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32
2.GrowthofABO
3andBO
2Crystals
kindsofparticles,thechemicalpotentialofaparticularkindofi-thcom-
ponentisgivenas
P
i
1
§
¨
©
w
w
N
G
i
·
¸
¹
TPN
,
ji
z
,
andthechangeinfreeGibbsenergyofasystemthatisheldatconstant
temperatureandpressureisgivenas
(2.8)
dG
1
¦µdN
i
1
n
1
i
i
InthermodynamicequilibriumwhendG=0
µdN
1
1
+
µdN
2
2
+}1
0
(2.9)
(2.10)
theaboveequationallowstoformulateconditionsofmassequilibrium
inaclosedsystem.Inthesimplestcaseofonecompoundexistingintwo
phasestheconditionofequilibriumofthesephasesis
µ
1
=
µ
2
(2.11)
andspontaneousmasstransportfromphase2tophase1occursattem-
peraturesorpressureswhenu
2>u
1.Itisalsopossibletochangethevalue
ofchemicalpotential,andthereforeinfluencetheequilibriumofthesys-
tem,byaddingacomponentthatdoesnotcrystallize[1].
Nucleationofthenewphase,inourcaseaclusterofatomsforming
solidmatter,canbediscussedintermsofthermodynamics,whenthedriv-
ingforceisΔu<0.TakingintoaccountGibbsfreeenergyofthevolume
ofagrowingclusteranditssurface,itwasshownthatforsomecritical
valueofradiusthesummaryfreeenergyreachesitsmaximum.Inoth-
erwords,whentheradiusofthegrowingseedcrystalisbiggerthanthe
criticalvalue,thecrystalbecomesstable[1,2].Thevalueofthiscritical
radiusdependsnotonlyonthermodynamicparametersofthecrystallizing
materialbutalsoonconditionsofcrystallization(formationoffreecrystal,
growthonsurfaceofthesamekindofcrystal-homogeneousgrowth,or
seedingonothersurface-heterogeneousgrowth).Thekineticsofseeding
(nucleationrate)canbeexpressedtakingintoaccountthefrequencyand
thecross-sectionofcollisionsofmonomerswithgrowingclusters,and
theirdissociationrate.Thenucleationrateisexpressedasthefrequency
oftheappearanceofaclusterwithdimensionsbiggerthanthecriticalra-
dius.Thefurthergrowthontheperfectcrystalsurfacedemandshighsu-
persaturationthatisneededtoformtwo-dimensionalseedsformingsteps
onwhichfurthergrowthispossible.Ontheotherhand,crystalgrowth