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TransferringtheunitofmassbetweenweightskeptinairandinvacuumLarsNielsen,DFM

Abstract

InthecurrentwattbalanceandcrystaldensitymeasurementsusedtoassignavaluetothePlanckconstanthintermsoftheinternationalprototypeofthekilogram ,theunitofmasshastobetransferredfromairtovacuum.Inthefuture,whenthekilogramhasbeenredefinedintermsofafixedvalueofthePlanckconstant,theunitofmasshastobetransferredfromvacuumtoair.Thisreportdescribeshowthismasstransfermaybedoneinawaythatenablesevaluationofthestandarduncertaintyassociatedwiththemasstransfer.

1. IntroductionTheunitofmass,kilogramiscurrentlydefinedasthemassoftheinternationalprototypeofthekilogram immediatelyaftercleaningandwashingusingaspecifiedprocedureinvolvingsolventsandsteam 1 2 3 .Astheprototype iskeptandusedinhumidair,anaturallayerofadsorbedwateronthesurfaceoftheartefactiscontributingtotheunitofmass.WhenmeasuringthePlanckconstant inthecurrentwattbalanceexperimentsortheAvogadroconstant inthecurrentcrystaldensityexperiment,massstandards weights thataretraceabletotheprototype butusedinvacuumarerequired.OnceanewdefinitionofthekilogramintermsofafixedvalueofthePlanckconstantisinplace,thekilogramwillberealizedinvacuumandneedstobetransferredtoairbeforebeingdisseminated.

Agravimetricmethodisoftenusedtomeasurethemasschangeduetowatersorption/desorptionassociatedwiththetransferofweightsbetweenairandvacuum.Thismethodusesatleasttwomassstandards knownassorptionartefacts havingthesamenominalvaluesofmassandvolume,thesamesurfacefinish,butalargedifferenceintheirsurfaceareas.Basedontheassumptionthatthesorptionperunitareaisthesameforthetwosorptionartefacts,thechangeinsorptioncanbecalculatedfromameasuredchangeinmassdifferencebetweenthetwosorptionartefactswhentransferredbetweenairandvacuum.

Experimentshaveshownthatthesorptionisnotalwaysreversible.Themodellingdescribedinthisreportaccountsforthat.

2. ModellingAssumethatsorptionartefactSiscycledrepeatedlybetweenthetwomediaair A andvacuum B .Thecyclingisdescribedbyamediasequence A, B, A, B, … ,sothat A, B, A,etc.Whenthesorptionartefactistransferredfrommedium tomedium themassoftheartefactisassumedtochangeaccordingtothemodel:

, , , 1

where

, isthemassoftheartefactSinmedium ,

, isthemassoftheartefactSinmedium ,

isthegeometricalsurfaceareaoftheartefactS,


isthechangeinsorptionperunitsurfaceareaoftheartefactwhentransferredfrommedium tomedium ,

isafactorforthesorptionartefactSconvertingitsgeometricsurfaceareatoaneffectivesurfaceareaforthesorption.

Notethatthequantity maybenegativeorpositivedependingonthemediabetweenwhichtheartefactistransferred.

Ifasetof sorptionartefactsS , S , . . . , S arecycledbetweenthetwomedia,equation 1 isreplacedby

, ,

, ,

⋮, ,

2

Thesuccessfuluseofsorptionartefactsrelyontheassumptionsthatthechangeinsorption isthesameforallsorptionartefactsinvolvedinthesametransferbetweenmediaAandB.Inordertotestthatassumption,atleastthreesorptionartefactsneedtobeused.Uptonowithasbeenusualpracticetoassumethatthefactors , … , areidentical andequalto1exactly .Thisimpliesthatthe

sorption isdefinedintermsofgeometricarearatherthanintermsofeffectivearea,andthattheratioofeffectiveareatogeometricalareaisthesameforallsorptionartefactsinaset.Thefactors

, … , canbeinterpretedastheratiosofeffectiveareatogeometricareasofthesorption

artefacts.Ifthesurfaceroughnessesoftheartefactshavebeenmeasured,someinformationabouttheseratiosisavailable.Inthatcase,bestestimates largerthan1 andassociatedstandarduncertaintiesmaybeassignedtothefactors , … , .Ifnoinformationoftheeffectiveareasofthe

artefactsisavailable,variationinthesurfacefinishofthesorptionartefactsinasetmaybetakenintoaccountbyassigningapriorvalue1toallfactors , … , butwithnon‐zerostandarduncertainties.

AssumethatoneweightM ispermanentlystoredinair mediumA ,andanotherweightM ispermanentlystoredinvacuum mediumB ;bothweightshavethesamenominalmassvaluesasthesorptionartefactsS , S , . . . , S .Thegoalistomeasurethemass oftheweightM storedinvacuumintermsofthemass oftheweightM storedinair,orviceversa.ThevolumesoftheweightsM andM aredenoted respectively ,andthevolumesofthesorptionartefactsS , S , . . . , S aredenoted , , . . . , .

Atstage inthemediasequencethesorptionartefactsS , S , . . . , S arecomparedwitheachotherandwitheithertheweightstoredinairortheweightstoredinvacuum.Thiscomparisonismodelledby

∆ 3

where

,, , , , … , , isavectorofthemassvaluesoftheweightsandthesorption

artefactsatstage inthemediasequence, , , , … , isavectorofthevolumevaluesofthemassandsorptionartefacts

involvedinthecomparisons, isthedesignmatrixaccordingtowhichthemassandsorptionartefactshavebeencomparedat

stage inthemediasequence, isthevectorofairdensitiesmeasuredforeachmasscomparisonatstage inthemedia

sequence,


∆ isthevectorofindicationdifferencesmeasuredforeachmasscomparisonperformedatstage ofthemediasequence,

isthescalefactorofthemasscomparatoratstage ofthemediasequence.

Whenweighingisdoneinair A ,theweightM storedinvacuumisnotinvolvedinthemasscomparison,soif A,thentheelementsofthesecondcolumnof areallzero.Similarly,whenweighingaredoneinvacuum B ,theweightM isnotinvolvedinthecomparisons,soif B,thentheelementsofthefirstcolumnof areallzero.Themasses and areassumedtoconstant;thisassumptionisplausibleonlyiftheweightM iskeptpermanentlyinair,andtheweightM iskeptinpermanentvacuum.Whenweighingisdoneinvacuum B ,theairdensityisnegligible,soif B,then .

Theequations 2 and 3 formthenecessarybasisfortransferringtheunitofmassbetweenairandvacuum.TheseequationsareeasilysolvedbythegeneralmethodofleastsquaresdevelopedatDFM 4 5 ,knownasDFM‐LSQ.Toillustratethis,considerthatthemassoftheweightM keptinairisknown,andthatwewanttomeasurethemassoftheweightM keptinvacuumusingtwosorptionartefactsS andS .Iftheshortestpossiblemediasequence A, B isapplied,threemassdifferencesmaybemeasuredinair,

Δ ,

Δ ,

Δ ,

1 1 01 0 10 1 1

,

,

, , 0

, 0 ,

0 , ,

,

,

, 4

andthreemassdifferencesmaybemeasuredinvacuum:

Δ ,

Δ ,

Δ ,

1 1 01 0 10 1 1

,

,

. 5

Themassesofthesorptionartefactsinairandinvacuumarerelatedtoasingle negative sorptioncoefficient throughtheequations

,

,

,

,6

Inthismeasurementthereare 6quantities,forwhichnopriorinformationisavailable,

,,,

,,

,,

,, , 7

and 16quantities

, , Δ , , . . , Δ , , , ,,,

,, , , , 8

forwhichwehavebestestimates andanassociatedcovariancematrix .Amongthequantities andthereare 8constraintsontheform

, , 9

whicharederivedfromequations 4 , 5 ,and 6 bysubtractiontherighthandsidesfromthelefthandsidesoftheequations.Followingref. 5 ,bestestimates and ofthequantities and ,aswellastheassociatedcovariancematrix,isfoundbyminimisingthechi‐squarefunction

10

subjecttotheconstraint , .Theconsistencyofmodelanddatashouldbecheckedbycomparingtheobservedchi‐squarevalue


11

withitsexpectationvalue,whichisequaltothedegreesoffreedom oftheleastsquaresadjustment:

, 12

where isthenumberofconstraintsand numberofquantities,forwhichnopriorinformationisavailable.Assumingthat istheoutcomeofachisquaredistribution ,consistencybetweendataandmodelmaybequestionedif

Pr , 13

where istheprobabilitythat bychance theoutcomeofachi‐squaredistribution islargerthanthevalueactuallyobserved,and isachosenlevelofsignificance.Itisrecommendedtousealevelofsignificanceintherange0.1 0.2.

Thelargerthenumberofdegreesoffreedomare,thestrongeristheconsistencytestexpressedbyequation 13 .Inthecasedescribedabove,wehave 8 6 2degreesoffreedomfortheleastsquaresadjustment;thesetwodegreesoffreedomwouldreducetozero,ifthetwo redundant masscomparisonofthesorptionartefactsinequations 4 5 wereomitted.Evenifthetworedundantmasscomparisonswereincluded,therewouldbenoredundancyinthedeterminationofthesorptioncoefficient.Inordertoobtainsuchredundancy,athirdsorptionartefactwouldhavetobeincluded.Assumingthatthemassvalues and areconstant,furtherredundancywouldbeobtainedbyexpandingthemediacyclefromjusttwostagestothreeormorestages.

Inadditiontothegeneraltestforconsistencydescribedabove,itisrecommendedtotestforconsistencybetweeneachmeasuredvalue inthearray withthecorrespondingadjustedvalue inthearray .Thisisdonebycalculatingthenormaliseddeviations

, 1, … , . 14

Measuredvalues forwhich| | 2arepotentialoutliers,althoughthereisaprobabilityofabout0.05thatthiscouldhappenbychance,assumingthat followsanormaldistributionN 0,1 .

3. AnalysisofsorptiondataUnfortunatelytherearenodataavailableforamasstransferbetweenmassandvacuum,whichhasbeencarriedoutusingtherecommendedprocedure.However,inasorptioncomparisoncarriedoutinthetaskgroupCCMWGMTG1,datawerereportedthathavebeenusedtodemonstratehowthemethodworksandhowaccurateitis.

IntheTG1sorptioncomparisonthreesorptionartefactS , S ,andS werecirculatedamonganumberofparticipants.ThesorptionartefactS wasanintegralweightofstainlesssteelidentifiedas‘71DD’,S wasastackoftwostainlesssteeldiscsidentifiedas‘Stack1’,andS wasastackoffourstainlesssteeldiscsidentifiedas‘Stack2’.Themeasuredvolumesandareasarelistedintable1.

Table1.Measuredvolumesandsurfaceareasofsorptionartefacts

S1 S2 S3V /cm3 126.7730 126.7737 126.7700

u (V )/cm3 0.0011 0.0014 0.0010

A /cm2 140.0 188.3 285.4

u (A )/cm22.3 2.6 3.3

K48(m1 kg)/mg 0.160

u (m )/mg 0.008

V /cm346.4879

u (V )/cm30.0005


Table2.MasscorrectionandvolumeofDanish nationalprototypeK48Eachparticipatinglaboratorydidmeasurementsinair mediumA andvacuum mediumB usingamediacycle A, B, … , withatleastonemeasurementinvacuumandtwomeasurementsinair.Thelaboratoriescomparedthethreesorptionartefactswitheachotherinairandinvacuum,andwithanunspecifiedreferencestandardkeptinair.Thesorptionartefactswerenotcomparedtoaweightkeptincontinuousvacuum.Ateachstageinthemediasequence,thelaboratoriesreportedthemeasuredmassvalueof andthethreemassdifferencesmeasuredbycomparingS , S ,andS .Theaverageairdensitiesassociatedwiththemasscomparisonsinairwerealsoreported.

Inordertoassesstheuncertaintyatwhichthemassofanationalprototypekeptinaircouldbetransferredfromairtovacuum,theindicationdifferences,whichwouldhavebeenobservedbycomparingtheintegralsorptionartefactS withtheDanishprototypeK48inair,werecalculatedusingtheairdensitiesandmassvaluesofS reportedbythelaboratories,thevolumeofS listedintable1andthemass andvolume oftheprototypelistedintable2.Thecalculationwasdoneusingequation 3 with 1 exactly .Similarly,allreportedmassdifferencesmeasuredinairwereconvertedintoindicationdifferences.

Measurementdatawerereconstructedthiswayforfiveparticipatinglaboratoriesthatprovidedthenecessaryinformation:BIPM,LNE,PTB,METAS,andINRIM.ThereconstructedmeasurementdatawereanalysedusingDFM‐LSQ.Acommonstandarduncertainty Δ 0.001mgwereassignedtothereconstructedindicationdifferencesΔ ,andforthefactors , , usedtodescribevariations

inthesorptionefficienciesofthesurfacesofthesorptionartefacts,apriorvalueequalto1withstandarduncertainty 0.1wasassignedtoeachsorptionartefact.

Theresultsoftheanalysisofthereconstructedmeasurementdataareshowninfigure1inthecaseofINRIM.Thefigureindicatesthatthesorptionisnotreversible;whatisremovedfromthesurfaceinthefirsttransitionfromairtovacuumislessthanthatremovedinthesubsequenttransitionsfromairtovacuum.Furthermore,moreisaddedtothesurfacebythetransferfromvacuumtoairthanthatremovedbytheprecedingtransitionfromairtovacuum.Asaresult,themassinairofthesorptionartefactsincreasesfromonestageinairtothenext,andthesameistrueforthemassinvacuum.Thisiswhyitisrecommendedtocomparethesorptionartefactswithaweightpermanentlykeptinvacuumaswellaswithaweightpermanentlykeptinair.Bydoingsoitcanbetestedifthemassdifferencebetweentheweightkeptinairandtheweightkeptinvacuumisconstantwithinthemeasurementuncertainty.

InthecaseofINRIM,thedegreesoffreedomoftheleastsquaresanalysisis 21,andtheobservedchi‐squarevalueis 2.5.Inthiscase,thereisnoindicationofinconsistenciesbetweendataandmodel.Thisisconfirmedbythenormaliseddeviationsshowninfigure2,whichallfallswellwithintherange| | 2.Iftheprioruncertaintiesofthefactors , , arereducedbyafactorof10 from

0.1to 0.01 ,theobservedchi‐squarevalueincreasesfrom 2.5to 11.1,whichalsonotlargerthanexpected.However,thenormaliseddeviationsassociatedthesurfaceareas

, , andthefactors , , failtomeetthecriteria| | 2byasignificantmargin

2.4 | | 3.1 .Thisindicatesthatthereisasignificantdifferenceinthesorptionefficienciesofthesurfacesofthethreesorptionartefacts.Hadonlytwosorptionartefactsbeenused,suchadifferencecouldnothavebeendetected.

Thestandarduncertaintiesofthemassvaluesfoundforthesorptionartefactsinairisabout0.010mg,whereasthestandarduncertaintiesofthemassvaluesfoundinvacuumareabout0.008mg.This


meansthattheuncertaintyassociatedwiththemasstransferfromairtovacuum orfromvacuumtoair isabout0.006mg.

FormtheanalysisofthedataprovidedbyBIPM,LNE,PTB,METAS,andINRIMintheCCMWGMTG1sorptioncomparison,fivesetsoffactors , , werefoundforthecirculatedsorptionartefacts

S , S ,andS .Thesefivesetsareshowninfigure3.Itisnotedthatthefivesetsofvaluesareconsistenttakingintoaccounttheassociatedstandarduncertainties,andthatsorptionartefactS seemstohaveasorptionefficiencythatisabout10%higherthanS andS .

Figure1.Thesorptioncoefficients andmasses , , , , , ofthesorptionartefactsS , S ,andS measuredbyINRIMattheninestagesofthemediasequence A, B, A, B, A, B, A, B, A ,wheremediumAisair oddvaluesofstageindex andBisvacuum evenvaluesofstageindex .Theerrorbarsindicatestandarduncertainties.

Figure2.Thenormaliseddeviationsassociatedwiththe48measuredquantitiesinvolvedintheleastsquaresadjustment.

1.50

1.52

1.54

1.56

1.58

1.60

1 2 3 4 5 6 7 8 9(m

i1 kg)/m

gStage in media sequence, i

Mass of S1

‐0.0003

‐0.0002

‐0.0001

0.0000

0.0001

0.0002

0.0003

1 2 3 4 5 6 7 8

s i/(mg/cm

2)

Stage in media sequence, i

Sorption coefficient

1.64

1.66

1.68

1.70

1.72

1.74

1 2 3 4 5 6 7 8 9

(mi1 kg)/m

g


Mass of S2

1.40

1.42

1.44

1.46

1.48

1.50

1 2 3 4 5 6 7 8 9

(mi1 kg)/m

g


Mass of S3

‐2.00

‐1.00

0.00

1.00

2.00

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47

d

Measured quantity no.

Normalised deviations


Figure3.Thefactors , , measuredbyfivelaboratoriesforthesamesetofsorptionartefactsS , S ,andS .Theerrorbarsindicatestandarduncertainties.

4. SimulationstudyInordertovalidatethemasstransferfromairtovacuumusingthemodeldescribedinsection2,asimulationstudywascarriedout.ThestudyinvolvedoneweightM permanentlystoredinairmediumA ,oneweightM permanentlystoredinvacuum mediumB ,and 3threesorptionartefactsS , S ,andS cycledbetweenairandvacuumusingthemediasequence A, B, A, B, A, B, A, B, A .Ateachstageinthemediasequence,allsixpossiblemassdifferencesbetweenthethreesorptionartefactsandtheweightkeptpermanentlyinthemediawereassumedtobemeasured.

4.1. SimulationofdataInthefirststep,truevalueswereassignedtothefollowingquantities:

Themass oftheweightkeptpermanentlyinvacuum. Themass andvolume oftheweightkeptpermanentlyinair. Thevolumes , , ,geometricalareas , , ,conversionfactors , , andinitial

masses,,

,,

,ofthesorptionartefactsS , S ,andS .

Theairdensities , , , , duringmasscomparisonsin assumedtobeconstantduringmasscomparisonsatagivenstageofthemediasequence .

Thebalancescalefactors . Theeightsorptioncoefficients , … , .

Usingequation 2 ,truemassvaluesofthethreesorptionartefactsatstage2–9inthemediasequencewerecalculated,andusingequation 3 atotalof9 6 54truedifferences∆ , , … , ∆ , werecalculated.

Inasecondsteparealisticstandarddeviation wasassignedtoeachquantity exceptforthefactors , , ,andameasuredvalue wasdrawfromanormaldistributionN , ,where

0.80

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

BIPM LNE METAS CEM INRIM

Factors p of sorption artefacts

S1

S2

S3


isthetruevalueofthequantity assignedorcalculatedinthefirststep.Forthefactors , ,

measuredvaluesequalto1wereassigned.

Atotaloftwentysimulationssimulationwascarriedoutinfourcases:

Case1:Reversiblesorption/desorptionofthesameamountofwater 0.0001mg/cm2 ateachmediatransfer.

Case2:Reversiblesorption/desorptionasinCase1,butacontaminationof0.010mgwasaddedtoallthreesorptionartefactsinthetransferfrommediastage4 vacuum tostage5 air andnotremovedagain.

Case3:Irreversiblesorption/desorptionsimilartothatobservedbyINRIMinthesorptioncomparisondescribedinsection3,seefigure1.

Case4:Irreversiblesorption/desorptionasinCase3,butacontaminationof0.010mgwasaddedtoallthreesorptionartefactsinthetransferfrommediastage4 vacuum tostage5 air asinCase2.

Ineachofthefourcases,thesameseedoftherandomnumbergeneratorwasusedforthesimulations.Thismeansthesequenceofrandomcomponentsisthesameineachofthefourcases;onlythetruevaluesofthequantitiesaredifferent.

Forthemassoftheweightkeptinvacuum,atruevalue 1kgwasassigned,andforthemasskeptinair,thebestestimatesgivenintable2wereusedastruevaluesofthemass andvolume oftheweightkeptinair,whereasthestandarduncertaintiesintable2wereusedasstandarddeviationsforthesimulation.Forthesorptionartefacts,thebestestimatesgivenintable1wereusedastruevaluesofthevolumes , , andthegeometricalareas , , ,whereasthestandarduncertainties

intable1wereusedasstandarddeviationsforthesimulation;twentysetsofconversionfactors, , randomlyselectedfromanormaldistributionN 1, with 0.1wereusedastrue

values,onesetforeachofthetwentysimulations.Asinitialmassvalues,,

,,

,ofthe

sorptionartefacts,thebestestimatesfoundbyINRIMinthesorptioncomparisondescribedinsection3 seefigure1 wereusedastruevalues.ThefiveairdensitiesreportedbyINRIMinthesorptioncomparisonwereusedastruevaluesoftheairdensities , , , , ;thestandarddeviationusedforthesimulationofairdensitieswas 0.00005kg/m .Forsimplicity,thevalues 1 exactly wereselectedforthebalancescalefactors.Forthesimulationoftheindicationdifferences∆ ,thestandarddeviation 0.001mgwasused.

4.2. AnalysisofsimulateddataToeachsimulatedvalue ofaquantity exceptthefactors , , ,astandarduncertainty

wasassigned.Forthefactors , , ,thebestestimate 1andfivedifferentbut

commonstandarduncertainties wereassigned:

1,

, ∈ 0.2, 0.1, 0.05, 0.02, 0.01 .

Eachsimulatedsetofdatawasanalysedfivetimes,onetimeforeachstandarduncertainty assignedtothefactors , , .Onlyoneofthestandarduncertainties isequaltothe

standarddeviation 0.1usedtosimulatevaluesofthefactors , , .Analyseswithlargerand

smallervaluesof wereperformedinordertoassesstheimportanceofselectingarealisticvalueof .Asummaryoftheresultsoftheanalysesperformedareshowninfigure4‐7.


Incaseofcompletelyreversiblesorption/desorption Case1 ,theadjustedvalueofthemass oftheweightkeptinairisfairlyindependentofthevalueoftheuncertainty assignedtothefactors , , intheleastsquaresanalysis,seefigure4.Onlyinthecase 0.2,wherethe

uncertaintyisoverestimatedbyafactoroftwo,thereareafewcases simulationno.14and19 ,wheretheadjustedvaluesofthemass aresignificantlydifferentcomparedtothecases,where

isequaltoorsmallerthanthestandarddeviation 0.1usedforthesimulationofthefactors , , .Thestandarduncertainty ,thevalueofwhichisindicatedwitherrorbarsinfigure

4,decreasesas decreases,butasthereareothersignificantuncertaintycontributionsto ,e.g.fromthestandarduncertainty 0.0080mgassignedtothemass ofthereferenceweightkeptinair,thedecreasein ispredominantwhen isreducedfrom 0.2,where0.8mg 1.4mg,to 0.1,where0.0096mg 0.0103mg.Inthecase

0.2,thevariationintheuncertainty oftheadjustedvalueofthemass isverylarge,especiallyforthoseadjustedvaluesof thatdiffersmostlyfromthetruevalue.Whathappensinthesecasesisthattheadjustedvaluesof , , areratherextremeinsuchawaythatthe

differencesoftheeffectivesorptionareas , , becomesrathersmall,orevengetthe

wrongsign.Itshouldbenotedthatforallfivestandarduncertainties ,theadjustedvalueofthemass isconsistentwiththetruevaluewithin 2 .Thesameisnottrueforthefactors , , ;forthesefactorsthetruevaluesareonlyexpectedtoberecovered,ifthevalueof is

equaltoorlargerthanthestandarddeviation 0.1usedforthesimulationofthetruevaluesof , , .Thegraphsintherightcolumnoffigure4confirmthat.Thegraphsinthemiddleoffigure

4showthatthetestofconsistenceofdatawithmodelisalmostindependentofthevalueoftheuncertainty assignedtothefactors , , .Atfirstthismightbeseenasaparadox,sincefor

smallvaluesof theadjustedvaluesof , , differsignificantlyfromthetruevalues.The

explanationis,however,thatduetothesignificantuncertaintyintheairbuoyancycorrection,theinformationaboutdifferencesintheeffectivesorptionareas , , cannotbeextracted

fromthemeasuredmassdifferenceswhensorption/desorptionisreversible.


Case1:Reversiblesorption/desorption

Figure4.ResultsoftheleastsquaresanalysisoftwentysimulationsinCase1withstandarduncertaintya 0.2,b 0.1,c 0.05,d 0.02ande 0.01assignedtothefactors , , .Thegraphtotheleftshowstheadjustedvaluesfoundforthemass ofweightkeptinvacuumcomparedtoitstruevalue.Thegraphinthemiddleshowsobservedchi‐squarevalue comparedtotheexpectationvalue fullline andthecriticalvaluecorrespondingtosignificancelevel 10% brokenline .Thegraphtotherightshowstheadjustedvalues opendots ofthefactors , , comparedtotheirtruevalues closeddots .Errorbarsindicatestandarduncertainties.

a

b

c

d

e


Case2:Reversiblesorption/desorption+0.01mgcontamination

Figure5.ResultsoftheleastsquaresanalysisoftwentysimulationsinCase2withstandarduncertaintya 0.2,b 0.1,c 0.05,d 0.02ande 0.01assignedtothefactors , , .

a

b

c

d

e


Case3:Irreversiblesorption/desorption


a

b

c

d

e


Case4:Irreversiblesorption/desorption+0.01mgcontamination


a

b

c

d

e


Incase2,wherethereversiblesorption/desorptionisdisruptedbyadding0.010mgcontaminationtoallthreesorptionartefactsbetweenstage4and5inthemediacycle,itisonlypossibletoobtainconsistencybetweendataandmodeliftheadjustedvaluesofthefactors , , aresothatthe

differencesineffectivesorptionareas , , areclosetozero.Thisisachievedif

≅ 1.2, ≅ 1and ≅ 0.6asinthegraphtotherightoffigure5a .Thatis,toachieve

consistencyalargevalueof isrequired,suchas 0.2.However,asthedifferencesintheeffectivesorptionareas , , areclosetozero,theuncertainty becomes

extremelylargeasseeninthegraphintheleftinfigure5a .For 0.05,theuncertainty reachesausefullevel,buttheadjustedvaluesof aresignificantlybiasedcomparedtothetruevalue,seefigure5c .Forsmallervaluesof ,thisbiasdisappearsasseeninfigure5d ‐e .However,theresultswouldnot andshouldnot betrustedbytheexperimenter,astheconsistencetestshowsasignificantinconsistencybetweendataandmodel,whichisduetothefactthatacontaminationof0.010mgwasaddedbetweenstage4and5inthemediacyclewithoutaccountingforitinthemodelusedtoanalysethedata.

Contaminationofthesorptionartefactsisaproblemonlyifitisnotuniformlydistributedoverthesurfacesofthesorptionartefacts.Infact,uniformlydistributedcontaminationallowstheratiosoftheeffectivesorptionareastobemeasuredfairlyaccurate.ThisisillustratedinCase3,whereuniformirreversiblesorption/desorptionhasbeenassumed.Asseeninthegraphsintheleftcolumnoffigure6,theadjustedvaluesofthemass oftheweightkeptinvacuumarevirtuallyindependentofthevalueassignedto .Theassociatedstandarduncertainties alsohaveonlylittledependenceofthevalueassignedto intheleastsquaresanalysis;ontheaverageitdecreasesfrom

0.0096mgfor 0.2to 0.0087mgfor 0.01.Incontrasttocase1,thetestofconsistencydependsonthevalueassignedto ,asseeninthegraphsinthemiddlecolumnoffigure6.However,evenfor 0.01theobservedchi‐squarevaluefallsbelowthecriticalvalueforatestofconsistenceata10%levelofsignificanceinelevenoutoftwentysimulations.Asshowninthegraphsintherightcolumnoffigure6,theadjustedvaluesofthefactors , , areconsistent

withthetruevaluesaslongastheuncertainty isequaltoorlargerthanthestandarddeviation0.1usedforthesimulation.For 0.1,thestandarduncertaintiesassociatedwiththefactors

, , areabout30%smallerinCase3thaninCase1.Thereasonisthatthegaininmassofthe

sorptionartefactsfromonestageingivenmediatothenextstageinthesamemediaismeasuredbycomparingthesorptionartefactswiththeweightstoredinthatmedia,whichisassumedtohavethesamemassatallstagesinthemediacycle.Thelargerthesemassgainsare,themoreinformationtherewouldberegardingthevaluesofthefactors , , .

Incase4,theuniformirreversiblesorption/desorptionwasdisruptedbyaddinga0.010mgcontaminationtoallsorptionartefactsbetweenstage4andstage5inthemediacycle.Asseeninthegraphsintheleftcolumnoffigure7,thisleadtoasignificantbiasintheadjustedvaluesofthemass oftheweightkeptinvacuum,whichislargerthelargerthevalueassignedto is.For 0.2,theaveragebiasis0.0300mg,andfor 0.01,theaveragebiasis0.0061mg.Forcomparison,theaveragebiasesfoundinCase3were0.0034mgfor 0.2and0.0028mgfor 0.01.Asseeninthegraphsintherightcolumnoffigure7,theadjustedvaluesofthefactors , ,

seektoreducethedifferencesintheeffectivesorptionareas , , ,justasincase2.It

islesspronouncedinCase4,however,duetotheadditionalamountofinformationabouttheeffectivesorptionareas.Forthemajorityofthesimulations,thechi‐squaretestindicatestheinconsistency


betweendataandmodelcreatedbyaddingthe0.010mgcontaminationtothedatawithoutaccountingforitinthemodel;seethegraphsintherightcolumnoffigure7.

5. DiscussionWhentransferringtheunitofmassfromairtovacuum,atraditionalassumptionhasbeenthatonlywaterisadsorbedanddesorbed,andthatthattheadsorption/desorptionisreversible.ThesorptioncomparisoncarriedoutinCMMWGMTG1shows,thatadsorption/desorptionisirreversibleingeneral,sothatthesorptionartefactsgainmassesastheyarerepeatedlycycledbetweenairandvacuum.Theirreversibilityseemstobemostpronouncedifthesorptionartefactsarecleanedjustbeforethemediacyclingexperiment.Aftersomecyclesbetweenairandvacuum,thesorption/desorptionseemstoapproachreversibility,andthemassesofthesorptionartefactstendtostabilizeinairaswellasinvacuum.

Thesimulationexperimentshowsthatirreversibilityofthesorption/desorptionisinfactusefulasitenablesameasurementoftherelativeabsorptionefficiencies , … , ofthesurfacesofasetof

sorptionartefactsS , S , . . . , S .Thequantities , … , maythereforebedeterminedbyperforming

anair‐vacuumcyclingexperimentonfreshlycleanedsorptionartefacts.Oncethesequantitieshavebeenmeasured,theirvaluesandassociatedcovariancematrixcouldbeusedasinputtotheanalysisofsubsequentair‐vacuumcyclingexperimentswithoutcleaningofthesorptionartefacts,wheretheirreversibilityofthesorption/desorptionprocessmightbelesspronounced.

Incaseofreversiblesorption/desorption,atleastthreesorptionartefactsareneededinordertotestthehypothesisthatthemassofthesorptionisproportionaltothegeometricalareaofthesorptionartefacts.Ifasorptionartefactismadeupof discs withspacers havingthesamenominalmass,theuniformityofthesorptionefficienciesamongthediscscouldbetestedinaseparateair‐vacuumcyclingexperimentinwhichthemassdifferencesamongthediscs withspacers aremeasuredinairandinvacuum.

Thesimulationswereperformedonanair‐vacuumcyclingexperiment,inwhichtheweightstoredinairwasaplatinum‐iridiumstandard,whereastheweightstoredinvacuumandthesorptionartefactsweremadeofstainlesssteel.ThiswasdoneinordertoevaluatethestandarduncertaintywithwhichthemassofaweightkeptinvacuumcouldbemeasuredintermsofanationalprototypekeptinairusingthesorptionstandardscirculatedintheCCMWGMTG1sorptioncomparison.Thisuncertaintyturnedouttobeontheorderof0.010mgandwasactuallydominatedbythestandarduncertainty0.008mgassignedtotheprototypeitself.Thevalidityofthisuncertaintycanonlybeprovenbyperformingaratherlargenumberofair‐vacuumcyclingexperiments similartotheperformedsimulation thatleadstoconsistentmassvaluesoftheweightstoredinvacuum,takingintoaccountthemeasurementuncertainty.Insuchanexperiment,theweightkeptinairshouldhavethesamenominalvolumeasthesorptionartefactsinordertoreducetheuncertaintyduetobuoyancyeffectswhenmeasuringinair.Forthesamereasonshouldthevolumes orvolumedifferences ofthesorptionartefactsandtheweightkeptinairbemeasuredwiththesmallestpossiblestandarduncertainty.

6. ConclusionAprocedurefortransferringtheunitofmassbetweenaweightkeptinairandaweightkeptinvacuumhasbeenpresented.Theprocedureinvolvestheuseofsorptionartefactsthatarerepeatedlycomparedtotheweightkeptinairandtotheweightkeptinvacuuminanair‐vacuumcycle.Amodelofthemeasurementbasedonhomogeneousabsorption/desorptionhasbeendescribed.Themodel


hasbeentestedonadaptedmeasurementdataprovidedbyparticipantsinasorptioncomparisonandonsimulateddata.Thesensitivitytomodellingerrorshasbeentestedbysimulationaswell.

Ithasbeenshownthatamassvalueinairhavingastandarduncertaintyof0.008mgcanbetransferredtoamassvalueinvacuumwithastandarduncertaintyof0.010mg,whichmeansthatanuncertaintycontributionof0.006mgfromtheair‐vacuumtransfercanbeachieved,atleastinprinciple.Themodelisbasedonthecrucialassumptionthatsorption/desorptionishomogenousoverthesurfacesofthesorptionartefacts.Ifthisassumptionfailsinreality,themassvaluecalculatedfortheweightkeptinvacuummightbesignificantlybiased.Althoughtheproposedmethodforanalysingthedatafromanair‐vacuumcyclingexperimentincludesatoolfortestingtheconsistencyamongdataandmodel,thereisaratherlargeprobability,thataninvalidassumptionmightnotbedetectedinasingleair‐vacuumcyclingexperiment.

7. References1 BIPM2006TheInternationalSystemofUnits SI 8thedn,Sévres,BIPM2 DavisR2003TheSIunitofmassMetrologia,40 2003 299–3053 GirardG1990ThewashingandcleaningofkilogramprototypesattheBIPM,Sévres,BIPM4 NielsenL1998LeastsquaresestimationusingLagrangemultipliersMetrologia35,115‐18

NielsenL2000Metrologia47,183 erratum 5 NielsenL2002EvaluationofmeasurementsbythemethodofleastsquaresAlgorithmsfor

ApproximationIVedJLevesleyetal UniversityofHuddersfield pp170‐86RecommendationG1 2010 oftheCCM,Considerationofanewdefinitionofthekilogram.