamorphousstate(T0):zerofreevolume,atomicvolumeΩ0• Increaseinvolume(e.g.T >T0):Ω >Ω0
à averagefreevolumeperatom:vf =Ω –Ω0• Characteristicforliquid:vf redistributedwithoutenergychange• Probability,P(v)dv,ofanatomhavingaparticularfreevolumebetweenv andv +dv,isdeterminedentirelybyentropy
DerivationofP(v)?• N atoms;totalfreevolumeVf à averagefreevolumeperatom,vf =Vf/N• Discretetreatment:Ni atomshavefreevolumevi (=iΔv)• Entropyconsideration:maximizingnumberofways(W)todistributeNatomsovercategoriescontainingNi atoms,
𝑊 =𝑁!
∏ 𝑁R!Ror
underthefollowingconstraints:S𝑁RR
= 𝑁, andγS𝑁R𝑣RR
= 𝑉Z
ln𝑊 = 𝑁 ln𝑁 −𝑁 −S𝑁R ln𝑁RR
+S𝑁RR
γ:overlapfactorbetweenadjacentcells,0.5<γ <1(µ) (ν)
CourseInformationAtomictransportinglassMaximizelnW undertheconstraints(µ,ν :Lagrangianmultiplier):
𝑃 𝑣 𝑑𝑣 =𝑁R𝑁 = 𝜈𝛾 exp(−𝜈𝛾𝑣)𝑑𝑣
,or
𝑁 =S𝑁R = 𝐴Sexp −𝜈𝛾𝑖∆𝑣f
RgE
=𝐴
1 − exp −𝜈𝛾∆𝑣
Fordiffusionandflowtotakeplace,thefreevolumemustbelargeenoughtoaccommodateanotheratom:v >v*(criticalfluctuation)
− ln𝑁R − 1 − 𝜇 − 𝜈𝛾𝑣R = 0 𝑁R = 𝐴exp(−𝜈𝛾𝑣R)
DetermineA andν fromtheconstraints
𝐴 = 𝑁[1 − exp −𝜈𝛾∆𝑣 ] ≈ 𝑁𝜈𝛾∆𝑣
Continuumlimit:
Secondconstraint: l 𝑃 𝑣 𝑣𝑑𝑣f
E= 𝑣Z 𝑃 𝑣 𝑑𝑣 =
𝛾𝑣Zexp −
𝛾𝑣𝑣Z
Probabilityofanatomhavingv >v*: 𝑃∗ = l 𝑃 𝑣 𝑑𝑣f
n∗= exp −
𝛾𝑣∗
𝑣Z
Δ𝑣 → 𝑑𝑣,𝑣R → 𝑣,
CourseInformationAtomictransportinglass𝑃∗ = l 𝑃 𝑣 𝑑𝑣
f
n∗= exp −
𝛾𝑣∗
𝑣Z
Temperaturedependenceofvf:(empirical)𝑣Z = 𝛼ΩE(𝑇 − 𝑇E)
𝑃∗ = exp −𝛾𝑣∗
𝛼ΩE(𝑇 − 𝑇E)= exp −
𝐵𝑇 − 𝑇E
Diffusion: 𝐷 = 𝑃∗ 𝑔Γ𝜆> = 𝑃∗ 𝑔ΓE𝜆> = 𝐷E exp −𝐵
𝑇 − 𝑇E
Flow: �̇� = 𝛾E𝑃∗Γ′
macroscopicstrainrate
localstrain
netjumprate(mustbedependentonstress)
�̇� = 𝛾E𝑃∗Γu = 𝛾E𝑃∗ΓE 1 − exp −Δ𝜇𝑘w𝑇
≈ 𝛾E𝑃∗ΓE𝜎𝛾EΩE𝑘w𝑇
Δµ:chemicalpotentialchangeinducedbystressΔ𝜇 = 𝜎𝛾EΩE
ifstressissmall
(workdoneuponrearrangement)
CourseInformationAtomictransportinglass
𝜂 =𝜎�̇� =
𝑘w𝑇ΓE𝛾E>ΩE𝑃∗
= 𝜂E exp𝐵
𝑇 − 𝑇EFulcher-Vogel
Stokes-Einstein:
Relationb/wdiffusivityandviscosity:
𝐷𝜂 =𝑔𝑘w𝑇𝜆>
𝛾E>ΩE≈𝑔𝑘w𝑇𝜆
(𝛾E ≈ 1, ΩE ≈ 𝜆x)
𝐷𝜂 =𝑘w𝑇3𝜋𝜆
Jumpfrequencyperatom: 𝑃∗ΓE =𝑘w𝑇𝜂ΩE
≈10{E
𝜂 poise /sec
Example:η =1013 poiseà eachatomjumpsevery1000secondsà configurationalfreezingà conventionalvalueforTg
CourseInformationEmpiricalsurveyofdiffusion
CourseInformationEmpiricalsurveyofviscosity