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1、3926J.Phys.Chem.1992,96,3926-3930besosincethederivativesoftheGreenfunctionarecontinuouswithj3,h5/N3.Asaconsequencewefindtailsdecayingasexceptnearthewall.NearthebrushtipthechainsbecomemoreTheexponentofthisdecayisthesameasthatoftheprofileofdiffusesothattheprofilete
2、ndstobesimilartothatvalidfortheanongraftedpolymersolutionincontactwithaadsorbingwallusualadsorbedpolymer,atleastwithinthelimitationsofourasgivenbydeGennesI3assumingaself-similarstructure.perturbationanalysis.WhenthetailsareappreciableweexpectourperturbationSinceo
3、urstartingeqs8and9areofthemean-fieldtype,thetreatmenttobefairlyreliablebeyondtheinflectionoftheprofile.decayofthetails(=l/j3z*2)isalsoameanfieldresult.InorderWeregardtheOccurrenceofalargedepletionzoneatsmall/3toheuristicallyimproveuponthisonemayfollowamethodtobea
4、nartifactsinceweusethe“bare”GreenfunctionfortheproposedbydeGennes.13Weknowthatthemean-fieldcorre-first-ordercorrectionwhichdoesnotaccountforthepresenceoflationlength5,=(wN~~)-’/~mustbereplacedby&z(wN~~)-~/~thewall.AgainwestressthattheAlexander-deGennesansatzval
5、idinscalingtheory(I=1).Hence,anexpressionlikeeq27isatbestapproximateandcertainlynottheonlyviablezereorderbecomestheoryusefulinevaluatingtails.a2-k*=Acknowledgment.Thispaperwaswrittentocelebratethe60th&k*4forz*>1(33)az*2birthdayofMarshallFixman,physicalchemistsu
6、premeandoneofthefoundersofpolymerstatisticalmechanics.Wearegrateful(13)DeGennes,P.G.Macromolecules1981,14,1637forfinancialsupportfromDSMResearch.DiffusionPastanEntropyBarrierRobertZwanzigLaboratoryofChemicalPhysics,Building2,NationalInstituteforDiabetesandDigesti
7、veandKidneyDiseases,NationalInstitutesofHealth,Bethesda,Maryland20892(Received:September3,1991)Diffusioninatubeorchannelofvaryingcrosssectionhasbeentreated,byFickandJacobs,asone-dimensionaldiffusionpastanentropybarrierdeterminedbythetubecross-sectionalareaorchann
8、elwidth.TheFick-Jacobsdiffusionequationisrederived,alongwithcorrectionsinvolvingthecurvatureofthetubeorchannel.Oncomparisonwithtwoexactlysolvableexamples,theFi
