Table Of ContentCHEMISTRY OF FUNCTIONAL MATERIALS
SURFACES AND INTERFACES
CHEMISTRY OF
FUNCTIONAL
MATERIALS
SURFACES AND
INTERFACES
Fundamentals and Applications
A H
NDREI ONCIUC
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Preface
Inthesetimesitisundeniablethatmostindustriesdeal evenlesssoacquirearelevantpracticalexperience.Nav-
increasinglymoreoftenthaneverwithsurfaceandinter- igatingthroughthemazeofscientificliteratureandinfor-
facialphenomena.Chemists,physicists,andmaterialsci- mation hard to decipher can be extremely intimidating
entists, with background and training in materials, for the future chemists. The same is true in industry;
surfaces, and interfaces are in great demand. From my working in industry one realizes that time is of the
experience in both industry and academia, I have essence. Chemists and laboratory technicians are
observed that students attending courses of a general expected by the company to be innovative and thrive
chemistry degree program encounter rather late in their in interdisciplinary fields, learn on the go, and become
curriculum courses dealing with interfacial phenomena expertsintheshortestamountoftime,onthejob.There-
and chemistry of interfaces. Some curricula have fore,Ifeelthatthisbookwouldbeusefulasatextbookfor
included the chemistry of interfaces under different for- students, chemists working in industry, and laboratory
mats, at the undergraduate level, some only at master’s techniciansfirstencounteringthechemistryofinterfaces,
level, in specialized modules. This can in part be interfacial phenomena, colloids, nanotechnology, poly-
explainedbythefactthatChemistryhasbecomeanenor- mer nanoparticle synthesis, etc. I have used myself part
mously vast array of scientific domains, branching into of this material in my teachings both in academia and
biochemistry, organic chemistry, physical chemistry, trainingoftechniciansfromindustry.Whilethismaterial
catalysis,industrialchemistry,inorganicchemistry,ana- used as a coursework material at master’s level has ini-
lytical chemistry, materials chemistry, nanotechnology, tially included much more theory and formula, I could
polymer chemistry, petroleum chemistry, etc. Due to feel the students had difficulties grasping these, due to
fecund research in the past two decades Chemistry of the pressure, lack of time, and an extremely burdening
functional materials and interfaces covers a multitude curriculum. I thus preferred to make the hard choice of
of intertwined interdisciplinary subjects from the nano- reducingthematerialonlytoessentialtheoriesandadopt
scale,suchassynthesisofpolymericandinorganicnano- amoredescriptiveandintuitivepresentation.Oneofthe
particles, to macroscopic phenomena such as leitmotifsofthebookistheemphasisonpracticalappli-
manufacturing of functional surfaces, food, and con- cations of such theories. After several years in refining
sumer products such as cosmetics,detergents,heteroge- this material I believe it came to a format well received
neous catalysts, etc. The amplitude and the amount of by the students. In addition, to make it more useful for
information in each of these fields put pressure on stu- chemistsperforminginterfacialexperiments,inindustry
dents,moresothanseveraldecadesago;itisnowharder or academia, I have tried to add experimental details or
for the students to keep track of the newest advances, hints on data interpretation from my own experience.
vii
C H A P T E R
1
Introduction
Interfacesaretheboundariesseparatingtwophasesanddefineallobjectsinthethree-dimensionalworld.Depend-
ingon thestrengthof cohesionforces andbinding energiesbetweenatomsandmolecules, thephasescanbegases,
liquids,andsolids,definingthephysicalstatesofmatter.Whenthecohesionenergiesbetweentheconstitutingatoms
andmoleculesarestrongerthanrandomizingeffectsofthethermalenergy,thephysicalstatechangesfromgastoa
condensedphaseofmatter,liquid,orsolid.TheBoltzmanndistributiongivestheprobabilityPthatasystemwillbein
a certain state as a function of the state’senergy and temperature:
P(cid:1)e(cid:3)E=kT
kTfactorisoftenusedasascaleenergyfactorinthemolecularinteractions.Thecohesiveenergiesperatomormolecule
at298KcanvaryfromseveralkTbetweengasatoms,between(cid:1)9and(cid:1)23kTinliquidHg(theliquidwiththestrongest
cohesiveenergy,57.9kJ/mol[1]),and>50kTinsolidsupto342kTinW(1kT(cid:4)4.05(cid:5)10(cid:3)21J),themetalwiththehigh-
est melting point. The kT energy scale factor is introduced and discussed in detail in Chapter 2. Because the most
importantinteractionsbetweenmaterialinterfacestakeplaceintheliquid,orbetweenmaterialinterfacesandliquids,
the solid-liquid, liquid-liquid, and liquid-air interfaces deserve special attention. The overall balance between the
repulsiveandattractiveforcesbetweensolutesandcolloidalobjectsinliquidsmustbecomparativelyequalorlarger
than9–23kTtohaveaggregation,adsorption,self-assembly,etc.,andbelow9kTtoobtainstabledispersionsandcol-
loids.Asmentioned,liquidsformatT¼298K,whenthecohesiveenergybetweentheconstitutingatomsandmole-
culesislargerthan9kT.Whileinthebulkofaliquidtheinteractionforcesofamoleculeoratomarefullysymmetricat
interfaces, in contrast, in the topmost layer of molecules or atomsthe interaction forcesare asymmetric.Dueto this
asymmetry,acertaintension/forcearisesintheplaneoftheinterface.Thestrongertheinterfacialtension,thestronger
theasymmetry.Atcontactbetweentwophases,thetopmostlayerofmoleculesatthephaseboundaryalsointeracts
withthemoleculesfromtheotherphase,thisiscalledadhesion.Theadhesionforcesandenergiescounterbalancethe
asymmetry of the forcesacting on the topmost molecular layer, i.e., the stronger the adhesion force, the smaller the
interfacial tension. If the adhesion force is stronger than the cohesion force, then the interfacial tension disappears,
the interface disappears, and the phases become fully miscible, as discussed in Chapter 3. This interfacial tension
hasalsothecharacterofanenergydensity,andforpristineinterfacesthisiscausallyrelatedtothecohesionenergy
inthebulkmaterial;interfacialenergydensityisabouthalfthecohesionenergyinbulk.Surfaceandinterfacialtension
ofliquid-gasandliquidinterfaces,aswellasinterfacialandsurfaceenergyofsolids-liquidandsolid-gasinterfaces,are
thoroughlydiscussedinChapter3.Theeffectsoftheinterfacetensioncanbeseeninsmallliquiddropletsormolten
metals,astheshapeofthedropletitselfismodeledbythisinterfacialtension.Thesmallworldofinsectsandbugsare
particularlyaffectedbytheinterfacialtension.Becausetheirsizeiscomparabletothecapillarylength,whentheshape
oftheliquidsisfullydeterminedbyinterfacialtension,notbygravitation,theyhaveadifferentperceptionofthesur-
roundingworldthanhumansdo.Interfacialtensioncanhaveadevastatingeffectoninsects;somedrownastheycan-
notescapethesurfacetension,butsomehaveadaptedtotakefulladvantageofit.Forexample,smallwaterdroplets
canbemanipulatedandtransportedbyantswithoutanyneedforbottlesorglasses,andsomemosquitoshaveadapted
on water to straddle along the smooth watersurface, etc. (Fig. 1).
Intuitively,theinterfacialtensionisthe2Dequivalentofthecohesionenergyin3D.Interfacialtensionisdiscussedin
detail in Chapter 3.
However, when the surface and the interface are chemically modified, e.g., withsurfactant adsorbates, the inter-
facial tension and energy density of interfaces do not reflect anymore the cohesive energy between the molecules
in the bulk phase. Thus, the interface itself can be treated as a thermodynamic system on its own, as discussed in
Chapter 7. The interfacial tension and interfacial energy density between phases are now an exclusive reflection of
1
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https://doi.org/10.1016/B978-0-12-821059-8.00004-1
2
1. Introduction
FIG.1
(A)Antdrinkingwater(https://www.shutterstock.com/image-photo/ant-drinking-water-505718482);(B)mosquitostridingonthesur-
faceofwater(https://www.shutterstock.com/image-photo/water-bug-standing-on-surface-calm-1732352752).
thelateralinteractionsbetweensurfactantmolecules,polymers,orparticlesadsorbedattheinterface.Infact,under-
standinghowtochange theinterfacialtensionandenergydensitybetweenphaseswasoneofthekeyenabling ele-
mentsinthedevelopmentofmosttechnologicaladvancesinthe20thand21stcenturies,rangingfromdetergency,oil,
and ore extraction to the advancedmanufacturing of processors and advanced electronic devices(seeChapter15).
Surfactants andamphiphiles aremolecules, polymers,andotherbuildingblocksof matter thatadsorb spontane-
ouslyatinterfaces.Surfactantslowertheinterfacialtensionandenergydensitybetweenphases(water-oil,water-gas,
solid-water)independentlyoftheircohesionenergy.Thisenablestheformationofemulsionsandfoamsandincrease
insurfacewettability.Earlier,itwasmentionedthatwhentheadhesionforcesarestrongerthanthecohesionforces
betweentwoliquids,theinterfacialtensionvanishes,andtheliquidsbecomemiscible.Thefactthat,inthepresenceofa
surfactantatinterfaces,theinterfacialtensionisnotanymoreatruereflectionofthebulkcohesionenergyofthephases
canbeunderstoodfromthefollowingexample.Iftheinterfacialtensionbetweentwowaterandoilphasesbecomes
vanishinglysmallduetotheadditionofasurfactant,thenthetwophasesdonotmix,butthistimetheyformemulsions
consistingofveryfineoildropletsdispersedintowater.Chapter4givesanintroductionintothevastfieldofsurfactant
chemistry.Emphasisisgivenonsurfactantclassification,surfactantdesign,andstructureactivityrelationship.Insim-
ple words, what makes a surfactant effective and how is this reflected in different physicochemical parameters?
Chapter4alsointroducesotheramphiphiles,suchasJanusnanoparticlesandsupra-amphiphiles,notingthatamphi-
philicity is a scalable property, being active well beyond the molecular scale, well into the nano- and microscales.
Amphiphilesandsurfactantshaveanimportantproperty,whichistoself-assembleintosuprastructures.Thisenables
thecreationofsmart,reconfigurable,or“environmentallyaware”materials,bottomup,viaself-assemblyprocesses.
Most of the surfaces we interact with on a daily basis are solid, such as the screen of the smartphone, the cup of
coffee,thewheelofthecar,etc.Thetactilefeel,theadhesion,isdeterminedbytheinterfacialenergybetweenourskin
andthesesurfaces.Inthemodernworld,theconceptoffunctionalsurfacesisgainingmorepopularityanditbecomesa
requirementintheconsumerproducts.Functionalsurfacescanbedefinedassurfacesthatperformafunction,suchas
self-cleaningwindows,orhaveasuperiorproperty,suchasantiadherent,omniphobicantifingerprintinsmartphone
screens,forexample,whileothersareicephobic,orantifogging,etc.Thekeyconceptsinunderstandingthephenom-
enabehindfunctionalsurfacesandinterfacesareadhesionandwetting.Surfacewettingrefersmainlytotheinteraction
ofaliquidwithasolidsurface.Earlier,itwasmentionedthatwhentheadhesionforcesarestrongerthanthecohesion
forcesbetweentwoliquids,theinterfacialtensionvanishes,andtheliquidsbecomemiscible.Theinterfacialtensionor
energybetweenasolidandaliquidcanalsobealtered,forexample,withsurfactants;however,whentheinterfacial
energybetweenasolidandaliquidbecomesvanishinglysmall,thesolidsurfacebecomesfullywettedbytheliquid.
Theconverseistrue:whentheinterfacialenergyislarge,thesurfacebecomesnonwetted,andtheliquidpearlsupon
thesurfaceofthesolid.Scientistshavelearnedthat,inadditiontointerfacialenergybetweenthesolidandliquid,the
geometryoftheinterfaceiskeytodesigningfunctionalsurfaces.Findinginspirationinnature,scientistsfoundoutthat
hierarchicalstructuringofthesurfaceofthesolidcanleadtoavarietyoffunctionalsurfaces,suchassuperhydropho-
bic, superhydrophilic, icephobic, omniphobic, self-cleaning, etc. Chapter 5 gives an overview of the phenomena of
3
Reference
wetting, wettability, and contact angle as the main measurement methods for macroscopic and nanoscale surfaces.
Chapter 5 also introduces the several functional surfaces.
InChapter6,aseriesofequationspermittingthecalculationofunknownsurfacetension,energy,workofadhesion,
etc.fromknownmeasurablemacroscopicparametershavebeengroupedunderthename“fundamentalequationof
interfaces.”Theirversatilityinpredictingthevaluesofmanyinterfacialparameters,forexample,interfacialtension,
wettability,polarityofthesurface,etc.fromcontactanglemakesthemextremelyusefulinpractice.InChapter7,the
surface and interfacial tension are introduced via thermodynamic treatment of the interfacial layer. Although this
treatment has no direct practical implications, it gives the theoretical background necessary for the interpretation
of interfacialadsorption isotherms andinterfacialtension vs concentration curves for surfactantsand amphiphiles.
Chapter 8 treats surface functionalization that can be achieved in different ways, by physical methods such as
rougheningofthesurface,orphotolithographicnanopatterning,andbychemicalmethods,byadsorptionofsurfactant
molecules. The adsorption of surfactant molecules on solid surfaces involves either chemical or physical bonding,
resulting in the formation of a self-assembled monolayer. Several types of chemical bonding and substrates are
reviewed.Inaddition,asurfactantmonolayercanbepreparedfirstatthewater-airinterfaceandthentransferredonto
the surface of the solid viathe Langmuir-Blodgett and dip-coating methods.
Solid-solidinterfacesalsohavepracticalrelevance,especiallyinlayeredelectronicdevices.Solid-solidinterface,in
particularthemetal-organicinterface,isthelocusofanothertypeofphenomenaofpracticalimportance,namelythe
electrontransfer.Inthepreviouschapters,theinterfacesweretheplacewheredifferentforcesmet.InChapter9,the
metal-organicinterfacesaretreatedasthecontactpointbetweenelectronenergylevelsofametal,materialwithdelo-
calizedelectronenergylevelscalledbands,andtheorganicmoleculesandpolymerswhoseenergylevelsarediscrete
andlocalized.Understandingelectrontransferbetweenmetalelectrodesandorganicconductorsisofpracticalimpor-
tance,especiallyforthemanufacturingoforganicphotovoltaics,organiclightemittingdiodes,andotherorganicelec-
tronicdevices.Anyofthesedevicesrequiresatleastseverallayersofelectroactiveorganicmaterials,andknowledgeof
adhesion,wettability, and interfaces is requiredfor their developmentand manufacturing.
Chapters10and11dealwiththeinteractionforcesandenergiesbetweeninterfacesindifferentmedia.Theseinter-
actionforcescanberepulsiveorattractiveandtheyarethesameforcesgoverningthemolecularinteractions.Thebal-
ancebetweentheattractiveandrepulsiveinteractionforcesisofpracticalimportance,controllingthephenomenaof
particleaggregation,colloidstability,particleadsorptiononsurfaces,self-assemblyofnanoparticles,etc.Chapter12
introducescolloids,whicharetheoldesttypeofnanomaterialsknownandaretodayencounteredinthefoodindustry,
pharma,andmanyotherconsumerproducts.Colloidsareconstitutedfromfinelydividedparticles,nanoparticles,or
liquiddropletsdispersedintoacontinuousmedium.Becausetheirsurface-to-volumeratioisveryhigh,theirbehavior
isgovernedalmostexclusivelybytheirsurfaceandinterfacialproperties.Synthesisofcolloidsaswellasstabilitycri-
teria is discussed.
Asacontinuationonthetopicofcolloids,butdeservingspecial attention,Chapter13introducesthesynthesis of
polymeric nanoparticles and polymeric nanostructured interfaces via emulsion polymerizations. As expected, the
interfacialaspectsdeterminethetypesofemulsionsandnatureofthenanomaterialsthatcanbesynthesized.Thetypes
of emulsions and conditions of formation are briefly reviewed. A case study covers some examples of synthesis of
nanostructured interfaces, polymerization of the emulsionsstabilized by amphiphilicparticles.
Some nanoparticles, depending on their surface properties,canalso spontaneouslyadsorb atinterfaces; they can
formmonolayersandstabilizeemulsions.Thefactorsresponsibleforwhysomeparticlescanadsorbatliquid-liquid,
liquid-gas,andsolid-liquidinterfacesarediscussedinChapter14.Onceadsorbedattheinterfacestheparticle-particle
interactions leads to the decrease in the interfacial tension. Responsible for this is their lateral interaction, which is
governedbythesametypesofforcesasincasesurfactants,andinadditionbyparticlespecificinteractions,capillary
floatation,orimmersionforces.Infact,inrecenttimes,nanoparticleshavebeenusedinthesynthesisofphotoniccrys-
tals via the Langmuir-Blodgett method andother self-assembly structures.
Thelastchapterofthisbookdiscussestheroleofinterfacesinintegratedcircuitmanufacturingviaphotolithogra-
phy.Photolithographyistheonlytop-downpreparationmethodofnanomaterialsandnanostructuredsurfaces.Inthe
pastfewyears,itevolvedintothemostprecisetechniquetopreparewithlargemachines,structuresassmallas7nm
(thegateofthefield-effecttransistor).Inpractice,thephotolithographicmanufacturingprocessofchipsandprocessors
requires in-depth knowledge and control of interfacial phenomena such as adhesion, wetting, capillary forces, and
interfaces.
Reference
[1] G.Kaptay,G.Csicsovszki,M.S.Yaghmaee,Anabsolutescaleforthecohesionenergyofpuremetals,Mater.Sci.Forum.414–415(2003)235–240.
https://doi.org/10.4028/www.scientific.net/MSF.414-415.235.
C H A P T E R
2
Thermal energy scale kT
Atthenanoscale,theinteractionenergiesaregenerallyexpressedinmultiplesofkT,alsoreferredtoasthethermal
energyscale.Theaveragekineticenergyofagasatomwiththreedegreesoffreedomis3/2kT,isroughlytheenergyof
thermal fluctuations at a given temperature (cid:1)1kT. The thermal energy has a randomizing effect contributing to an
increaseintheentropyofthethermodynamicsystem.Byexpressingtheenergyofintermolecularinteractionsornano-
particle interactions as multiples of kT, the interaction strength can be compared with the randomizing effect of
temperature.
Next,itisinstructivetofollowthekTinseveraldifferentcontextsaswellasitsorigin.Athermodynamicsystemwill
tendtomovetowardalowerenergystatewhenavailable.Whenappliedtochemicalsystems,forexample,asolutein
asolutionoragashasachemicalpotentialdefinedastherateofchangeoftheGibbsfreeenergywiththenumberof
species in thesystem, atconstant temperature and pressure:
(cid:1) (cid:3)
δG
μi¼ δN (2.1)
i T,P
Therefore,thechangeinchemicalpotentialofagasorasoluteinasolutionchangeswiththechangeinconcentration.
Chemical potentials are important in describing the equilibrium in physicochemical processes such as evaporation,
melting,boiling,solubility,interfacialadsorption,liquid-liquidextraction,etc.Thereasonwhythechemicalpotentials
aresoimportantintheequilibriumchemistryisthatwhenthetwochemicalsystemsareopenandcanexchangemol-
eculesoratoms,therateofchangeoftheirfreeenergywouldbeequalwhenequilibriumisestablished.Take,forexam-
ple,themoleculesinthevaporandtheliquidphaseatequilibrium;byequatingthechemicalpotentialsofthemolecule
of type iin two phases at equilibrium, or two regions 1and 2, we obtain
μ1i +kT lnXi1¼μ2i +kT lnXi2 (2.2)
At equilibrium between n different phases, the aboveequality must besatisfiedforall phases:
μni +kT lnXin¼constant (2.3)
where Xin is the molecular fraction, volume fraction, or concentration of solute in phase n. For pure solution, this is
usuallytakenasunity.ThefactorklnXisknownunderdifferentnames,suchastheentropyofmixing,configuration
entropy, entropy of confining themolecules, etc.
Eq. (2.2) gives us the possibility to calculate the distribution of molecules between two phases, or two regions of
space at equilibrium, for example, a liquid in equilibrium with its vapors, or the distribution of the molecules of
gas in the atmosphere due to changes in the gravitational potential with altitude. For example, the number density
ρzof themolecules of gas in theEarth’satmosphere changeswith the altitude z and themathematical functionthat
gives us the possibilityto predict this change is
μzi +kT lnρzi ¼μ0i +kT lnρ0i (2.4)
where ρiz is the number density of molecule i at altitude z and ρi0 is the number density of molecules of gas i at the
surfaceoftheEarthz¼0.Rearrangingtheaboveformulagivesusthebarometricformulaorbarometriclawthatgives
the density at the altitudez as afunction of the number density of air molecules at the sea level ρi0:
5
ChemistryofFunctionalMaterialsSurfacesandInterfaces Copyright©2021ElsevierInc.Allrightsreserved.
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6
2. ThermalenergyscalekT
(cid:1) (cid:4) (cid:5)(cid:3) (cid:1) (cid:3)
(cid:3) μz(cid:3)μ0 (cid:3)ðmgzÞ
ρzi ¼ρ0i exp kiT i ¼ρ0i exp kT (2.5)
where misthemolecular massand gisthe gravitationalacceleration.Note that thepotentialenergyofthe air mol-
eculesmgzis“compared”tothekTatanyheightabovetheEarth’ssurface.Withtheincreaseinthepotentialenergyof
themoleculescomparedtokT,lessmoleculesarefoundathigheraltitudes(Fig.2.1).Inotherwords,ifmgzissmall
comparedtokT,thenthethermalenergywoulduniformizethedistributionofmoleculeswiththealtitudesuchthat
little variation in the number density of air molecules wouldbe registered.
The same distribution applies to ions that, for example, carry a charge e between two different regions that have
differentpotentialsψ and ψ :
1 2
(cid:1) (cid:3)
(cid:3)eðψ (cid:3)ψ Þ
ρ2i ¼ρ1i exp k2T 1 (2.6)
andthisisknownastheNernstequation.Itisnonethelessimportanttonotethatinteractionsareadditive;forexample,
ifthedifferenceinenergybetweentworegionsisgivenbypotential,potentialenergy,andchemicalpotential,thenthe
exponentwill be the sum of all these contributions.
Theaboveequationsalsogiveusthepossibilitytogaugethestrengthofinteractionbetweenmolecules.Forexam-
ple,ifaliquidisinequilibriumwithitsvaporsatstandardconditionsofpressure1atmandtemperature298K,then
1mol of gas will occupy approximately 22.4m3 and a mole of liquid approximately 0.02m3. Then the difference in
energy between the liquid and gasstates willbe [1]:
Xgas 22:4
μ0gas(cid:3)μ0liquid(cid:1)kT ln i (cid:1)kT ln (cid:1)7kT (2.7)
i i Xliq 0:02
i
where 1kT is approximately the energy of the thermal fluctuations. Therefore, it can be said that if the interaction
strength between molecules in a gas phase at temperature T is larger than 7kT, then it condenses into liquid. Con-
versely,ifthecohesionstrength betweenthemoleculesofaliquid becomesmallerthan7kT,thenittransformsinto
gas as the cohesion energy is simply too low to hold the molecules together. This alludes to what is known as the
Trouton rule, which states that the entropy of vaporization isroughly the same for different kinds of liquids, about
85JK(cid:3)1mol(cid:3)1, which is roughly9.5kT.
FIG.2.1
Thebottlewascapped(left)inthemountainandbroughttothegroundlevel(right).
7
2. ThermalenergyscalekT
FIG.2.2 VariousinteractionenergiesonthekTscale.
The kT criterion can be generalized to gauge the interaction strength between molecules; as stated above, if the
interaction between molecules in a medium is larger than 9.5 kT at a given temperature, then this interaction will
dominateoverthethermalfluctuationsandformacondensedphase,duetoaggregation,adsorption,orself-assembly.
Forinteractionenergies,theuseofthekTenergyscaleisconvenient,as1kTequalsthethermallyinduced3DBrownian
motionenergyofamolecule(surfactant,orsolute,orparticle),whichprovidesareferencevalueofinteractionenergies
formoleculessticking together vsflyapart,binding vsunbinding,etc.(Fig.2.2).Fig.2.2providesavarietyofinter-
action energiesrepresented on the kT energy scale.
Similarly,the kT factoris also met in kinetics. For example, the Arrhenius equationper molecule is
E
k¼Ae(cid:3)kTa (2.8)
where Ea isthe activation energy barrier and k is therate constant of the reaction. If, for example, Eais muchlarger
thankT,thenthereactionrateisalsoverysmall.Ontheotherhand,iftheenergybarrieriscomparabletokT,thenthe
reaction rate ishigh,and the reaction can beactivated by the thermal energy.
The kT factor is also encountered in the Boltzmann distribution, which is a probability distribution that gives the
probabilityof a state to existfunction of the state’senergy and temperature and it is given by
(cid:1) (cid:3)
E
exp (cid:3) i
kT
P¼Xn (cid:1) E (cid:3) (2.9)
exp (cid:3) j
kT
j
wherePistheprobability ofstatei,oftheenergy Ei,andnisthetotalnumberofaccessiblestatesofcorresponding
energiesEj(j¼1(cid:3)n).TheBoltzmanndistributiondescribesthedistributionofparticles,suchasatomsormolecules,over
allaccessibleenergystates.Inasystemconsistingofmanyparticles,theprobabilityofpickingarandomparticlewith
theenergyEiisequaltothenumberofparticlesinstateidividedbythetotalnumberofparticlesinthesystem,thatis,
the fraction of particles occupying the state i:
(cid:1) (cid:3)
E
exp (cid:3) i
N kT
Pi¼N i ¼Xn (cid:1) E (cid:3) (2.10)
total exp (cid:3) j
kT
j
