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