LECTURE 1: Disordered solids: Structure properties · Diffusion-limited aggregation: crystal vs amorphous structures Meakin, Jamtveit, PNAS 2009 •Limited –a seed particle is placed

LECTURE1:Disorderedsolids:Structureproperties

• Whatisandisorderedsolid• Howdowemeasuredisorder• Howareamorphoussolidstypicallyformed

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Whatareamorphoussolids?

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Nicolasetal,arXiv:1708.09194 (2017)3

Nicolasetal,arXiv:1708.09194 (2017)

• inorganiccompounds(e.g.SiO2/silicates,B2O3/borates,GeO2/germanates,etc)

• organiccompounds(e.g. polymers)

• elements(e.g.sulfur)• metalalloys (e.g.Fe80B20)• Granularmatter

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Wheredoweencounteramorphousmaterials?

• Foodproducts:mayonnaise,chocolatemousse,ketchup

• Houseproducts,cosmetics• Plastic• Transformerscorematerial,• Biomedicalimplants,• Watches• fiberoptics• etc

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Crystals

• Long-rangeorder• Regularatomicarrangement

Amorphoussolids

• Short-rangeorder• Randomatomicarrangement

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BorateglassB2O3 (crystal)

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Howdowemeasuredisorder?

• Statisticalmeasures:Probabilitydistributionfunctions

• Thermodynamicsmeasures:Excessvolume

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Howdowemeasuredisorder?

• Statisticalmeasure:Probabilitydistributionfunctions

• Statisticaldescriptionofatomicpositions

• Describesimportantfeaturesoftheatomicarrangements

• Accessibleexperimentallythroughscatteringexperiments

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Singleparticledistributionfunction• NatomsinthevolumeVofthematerial

• Averagedensity𝑛" =$%

• Singleparticledistributionfunction

𝑛& 𝒓 = ⟨)𝛿(𝒓 − 𝒓-)�

-

⟩

• Ensembleaverageoverpossibleconfigurations(statisticaldescription)

• 𝑛& 𝒓 𝑑𝒓 istheaveragenumberofatomcentersinavolumeelementdr aroundr

• 𝑛& 𝒓 𝑑𝒓 canbeinterpretedastheprobabilitytofindanatomcenterbetwen randr+dr

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Two-particlesdistributionfunction

• NatomsinthevolumeVofthematerial

𝑛2 𝒓&, 𝒓2 = ⟨∑ ∑ 𝛿 𝒓- − 𝒓& 𝛿(𝒓5 − 𝒓2)⟩�56-

�-

• Probability tofind atomcenters simultaneously inashell of thickness𝑑𝑟&around𝑟& andashell 𝑑𝑟2 around𝑟2 is𝑛2 𝒓&, 𝒓2 𝑑𝒓&𝑑𝒓2

∫ 𝑑𝒓2𝑛2 𝒓&, 𝒓2 = 𝑁 − 1 𝑛&(𝒓&)

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Atomiccorrelations

• Probabilityofhavinganatomat𝒓2iscorrelatedwiththeprobabilityforanatomat𝒓&,ifthesepositionsarenottoofarapart.

• Thisisexpectedduetointer-particleforcesandconstraintsduetochemicalbonds.• Homogeneousmaterial:𝑛&(𝒓&) = 𝜌

• If 𝒓& − 𝒓𝟐 ≫ 1,atomsareuncorrelatedand:𝑛2(𝒓&, 𝒓2) ∼ 𝜌2

• Generallywecanwrite𝑛2(𝒓&, 𝒓2) = 𝜌2𝑔(𝒓&, 𝒓2)

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Pairdistributionfunction(PDF)

• Generaldefinition: g 𝒓&, 𝒓2 = BC(𝒓D,𝒓C)BD 𝒓D BD(𝒓C)

• Put𝒓2 = 0,thendefine 𝒓 = 𝒓& − 𝒓2

• Homogeneousmaterial

• 𝑔 𝒓&, 𝒓2 = 𝑔 𝒓 = 𝑛2(𝒓)/𝜌2 describesthecorrelationsintheatompositions

• 𝜌𝑔 𝒓 𝑑𝒓 isinterpretedastheprobabilitytofindanatomin𝑑𝒓,avectordistance𝒓 fromanotheratomattheorigin.

• 𝜌∫ 𝑔 𝒓 𝑑𝒓 = 𝑁 − 1

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Isotropicmaterials• 𝒓 isthedistancebetween𝒓& and𝒓2,𝒓 ≡ 𝒓𝟐 − 𝒓𝟏• 𝜌𝑔(𝒓) canbeintegratedtogivetheaveragedensityofatomsinasphericalshellaroundtheorigin

I 𝑑𝑟𝜌𝑔 𝑟JKLMJ

JK= 4𝜋2𝑟"2𝑔 𝑟" Δ𝑟 = 𝜌𝑅 𝑟" Δ𝑟"

• RadialdistributionfunctionR 𝑟

R 𝑟 = 4𝜋2𝑟2𝑔(𝑟)

• IntegrateR 𝑟 aroundthefirstpeak->Coordinationnumberofthefirstshell

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PairDistributionFunction(PDF)

• PDFshowoscillationsduetonearestneighbor,nextnearestneighborshells,etc.

• Oscillationsdampedoutas𝑟increases

• 𝑔 𝑟 ∼ 1 forlarge𝑟

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PDFforidealcrystalvsglasses

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http://www.globalsino.com/EM/page3097.html 17

http://www.globalsino.com/EM/page3097.html 18

http://www.globalsino.com/EM/page3097.html 19

http://www.globalsino.com/EM/page3097.html20

RadialdistributionfunctionThenumberofneighborstotheoriginatomsituatedbetweenr1 andr2 isgivenbytheareaintegralunderthefirstpeak

⋕ 𝑛𝑒𝑖𝑔ℎ𝑏𝑜𝑟𝑠 JD,JC = I 𝑑𝑟𝑅(𝑟)JC

JD

http://www.globalsino.com/EM/page3097.html21

Radialdistributionfunction

http://www.globalsino.com/EM/page3097.html22

Howdowemeasuredisorder?

• Statisticalmeasures:Probabilitydistributionfunctions

• Thermodynamicsmeasures:Excessvolume

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Measureofdisorder:Excessmolarvolume• Energeticsofbonddensity

• Orderedcrystalhashigherpacking->morebondsperunitvolume•• Bindingenergy𝐸[\]^ < 𝐸 ^]aa

• Bindingenergy<->molarvolumeV

• 𝑉 ^]aa > 𝑉[\]^

• ExcessmolarvolumeΔ𝑉 = 𝑉 ^]aa − 𝑉[\]^ isameasureofdisorder

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Glassformationfromliquid

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CoolingcurveHeatingcurve

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Glassformationfromliquid

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CoolingcurveHeatingcurve

𝑻𝒎

𝑻𝒎 functionofcomposition

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Glassformationfromliquid

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CoolingcurveHeatingcurve

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Glassformationfromliquid

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CoolingcurveHeatingcurve

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Glassformationfromliquid

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CoolingcurveHeatingcurve

𝑻𝒈 dependson

(atommobility)-1X

(complexityofcrystalstructure)

X(coolingrate)

X(composition)

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Glassformationfromliquid

OpenCourseMIT3.017

CoolingcurveHeatingcurve

ExcessVolume(disorder)

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SummaryLECTURE1:Disorderedsolids:Structureproperties

• Whatisandisorderedsolid• Howdowemeasuredisorder• Howareamorphoussolidstypicallyformed

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SummaryLECTURE1:• Whatisandisorderedsolid

• Solidswithshort-rangedisorder

• Howdowemeasuredisorder• Pair-correlationfunction• Excessvolume

• Howareamorphoussolidstypicallyformed• Glasstransition

Versus

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LECTURE2Atomicdiffusion:equilibriumandnon-equilibriumstructures• Diffusionofatoms

• Diffusionlimitedaggregation,crystalgrowth

• Fractalsandfractalmeasure

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Diffusion• Atany𝑇 > 0K,allatomsirrespectiveoftheiraggregation(gas,liquid,solid,etc)areinconstantmotion->thermalmotion

• Movementofatomsisassociatedwithcollisions->singleparticletrajectoryisazigzag– diffusiveparticle

• Acollectionofdiffusiveparticleshasanobservabledriftfromhightolowerconcentrations

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Source:wikipedia36

Flow anddispersion

Molecular dispersion(Diffusion)

Dispersion by laminarchannel flow

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Molecularmixing- diffusion

D" =12𝑑

lim\→n

⟨𝑥2⟩𝑡

𝜕\𝐶 = D"𝛻2𝐶

Diffusioncoefficient

MacroscopicLaw

D" =𝑎2

2𝜏

Relatedtomicroscopicconstants(lengthofrandomsteps,timebetweensteps)

𝐶(𝑟, 𝑡) =14𝜋𝐷𝑡� 𝑒y

JyJK C

z{\

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Mean-squaredisplacement

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Fick’slaw• Flux=(conductivity)x(drivingforce)• Foratomicormoleculardiffusion:“conductivity”=diffusivityconstant𝐷• Dreflectsthemobilityofthediffusingparticlesanddependsontheenvironment:e.g.Dislargerforparticlesingases,thaninliquids.

• “Drivingforce”=- concentrationgradient

𝐽 = −𝐷𝛻𝐶

𝐽𝑚𝑜𝑙𝑒𝑠𝑐𝑚2𝑠

= −𝐷𝑑𝐶𝑑𝑥

𝑚𝑜𝑙𝑒𝑠 ⋅ 𝑐𝑚y�

𝑐𝑚[𝐷] = 𝑐𝑚2/𝑠

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Fick’ssecondlaw(dynamics)

• 𝐽[ − 𝐽[L�[ = ���\𝑑𝑥

• Taylorseriesfor𝑑𝑥 → 0:𝐽[L�[ = 𝐽[ +

����[𝑑𝑥+. .

• ��[

𝐷 ���[

𝑑𝑥 = ���\𝑑𝑥

• ���\

=D�C�

�[C

x x+dx

𝐽[ 𝐽[L�[C

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Steady-stateconcentration

• Equilibrium concentration satisfies

𝛻 ⋅ 𝐷𝛻𝐶 = 0• Boundary conditions

C

X

C

X

constantflux→ 𝐷𝛻𝐶 = 𝑐𝑜𝑛𝑠𝑡. zerofluxatthe𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑖𝑒𝑠→ 𝐷𝛻𝐶 = 0

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Example of diffusion on acrystal lattice

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Diffusion mechanims insolids

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Interdiffusion:e.g.alloysAtomsmigratefromhighconcentrationtolowconcentrationregions

https://www.slideshare.net/IbrahimAbuawwad1/ch05-48057054

Diffusion mechanims insolids

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Vacancydiffusion• Atomsexchangepositionswithvacancies• Appliestosubstitutionalimpuritiesatoms

https://www.slideshare.net/IbrahimAbuawwad1/ch05-48057054

Diffusion mechanims insolids

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Self-diffusion:singleelementsolids,eg monoatomicmetals• Atomshopinthecrystallatticeanddiffusewhentheirthermalmotionexceedsanactivationenergyforself-diffusion

https://www.slideshare.net/IbrahimAbuawwad1/ch05-48057054

Diffusion mechanims insolids

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Interstitialdiffusion• Interstitialatomsdiffusebetweenatoms

https://www.slideshare.net/IbrahimAbuawwad1/ch05-48057054

Diffusion-limitedaggregation:crystalvsamorphousstructures

Meakin,Jamtveit,PNAS2009

• Limited– aseedparticleisplacedatthecenterandcannotmove• Diffusion– anotherparticleisaddedatanarbitrarypositionandisdiffusingtowardstheseedparticle• Aggregation– thediffusingparticlecanstickwiththeseedparticleoranyparticlewithintheaggregatecluster(probabilityofaggregation)• Theprocessofdiffusion-aggregationisrepeatedseveraltimes.

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Diffusion-limitedaggregation:crystalvsamorphousstructures

Meakin,Jamtveit,PNAS2009 49

Diffusion-limitedaggregation:crystalvsamorphousstructures

Meakin,Jamtveit,PNAS2009

PreferredlatticegrowthRandomclustergrowth

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Diffusion-limitedaggregation:amorphousstructures

Theamorphousclusterstendtobefractalsandhaveafractaldimension.

Whatisafractal?

Whatisafractaldimension?

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Whatisafractal?

• Self-similarityacrossmanyscales(scale-invariant)• Lackofanintrinsic

lengthscale• Don´tfilltheEuclideanspace

whichembedsthem

• Theirintrinsic(fractal)dimensionissmallerthantheEuclideandimensionofthespaceinwhichtheyareembedded

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DimensionsandrescalingobjectsDoublethesize,doublethemass

𝑀 = 𝑘𝐿&2𝑀 = 𝑘(2𝐿)&

𝐷 = 1

Doublethesize,quadruplethemass

𝑀 = 𝑘𝐿24𝑀 = 𝑘(2𝐿)2

𝐷 = 2

Doublethesize,massincreasesbyafactorof2� = 8

𝑀 = 𝑘𝐿�8𝑀 = 𝑘(2𝐿)&

𝐷 = 3

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Dimensionsandrescalingobjects𝐷 = 1

𝐷 = 2

𝐷 = 3

𝑀 = 𝑘𝐿�

𝑎𝑀 = 𝑘 𝑠𝐿 �

𝑎 = 𝑠�, 𝑠 > 1

𝑑 =log(⋕ 𝑜𝑓𝑝𝑖𝑒𝑐𝑒𝑠)log(𝑠𝑐𝑎𝑙𝑒𝑓𝑎𝑐𝑡𝑜𝑟)

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Fractaldimension:Sierpinski Gasket

• a= 𝒔𝑫

• Ateachiteration,twosidesofthetrianglearedoubled• 𝑠 = 2• Thenumberofnewtrianglesincreasesbyafactorof3• 𝑎 = 3

𝐷 =log 3log 2

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Fractaldimension:theKochcurve

• What’s the fractal dimension?

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Fractaldimension:theKochcurve

• a= 𝟏𝒔

𝑫

• 𝑠 = 1/3• Thenumberofsegmentsincreasesbyafactorof4• 𝑎 = 4

𝐷 =log 4log 3

> 1

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Kochsnowflake

• Equilateraltriangle

• ApplyKochcurvetoeachedge

• Perimeterincreasesby4/3ateachiteration→ ∞

• Areaisbounded

https://en.wikipedia.org/wiki/Koch_snowflake

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Fractaldimension• Amountof mass (orother measure that scales with the size)of

anobject inside acircle of radius𝑟hasapower-law relation

• DLAclusterin2D,𝐷 ≈ 1.7

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Arerealsnowflakesfractal?

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Arerealsnowflakesfractal?

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Nittmann andStanley,J.Phys.A1987

Arerealsnowflakesfractal?

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Nittmann andStanley,J.Phys.A1987