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Hydrodynamic and dispersion behavior in a non-porous silica monolith through fluid dynamic study of a computational mimic reconstructed from sub-micro-tomographic scans

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Hydrodynamic and dispersion behavior in a non-porous silica monolith through fluid dynamic study of a computational mimic reconstructed from sub-micro-tomographic scans
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   JournalofChromatographyA, 1274 (2013) 65–76 ContentslistsavailableatSciVerseScienceDirect  Journal   of    Chromatography   A  journalhomepage:www.elsevier.com/locate/chroma Hydrodynamic   and   dispersion   behavior   in   a   non-porous   silica   monolith   throughfluid   dynamic   study   of    a   computational   mimic   reconstructed   fromsub-micro-tomographic   scans Kai-Chee   Loh ∗ ,   Vivek   Vasudevan DepartmentofChemicalandBiomolecularEngineering,NationalUniversityofSingapore,4EngineeringDrive4,Singapore117585,Singapore a   r   t   i   c   l   e   i   n   f   o  Articlehistory: Received18April2012Receivedinrevisedform19October2012Accepted26November2012 Available online 8 December 2012 Keywords: HydrodynamicsDispersionHETPComputationalfluiddynamicsSilicamonolithComputedtomography a   b   s   t   r   a   c   t An   analysis   of    the   transport   properties   of    the   bulk   homogeneous   core   of    acommercially   available   silicamonolith   (Chromolith ® )   ispresented   via   direct   numerical   simulations   in   a   topological   model   recon-structed   from   3D   nanotomographic   scans   at   isotropic   resolutions   of    390   nm,   290nm   and   190   nm.   Thepore   and   skeleton   size   distributions   were   calculated   from   image   analysis   anda   representative   unit    cell from   each   resolution   was   reconstructed   to   simulate   the   hydrodynamic   transport   properties   using   Com-putational   Fluid   Dynamics   (CFD).   A30  m   × 30  m   × 30  m   unit    cell   extracted   at   190   nm   resolution   wasfound   to   be   representative   of    hydrodynamic   permeability.   Numerical   peak   parking   simulations   yieldedanaxial   external   obstruction   factor   (   e )of    0.8.   Mass   transfer   characteristics   of    a   large   non-penetratingmolecule   (BSA)   were   evaluated   under   non-retained   conditions   so   as   to   elucidate   the   eddydispersioncontribution   to   total   HETP.   Transverse   and   axial   dispersion   length   scales   in   the   reconstructed   modelwere   resolved   and   related   to   the   structural   heterogeneities   in   the   silica   monolith.   Deviations   of    simu-latedHETP   from   experimental   measurements   were   attributed   to   atranscolumn   dispersion   contribution,which   amounted   to   about   90%   of    the   total   HETP.   The   presented   approach   provides   great   scope   to   analyzethe   contributions   of    pore   network   topology   to   separation   performance   of    silica   monoliths   (and   otherporous   media)   in   HPLC   applications.   Asignificant   reduction   insimulation   time   and   memory   resourceshas   been   observed   due   to   the   lower   scanning   resolution,   without   significant   loss   in   prediction   accuracy. © 2012 Elsevier B.V. All rights reserved. 1.Introduction Silicamonolithsareaclassofchromatographiccolumnswhichgenerallyprovidehigherperformancethanconventionalpartic-ulatecolumnsinpressure-drivenliquidchromatography[1].   Animportantcharacteristicofsilicamonolithsistheirhighexternalporosityresultingfromanetworkofthrough-macropores[2].   Thisregularstructureofmacroporechannelsislessconstrictedandlesstortuousthanthatinpackedbeds.Thestationaryphaseskeleton,consistingofanetworkofsmall,thinthreadsofporoussilica,hasnoeffectonthehydraulicresistanceandhence,canbetailoredtoenhancetherateofmasstransferofsmallmoleculesthroughitsmesoporousnetwork.Thesetwostructuralcharacteristicsprovideacombinationoflowhydraulicresistancetothemobilephaseandenhancedmasstransferratesofsamplemoleculesthroughthecol-umn.Monolithsowetheirversatilitytothefactthattheporeandskeletondimensionscanbecontrolledindependentlyduring ∗ Correspondingauthor.Tel.:+6565162174;fax:+6565771936. E-mailaddress: chelohkc@nus.edu.sg (K.-C.Loh). theinsitupolymerizationprocess[3].   Thisposesachal-lengeinmodelingmonolithssinceunlikepackedbeds,thereisnosinglegeometricalfeaturethatcanuniquelycharacterizeboththeirhydrodynamicandseparationperformance.Severalphysico-chemicalmodelsformonolithiccolumnsexistinlit-eraturethatdifferintheirdegreeofsimplification[4].   Thesemodelscanbebroadlyclassifiedinto macroscopic  and micro-scopic  modelsbasedontheapproachesadoptedtorepresentthephysicaldomaininwhichtheconstitutivetransportequa-tionsaresolved.Theformerinvolvessimplifyingtheunderlyingstructuralfeaturesoftheporousmediumbyincorporatingaveragedmorphologicalparametersonamacroscopic(Darcy)scale.Thelatterattempttoexplicitlyincludethestructuralandmorphologicalnon-idealitiesattheporelevelandderiveamacro-scopicbehaviorfromthedescriptionatthemicroscopicleveloverarepresentativeelementaryvolume(REV),ora unitcell [5].Macroscopicmodelsuse averaged parameterssuchasporosity,poresizedistributionandtortuosityindirectlyinferredfromaver-aginginformationobtainedfromexperimentaltechniquessuchasmercuryporosimetry,inversesize-exclusionchromatographyandnitrogenadsorptiontocharacterizetheporousmorphology. 0021-9673/$–seefrontmatter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.chroma.2012.11.085  66  K.-C.Loh,V.Vasudevan/J.Chromatogr.A 1274 (2013) 65–76 Thenon-idealitiesintroducedintotheseparationbehaviorduetotheinhomogeneityandanisotropyofthesemorphologicalfea-turesarelumpedintoanaxialdispersioncoefficientinthegeneralchromatographyratemodel[6].   Modelsbasedonamacroscopicapproacharewidelyusedbecauseoftheirconvenienceandfamil-iarityinengineeringpractices.Forexample,thestructureofsilicamonolithiccolumnwassimplifiedasabundleofequalnon-overlappingparallelchannelswithasilicaskeletonwallandalumpedadjustableparameter(axialdispersioncoefficient)was   uti-lizedtoaccountforadditionaldispersioneffectspresentduetothepolydispersityofthemonolithstructure[7].Inanothermodel[6], themonolithstructurewassimplifiedasananastomosedstructureofporouscylindricalskeletonsurroundedbythroughporespace.Thedegreeofsimplificationinthesemodelsreflectseitherthepur-poseofthemodeltoeasilyfacilitatepredictionofexperimentaldatathroughlumpedparametersorpaucityintheavailabilityof morphologicaldetailsforanaccuratereconstruction.Aninherentdisadvantageofthesemodelsistheempiricalorsemi-empiricalestimationofseveralmacroscopiccharacteristicsofmasstransportinporousmedia[4].   Themodelsfail(ordonotattempt)torelatetheobservedtransportbehaviortothemorphologicalcharacter-isticsoftheporousmedium.Inordertodoso,themicrostructureoftheporousmaterialshouldbeknownsothatforagivenphys-icalphenomenon,thecorrespondingeffectivepropertiesofthematerialcanbedirectlycomputedbyanalysisofthe unitcell [5].Apore-scalemodelformonolithswasfirstdevelopedbyMey-ersandLiapis[8]asanextensionoftheporenetworkmodeldevelopedbyLohandWang[9]f orperfusivecolumns.Thepore spacewithintheporousmediawasrepresentedasa3Dlatticenetworkofnodesconnectedbybondsrepresentingthepores.ThenetworkwasgeneratedbyaMonteCarloapproachandasizedistributionwasattachedtotheporediameterstosimulatemercuryintrusion.Althoughaporenetworkmodelmay   function-allyrepresenttheporousmedium,itdoesnotprovidearealisticpictureofthesizedistributionandconnectivityofthemorpho-logicalnetwork.ThefirstattemptatrelatingthemicrostructuremorphologytothetransportmechanisminsilicamonolithswasperformedbyVervoortetal.[10],inwhichthestationaryskele- tonphasewasidealizedasanetworkofporoustetrahedralunitsbasedonthecrystallatticeofdiamond.Theflow-fieldinthepore-spacearoundtheskeletonswassimulatedusingComputationalFluidDynamics(CFD).Thedeviationofthehomogenoustetrahe-dronskeletonmodel(TSM)fromexperimentalpressuredropdatawasattributedtoaconsiderablesurfacecontourheterogeneityof theskeletonmembersasobservedfromLaserScanningConfo-calMicroscope(LSCM)images[11].TheTSMmodelwas   modifiedbyaddingdifferentdegreesofheterogeneitiestothestatisticalsizedistributionoftheskeletons,therebycapturingtoanextenttheheterogeneousmorphologyinsilicamonoliths.ThoughthemodifiedTSMmodelswereabletopredictalowerandmorereal-istichydraulicpermeabilityforthesilicamonoliths,theywereunabletoaccuratelycapturetheexperimentalbandbroadeningprocesses[12].TheTSMmodel,however,wasabletoprovide anestimateofthemaximumgaininperformancethatcouldbeachievedfromahomogenoussilicamonolith[13].Theinability toaccuratelycapturethethree-dimensionalheterogeneityinthemonolithicstructureseemstobetheprimaryreasonforthefailureoftheaforementionedmodelstorelatemonolithperformancetoitsmicrostructure.Asopposedtotheindirecttechniquestoinfermorphologicaldetails,directvisualizationtechniquesoffergreatpromisetocap-turethetruemorphologyofporousmaterialswithouthavingtointroducemodeldependencies[14].Scanningelectronmicroscopy (SEM)andtransmissionelectronmicroscopy(TEM)onlyallowfordirectvisualizationofporousmorphologyintwodimensions.Theaccuracyofinformationextractedthroughanassemblyof reconstructedTEMimagesfromserialsectionsofporousmediaislimitedbythethicknessofthesectionsattainable,whichmaybeinadequatetoresolvethemeso/microporouslengthscalesinchromatography[15].Recently,TEMsectionsatlessthan100nm transverseresolutionwereusedtocharacterizethemacroporesizedistributioninmonoliths[16].Artificialartifactswereobserved intheimagesduetophysicalsectioningandnoeffortwasmadetorelatemicrostructureinformationtotheseparationperfor-manceofthecolumns.Rapidadvancesinnon-invasive3Dscanningtechniqueshaveledtotheuseofnuclearmagneticresonanceimaging(MRI)[17–19],   laserscanningconfocalmicroscopy(LSCM)[20,21],X-raycomputedmicrotomography(  CT)[22]andelec- trontomography[14]tocapture as–is theinherentmorphologiesofporousmaterials.Accuratereconstructionof3Dmorphologiesofporousmaterialsfromnon-invasivescanshasenableddetailedstudyofstructuralparameters[23]anddevelopmentofimproved microscalemodels[24,25].Directuseofreconstructed3Dimages asboundinggeometriesinanenvironmentamenabletosolv-ingfundamentaltransportequationshasfacilitatedmorerealisticmicroscalemodelsandobviatedtheneedformodel-dependentfittingparameters[26–32].Recently,non-invasive3Dscanningtechniqueshavebeenemployedtocapturethemacroporousnetworkinmonoliths.LSCMwas   usedtocapturetheinter-skeletonmacroporespaceinabare-silicamonolithic(ChromolithCapRod TM )column[33],whilebackscatterSEMwas   employedtoimageandseriallysectionaportionofacommercialpolymeric(CIM TM DEAEanion-exchanger)monolithicdisktogeneratetheflow-throughporenetwork[34].Thecapillarymonolithwas   scannedataresolu-tionof30nm × 30nm × 120nm,   whilethepolymermonolithwasimagedataresolutionof18.5nm × 18.5nmin50nmserialsec-tions.We   focusourattentionatreconstructingthe3DmorphologyofthemacroporenetworkinacommerciallyavailableChromolith ® Performancecolumn(RP-18e)atasignificantlylowerresolutionthanthoseinliterature.Ourresultsarecomparedagainstthoseobtainedforthecapillarysilicamonolith[21],primarilybecauseto ourknowledge,thisistheonlynon-invasiveCLSMstudyforsilicamonolithsandhenceformsabasisforourcomparison.Hlushkouetal.[21]employedlattice-Boltzmanequation(LBE) approachtodirectlymodeltheporescalehydrodynamicflowinthereconstructed3Dmatrix.Althoughthesimulatedandexper-imentalpermeabilitieswereinexcellentagreement( ≤ 4.1%),thehighscanningresolutionresultedin320millionvoxels(225millionvoxelsinmacroporespace).Thenetresultwasalongercompu-tationaltime(40h)aswellashighutilizationofcomputationalresources(64processors)tosimulatethesteady-statevelocityfield.Itisourbeliefthatalowerscanningresolutionissuffi-cienttoadequatelycapturetheinhomogeneitiesinthemacroporenetwork.Alowerscanningresolutionwould,withoutsignificantlossofaccuracy,appreciablyreducethecomputationalexpenseandatthesametime,beabletorelatethehydrodynamicanddispersionperformanceofthesilicamonolithtoitsporestruc-ture.Inthisresearch,wefocusedonreconstructinganaccurate3Dmodelofthemacroporenetworkinacommerciallyavailablesilicamonolith(Chromolith ® RP18-e)fromnon-invasiveX-raycom-putedmicrotomographyscansatisotropicresolutionsof390nm,290nmand190nm.   Further,hydrodynamicanddispersionchar-acteristicsofthereconstructedporousmediumwereanalyzedinacommerciallyavailablesimulationpackage(AnsysFluent ® )andcontributionstotheoverallHETPperformancefrommoleculardiffusionandeddydispersionwereelucidatedundernon-porousconditions.Thetimeandlengthscalesinvolvedineddydispersionwerealsoresolvedtoaddressthestructuralheterogeneitiesofthe bulk macroporespaceinasilicamonolith.  K.-C.Loh,V.Vasudevan/J.Chromatogr.A 1274 (2013) 65–76 67 2.Morphologyreconstructionandimageanalysis  2.1.Monolithsamplepreparationand  CTscan A4.6mm × 100mmChromolith ® Performancecolumn(RP-18endcapped)fromMerck(Darmstadt,Germany)wascleansedwithmethanolfollowedby60/40acetonitrile/water(v/v)mix-turebeforebeingcutintofive20mmtransversesectionsusingamicrotome.Thecutsectionswerestoredinmethanolvials.Thepolyetheretherketone(PEEK)claddingaroundacoupleofsectionswascutopenwithamicrotome.SamplesforCTscanswerepre-paredbygentlybreakingtheporoussilicapiecesintofragmentsandsealingthemin4mmcapillaries.  CTscanswereperformedatSkyscan,BelgiumwithSkyscan2011X-raynanotomographatisotropicresolutionsof390nm,   290nmand190nm.  2.2.CTimageanalysis Theseriesof2DimagesreceivedfromSkyScanwerereconstructedandsubsequentlyanalyzedusingAnalyze ® 9.0(BiomedicalImagingResource,MayoClinic,USA).Therecon-structedimagesweresubjectedtoalow-passmedianfiltertoenhancetheimagequality.Thevoxel-intensityhistogramofthereconstructedvolumewassegmentedwithasingle automatic threshold functionintoabinarizedvolumerepresentingtheporeandskeletonnetwork.Porosityoftheextractedporenetworkwascomputedasaratioofthenumberofextractedvoxelstothenumberofvoxelsintheoriginalimage.Thesurfacedefiningtheboundarybetweentheskeletonandporenetworkwas   approxi-matedasasurfacemeshconsistingoftriangularelementsusingthe SurfaceExtractor  moduleandwassavedasastereolithographic(STL)file.STLfilesofvarioussizeswereextractedtodeterminetherepresentativeelementaryvolume(REV)byporosityanalysisandsteady-statehydrodynamicsimulations.Atext(CSV)filedescribingthespatialvoxeldistributioninthebinarizedsegmentedimagerepresentingtheporeandskeletonnetworkwasgenerated.Thisservedasatemplatetoformulateafunctiondescribingtheporesizesbyspecifyingthedistanceof every  pore voxelfromthenearest skeleton voxelandviceversa,fortheskeletonsizes[32].Consecutive  zeroes and ones alongthe  x ,  y and  z  directionswerecounted,thusestimatingtheporeandskele-tondiameters,respectively.Theporeandskeletondiametervaluesthatprojectedoutsidetheedgesofthetruncatedcubicmodelwerediscarded. 3.DevelopmentofCFDmodel Anunstructuredtetrahedralmeshwasgeneratedinthevolumeenclosedbythesurfacesrepresentingtheskeletonandporenet-workusingTGrid ® 5.0.6–agridgenerationpre-processortothecommerciallyavailablesimulationpackage,AnsysFluent ® 12.1.4.Thetruncationofthepore-skeletonnetworkinterfacealongthreemutuallyperpendicularaxescreatedartificialboundarieswhichservedasboundingfaceswithinwhichtheporevolumewasmeshed.Cubicvolumesofdifferentsizeswereextractedtodeter-minetherepresentativeelementaryvolume( unitcell ).Sincethereisnegligiblebulkflowinthemesoporousskeletonnetwork[35],onlytheporevolumewasdiscretizedforhydro- dynamicsimulations.Flowwassimulatedinbothforwardandreversedirections.Ineachcase,boundaryfacesparalleltotheflowdirectionweresetas symmetry facessothatthe unitcell behavesasifitisembeddedinaninfinitelyreplicatedstructure.Thesymme-tryboundaryconditionbasicallyservedtosetazeromomentumgradientacrossthatboundary.Thepore-skeletoninterfacewassetasa wall tosimulatethenoslipboundarycondition.Anoperatingpressureof1atmwas   setattheflowoutlet,whiletheinletvelocitywas   settotheinletinterstitialvelocitytosimulateaflowraterangeof0.5–10ml/minthroughthecolumn.Water(den-sity1000kg/m 3 andviscosity1.003cP)was   chosenasthemobilephase.ThecorrespondingReynoldsnumber(basedonamacroporediameterof2  m[4])   was   oftheorderof0.001–0.01suchthatflowconditionswerestrictlycreepingflow.ThesimulationswerecarriedoutattheHighPerformanceComputingLinuxclustersattheNationalUniversityofSingaporeComputerCentre.Two   typesofclusterswereutilizedinthesimulations:  2 × IntelHarpertownE54302.66GHzQuad-coreProcessors(16GBmemory) and  2 × IntelNehalem-EPX55502.67GHzQuad-coreprocessors(48GBmemory) .Ateachsurface-gridresolution,thesteady-statevelocityfieldwassimulatedbymonitoringtheresidualsumofthevelocityfieldateveryiteration,thusfollowingtheconvergencehistory.Ascaledresidualof1e − 04wasemployedtoensureconvergence.Thevolumemeshwas   thenrefinedbasedonthevelocity-gradientsusingagradientadaptiontechniqueandthesimulationwasrunforanother300iterations.Thedifferencesinthevolume-averagedvelocitiesbeforeandaftergradientadaptionwerelessthan1%.Theequationofcontinuity(Eq.(1))   andNavierStokesequation(Eq.(2))weresolvedsimultaneouslyatthegeneratedtetrahedralmeshnodesinthemacroporevolumesubjecttoinletvelocityandoutletpressureboundaryconditionstodeterminethe3Dsteady-statevelocity-fieldofanincompressibleNewtonianfluid. ∇  .  v  = 0 ∂ v   x ∂x  +  ∂ v   y ∂y  +  ∂ v   z  ∂z   = 0(1)   v  .  v  =− ∇   p +  2  v  +    g   v   x ∂ v   x ∂x  + v   y ∂ v   x ∂y  + v   z  ∂ v   x ∂z   =− ∂p∂x  +   ∂ 2 v   x ∂x 2  +  ∂ 2 v   x ∂y 2  +  ∂ 2 v   x ∂z  2  + g   x   v   x ∂ v   y ∂x  + v   y ∂ v   y ∂y  + v   z  ∂ v   y ∂z   =− ∂p∂y  +   ∂ 2 v   y ∂x 2  +  ∂ 2 v   y ∂y 2  +  ∂ 2 v   y ∂z  2  + g   y   v   x ∂ v   z  ∂x  + v   y ∂ v   z  ∂y  + v   z  ∂ v   z  ∂z   =− ∂p∂z   +   ∂ 2 v   z  ∂x 2  +  ∂ 2 v   z  ∂y 2  +  ∂ 2 v   z  ∂z  2  + g   z  (2) Numericalpeakparkingsimulations[36]wereperformedbypatch- ingarectangularplug(0.3  m)   ofananalyteatthecenterofthesimulationdomainintheconvectionzone,intheabsenceofaveloc-ityfield.Theaxiallyspreadingconcentrationprofilewasmonitoredasafunctionoftimeatvariouscross-sectionsofthemacroporevolumealongtheaxialdirection.Diffusion(underisothermalcon-ditions)intheabsenceofconvectionwassimulatedbysolvingthemassbalanceequation(Eq.(3))fortheinjectedanalytespecies. ∂C ∂t   = D m ∇  2 C ∂C ∂t   = D m  ∂ 2 C ∂x 2  +  ∂ 2 C ∂y 2  +  ∂ 2 C ∂z  2   (3)Asaninitialestimate,thesimulationtimestepwascalculatedsuchthatitwas   lessthanthatrequiredtodiffusethroughhalfthegridresolution.Atregulartimeintervals,themeanlocation ¯  z    andspatial(axial)variance   2  z   ofthespeciesdistributionwascalculatedusing:¯  z  =    Czdz     Cdz  (4)   2  z   =    C  (  z  − ¯  z  ) 2 dz     Cdz  (5)where, C  isthespeciesmassfractionand  z  istheaxialcoordi-nate.Simulationswereperformedtillthecalculatedvarianceof thedevelopedpulsevariedlinearlywithsimulationtimeatevery  68  K.-C.Loh,V.Vasudevan/J.Chromatogr.A 1274 (2013) 65–76 instant.Thesimulationswerecontinuedtilltheoverallregressioncoefficientforthelinearfitofcalculatedvarianceversussimula-tiontimewas>0.98.Theeffectivediffusioncoefficient( D eff  )andtheexternalobstructionfactor(   e )werecalculatedas: D eff   =  12 d  2  z  dt   (6)   e  = D eff  D m (7)where D m  isthemoleculardiffusivityoftheanalyte.Thesimu-lationswereperformedundernon-retainedand non-penetrating  conditions,i.e.theanalytemoleculewasexcludedfromthemeso-porenetworkintheskeleton. Non-retained dispersionsimulationswerethenperformedforthecaseofa non-porous skeleton.Axialdispersionsimulationswereperformedbypatchinganinstantaneousrectangularpulse( dx × dy × dz  =30  m × 30  m × 0.15  m)   attheinlet(  z  =0),whilefortransversedispersionsimulations,aninstantaneousrectangularpulse( dx × dy × dz  =0.3  m × 30  m × 0.15  m)   was   patchedattheinlet(  z  =0)centeredat  x =15  m.   Theanalytemassfractionwassetto0.1inthepatchedregioninbothcasestosimulatelocallylinearconditions.Theinletpulsewassuperimposedonthepre-calculatedvelocityfieldandonlyaspeciesbalancewasperformedtotrackthedispersionoftheinjectedpulse.Itshouldbenotedherethatthecalculatedvelocityfielddoesnotdependonthechoiceofmobilephasebutonlyontheimposedflowrate(inletvelocity).Itwasassumedthattheinjectedpulsedidnotalterthevelocityfieldinanyway.Diffusioninthepresenceofthesteady-statevelocityfieldwassimulatedbysolvingthegeneralcontinuumconvection-diffusionequation(Eq.(8))   underisothermalconditions. ∂C ∂t   + v  . ∇  C  = D m ∇  2 C ∂C ∂t   + v   x ∂C ∂x  + v   y ∂C ∂y  + v   z  ∂C ∂z   = D m  ∂ 2 C ∂x 2  +  ∂ 2 C ∂y 2  +  ∂ 2 C ∂z  2   (8)Theaxialdispersionoftheinjectedpulsewasmonitoredbycalculatingtheaveragespeciesmassfractionatseveraldetectionplanes(normaltotheflowdirection)intheconvectionzoneasafunctionoftime.Thetransversedispersionwasalsomonitoredatregularlyspaceddetectionplanesparalleltotheflowdirection(inthe  yz  plane).Forbothpeakparkinganddispersionsimulations,aninitialestimateofthesimulationtime-stepsizewascalculatedasavaluelessthanthatrequiredtodispersetheanalyteoverhalfthegridresolution(   x )byconvection(Eq.(9))ordiffusion(Eq.(10)). Simulationswererepeatedatdoubleandhalftheestimatedtimestepsizetodeterminetheeffectoftime-steponsimulationaccu-racy.Itwasobservedthattheinitialestimatemadewasaccuratetocapturethediffusion/dispersionphenomenawithoutsignificantlossofaccuracyandwithinmanageablesimulationtimes.A down-streamboundarycondition ( outflow condition)was   appliedattheflowoutletimplyingthatspeciesgradientintheflowdirectionattheexitwaszero.The symmetry boundaryconditionsservedtosetazeroconcentrationgradientacrosstherespectiveboundaries. t  sim  <  12 xu a v   z  (9) t  sim  <  12( x ) 2 D m (10)Forbothpeakparkingaswellasdispersionsimulations,thepore-skeletoninterfacewassetasanimpermeablewall.Amixtureofacetonitrileandwater(55/45%,v/v)(density904kg/m 3 andvis-cosity0.79cP)waschosenasthemobilephase,whilebovineserumalbumin(BSA)withamoleculardiffusivity[37]of7.5 × 10 − 11 m 2 /swasselectedasthesolute.Fordispersionsimulationsat  sf  ≤ 5,solutewithmoleculardiffusivityof2 × 10 − 9 m 2 /s(diffusivityof phenolinacetonitrile)was   selectedtoinvestigatethelowerrangesofreducedsuperficialvelocities.Dispersionsimulationswereperformeduntiltheinjectedpulseremainedwithinthesimulationdomain[38].Thedispersioncoef- ficientvalueunderwentatransientregimewheretheinjectedplugtransitedfromitsinitialrectangularshapeintoaGaussianshape.Whenthespecieshaddispersedthroughasufficientlyrepresen-tativepartoftheentiremedium,aconstantdispersioncoefficientwasobtained.Thissufficientlyrepresentativepartofthesimulationdomainwas   quantifiedintermsofatransversedispersionlength,beyondwhichthedispersioncoefficientremainedfairlyconstant.Continuedsimulationsinthefinitedomainledtoadecreaseinthedispersioncoefficientassoonasasignificantamountofana-lytereachedtheendofthesimulationdomain.Dependingonthemobilephaseflowrate,thesimulationswereterminatedeithertilltheoverallregressioncoefficientforthelinearfitofvarianceversussimulationtimewas>0.98,or<0.95,whicheverwas   earlier.TransientlongitudinaldispersioncoefficientandthecorrespondingHETPwerecalculatedfrom: D L ( t  ) =  12 d  2  z  dt   (11) H  ( t  ) =  2 D L ( t  ) u a v  (12)where   2  z   isthespatialvarianceoftheanalytepulseand u av  istheaveragelinearinterstitialvelocity.Likewise,thetransversedis-persioncoefficientwasdeterminedfromtheaverageslopeofthetransversespatialvarianceoftheinjectedpulsewithtime. D T  ( t  ) =  12 d  2  x dt   (13)Theretentiontimesateachdetectionplaneintheaxialdirectionwerecalculatedfromthezerothandfirstorderabsolutemomentsoftheobtainedbreakthroughcurvesusing: t  R,i  =   ∞ 0  C  i tdt    ∞ 0  C  i dt  (14)AllnumericalintegrationswereevaluatedinMicrosoftExcel ® usingSimpson’s 1   ⁄   3rdrule. 4.Experimentalprocedure 4.1.Apparatusandmaterials ChromatographicexperimentswereperformedonaWaters(Milford,MA,   USA)HPLCsystemequippedwitha600Emulti-solventdeliverypump,in-linedegasserAF,a717plusauto-sampleranda2487dual-wavelengthabsorbancedetector.ThesystemwasoperatedusingtheEmpower2ProsoftwaresuppliedbyWaters.Silicamonolith(Chromolith ® RP-18e,4.6mm × 100mm)   wasobtainedfromMerck(Darmstadt,Germany).De-ionizedfilteredwaterfromMilli-Qplus185(Millipore,Bedford,USA)was   usedintheexperiments.HPLC-gradeacetonitrile(ACN)andreagent-gradetrifluoroaceticacid(TFA)wereobtainedfromFisherScientific(FairLawn,NJ,USA).BSA(67,000Da)was   purchasedfromSigma(St.Louis,MO,   USA). 4.2.Methods4.2.1.Mercuryintrusionporosimetry Allmonolithsamplesweredriedat100 ◦ Cfor5–6h.TheporosimetrytestswerecarriedoutonaMicromeritics(Norcross,GA,USA)AutoPoreIII9420mercuryintrusionporosimetercapable  K.-C.Loh,V.Vasudevan/J.Chromatogr.A 1274 (2013) 65–76 69 ofgeneratingpressuresupto414MPa,correspondingtoamini-mum   cylindricalporediameterof3nm.   Themercurycontactanglewassetto130 ◦ andintrusionwasperformedupto400MPa.DataacquisitionwasdoneusingAutoPoreIIIsoftware,v2.00. 4.2.2.Pressuredropmeasurements SystempressuredropmeasurementsweretakenfromtheHPLCinstrumentpressuregaugeformobilephaseflowratesof 0.5ml/minto10ml/min.Thesystempressuredropwas   correctedfortheextra-columnpressuredropbyreplacingthecolumnwithazero-volumeconnectortoobtainthepressuredropacrossthecolumn.Allexperimentswereperformedatroomtemperatureof 22 ◦ C.Theexperimentswereconductedatleastinduplicatesforeachflowrateunderthesameexperimentalconditions.Columnpermeabilitywasthencalculatedbasedonthesuperficialvelocityaccordingtotheequation: K  = u sf  LP   (15)where  ismobilephaseviscosity, L islengthofthecolumn, u sf   isthesuperficialvelocityand  P    isthecolumnpressuredrop. 4.2.3.HETPmeasurements TheHETPdatafornon-retained non-penetrating  experimentswereobtainedbyinjecting10  lof1mg/mlBSAsolutionusingtheauto-sampler.Theflowrateofmobilephaserangedfrom0.5ml/minto10ml/min.The2487dual-wavelengthabsorbancedetectorwas   setat280nmwithsinglechanneldetection.Themobilephaseusedwas   55%ACN-water+0.1%TFA(%,v/v).Thisensurednon-retentionofBSAonthemonolith.Priortoinjection,thecolumnwasequilibratedwithatleast20columnvolumesof themobilephase.Extra-columneffectsweretakenintoaccountbyrepeatingtheexperimentswithoutthecolumnunderthesameexperimentalconditions.ThechromatogramswereexportedtoMicrosoftExcel ® byEmpower2Prosoftwareforfurthernumer-icalanalysis.HETPdatawereobtainedfromthehalf-heightpeakbandwidthmethodassuggestedbyGrittiandGuiochon[39]where theauthorsperformedapore-blockingexperimenttoshowthatthismethodgavemorephysicallymeaningfulvaluesthanthosemeasuredbymomentanalysisduetothesensitivityofthesec-ondmomenttopeaktailingandintegrationlimits.Thismethodconsistedofmeasuringtheelutiontimeofthechromatogramatitsapexanditsbandwidthatexactlyhalfitsheight.Themethodignoredtheconsequencesofpeakasymmetryduetotailingand/orfrontingofthepeakclosetoitsbase.Thethreecorrespondingelu-tiontimesare t  R , t   f  ,1/2  (adsorptionfront)and t  r  ,1/2  (desorptionrear).Thesameprocedurewas   repeatedfortheanalysisofextra-columnbandprofilesandthemeasurementofelutiontimes t  R , ex , t   f  ,1/2, ex  and t  r  ,1/2, ex . HETP  = L ( t  r, 1 / 2 − t   f, 1 / 2 ) 2 − ( t  r, 1 / 2 ex − t   f, 1 / 2 ex ) 2 5 . 545( t  R − t  R,ex ) 2  (16)Allexperimentalvalueswereobtainedasanaverageofatleastthreeinjectionsperformedunderidenticalconditions. 5.Resultsanddiscussion 5.1.Morphologyreconstruction Thelargestmodel,intermsofphysicalsize,aswellasinvoxeldimensions,thatwasextractedfromthethreescannedresolu-tionsissummarizedinTable1.Thevoxelsizesateachresolution isanindicationofthecomputingmemoryrequiredtorunthecorrespondingsimulations.Itistobenotedthatatthescanningresolutionsemployed,onlythemacroporenetwork,andhenceonly Fig.1. Variationofbulkexternalporositywithmodelcubesize. theexternal(inter-skeleton)porosity,couldbecapturedbytheCTscans.Cubesofdifferentsizesweresectionedfromvariousloca-tionsofthereconstructedmodelateachresolutiontodeterminethesizeofarepresentative unitcell .   Theporosityfortheextractedcubewasobtainedfromimageanalysisasaratioofthenumberofvoxelsinthemacroporespacetothetotalnumberofvoxelsinthe unitcell (Fig.1).Itwasfoundthatirrespectiveofmodelloca- tion,acubesizeofabout50  mwasrepresentativeofthescannedmonolithsampleintermsofitsoverallexternalporosityforreso-lutionsof390nmand290nm.   However,asmallercubeof30  mwasfoundtoberepresentativeoftheoverallporosityat190nmresolution.Anexternalporosityof0.6–0.65wascalculatedforallresolutions,whichistypicalforsilicamonoliths[35,40,41]. 5.2.Poreandskeletonsizedistributions Spatialco-ordinatesofthevoxelsintheporenetworkinthebinarisedsegmentedimagesateachresolutionwereprocessedinMS   Excel ® todeterminethenumberofconsecutive  zeroes inthethreemutuallyperpendiculardirections.Similaranalysiswasper-formedtoestimatetheskeletonsizedistributions.TheporesizedistributionsatdifferentresolutionsareshowninFig.2A.Itis apparentthatthedistributionsbecomenarrowerandwell-definedathigherresolutions.ThePSDofalargercell(40  m,   at190nmresolution)isalsoshown(Fig.2A)toillustratethenegligibledif- ferenceindistributioncomparedtothatofthe30  m unitcell .Fig.2BcomparesthePSDofthe30  mcellagainstthedistribu-tionsobtainedfromthegraphicalreconstructionfromLSCMscans
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