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Vacuum and Ultravacuum Physics and Technology Igor Bello

Thermodynamics of Gases at Low Pressures

Publication details https://www.routledgehandbooks.com/doi/10.1201/9781315155364-3 Igor Bello Published online on: 09 Nov 2017

How to cite :- Igor Bello. 09 Nov 2017, Thermodynamics of Gases at Low Pressures from: Vacuum and Ultravacuum, Physics and Technology CRC Press Accessed on: 25 Sep 2021 https://www.routledgehandbooks.com/doi/10.1201/9781315155364-3

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The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The publisher shall not be liable for an loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 function. but it only depends on the initial and final gas states. Therefore, the energy is a state the energy changedoesnotdependonthewayofgaschangesfrom onestatetoanother, energy. Boththeenergy andworkhaveequivalent effects. Theexperiments showthat Equations 3.1and3.2represent thelawofconservationenergy, includingmechanical used forincreasing theinternalgasenergy andperformingthemechanicalwork.

saturated vaporpressures ofsolidandliquids,evaporationprocesses atvacuum chapter. Thelawsofthermodynamicscanbeappliedtophasetransformation,inherent gases, gasheatcapacities,andtheirrelationships atlowpressures are discussedinthis gases at low pressures. Among them, internal energy of gas, enthalpy, work performed by Thermodynamics assistsinelucidatingimportantphysicalquantitiesandbehaviorof Thermodynamics ofGases at LowPressures 3 For thepracticalreasons, wecanintroduce anewstatequantity which which canberewritten totheequation that is, equivalent tothedifferential portionofmechanicworkdWperformedbythegassystem, a differential valuedUisequaltothesuppliedthermalenergy dQsubtractedinanenergy of individualmoleculesandtheirownenergies. Theincrease oftheinternalgasenergy by If gasmoleculesare invariable,theinternalenergy isequaltothesumofkineticenergies 3.1 conditions inthin-filmtechnology. By addingandsubtractingVdp,thefirstlawofthermodynamicscanfurtherbe recast to

First LawofThermodynamicsandEnthalpyAppliedtoIdealGases is thefirstlawofthermodynamics. Qd pdV dU dQ =+ Qd Wd pdV dU dW dU dQ +=+ =+ += Ud dW dQ dU Vd H 71 Accordingly, thesuppliedheatcanpartially be pV = =+ –– Up pdU dp – V

+ dp ()

VV dp

(3.4) (3.3) (3.2) (3.1) 115 Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 and theworkdWisneithertotaldifferential northestatequantity. work performedatatransitionfrom onestatetoanotherdependsonthetransitionpath, the internalenergy isthestatefunction,anditther gives thefirstthermodynamiclawinform equivalent totheheatabsorbedbygas.Then,introducing enthalpyintoEquation3.3 Hence, thechangeinenthalpycanalsobeexpressed byequationdH=dU+dW,being volume bythedifferential portiondVimpliesthatthegasperformsworkdW=pdV. at constantpressure isdH=dU+pdV. Accordingly, atconstantpressure, theincrease of enthalpy byfulldifferential ofenthalpy, thatis,dH=dU+d(pV)VdppdV,which systems. When we consider chemical reactions inside a gas, we can express the change in Enthalpy isthemeasurablequantitythatdescribesinternalenergy ofextendedgas additional energy equivalent tothemechanicalworkpVthatisstor does contain not only the internal energy of gases that is stored in molecules but also called 116 physicist andchemistNicolas Clément-Desormesintroduced thisheatunit Using thisapproach, theunitofcalorie(cal)wasintroduced. Forthefirsttime,French ties thatraisethetemperature ofdefinedamounts of asubstancebyequaltemperatures. further discussedinChapter5ontransferphenomena. are greater than100Pa. transfer byconventionbecomesnotableonlyintherough vacuumregion whenpressures the transferofheatbyconvectionprocess cantechnicallybeenforced. However, heat the gasintomotion.Innature, theconvectionphenomenonisknownaswind. Inpractice, local gas temperature, which violates the mechanic balance of the gas system and drives temperatures. radiation mayoverwhelmothermechanismsof theheattransferevenatlower In vacuumconditionsandevenatalowertemperature differences, transferofheatby heat betweentwoobjectsbyradiationisveryeffective atlarge temperature differences. the heat transfersexistsinceall objects can emitand absorb radiation energy. Transfer of tion, convection,andconduction. and finaltemperatures ofthesegassystems.Ingeneral,heatcanbetransferred byradia- tures viaheattransferphenomena.Themeasure oftheheattransferdependsoninitial Two gassystemswith different temperatures haveatendencytoequalizetheirtempera- 3.2 15.5 to raisethetemperature of1gchemicallypure wateratstandard atmosphere from 14.5to calorie in1824.Oneoftheseveral introduced calories isdefinedastheheatquantityneeded Since thewayofpassinggassystemfrom onestatetoanotherdoesnotplayanyrole, In heatmeasurement, itispossibletochooseaheatunit basedonequalityofheatquanti- The third formoftheheattransferbyconductiondependsongasconductivity, whichis In heattransferbyconvection,themassdensityofa gas altersuponthevariationof Heat transferbyradiationdependsonlyonthetemperature ofobjectsbetweenwhich

Definition ofGasHeatCapacities °C. Thisisknown as15°Ccalorie.Calories were alsodefinedat othertemperatures, enthalpy. The quantity of enthalpy is introduced because the energy of the system QdH dQ = Vacuum andUltravacuum:Physics and Technology – Vd p

efore atotaldifferential. However, the ed intheenvironment. 72 askilogram- (3.5) Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 use 15°Ccalorieforwhichtheconversionis 4.184 spheric pressure from 0°Cto1isalsocalledcaloriebutwiththerecalculation of1cal= calories. For example, heat needed to increase temperature of 1 recalculation factorfrom caloriestojoulesmaydeviatebecauseofdifferent definitionsof Despite the intensionof the definition of theheat inJouleswithoutany reference towater, with theSIunitsystem,heatisJoule(J),and15°Ccalorieequalto4.1855. of heatasaformenergy ledtothesameunitsforbothheatandenergy. Inconsistence the abilitytoincrease temperature ofthedefinedsubstanceamounts.Theunderstanding pound ofwaterfrom 63to 64°F.Thesedefinitionscomefrom theunderstandingofheatas duced asaheatamount.TheunitBtuistheamountthatraisestemperature ofa used frequently. IntheBritishImperialUnitSystem,athermalunit(Btu ) wasintro- including 4°C,17and20calories. A multipleunitofkilocalorie( kcal) hasalsobeen Thermodynamics ofGases at LowPressures where which iscalledmolarheatcapacity.Thus,definedbyequation many calculations,itisalsousefultorefer theheatcapacitytoagasamountofkmol(mol), which is r Since twogassystemsofthesamekindhavecapacitiesproportional totheirmasses,for called theheatcapacity,whichinmathematicalformis heating conditionsofgassystems. Accordingly, atconstantvolume, theheatcapacityis pressure ofagassystem ismaintainedconstant.Thus,theheatcapacityhasto refer tothe refers tothechemicalnature ofagas. The magnitudeoftheheatcapacityappliedtoaquantity of1kmol(mol)isuniqueand

eference, theheatcapacityrelated tothegasmassofunityisdefined: M n The neededheattoincrease thetemperatur The heatcapacitycanbedetermined undertheconditionsatwhicheithervolume or M isthemassofgasinkg m a =M/ isthemolarmassinkg/kmol J. Theamountoftheheat4.184standsforthermochemicalcalorie.Herein, we called specific heat capacity. The specific heat capacity is unique for each gas. In a isthegasquantityinkmol 0415 6 10 967 3 1855 4 10 1 ca lk == C V ¢ C -- = 33 == () ca nn dQ C dT c mm lJ ¢ == VV C MM C = .. 1 ¢ ¢ = () dQ dU dT dQ dT dT e ofagaswithmassMbydegree is 1 =

dQ dT =´ M = M

è ç æ a ¶ ¶ dQ dT U T ö ø ÷

Bt g water at standard atmo- u

(3.10) (3.9) (3.8) (3.7) (3.6) 117 Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3

environment andviceversa. decrease inenvironmental temperature results inaheattransferfrom agassystemtothe conditions canoverturnthedirection oftheprocess. Forexample,anextremely small These processes relate toidealbalancedsystems,where eventinychangesofthesystem respectively, followingfr and atconstantpressure, theheatcapacityis 118 from where themolarheat atconstantvolumeis described byequation nal energy ofthegas.Then,intermsthermodynamics,isochoric process is only raisesitstemperature, andaccordingly increases thegaspressure and thustheinter an idealgashastobezero aswell(dW=pdV0). As aresult, theheatsuppliedintogas Thus, volume change is invariable (dV = 0), and consequently the work performed by such heated orcooledwhilethegasvolumeVismaintainedconstant,asillustratedin Figure 3.1. sure In general,idealgassystemsare describedbythree macroscopic parameters,thatis,pres- 3.3 where temperatures from T For instance,theintegrationofEquation3.8withinboundariesheatfrom 0toQand of temperatures allowsustoseparatevariablesandintegrateequationsforheatcapacities. increases itstemperature. Theassumptionofconstantheatcapacityinaconsidered range the definitionofheatcapacitiesmayimplythatsuppliedintogassystem energy UandenthalpyH . Now, intermsofthermodynamics,wemayanalyzethesimplestreversible processes. T isthefinaltemperature ofasubstance T M isthemassofsubstance c isthespecificheatcapacity Unless heatisusedforperformingworkortransformingasubstancetoanotherphase,

0 istheinitialtemperature Isochoric Processes:MolarHeatofIdealGases atConstantVolume p, volumeVandtemperature T.Intheisochoricprocess, thegassystemiseither 0 toTshowsthatheatsuppliedintoagassystemis om themeasurement conditions andthedefinitionofinternalgas C C ¢ P V = == () dQ n dT Q 1 m =- dQ dT pp QdU dQ cM Vacuum andUltravacuum:Physics and Technology = () = () dH dT TT dU dT

= 0 = è ç æ

è ç æ ¶ ¶ ¶ U ¶ T H T ø ÷ ö V ø ÷ ö p

(3.14) (3.13) (3.12) (3.11) - Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3

If theinternalenergy of1kmoltheidealgasisU Thermodynamics ofGases at LowPressures tions canmake differences athighertemperatures. Inthecaseof triatomic(molarheatis additional quadraticvibration components(potentialandkineticenergies). Thesevibra- diatomic gases at constant volume is equal to 20.786 kJ/(K kmol), though there are two same forallmonatomic ideal gases(12.47kJ/(Kkmol)).Similarly, themolar heatof heat supplied into the gas is equal to 12.7 mechanical work.Iftemperature of1kmolamonatomicgasisincreased by1 K These heatsper1kmolatconstantvolumesrefer tothegassystemsthatdo notperform respectively, thenthecorresponding molarheat capacities are diatomic, triatomic,andpolyatomicmoleculesare i is themolarheatcapacityatconstantvolume.Ifdegrees offr Hence, Isochoric process whentheheatsuppliedonlyincreases thegastemperature. FIGURE 3.1 C VV == 32 Rk 00 786 4 ./ p C JK C v () V == == kmol 2 5 dU 3 2 Rk Rk 0 == 0 C 2 i 0786 20 V 247 12 Rd and kJ/(K kmol). Accordingly, the molar heat is the = ./ 0 ./ V 2 TC i = R const CR 0 JK

= () V () == dT

iR 43 Is kmol = 3,5,6,and8(seeChapter2.11), kmol 0

oc T/2, thenitsderivativeis hor

2578 3 V ./ eedom formonoatomic, kJ () Kk mo l

(3.19) (3.18) (3.17) (3.16) (3.15) , the 119

Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 considered temperatures, whichgivesthechangeingasinternalenergy: independent ofthetemperature. ume, themolarheatcapacitydependsondegrees offreedom ofmolecules,butitis numerical data, capacities. Theplotsofheatcapacitiesatconstantvolumeconstructed from selected parameter amplifiesthediscrepancies betweenexperimentalandtheoretical dataofheat observed in the case of polyatomic molecules. In particular, increasing the temperature some inconsistency. Larger deviations of the empirical values from those predicted are above room temperature, the heat capacities and Physics,72ndedn.,CRCPress, BocaRaton,FL,1991–1992,pp.6-18–6-27.) where theidealgasisheatedfrom temperature C 120 Heat capacitiesC FIGURE 3.2 capacity states are progressively activatedwithincreasing temperature, theinconsistencyinheat because the number of vibrationalstates depends onthe molecular structures. Since these vibrational componentsofenergy causemore obviousdiscrepancies in molar heats V In assumptionoftheconstantmolarheatcapacity, Equation3.15canbeintegratedover Following thetreatment of thediscussedcase,itcanbesuggestedthatatconstantvol- Nevertheless, comparingthetheoretical andempiricalmolarheatcapacitiesindicates

= 24.786 C V ismore noticeable,particularly forpolyatomicmolecules. kJ/(K kmol))andpolyatomicmolecules(molarheatis33.258)), V ofsomegasesatconstantvolume. (Numericaldata:From Lide, D.R.,CRCHandbookofChemistry 73 inFigure 3.2,couldserveasanexample. At temperature ofafew100°C

Molar heat capacity CV, kJ/(K kmol) D 55 10 15 20 25 30 35 40 45 50 UU 10 =- 2 UC 00 == Vacuum andUltravacuum:Physics and Technology T ò Te T 0 mp C VV V dT erature significantly exceed the theoretical values, T 0 10 totemperature T. 3 CT ( K () ) - He H N O Cl H CO T 2 2 2 2 2 O 2

(3.20) Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3

temperatures andformolecules thatare more complex. may fairlydeviatefrom theempiricalheatcapacitiesespeciallyobtainedathigher conditions. Otherwise, the heatcapacitiesdeterminedbasedonequipartition theorem experimental data only for rigid gas molecules and at the temperatures close to the room degree offreedom, definedintheequipartitiontheorem, is reasonably consistentwith except forhelium.Thus,theassumptionofequaldistributionenergy basedonthe Thermodynamics ofGases at LowPressures on thefreedom numberofmolecules,andthey are temperature-independent. In view of that,thetheoretical molarheatsofidealgasesatconstantpressur the molarheatcapacityatconstantpressure istransformedto formula(3.22)andEquation3.16, ume insteadofmolarheatcapacities.UsingtheMayer’s The expression comprises constant. smaller thanthemolarheatcapacityatconstantpressure bythevalueofuniversalgas volume. Accordingly, themolarheatcapacityatconstantvolumeofanarbitrarygasis mula gives the named afterGermanphysicianandphysicist,JuliusRobertvonMayer.for TheMayer’s where applied totheisobaricprocess leadstotheequation gas expansion,thealsoperformsawork. Accordingly, thefirstlawofthermodynamics causes avolumetricexpansionofthegas.Sincepartsuppliedheatisusedfor ing ofidealgasraisesthetemperature andinternalenergy ofthegas,whichconsequently the isobaric process. In the isobaric process, the pressure change is zero (dp = 0). Thus, - heat Heating orcoolingagaswhilepressure ismaintainedconstant(p=)called 3.4 ideal gaslawappliedto1kmol: formula: The division of the Mayer’s formulabythemolarmassM The divisionoftheMayer’s

Isobaric ProcessesofIdealGasesandMayer’sFormula C p is the molar heat capacity at constant pressure. The substitution of relationship between the heat capacities at constant pressure and constant specific heat capacities at constant pressure and constant vol- pdV =R c C dQ p p =+ =+ == M C 0 2 i C V dT intoEquation3.21givesthewell-knownMayer’s Cd aa RR pV p 00 =+ Td M R CR 00 =+ = pdV U c i V 0 + +

2 2 M R R 0 a

a oftheidealgasyields dU = C e only depend V dT and the (3.24) (3.23) (3.22) (3.21) 121 - Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 C H O 1.82×10 N 28.17 CH HCl 6.97×10 H Cl 26.86 CO CO Equation 3.24. Another polynomialapproximation C the dataobtainedbypolynomialapproximation andthoseacquired theoretically using racy of about 5% when temperature is in the range of 273–1500 be fittedtotheempiricalmolarheatcapacityatconstantpressure (101,325Pa)withaccu- increase from T change ingasinternalenergy oftheextendedsystem,enthalpy, duetothetemperature gases canbefoundinReference 74. E.T. –molar heatcapacitycalculatedusingequipartition theorem (Equation3.16 ). (C where temperature. Therefore, theempiricaldataar more complexmolecular structures andgaseswithtemperatures farfrom theroom pressure indicatesconsiderablediscrepancies. Thelarger deviationsexhibitgaseswith However, comparison ofexperimentalandtheoretical dataofheatcapacitiesatconstant 122 Gas Constants forPolynomial Approximation TABLE 3.1 Br integration: p 2 2 2 2 –molarheatcapacity atconstantpressure p=101,325Paanddifferent temperatures. 2 2 Considering theheatcapacityatconstantpressure tobetemperature-independent, the The constantsa, C T istheabsolutetemperature inkelvins 03 9.61×10 30.36 O p 2 4

= p isthemolarheatcapacityatconstantpressure inkJ/(Kkmol)units

+ 57 12.98×10 5.23×10 25.72 75.50×10 27.30 14.15 10.14 ×10 29.07 43.50 ×10 31.70 26.00 4.075 ×10 35.24

b A bT

cT −0.836 ×10 2 C ) are giveninTable 3.1forsomegases.Thispolynomialapproximation can p

0 = toTmaybecalculatedatconstantpressure usingthefollowing

bT b, −3 −3 −3 −3 −3 −3 −3 −3 −3 −3

+ c,

cT −38.6 ×10 −0.04 ×10 −2.72 ×10 −14.9 ×10 148.3 ×10 2 −180 ×10 −8.2 ×10 d canbetabulated. Approximations withthree constants 15.5 ×10 20.1 ×10 11.8 ×10 D HH c C =- p −7 −7 −7 −7 −7 −7 −7 −7 −7 −7 =+ ab 75–78 HC 00 273.15 K (Equation3.25)ofMolarHeatCapacitiesC 897 1253.4 29.1006 29.1006 34.840 29.1006 32.526 29.1006 29.1006 31.540 31.245 29.914 30.244 29.1006 41.568 47.75 28.9775 29.4675 28.7282 29.1545 37.825 33.4298 36.702 28.7827 51.46 28.9916 30.14 34.4494 36.905 39.1472 28.7027 36.2419 Tc == Vacuum andUltravacuum:Physics and Technology — ++ T ò T 0 Td pp e approximated byanexpansion series: dT 23 T CT Heat CapacityC + 5404.5 33.2578 41.150 35.460 () 500 K p

= -

a T

K. The table also compares

cT p (kJ/kmolK) 2 1000 K 20033.2578 72.000 33.2578 29.1006 84.330 33.010

dT 3

eT 4 formany p E.T. (3.25) (3.26) Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 is thespecificheatcapacityatconstantpressure, C gases thefreedom numberofmoleculesisi=3,thentheratioheatcapacitiesyields ber ofmolecules,thisquantityisrelated tothemolecularstructure. Whenformonatomic ity atconstantvolume.Sincetheratioofheatcapacities isthefunctionoffreedom num- pressure, in thepVdiagram(Figure 3.3).Theshadedarea Thermodynamics ofGases at LowPressures ute tothetotalgasenergy insignificantly. Hence, the ratioofheatcapacitiesis degrees offreedom arising from stretch vibration(kineticandpotentialenergies) contrib- have thenumberofdegrees offreedom ofi=5,assumingthecondition at whichtwo In accordance withtheequipartitiontheorem (seeChapter2.10 describing theproperties ofgasesisaveryusefulquantity. represents theperformedworkbygasatanisobaricprocess. Work performedatisobaricprocess. FIGURE 3.3 The isobaricprocess ispresented byanabscissacalledisobar, whichisparalleltoVaxis In It caneasilybefoundthatthe Equation 3.28, c V is the specific heat capacity at constant volume, and i isthefr p eedom numberofmolecules,R k= p ratio ofheatcapacities = W const k= C C V k= V 0 p == C C == V ò V C C 0 V p pdV c c V p V p = = 32 52 + 3 i + 5 + pV 2 2 i () 2 R = R 0 = 1 p - 0 14 . is the molar heat capacity at constant isthemolarheatcapacityatconstant 666 V . = 0

+ i 2 0 istheuniversalgasconstant,c

Isobar VV ), diatomiclinearmolecules C V is the molar heat capac- (3.30) (3.29) (3.28) (3.27) 123 p

Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 columns according totheirmolecularstructures. specific capacityatconstantpressure andthespecificcapacityatconstantvolume.Thegases are sortedtothe The theoretical dataare calculatedusingEquation3.28andempiricaldataare determinedfrom theratioof 1.401 at–40°Ctoκ=1.3211000. does notchangesignificantly. Forexample,theheatcapacityratioofairchangesfrom κ= kinetic theoryofgases. Aside alarge temperature variation,theheatcapacityratioofgases indicated byheatcapacities,whichisoneofthemost surprising outcomesofthemolecular tions oftheoretical heatcapacityratiosfrom experimentalvaluesare notassignificant weak dependencescanstillbeapproximated bypolynomialfitting. of heatcapacitiesweaklydependsonthetemperature. Thespecificheatcapacities with Theoretical andExperimentalValues oftheHeatCapacityRatioκ=c TABLE 3.2 124 can beconsidered asisothermalbecause theprocess isslowandgas moleculescanexchange lation ofstatequantities. At lowpressure, smaller than 10Pa,theprocess ofexpanding gas may reduce. Thevariationoftemperature shouldbetakenintoconsideration atthecalcu- In pumpedvacuumsystems, agasvolumeexpands,andasresult, thegastemperature 3.5 heat capacitiessummarizedinTable 3.2. heat capacityratiobasedonthefreedom numberandthatdeterminedfrom thespecific values calculatedusingtheprincipleofenergy partition.Now, wecancompare the We havealready foundthattheexperimentalheatcapacitiesdiffer fr dom ofi=6,whichgivestheratioheatcapacities In thesamecondition,triatomicnonlinearmoleculeshavenumberofdegrees offree- from Table 5.3 e163 9.5H 293.15 1.6634 He Gas g160 3.0N .80231 SO 293.15 1.3860 NO 633.00 N 1.6700 293.15 Hg 1.6667 Ne r165 7.5Cl 273.15 1.6556 Kr .3012. C .10231 C NH 293.15 H 293.15 1.4100 293.15 1.4166 HCl 1.4067 CO 1123.0 Air 1123.0 1.7300 O 293.15 1.6800 K 293.15 1.6494 Na 1.6667 Xe Ar d169 B .30—CH — 1.4300 HBr — 1.6690 Cd Despite inconsistencyintheoretical andexperimentalheatcapacities(C

Monoatomic Molecules Isothermal Processesof IdealGases Theoretical κ=1.666 κ Gas T (K) 2 2 2 2 k= Diatomic Molecules Theoretical κ=1.4 79 C C At 50°C, V p .33231 NO N CO 293.15 H 293.15 293.15 1.3333 293.15 1.3945 1.3997 1.4094 = Vacuum andUltravacuum:Physics and Technology κ 62 + 6 = κ =1.4andat200°C, 1 . Gas T (K) 333

p /c 2 2 2 2 H .76293.15 1.2756 O .22273.15 1.3262 322-589 S 1.3200 O V 2 2 4 2 3 forSomeGases 2 Polyatomic Molecules Theoretical κ=1.333 80,81 κ =1.399.Theratio om thetheoretical .13293.15 293.15 1.3193 293.15 1.2549 293.15 1.3390 1.2885 .08293.15 293.15 1.3058 1.2336 V κ , C p ), thedevia- (3.31) T (K) Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 of p=R pV diagramisthearea constrainedbythevolumetricchangeofgasandisotherm isotherms forhighertemperatures are abovethoseforlowertemperatures. previously, thepVplotofBoyle’slawinFigure 2.3showsahyperbolicisotherm.The

temperature isconstant,thenfrom thedefinitionofheatcapacity tions? Ifintheisothermalprocess, theacceptedheatisdQ,whichgreater thanzero, and energy. Then,thequestionis,whatisheatcapacityofgasattheseparticularcondi- mal process, anidealgasacceptstheheat,butitneitherheatsupnoraltersitsinternal compression changes to heat,which is then transferred to theenvironment. In theisother- However, ataconstanttemperature, whenthegasiscompressed, theentir dU =C the temperature differential andthedifferential ofinternalenergy havetobezero (dT=0; work. Assuming thetemperature andinternal energy ofagassystemtobeconstant,both temperature andinternal energy ofgasare constant,whilethegasperforms amechanical constant. Thisprocess can bedescribedonthethermodynamicprinciplesatwhichboth heat withsurrounding walls andmaintainpracticallythetemperature ofchamberwalls Thermodynamics ofGases at LowPressures Work performedatanisothermalprocess. FIGURE 3.4 ideal gaslawshouldbeidenticalwiththeBoyle’spV Thus, thegasexhibitsinfiniteheatcapacity. Inisothermalexpansionorcompression, the conversion describestheequality an external source entirely converts to mechanical work. In thermodynamics, this energy Since the heat supplied into the gas entirely changes to mechanical work, the work in the V 0 dT =0).Thisimpliesthatatconstanttemperature thesuppliedheattogasfrom T/V (seeFigure 3.4).Then,thedifferential workcarriedoutbythegasis p p p 0 C TT =® V 0 dQ dT Isotherm dW fo QdW dQ ri = TC dT = RT V 0 dV V

®¥ =R T = const

0 . As illustrated T =constant.As V e workusedfor (3.32) (3.34) (3.33) 125 Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3

3.6 ible process. Example 3.5 illustratesthatthisworkhastobedistinguishedfrom theworkatanirrevers- ratio mayyield from different initial and finalvolumetric andpressure variables. absolute valuesofvolumetricandpressure parametersdonotmatterbecausethesame the isothermalworkisfunctionalityofrelative volumetricorpressure changes,andthe initial andfinalpressures. Ina reversible process, boththealternativeequationsshowthat constant. Thisisothermalworkcanalternativelybedeterminedfrom therelative changeof ric ratioofanidealgasinitsfinalandinitialstates,whiletemperature ismaintained The derivedequationrepresents themaximalwork,whichisproportional tothevolumet Introduction of V or Since attheinitialvolumeV 126 the gasis neither acceptsheatfrom environment norgivestoit. Accordingly, theheatchangeof Strictly speaking,theadiabatic process isathermally isolatedprocess, atwhich agas of energy betweenthewallsandmoleculesisnegligible istheadiabaticprocess. ecules andwallcannotequalize theirtemperatures. Theprocess atwhichthe transfer At higherpressures invacuumsystems,theexpansionofagascanbesofastthatmol- thermal expansion.Then,forn which istheworkperformedbyanidealgaswithaquantityof1kmol( equation:

work is

ume V

Adiabatic ProcessesofIdealGases 0 tothevolumeV,workperformedbygasisgivenintegral = R 0 T/p and V Qd Wd dU dW dW dU dQ 0 == W , theworkisequaltozero, atthegasexpansionfrom thevol- W 0 == m W =M == RT 00 nR == 0 mm = R RT +® V ò V / W 0 0 0 M T dV V = ln a nln ln 0

Vacuum andUltravacuum:Physics and Technology T/p kmol oftheidealgas,reversible isothermal RT V V V V 0 RT 0 0 0 ln into Equation 3.36 yields an alternative nR RT () V V -=- =- nln ln 0 0

VV 0 T - ln p p 0 p

p 0 Cd 0

mol) iniso (3.39) (3.38) (3.36) (3.35) (3.37) - - Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 gas volume.ThisimpliesthatforV V R of anidealgasundergoing anisothermalprocess, itisvalidthatthegaspressure isp= bat, seeninFigure 3.5,differs from theisothermwithitssteepercourse.Forasinglekmol Figure 3.5 showsthecoursesofbothcalculatedadiabatandisotherm.Theadia - substance withzero heatcapacity(dQ=0atdT≠0). diesel engines. ). However, theheatdevelopedinadiabaticcompression ignitesfuelin Chapter 4.7 Another exampleoftheadiabaticexpansionisaphaseliquefactiongases(see circuits, where temperatures aslow15–6Kare induceduponheliumgasexpansion. films. undesirable. The surfaces of the walls are polished and coated by radiation-reflective is reduced byspecialpreparation ofthewallsurfacesbetweenwhichheattransferis vacuum does not restrict heat transfer by radiation. The transport ofenergy by radiation using averygoodthermalinsulation.Thebestinsulatorishigh vacuum, but the changeofgasinternalenergy ispositiveandthegasheated. temperature. Conversely, ifthegasiscompressed, thegas gas expands andthechangeininternalenergy isnegative,indicatingreduction ofthe gas riment of internal energy. Ifthegassystem performsthepositivework,then Thermodynamics ofGases at LowPressures Comparison ofthe adiabatandisothermfor1mol ofanidealgasatK. 300 FIGURE 3.5 decreases as the volume increases from processes, three variables,pressure p,volumeVandtemperature T,havetobetakeninto 0 adiabatic process, the gas behaves as a The phenomenon of adiabatic expansion is used in cryogenic pumps with closed helium It shouldbenotedthatlikeidealgas,theadiabaticprocess isideal.Itcanbeimitated Since thesystemisthermallyisolated,workperformedbygastodet- T/V andtheproduct oftemperature TandtheuniversalgasconstantR 0 , theadiabathastobeaboveisotherm. This analysis indicates that inadiabatic

Pressure (kPa) 20 40 60 0 0.0 >V 0 , theadiabathastobebelowisothermandfor 0.1 V Ide 0 Volume ( to Ad Isotherm at300 al ga V ia 0 V ba s (1 , and it increases with a reduction in the t m mol 3 0.2 ) ) K performs negativework,while 0. 3 0 isinvariable. 127 0 T

Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3

differential form

Then, thesubstitutionofEquation3.41into3.40 Equation 3.40usingtheidealgaslawappliedtoaquantityof1kmol(pV=R For thederivationofanalyticalexpression oftheadiabat,weexcludevariableTfr the firstthermodynamiclawis applied totheadiabaticprocess atwhichtheheatchangeisequaltozero. Assuming dQ = 0, able, whiletemperature T ismaintainedconstant. account, contrastingisothermalprocesses atwhichonlypressure pandvolume V are vari- 128 volume, because from where thefinalanalyticalformofadiabataffiliatedwith reversiblepr The integrationofEquation3.44andsubstitutiontheheatratio κ Since C point ofexpansion,theexternalandinternalpressures differ byaninfinitesimalvalue. trolled expansion The performedsubstitutionisonlypermittedforthedescriptionofgassystemswithcon-

The analyticalformoftheadiabatcanbederivedfrom thefirstthermodynamiclaw Further recasting thelastequationleadsto V +R V ρ 0 0 —mass density) denote the quantities in an initial state and those without indices —mass density)denotethe quantitiesinaninitialstateandthosewithoutindices

= 0

M =C / ρ p 0 , theequationcanbetransformedto and satisfying the conditions of reversible processes. This means that at any V

M pV / Qd dW dU dQ kk ρ , where the symbols withzer == C =+ R è ç æ cons l V 0 C R nl dT () VV 00 pV pd dp + tp += p =+ 10 VV k R ø ÷ ö == 1 + 0 nl dV Vacuum andUltravacuum:Physics and Technology 00 V C C V () =+ pd V p ppdV dp Cd += dV V VV V C R or n += () pdV T cons dp p 0 dp p

p 00 t

è ç æ gives

= r r 0

ö ø ÷ o indices (p k

=C p /C 0 —pressure, V gives ocess is 0 T) inthe (3.41) (3.40) (3.46) (3.45) (3.44) (3.43) (3.42) V om 0 — Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 ratio into we may obtain adiabatic process, we that is,dW=–C written inotherforms. sure andvolumeiisthenumberoffreedom degrees.) Since foradiabaticequationitisvalid gas states as correspond toparametersin thefinalstate. initial We maywriteidealequationsforthefinaland Thermodynamics ofGases at LowPressures the ideal gasisdefined. adiabatic conditions.However, theidealgaslawisvalidatanycondition under which The adiabaticequationsupplementstheidealgaslawanditcanbeappliedonlyat work performedatadiabatic process isthen due tothetemperature changesfrom initialtemperature

system is system be steeperthanthecoursesofisotherms.(C change isexpressed bytheequations p = At adiabaticreversible process, theworkindifferential formisgivenbyEquation 3.39, Using theidealgaslawfor1kmol(p=R Since theratioofheatcapacitiesκ Similarly, from theidealgasequationforfinalandinitialstatewecan write M r a RT 0

T T and 00

p = =

è ç æ T T

MV p p 00 p V M 0 dT, from where theintegralvalueofworkperformedbygas = a ö ø ÷ = è ç æ k -k may write RT M r 1 r r

0 0 gives a ö ø ÷ RT k and - 00 1 . Accordingly, in the adiabatic process, the relative temperature , andfortheirratio T W

T p 00 00 T T = =- = 0000 è ç æ CT T T == MV r 00 VV =C r M è ç æ p a () p = T T V V 0 W 0 è ç æ 00 V V ö ø ÷ p RT ø ÷ ö p /C p k k =- k - TC 1 =® ö ø ÷ 0 0 V andthenbyacoupleofalgebraicoperations, T/V), theequationofadiabatmayalsobe , then their ratio gives p k Cd =(i+2)/>1,thecoursesofadiabatshaveto p andC k p - p p =- 1 V 0 è ç æ = T ò T 0 p è ç æ p 0 T p r T p V V r ö ø ÷ 00 V 0

0 1 are molarcapacitiesatconstantpres- / ç è æ = ö ø ÷ k = 1 k - r = r è ç æ 1

æ è ç p T p T T T T 0 0 p p 0 0 ö ø ÷ 0 toT.Theintegralvalueofthe ø ÷ ö . Substitution of thispressure 1

ö ø ÷ / k k , fortemperature ratiodue to k - 1 . p p 0 =® V V 0 T T 00 T T = p (3.47) (3.49) (3.48) p 00 129 V V . Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3

reversible. mum work of thegas system canbe obtained when anychange in the system is fully Since the temperatures Tand ried outatthereversible adiabatic expansion.Sinceforagasquantityof1kmol,itisvalid which is equal to lowering the content of gas energy. It represents the maximum work car 130 where process is Hence, thefinalequationforworkperformedby anidealgasatadiabatic reversible as follows: done byexcludingpressur The finalmodificationoftheanalyticalexpression fortheworkatadiabatic process maybe the gasworkatanadiabaticprocess canberewritten to The lastequationrepresents themaximumworkperformedbyagassystem. Themaxi- κ istheheatcapacityratioat constantpressure andvolume,respectively V isthefinalvolume V p 0 0 istheinitialpressure istheinitialvolume T 0 maybeexcludedfrom Equation3.49 asfollows: e viaintroduction oftheequationadiabat W pV W pV 00 C W R VV =- = 0 pV = C = T == k R pV 00 pV 0 00 V k pV 0 - CC V V 00 - pV Vacuum andUltravacuum:Physics and Technology 1 0 pV kk == C 00 1 - ë ê ê é V V = 1 è ç æ pV 00 T k k - 1 pV ç è æ 00 - 1 è ç æ = V V è ç æ pV pV 0 V R 00

V pV pV ø ÷ ö 0 00 1 -k k

ö ø ÷ - 1 ø ÷ ö k 1

- û ú ú ù

ø ÷ ö 1

(3.51) (3.50) (3.56) (3.55) (3.54) (3.53) (3.52) - Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3

where theexponentofpolytrop is they are describedbythe equation ofpolytrops The processes betweenthe twoboundarycasesare knownasthepolytropic processes, and whenever the heat exchange between the gas and the container enclosure is incomplete. ideal gases. The general process, however, takes place between these boundary cases Equation 3.46,refer totwolimitingideal casesthatare characteristicofthebehavior Both the isothermal process, given by Equation 2.31, and the adiabatic process, given by In practice,theheatexchangebetweenagasandwallsofvacuumsystemmayoccur. 3.7 Thermodynamics ofGases at LowPressures adiabatic processes are boundarycasesofthepolytrophic pr ses ofpolytropic process candemonstratethattheisochoric,isobaric,isothermal,and coincident withtheisochoric,isobaric,isothermal,andadiabaticprocesses. Simpleanaly- general process. Itcanbeshownthat,incertainidealcases,thepolytropic process canbe the molarheatatapolytropic process. cess, to κreferring to adiabatic process (1 The exponent of the Up tonow, alltheprocesses discussedabove are polytropic. The polytropic process is a

2. 3. 1.

Polytropic ProcessesofIdealGases

pV to isobarp Polytropic—isobaric process: Theequationofpolytr Polytropic—isothermal process: Isothermalpr process, ifthemolarcapacityofpolytrop C This equationcanfurtherbetransformedtoan describingisochoric (3.57) to Polytropic—isochoric process: (Equation 3.58) istransformedto tropic process, which can be illustrated when the exponent of polytrop capacity ofpolytrop C oyrp is polytrop volume (C= 0

const. Theexponentofthepolytrop (Equation3.58)iszero whenthemolar =const,whentheexponentofpolytr pV 0

= V polytrop can attain values from 1, corresponding to isothermal pro- ). At thiscondition,theexponentχ

const, whichisequaltotheisochorV=. isequaltothemolarcapacityC At thisanalysis,firstwe recast equationofpolytrops pV pV 1/ c= c= c

< c

CC χ CC 1 1 = =

< - - - - cons

cons κ). In the exponent C C isequaltomolarcapacityC C C V V p p ocess isthelimitingcaseofpoly- t

op χ op p V = p 0,thatis,whenthepolytrop atconstantpressure (C

→ χ ocess.

= ∞. Thus,forC=

can be transformed const canbetransformed χ , the capital Cdenotes V at constant C =C (3.60) (3.59) V , the (3.57) (3.58) p ). 131 Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 gas entersthesecondcoppercoilimmersedinaknown amountofthermallyinsulated located inawaterthermostat6,whichprovides aconstantgastemperature T.Then,the lary toserveasaflowmeter. Themeasured gasthroughput Q′passesviaacoppertubecoil capillary 4withtwoabsolutecapacitancegauges(3and5)placedateachendofthecapil- the flowmeter9measures aconstantgasflowQ′.Theassemblycomprisescalibrated 132 Measurement ofheatcapacitiesgasesat constantpressures. FIGURE 3.6 ferred totheprocessing chamber. capacity andtherecorded increase oftemperature allowustodeterminetheheattrans- ture increase canthenberecorded. Knowledgeofthemasswateranditsspecificheat increases intheprocessing chamber, thentheheatistransferred tothewater. Thetempera- isolated containerfilledwithwaterinwhichthere isaprocessing chamber. Iftemperature determine reaction heat,enthalpy, andentropy. Thecalorimeterconsistsofathermally Heat capacitiesare measured incalorimeters.Calorimetricmeasurements enableusto 3.8 while thespecificheatcapacity(c

is thermallystabilizedattemperature T Figure 3.6. At apreset pressure, agascanflowviathemeasurement system.First,thegas A measurement principleofthespecificheatcapacityatconstantpressure isillustratedin The specificheatcapacityofgasesisconvenientlymeasured atconstantpressure

Measurements ofGasHeatCapacities

Hence, whentheheatcapacityC the equationofadiabatpV equal totheheatcapacityratio(χ pr polytropic process. Thisstatementcaneasilybeproved becuaseattheadiabatic Polytropic—adiabatic process : Theadiabaticpr polytrop isidenticalwith theequationofisothermpV=const. tend tozero andthusthe exponentofpolytrop χ=1.Then,theequationof ocess, the heat capacityisC=0. Accordingly, theexponent of thepolytrop is 1 V ) ofsolidsandliquidsisdeterminedatconstantvolume. κ

10 const. →

= 0 inagascontainer10.Whenthevalve2isopened,

κ), sothattheequation of thepolytr ∞, theratiosofheatcapacitiesC Vacuum andUltravacuum:Physics and Technology Q ΄ 5432 9 6 ocess isthelimitingcaseof 8 7 Q ΄ T 1 , T 2 p /C and op turnsto 82,83 V /C (c p ), Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 V container. So,weobservetwocharacteristicsteps.Firstly, theconfinedgas,involume fined gasstartsto exchange heatwithsurroundings, whichcauses the pressure drop in the are equal,buttemperature oftheconfinedgasdiffers from thatofsurroundings. Thecon- When pressures equalize,thegatevalveisclosed.Thus,internalandexternalpressures sure. The pressure equilibrium occurs, but the temperature of the expanding gas alters. suddenly, thegasflowsfrom theconfinedvolumetosurroundings withatmospheric pres- rounding atmosphere. When a gate valve that is mounted on the gas container is opened system, with absolutely determined pressures, is in a thermal equilibrium with the sur initially confinedinacontainerwithhigherpressure thanatmosphericpressure. Thisgas measurement oftheheatcapacityratiocanbeexplainedwithasimplegassystem. A gas is eter is composed. Thus,specific heatcapacities pressure c are measured at constant pressure, then the specific heat capacity of the gas atconstant total mass enables ustocalculatetheheatcapacityofagasatconstantpressure. InEquation3.61 then thefollowingequation cific heatcapacitiesofallcomponents,from whichcalorimeteriscomposed,are known, the calorimeterformeasured time,massofwater, andindividual massesandthespe- temperature changeofwater inthecalorimetriccontainer8,gasmassMthatpassedvia ture atwhichitexitsconstant pressure, forinstance,tothestandard atmosphere. Ifthe increases thewatertemperature, whichismeasured. Thegascoolsdownto the tempera- water incalorimetriccontainer8.Theheattransferred from theflowinggasintowater Thermodynamics ofGases at LowPressures A numberofarticlesillustratemeasurements ofheatcapacityratios. 3.9 includes themassofwaterM the gaspressure alsoattainsaconstant valuep and surroundings. Whenthegastemperature stabilizesattemperature ofsurroundings, step oftheresidual gasisfollowedbytheheatexchange betweentheconfined residual gas expansion describedbythe equationoftheadiabat, changes itstemperature whenthegatevalveisopened. Thisisconsistentwiththeadiabatic ture them absorbheat.Insummary, whenthegasmassMpassedviacalorimeter, tempera- the corresponding materialcomponentsofthecalorimeterhavetobeknownbecauseall 1 , suddenlyexpandsfrom theinitialpressure p

T oftheflowinggas,initialtemperature T Measurement oftheHeatCapacityRatio p canbedeterminedfrom Equation3.61. W M andmassesM Cp () TT M 21 CW - =+ Mc =- Mc W 1 i ofallcomponentsfrom whichthecalorim- , andfinaltemperature T 2 c . Thepressure p è æ ç W å i T = ofwaterandspecific heatcapacities s 0 1 totheatmosphericpressure p Mc TT pV ii 21 11 + 2

kk = ø ÷ ö pV

a 2 canalsobereached when . Secondly, theseparation 84, 85 The principle of the Theprincipleofthe 2 inthecalorimeter a and it andit (3.61) (3.62) c 133 i of - , Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 capacity ratio temperature change. In temperature change. where massdensityis expansion from theinitialpressure p the gas 134 where wesubstitute R from where speedofsoundis Equation 3.66into The substitutionofmassdensitygivenbyEquation3.64 andelasticbulkmodulusgivenby equation pressure changesare considered tobeadiabatic,andtheadiabaticprocess isdescribedby sions andexpansionsatwhichheatcanbarely beexchanged.Therefore, thevolumeand In anenvironment ofanidealgas,thepropagating soundwavescauseveryfastcompres and elasticbulkmodulusis ume V sound inagasenvironment. measurements. air. Becauseoffastchanges, the equation of specifically describedbyequation by elasticproperties ofa material,namely bulk modulusandmassdensity, whichare

The othermethodtodeterminetheheatcapacityratioisofspeed from thesetwoequations leadstothe equation (p V κ p

const, where κistheratioofthermalheatcapacities,whichabout1.4for k= Equation 3.63yields nln ln nln ln E pp pp B u 0 12 1 u =- s T =pV/n - - s this case, Boyle’s law (p = = V è ç æ è ç æ cons 86 dV nM E a r d Inahomogeneous environment, the speedofsound is given ma B

can becalculatedonabaseofsimpleabsolutepressure ø ÷ ö tV () 12 m cons ×× / from theideal gasequation. k V E r= = k B tV è ç ç ç æ = × ø ÷ ö cons mn 12 u nM - // k s V ma VV dV V dp = = Vacuum andUltravacuum:Physics and Technology == k t =× the adiabatcan be recast to p / è ç æ æ è ç × M V cons k E k r B ö ø ÷ ÷ ÷ nM =- 12 1 ø ÷ ö ma pV / 1 is carried out very slowly without the iscarriedoutveryslowlywithoutthe 12 tV V / nM V = ma 1

V = p è ç æ k dV dp ø ÷ ö V cons 12 nM

2 -+

ma V () 1 = k / tV ) can be applied. Excluding vol- æ è ç 1 p ×× k 2 k V = ) M RT κ kk =p 0 cons a ø ÷ ö V 12 ø ÷ ö 1 / 1 / k t //2

× p

k a from where theheat

const

⋅

V −κ . Hence, (3.63) (3.68) (3.67) (3.66) (3.65) (3.64) - Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3

can bedeterminedfrom equation Thermodynamics ofGases at LowPressures gate valveleads topressure equalization. Thisprocess isirreversible sincethere is no ple is two vacuum systems with different pressures isolated by a gate valve. Opening the energy is supplied to the gas system. For irreversible process at low pressure, a good exam- state toother, butthegassystemcannotbereturned toitsinitialstateunlessmechanical ried out in ideal gases. Irreversible processes are those in which gas systems pass from one useful tohighlighttheuniqueness ofreversible and irreversible processes thatcanbecar Prior tothediscussionofsecondlawthermodynamics appliedtoidealgases,itis 3.10 where tions isgiveninTable 3.3 . less rigidbonding.Forillustration,thespeedofsound insomegasesatstandard condi- with more complexmolecularstructures and gaseswithmoleculesformedofatoms higher temperature. Deviationsfrom thissimplemechanistictheoryare observedforgases with empiricaldataofmanygasesatnearroom temperatures andsomeofthemevenat structure. molecules. Thus,themeasurement oftheheatcapacityratiopointsatmolecular of mercury vaporsitwasdiscovered thatthemercury vaporsconsistonlyofmonatomic gases donotcombinewithanyotherelements.Inmeasurements oftheheatcapacityratio It shouldbenotedthatthelastequationrefers toidealgases. M T istheabsolutegastemperature M Since speed of sound, u The simplestkinetictheorypredicts theheatcapacityratiosbeingremarkably consistent In early years, the measurement of the heat capacity ratio led to conclusions that noble R 0 a a istheuniversalgasconstant

/R isthemolarmassofgasinterest The SecondLawofThermodynamicsApplied toIdealGases 0 T =ρ/pisthemassdensity(seeEquation2.24)atpressure ofunity u u Gas Speed ofSoundinSomeGasesattheSTP Conditions(273.15K,101,325Pa) TABLE 3.3 a H Gas Data are calculatedusingEquation3.68. s s 9.9218 3. 3.1473 1.8234 168.91 213.43 315.28 417.38 336.41 433.1 211.83 297.69 (m/s) (m/s) 3.0378 3.7291 3.1251 7.71262.28 972.57 205.14 337.01 259.15 430.47 307.82 331.10 i rCH Ar Air 2 rN N Ne Kr S s , is a measurable parameter, the ratio of thermal heat capacity k == 4 uM RT sa 2 0 CO 2 2 u s 2 r p NH OCl CO

3 O 2 2 SO eH He 2 Xe 2 (3.69) 135 - Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3

cates thefollowingelementaryprocesses: Carnot, iscalledtheCarnotcycle.Thecycle,inpVdiagram(Figure 3.7),indi- originally investigated by French physicist and military engineer Sadi Nicolas Leonard therms andadiabats. A cyclicprocess withtwoisothermalandadiabaticprocesses, and adiabaticprocesses provide alooptransition,whichisgraphicallypresented byiso- which there is a transition that can be bridged by adiabatic processes. Thus, two isothermal the heattransferatheaterandcoolersidesare twoindividualisothermalprocesses between cycle process inasystemcomprisingheaterandcoolerwithdefinedtemperatures. Both the gassystemtoanenvironment oracooler. Accordingly, wecouldimagineareversible method atwhichtransitioninthereverse direction isinducedduetotheheattransferfrom with thesecondlawofthermodynamicsandreversibility oftheprocess, there hastobea ment oftheexternalenergy, whichissuppliedintothesystem.However, inconsistence ing through itsinitialstate incycles. A gassystemmayperformpositiveworktothedetri- interstate transitions. cal systemsfrom theirinitial statestootherandback,eitherdirectly orviaother to theirinitialstates.However, reversible processes are thosethatallowtransitionofphysi- spontaneous process, without supplyingmechanicalwork,whichcanreturn thesystems 136 Carnot cycleofanidealgas. FIGURE 3.7

We maypresume athermodynamicgassystemthatperformsreversible process, pass- 2. 1. 3.

temperature volume The adiabaticexpansionfrom thevolumeV ideal gas.Thisworkisgivenbythearea ABB′A. is givenbythearea CDD′C. The isothermalexpansionfrom avolumeofV The subsequentisothermalcompression ofthegasfrom thevolumeV which isgivenbythearea BCC′B. of temperature from avalueoftheheatertemperature T temperature V D atthecoolertemperature T T T C H p . Inthisprocess, thegasperformsworkW , which is associated with performing positive work W A΄ V Ad A A D΄ ia D ba V t D Vacuum andUltravacuum:Physics and Technology C atwhichgasperformsnegativeworkW B΄ Ad V B B tothevolumeV ia B C΄ C ba V t T C A H toavolumeofV T T H C > Isotherm T C Isotherm H toavalueofthecooler C , inducingadecrease V BC =C B attheheater V ′(T AB C H tothe by an

– T C CD ),

Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3

the efficiencyofheatsystemisratioperformedworkandsupplied of theworkWperformedandtotalheatQ Thermodynamics ofGases at LowPressures It canbeproved that Introducing Equation3.75and and thenegativeworkW If fortwoadiabaticprocesses itisvalid performed bythegasatCarnotcycleandQ the performedworkbygassystem(Q temperature

to thecooler. Since the heatissuppliedtoperformmechanicalworkbygas,whilerest heatisdelivered Then, thetotalworkis The efficiencyoftheheatsystemisthenratio The efficiencyoftheCarnotcycleisgivenbyEquation3.70,where 4.

temperature increases from thevalueT negative workW The finaladiabaticcompression from thevolumeV T H , attheisothermalexpansion.Theheatacceptedfrom theheaterisequalto DA W = CD BC C isrewritten toW V ¢ += Q (T WC HC Q W DA C H =+ –T =W =+ WQ WW H Q Q W h ) corresponds tothearea DAA′D. BBC AB VH AB H h C = () =- intoEquation3.70givestheCarnotefficiency TT = == h WW H WW QQ H =W W W = - AB suppliedtothegasfrom theheater. A partof or AB HC CD W DC C AB Q Q W tothevalueT - - CV ++

AB H H = AB WW WQ

+- − H ). DC T CD istheheatacceptedfrom theheaterat DC T

CT W

=- H C

() HC , thenthetotalworkis HC DA Q D

T tothevolumeV A

andthegasperforms = 0

W isthetotalwork A atwhich (3.71) (3.70) (3.77) (3.76) (3.75) (3.74) (3.73) (3.72) 137 Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 increase isgivenbyequation introduced byGermanphysicistRudolfJuliusEmanuelClausiusin1850.Theentropy term “entropy” comesfrom a Greek word that means “togive a direction.” It was first thermodynamics andherein. Entropy isthequantityunavailableformechanicalwork.The

T Carnot cyclescanbefoundinthebooksdevotedtothermodynamics. systems andidealadiabaticisothermalprocesses. More detailsonthereversible theoretical Carnotcyclebecausetheefficiencyofisderivedforidealgas for determinationofsaturatedvaporpressures andevaporationratesofmaterials. of gasesandvaporswithotherphasesderivationtheClausius–Clapeyron equation tures. EfficiencyoftheCarnot cyclehasimportantimplicationforthetheoryofequilibrium Since the ratio of the heats 138 When thesystempassesfrom thestateAto given by the expression dQ/T. This quantity is called This simpleanalysisindicatestheexistenceofastate functionquantitywhosechangeis final andinitialvaluesare equal.Thequantitieswithsuchpr After thecycleprocess, thesumofarbitraryquantityisequaltozero, implyingthatthe from where responding temperatures, thatis, is negative, then on the base of the Carnot cycle, the heat ratio is equal to the ratio of cor If theheatacceptedbyagasispositiveanddelivered byagastotheenvironment 3.11 H It should be noted that the empirical efficiency of real heat cycles is lower than that of the Accordingly, the /T

C Entropy ofGasSystems , theefficiencygivenbyEquation3.72canbetransformedto Carnot heat efficiency is only a function of the heater and cooler tempera- Q H Q T /Q A A -= C is directly proportional to the ratio of temperatures Q T B S B BA h -= Q dS Q = 00 Vacuum andUltravacuum:Physics and Technology S A B TT = HC = or T dQ - T T T C ò A B A B dQ

T å iA =

B, thetotalchangeinentropy is Q T entropy, denoted by the symbol i i =

operties are statefunctions. 87–89 (3.78) (3.82) (3.81) (3.80) (3.79) S in - Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 tion givesthechangeofentropy intheform

change ofentropy:

of Substances(1923), tion ofthethird lawofthermodynamicsisgiveninalaterbookThermodynamicsandEnergy the third law(alsocalledzero law)ofthermodynamics.However, asatisfactoryformula- entropy is erally onlyincreases orremains thesame,S which meansthatinreversible processes, theenergies oftheisolatedsystems do notchange. In thecaseofCarnotcycle,totalchangeinentropy is Thermodynamics ofGases at LowPressures which followsfrom Equation3.85. and for the isochoric Thus, fortheisothermalexpansion,whenT=constant Assuming heatcapacityC of ahomogeneous substanceapproaches zero iftemperature goes totheabsolutezero (0 K constant intheequationfor entropy onthebaseofboundary conditionsatwhichentropy substances. positive value.Thisstatement, onentropy equaltozero at0K,ismetforperfectcrystalline crystal elementistakento be zero at0K,thenentropy ofeachsubstancewillhaveafinal modynamics The substitutionofastatechangetheidealgasformulatedbyfirstprinciplether different processes, butitdoesnot allow ustodetermine the absolute values ofentropy. heat perunitoftemperature andhenceitsunitisjouleperkelvin(J/K). second lawofthermodynamics.Obviously, thechangeinentropy isequalto thechangein However Based onthestudyofheatcapacities This waytheformulatedthird lawofthermodynamics allowsustofindtheintegration The mathematicalexpression forentropy enablesustofindonlytheentropy changesin , entropy increases inirreversible processes. Entropy ofanisolatedsystemgen- dQ = dU + 90 process when V = where thestatementonentropy isasfollows:Ifentropy ofasingle pdV = C V D independentofthetemperature, integrationofthelastequa- SS =- S D BA Sn V -= BA dT + n dS == S SC =+ m R D Cd at lowtemperatures, Walter Nernst (1906)revealed constant, the simplified expression of the change in =+ SC 00 ò A m B V T nln ln R = dQ T 0 V T TdV/V into Equation3.81yieldsthefollowing V V nln ln B V A B =-

≥ ln nR T T

Q S T m A B nR A B T T B , whichisthecommonstatementof m A B 0

nR Q T dV m V A , theentropy changeis A 0 =

p p A B 0

V V A B

(3.83) (3.87) (3.86) (3.85) (3.84) 139 ). - Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 volume andatthezero exchangeofwork,theHelmholtzfree energy reaches minimum. corresponds toreversible processes.) corresponding tocomposition,weusethestatefunction nal energy (dU=dQ+dW)intothetotaldifferential oftheHelmholtzfree energy, yields Helmholtz. energy is named after German physicist and physician, Hermann Ludwig Ferdinand von ment withtemperature T equal totheinternalenergy Ubecauseapartofenergy TScanbetransferred from environ- The formulaindicatesthatforthesystemformationthere isnotnecessarilyneededenergy where function calledtheHelmholtzfree energy definedbyformula quantities neededfordeterminationofcomposition,itisveryappropriate touseastate cases where processes are investigatedatstatevariablesofvolume,temperature, andother However, theminimalinternalenergy canbeobtainedwhen theentropy isconstant. the maximalentropy. Themaximalentropy isreached whentheinternalenergy isconstant. of these equilibrium states may attain the minimal internal energy and the other may reach Intrinsic processes inmaterial systemsdrivethesetotheirequilibriumstates.One 3.12 140 entropy, for thesystemformationif there are notemperature andvolumetricchanges,Sisthefinal retical physicistandchemist,JosiahWillard Gibbs. Again, Uistheinternalenergy needed called At constanttemperature andvolume, Helmholtz fr Helmholtz free energy is If thesystemcanonlyexchangevolumetricworkwithsurroundings, thenthe means thatforanychangeoftheindependentstatevariables itisvalidthatdA dQ For studyingprocesses inasystemdefinedbytemperature, pressure, andother The introduction ofthefirstlawthermodynamics,expressing thechangeofinter S isthefinalentropy T istheabsolutetemperature U istheinternalenergy In practice,thesystemsare usuallystudiedatconstanttemperature orpressure. Forall ≤

ered system Thermodynamic FreeEnergy TdS consideringbothreversible andirreversible processes. (Thesignofequality Gibbs free energy, originallyintr H is the enthalpy, AdU dA , whichcouldbetakenastheenergy required toformtheconsid- h inaspontaneousprocess toformasystem.Helmholtzfree =- p is the pressure, G =- AdQ dA UT h Td =- Sp SS At equilibrium,constanttemperature andconstant oduced by American mathematical-engineer, theo- +=+= -= A h VA Td dW dQ dT Vacuum andUltravacuum:Physics and Technology =- TdS UT V is the final volume, and -- h pdV S +- pV

ee energy isdA Sd HT T Td -

SS S -

h =dQ–SdT,which T is the absolute h

≤ 0,where variables (3.88) (3.91) (3.90) (3.89) - Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3

these functionsisthedrivingforce ofphysicalandchemicalprocesses. the changeofGibbsfree energy asfollows: Wherefore, thechangeofathermodynamicfunctionGibbsfree energy is tion ofonesubstancephasetoanotheroccurringatconstantpressure andtemperature. mitted toexchange.TheGibbsfree energy isavitalquantitythatexplainsthetransforma- constant temperature andpressure andconditionsatwhichonlyvolumetricworkisper a casewhenonlythevolumetricworkispermittedtoexchange. smaller orequaltozero (dG change ofanindependentvariablethesystem,inGibbsenergy hastobe Since for the r at constantpressure andtemperature, thentheGibbsfr energy needed additionalenergy pVtoperformworkinconsistencewiththecontentofenthalpy. temperature. However, wheneverthesystemisformed from asmallvolume,there is Thermodynamics ofGases at LowPressures surface ofwater doesnotdependonthe volumevesselwhere water isplaced. ponents inanyheterogeneous system.Forexample, pressure ofwatervaporabovethe between phasesisthatall the phasesare independent ofthesubstancequantitiescom- and thedensityofindividual phasecomponents.Theimportantfeature ofequilibrium we needsomenumericalvalues ofchosenvariablessuchaspressure, temperature, volume, For instance,waterthatcontains icepiecesisatwo-phasesystem. can bemixedinanyratios. The systemwith more phasesiscalled a heterogeneous system. depending onthesystemvariables.Gasmixture is,however, singlephasebecausegases independent phase.Matterscanthusexistindifferent phases,thatis,solid,liquid,orgas, Each physically, chemically, ormechanicallyresolved partofthesystem represents an 3.13 given asfunctionsofindependentstatevariablesA namic potentialsoftheinitialandfinalstatesare equalatthephasetransition. dT =0,andthusdifferential ofGibbfree energy dG=0,whichmeansthatthethermody- At phasetransitionofmatter, thedifferentials ofpr The substitutionofthechangefree energy dA At equilibrium, the Gibbs free energy is at its minimum and system is characterized by If thefirstthermodynamiclawissubstitutedintototaldifferential oftheGibbsfree For a description of a phase system in its thermodynamic equilibrium with other phases, For adescriptionofphase systeminitsthermodynamicequilibriumwithother phases, The HelmholtzandGibbsfree energies are infactthermodynamicpotentialsiftheyare

Thermodynamic EquilibriumofGaseswith Their OtherPhases eversible process ≤ 0),specificallyatconstanttemperature andpressure andin Gd pdV dU dG dQ = TdS and for irreversible process Gd WpdV dW dQ dG =+ Gd pdV dA dG =+ dG =+ =- h Sd +- Vd TV +- + pT + dp h essure pandtemperature Tare dp=0, Vd (V,T) andGp).Then,thegradientof h gives =–pdV–SdTintoEquation3.94gives dS

Td p ee energy canbewrittenin theform -

S Sd

dQ < TdS, then for any (3.93) (3.92) (3.95) (3.94) 141 - Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 independent, whichmeans thatnoneofthethree phases increases tothedetriment ofot in astateofthermodynamic equilibrium.Thestateofthesystemisstable and time- The triplepointofwateris at611.73 Paand273.16K.Inthetriplepoint,allthree phasesare serve itsnutritionvalues.The lines(a),(b),and(c)meetinasinglepointcalled triple point. food isfrozen andthenwatersublimatesundervacuum conditionstolastlongerandpre- vacuum. This phenomenon is widely used infoodand pharmaceutical industries. First phase whenpressure isreduced orviceversa.Obviously, icemayrapidly sublimatein in equilibrium. Any changeinpressure leadstothetransitionfrom thesolidtogas p andviceversa.Thelastline(c)iscalledsublimationline, where bothsolidand gas coexist its liquidphase,passestogasphase(vapor)whenthe pressure decreases belowthevalue vapor, existingatsaturatedpressure p(seeFigure) inthermodynamicequilibriumwith 3.8 sure pointsatwhichwaterandvaporare inthermodynamicequilibrium.Thewater water ortoice.Thecurve(b)istheboilingline comprisingpairstemperature/pres- to theline(a),anychangeintemperature orpressure leadstotransitionofeithertheice pressure abscissauptotheatmosphericpressure. Ifthesystemisinstatescorresponding the denotedtriplepointat273.16Kand611.73 Pa, in is at273.15K,theline(a)betweenfreezing point(273.15K)atatmosphericpressure and (<101,325 and liquidphases(water).Theline(a)isplottedatpressure belowthestandard pressure (a) isthelineoffusionrepresenting thethermodynamicequilibriumbetweensolid(ice) behavior ofwateranditsotherphasesatvacuumconditions,thatis,below10 substance. ent phases,waterisasuitableexamplebecauseofourextensiveexperiencewiththis describe phasediagrams.Fordescriptionofthermodynamicequilibriumbetweendiffer 142 Phase diagramofwateratvacuumconditions. FIGURE 3.8 The Behaviors ofgasesandvaporsinthermodynamicequilibriumwiththeirotherphases p –TplotinFigure 3.8isthephasediagramofwater. Note,thediagramdisplays Pa). Taking intoaccountthatthefreezing pointofwateratatmosphericpressure

Pressure (Pa) 10 10 10 10 10 10 10 10 10 –3 –2 –1 –100 0 1 2 3 4 5 Solid c –50 Temperature (° Vacuum andUltravacuum:Physics and Technology ab 0 Liqui Tr iple po C Va Figure 3.8, isnearlyparallelwiththe ) d po 50 r int 100 5

Pa. The curve hers. hers. - Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 from liquid to vapor phase, the water volume increase is ΔV = Q water attemperature T may describemanysubstancesincludingsublimatingmaterials.Vaporization of1kmol transition andsaturatedvaporpressure ofwater. However, thederivationanditsoutcome water withanamountof1kmolinthethermodynamicstate(1)Figure 3.9toderivephase tion heat, and volumetric change of the substance at vaporization. For simplicity, consider vapor pressure andtemperature of a substance can be found from a Carnot cycle, vaporiza- Chapter 3.10. Using the second law ofthermodynamics,therelation between the saturated major amountofadsorbedwaterisremoved bythepumpingaction. white fume from the internal surface can be observed. The fume quickly diminishes as the atmospheric pressure andthechamber is wellilluminated,an“explosive”evolution of a water moleculescontinuetoescapebysublimation.Whenrapidpumpingstartsfrom vacuum ispoor. Theintensecoolingeffect finallycauseswatertoturnintoice,from which loss carriedawaybyrapidlyevaporatedwatermolecules,whiletheheatflowbackvia When pumpingcontinues,water“boiling”stopsandcoolsdownduetotheheat process leadsfirsttoan increase ofthewatertemperature. Thecontainer isindeedhot. sively evaporates, as observed at the boiling point given at atmospheric pressure. This down from atmospheric pressure, waterstartstorelease absorbedgaseswhileitinten- water in an opened container is loaded into a vacuum chamber and the chamber is pumped librium withitsvapor. Theevaporationandcondensationratesare equal. rated vaporpressure ataninvariabletemperature. Theliquidisinathermodynamic equi- enclosed container, itevaporates,butitspressure doesnotincrease abovethevalueofsatu- certain pressure calledsaturatedvaporpressure. Forinstance,ifaliquidislocatedinan At giventemperature, the transitionofsubstancefrom onephasetoanothertakesplaceat Thermodynamics ofGases at LowPressures Carnot cycleforthederivationof Clapeyron equation. FIGURE 3.9 subsequent adiabaticcompression returns thewatertoitsinitialstate(1). is transition from thestate(3)to(4), attemperature T–dTinsuchawaythatthe represents thethermodynamictransitionofwaterfrom thestate(2)to(3). by volume ofwatervaporandV H The saturatedvaporpressure canbedeterminedonthermodynamicprinciplesderivedin We are familiarwithheatingandcoolingwaterwhenitisexposedtovacuum.When Then, weproceed withliquefactionofwatervaporbyanisothermalcompression, which The subsequentadiabaticexpansioncausesdropping ofbothpressure andtemperature equaltotheenthalpychangeΔH dp and dT, respectively. In the Carnot diagram (Figure 3.9), the adiabatic expansion , from theinitialstate(1)to(2),requires vaporizationheat dp p L isthevolumeofwaterinliquidphase. V 1 L of 1 kmolwater. At thisisothermalphasetransition 43 T T– W dT V 2 G V G – – V L , where V G is the isthe 143 Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 equation the volumedifference issimplifiedtoΔV= vapor is V

Then, integrationofEquation3.99yields efficiency oftheCarnotcycle: which can be r where 144 If wedenoteA=ln(const where volume perkmolofwatervaporV is muchgreater than that initsliquidphase.Forexample,atatemperature of100°C,the temperature considerablyhigherthanthecriticaltemperature, thevolumeofgasphase Rudolf JuliusEmanuelClausiusandFrench engineer, BenoitPaul-EmileClapeyr The Clausius–Clapeyron equationisnamedafterGermanmathematicianandphysicist Then, substitutionforthevolumetricchangeΔVinClausius–Clapeyron equationgives

ized waterof1kmolbehavesasanidealgas,thevolumetricincrease isΔV ing asingleCarnotcycleisgivenbythearea oftheCarnotloop,whichisdW=(V which isequaltothechangeofenthalpyΔH Δ

Vdp. SinceinthiscaseT L ΔH is the changeinenthalpy, vaporization heat, which is assumedto be temperature- Now, usingpreviously derivedEquations3.70and3.78,wecananalyticallyexpress the R T isthevaportemperature p isthesaturatedvaporpressur . Thus,theliquidvolumemaybeneglected.Hence,inClausius–Clapeyron equation, 0 istheuniversalgasconstant independent W istheworkperformedbysystemwhileQ ecast to the well-known Clausius–Clapeyron equation, l nl H pD =T ) =ln(D′andBΔH/R = andT n h ¢ l == -= n G e is1606.73timesgreater thanthatforliquidwaterphase dW C D RT pA D =T–dT,theefficiencyofCarnotcycleis H h 0 H =- == dp p dT dp Q W or dT T Vacuum andUltravacuum:Physics and Technology , atthephasetransition.Theworkperformeddur H = T B = D R Þ= G ®= TV pD H

D 0 TT – V D H HC D pe dT T T D Vd 0 - H , thenthenaturallogarithm ofsaturated 2 L H

¢¢

≈ p -- AB D V

- HR H // G / is the heat supplied to the system, istheheatsuppliedtosystem, dT ; (V T T 0

TB =

≫ e V L ). Assuming thatvapor T 91

also called Clapeyron ≈ V G G

=R – V (3.101) (3.100) on. At L (3.97) (3.96) (3.99) (3.98) 0 ) T/p. = dp = - -

Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 A andB.Sincetheconstant components suchasscrews andwashersare notrecommended in and where Thermodynamics ofGases at LowPressures pressures pressures inassumptionthattheenthalpyistemperature-independent kmol ( tion permitsustodeterminethechangeinenthalpyreferring toasubstanceamountof reversible processes andarelatively narrow temperature range.Theoretically, thisequa- be determinedtoo. the orders of10 ments. Thefigure showsthatthesaturationvaporpressures ofzincandcadmiumare on vapor pressures plottedagainsttemperature are inFigure 3.10forsomechemicalele- Equation 3.101. range oftemperatures. Below~100Pa,vaporpressures canbecalculatedusingthederived The Clausius–Clapeyron equation is derived from thermodynamic principles for The constantsA Cu Cs K Na Li from data givenbyRoth The pressures calculatedfrom theirnaturallogarithmare inpascals;theAandBvaluesare recalculated Zr Ti Si C In Ga La Al B Cd Zn Ba Sr Ca Mg Be Au Ag Element Materials with Aid ofEquation3.101 The ConstantsAandBforCalculationofSaturatedVapor Pressures ofSingleElement TABLE 3.4 B havebeenfoundformanymaterialsempirically. Obviously, themeasurement of mol) basedonthemeasurements ofthetemperatures andcorresponding vapor A and B are constants that are characteristic of particular materials. The constants p of a substance at two different temperatures –3 and10 Constants inEquation3.101 25.519 20.800 21.652 22.665 23.286 26.371 26.762 27.269 34.198 23.839 24.253 24.690 25.128 28.080 24.598 24.776 22.618 22.642 23.816 24.782 25.634 25.358 25.266 andB A 93 ; datausedwith permission ofElsevier. –1 are inTable 3.4forsinglechemicalelements,whilethesaturated

Pa at500K

ΔH/ 39,091.02 10,313.77 12,638.97 18,578.59 69,756.05 53,410.58 49,036.43 92,087.20 28,731.21 31,862.17 48,000.45 36,696.75 68,190.57 13,168.47 15,056.05 20,161.10 18,026.07 20,581.49 17,611.68 37,916.90 40,472.32 32,852.11 8,748.28 R B 0 , respectively. Thus,usingzinc-andcadmium-plated , thechangeinenthalpy, thatis,evaporationheat,can Element Sb Pb Sn Ge Th Ta V Pt Ir Os Pd Rh Ru Ni Co Fe Mn U W Mo Cr Bi 2 T enables us to determine constants enables us to determine constants Constants inEquation3.101 302342,393.42 22.800 23.032 24.944 26.808 28.005 28.075 26.83 20.075 29.272 25.105 27.775 29.064 27.338 27.223 26.624 25.934 24.66 26.532 24.782 27.775 23724 23.66 A vacuum application. 92 inthegiven 536,763.02 22,354.17 41,508.31 65,381.91 92,570.66 59,212.07 62,803.47 71,897.08 45,375.97 63,816.43 77,813.68 48,253.69 48,599.02 45,974.53 31,631.95 93,652.68 71,022.25 46,043.60 21,939.78 19,867.81 8,580.66 B 145 A Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3

vaporization heatinEquation3.99 gives a vaporizedmaterial).Then, substitutionofthislinearfunctiontemperature forthe 146 ture Δ dependent. Thesimplestcaseiswhenthevaporization heatisalinearfunctionoftempera- temperature-invariable. However, inmanycases,thevaporization and definedtemperature rangesinwhichtheenthalpychange(vaporization heat)is Clapeyron equation.Thus,theillustratedconclusionscanbeappliedtovarious substances systems. smallincrease intemperature, anddepositedinundesirableareas invacuum tively addition tovacuumcontamination,thesematerialscanbeevaporatedbyarela- In Saturated vaporpressures ofsomepure chemicalelementsindependenceontemperature. FIGURE 3.10

Equation 3.101isvalidfortheconditionsasassumedatderivation oftheClausius– Vapor pressure (Pa) 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 H = ΔH –9 –8 –7 –6 –5 –4 –3 –2 –1 4 5 0 1 2 3 200 0

+ cT (Δ H 300 0 is the evaporation heat at T = 0 and c is a constant characteristic of 500400 600 dp p P S x = Te Vacuum andUltravacuum:Physics and Technology As 900800700 Hc mp RT 0 0 1000 Cd erature +D 2 T Zn Te dT (K) 2 1500

Ca Bi 2000 x In Ag Al Cu heat istemperature- 3000 Au Fe La B C x 50004000 Ir (3.102) 6000 Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 of the total numberparticles of v particles thatmayescapefrom aunitarea persecondwithvelocitiesfrom theintervalof Thus, three constantsA=ln or thesaturatedvaporpressure is from where byintegration,naturallogarithmofsaturatedvaporpressure is Thermodynamics ofGases at LowPressures Maxwell–Boltzmann distribution,someparticlesmay havevelocitycomponentv particles formfieldsofsurfaceforces witharangeofcouplemonolayers. the material,boundingforces onthesurfaceinterfaceare unbalanced.Thesurface particles haveenergy proportional tothematerialtemperature T.Unliketheforces inside of moleculesaccording tovelocitiesorenergies. namic parameters.SimilarequationcanbeobtainedusingMaxwell–Boltzmanndistribution modynamic equilibriumofvaporswiththeirotherphasesusingmacroscopic thermody- The equationforsaturatedvaporpressure aboveisderivedbasedontheconceptofther 3.14 from 10 surement atlarger numbersoftemperatures andaveragingleadstomore accuratedata. tal measurements of saturated pressures at three different temperatures at least. The mea-

tion function w direction of the surface normal and energies exceeding a threshold velocity u Rasmussen. fitting constants. fitting x to escape from thefieldofattractivesurfaceforces. Then,thedn tov When the ener Particles inliquidsorsolidsare heldtogetherbybondingforces. Onaverage,individual An equationforcalculation of vaporpressure ofmetallicmaterialsinthepressure range The saturated vapor pressures are alsodiscussed in articles by Flatau et al. v y

tov Equilibrium ofGaseousPhasesfromKineticTheoryGases x +dv –10 y +dv to100Paisalsogivenby Alcock etal. x 95 is dn y , sy which isexpressed byaone-dimensionalMaxwell–Boltzmanndistribu- () vn gy distribution of particles at the solid–vapor interface describes l nl = pD dd s = æ è ç F 2 nl ss p D′, m = n kT s ¢ ofthesolid may have velocities(energies) from therange -+ B =ΔH nv ø ÷ ö 12 D RT / () H 00 ed yy 0 - mv 2 vn pD 0 kT /R y 2 R = c = v 0 , andC=c/R y nl ¢ ; (see Equation 2.103 ). The number of surface s Te æ è ç Tp CB 2 Þ= p - m kT / T nl 96

ø ÷ ö However, theequationcontainsfive 12 / ev A 0 - are determinedfrom experimen- mv 2 -+ kT x 2 T B yy dv CT

s ( v y ) fractionfrom and energy y (3.104) (3.103) (3.105) inthe 94 and 147 -

Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 with thesamevelocity. Accordingly, thevapordensityis

where By acoupleofarithmeticoperationsandsubstitutionaveragethermalvelocityv Hence, thefluxdensityofparticleswithvelocitiesgreater thantheescapevelocityuis 148 two equations. These equations can simultaneously be satisfied only when velocities where massesmofindividualmoleculesand constants1/4and1/2are canceledinthelast and momentum andenergy are respectively expressed byequations rium andenergy equilibriumofevaporatingandcondensingparticles.Theequilibria for However, and inadynamicequilibrium,thetwofluxdensitiesshouldbeequal( v πm solids. Ifthevolumedensityofvapormoleculesisn v v v , thenthefluxdensityimpingingonaunitsurfaceofsolidis are equal(v W The symbolwdenotestheescapeenergy (vaporizationenergy) forasingleparticle R However, the vapors above the surface may also condense on the surfaces of liquids or ) 0 1/2

istheuniversalgasconstant =

weobtaintheparticlefluxdensityleavingsurfaceinform w

N a a thermodynamic equilibrium state is also characterized by momentum equilib- isthevaporizationenergy referring totheparticleswithanamountofkmol F ss = s =v n è ç æ v 22 ), whichmeansthatatthe interface, moleculesvaporizeandcondense pp FF m kT ss == ø ÷ ö 4 1 12 4 1 // nv nv ò ss ¥ u ev -- s en en mv 2 -- -- kT RT mu W 2 x 2 00 kT 2 nn yy == vn dv vs nv nv == 4 1 4 1 ss ss F = 33 22 vv en vv en ss en vn Vacuum andUltravacuum:Physics and Technology ve - - - = s RT RT RT W W è ç æ 0 W 0 0 4 1 or kT w nv = = m kT v vv vv or s

v v e ø ÷ ö ss 12 - ve

kT v w andtheaveragethermalvelocityis

kT m RT ss W = e 4 1 mmu 22 kT nv nv 22 vv = s e

n - s RT W è ç æ 0 2

kT p Φ m s

= ø ÷ ö

e Φ - v mu ), thatis, kT

= (3.107) (3.106) (3.109) (3.108) (3.112) (3.110) (3.11 v

(8kT/ s and 1) Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 against thetemperature inFigure 3.11, usingnumericaldatafrom HonigandHook. vacuum systems.Forreference, thesaturatedpressures ofcommongasesare plotted rated vaporpressures ofcommongasesare limitingfactorsoftheultimatepressure in solid form at –78.5 °C. At production of low pressures using low temperatures, the satu-

elsewhere, forexample,to plasmaprocesses. and theClausius–Clapeyron equation. heat (ener which isindeedsimilartoEquation3.100,beingexponentiallydependentonvaporization Introduction ofn Thermodynamics ofGases at LowPressures Hock (1960). Saturated vapor pressure ofcommon gasesatlowtemperatures, plottedfrom thenumericaldatabyHoningand FIGURE 3.11 in Table 3.5,andcanbecompared withthosebyLide(1991–1992). The boiling andmeltingpointsofthesegasescompiledfrom individualgasproperties are liquid orsolidformsatlowtemperatures; forexample,carbondioxide,CO increasing theirpressure. Particularly, unstablegasesandvaporscanbeconvertedinto at standard conditionscanbeliquefiedorsolidifiedby reducingtemperature and/or converted entirely tovaporsbyincreasing theirtemperature. Mattersthatare ingasphases All solidandliquidmaterialshavevaporpressures abovetheirsurfacesandtheycanbe 3.15 The derivationleadingtoEquation3.112 hasagreater impact,since itcanbeapplied

in Vacuum T Saturated Vapor PressureofSomeMaterialsUsed gy), orchangeinenthalpy, ΔH,followingfrom thethermodynamic principles

Pressure, p (Pa) 10 10 10 10 10 10 10 10 10 –10 –8 –6 –4 –2 v 2 4 6 0 =p echnology v kT yieldsvaporpressure H 2 Ne pn vs = Temperature, T(K) 10 kT N e 2 - RT W 0

CH 4 CO Xe CO Ar

2 100 98 O 2 2 , canbeina (3.113) 149 97

Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 vapor pressures are asfollows:isopropyl alcohol—4.1kPa,ethanol—5.85 have saturatedvaporpressures from unitstotensofkPaat20°C.Forexample,, and someofthemare employed inleaktestingonvacuumsystems.Mostofthesesolvents pressures ofmercury at20°C in diffusion pumps,andsealingsubstanceinvacuumapplications.Thesaturationvapor chemical stabilityathighertemperatures, mercury wasalsousedasapumpingmedium manometric liquid for low-pressure measurement. Owing tothe mass uniformity and 150 Saturated vapor pressure ofmercury asafunctionoftemperature. FIGURE 3.12 chemical pr acetone—24.59 methanol—12.93 at temperature of92.15K presumed tobebarely measurableattemperature lowerthanliquidnitrogen (77K perature reduces, asillustratedinFigure 3.12.Thesaturatedvaporpressure ofmercury is 1.33 × 10 Organic solvents are often used for cleaning material surfaces in vacuum applications, Mercury isaliquidmetalatnormal(NTP)conditions.Ithaswell-definedphysicaland 73 16 9.5191 .22.8199 70 73 02165.1 161.4 90.2 54.8 77.35 63.05 27.07 24.56 119.93 115.79 20.28 14.01 4.22 0.95 109.15 91.15 195.15 81.65 216.15 87.35 68.15 83.79 b.p. (K) m.p. (K) a rC CO CO Ar Gas from Individual Properties ofChemicalElementsand Gases Melting Points(m.p.)andBoiling(b.p.)ofCommonGases Abstracted TABLE 3.5 –6

Pa, respectively. Obviously, itsvaporpressure rapidlydecreases whenthetem- operties, andcanbepurifiedbyevaporation,thenitusedas a kPa. kPa, trichloroethylene—14 kPa

Saturated vapor pressure (Pa) , thetheoretical estimateofmercury pressure is4.63×10 10 10 10 10 10 10 10 10 10 10 10 10 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 2 , 0°C Mercury CH , and–78.5°C 4 Temperature (°C) Vacuum andUltravacuum:Physics and Technology eH He , carbontetrachloride—20.7kPaand 2 are 1.87×10 150100500–50–100 rN N Ne Kr 200 250 –1

Pa, 2.92×10 2 O 2 –2

Pa, and ). Since Xe –30

Pa, Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 Figure 3.13. lated theconstantstoSIunits,computedsaturationvaporpressures, andmadeplotsin ultimate pressure intheorder of10 additional accessories(low-temperatur diffusion pumpsoperating withmercury asapumpingmediumandequippedwith Thermodynamics ofGases at LowPressures pumps, plottedfollowing Equation3.114 . Saturated vaporpressure ofdimethylpolysiloxanefluids(DC702,DC703,DC704, andDC705)fordiffusion FIGURE 3.13 ments. Using this equation and ABconstants, in Leybold’s Taschenbuch, material properties andtheyare determinedfrom thepressure andtemperature measure- where “ln”isthenaturallogarithm,pressur [(CH diffusion pumps.Forexample, thedimethylpolysiloxanefluidsare basedon(CH health andsafety issues. Therefore, different specialfluids are employed inejectorand mercury atthestandard conditionshasalwaysbeenofconcernparticularlybecause molecules. For instance, pirical Equation3.101,asdeducedfrom theClausius–Clapeyron equation: diffusion pumps. 175 °C temperature, theyhavelowsaturatedvaporpressures, whileattheboilertemperature of Silicons undertrademarkMS702–MS705.Thesesiliconpolymerswithn marks DC702, DC703, D704,and DC 705produced by the Dow CorningorMidland The saturatedvaporofDCdimethylpolysiloxanefluidscanbeestimatedusingsemiem- 3 ) 2 , theyprovide sufficientlyhighpressure toformsuitablemolecularjetbeamsin SiO] n –Si(CH

Pressure (Pa) 3 10 ) 10 10 10 10 3 10 10 10 –10 molecularstructures. Theyare commercially availableundertrade- –8 –6 –4 –2 0 2 4 250 the average molecular number of DC702 is 530 for DC703 DC702 DC704 DC705 300 l n pA –11 =-

Pa. However, thehighsaturatedvaporpressure of e trapsattheirinlets)were abletoobtainthe 350 Te T B e isinPa,andconstantsABare related to mp ®= erature pe 400 ( A K - ) T B

450 500

≥ n = 5. At room

5 haveheavy 99 we recalcu- 3 (3.114) ) 3 SiO– 151 Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 contributes to thebackground pr area ofthesubstrate,whileremaining fraction(1 on acoldersubstrate. A fractionαofthefluxdensitycanbeadsorbedoncoldersurface under highvacuum,evaporatedmoleculestravelalongstraightpathsandmayimpinge calculated from thedifference oftwomolecularfluxdensities(seeEquation2.111): the fluxleavingevaporantsurface.Then,evaporatedmoleculardensitycanbe evaporant massreduces because themolecularfluxfrom thebackground is smallerthan

can bemadeatevaporationoftheorganic liquidswithhighboilingpoints. can bewrittenintheform theory, thecoefficient where surface ofanevaporantisgreater thanthebackground pressure p and thecondensingmoleculesare equal.However, ifthesaturation pressure pabovethe sure doesnotchange,becuase thefluxdensitiesofevaporating/sublimatingmolecules volume. At equilibrium conditions and invariable temperature, the saturation vapor pres- saturation vaporpressure, whichcanbeattainedinanysufficientlysmallandenclosed Thermodynamic equilibriumofagaseousphasewithitsotherphasestakesplaceatthe 3.16 152 are adsorbed.Thestickingcoefficientisthenpracticallyequaltoone. atoms impingingonasubstratesurfacewithtemperature closetotheroom temperature introduced astheHertz–Knudsenequationfordeposition,previously molecular fluxdensity, seethederivationofthisquantitygivenbyEquation2.111. For furtherinsightintothephysicalphenomenonandrelation betweenpressure and mass from aunitarea of theevaporantpersecondis sition fluxdensityisequal to evaporationdensity(seeEquation3.116). Then,theevaporated tion thatallmoleculesare adsorbedattheirimpacton thesubstrate,thatis,α=1,depo-

N isthenumberofmoleculesevaporatedformarea A A At vacuumevaporationofmetals,itisreasonable toassumethatvirtually all metal When thebackground pressure p k istheBoltzmannconstant m isthemassofsingleevaporatedmolecule p p isthesaturatedvaporpressure abovetheevaporantsurfaceattemperature T b E isthebackground pressure oftheevaporatedsubstance istheevaporantsurfacearea

Vacuum ThermalEvaporation m M EE == F α istermedthestickingcoefficient. Hence,thedepositedfluxdensity mN At EE FF == DE At M == essure, p F b ofanevaporantislowerthatitssaturationpressure p E aa æ è ç == 22 At pp N m Vacuum andUltravacuum:Physics and Technology At kT E N b , oftheevaporatedsubstance.Insorption E ø ÷ ö 12 / = 2 pp () p pp - mk - 2 pp p b -

T − mk b

α) leavesthesubstratesurfaceand

E b = T fortimet è ç æ a

M RT a 0 ö ø ÷ 12 b / ofthatsubstance,the 100 () . pp Similarconjecture - bb 101

Inassump- (3.115) (3.117) (3.116) Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3

Then, thesaturatedvaporpressure canbedeterminedusingequation where Hence, thetotalevaporatedmassfrom thesource withsurfacear However, atvacuumevaporationofmaterials,thepressure ofevaporantp Thermodynamics ofGases at LowPressures Dushman authors later. Thedataonvaporpr The illustrateddeterminationofvaporpressures wasusedbyLangmuirandseveralother mass evaporationrategiveninkgm tion ofthesaturatedvaporpressures ofmaterials,butitalsoallowsustodeterminethe derived Clausius–Clapeyron equationprovides usnotonlywiththebasefordetermina- are rapidlyadsorbedathighvacuumconditionsandlowtemperatures. Hence,the ground pressure ofevaporant isusuallynegligiblebecausethevaporsofsolidevaporants rant, anditisaffected bythebackground pressure ofevaporant.However, theback - vacuum background pressure p much smallerthanthesaturatedvaporpressure pabovetheevaporantsurfacebecause

substrate, thatis,α=1,thelastequationcanbetransformedto ration/condensation process atwhichallmoleculesare adsorbedattheirimpactonthe

A a radiusr gate radiallyalongstraight pathsviaasphericalvolumewithoutscattering.Obviously, at are isotropic, whichmeans thatathighvacuumconditions,thevapormolecules propa- method, thatis,byweighting theevaporantbefore and afterdeposition.Thepointsources by Equation 3.117.

= M M m isthemassofasingleevaporantatom/molecule N isthenumberofmoleculescorresponding tothemassM T istheabsolutetemperature ofevaporatedmaterialandthusalsoitsvapors M isthemassevaporatedfr The evaporatedmassM R k istheBoltzmannconstant Now, let usconsiderapointsource (Figure) andtotalevaporatedmass 3.14aM,asgiven

Equation 3.119. Theevaporatedmass M 0 4π a E istheuniversalgasconstant isthemolarmassofevaporatedsubstanceinkg/kmol istheevaporatedmassfrom anarea unitypersecond r 2 . So,atr , theirmasssurfacedensities are equalinanypoint of thesphere withanarea of 102 andHonig. , themassflux densityis m M M EE == == 103 MA isproportional tothesaturatedvaporpressure p F EE om thesource area A tm p B mN == At isverylow. Thus,atsuchconditionsanddynamicevapo- F EE At M EE EE == At –2 At M essures ofvariousmaterialswere compiledby 22

s pp m

–1 == kT At M ifthesaturationpressure issubstitutedinto æ è ç 22 =F 22 canalsobedetermined by agravimetric pp m pp m m kT At kT M ptA , andatconditions oflowbackground E ø ÷ ö fortimet 12 / E p M RT = a 0 è ç æ

M M RT RT a 0 a 0 ea A ptA ö ø ÷ 12 / E E p

fortimetis

oftheevapo- b isusually (3.120) (3.119) (3.118) 153 Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3

given conditions,themolecularmassflowcanbe rewritten as refers tothemassflowdensitymΦ,andmolecularatradius r, atthe 154 and hence, the gas phase As said,thisdepositedmassfluxdensityisequalto the massfluxdensitydeposited from where pressure Evaporation sources: (a)pointsource and(b)surfacesource. FIGURE 3.14

deposited fortimet a distance (radius) r such circumstances, theevaporatedandcondensedmassfluxdensitiesare equal.Since at Thus, withtheassistanceofEquation3.121,wecanfindmassfluxdensity, thatis ρ isthemassdensityofdepositedfilmoversphericalarea, d isthefilmthicknessatrdepositedfortimet p b oftheevaporantunderhigh-vacuumconditions,stickingcoefficientα (a) on an area of unity per second. If we denote deposition rate, , mass flow density is equal around the sphere, and , wecandeterminethedepositedfilmthicknessfrom equation P At M oint source == rF rm dd r θ r or d M At = M dA dA rp p ==rr r ==rr Vacuum andUltravacuum:Physics and Technology r è ç æ T == 2 VA d M t RT (b 2 a 0 ) p mp mk r ö ø ÷ d d 12

T Su

rfac rp p e source æ è ç 2 M φ RT a 0 r ö ø ÷ 12 θ / A

= M

4πr dA is the total mass dA r 2 T r , atdistancer d = d/t, which = (3.124) (3.123) (3.122) (3.121) 1. At Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3

to aunitarea ofthesphere with radius r is evenly overanyconcentricsphere witharadiusofr.Thus,theevaporatedmassreferring Table 3.5forpure chemicalelements. based onpressure andtemperature measurements. TheconstantsAandBare listedin empirical materialconstantsfittedtoEquation3.101foraparticularevaporated diffusion coefficient, butamaterial constant (A=lnD′),asfollowsfrom the filmthickness where we substitute The substitutionforpressure pfrom Equation3.100givesthedepositionrate Thermodynamics ofGases at LowPressures In contrast to spherical Then, thedivisionoflastequationbymassdensity ρandthetiltedarea where tilted area dA Now, if thesurfacearea istilted byanangleofθwithrespect totheradialdirection, the area ofasphere withradiusris sure oftheevaporantissignificantlysmallerthanitssaturationpressure (p

is equaltothedepositedmass,forwhichwemaywrite all theevaporatedmoleculesare adsorbed( to theradiusr)becauseitisvalidthatdA pressure surface isnotuniform.In thiscase,thefilmuniformityonaplanarsubstrate canbe In thecaseofapointsource, asseeninFigure 3.14,theevaporatedmassisdistributed dV ρ isthemassdensityofdepositedfilm d isthefilmthickness dA r r istheelementaryarea istheelementaryvolume p

e A T

isdepositedwiththesamemassofevaporantas area dA −

B/T asgivenbyEquations3.100and3.101.ItshouldbenotedthatD′isnot for molecular flux density r d surface, the thickness deposited from a point source on a planar = dM De ¢ - =® D rp HT 44 dM / M p r R d 0 Md hdA dV dM 22 = è ç æ == dA 4 22 r M == p T dM dA M r Rk

co r rr

= 4 0 a T dA M p s

α =1),wemayconsiderthatevaporatedmass dA T r q rr 2 r ø ÷ ö T . At conditionswhenthebackground pres-

12 M and the massevaporated on an elementary

/ cos 4

pr dA co F dM = r

θ. Hence, 2 s == eM T q AB At = -

rp N

/ M T æ è ç co p r 2 s p RT q p mk a 0

T ø ÷ ö

and saturated vapor b

≪ r A andBbeing (beingnormal p), andwhen dA T (3.125) (3.128) (3.127) (3.126) gives 155 Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 156 Thus, ifthefilmthickness is100nminthecenterofwafer, thenthefilmthicknessis center is0.3m.Employingthelastequation,relative changeofthefilmthicknessis wafer withadiameterof80mm,whenthedistancebetweenpointsource andwafer As an example, we Then, therelative changeinfilmthicknesswiththeℓ Apparently, themaximalfilmthicknessisatℓ and cosθ=h/r,thelastequationcanberewritten to determined trigonometrically as illustrated in Figure 3.15. Sincer Schematic forthicknesscalculationoffilmdepositedonaplanarflatsurface from apointevaporationsource. FIGURE 3.15 the substrate. 94.7 nm ataradius of 40mm,when evaporated from a pointsource thatis 30 cmapartfrom d = 4 d d p 0 can calculate the relative change of the thickness, for instance, for a () = h M 22 è ç æ 1 + + h 1 d d 2 2 0 r h ö ø ÷ r = 32 // = 4 pr = 44 4 pr () æ è ç pr h d () 1 () h 22 0 hh h + Mh + 0 M 2222 = 2 00 + 03 4 Vacuum andUltravacuum:Physics and Technology 32 . φ . 1 / pr 4 32 3 () Mh / 2 h

Mh () = ℓ ö ø ÷ r 2

32 0, which is which 0, 32 = + / θ =® æ è ç 1 7 947 0 974 0

-displacement is -displacement + .. 12 // h 1 2 2 ö ø ÷ = 32 / dd pr

= () h 22 Mh + 2

h 0 32

ℓ

2 , r

(h 2

+ (3.132) (3.131) (3.130) (3.129)

ℓ 2 ) 1/2 ,

Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 which canbetransformedto By recasting Equation3.133 anditsdivisionbymassdensity Figure 3.14b. deposited onaplanarsurfacesubstratefrom asurfaceevaporationsource asseenin Thermodynamics ofGases at LowPressures loaded with different materialscannot beplacedtothesame locationunlesstheyare multilayer structures andfilmthicknessasthin as60nm.Two andmore individual sources account. Forexample,organic electroluminescence devices(OELDs)are designedwith the substrate as well as its shape are geometrical parameters that should be taken into is avitalparameterinthin-film deposition.Thesource-to-substrate distanceandsizeof riers, whichmayberelated tothedeviceperformance. Therefore, thethicknessuniformity tronic structures, the thicknessnonuniformitymayaffect thetransienttimeofcharge car to thelightinterference, andvariationsindepthofcolorontransparent substrates.Inelec- an ununiformedfilmthicknessisvisualizedascolor differences overopaquesurfacesdue on thegeometricalconfigurationofasource andsubstrate.Inthecaseofvery thinfilms, The theoretical sectionaboveindicatesthatuniformityinfilmthicknessdepends largely 3.17 last equationwhenthefilmisdepositedfrom asurfacesource. The relative changeinthicknessornonuniformityofcanbeestimatedusingthe Then, therelative changeinthicknesswithℓ-displacementis Obviously, themaximalthicknessisatℓ Based on similar considerations, we may deduce the mass and thickness of films

Thermal EvaporationfromMultipleSources d d d == dM d == 0 22 rp = pr dM M dA = r () 2 h 2 T d 22 M p

= 0 h + h r r

0, that is, that 0, 4 = 2 2 2 2 dA 2 M M pr 2 r T E Mh 2 = pr r co h 1 æ è ç co 2 sc 1 ()

h fq sc + 22 fq 1 h + os 2 2 2 os ö ø ÷ 2

, wereceive (3.134) (3.133) (3.137) (3.136) (3.135) 157 - Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 158 arsenic, whichevaporatesas As effect. Theimpactangleofmoleculesonsubstratemayalsoaffect adsorption. at whichmolecularfluxarrivestothesubstrateand maskthicknessduetoshadowing strates rotates. Figure 3.16b,where thenormalofeach source intersectsthesubstratecenterandsub- with thesubstraterotation. A more favorablearrangementistheconfocalconfigurationin and source-to-substrate distance,source 1 maynotprovide satisfactory uniformity even impurities from vacuumbackground. Inaddition,dependingonthesizeofsubstrate ment distance,thedepositiontimeisprolonged, whichmayincrease theincorporationof ununiformed (indicatedbyFigure 3.16c)unlessthesubstraterotates. Uponthedisplace- in geometrical configurationsofevaporationsources andasubstrate.Thefirstconfiguration, placed onarotatable carousel andusedinsequence.Figure 3.16schematicallydepictstwo strate rotation; (c)thin-filmprofile withoutsubstrate rotation;1,2,twosources; 3,substrate;4,5,films. Source—substrate configuration:(a)nonrecommended; (b) recommended confocalconfigurationwithasub- FIGURE 3.16 varies uponthe evaporationtime.We candeposit filmswithdesired compositionifthe tion thusdiffers from that ofthesource material.For thesamereason, thefilm composition of onechemicalelementover anothercommonlytakesplace.Theresulting filmcomposi- constituents exhibitdissimilar saturationvaporpressures. Thus,preferential evaporation different massrates,since theysegregate upontheir melts,andthesegregated elementary dard singlesource deposition. With afewexemptions,thealloyconstituents evaporateat ple deposition of alloys. The composition of alloy source is usually unpreserved by a stan- elements havewidetechnicalapplications,butthere isnomethod,whichenables ussim- ties thatcaneasiermigrateoversurfaceswhenadsorbed. zone effusion cracker cells, to break molecular clusters to atoms or smaller evaporant enti - form ratherthaninlarge molecules.Therefore, there are availablespecial sources, two cules. However, itcanbemore beneficial insomecasestoevaporate materialinatomic Some chemicalelementsevaporateinmolecularrather thanatomicforms.Examplesare When usingmaskstodepositpatternedstructures, weshouldpayattention totheangle Alloy deposition:Thin-filmalloysandvariousstructures consistingofdifferent chemical Figure 3.16a,isunsuitablebecausethedepositedfilmespeciallyfrom source 2ishighly (a) 1 (c) 2 4 , andsulfurs,whichevaporateasS 3 4 5 Vacuum andUltravacuum:Physics and Technology 1 (b) 2 2 , S 4 , S 6 , andS 8 mole- Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 imparted onevaporatedmolecules.Thusthetotalsupplied heatintoevaporantisgivenby than itsboilingpoint. At 1082°C,whensaturatedvaporpressure is1.33×10 is ~100Pa.So,atvacuumconditions,aluminumdepositedmuchlowertemperatures vacuum conditions,itintensivelyevaporatesat1547°Cwhenitssaturatedvaporpressur example, althoughaluminumboilsat2519°Cunderatmosphericconditions,high- related totheevaporationrateatpointwhere thephasetransformationoccurs.For evaporant surfaceisintheorder of10–100Pa(~0.1to~1Torr). Thesaturatedpressure is at atmosphericpressure. Aluminum rapidlyturnstovaporat2519°C,whichisitsboilingpoint(Table 3.8)defined sufficiently hightemperature where thematerialsmeltandspontaneouslyevaporate. requires alloy materials in the form of well degassed wires fed to the heated surface with a evaporated employingflashedtechniquesfrom ahotanddrysource. Thistechnique individual sources inconfocal configurations,aspreviously described.Theycanalsobe tion time. depletes inthesource. Therefore, thesource alloyhastobereplenished aftersomeopera- spondingly increased. However, the more volatile component of the alloy gradually content ofthecomponentwithlowersaturationpressure inthesource alloy iscorre- Thermodynamics ofGases at LowPressures mately We cansaythattheaverageenergy carriedbyasingleevaporated moleculeisapprox ture of1082°C=1355.15K,theaverageenergy carriedbyanevaporatedmolecule is kinetic energy ofmoleculesatevaporationtemperature. So,ifevaporationisattempera and paraziticheatlossinradiationbyconduction viaelectricalleads. tions (~10 thin-film depositionbyconventionalevaporationmethods,thatis,athigh-vacuumcondi- tional evaporation because it involves mediated chemical reactions. In this chapter, only are suppliedtomediatedesired reactions. However, thisprocess differs from the conven- ferent gasesandelevatedtemperatures. At suchconditions,specificgasesandcatalysts synthesis ofnanomaterials,whichcanbecarriedoutatsubatmosphericpressures ofdif- solids totheirevaporationtemperatures athigh-vacuumconditions.Theexceptionisthe Thermal evaporationofsolidmaterialstoformthinfilmstakesplaceuponheating 3.18 energy forfusion(W minum depositionrateis8.23×10 In thin-filmdepositionbyconventionalevaporation,thesaturatedvaporpressure over For betterelucidation,letusconsiderthespecificevaporationcaseofaluminum. The composition of film alloys can be controlled well using evaporation from two or few The heatimpartedontheevaporatedatomicmolecules isproportional tothe average

Conditions atVacuum ThermalEvaporation w KE –4

Pa), isbrieflydiscussed. w ≈ KE 0.2eV=3.2×10 == 3 kT 2 Fusion ), andvaporization(W 3 2 13 .. 81 ´´ W 351 0 00175 0 10 805 2 15 1355 0 -- −20 =+ 23 –5 WW

kg m J Fusio atevaporation temperature of~1000 °C nV –2

s –1 . Forevaporation,weneedtosupplythe apor Vapor =´ + ), aswelltheenergy (W W .. KE

20 Je = V –1

Pa, thealu- KE ) thatis (3.139) (3.138) , and , and 159 i e - Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 erable attention. At depositionofcompactsolidfilmsandthefirst approximation, we its effect ontheevaporantvaporfluxandgrowing filmstructures. thin-film depositionshouldbedonewithattentivenessonthevacuumenvironment and ment, whichisneitherfree ofgasmolecules,norchemicallyinactive.Therefore, any acteristics atevaporationprocesses. We needtoconsidervacuumasacomplexenviron- omic moleculeswithmolarquantityofamole,whichgivesthevalue molecules tothesubstrate,wecancalculateheattransferred byevaporatedmonoat- For comparisonoftheheatneededforfusionandvaporizationwithtransferby cm Hence, theestimateofspecificenergy impartedonevaporatedmoleculesin reference to w 160 quality aswelltransportprocesses thatmayoccurunderdifferent vacuumconditions. and thesubstrate.Themeanfr compare themeanfree pathofmoleculeswiththedistancebetweenevaporantsource into account,too. the source andsubstrateare inthelineofsight,transferheatbyradiationhastobetaken ited onaunitarea forthetimeofunity. Thisheatcanbeconsiderableinsomecases.Since heating effect from depositedmoleculesisproportional tothenumberofmoleculesdepos- aluminum. As seeninTable 3.6,thisvalueiscomparabletothefusionener u mass num ature-limited, andtheyare themeasure ofthevelocitymolecules.Since thealumi- density is KE In conventionalevaporationprocesses, there are severalparametersthatdeserveconsid- Although wedefinevacuuminChapter1.6,needtohighlightspecificchar The kineticenergy isgivenuptothedepositedsubstrateinformofheat.Thus, 3 ofsolidaluminumintheformkineticenergy is

≈ n

0.4 == m r eV

density is2700kg/ = oln on 2519°C kJ/mol 10.71 660.32°C Boiling point Melting point 26.98kg/kmol Fusionheat Molar mass 24.30kJ/(kmolK) Molar heat Aluminum Some Aluminum Properties atEvaporation TABLE 3.6 698 26

6.41 2700 W . W EKE KE EKE KE

× kgkmol

10 == kg == wN −20 wn m

- J 3 atevaporationtemperature of~2000°C - 1 a 60 .. 0 10 805 2 0 06021 689 16 10 022 6 10 805 2 .. 21 ee pathistakenasacriterionfortheassessmentofvacuum .. m ´= 3 andthemolarmassis26.98kg/ ´´ 06 ´´ 26 kmol -- 20 -- 20 Jc Jm Vacuum andUltravacuum:Physics and Technology -- iei nry16.89kJ/mol@1082° C Kinetic energy 284kJ/mol Vaporization heat 12 60 21 ´» 24 10 02446 ´= Transformation toVapor 01 22 23 ´ mJ 33 lkJ ol 83 m 11 10,526 kJ/kg 396 kJ/kg . 910 69 »»´ . Thesevaluesare temper ´ . 60 . gy neededformelting , the atomic mass kmol, theatomicmass 21 cm mo 0 - l 23 22 - 3

cm -

(3.141) (3.140) (3.142) - - Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3

the referenced equation distances thanthedistanced how many molecules n many moleculespassthegivendistancewithoutcollisions.Theanswertoquestion 10 p = difference. in this publication(seeChapter 2.21 ). However, this will not make a tangible performed withspecificequationsformeanfree pathingasmixtures, whichare derived molecules ofanevaporantathigh-vacuumconditions.More suitablecalculationcanbe and thoughtheresidual moleculesar This estimateisacceptablethoughthelastequationappliedtoairmoleculesat20° d background invacuum equal tothesource-to-substrate distanceof,forexample, that we need vacuum conditions that provide the mean free pathL cient totakeintoconsiderationjustmoleculesintheresidual vacuumenvironment. Say, with background molecules. Fortheestimateofmolecularmeanfree pathL from the source to the substrate. They should not be scattered, and they should not react Thermodynamics ofGases at LowPressures paths, thatis,n can befoundoutfrom thedistributionlawofmoleculesinaccordance withtheirfree Example 2.23). Thus,for L e path distance (x respondingly standforthemolecularfluxdensities.Inourcase,x orders ofmagnitudeswithrespect tothatof2.47×10 order ofpressure magnitude to10 ular collisionsandscattering.Whenthebackground pressure isfurtherreduced byan pressure—101,325 Although atpressure of10 collisions onthe shorter distances than the source-to-substrate distance,d − 662 = L Since the meanfree path isthe average valueof the free molecularpaths,atpressure In vacuumdepositionsystems,evaporatedmoleculesshouldhavestraightpaths / L

= L

e –2 , thefractionmoleculesthattraveladistancegreater thand −1

Pa, there isaconsiderablenumberofmoleculesthatmakecollisionsonshorter

≈ mm. Themeanfree pathcan beestimatedusingEquation2.135(seealso Ld 0.37. Obviously, at the pressure of 10

010 10 n (m p (Pa) and Their RelationwithMolecularFluxDensity, andtheRatio,Φ Gas Pressure p,MolecularDensitynMeanFree PassofMolecules TABLE 3.7 Herein, we presume that the ratio 10 Φ L (m)6.62× of Evaporant-to-Background MolecularFluxDensitiesat20°C mean free pathn=2.47×10 stant evaporationrateatallbackground pressures; read molecular densityand = d == E /Φ = 662 mm). Because of the formulation of task, –3

= B .7 10 ) 2.47×

n 66 0 .. Pa), themoleculardensityisstillhigh(see e − 21 x ´ / p L orΦ of the total number 06 10 -- –4 –4 21 33 =d () –2

= Pa =662mm.Theabovecalculationsdonotdirectly showhow

10 10 10 Pa, themoleculardensity(2.47×10 Φ =662mm,wecalculatethecorresponding pressure using ,, mp –3 –3 20 0 0 e − 21 x –3 ®= / m 10 10 10 10 L

(seeEquations2.153and2.154),where Φ Pa, thenbythesamecalculationitcanbefoundthat –3 –2 –2 19 –1 e mostlywater(60%–90%)interactingwithatoms/ and6.62×10 Φ E /Φ 10 10 10 10 210 62 –1 –1 18 –2 B = 10 at 10 n ´ d 0 10 10 10 10 –2 actually travel the distance longer than

0 0 17 –3 Pa, the majority of molecules (63%) make –4 m,respectively. () Pa –4 010 10 10 10 10

Pa, while maintaining the con- 1 16 –4 m 25 10 10 10 =

2 2 15 –5 66 m 662 0 . Table 3.7 ) intermsofmolec- –3 . d 21 at10 10 10 10 10 is equal to the mean free isthesource-to- ´ 3 3 14 –6 18 0 m isn

- m 3 10 10 10 10 5 E

–3 /Φ = 4 4 13 –7 in the molecular Pa (~atmospheric ) isreduced by7 10 / B n , -- 10 10 10 10 0 2

= 5 5 12 –8 Pa

= 622mm. 0 e andΦ , itissuffi- −

x / substrate L

= (3.143)

e − cor d 161 / L

C = x - Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 Substitution for the pressure p = 10 molecules is 66.2 m. First, we determine the molecular density, using equation n = (molecular density) aluminum oxideinsteadof pure aluminummetal. Forsuchreasons, theknowledge of high-vacuum environment ataverylowdeposition rate,whichleadstotheformationof desired filmproperties. A good illustrative example istheevaporation ofaluminumin films inlarge quantitiesandaffect theircompositions,microstructures, anddeterioratethe cially atlowdepositionrates, whenthereactive gases canincorporateintothegrowing as permanentgassources. Theknowledgeonthepresence ofreactive gasesisvitalespe- gas environment with70%–90%ofwatervapor. Thesedesorbinggasesandvaporsbehave desorption ofgasesfrom theinternalsurfaceofvacuumsystemsmakingup residual ticular concernatthinfilmdeposition.Inhigh-vacuum systems,themajorgassource isin species foundinvacuumenvironment. However, chemicallyreactive speciesare ofpar evaporant takesplace. surface area of any vacuum system and, thus, the substrate where the deposition of an Nevertheless, thisisstillanunsatisfactoryunderstanding becausethebackground prerequisites forvacuumevaporationare understoodintheframeofdiscussionabove. we calculatethemeandistanceofmolecules,whichis k =1.38×10 consideration thesizeofmolecules(inorder of10 par molecules givestheimpression thatthemeanfree pathofmoleculesistoolongwhencom- Comparison ofthemeanmoleculardistance(3.43μm)andfr At pressure of10 Hence, thepressure is substrate distance,d=0.662mm rest (9.5%)are stillscattered. more than90.5%ofmolecules travelthedistanced 162 ity, wemayfindthemeanfree distanceofmoleculesat10 which is100timeslongerthanthegeometricalsource-to-substrate distance.Outofcurios- The molecularfluxdensityfrom vacuumbackground includesallkindsofmolecular So, thetransitandscatteringofmoleculesinavacuumenvironment andfollowing If itisdesired that99% of moleculestravelthedistancelongerthansource-to- ed tothemeanfree distanceamongneighboringmolecules.However, takinginto 0 .l –23 99

J/K givesthemoleculardensityn=2.47×10 =® e –4 -

x xL Pa (7.5×10 / is the source of molecules that unceasingly impinge on the internal == 3 11 n n. 09 3 90 24 . –7 =- , withoutcollisions,thenitcanbewritten

pP 71 Torr), meanfree pathisalready –4 ´ =- L x

Pa, temperature T = 293.15 K, and Boltzmann constant 0 ®= 63 16 ln 100 ln m 09 . Vacuum andUltravacuum:Physics and Technology - 9 . 99 =´ =´ 3710 4337 3 11 .. - =662mmwithoutcollisions,whilethe 66 0 662 0 . - . 4 –10 21 ´ p m)virtuallyclarifiestheproblem. a

- 0 –4 m 6 -

mm 3 Pa, whenthemeanfeepathof 16 ®= »

m 34 –3 ln (seeTable 3.7),andthen 3 09 Lm . m = 100 9 ee path(66.2m)of 66

. - 10 21 ´ - 4 0 pp -

3 = pressure 62 66 (3.146) (3.145) (3.144) p/kT. . - , Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 be depositedindetrimentalquantities.TheirfluxdensityΦ from thevacuumbackground, asindicatedinFigure 3.17.Thebackground moleculescan technological andanalyticalprocesses carriedoutundervacuumconditions. potential chemical activity of a high vacuum environment should be kept in mind at any Thermodynamics ofGases at LowPressures from vacuumbackground. Elucidation of mixing the molecular flux density FIGURE 3.17 grow from theevaporantfluxΦ 3.0–3.5 molecules are deposited,theevaporationtakesalongertime.Ittypicallyaround These considerably affect thefilmstructures andtheirproperties. well asmaterialproperties suchassurfaceenergies withrespect todepositedfilms. and high-valueproducts/devices. high price.ThedepositioninUHVisonlyeconomicalforrelatively smallareas inresearch lem canbesolvedby lowering the pressure tomuch lower pressure (UHV) but atavery growing filmstructures from thevacuumbackground indetrimentalquantities.Thisprob- clusters. However, thedrawbackalsoisfluxofimpuritiesthatcanincorporateinto configurations instantoffastpilingmoleculesandquickformationimmobilemolecular time-space tomigrateoverthesurfacebeassembleinthermodynamicallyfavorable single crystalstructures. At lowevaporationrate,theadsorbedatoms/moleculeshavea Low evaporationrateisneededforthedepositionofhigh-qualityfilmsandwith longer underthediscussedconditions,materialidentityoffilmsbecomesproblematic. due toadsorbedgasesandvaporsonsubstrateinterfaces.Whenevertheevaporationtakes even easilyevaporablematerialslikealuminumformislandsratherthanuniformfilms the residual vacuumatmosphere. A higherconcentrationofreactive speciesinvacuum, need tobeconcernedifthedetrimentalquantitiesofchemicallyreactive moleculesexistin impurities proportionally reduces byshorteningtheevaporationtime.However, westill evaporation takesasecondorfractionofsecond.Theincorporationbackground At aflashevaporation,thepressure isusuallyintheorder of10 Let usbemore specific.Whenfunctionalfilmsmadeof organic materialswithsmall Other importantparametersare substratepurity, homogeneityinadsorptionsites,as min togrow afunctionallayerwiththicknessof60nm.Obviously, thesefilms E , butthere are alsomoleculesarrivingtothegrowing films Φ B Φ E from the evaporation source and molecular flux density Φ E B canbeestimatedusingthe –4

Pa (~10 –6

Torr) andthe 163 Φ B

Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 where By substitutionfortheconstants,andgasparametersatbackground temperatur ture of293.15K,thedepositedmolecularfluxdensityis Hertz–Knudsen equation.Taking intoaccount70%ofwater vaporandthegastempera- 164 which canberewritten as by multiplicationwithmassdensity, weobtainthemassgr growth rate The product oftheuniformlydepositedar T = 293.15 K(20°C),weobtain deposited withthicknessof60nmfor3min(180s),whichgivesthedepositionrate over anarea ofunitypersecond. Assume thatafilmmadeofsmallorganic moleculesis ratio ofthemolecularfluxdensities. please ourselves”whenwedealwithharshreality, referring totheunfavorablearrival

Φ If wetakethestickingcoefficientα=0.3,whichisalready lowvalue,wereceive B m =M T istheabsolutetemperature ofgasinvacuumbackground k istheBoltzmannconstant α =1.0isthestickingcoefficient m isthemassofasinglewatermoleculeinbackground p The depositedmolecularfluxdensitycanalsobefoundoutfrom thedepositedthickness N M

≈ B a isthetotalpressure, 0.7refers to70%ofwatermolecules a

isthe Avogadro constant 7.62 =18kg/kmolisthemolarmassofwater F B a

= × /N

10 a 2 isthemassofasinglewatermolecule 17 p

2 10 022 6 m . −2 18

s ´ −1 kgkmol . Below, weuseevenalowervalueforthestickingcoefficient“to 61 26 kmol r 07 - == .. 1 h t ´´ - 10 ´´ 010 60 - 13 4 . A Pa 180 ´ F 81 r A BD rr r M - 10 == t s 0 9 == = Vacuum andUltravacuum:Physics and Technology - m =r 23 hA 07 ea Aanddepositionrateristhevolumetric hA t . t 2 =´ JK p hA ´ 3 10 333 3 mk t -- ./ p 1 V r

B t ´ T a

V 9 15 293 t

. - 10 K owth rate ms »´

25 . 41 0 18 ms -- 21

(3.148) (3.147) (3.152) (3.151) (3.150) (3.149) e of Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 vented and therefore often poorly degassed, which makes a difference in incorporation of thedeposition systemsoperateforashorttime,andthey are periodically laboratories, continuously andtherefore thesources andsystems are degassedwell. However, in in degassingthedeposition system.Intheindustrialplants,evaporantsources operate fabricated in many laboratories at “similar deposition conditions.” The major difference is and theirultimatefailure. may result inunsatisfactoryperformanceoffabricateddevices,shorteningtheir lifetime sing ofanyinternalpartsthedepositionsystem. Consequently, poorer filmproperties properties oforganic filmsmaymeasurablybeaffected evenatsmallchangesinoutgas- films withcrystallatticestructures. ular beamepitaxy(MBE).InMBEsuchahighvalue isneededforgrowing high-quality

Assuming thatthemolarmassoforganic moleculeishypothetically100kg/ where N/ Since themolaramountis Thermodynamics ofGases at LowPressures W Accordingly, onemolecule ofwaterincorporatesper20depositedorganic molecules. growing film.Then,theincorporatedratiooftwofluxesis for water, and100%adsorptionoforganic moleculesattheirimpactonthesurfaceof certainly isnotsmall.Inaveryfavorablecase,considertheadsorptioncoefficienttobe0.1 adsorption coefficientofwatermoleculesonaparticularsurface,andthis that hasarrivedtothegrowing or which meansthatforeachtwodepositedorganic molecules,there isonewatermolecule Thus, theratioofevaporant-to-background moleculesfluxdensitiesis mass densityisestimatedtobe2.5g/cm evaporant fluxdensity

Usually thedevicesproduced inindustrialplantshave higherperformancethanthose At thedepositionwhere awatermoleculeincorporatesper2–20organic molecules,the The amount of incorporated impurities into the organic film structure depends on the e may compare thisvaluewiththepracticalarrivalrateratio(being>10 At isthedepositedevaporantfluxdensity. ThusEquation3.153takestheform F= 5033310 333 3 2500 m kg 3 F F M . M B F F aa = B =® ´ 25 = NA NM .. N N 50 ganic film. 25 50 a 41 . . - . ´´ 10 21 a t 41 21 ´ ´ =® 00 ´ m M s 18 Fr rr 0 0 0 2 10 022 6 18 ms h = t 18 3 18 = . =2500kg/m -- ms 100 21 N ms ms N -- r 21 a -- -- ´ 21 M At 21 N N M kgkmol a a a = , thelastequationcanberecast to

61 26 1 =» kmol =» r 976 1 . - M 976 19 N 1 3 a a . , wecancalculatethedeposited -

»´ 2 50

20 . 21

0 118 ms -- 4 21 ) usedinmolec- kmol, whilethe

(3.154) (3.153) (3.157) (3.156) (3.155) 165 Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 MBE, weneededpressure of~10 pressure acting counterparts.Forconvenience,wepresent Table 3.7,whichlistsrelationships of partial pressure ofreactive vacuumconstituents and mutualchemicalreactivity ofinter layers shouldbedepositedby ALD forthefilmuniformityandminimizationofdefects. suppressed usingeffective capsulationandgettertrapping. Again, thefirstseveralcapsulation gases is surrounding atmosphere from where gases permeate or leak. These processes can be due tostrong electricfieldsandlarge current densities,isobstructed. Othersource of reactive which are thereasons forshortcontactsupon metallization andlocalizedburningorganic films adsorption sitesforfurtherdeposition.Thiswaytheformationofsurfacepinholesandspikes, ally, layer-by-layer. Moleculesdonotpileoneonanotherandprovide uniformlydistributed graphic structures. sition canbecarriedoutinshortpulses(interrupted deposition)toimprove crystallo- over the solid surfaces to form thermodynamically favorable structures. Therefore, depo- rate allowstheadsorbedmoleculesandparticles(ad-atoms/ad-molecules) tomigrate Al they shouldbepassivatedbyauniformbarrierlayer, forexample,bySiO the substrates(glassorplasticweb)havetobedegassedundervacuumconditionsandthen diffusion from substratesandbypermeationviapassivationcapsulationbarriers.Therefore, molecules duetothelongheatingprocess. only becauseoftheirevaporation,butalsothedegradationsmallorganic film anddevicequality, smallorganic moleculeshavetobe replenished continuouslynot impurities intothefunctionallayers.However, inindustrialoperations,forthesakeof 166 not straightforward. Thetotalmolecularflux(denoted justhere asΦins Deeper insight into evaporationand systems shows thatthesystemscalingis 3.19 Pa, thearrivalrateratioisΦ At the distanceofrandR,thefluxdensitiesare surface ofapointsource propagates radiallyviaahighvacuum,asseenin Figure. 3.18 rate ratiobythesamefactor. Assuming theevaporationrateconstantasthatat10 cates thatreducing thepressure byanorder ofmagnitudemeanstheincrease ofthearrival impinging molecularfluxes(Φ only waytoincrease theratioΦ ing webs.Iftheincrease ofevaporationrateisimpossible,oritinherently limited,the for averyshorttime(flashevaporation)orbycontinuousevaporationoffilmsonfastmov- prepared whenthearrivalrateratioisgreater. Theevaporatedfilmsare usuallydeposited there isamoleculefrom thevacuumbackground. Obviously, thehigherfilmpuritycanbe vacuum background at10 the from only onemoleculeper10,000evaporantmoleculesarrivestothegrowing film The incorporationofmoleculesfrom thevacuumbackground dependslargely onthe The formationoffilmstructures alsodependsonthedepositionrate.Thelow The reactive species(water andoxygen)mayalsoincorporateintothedevicestructure by 2 O

3 possiblyusinganatomiclayerdeposition(ALD).The ALD filmsgrow two-dimension- Scaling ofEvaporationSystems p, moleculardensitiesnmeanfree pathsL,andarrivalrateratiosΦ –7

Pa. Accordingly, atevaporationrateaslow0.1Å/s,usedin E /Φ F E E –evaporant, andΦ B rR –8 /Φ

= 10,whichmeansthatforeach10evaporantmolecules == Patoprovide highfilmpurity. 44 B istoreduce thebackground pressure. Table indi- 3.7 p F rR 22 Vacuum andUltravacuum:Physics and Technology and F B –background). Say, atpressure of10 p F

2 orbetterbyanSiO –1 ) leavingthe E /Φ –4

B Pa, then oftwo (3.158) 2 –4 – -

Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 radius on theevaporatedparticles thatheatthegrowing films andsensorofthicknessmonitor. ture. Inaddition,duetothe increase oftemperature, the higherkineticenergy is imparted consequently leadstothe higher incorporationofmoleculesintothegrowing filmstruc- increase in the degassing rate raises the background pressure of reactive molecules, which tion heattransfer, whichsubsequentlyinducesthehigherdegassing rateofmaterials.The higher temperature alsoincreases thetemperature ofsurrounding componentsviaradia- the highertemperature mayaffect manyotherparameters.Theevaporationsource with the same asthatatr.Thisallowustowrite total fluxdensityΦhastobeincreased tothevalueofΦ′,atwhichfluxdensityis The possiblesolution is to provide thesamefluxdensities atbothradiiRandr.Hence, the invariable. Consequently, theincorporationofimpuritiesfrom thebackground ishigher. increasing the evaporationrate.However theevaporation is typicallycarriedout at with unitsofm Thermodynamics ofGases at LowPressures the corr Since themassfluxdensityisproduct ofmolecularfluxdensityandmass ofmolecule, from wher Scaling ofevaporationsystems. FIGURE 3.18 density hastoincrease bythesquare ofradiiR of flux densityissmallerduetothelarger sphericalsurface. Asa result, thearrivalrateratio An alternativewouldbe reducing thepressure tothelevelofp Φ E /Φ r. Thisalsomeansthatthetemperature hastoincrease correspondingly. However, esponding massfluxdensitiesare intheratioofM′/ B issmallertoo,whenthesaturationpressure andevaporationtemperature are e –2 s –1 . Accordingly, atthelongersource-to-substrate distance,theevaporant F r == FF 44 ¢ p F = rR Φ 22 R r 2 2 2

/r p F ¢ 2 tokeepthesamedensityasthatat R

r Φ Φ r R

R 2 /r 2 . Thus,thetotalflux /(R 2 / r 2 ) insteadof (3.160) (3.159) 167 Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 time the same mass on an area of unity at the same evaporation temperature, the deposition when thefilmisdepositedonsurfacesphere withalarger radius.Forthedepositionof Since R are masses wecanwrite evaporated fluxfrom apointsource isΦ prevents thedepositionofanevaporantonundesirableareas. the undesirable heating of the shield and consequential gas desorption. The shield also jackets, especiallyinthecasesoflongerdeposition.Thecooledshieldingalsosuppresses incorporate intothegrowing films. rounding componentsandtheiroutgassing.Thedesorbingreactive gasesthenmay rity incorporation,becauseatshortendistances,radiationheatcancauseheatingofsur ing speedsforwaterfarexceedthoseordinary gases. ing films.Inthisaspect,itisalsosuitabletousecryogenicpumpsinwhichnominalpump- magnitude, andthusproportionally lowertheincorporationofimpuritiesintogrow- environment. also be facilitated by different types of getters and additional gas activation in a deposition cryosorption pumpconnectedtothechamber. Theeffective capturingofreactive gasescan trapoperating at temperature of liquid nitrogen (–196 °C) or possible using a Mixner’s background. same molecular arrival rate ratio of fluxes from the evaporation source and vacuum raising thedepositionratetwiceorreducing thepressure byafactorof2to maintainthe facilitate thisapproach. nearly ultimatepressure ofthedepositionsystemthatusuallymakesdifficulties to 168 drift inthecrystal ofthethicknessmonitor, which consequentlygivesincorrect reading of ferred byevaporantare different. Thesedifferences inheattransfercause anelectronic ration, andthenusedinthe othergeometrywher This calculationisusefulwhen thethicknessmonitoriscalibratedforaparticular configu- from wher sition timetare Now, letusconsiderthatthesource-target distanceRisreduced tothedistancer.Iftotal Because oftheradiationheat,evaporationsources are shieldedusingwater-cooled Reducing thegeometricaldimensionmaynothaveapositiveeffect withregard toimpu- The lasttwomethodscanreduce thepressure ofreactive gasesbynearlyanorder of The reduction ofpartialpressures ofcondensablespecies,particularlywater vapors,is For illustration, increasing the distance from 50 to 70.5

t 4 R >r,themassandthicknessoffilmdepositedonaunitarea fortimetissmaller r F hastobeshorterforthesphere withasmallerradius.Thus,forequalityof two p To R t 2 e thedepositiontimeatshorterdistanceris and 4 F 4 F p p To r To R t 2 t 2 , respectively, whilethecorresponding depositedmassesforthedepo- mt

and 4 F p To r t 2 mt 44 F p , where misthemassofasingledepositedmolecule. To R t 22 Tot t mt rR Vacuum andUltravacuum:Physics and Technology , thenthefluxdensitiesindistancesofrand = R R = r 2 2 F t p To r

e theradiationheatand heattrans- t mt r

cm, approximately, requires either (3.162) (3.161) - Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 3.20.1 controlled deposition. folds and dimples, which prevent dripping of the melted evaporants. Evaporation from a seen in Figure 3.19 denoted by the positions 3 and 4. Obviously, boats are constructed with for manyapplicationsincludingpackingmaterialsand capacitors. The large volume crucibles are used in continuous roll-to-roll aluminum coating of plastics properties have also graphite sources. Crucible can be of sizes from 1 ml to as large as 1 l. sources are highlyinert,buttheyare hydroscopic, andtheyare difficulttodegas.Similar boron-nitride (PBN),orevenelectricallyconductivematerials.Hexagonalboron-nitride them. Thecrucible canbemadefrom different materials:pyrex, quartz,alumina,pyrolytic upon their melting. Basket coils can hold various crucibles with an evaporant and heat coils canholdasmallamountofevaporantwhiletheyminimizedrippingthe Multistrand coilscanbeusedinlarge drum coatersasisotropic sources. Conicalbasket rather highloadcapacity. Theweightofevaporantshouldbe10%thecoilmass. ral shapetoconicalbaskets.Figure 3.19depictsamultistrandhelicalcoilsource (1)having electric current. Thecoilsare madeinmanyshapes,from asimpleV-shape viahelicalspi- In resistance evaporation, we can use coils, boats, and special sources heated by passage of high vacuumcondition(10 a refined formoftheevaporationtechniquethatusesmaterialsunderultra- (d) evaporation byinductionheating.MBEcouldalsobe classifiedinthisgroup, sinceitis (RE), (b)electron-assisted evaporation(EAE),(c)electron beamevaporation(EBE),and tions. The thermal evaporation techniques can be classified into (a) resistance evaporation rant properties, expected film properties, coating flow process, and economical motiva- vacuum. Thetechniquesare chosendependingonthetypeofcoatedworkpieces,evapo- Thermal evaporationcanbecarriedoutusingdifferent evaporationtechniquesinhigh 3.20 is nonfunctional. ture ofOELDs,thenthedriving voltageofthedevicecanbehighoreventually the filmthickness.Forexample,ifthicknessis greater thanintheoptimizedstruc- Thermodynamics ofGases at LowPressures boat; 4,dimple 5,buffer source. Sources for resistance evaporation: 1,multistrandwire coil; 2,rod overwoundbyan evaporantwire; 3,folded FIGURE 3.19 Boats are otheralternativesofresistance sources thatcanbeprepared inmany shapes,as

Different ThermalEvaporationTechniques

eitne Evapor Resistance 5 2 1 ation –8

Pa) andhighlycontrolled depositionrateincludingpulse- 4 3 169 Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 molecular fluxpassingthrough theaperture isproportional totheaperture area (Φ passing theaperture iscalculated from thesaturatedvaporpressure inthecell.Thetotal environment andmaintains theconstantpressure inthecell.Themolecularflux densityΦ walls. A smallaperture ofthecell(Figure 3.21)separatestheultrahighvacuumandcell Maxwell–Boltzmann distribution and where molecular motion is unaffected by the cell when amaterialisevaporatedfrom amolecular cell, where molecularvelocities follow specific conditions,anevaporationcell (Figure 3.21 ) used inMBEcan be treated similarly, (Figure 3.20). The flux density around the illustrated surface sphere is practically equal. At boat withasurfacepointsource hasatypicalangularcosinedistributionofthefluxdensity 170 Molecular flux density from anevaporation cell;thinwallaperture. FIGURE 3.21 Evaporation fluxdensityfrom asurfacepointsource. FIGURE 3.20 A a Vacuum andUltravacuum:Physics and Technology Φ=Φ Φ 0 0 pd A a 2 d T

Φ 0 A a ). 0

Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 materials withhighermeltingpointslikechromium (M.P. = materials. Duetothegoodthermalcontact,suchsources are usedfortheevaporationof position 2.Uniquesources are alsorods madeofrefractory metalsplatedwithevaporant turns ofanevaporantwire around arefractory metalrod, seeninFigure 3.19denotedbythe sing duringtheevaporationprocess. They are verywellthermallyshieldedandexternallycooledtoprevent undesired degas- vapor pressure in the cell interior. However, practical evaporation cells have conical shapes. used, asillustratedinFigure 3.22.Thelinearevaporationsources canbeconstructed with vapors toexitachimneyzone,butfilteroutmicroparticles from thedepositionvaporflux. of loaded byanevaporantpowderandzonewithbaffleplates/finsthatpermitentrance illustrated inFigure 3.19denotedbytheposition5.Thebafflesources compriseaheatedzone rials) thatintake microparticles into evaporated vapor flux,there are availablebafflesources, powder materials,forexample,siliconmonoxide(SiO),andsimilarmaterials(spittingmate- The fluxdensityΦ of thesphere is molecular fluxfollowscosineangulardistribution.Thedensityinanypoint Figure 3.21,whentheaperture issmall andworksasapointsource ofmolecules. Hence, Then, themolecularfluxisevenlydistributedoversurfaceofasphere, asillustratedin Thermodynamics ofGases at LowPressures Example ofalinear evaporationsource. FIGURE 3.22 Special resistance evaporationsources are heatedrods thatare overwindedbyseveral For larger area of uniform coating and moving objects, linear evaporation sources are 0 isdeterminedusingtheequation FF = 0 p A d e 2

F 0 = 1907°C 2 p p mk T , where pissaturated ). For evaporation of ). Forevaporationof (3.163) 171 Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 evaporation sources are heatedbyelectron 4is emissioncurrent (Figure). The filament 3.23 Electron-assisted evaporationsources are alternativestotheresistance thermalsources. These 3.20.2 ties intheprecision control ofevaporationtemperatures. current sources forthelow-current evaporationofsmallorganic moleculescausesdifficul- much smaller(unitsofamperes) becauseoflowerevaporationtemperatures. Usinghigh- amperes at relatively low voltage. At evaporation of small organic molecules, currents are melting points,particularlysuchassmallorganic molecules. chemically stable,andsuitableforheatingofcrucibles andevaporationmaterialswithlow ing Al mended. Kanthalisinherently passivatedbyoxidationofanaluminumconstituentform- these sources require higherdepositiontemperatures. to prevent alloyingandthus the premature failure andfilmcross-contamination. However, many evaporants. A source coated by inert materials (e.g., alumina) is often a good choice practical tablesthatare availablewithrecommended source materialsandtechniquesfor should becarefully selected.Theselectionsare basedonexperienceofanoperatorand traction. Forprevention/minimization oftheseeffects, materialsforevaporationsources these sources brittle, which leads often to their cracking upon thermal expansion and con- alloying ismostlythereason forfailure ofresistance evaporationsources. Alloying makes may result incross-contamination ofthedepositedfilmby refractory metals. Inaddition, evaporation temperatures thanthepure evaporantandpure refractory materials,which num). Thesematerialsoftenalloywithevaporant.Thealloyedmayhavelower of evaporantcondensationandthusprevents theblockingoforifices. the surfacetemperature of theevaporantload.Thisarrangementmaintainsorificesfree surface whileitmaintainsthetemperature oftheshieldswithorificesjustalittleabove which are individuallycontrolled. Thetopheaterprovides evaporationofmaterial from its puter simulation.Thesource canbedesignedwithabottomandtopresistance heaters, multiple orificeswithspacingbetweenneighboringopenings,usuallydesignedbycom- 172 1,2, contactsforheated filament4;2,3,high-voltage contacts;5,evaporantwire; 6, TiB Electron sources: (a) electron heated rod source; (b) electron heated crucible source; (c)induction heated crucible: FIGURE 3.23 heated crucible with evaporantload,9,high-frequency inductioncoil. In laboratorymetalevaporation,electriccurrent isoftenfrom tenstohundreds of For lowertemperature evaporationfrom crucibles, kanthal(FeCrAlalloy)isrecom- The heatedcomponentsare madeofrefractory metals(tungsten,tantalum,molybde- 2

O lcrnAsse Ev Electron-Assisted 3 onitssurface.Kanthalhasameltingpointofabout1500°C.Itisdurable, (a) e 4 6 Wi re 5 aporation 1 3 2 (b 4 e ) Vacuum andUltravacuum:Physics and Technology 7 1 3 2 (c) 9 8 2 –BN intermetallic bar;7,8, ≈hf Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 of 10keV.Theseparametersthenrepresent thedepositedpowerdensityof60kW/cm heat. Theelasticallyrecoiled electrons carrytheirenergy awaytothepointsoftheir However, electrons alsomakeelasticcollisions,andtheirenergy isnotconvertedtothe by conductionthrough asolidmaterialtothewater-cooled hearthandthermalradiation. rated molecules.Theotherpartofthermalenergy inducedbyelectron beamisdissipated the energy is used for fusion and evaporation and some portion is imparted on the evapo- secondary impacts. Stable evaporation/deposition evaporant area of0.25–2cm reaches ~10Pa. At highevaporationrate,thevaporpressure canbeashigh1000Pa.The evaporant volume with a depth of a fraction of mm. At evaporation, the saturation pressure electron beamisbendedby270°andfinallyitsenergy ismostlyabsorbedinarathersmall netic fieldbetweenthepolesisinducedbyanelectriccurrent passingviaacoil9.The netic steeringpoles(5,6)priortotheirimpactontheevaporant8inhearth3.Themag- than thatofthefilament.Theacceleratedthermionicelectrons thusenterbetweenthemag- from reaching thegrounded bodybyshieldingelectrode atahighernegativepotential a hotfilamentfloatingathighpotential(10kV)againstthehearthanodeisprevented Electron beamissuppliedbyanelectron gunlocatedinhousing7.Electrons emittedfrom energy onlytothetopsurfaceofevaporant8,whichisplacedinawater-cooled hearth3. see 0.2 to0.4eV,which is similar to thatat the resistance evaporation. For abriefclarification, evaporant byanelectron beam.Theenergy ofevaporatedatoms/moleculesrangesfrom In electr 3.20.3 be appliedtoheatacrucible withaloadedevaporant,asseeninFigure 3.23b. g BN intermetallicbarsources canevaporatealuminumwire withamassflowrateofseveral terminals 1and2.Thefilamentfloatsonahighpotentialwith respecttotheheated TiB heated toanincandescenttemperature (Figure 3.23a)byanelectriccurrent passing between Thermodynamics ofGases at LowPressures 5,6, polesofanelectromagnet; 7,housingof anelectron source; 8,evaporantload;9,hiddencoil ofelectromagnet. Schematic ofasystem forelectron beamevaporation: 1,2,watercooling;3,hearth forevaporant;4,electron beam; FIGURE 3.24 when thearea exposedtotheelectron beamis0.25cm the product ofhighvoltage placedontheheatedbarand intermetallic barthatisattheground potential.Then,forevaporation,theelectricpoweris wire is / The depositedpowerinduceslocalizedmeltandvaporizationofevaporant.So,apart min onaplasticwebmovingwithspeedof~3.4 Figure 3.24.Unlikeinresistance evaporation,anelectron beam4suppliesthermal

automatically supplied to the heated bar as the wire melts and evaporates. The TiB Electron BeamEvaporation on beamevaporation(EBE),thermalenergy forevaporationissuppliedtoan 2 isexposedtotheelectron beamcurrent of1.5Awithenergy 2 7891 6 takes place only at stable energy losses. m / 2 . s . Electron-assisted heating canalso emission current. Theevaporant 543 2 –BN –BN 173 2 2 – , Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 diamond inhydrogen, hydrogen/argon, andhydrogen/argon/3% O the meltingpointofmetallicMo(2623°C). As averygoodexamplecanserveetchingof vacuum conditionare listedinTable 3.8 . of pure refractory metals.Examplesofrefractory metalswhoseoxidescanbeformedat Similarly, somealloys,oxides,andsulfidescanhavemuchlowermeltingpointsthanthose points of pure gold and pure germanium are 1064.18°Cand 938.23 °C,respectively. gold/germanium alloy(Au/12%Ge)hasameltingpointatjust356°C,whilethe the source. Somealloyscanexhibitextremely lowmeltingpoints.Forexample,eutectic contamination canbecausedbyalloyingofanevaporantwithaconstruction materialof from themelts. getters extensivelyinthefinalstepsofproduction process, aswellforcrystalgrowth netic induction. wire isevaporatedfrom thesurfaceofanelectricallyconductivebarheatedbyelectromag- deposits onasubstrate. Alternatively, evaporantcontinuouslysuppliedintheformofa cible, causing thermal heating ofthecrucible loaded by anevaporant. Thus, the evaporant electromagnetic fieldinduces eddycurrents, alsoknownasFoucaultcurrents, inthecr (200–500 An electricallyconductivecr 3.20.4 the potentialcross-contamination ofgrowing filmsare minimized. 174 heated to1897°Chasasaturatedvaporpressure of1.33×10 measurable quantitieswhenheatedtomediumtemperatures. Forexample,molybdenum sten, tantalum,andmolybdenum,theycanreadily bedepositedascross-contaminants in Although forconstruction ofresistance sources weuserefractory materialssuchastung- 3.21 few turnsofthickcopperconductorsisaparthigh-powerandhigh-frequency ). Thecrucible 8isplacedinthecenterofaninductioncoil9.Thewitha (Figure 3.23c a massdepositionrateof1.20×10 Molybdenum trioxidehasmeltingpointofonly795° C, whichcontrastsverymuchwith The inductionheatinghasbeenusedfordegassingofelectron tubesandevaporationof Since onlyasmallvolumeofevaporantnearthesurfaceregion melts,thealloyingand WO Re MeltingPoint(° C) WS Material MeltingPoint(°C) W Material W MeltingPoint(°C) Material Melting PointsofSelectedRefractoryMetalsandTheirComposites TABLE 3.8

Cross-Contamination atThermalEvaporation 2

C 2 Thermal InductionEvaporation 3 kHz) and is 1250 1473 3186 2960 3422 inductively coupled with the crucible. Via this coupling, alternative ucible isheatedbyacontactlessinductionmethod MoSi MoO ReB Mo Mo –6

kg m 2 2 C 3 2 –2

Vacuum andUltravacuum:Physics and Technology s –1 (1.20×10 2697 2400 2050 2623 795 –7

g cm –2 –3

TaC TaS Ta TaN Ta

s Pa, anditisdepositedwith –1 2 O ). However, seriouscross- 2 5 2 usingahotfilament ~1300 3880 1872 3360 3017 oscillator u- Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 Accordingly, thetransportoftungstenmassismediatedbysmalloxygenadditives. of 1700°C deposited on the diamond substrate. The fairly low melting point of 1473° plasma reactor. Etching in an H Thermodynamics ofGases at LowPressures source with loadedevaporantfollowedby evaporation. Two-step degassingprocess: (a)degassingofsource attemperature abovethatforevaporation;(b)degassing the FIGURE 3.25 and bulkofevaporantare removed inquantitiesashighpossible(Figure 3.25b). Thisis into degassedsources, intheseconddegassingprocess, onlygasesfrom thesource surface room temperature, thediffusion ofgasesinsolidisverylow. Thus,afterloadingevaporant onds), butthesegasescannotenterthematerialbulkindetrimentalquantities,sinceat pressure, the environmental gases are adsorbedonthesource surfacesinstantly(nanosec- atmospheric pressure forloadingan evaporant material.Whenexposedto atmospheric face andbulkofanevaporationsource. After cooling,thecellsare exposedtonitrogen with intended evaporation.Inthefirststep(Figure 3.25),gasesare removed from boththesur of gasesintheheatedsource andobtainasufficientlycleanevaporationsource forthe perature iswellabovetheevaporationtemperature ofevaporanttominimizethecontent evaporant. Thetwo-stepprocess isessentialbecause,inthefirststep,degassingtem- without loadedevaporantfirst.Then,inthesecondstep,itisdegassedwith Degassing isperformedinatwo-stepprocess. Eachnewevaporationsource isdegassed 3.22 mond toformtungstencarbide. As aresult, tungsten,tungstencarbide,andWO ing hydrogen atmosphere. A portionofreduced tungstenreacts withcarbonatomsofdia- temperature of~850°C which rapidlyevaporatesatthefilamenttemperature of2100°C may converttodeposition.Thetungstenfilamentsoxidizeandformtrioxide(WO morphology isdrasticallychanged. When oxygen is added in an amount of 3% to a hydrogen/argon gas mixture, the surface faceted surfacesofdiamondpolycrystallitesturntothesurfacewithevidentetchingpits. perature of2100°Candasubstratebiasclearlyindicatesetchingprocess atwhichtheflat Although oxygenisinatraceamountthedepositionenvironment, theetchingprocess

Degassing ofEvaporationSources forWO 3 withrespect tothemeltingpointoftungsten(3422°C (a) ondiamondsurfaces,where itispartiallyreduced inastrong reduc - D egassing U 1 2 /Ar gas mixture activated by tungsten filaments with tem- I I 1 (b ) D Ev egassing II ap U oratio 2 n and then it condenses at andthenitcondensesat I 2 C and boiling point and boiling point ) are explanatory. 104 3 are are 175 3 - ) Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 methodical increasing powerinsteps,considerableinstabilitiesmayappear. these materialsrequires someskillandpatienceoftheoperatorbecausedespite causes increase ofthesaturationpressure byseveralorders ofpressure magnitude,to~10Pa. occurring at1245K.Therelatively smalltemperature increase ofaluminumto1640K of aluminum for which the measurable evaporation starts at saturated pressure of 10 means thatdepositedpowerdensityisreduced. Fordemonstration,wemayusetheexample increase inevaporationrateisonlypossiblebyincrease inevaporationarea. However, this energy isabsorbedinionizationofvaporabovetheevaporantsurface.Therefore, further vapor density above the evaporant surface. Beyond thispoint, the excess electron beam increase considerably, whiletheevaporationrateincreases rapidlyuntilitislimitedbythe steady evaporation, the thermallossesbyconductionto the hearthandradiationdonot Since thechangeintemperature issmallfrom thepointofmeasurableevaporationto viscous flow. Thisstateoccursat~10Pa,whenmeanfree pathisafractionofmillimeter. increases untiltheevaporantpressure overthesurfaceofelectron beamimpactreaches a above 5×10 and heldforbetterdegassingbefore openingtheshutter. to depositafilmonthesubstrate. Atslowdeposition,theevaporationtemperature issetup the degassing.Finally, weopentheshutterandrapidlyincrease theevaporant temperature the just a little below the evaporation with the closed shutter. In each step, we maintain carried outbyincreasing the temperature insteps(whilemonitoringpressure) uptovalue 176 3.23 Cu, Ag, Au), notable evaporation starts at the vapor pressure of 10 The conditioningisperformedwithclosedshutter. power continuesuntilthegunoperatesatmaximumwithnoinstabilitiesobserved. power isreduced downjustbelowthefirstinstability. Theincremental increase inthe with instabilitiesandelectricsparks.Whentheseeffects are observed,thesuppliedelectric accompanied byconsequentialgasdesorption.Larger gasdesorptionmaybeobserved power slowlyincreases insmallincrements. Thetemperature increase ofevaporantis conditioning, theelectron beamscansoveradefinedsurfacearea ofevaporantwhile oration is unstable and intense sparks can be induced due to the huge gas evolution. At degassing process istermed“conditioning.”Unlesstheevaporantconditioned,evap- identity ofthedepositedfilmsbecomesaconsiderableproblem. InthecaseofEBE, evaporation source andtheotherheatedcomponentsinchamber. Thus,compositional Failure ofdegassingleads toextensivedirect incorporationofgasesreleased from the temperature, thepressure drops backclosetothevalueatwhichdegassingstarted. Conditioning ofsemimeltingandsublimatingmaterials isnoteasy. Conditioningof During aslowincrease in temperature, thepressure rises,butitisnotallowedtoincrease In the caseofso-calledideal evaporant, whichare lowmeltingpoint materials(suchas Al, temperature untilpressure stabilizeatthevalueclosetothatmeasur

expansion process? is theinternalenergy ofthegas,whenheat30kJissuppliedintogasduring the Gas undergoing anisobaricexpansion at 1atmincreases itsvolumefrom 20to80l.What Example 3.1 and Thin-FilmDeposition Examples AppliedtoThermodynamics ofGases –3 –10 –2

Pa. Thedegassingisconsidered tobecompletedwhen,attheintended Vacuum andUltravacuum:Physics and Technology

–2

. Evaporation rate Pa. Evaporation rate ed before starting then then –2

Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 Thermodynamics ofGases at LowPressures W approximately The molarmassofNOfound from theperiodic table ofchemicalelementsis Solution when areversible andisothermalprocess iscarriedoutat300K? What istheworkneededtocompress 30gofnitricoxide(NO)from 80kPato0.8MPa, Example 3.3 by heliumatthisprocess isabout26.147kJ. where molargasquantityisn When thegasincreases itsvolumefrom 20l=0.02m Solution accepted from theenvironment givenbyEquation3.38: and the work performed at this isothermal expansion is maximal and equal to the heat increases tothevalueof At isothermal reversible expansion, calculated from the Boyle’s law, the helium Using theidealgaslaw, theinitialvolumeis Solution process? helium volumeafteritsexpansionandwhatistheworkperformedbyatthis this process, theheliumpressure dropped from 101,325Pato10,132.5.Whatisthe Helium withmassof20gunderwentareversible isothermal expansionat0°C.During Example 3.2 The internalenergy ofthe gasincreased by23.92kJ. Equation 3.2,thechangeofgasinternalenergy is If during this isobaric expansion, the heat of 30,000Jis supplied to the gas, according to it performstheworkasgivenbyEquation3.27:

== nR m 0 T V ln 0 == V V M 0 M D a Wp 51 RT D ´´ p 0 =- EQ 08 V =- () -- VV 31 DD 4 == kmol 00 V kgkmol ., 0 2 0 p m p 0

WJ kg M = 0.02kg/(4 =´ 0 325 101 ,. a - =- 112 0 1 47 314 1 =+ . 0006079 6 000 30 1 47 314 8 41 0/ 30 16 14 ,. ,, m JK .. Pa 33 0125 132 10 0 325 101 = kg/kmol)= 0.005kmol.Theworkperformed Jk ,. 0 32 101 kmol () , 00 mo gkmol kg , 80 lK - 3 -- 1 to80l=0.08m - 11 ., Pa 55 Pa 52 ´´ 7 15 273 .. Pa JJ 267 5 6079 02 = = ´ 11 .l 7 15 273 mJ . 2 5 920 3 3 2 K = m . K n 3 at1atm=101,325Pa, 112 0 = 11 . . 112 0 . 2 m m 33 3 m = 3 6169 146 26 volume ,. J 177 Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 178

conditions, Equations1.4and1.5canbeusedtocalculatetheinitialargon volumeV pressure afterthereversible adiabaticexpansionis The reversible isothermalworkoncompression isgiven byEquation3.38: Thus, 30g=0.03kgofNOcorresponds togasmolaramount using Equation3.53 , whichis The workperformedbythegas, ontheadiabaticreversible expansion,canbecalcu heat capacityratiois where For reversible process, wemaywritetheequationofadiabat where Since bothinitialargon volume V Solution temperature oftheargon andtheworkperformedbyit? 101,325 Argon withtheamount of 20gisinacontaineratstandardK, conditions(273.15 Example 3.4 Hence, therequired worktocompress NOisaboutW=5743.11 J. Thus, theargon volumebefore theexpansionis value ofκispracticallythesame(seeTable 3.2). The gasmolaramountis M M isthetotalmassofargon After thegasexpansion,temperature canbecalculatedfrom theidealgaslaw a istheargon molarmass V κ istheheatcapacityratio.Formonatomicgasargon, thetheoretical valueofthe Pa). At areversible process, argon expandsadiabaticallyto33.6l.Whatisthe == nV W ma p == = nR m T è ç æ 51 V V == ´´ 0 0 T

nR ø ÷ ö nM k= 02 pV k m ma ln -- p 43 == 0 0 p C kmol n C p = 0 m V p æ è ç ç /. == 51 = 11 33 Mk

0834300 314 8 10 W ´´ . . M 32 M 6237 16 21 61 - + 3 a 19110810 12068 1 4139 2 08 = 3 ´ ´ ,. ./ 0 - kmol andstandard molarvolume V 00 4 k pV 0 0 = 00 010 20 - kmol 40 - - 33 1 23 23 1 . ´ Pa ´´ m 666 m è ç ç æ mk gkmol kg gk Vacuum andUltravacuum:Physics and Technology 1 // ´´ ,l / - () ö ø ÷ ÷

- 33 (see 3 666 1 ,/ pV . mo 01 pV 341 00 kg 61 kmol lm ´ gk Equation 3.29).For Ar, theexperimental =´ =´ 0 325 101 ø ÷ ÷ ö Jk J . 51 0 mo () .. K - , 23 mo lk m 0 - lK = 4 PPa kmol 0 K - » 3 = ´ 62745 237 16 131 , pV 23 mo n ,. çç è ç æ 0 kk a ..24 l are givenatthesame 81 81 == »´ ´ ´ cons 11 K 0 0 4 5 21 Pa tp Pa Pa 0 ø ÷ ÷ ö - V 2 m . Hence, 33 lated 0 .

Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 Thermodynamics ofGases at LowPressures Then, thegastemperature aftertheexpansionis

is law ofthermodynamicsappliedtotheadiabaticprocess isgivenbyEquation3.40,which expansion. Therefore, theequationofadiabat(3.46)cannotbeusedinthiscase.Thefirst At suddenchangeoftheexternalpressure, argon undergoes irreversible adiabatic Solution work byargon atthisadiabatic expansionprocess? expands against this external pressure. What is the argon temperature andperformed the externalpressure suddenlydrops tothevalueof16,237Pa,argon adiabatically does atthereversible process. cools less and performs less work at theadiabatic irreversible process thanit The gas batic processes, theargon stateswere setup equally, disparateresults are obtained.

subsequent algebraicoperationsgivethefinaltemperature tion forVand

states ofthegas.Thus,atconstantheatcapacity:–p(V– molar amount;andpistheexternalpressure againstwhichgasexpands). (see Equation 3.39).Thus,−W The change of the gas internal energy is equal to the work performed by the gas where Argon withtheamountof 5×10 Example 3.5 batic process inthefollowingexample. reversible expansion can be compared with the values calculated for irreversible adia - The calculatedvaluesoftemperature andworkperformedbythegasatadiabatic at constantvolume;C Hence, −W Although in C p R Thus, theworkperformedbygasis The changeoftheinternalgasenergy andworkdependsonlyontheinitialfinal T dQ =0C′ 0 0 V 0 istheinitialpressure istheinitialtemperature istheuniversalgasconstant

= constant volumeformonatomicmolecules

3R W T 0 = /2 = 0 325 101

2471 12 =

= V

dT +pdV(where C' ,/ ,. Examples 3.4 and3.5,representing both reversible and irreversible adia- 5

0 × followingfrom theideal gas law(V=n

8,314

10 2471 12 Jk Pa −4 6 1 666 1 ,/ () . kmol V ´´ mo isthemolarheatcapacityatconstantvolume;n J/(K kmol)/2 11 Jk - lK

21 () ×

12, mmo

= + 0

lK − 471 0 325 101 –4 -- 6237 16 23

ΔU kmol isinacontainerat273.15Kand101,325Pa.When V mP , =n , T J +

= = kmol = 341 8 è ç ç æ

12, n ,/ 1 m C m Pa C - Pa C CR V

−1 471 V 62745 237 16 V V + istheheatcapacityofgivengasamount

101 K 314 8 (T Jk + ,. ,/ −1 p p () 0 J/(K kmol 0 (273.15 ,,

− mo 325 R 0

0 T) Jk lK T () 0 Pa mo

am ´´ K ´´ ) isthemolarheatcapacityat lK 11

− m 33 . 0 R

181.17 . ) =n 21 0 61 T/p andV 7 51117 181 15 273 m 0 .. C 0 - K) 23 V 23 m (T – KK

573.54 J = 0 ø ÷ ÷ ö =n » 0 ). Thesubstitu- 8 8 883 m m istheargon R . 0 T/p J 0 ) and 179 Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 180

10 is theuniversal gasconstant.ΔH=BR

from where where thevaporizationheat isequaltotheenthalpychange:W=ΔHandR At T Solution aluminum, respectively. Whatisthevaporizationheatofaluminum? Saturated vaporpressures of1μand100were measured at1142 °Cand1427for Example 3.7 The changeofentropy is10.12 J/Katthisisothermalandreversible process. Then, thevaporizationheatis approximately ΔH=3.05×10 4 Solution expansion isperformedwith4gofhelium? a reversible isothermal expansion at 300 K.Whatis the change in entropy when the The heliumpressure drops from atmosphericpressure to30kPawhenheliumundergoes Example 3.6 10

(kmol K)istheuniversalgasconstant.Hence, At T By subtracting(E.3.7.2)from ( This value can be calculated from Bconstant in Table 3.4 becauseB=ΔH/ R Then, usingEquation3.86,thechangeofentropy is g ofHecorresponds toHemolaramountn −3 8 J/kmol = 307kJ/mol. kmol. 1 =1142 +273.15=1415.15Kvaporpressure isp l 2 D nl =1427+273.151700.15Kvaporpressure isp Sn pp D 21 == HR -= m = nl R 0 0 ln TT TT 21 A p p 12 - A B -- DD RT 08314 8 10 ln H 02 l -- p p nl 31 11 1 2 kmol pA = -=- =- E.3.7.1), weobtain 8314 è ç æ A ´´ l - l n n ,l T B kmol D R pA pA H 2 1 01 J =- 0 =- Vacuum andUltravacuum:Physics and Technology or =36,966.75K ×8,314=Jkmol T Jk K mo ø ÷ ö 701 1 15 1700 451 7015 1700 15 1415 ®= n D D lK RT RT pA H H m 01 02 n =M/ . .. p p - 1 1 1 10 2 1

K = 3.4μ×10 KK 2 n D ´ - =260μ0.26Torr =34.58Pa. RT RT è ç ç ææ H H 0 0 325 101 a 4415 15 0000 30

è ç æ 1 = 4×10 8 , J/kmol=305kJ/mol. , 11 . 12 - K T Pa Pa ln øø ÷ ö −3 ø ÷ ÷ ö = 458 34

−3 04 = kg/(4 . Torr =0.452Pa. R . 012 10 H 5 0 –1 ./

Pa TT K Pa TT 21 kg/kmol) –1 0 12 0 -D =8341J/ JK where R

= 3.07× (E.3.7.2) (E.3.7.1) = 0

Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 Thermodynamics ofGases at LowPressures may write from where because base is can berecast tothedecimal logarithm.Fortheconversiontodecimallogarithm, The equationwithnaturallogarithm Solution to theother. version oftheseequationstoyieldSIunitsaswelltheconversionfrom onelogarithm tions containingdifferent logarithmsandnon-SIunits, itisusefultoillustratethecon- Since saturationvaporpressures ofmaterialsare oftenexpressed bysemiempiricalequa- Example 3.8 from where and hence, rithm, wemayusesimilaralgebraicoperations: tion withthedecimallogarithm. have tobemultipliedbythefactorof0.4342945tabulatedasA′andB,forequa- formed totheSIpressure units(Pa).Since1μ then beusedintheequationwithnaturallogarithm. rithm havetobemultipliedbythefactor2.302585obtain A andBconstants,whichcan Since ln10 Accordingly, theconstants Alternatively, thetransformationofequationfrom thedecimaltonaturalloga- If theequation gives thevalueofpressure pinthe units ofmicrons, it can betrans- adapted to be 10. Then, the last equation can be transformed to

= l og

2.302585, theconstantsA′ pA =+ lo g. l og 1333 0 l pA og =- l n. 4342945 0 pP pA 100 A andBgiveninEquationE.3.8.1withthenaturallogarithm pA . =- = =- x 01 1333 0 10 10 ¢ =® 302585 2 -= ¢ l AB n xex ex B T ¢¢ -- pA ¢ B T // ¢ =- TA (. ®= A andB′inEquationE.3.8.4withthedecimalloga- [] =® m ¢ pe -- 4342945 0 lo ¢ . = 875169 0 T B g. 10 ®= 302585 2 T . AB

¢ = ¢¢ - pe BT

10 ¢ T / T = B ) −3 = AB . ®=

- B Torr B T 4342945 302585 2 ¢ ¢ / ./ lo =- T =-

g A = A [] ¢¢

pA 10 () AB a ¢¢ −3 - B T T B ¢

, where A T ¢

133.3 -

B T ¢

Pa p ″

= =

0.1333

10 A ′ 0.43429 −0.875169. (E.3.8.3) (E.3.8.1) (E.3.8.4) (E.3.8.2)

Pa, we (A

−

B/T) , 181 Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 182 For theatomicratioB/N=1/1,averagemolarmassis Solution the boron-nitride film composed ofcBN80%/hBN20%hasthemassdensityρ the boron-nitride 3.48 g/cm a′ = 0.64, b=5.27×10 sure (1.8×10 find thatCdisdepositedonasample.Cadmiumalsohas relatively highsaturationpres- if there isascrew thatiscadmium-platedclosetoathermalsource invacuum, wemay Solution

and Table 3.4: For theestimationofsaturatedvaporpressure ofcadmium, wemayuseEquation3.91 Solution What isthesaturatedvaporpressure ofcadmiumat 273.15and500K? Example 3.9 Since themassdensityofhBN isρ strate biasof60Vwhentheioncurrent densitywas4mA/cm agonal (hBN)phaseis80%/20%.Whatwastheion-to-deposition fluxratioatthesub- (Fourier transforminfrared) analysisindicatesthattheratioofcubicphase(cBN)to hex- compositional analysis showsthatboron-to-nitrogen atomicratiois1/1,whiletheFTIR tering. Thedepositionratewas200nm/h.XPS(X-rayphotoelectron spectroscopy) A boron-nitride (BN)filmwasdepositedbya reactive radiofrequency magnetron sput- Example 3.11 From which,thesaturatedvaporpressure ofred phosphorus isabout1.02Paat450K. What isthesaturatedvaporpressure ofred phosphorus attemperature of455K? Example 3.10 vacuum applications. the samereasons, brass,being analloyofcopperandzinc,isunsuitableforultrahigh Zinc-plated surfacesare therefore notrecommended forvacuumapplications,either. For constants mended forvacuumapplications.Very highsaturationpressure alsoexhibitszinc(the A ¢ Practically, thesamevaluecanalsoberead from thegraphsforCdinFigure 3.10.So, For calculationofvaporpressure ofphosphorus, useequation -+ l og B T ¢ p F or =- () Fo M ab 3 012 10 A =24.776andB15,056.05),which is oftenusedforplatingmetalsurfaces.

¢¢ + Cd rC + a

.. 0.2 = –9 t2315 273 at da TT 0811 0067 14 811 10

× Pa) atroom temperature. Therefore, Cd-platedsurfacesare notrecom -

2.50 ./ 5600 lo t: 450 500 g .: g/cm gko gkmol kg kmol kg –4

-+ Kp where thefittingconstants A′ are empiricalconstants,and pressure pisinpascals. Kp () 06 3 ln

= ln 45

3.224 + 2 =- =- .l A 71 5 5 01 24 2 44 12 12 10 450 450 10 27 ./ A g/cm ´ H TK

B = TK B

2.50 =- - =- 4 3 Vacuum andUltravacuum:Physics and Technology 4598 24 . 4598 24 . g/cm . og == 3 2481 408 12 408 12 31847 168 13 andthatforcBNisρ 31847 168 13 9 15 293 500 ./ ,. ,. =- . = 10.12,B′ gkmol kg .. ®= ®= 2 ? p

p l og 17 T, = 5600,Ca′+bT, 9 10 498 1 . pA + . ./ 61 =- ´ .. H ¢ ´ gm 0

= -

1 = 3.480 B T -- PPa ¢ ol 9 += .327 007 0 Pa CT

H lo g/cm

g 0.8

× 3 , Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 Thermodynamics ofGases at LowPressures

deposited for an hour. Thus, the volume grown over an area of 1 cm

ited withthegrowth rates of18μm/hover1cm cyclotron resonance microwave chemical vapor deposition). The cBNfilmswere depos- Jet CVD(direct current jetchemicalvapordeposition)andanECRMWCVD(electron Compare themassdeposition ratesofcubicboron-nitride (cBN)filmsprepared byaDC Example 3.12 approximately 29energetic ionsperdepositedatomare needed. Hence, theiontodepositedfluxratiois If current densityj=4mA /cm 10

Mass densityofcBN:ρ=3.48g/cm Solution an area of3in.indiameteremployingtheECRMWCVD. 1 cm

A M The numberofdepositedatomsfoundfrom molaramountis At thedepositionrateof200nm/h=2×10 To formthe boron-nitride filmwithaphasecompositionofcBN/hBN=80%/20%, −5 == ¢¢ b. M M a. cm=2×10 == 2 p

= 4×10

¢ ¢¢ Vc 4 DC JetCVD:growth rate ECR CVD:growth rate Volume grown perunittime:V′=Ar1cm Diameter ofarea: d d == == 2 r Vc 333 3 31 0313 0 ./ r p 4 10 648 3 ´ .. −3 18 M 76 M .. A,thenthenumberofincidentionsonanarea of1cm . Cara −5 aa 4 ´´ ´´ 2 cm = =® 22 10 21 N th cm N 3 -- - ´´ . 33 33 M N 03 mh mh = ¢ -- 1 =3.0in.in.×2.54(cm/)7.62; + 53 == N 45 // // cm ./ == 6456 475 63 063475 0 .; == r ./ 64 It =0.8μ iiaa h miliCarat e 1 MN cm r 34 34 M 2 =18μ , thecurrent overthearea of1cm 22 a 86 80 41 a .. 2408 12 224 3 g gc 0 10 602 1 ´´ ; 1Carat=200mg m VA . ./ Cara m / cm VN ¢ 3600 0 mg == h gc M

/ rr - =0.8×10 N N 3 33 3 ´ h th m As gm a =´ =1.8×10 == 2 + 2 rc a - - 3 ®= −5 ./ - .. 610 26 ´ ol 19 1651 695 12 012695 8 10 988 8 cm/h,thefilmwiththicknessof200nmis 31 2 60 56 . N usingtheDCJetCVDand0.8μm/hover . As ./ ./ 2 31 2 −6 ×1.810 ´ mc ´´

´ VN m −5 =´ 10 M

0 / ilCrth milliCarat m gh ´´ 8 10 988 8 // 18 h 31 23 19 a . hm 81 / =8×10 a h mo = == =1.8×10 873 28 62 03 −3 l ./ - -

5 .. cm/h =1.8×10 6 =´ 19 −7 mh mg 31

m . 2 gh isI=(4mA/cm / −3 31 h =´ 2 h

is: V = 1 cm cm/ 2 =8×10 for1his .1 = 200 648 0 18 62 200 h 12 . ; mg A 6 .. 10 695 −3 =1 m mg −5 /

cm - Cara

ggh cm/ 33 / / mg cm 2 cm Cara 3 × 2 × /h t h / 2 2 / h ) × h t 183 Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3 184 energy ofthesurfaceparticleandthosesputtered, someelectrons canbetransferred from chemical environment ofthetarget. Dependingontheelectron affinityandionization complex becauseitisstrongly affected bytheso-calledmatrixeffect, thatis,bysurface tering yields (positive or negative ions), which in contrast to sputtering yield is much more interested inionsputteringyieldanditsenhancement. Therefore, wealsodefineionsput- the depositionandformationoffilmmicrostructures. Inmaterialanalyses,weare Although, the neutral particles absolutely dominate, ions may play very important role at The sputterparticlesincludeelectricallyneutral andions(positivenegative). sputtered perincidentiscalledsputteringyield,thatis, the surfaceparticlesare ejectedtovacuumandthendeposited.Thenumber of particles target structure isstr collision cascades, and displacement of surface particles. Due to the displacement, the where theytransfertheirmomentatothesurfaceatoms.Theimpactionsinducecollisions, of sputtering,whichistheprocess atwhichenergetic ionsimpingeonacathode(target) lar, whichmayhaveverylarge sputtered areas. All thesedevicesoperateontheprinciple planar orcylindricalconfigurations.Theconfigurationcanbecircular or rectangu- with magneticfields.Thelatterdevicesare knownasmagnetrons. Magnetrons canbein from hotfilaments. using ionneutralizationbyelectrons, whenionbeamspassviaelectron cloudsemitted also useenergetic ionandmolecularbeams.Energetic molecularbeamsare produced pulse electricdischarges. Theircombinationsare alsoemployed.Forsputtering,wecan briefly herein. the sputteringpumpinChapter15.8.Therefore, weelucidatesputteringprocess very ing surfaceinanalyticalmethodsaswellsputteringionpumps.We discuss oxides, nitrides, or a vast number of compositematerials.We also use sputteringfor clean- tors, andinsulators.Thedepositedfilmssource materialscanbepure elements, the depositionofthinfilmsmademanymaterialsincludingconductors,semiconduc- ions withthesurfacesofsolids(seealso Sputtering is one ofseveral processes that can take place at the interaction ofenergetic 3.24 Sputtering devicescanbedesignedindiode,triode,andelectrode systemsequipped Generally, insputteringprocess, weusedirect current (d.c.),radiofrequency (r.f.), and

metallic contamination. at temperatures below 1000 °C, and the cBN films do not show measurable traces of trast, ECRMWCVDallowstogrowth cBNwithlowgasconsumption.Thedepositionis consumption. ThecBNbyDCjetisalsocontaminatedwithelectrode materials.Incon- is larger. Inaddition,DCJetCVDiscarriedattemperatures above1000°Candhighgas than thatbyDCJetCVD,anditseemstobemore effective becausethedepositionarea At thegivendepositionrateandareas, themassdepositedbyECRMWCVDisgreater Sputtering, Etching,andDeposition essed andcompressed tosuchextentthatatthestructure relaxation, Appendix A.4).Sputteringistheprocess usedfor Y Vacuum andUltravacuum:Physics and Technology = N N + Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3

for molarquantity: sputtering yield. trometry (SIMS).However, thesputteringrateofneutralparticles isfarhigherthantheion of ionsputteringyieldsviatheelectron transferisemployedatsecondaryionmassspec- can beconvertedtonegativeorpositiveionsaconsiderableextent.Suchenhancement the surfaceparticles to theescaping neutral particles and vice versa. Thus,neutral particles Thermodynamics ofGases at LowPressures cally estimatedandexperimentally measured, too. able forcalculations. materials in the forms of tables, graphicaldependences,as well as there is software avail- masses. Sputteringyieldsare availablefordifferent ionsandtheirenergies andtarget nature ofmaterial,andsputteringyieldthatisdependentontheenergy ofionsandtheir This indicatesthatthesputtering(etching)rateis dependentonthecurrent density, ing rateis The volumeofsputtered materialcanbemeasured byaprofilometer. Hence,thesputter where The substitutionforNand while thenumberofionsN carries elementarycharge e.Thus, ing onanarea Aofthetarget fortimetandelectriccurrent Iinassumptionthateachion The number of sputtered particles The massanddepositionrate are proportional totheetchingrateandcanbe theoreti- r =h/tisthesputteringrate ρ isthemassdensityofsputtered material A isthesputtered area h isthedepthofsputtered volumeV Y == N N M M aa jA a e =® N t M a M + canbedeterminedfrom thetotalelectriccharge eN + intothesputteringyieldgives MN M e NI N a a ++ =® =® jA N can be expressed by their mass e M M rY t tN a = == NN Ah jA a r j rr t e == M N eN M e It a a a a a == V M jA h rr t a e N t j e a M N a a Ah M = a r r N j e a M N a a M using equation + imping- 185 - Downloaded By: 10.3.98.104 At: 15:07 25 Sep 2021; For: 9781315155364, chapter3, 10.1201/9781315155364-3