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LECTURE1:Disorderedsolids:Structureproperties
• Whatisandisorderedsolid• Howdowemeasuredisorder• Howareamorphoussolidstypicallyformed
1
Whatareamorphoussolids?
2
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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OpenCourseMIT3.017
BorateglassB2O3 (crystal)
7
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
10
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&)
11
Atomiccorrelations
• Probabilityofhavinganatomat𝒓2iscorrelatedwiththeprobabilityforanatomat𝒓&,ifthesepositionsarenottoofarapart.
• Thisisexpectedduetointer-particleforcesandconstraintsduetochemicalbonds.• Homogeneousmaterial:𝑛&(𝒓&) = 𝜌
• If 𝒓& − 𝒓𝟐 ≫ 1,atomsareuncorrelatedand:𝑛2(𝒓&, 𝒓2) ∼ 𝜌2
• Generallywecanwrite𝑛2(𝒓&, 𝒓2) = 𝜌2𝑔(𝒓&, 𝒓2)
12
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
13
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
14
PairDistributionFunction(PDF)
• PDFshowoscillationsduetonearestneighbor,nextnearestneighborshells,etc.
• Oscillationsdampedoutas𝑟increases
• 𝑔 𝑟 ∼ 1 forlarge𝑟
15
PDFforidealcrystalvsglasses
OpenCourseMIT3.017 16
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
23
OpenCourseMIT3.01724
Measureofdisorder:Excessmolarvolume• Energeticsofbonddensity
• Orderedcrystalhashigherpacking->morebondsperunitvolume•• Bindingenergy𝐸[\]^ < 𝐸 ^]aa
• Bindingenergy<->molarvolumeV
• 𝑉 ^]aa > 𝑉[\]^
• ExcessmolarvolumeΔ𝑉 = 𝑉 ^]aa − 𝑉[\]^ isameasureofdisorder
OpenCourseMIT3.017 25
Glassformationfromliquid
OpenCourseMIT3.017
CoolingcurveHeatingcurve
26
Glassformationfromliquid
OpenCourseMIT3.017
CoolingcurveHeatingcurve
𝑻𝒎
𝑻𝒎 functionofcomposition
27
Glassformationfromliquid
OpenCourseMIT3.017
CoolingcurveHeatingcurve
28
Glassformationfromliquid
OpenCourseMIT3.017
CoolingcurveHeatingcurve
29
Glassformationfromliquid
OpenCourseMIT3.017
CoolingcurveHeatingcurve
𝑻𝒈 dependson
(atommobility)-1X
(complexityofcrystalstructure)
X(coolingrate)
X(composition)
30
Glassformationfromliquid
OpenCourseMIT3.017
CoolingcurveHeatingcurve
ExcessVolume(disorder)
31
SummaryLECTURE1:Disorderedsolids:Structureproperties
• Whatisandisorderedsolid• Howdowemeasuredisorder• Howareamorphoussolidstypicallyformed
32
SummaryLECTURE1:• Whatisandisorderedsolid
• Solidswithshort-rangedisorder
• Howdowemeasuredisorder• Pair-correlationfunction• Excessvolume
• Howareamorphoussolidstypicallyformed• Glasstransition
Versus
33
LECTURE2Atomicdiffusion:equilibriumandnon-equilibriumstructures• Diffusionofatoms
• Diffusionlimitedaggregation,crystalgrowth
• Fractalsandfractalmeasure
34
Diffusion• Atany𝑇 > 0K,allatomsirrespectiveoftheiraggregation(gas,liquid,solid,etc)areinconstantmotion->thermalmotion
• Movementofatomsisassociatedwithcollisions->singleparticletrajectoryisazigzag– diffusiveparticle
• Acollectionofdiffusiveparticleshasanobservabledriftfromhightolowerconcentrations
35
Source:wikipedia36
Flow anddispersion
Molecular dispersion(Diffusion)
Dispersion by laminarchannel flow
37
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{\
38
Mean-squaredisplacement
39
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
40
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
41
Steady-stateconcentration
• Equilibrium concentration satisfies
𝛻 ⋅ 𝐷𝛻𝐶 = 0• Boundary conditions
C
X
C
X
constantflux→ 𝐷𝛻𝐶 = 𝑐𝑜𝑛𝑠𝑡. zerofluxatthe𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑖𝑒𝑠→ 𝐷𝛻𝐶 = 0
42
Example of diffusion on acrystal lattice
43
Diffusion mechanims insolids
44
Interdiffusion:e.g.alloysAtomsmigratefromhighconcentrationtolowconcentrationregions
https://www.slideshare.net/IbrahimAbuawwad1/ch05-48057054
Diffusion mechanims insolids
45
Vacancydiffusion• Atomsexchangepositionswithvacancies• Appliestosubstitutionalimpuritiesatoms
https://www.slideshare.net/IbrahimAbuawwad1/ch05-48057054
Diffusion mechanims insolids
46
Self-diffusion:singleelementsolids,eg monoatomicmetals• Atomshopinthecrystallatticeanddiffusewhentheirthermalmotionexceedsanactivationenergyforself-diffusion
https://www.slideshare.net/IbrahimAbuawwad1/ch05-48057054
Diffusion mechanims insolids
47
Interstitialdiffusion• Interstitialatomsdiffusebetweenatoms
https://www.slideshare.net/IbrahimAbuawwad1/ch05-48057054
Diffusion-limitedaggregation:crystalvsamorphousstructures
Meakin,Jamtveit,PNAS2009
• Limited– aseedparticleisplacedatthecenterandcannotmove• Diffusion– anotherparticleisaddedatanarbitrarypositionandisdiffusingtowardstheseedparticle• Aggregation– thediffusingparticlecanstickwiththeseedparticleoranyparticlewithintheaggregatecluster(probabilityofaggregation)• Theprocessofdiffusion-aggregationisrepeatedseveraltimes.
48
Diffusion-limitedaggregation:crystalvsamorphousstructures
Meakin,Jamtveit,PNAS2009 49
Diffusion-limitedaggregation:crystalvsamorphousstructures
Meakin,Jamtveit,PNAS2009
PreferredlatticegrowthRandomclustergrowth
50
Diffusion-limitedaggregation:amorphousstructures
Theamorphousclusterstendtobefractalsandhaveafractaldimension.
Whatisafractal?
Whatisafractaldimension?
51
Whatisafractal?
• Self-similarityacrossmanyscales(scale-invariant)• Lackofanintrinsic
lengthscale• Don´tfilltheEuclideanspace
whichembedsthem
• Theirintrinsic(fractal)dimensionissmallerthantheEuclideandimensionofthespaceinwhichtheyareembedded
52
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
53
Dimensionsandrescalingobjects𝐷 = 1
𝐷 = 2
𝐷 = 3
𝑀 = 𝑘𝐿�
𝑎𝑀 = 𝑘 𝑠𝐿 �
𝑎 = 𝑠�, 𝑠 > 1
𝑑 =log(⋕ 𝑜𝑓𝑝𝑖𝑒𝑐𝑒𝑠)log(𝑠𝑐𝑎𝑙𝑒𝑓𝑎𝑐𝑡𝑜𝑟)
54
Fractaldimension:Sierpinski Gasket
• a= 𝒔𝑫
• Ateachiteration,twosidesofthetrianglearedoubled• 𝑠 = 2• Thenumberofnewtrianglesincreasesbyafactorof3• 𝑎 = 3
𝐷 =log 3log 2
55
Fractaldimension:theKochcurve
• What’s the fractal dimension?
56
Fractaldimension:theKochcurve
• a= 𝟏𝒔
𝑫
• 𝑠 = 1/3• Thenumberofsegmentsincreasesbyafactorof4• 𝑎 = 4
𝐷 =log 4log 3
> 1
57
Kochsnowflake
• Equilateraltriangle
• ApplyKochcurvetoeachedge
• Perimeterincreasesby4/3ateachiteration→ ∞
• Areaisbounded
https://en.wikipedia.org/wiki/Koch_snowflake
58
Fractaldimension• Amountof mass (orother measure that scales with the size)of
anobject inside acircle of radius𝑟hasapower-law relation
• DLAclusterin2D,𝐷 ≈ 1.7
59
Arerealsnowflakesfractal?
60
Arerealsnowflakesfractal?
61
Nittmann andStanley,J.Phys.A1987
Arerealsnowflakesfractal?
62
Nittmann andStanley,J.Phys.A1987