Kubelka-Munk-Theory

TheKubelka-MunkTheoryofReflectanceThistheorywasoriginallydevelopedforpaintfilmsbutworksquitewellinmanycircumstancesforpaper.Itisnot,however,terriblygoodfordyedpapers(orverydark,unbleachedpapers)whenlightabsorptionreachesahighlevel.Alimitingassumptionisthattheparticlesmakingupthelayermustbemuchsmallerthanthetotalthickness.Bothabsorbingandscatteringmediamustbeuniformlydistributedthroughthesheet.Ideally,illuminationshouldbewithdiffusemonochromaticlightandobservationshouldbeofthediffusereflectanceofthepaper.Thetheoryworksbestforopticallythickmaterialswhere50%oflightisreflectedand20%istransmitted.KubelkaandMunkZeit.FürTekn.Physik,12,p593(1931).Figure1ConsiderlightofintensityI0incidentonanon-glossypieceofpaperofthicknessXandreflectanceR.BehindthispieceofpaperisasurfaceofreflectanceR’.Thelightwhichre-emergesfromthetopsurfaceofthepaperafterscattering,absorptionortransmissionhasintensityI.AtadistancexfromthebottomsurfaceofthepaperthereisathinlaminaofthicknessdxandscatteredlightisincidentonitwhichistravellingbothupwardsanddownwardsthroughitwithintensitiesiRandiT,respectively.Define:KistheAbsorptionCoefficient≡thelimitingfractionofabsorptionoflightenergyperunitthickness,asthicknessbecomesverysmall.SistheScatteringCoefficient≡thelimitingfractionoflightenergyscatteredbackwardsperunitthicknessasthicknesstendstozero.TheeffectofthematerialinathinelementdxoniTandiRisto:decreaseiTbyiT(S+K)dx(absorptionandscattering)decreaseiRbyiR(S+K)dx(absorptionandscattering)increaseiTbyiRSdx(scatteredlightfromiRreinforcesiT)increaseiRbyiTSdx(scatteredlightfromITreinforcesIR).So:[1][2]wherexismeasuredfromthebottomofthesheet,i.e.upwardsinfigure1,whichaffectsthesigns.Divide[1]byiTand[2]byiRandaddtogether:[3]DefineR=I/I0asreflectanceofsheetandr=iR/iTasreflectanceofincrementand:and,rearranging:[4]whichgivesRintermsofS,KandR’.[5]TheLHS(LefthandSide)integrationisachievedbypartialfractions:[6][7]ConsiderthelimitingconditionwhereX=∞,R=R∞andR’cantakeanyvalue,sincenolightgetstoit,sowecansetR’=0.TheLHSof[7]mustequal∞,whichmeansthatthedenominatormustequal0and:[8][9]Notethatequation[9]canbeapproximatedto.Equation[9]impliesthatR∞canonlybe1ifKisnon-zero.Thisisreasonablebecause,ifthereisnoabsorption,alllightmustbescattereduntilitreappearsfromthetopsurfaceofthepaper![10][11]UptonowXisinlengthunitsandKandSareinappropriatedimensionstoleaveSXandKXdimensionless.VandenAkker(see,forexample,HandbookofPaperScience)sayswecanuseanincrementalgrammagelayerdWandredefine:•kisfractionalabsorptionlossofradiantfluxperunitbasisweight,•sisfractionalscatteringlossofradiantfluxperunitbasisweightandreplaceKXandSXwithkWandsWinallsolutionsandgraphicalaids.kistypically2m²kg-1forcoatedanduncoatedfinepapersmadefrombleachedchemicalpulps,is3k6m²kg-1formechanicalpulpsandisaround14m²kg-1forunbleachedkraftpulps.sis50m²kg-1forfilledandcoatedfinepapers,20s40m²kg-1forbleachedandunbleachedchemicalpulpsandistypically40s70m²kg-1formechanicalpulps.TodifferentiatesandkfromSandK,weoftencallthemspecificscatteringcoefficientandspecificabsorptioncoefficient.ItisalsousefultomakeRthesubjectofequation[11]:[12]Bothequations[11]and[12]areusedtodeterminebasisweightcorrectedopacity.R0dependsonbasisweight,sowhenhandsheets,whichdifferfromthestandardweightareused,R0mustbecorrected.Obviously,R∞doesnotchange.Fromequation[11],scatteringcoefficientscanbecalculatedbysubstitutingR=R0andR’=0:[13]Now,usingequation[12],R0forthestandardgrammage,Wstdcanbecalculated:[14]