Analytical and numerical computations of the van der Waals force in complex geometries: Application to the filtration of colloidal particles

To link to this article: DOI: 10.1016/j.colsurfa.2012.07.033

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Eprints ID: 8613

To cite this version:

Hallez, Yannick Analytical and numerical computations of the van der Waals

force in complex geometries: Application to the filtration of colloidal particles.

(2012) Colloids and Surfaces A: Physicochemical and Engineering Aspects, vol.

414 . pp. 466-476. ISSN 0927-7757

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Analytical

and

numerical

computations

of

the

van

der

Waals

force

in

complex

geometries:

Application

to

the

filtration

of

colloidal

particles

Y.

Hallez

a,b,∗

a_{UniversitédeToulouse,INPT,UPS,LaboratoiredeGénieChimique,118routedeNarbonne,F-31062ToulouseCEDEX9,France} b_{CNRS,LaboratoiredeGénieChimique,F-31030Toulouse,France}

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◮ An exact solution for the sphere/wedge vdW interaction energyisderived.

◮ Thevalidityofbothsimpleandnew modelsforsphere/poreinteractions isassessed.

◮ Anewnumericaltooltocomputethe vanderWaalscolloidalinteractions ispresented.

◮ Anadaptivemeshrefinement strat-egyisusedtodiscretizethesolids.

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Keywords: VanderWaals Analytical DLVO Filtration Colloid Sphere Pore Numerical Computation Adaptivemesh

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Particlecaptureduringthefiltrationofcolloidaldispersionsdependsonacomplexbalancebetween repulsiveforces,suchashydrodynamicorelectrostaticeffects,andattractiveforces,amongstthemthe vanderWaalsinteractionforces.Satisfactoryexpressionsforthelatterarethusrequiredincomplex geometries.ExactexpressionsforthegeometricalfactorinvolvedinthevanderWaalsinteractionenergy basedonHamaker’sadditivityhypothesisarederivedforasphereininteractionwithasquarewedge,a semi-infiniteorfiniteslit,asemi-infiniteslab,a2Dpillar,arectangularrod,acornerandarectangular channel.Anumericaltoolbasedonanadaptivemeshrefinementstrategyispresentedandusedtovalidate theanalyticalresults.Theanalyticalresultforasphere/wedgesystemisusedtoassessthedomainof applicabilityofthesphere/planemodelinthevicinityoftheedge.Theinteractionbetweenasphere andacylindricalporeinaplateoffinitethicknessisthensimulatedandtherangeofvalidityofthe sphere/wedgesystemasamodelofthesphere/poresystemisdeducedfromthenumericalresults.

1. Introduction

The evaluation of the van der Waals force or interaction energy betweentwo ormore solidsis of crucial importancein manyacademicandindustrialproblems.Someexamplesarethe

∗ Tel:+33561556064;fax:+33561556139. E-mailaddress:hallez@chimie.ups-tlse.fr

(de)stabilizationofcolloidalsuspensionsinwhichanequilibrium betweentherepulsiveelectrostaticforcesandattractivevander Waalsinteractionsisinvolved(e.g.[1]),thedryingofcolloidalfilms toproducesurfacecoatings,thepredictionofproteininteractions [2]orthefiltrationofsub-micronicparticlesinwatertreatment [3,4].

Inthelatterproblem,particlecaptureistheresultofacomplex interplaybetweenhydrodynamic,electrostaticandvanderWaals forces.Letussupposecolloidsarenottoosmallandtheinfluenceof

Brownianmotionismoderate.Mostofthetimeelectrostatic inter-actionsarerepulsiveandtheStokes flowtransporting acolloid cannotbringitintocontactwithawallinafinitetimebecause ofrepulsivelubricationforces.Inshort,anexternalbodyforceis necessarytobringthesurfacesintocontact.Itmaybepositiveor negativebuoyancybutitseffectislowcomparedtootherforces forsub-micronicobjects,oritmaybeacentrifugalforce.Often thesetwoeffectsarenegligibleifnotabsentandtheattractiveforce ismainlyduetovanderWaalsinteractionsbetweenthecolloids andthesurface.Hencetheparticleretentionproblemisabalance betweentheattractivevanderWaalsinteractionsandalltheother repulsiveforces.Thesolutionofsuchaproblemisthus dramati-callydependentinthequalityoftheevaluationofvanderWaals interactions.

Ifexacttheoreticalsolutions fortheseinteractions are avail-ableforsimplifiedgeometries(e.g.[5]), problemsdependingon thecomplexshapesofthesolidsarenottractableanalytically.It isnecessaryeithertoresorttonumericalcomputationsortodo anadditivityhypothesisandmodelthesolidsassimplespheres, infinite rods,slabs... Concerningtheformer solution,frequently encounteredmethodsfornumericallycomputingaccuratevalues ofthevanderWaalsinteractionenergyaredirectmolecular simu-lationsincludingN-bodyinteractions,explicitcomputationofthe meanelectromagnetic(EM)stresstensor[6],andmanyother meth-odsdiscussedinRef.[7].Thesemethodsprovideaccurateresults, takingintoaccountretardationandnon-additiveeffects,butare howeveroftenextremely CPU expensiveand/orlimited tovery simplegeometriesand/orunabletodealwithlargesolids.The addi-tivityapproach,usedintheubiquitousDLVOtheory[8,9],hasthe advantageofrapidlyprovidingasimpleresultthatmaybe inte-gratedasanelementarybrickinamorecomplexframework(e.g. [10–12]).Itmayintroduce,however,asignificanterroronthe quan-titativeevaluationoftheinteractionsforatleasttworeasons:the additivityhypothesisbyitself andthesimplifiedgeometriesfor whichanalyticalexpressionsareavailable.Theadditiveapproachis notadaptedforDebyeandKeesominteractions,anditshouldthus berestrictedtocasesinwhich(London)dispersioninteractionsare dominant.Intheframeworkofdispersioninteractions,this approx-imationmayleadtolargeinaccuraciesformetalsbutisexpectedto providegoodestimatesforliquidsormolecularsolids[13]. Cova-lentsolidslikediamondorquartzareconsideredanintermediate case.Generally,thisapproachisconsideredtoprovideresultswith anerrorlessthan20%whencomparedtoexactsolutions[14].On theotherhand,theinabilityofthestandardDLVOsphereorplane modelstoreproduceexperimentalornumericalresultsforcomplex solidgeometrieshasbeenrecognizedbymanyauthors.Forinstance thestabilityofasynthetictitaniacolloidalsuspensioncouldnotbe predictedwithouttakingintoaccountthesurfaceroughnessofthe particles[15],themolecularrecognitionprocesses(linkedtovan derWaalsforces)stronglydependonproteinconfigurationsand relativeorientations [2],thedepositionofparticlesonroughor nano-patternedsurfacesisgreatlyaffectedbythegeometricdetails ofthesurfaces[16–19].

Iftheaccuratenon-additivemethodsareofgreatinterestfor understandingthephysicsofvanderWaalsinteractions,theiruse isactuallyanimpracticablepathwhenanumberofobjects inter-actwithinaflowingfluid.Indeedthehydrodynamicsinafiltration system(andinmostengineeringsystems)aregenerallynotknown analyticallyduetothecomplexgeometryevenformodelspherical particlesandmodelmembranesdesignedasarraysofwell con-trolledcylindricalpores.Inthebestcasesemi-analyticalmodelsof trajectoriescanbeintegratednumerically[3]andintheworst sit-uationsthefullStokesequationsmustbesolvednumericallyinthe filtrationgeometryatgreatcomputationalexpense.Thereforein studiesinvolvingbothhydrodynamicandsurfaceforcesasa cou-pledproblem,itisnecessarytousesimple(andfast)modelsforvan

derWaalsforces,evenifitmeanssometimescrudermodelsthan wewished.Inthecontextoffiltration,eitheraneffectivedistance ofcapture[3]orasphere–planeorplane–planeDLVO approxima-tion[20]isgenerallyinvoked.Theadditivityhypothesisisintrinsic tothisapproachanditseemsweneedtolivewiththisuncertainty. Thusamongstthetwoaforementionederrorsourcesinvolvedin DLVO computationsone degreeoffreedom left toimprovethe vanderWaalsinteractionenergyestimationistoreducetheerror introducedbysimplificationsoftheactualgeometriesofthesolids. ThemostclassicalclosedformsolutionsofvanderWaals interac-tionswereobtainedforspheres,cylinders,flatplates,disks,parallel slabsorcombinationsofthesegeometries(e.g.[26,14,23,28,29]). Examplesofgeometriespreviouslytreatedsemi-analyticallyare areproteins(asacollectionofspheres)[2],roughsurfaces[16], asphereinatube[23],asphereonanano-patternedplate[19] amongmanyothers.

Theaimofthepresentworkistwofold:firstlyitistopresent brieflyinSection2anewnumericaltoolabletocomputethevan derWaalsinteractionenergywithintheadditivityhypothesisbut forarbitrarycomplexgeometriesandwithoutanyother approxi-mation.Thescopeofthisarticleistopresentthisusefulnumerical methodtothecolloidcommunityandtoillustrateitbyan exam-ple.Thedetailsofthenumericaltechniquewillthusbedeveloped moreindepthinajournalspecializedinnumericalanalysis.The secondaimofthepresentworkistoassesstherangeofvalidity ofbothwell-knownandnewDLVOapproximationsadaptedthe contextofcolloidfiltrationbyusingthenumericalcodeasa vali-dationtool.Hencetheinteractionenergybetweenasphereanda slitoffinitedepthiscomputedanalyticallyandvalidated numeri-callyinSection3.Theinteractionenergybetweenasphereanda cylindricalporeisthencomputednumericallyinSection4andthe validityofthesphere/slitapproximationasamodelofathereal sphere/poresolutionisexamined.

2. Numericalmethod

Thissectiondescribesthenumericalapproachesunderlyingthe resultspresentedin therestof thearticleand implemented in theWITScodedevelopedatLGC.ThevanderWaalsinteraction betweentwosolidsisestimatedwiththeuseofHamaker’s additiv-ityhypothesis.Inthiscontext,theinteractionenergyiscomputed asageometricalintegraloverthevolumesV₁andV₂ ofthetwo particles: E∗₌₋ A 2

Z

V1

Z

V2 f(r)dV2V1, (1)

where A is an effective Hamaker’s constant, f(r)=1/r6_=(x2_+y2_+z2₎−3 _{(for non-retarded interactions),}

x2_=(x

2−x1)2, y2=(y2−y1)2 and z2=(z2−z1)2, and (x1, y1, z1)

and(x2,y2,z2)aretwopointsinvolumesV1andV2respectively.

Inthepresentwork,itisassumedthattheretardation/screening effectsof thesolventaretakenintoaccountatleast ina crude wayintheHamakerconstant(asadvisedin[21,14])orintheform off(r)([22]andreferences therein)and thefocus isputonthe evaluationofgeometricaleffects forthereasonsdetailedinthe introduction.HenceresultswillbepresentedintermsofE=E*_/A.

TheintegralinEq.(1)canbeevaluatednumericallybystandard quadraturemethodsbasedonthediscretizationoftheparticlesin smallvolumesandsummation.Thestandardsecondorder mid-point approximationis usedhere.To obtaincorrectresultsthe volumesoftheseelementsmustbesmall,whichleadstoagreat numberofintegrationelementsandaheavycomputermemory use.AstheintegralinEq.(1)consistsinsixonedimensional inte-gralstheCPUcostisalsoveryhigh.

Fig.1. Octreemeshgeneratedtocomputetheinteractionenergybetweenasphere andacylindricalpore.Thesurfacesofthesolidsarerepresentedasathickblack line.Thecolormaprepresents,atanypointofagivensolid,theinteractionenergy betweenthispointandtheentireothersolid.Darkerzonesaresignsofhigher interactionenergies.Thecolorscaleisskewedtovisualizesmallvalues.

Thereareessentiallythreeclassicalwaystoreducethis com-putationalcost,plusanewonedescribedinthenextparagraph. Whenthegeometryisrelativelysimple,itmaysometimesbe pos-sibletocomputeanalyticallypartoftheintegralsinvolvedinEq. (1),thusreducingthenumberofintegrationdimensions(e.g.[23]). Inthepresentwork,oneofthesolidsisasphereandsothree inte-gralshavebeencomputedanalytically,leavingthreeotherintegrals todealwithnumerically.AsecondwaytoreducetheCPU load istodecreasethenumberofintegrationdimensionsby convert-ingtheintegralsonvolumesinintegralsonsurfaces.Thisisdone forexampleintheSurfaceFormulationMethod[24,25]andinthe SurfaceElementIntegration(SEI)method[17,19].Thethirdway toreducetheCPU costistokeepthesixintegralsbutsplitthe solidsinsmallelementsinanadaptiveway,i.e.thesmallest ele-mentsarelocatedinregionswhereahighprecisionisdemanded andlargeronesareusedelsewhere(Fig.1).Ithastheadvantage of keeping thecode able to deal with anygeometry, any sur-facecurvature,andwithoutintroducinganyapproximationonthe physics.

Anadditionalideatoreducethecomputationalcosthasbeen developedandimplementedinthepresentcodeforcasesinwhich thegeometry of thesolids isso complexthat thesix integrals havetobecomputednumerically. Itrelies ontheideaof map-ping thevolumesof theparticles withcubesof differentsizes, thecubesbeingaslargeas possiblefarfromanyinterfaceand smaller and smaller near the interface in order to fit it. This approach requiresan analyticalformfor theinteractionenergy between two cubes of arbitrary size and positioninstead of a simple quadraturerulesincetheelementsarenot infinitesimal anymore.Toourknowledge,thisresultisnotavailableinthe liter-ature.Thisiswhyitisderivedforthegeneralcaseoftwoslabsin AppendixA.

ThisapproachisphilosophicallyclosetotheSEImethod[17,19] whichsplitsthesurfaceofparticlesinsmallflatplateelementsand thensumstheinteractionenergiesforthedifferentplatesassuming theseenergiesarecloselyrelatedtotheanalyticalsolutionfortwo

parallelsemi-infinitesolidsseparatedbyagap.Sincethepresent methodperforms asummationonallthecubecouples,it could betermeda“VolumeElement Integration”(VEI)method. How-ever,animportantdistinctionisthatintheSEImethodtheflat plateapproximationismadeforeachsurfaceelement,whichcould deterioratethecomputationforhighlycurvedsurfaces,whereas inthepresentapproachnosuchapproximationismadeandany geometrycanbehandledwithanequivalentprecision.Amethod using spheres instead of cubes as mapping elements has also been presented recently (the “Sphere Packing Approach” [27]). Evenifittendstoleaveholesinsidetheparticle,itmaybe effi-cientforsomecases,especiallyforparticleswithmicro-asperities. Anhybrid approach usingour methodin thecore ofthe body andthespherepackingapproachnearthesurfacecouldalsobe designed easilyfor anoptimalprecision/CPU costratiointhese cases.

Ashortcommentontheadaptivemeshingstrategyemployed hereisinordertounderstandthedegreeofprecisionofthe numeri-calcomputationsunderlyingtheresultspresentedintherestofthis article.Themesh,whichisbasedonan“octree”datastructure,is refinedonlyinregionsofinterestandleftverycoarseelsewhere,so thatthecomputationalcostandthememoryuseareoptimized.The qualityoftheresultsisthendirectlylinkedtotherefinement pro-cedure.Inthepresentproblemconsistingincomputingtheintegral of1/r6_{,therefinementprocedureisthefollowingone:}

1. createauniformcoarsemeshoflevelnmin(usually4,5or6in

thepresentwork)byrefininganycelluptothislevel,

2. computeanestimateoftheinteractionenergyobtainedfor cur-rentmeshlevelwithaprescribedquadratureruleortheVEI, 3.computethemaximumvalueof1/r6_{forallthecouplesofcells}

ofsolid1andsolid2,

4.scanallcellsiofsolid1andforeachonecomputethevalueof 1/r6_{withallcellsjofsolid2;if1/r}6_{>tol×max(1/r}6_)tagcellsi

andjforrefinement,

5.refinethecellstaggedforrefinementtoreachthenextmesh levelandgobacktopoint2.

Theonlyparameterthatcanbespecifiedbytheuseristhe toler-ancetol.Ifitisclosetozero,allthecellsinsidethesolidswillbe refined.Nobiaswillbeintroducedinthecomputationsbutthey willbeexpensive.Whentolislarger,onlycellsofsolid1closer tosolid 2willberefined,and viceversa. Computationswillbe cheapersince cellsoutsideofthezoneofinfluenceofonesolid ontheotherwillnotberefined.Iftolistoolarge,importantcells canbeleftcoarsewhereastheyshouldhavebeenrefinedandthe resultswillbepoorwhateverthefinalmeshlevel.Alittletesting mustbedonetoestablishareliablevalueoftol.Inthepresentwork, tol=10−2_{permittedtoobtaingoodresultsforalmostall}

configura-tions,butnotall.Hencetol=10−3_{hasbeensetandallthenumerical}

resultsalwaysconvergedtowardstheoreticalvalues,when avail-able,withinafewpercent.Anexampleofmeshobtainedwiththis valueisdisplayedonFig.1,togetherwithacolormaprepresenting atanypointofagivensolidtheinteractionenergybetweenthis pointandtheentireothersolid.Itcanbecheckedthatthisvalue almostvanishesbeforethetransitiontolargercells,whichmeans theresultswillbeobtainedwithan(high)accuracylinkedtothe smallestcellssize.Inpractice,theinteractionenergywascomputed forincreasingmeshlevelsautomaticallyandthesimulationwas stoppedwhenestimationsofEfortwoorthreeincreasingmesh levelsdifferedbylessthan1–2%.Allthesimulationspresentedin therestofthisarticlewereperformedwiththethreeintegralson thespherevolumecomputedanalyticallyandtheadaptivemesh refinementactivatedfortheothersolidtospareamaximumof computationaltime.

3. Interactionbetweenasphereandaslitoffinitedepth 3.1. Interactionenergybetweenasphereandasemi-infinite squarewedge

Thenon-retardedinteractionenergybetweenasphereand a pointis Es/point=− 4 3Qpˇ a3 (d2_−a2₎3,

whereQpisthenumberofatomperunitvolume,ˇisthe

Lifschitz-vanderWaalsenergyconstant,aisthesphereradiusanddisthe distancebetweenthespherecenterandthepoint[23].The interac-tionenergybetweenthesphereandthesemi-infinitesquarewedge (a“quarter-space”)asdepictedonFig.2(a)isthus

Es/w(a,b,c)=− 4Aa3 3

Z

∞ x=b

Z

∞ y=c

Z

∞ z=−∞ dxdydz (d2_−a2₎3, (2) whered2_=x2_+y2_+z2_,A=2_Q

pQwˇistheeffectiveHamaker

con-stantandQwisthenumberofatomsperunitvolumeinthewedge.

TheintegralinEq.(2)canbecomputedanalytically.Thefirst integralagainstzleadsto

Es/w(a,b,c)=− Aa3 2

Z

∞ x=b

Z

∞ y=c dxdy (x2_+y2_−a2₎5/2, (3)

wherex2_+y2 _{isthesquareddistancebetweenthespherecenter}

andapointinthewedge,whichisalwaysstrictlylargerthana2_.

Thenextintegrationagainstygives

Es/w(a,b,c) =−Aa 3 2

Z

∞ x=b 1 (x2_−a2₎2



2 3− c(3(x2 −a2₎ +2c2₎ 3(x2_−a2_+c2₎3/2



dx, (4)

Notethatthisexpressionholdsfor|x|=/ a.Thecasex=acanbe encounteredintwoverydifferentconfigurations.Inthefirstone, c_≤0andb=a(sphereatcontactwiththewedge“left”plane,see Fig.2(a)).Inthiscasetheinteractionenergyeffectivelydiverges butthiscanbesolvedassumingthecontactisnevercompletely achieved.Thesecondconfigurationiswhenc>0andb<a,i.e.the sphere’sforemostpointisatleastpartiallyontherightsideofthe wedge’sleftplane.Inthiscasethepointsinthewedge correspond-ingtox=ahavenothingspecialanditcanbesafelyassumedona physicalbasisthattheintegralbetweenband_∞canbesplitina partbetweenbanda−_{andapartbetween}a+_{and∞,thetwoparts}

beingfinite.Themissingpartx=ahasazerovolumeanddoesnot contributetotheintegral.Hencewecancontinuewithexpression (4).Thelastintegralagainstxleadsto

Es/w(a,b=/ ±a,c) = − A 12







ln







_|c|−a |c|+a



sign(c) _c2_−a2_+|c|

p

_b2_+c2_−a2_+ab c2_−a2_+|c|

p

_b2_+c2_−a2_−ab

!

sign(c)



_|b−a| |b+a|



1−sign(c)





− 2abc(b 2_+c2_−2a2₎

p

b2_+c2_−a2_(b2_−a2_)(c2_−a2₎+ 2ac c2_−a2 + 2ab b2_−a2

)

(5)

Asexpected,thisexpressioncannotbeusedasiswhenb=±a.If c_≤0,thelastterminthelogarithmmakesitdiverge,whichwas expectedphysically(thespheretouchestheleftmostplaneofthe wedge).Ifc>0,thisproblematictermvanishesandthesecondand lasttermscanceloutexactly,whichmakesthisformulausablein practice.Ifahumanbeingisabletoeliminatetheproblematicterms analytically,caremust betakenwhenusingtheformulawitha

calculatororacomputerprogram.Ifb=±aandc>0itshouldeither bereplacedbythecorrespondingform

Es/w(a,b=±a,c>0)=− A 6

n

ln

h

c c+a

i

+ ac c2_−a2

o

(6) orthevalueofbshouldbeperturbedslightlytoeffectivelycompute lim

b→aEs/w(a,b=/ ±a,c)with(5).

Fortunately,formula(5)degeneratestothesphere–planeone for_{b→− ∞}andc>aorfor_{c→− ∞}andb>a.Ifc=0,thisexpression shouldbereplacedbyhalfthesimplersphere/planevalue,which istheexactsolutionforthissymmetriccase.

Expression(5)hasbeenvalidatedusingthenumericalmethod brieflydescribedinSection2.AcontourmapofEobtainedwith Eq.(5)isdrawnonFig.3togeta qualitativeideaonits behav-ior.Theinteractionenergiesbetweenasphereandasquarewedge obtainedanalyticallyand numerically are reportedonFig. 4to permitaquantitativecomparison.Tobeperfectlycompletesome precisionsconcerningthenumericalcomputationsareinorder.In allthesimulationsthesphereradiusisa=0.1(thisvalueis com-pletelyarbitraryandnormalizedbythenumericalboxsize).Inthe sphere/planesimulationtheplaneisnotinfinitebutisactuallya finiteslabwithasurfaceareaof1facingthesphereandadepthof 0.5.Ithasbeencheckedthatlargervaluesoftheslabtospheresize ratiodidnotleadtoimprovementsoftheresults.Thecomparison oftheanalyticalresults(5)andofthenumericalresultspresented onFig.4isverygood.Forlargenegativevaluesofc(sphereinfront ofasolidwall,seeFig.2(a))thesphere/planemodelisrecovered. Forthespecialcasec=0halfthesphere/planevalueisobtainedas expected.Finallyforpositivevaluesofc(spherenomoreinfront ofasolidplane)theinteractionenergyisdrasticallyreduced.

Expression(5)forasphere/wedgeinteractioncanbeusedto estimatetherangeofvalidityofthesphere/planeexpressionfora particle“far”fromtheedgeofthewedgeatc_≪0.Amapofthe rela-tiveerrorontheinteractionenergyintroducedbythesphere/plane modelisrepresentedonFig.5.Asphereisdrawnasanexample oflocationcorrespondingtothe10%error.Qualitatively,farfrom thewedgeandclosetothewallthesphere/planemodelprovides veryaccurateresults.Foranylateralpositionc,itisalways possi-bletofindadistancetothewallbsufficientlysmalltomakethe sphere/planeapproximationvalid.Ontheotherhand,forpositions fartherandfartherapartfromthewall(largeb)thesphere/plane approximationislessandlessaccurate.Thisisduetothefactthat thevalueof1/r6 _{betweenonepointofthesphereandthe}

clos-estpointinthewallisnotmuchdifferentfromthatofthesame pointinthesphereandthecornerofthewallsothatthezone ofthewallinducinganon-negligibleinteractionenergyis some-what“truncated”bythepore.ThiseffectcanbevisualizedonFig.1. Whenthesphereisveryclosetothewallthedistanceofclosest approachisverysmalland1/r6_{decreasessharplysothatonlyavery}

localizedpartofthewallcontributestoasignificantpartofthetotal

interaction.Hencethespherecanapproachtheedgequiteclosely (smallc)withoutanysubstantialdiscrepancyinthesphere/plane model.Asaruleofthumb,Fig.5showsthesphere/plane approxi-mationtobevalid(lessthan10%error)for−c/a>1andb<−0.86c. For−c/a≤1theerrorgrowsrapidlywiththedistancetothewall andtheexactsolution(5)shouldbepreferredtothesphere/plane approximation.

Fig.3.Contourmapofln(E)obtainedwithexpression(5)foraspherecenterplaced atcoordinates(b,c).Thesurfaceofthewedgeisdrawnasathickblackline.No datawasgeneratedforb≤aandc≤aforpost-processingreasonssothehigher levelcontoursarediscontinuousneartheedgebuttheyareofcoursecontinuousin reality.

Withtheexpressionoftheinteractionenergybetweenasphere andasquarewedgeknown,itispossibletoobtainexactrelations forseveralothergeometriesusingtheadditivityhypothesis.Some ofthemarelistedinthenextparagraphs.

3.1.1. Interactionenergybetweenasphereandasemi-infinite slab

Theinteractionenergybetweenasphereandasemi-infiniteslab ofwidthL(Fig.2(b))isdirectlyrelatedtoEs/was

Es/islab(a,b,c,L)=Es/w(a,b,c)−Es/w(a,b+L,c) (7)

3.1.2. Interactionenergybetweenasphereandasemi-infiniteslit Theinteractionenergybetweenasphereandasemi-infiniteslit ofwidthl(Fig.2(c))canbeexpressedbyconsideringthe semi-infiniteslitasasetoftwosquarewedges:

Es/islit(a,b,c,l)=Es/w(a,b,c)+Es/w(a,b,l−c) (8)

Fig.4.Interactionenergybetweenasphereandasquarewedgeasdepictedon

Fig.2(a).Continuouslines:expression(5);dashedline:sphere/planemodel; dot-dashedline:halfthesphere/planemodel;symbols:numericalcomputations.From toptobottomcurve:c/a=−∞,−4,−2,−1,−1/2,−1/4,0,1/4,1/2.

Fig.5.Contourmapoftherelativeerror(%)betweenthesphere/planemodeland theexactexpression(5)inthevicinityofthewedge.Theplaneclosesttothesphere isdrawnasablacklineandextendsfromc=0to−c→∞(theedgeofthewedgeis locatedat(0,0)).Theequationofthedashedlineisb=0.86(−c).

3.1.3. Interactionenergybetweenasphereandaslitoffinite depth

Theinteractionenergybetweenasphereandafiniteslitofwidth landdepthL(Fig.2(d))canbededucedfromthepreviousresultas Es/fslit(a,b,c,l,L)=Es/islit(a,b,c,l)−Es/islit(a,b+L,c,l) (9)

ThevalidationofEqs.(8)and(9)againstthenumericalintegration resultsispresentedonFig.6.Theslitwidthisl=3a,thedistance betweenthesphere centerand the upperplaneis c=5a/4 and thatbetweenthespherecenterandthelowerplaneisl₋c=7a/4. ForthefiniteslitcasethedepthisL=a.Thecomparisonisvery good for both geometries.Considering theinfinite slitproblem (dashedline),theinteractionvanishesforlargepositivebsincethe sphereisfarfromtheslitentrance.Whenbisdecreased,the mag-nitudeoftheinteractionenergyincreasesprogressivelytoreach thatofaspherebetweentwoplanesseparatedbytheslit’swidth

Fig.6.Interactionenergybetweenasphereandeitheraninfiniteslit(Fig.2(c))or afiniteslit(Fig.2(d)).Theanalyticalrelationfortheinfinite(resp.finite)slitis(8)

(resp.(9)).Thetwoplanestheory(dottedline)isbasedontheuseofthesphere/plane interactionbetweenthesphereandtheinnerplanesoftheslit.

(dottedline).Thisapproximation,obviouslyvalidwhenthesphere isinsidetheslitandfarfromitsentrance,isseentoprovideresults within10%oftheexactsolutionprovidedthesphereisentirely insidetheslit(b/a<−1)inthepresentexample.Howeveritshould bestressedthatthisapproximationwouldbesatisfactorycloser totheslitentranceifthedistancebetweenthespheresurfaceand theslitinnerplanesweresmaller,forthesamereasonsasthose detailedforthesphere/wedgeproblem.Largervaluesof_−bwould berequiredtoachievethesameaccuracyforlargerl/aratios,i.e. largerslitscomparedtotheparticlesize.

Thefiniteslitcaseshowsvaluessimilartothoseofthe semi-infiniteslitcasewhenthesphereisbeforetheslitentrance.The interactionenergyreachesamaximumatthemid-pointbetween theslitentranceandexit(b/a=−0.5).Itistheoreticallysymmetrical againstthispointbutthenumericalintegrationhasbeenperformed onbothsidestocheckthevariabilityoftheresultsagainstthemesh details.Thedifferencebetweentherightandleftpointsis2–3% atmostandoftensmaller.Themaximumvalueoftheinteraction energyisonlyapproximatelyhalfofthatbetweentwoplanesin thepresentexampleduetothefiniteslitdepth.Howeveritshould bestressedthatthisresultdependsontheparticularvaluesofthe a/L,a/landc/lratios.

3.1.4. Interactionenergyforasphereabovea2Dpillar

Theinteractionenergyforasphereabovea2Dpillarofwidth l(Fig.2)canbededucedfromthepreviousresultandthesphere planeresultsas

Es/2Dpillar(a,b,c,l)=Es/plane(a,b)−Es/islit(a,b,c,l) (10)

3.1.5. Interactionenergybetweenasphereandaninfinite rectangularrod

Theinteractionenergybetweenasphereandaninfinite rect-angularrodofwidthLinthexdirectionandlintheydirection (Fig.2(f))canbededucedfromtheinteractionenergyforthesphere anda2Dpillaras

Es/irod(a,b,c,l,L)=Es/2Dpillar(a,b,c,l)−Es/2Dpillar(a,b+L,c,l)

(11)

3.1.6. Interactionenergybetweenasphereanda2Dcorner

Thewallofthesurroundingcornercanbesplitinthreewedges asdepictedonFig.2(g).Theinteractionenergybetweenthesphere andregionIisEs/w(a,c,−b),thatofregionIIisEs/w(a,b,c)andthat

ofregionIIIisEs/w(a,b,−c).Thetotalinteractionenergyisthen

Es/corner(a,b,c)=Es/w(a,b,c)+Es/w(a,c,−b)+Es/w(a,b,−c) (12)

3.1.7. Interactionenergyforasphereinsideaninfinite rectangularchannel

The rectangular channel (Fig.2(h)) can be constructed as a superpositionoffourwedges(regionsII,IV,VIandVIII)andfour 2Dpillars(regionsI,III,V,VII).Theresultinginteractionenergyfor achannelofwidthLinthexdirectionandlintheydirectionis:

Es/channel(a,b,c,l,L) = Es/w(a,b,c)+Es/w(a,b,l−c)+Es/w(a,L−b,c)

+Es/w(a,L−b,l−c)

+Es/2Dpillar(a,b,c,l)+Es/2Dpillar(a,L−b,c,l)

+Es/2Dpillar(a,c,b,L)+Es/2Dpillar(a,l−c,b,L)

(13)

Relations(12)and(13)arecomparedtonumericalintegration resultsonFig.7.Inthesphere/cornercasethedistancebetween thespherecenterandtheupperplaneisc/a=3/2andthedistance betweenthespherecenterandtherightplaneb/aisvaried.Inthe sphere/channelcase,thechannelwidthisL=6ainthexdirection anditsheightisl=2.75aintheydirection.Theconsistencybetween

Fig.7. Interactionenergybetweenasphereandeitheracorner(Fig.2(g))ora rectan-gularchannel(Fig.2(h)).Theanalyticalrelationforthecorneris(12).Theanalytical solutionforasphereinsidearectangularchannelis(13).

theanalyticalandnumericalresultsisgood,whichvalidatesEqs. (12)and(13).NotethatEq.(10)fortheinteractionbetweenasphere andasemi-infiniteslabisalsovalidatedindirectlysinceitisused inEq.(13).

Considering thesphere/cornertest (thick continuouslineon Fig.7),thesolutiontendstothesphere/topplaneapproximation forlargevaluesofb/a,i.e.whenthesphereisfarfromtheright plane.Ifthesphereismovedtowardtherightplane(b/adecreases) themagnitudeoftheinteractionenergyincreasesandreachesthat givenbythesphere/rightplanemodelforsmallvaluesofb/aas expected.Wemaywonderifthesphere/cornerinteractioncould bemodeledasthesumofthesphere/upperplaneandsphere/right planeinteractions.Theanswerisyesinthepresentexample.Itis notplottedonFig.7becausethecurvewouldbeindistinguishable fromtheexactsphere/cornersolution.Thedifferencecanhowever beseenwhenzoominginthe1.25<b/a<1.75zone(notpresented here).Thisdifferenceisduetothefactthatwiththeaforementioned “superpositionofplanes”modeltheinteractionbetweenthesphere andregionIIiscountedtwice.Letusassumeacharacteristic dis-tancebetweenthesphereandanyobjectisthedistanceofclosest approachbetweenthem.Thedistanceofclosestapproachbetween thesphereandtheupper(resp.right)planeisc₋a(resp.b₋a) andthatbetweenthesphereandregionIIis

p

b2_+c2_−a.Assume

b=cforclarity.Theratioofthedistancesofclosestapproachwith oneplaneandthewedgeisr=(b/a−1)/(√2b/a−1).Ifthesphere isclosetocontactwiththetwoplanes,r_→0andtheinteraction energyisgivenbythetwoplanes.Ifthesphereisatadistancelarge enoughsuchthatb/a_≫1,theratioisr=1/√2,whichisfiniteand closetounitysothattheadditionalwedgecontributionincluded inthesphere/twoplanesmodelwillnotbenegligibleanymore.

Inthesphere/channeltest,thesolutiontendstothesphere/right planeorsphere/leftplanemodelsolutionwhenb/aissmall(near contactwiththerightplane)orlarge(nearcontactwiththeleft plane),ashighlightedonFig.7.Whenthesphereisatmid-distance betweenthetwosideplanes,theinteractionenergyisclosetothat givenbytheinteractionwiththeupperandlowerplanesalone. ThisisduetothelargeL/lratiointhepresentexampleandisnot ageneralresult.A comparisonwithasphere/fourplanesmodel isprovided(thindashedline).Inthismodeltheinteractionwith thefourregionsII,IV,VI,VIII(Fig.2(h))iscountedtwicebutthe approximationisquitegoodanyway.Onceagain,itis linkedto thegeometricalparameterschosenforthepresentexample.This approximationcouldbelesssatisfactoryforasmallersphereinthe

middleofthechannel.InasquarechannelofwidthL=l=Na,andfor asphereatthecenterofthechannel,thediscrepancybetweenthe exactsolutionandthefourplanesapproximationis1.5,10.4,12.7, 13.1and13.1%forN=2.1,4,10,100and1000,respectively. Keep-inginmindtheinteractionwillbealmosttotallyreducedtozero atadistanceofsay10nm(atleastinwater)andthatthe small-estparticlestractablewiththepresentHamaker approachmay havearadiusoftheorderof1nm,themaximumvalueofN=L/a foranon-negligibleinteractionwillbeO(10)atmostandtheerror inducedbythefourplanesapproximationwillliewithin10%ofthe “exact”value.Sincetheadditivityhypothesismayinducebyitself discrepanciesaslargeas10–20%withthereality,thefourplanes approximationcanbeconsideredtobeanacceptablemodelfor practicaluse,althoughonemayprefertouseEq.(13)sinceitis nowavailable.

4. Interactionbetweenasphereandacylindricalpore The interaction energy between a sphere and a cylindrical porehasbeencomputedby[23]when thesphereis insidethe pore.Recently,ithasbeenshownthatasinglesphericalparticle approachingasinglecylindricalporewasmorelikelytobe cap-turedattwodistinctpositions[3].Oneisslightlyawayfromthe pore,abovetheplate,andtheotherisrightontheporeedge.Both experimentsandatrajectorymodelincludinghydrodynamicsand vanderWaalsforces(simplyasaneffectivedistanceofcapture) showedthesepreferentialpositionsofcapturebuttheexact loca-tionswereslightlydifferentwiththetwoapproaches.Theprincipal motivationofthepresentworkwastoobtainasatisfactory expres-sionforthevanderWaalsinteractionbetweenthesphereandthe membrane+poresystemwhenthesphereisstilloutsidethepore andveryclosetoitsentrance,orevenontheedgeoftheporewhere thesphere/planemodelisunusable.Moreover,theexperiments wereconductedusingamicrosieveasamembraneof1.4mm thick-nesswhereasthespherestypicallyhadaradiusof1mm.Thefinite depthoftheporethusneedstobetakenintoaccount.

Sincenoanalyticalsolutionhasbeenestablishedforasphere aboveaplateperforatedbyacylindricalhole,theinteractionenergy hasbeencomputednumericallyforaplatethicknessL=1.4aanda spherelocatedrightabovetheporeedgeatc=0.Forasufficiently largeporeradius(rp≫a)andaspheresufficientlyclosetothepore

entrance(smallb),thecylindricalporeedgeisseenbythesphereas asemi-infiniteslabstraightedgeandrelation(7)canbeusedsafely forthesphere/poreinteraction.Itishoweverdesirabletoassessthe rangeofvalidityofsuchanapproximationintermsofvaluesofrp/a

andb/a.

Thesphere/cylindricalporeinteractionenergy hasthusbeen computednumericallyfordifferentporeradiiandiscomparedto thesphere/semi-infiniteslabinteractionenergy(7)onFig.8. Sim-ulationshavebeenperformedforporeradiirprangingfrom0.1a

to5aand forvariousdistancestotheplatesurfaceb.One sim-ulationwithasemi-infiniteslab(circles)hasbeenperformedto checktheconsistencybetweenthenumericalresultsandrelation (7).Foranyporeradiusthesphere/plateresultisrecoveredwhen b_≫rp (dashed line). Indeed, when thesphere/plate distance is

increasedthezoneoftheplatecontributingtothetotalinteraction becomeslargerandlargercomparedtotheporesizeandthe exis-tenceoftheporecanbeomitted.Thisconditionismet,ofcourse, forsmallervaluesofthesphere/poredistancewhenrp/aissmall.

Closetocontact,ifrp/a≪1theinteractionenergytendtothatof thesphere/platesystembutlessrapidlythanforb_≫a.Onceagain closetocontact,ifrp/a&1(thespherefitsexactlyinside–orislarger

than–thepore)thesphere/semi-infiniteslabenergy(7)is recov-eredwithin5%andifrp/a&2itisrecoveredwithin1–2%whichis

oftheorderoftheprecisionofthenumericalresults.Hencethe

Fig.8.Interactionenergycomputednumericallybetweenasphereandacylindrical poreinamicrosieveofdepthL=1.4anormalizedbytheinteractionenergybetween asphereandasemi-infiniteslab(Fig.2(b),Eq.(7)).Thedifferentcurvescorrespond todifferentporeradiirp.Thesphereislocatedrightabovetheporeedge(c=0).The

dashedlineisthesphere/platevalue(Es/plane(a,b)−Es/plane(a,b+L)),whichistwice

thesphere/semi-infiniteslabvalueforc=0.

sphere/semi-infiniteslabapproximationisaverygoodoneforan applicationtoafiltrationproblem.Indeed,forfiltrationproblems wheretheporesizeissmallerthantheparticlesize,theretentionis duetostericeffectsandthevanderWaalsforcedoesnotplaya cru-cialrole.Theapplicationofthepresentworkconcernsfiltrationof particlessmallerthantheporesize,i.e.rp/a>1andthediscrepancy

betweenthesphere/semi-infiniteslabandtheactualsphere/pore geometryisalwayslessthan5%,atleastforasphererightabove theporeedge(c=0).

Ifthesphereisabovetheplate(c<0),theradiusofcurvature oftheporewillhaveevenlesseffectandtheapproximationisstill valid.Thesphere/plateapproximationmayevenbeusedunderthe conditionsdetailedinSection3.1.

Whenthespherecenterliesabovetheporeorifice(c>0),the influenceoftheradiusoftheporewillbemoredrasticthanfor thec=0casepresentedonFig.8.Inthemicrosieveexperiments ofLinetal.[3],themostprobablepositionofcaptureonthepore edgecorrespondedapproximatelytoasphereincontactwiththe poreedgeandb=c.Numericalcomputationshavebeenperformed withb=candadistanceofclosestapproachofa/64.Inthiscasethe discrepancybetweenthesphere/semi-infiniteslabmodelandthe computed“exact”resultis50,20,6,and1%forrp/a=1,2,5,and10

respectively.Henceinafutureexperimentitwouldbedesirable touseaporeradiusfivetimesaslargeastheparticleradiusifthe analyticalexpressionistobeused.Notethatanumerical computa-tionoftheexactinteractionenergytakesonlyafewsecondswith thecodepresentedinthisworksoitisalwayspossibletocoupleit directlytoahydrodynamictrajectorysolverforsmallerporesizes. 5. Conclusion

AnalyticalandnumericalevaluationsofthevanderWaalsforce betweensphericalcolloidal particlesand more orless complex geometrieshavebeenproposed.Theanalyticalformulasderived inthisarticlehavebeenvalidatedwiththenewsimulationtool brieflypresentedinthefirstsection.Therangeofvalidityof sim-plesphere/planemodelsasapproximationsofmoresophisticated orunknownmodelsforcomplexgeometriesinvolvedincolloidal filtrationprocesseshasalsobeenassessedbycomparisonwiththe simulationresults.

ThenumericalmethodimplementedintheWITScodeisbased onaclassicalcomputationofthesixintegralsinEq.(1)for com-pletely arbitrary geometries or on theintegration of the three remainingintegralswhenoneofthesolidsisasphereand inte-grationonitsvolumeisperformedanalytically.Theoriginalitylies intheuseofaso-calledoctreemeshtodiscretizethevolumesof thesolidsinanautomaticadaptivemannerwhichminimizesthe CPUcostwithoutintroducingassumptionsonthephysics. Succes-siveevaluationsoftheinteractionenergyforprogressivelyrefined meshesenabletomonitortheconvergenceofthenumerical inte-grationalgorithmandprovidesanestimateofthenumericalerror. Simulationswerestoppedwhen twosuccessiveevaluations dif-feredfromlessthan1–2%.

Ananalytical expression for theinteraction energy between asphereanda squarewedgehasbeenpresentedandvalidated againstresultsobtainednumerically.Ithasbeenshownthatthe sphere/planeapproximationcanbeusedwithina10%errorinthe vicinityoftheedgeofthewedgeprovidedthespherecenterisin frontoftheplane,oneradiusawayfromtheedge(c/a<−1),and providedthedistancebetweenthespherecenter andtheplane respectsapproximatelyb<c.

Exactexpressionsforthetheinteractionsbetweenasphereand asemi-infiniteslab,asemi-infiniteslit,afiniteslit,a2Dpillar,an infiniterectangularrod,a2Dcornerandarectangularchannelhave been derived from thesphere/plane and sphere/wedge results. Theyhavebeenvalidatedagainstresultsissuedfromnumerical simulations.Combinationsofsphere/planeapproximationscanbe usedtoreplacemore complexsphere/corner orsphere/channel interactionswithin10%erroratleastinwaterwherethevander Waalsforcesarerapidlyscreened.

Sinceparticlecaptureatthesurfaceofafiltrationmembrane dependsonthevanderWaalsforcebetweenthecolloidsandthe membrane,itwouldbedesirabletoknowananalyticalexpression fortheinteractionenergybetweenasphereandacylindricalpore inaplate(whenthesphereisoutsidethepore).Suchanexpression ishoweverunavailabletoourknowledge,hencenumerical simu-lationshavebeenperformedtoassesstherangeofvalidityofthe approximationconsistinginreplacingthecylindricalporeedgeby thestraightedgeofasemi-infiniteslab.Closetocontactandwhen thesphereisrightabovetheporeedge,thisapproximationfalls within10%oftheexactresultforaporeradiusaslargeasthe parti-cleradiusandisalmostexactforporesradiilargerthantwoparticle radii.Whentheparticleisarrestedontheporeedgebutpartially blockingtheporeasinRef.[3],theapproximationisvalidforpore radiilargerthatfiveparticleradii.

Theseresultsareveryencouragingtostudythemechanismsof captureofcolloidalparticlesonmicrosieves.Furtherexperiments willbeconductedincollaborationwiththefluiddynamicistsof IMFTinvolvedinRef.[3].Thepresentresultswillhelp dimension-ingthenew experimentalsetupsand, itis believed, enrichthe numericalcomputationoftheparticlestrajectoriesandpositions ofcapture.

Acknowledgements

IwarmlythankPaulDuru,MartineMeirelesandPatriceBacchin forfruitfuldiscussionsonthis topic.ThesupportoftheCALMIP project,whichprovidedaccesstothe“Hyperion”SGIICEandUV supercomputers,isalsogreatlyacknowledged.

AppendixA. Analyticalsolutionforthenon-retarded interactionenergybetweentwoslabslocatedatarbitrary positions

TocomputetheintegralinEq.(1)withV₁andV₂beingtwoslabs ofsizea1×b1×c1anda2×b2×c2respectively,andseparatedby

Fig.9.Generalconfigurationoftwoslabsofsizea1×b1×c1anda2×b2×c2

sepa-ratedbydistancesdx,dyanddzinthex,yandzdirectionsrespectively.

distancesdx,dyanddzinthex,yandzdirectionsrespectively(Fig.9),

themethodofRef.[26]isused.Eq.1canbespecializedin

I=−2E=

Z

a1 0 dx1

Z

dx+a1+a2−x1 dx+a1−x1 dx

Z

b1 0 dy1

Z

dy+b1+b2−y1 dy+b1−y1 dy

Z

c1 0 dz1

Z

dz+c1+c2−z1 dz+c1−z1 dzh(x,y,z), (A.1) whereh(x,y,z)=(x2_+y2_+z2₎−3_.

Thesecondantiderivativeofhagainstzis:

h2(x,y,z)= 3z 8(x2_+y2₎5/2arctan

"

z

p

x2_+y2

#

−_8(x2_+y2₊1_z2_)(x2_+y2₎ (A.2)

Thisexpressionistrueprovidedx2_+y2 _/

= 0.Inthepresent appli-cation there is alwaysat least one spacedirection,say x, with non-zerointegrationlimits(i.e.slabs areneitherincontactnor overlap).Therefore,withthisconstraintinmind,Eq.(A.2)isalways valid.Intheintegrationprocess,h2isevaluatedforfourvaluesofz

beforeotherintegrationsagainstyandxareconducted.Thesecond antiderivativeofh2againstyis h4(x,y,z=/0) = − 3y 16x3atan



y x



+ z

%

2y2_+x2

8x4

p

_x2_+y2 arctan

"

z

p

x2_+y2

#

+lasttermwithy,zinterchanged

(A.3) or h4(x,y,z=0)=− y 16x3arctan

h

_y x

i

(A.4) Onceagain,asx =/ 0thisexpressionisalwaysvalid.Thespecial casey=0isnowconsidered(whenonefaceofeachslabisinthe sameplanewithnormaldirectiony):

h4(x,y=0,z=/0)= z 8x3arctan

h

_z x

i

(A.5)

orify=z=0(whenonefaceofeachslabisinthesameplanewith normaldirectionyandonefaceofeachslabisinthesameplane withnormaldirectionz)

Fig.10.GeometriestreatedinRef.[26].(a)Twoparallelslabs;(b)twoslabswith onefacingedge.

Table1

Comparisonofthepresentresultswiththoseof[26]forthecaseoftwoparallelslabs (Fig.10(a))

(a,b,c,dz) Eq.(21)in[26] Present(A.1)and(A.7)to

(A.10)

(0.5,0.3,0.1,0.2) −1.60428336087276×10−2 _{−1.60428336087304×10}−2

(0.1,0.1,0.5,0.3) −5.71796733284981×10−5 _{−5.71796733296455×10}−5

(1.0,2.0,0.4,0.1) −4.15196970900943 −4.15196970900983

Table2

Comparisonofthepresentresultswiththoseof[26]forthecaseoftwoskew par-allelepipedswithsquarecross-section(Fig.10(b))

(a,c,d) Eq.(25)in[26] Present(A.1)and(A.7)to

(A.10)

(0.1,0.5,0.2) −3.290827854534256×10−4 _{−3.290827854534263×10}−4

(0.3,0.1,0.05) −4.144539374852781×10−3 _{−4.144539374851914×10}−3

(1.0,2.0,0.1) −0.133166827308585 −0.133166827308519

Thesecondantiderivativeofh4againstxis

h6(x,y=/0,z=/0)= 1 32ln

"

%

x2_+y2

3 x2

%

_x2_+y2_+z2

2

#

+ 3 32

h

_x y− y x

i

arctan



y x



+ x

%

y2_+z2

3/2 24y2_z2 arctan

"

x

p

y2_+z2

#

+1 24z

h

₁ x2+ 1 y2

i p

x2_+y2_arctan

"

z

p

x2_+y2

#

+lasttermwithy,zinterchanged

(A.7) or h6(x,y=/0,z=0)= 1 32ln



x2_+y2 x2



+₃₂1

h

_x y− y x

i

arctan

h

y x

i

(A.8) or h6(x,y=0,z=/0) = − 1 16ln



x2_+z2 x2



−₁₆1

h

_x z− z x

i

arctan

h

z x

i

(A.9) or h6(x,y=0,z=0)=x (A.10)

Thevalidityoftheserelationscanbecheckedagainsttheresults ofRef.[26]fortwoparallelslabs(Fig.10(a))andtwoskew paral-lelepipedsofsquare cross-section(Fig.10(b),a1=b1=a2=b2≡a,

c1=c2≡c,dz=−c1anddx=dy≡d).Someresultsobtainedforthese two configurationsarereportedin Tables 1and 2 respectively. ThepresentresultsareidenticaltothoseofRef.[26]tomachine accuracy.

References

[1] M.-L.Rami,M.Meireles,B.Cabane,C.Guizard,Colloidalstabilityfor concen-tratedzirconiaaqueoussuspensions,J.Am.Ceram.Soc.92(1)(2009)S50–S56,

http://dx.doi.org/10.1111/j.1551-2916.2008.02681.x.

[2] C.Roth,B.Neal,A.Lenhoff,Vanderwaalsinteractionsinvolvingproteins, Biophys. J. 70(2) (1996) 977–987,http://www.sciencedirect.com/science/ article/pii/S0006349596796418.

[3]J.Lin,D.Bourrier,M.Dilhan,P.Duru,Particledepositionontoamicrosieve, Phys. Fluids 21 (7) (2009) 073301, http://dx.doi.org/10.1063/1.3160732, http://link.aip.org/link/?PHF/21/073301/1.

[4] P.Bacchin,A.Marty,P.Duru,M.Meireles,P.Aimar,Colloidalsurface interac-tionsandmembranefouling:investigationsatporescale,Adv.ColloidInterface Sci.164(1–2,SI)(2011)2–11,http://dx.doi.org/10.1016/j.cis.2010.10.005. [5] J. Stenhammar, P. Linse, H. Wennerstrom, G. Karlstrom, An exact

cal-culation of the van der Waals interaction between two spheres of classical dipolar fluid, J. Phys. Chem. B 114 (42) (2010) 13372–13380,

http://dx.doi.org/10.1021/jp105754t.

[6]A. Rodriguez, M. Ibanescu, D. Iannuzzi, F. Capasso, J.D. Joannopoulos, S.G. Johnson, Computation and visualization of casimir forces in arbi-trary geometries: nonmonotonic lateral-wall forces and the failure of proximity-force approximations, Phys. Rev. Lett. 99 (2007) 080401,

http://dx.doi.org/10.1103/PhysRevLett.99.080401, http://link.aps.org/doi/ 10.1103/PhysRevLett.99.080401.

[7]R.H. French, V.A.Parsegian,R. Podgornik,R.F.Rajter,A.Jagota, J.Luo, D. Asthagiri, M.K. Chaudhury,Y.-m. Chiang, S.Granick, S. Kalinin, M. Kar-dar, R. Kjellander, D.C. Langreth, J. Lewis, S.Lustig, D. Wesolowski, J.S. Wettlaufer, W.-Y. Ching, M. Finnis,F. Houlihan, O.A. von Lilienfeld, C.J. vanOss,T.Zemb,Longrangeinteractionsinnanoscalescience,Rev.Mod. Phys.82(2010)1887–1944,http://dx.doi.org/10.1103/RevModPhys.82.1887, http://link.aps.org/doi/10.1103/RevModPhys.82.1887.

[8]L.D.Derjaguin,B.V.Landau,ActaPhys.Chim.U.R.S.S.14(1941)633. [9]E.Verwey,J.Overbeek,TheoryoftheStabilityofLyophobicColloids,Elsevier,

Amsterdam,Netherlands,1948.

[10]M.Fujita,H.Nishikawa,T.Okubo,Y.Yamaguchi,Multiscalesimulationof two-dimensionalself-organizationofnanoparticlesinliquidfilm,Jpn,J.Appl.Phys., Part143(7A)(2004)4434–4442,http://dx.doi.org/10.1143/jjap.43.4434. [11] M.Fujita,Y.Yamaguchi,Multiscalesimulationmethodforself-organizationof

nanoparticlesindensesuspension,J.Comput.Phys.223(1)(2007)108–120,

http://dx.doi.org/10.1016/j.jcp.2006.09.001.

[12]B. Schäfer, M. Hecht, J. Harting, H. Nirschl, Agglomeration and filtra-tion of colloidal suspensions with dvlo interactions in simulation and experiment, J. Colloid Interface Sci. 349 (1) (2010) 186–195,

http://dx.doi.org/10.1016/j.jcis.2010.05.025, http://www.sciencedirect.com/ science/article/pii/S0021979710005527.

[13] H.Wennerstrom,ThevanderWaalsinteractionbetweencolloidalparticles anditsmolecularinterpretation,ColloidsSurf.A228(1–3)(2003)189–195,

http://dx.doi.org/10.1016/j.colsurfa.2003.08.006.

[14]J.Israelachvili,IntermolecularandSurfaceForces,AcademicPress,1991. [15] D.R.Snoswell, J. Duan,D. Fornasiero,J. Ralston, Colloid stabilityof

syn-thetic titaniaand theinfluence of surfaceroughness,J. ColloidInterface Sci. 286 (2) (2005) 526–535, http://dx.doi.org/10.1016/j.jcis.2005.01.056, http://www.sciencedirect.com/science/article/pii/S0021979705000652. [16] L. Suresh, J.Y. Walz, Effect of surface roughness on the interaction

energy between a colloidal sphere and a flat plate, J. Colloid Inter-face Sci.183(1)(1996) 199–213,http://dx.doi.org/10.1006/jcis.1996.0535, http://www.sciencedirect.com/science/article/pii/S0021979796905354. [17]S.Bhattacharjee,J. Chen,M.Elimelech,DLVOinteractionenergybetween

spheroidal particlesanda fiatsurface,ColloidsSurf.,A165(1–3)(2000) 143–156,http://dx.doi.org/10.1016/S0927-7757(99)00448-3.

[18]E.Hoek,G.Agarwal,Extendeddlvointeractionsbetweensphericalparticles androughsurfaces,J.ColloidInterfaceSci.298(2006)50–58.

[19]E.Martines,L.Csaderova,H.Morgan,A.S.G.Curtis,M.O.Riehle,DLVOinteraction energybetweenasphereandanano-patternedplate,ColloidsSurf.A318(1–3) (2008)45–52,http://dx.doi.org/10.1016/j.colsurfa.2007.11.035.

[20]M.-M.Kim,A.Zydney,Effectofelectrostatic,hydrodynamic,andbrownian forcesonparticletrajectoriesandsievinginnormalflowfiltration,J.Colloid InterfaceSci.269(2004)425–431.

[21]W.Russel,D.Saville,W.Schowalter,ColloidalDispersions,Cambridge Univer-sityPress,1989.

[22]A.Anandarajah,J.Chen,Singlecorrectionfunctionforcomputingretarded vanderwaalsattraction,J.ColloidInterfaceSci.176(2)(1995)293–300,

http://dx.doi.org/10.1006/jcis.1995.9964, http://www.sciencedirect.com/ science/article/pii/S002197978579964X.

[23] S. Bhattacharjee, A. Sharma, Lifshitz-van der Waals energy of spherical particlesincylindricalpores,J.ColloidInterfaceSci.171(2)(1995)288–296,

http://dx.doi.org/10.1006/jcis.1995.1183, http://www.sciencedirect.com/ science/article/pii/S0021979785711836.

[24]C. Argento, A. Jagota, W. Carter, Surface formulation for molecu-lar interactions of macroscopic bodies, J. Mech. Phys. Solids 45 (7) (1997) 1161–1183, http://dx.doi.org/10.1016/S0022-5096(96)00121-4, http://www.sciencedirect.com/science/article/pii/S0022509696001214. [25] P. Yang, X. Qian, A general, accurate procedure for calculating

molec-ular interaction force, J, Colloid Interface Sci. 337 (2) (2009) 594–605,

http://dx.doi.org/10.1016/j.jcis.2009.05.055, http://www.sciencedirect.com/ science/article/pii/S0021979709006262.

[26]G.D.Rocco,W.Hoover,Ontheinteractionofcolloidalparticles,Proc.Natl.Acad. Sci.46(1960)1057.

[27]L.Zhang,J.Cecil,D.Vasquez,J.Jones,B.Garner,ModelingofvanderWaalsforces duringtheassemblyofmicrodevices,automationscienceandengineering,in: CASE’06.IEEEInternationalConference,2006.

[28]S.W.Montgomery,M.A.Franchek,V.W.Goldschmidt,Analyticaldispersion forcecalculationsfornontraditionalgeometries,J.ColloidInterfaceSci.227 (2000)567–584.

[29]V.A.Parsegian,VanDerWaalsForces:AHandbookforBiologists,Chemists, Engineers,andPhysicists,CambridgeUniversityPress,2006.