Energy Stored in Nanoscale Water Capillary Bridges between Patchy Surfaces - Tang et al. - Unknown - Unknown

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SupportingInformationEnergyStoredinNanoscaleWaterCapillaryBridgesbetweenPatchySurfacesBinzeTang1,SergeyV.Buldyrev2*,LimeiXu1,3?,NicolasGiovambattista4,5?1InternationalCenterforQuantumMaterials,SchoolofPhysics,PekingUniversity,Beijing100871,China2DepartmentofPhysics,YeshivaUniversity,500West185thStreet,NewYork,NY100333CollaborativeInnovationCenterofQuantumMatter,Beijing,China4DepartmentofPhysics,BrooklynCollegeoftheCityUniversityofNewYork,Brooklyn,NewYork11210,UnitedStates5Ph.D.ProgramsinChemistryandPhysics,TheGraduateCenteroftheCityUniversityofNewYork,NewYork,NY10016,UnitedStates*Correspondingauthor:E-mail:buldyrev@yu.edu?Correspondingauthor:E-mail:limei.xu@pku.edu.cn?Correspondingauthor:E-mail:ngiovambattista@brooklyn.cuny.eduNumberofpages:10Numberoffigures:3Numberoftables:3Contents:1.Fittingparametersofthewatercapillarybridges.2.EnergyDensityStoredinCapillaryBridgesbetweenHomogeneousSurfacesS1

11.Fittingparametersofthewatercapillarybridges.InFig.2ofthemanuscriptwefittheprofileofthewatercapillarybridgesobtainedfromourmoleculardynamics(MD)simulationsusingthecorrespondingexpressionprovidedbycapillaritytheory(CT).CTpredictsthattheprofiler(z)ofatranslationallysymmetriccapillarybridgeisgivenbyacirclecenteredat(rc,zc)wherezc=0,rc=r0-R2,andr0ishalfthethicknessofthebridgeatz=0(seeFig.1dofthemainmanuscript),i.e.,[r(z)?rc]2+(z-zc)2=(R2)2TableS1containstheparametersr0andR2correspondingofthetheoreticalprofilesofthecapillarybridgesshowninFig.2ofthemainmanuscriptindicatedbysolidlines.Inthiscase,allthepointsoftheprofileobtainedfromMDsimulationsareincludedinthefittingprocedure.TableS2containstheparametersr0andR2correspondingofthetheoreticalprofilesofthecapillarybridgesshowninFig.2ofthemainmanuscriptindicatedbydashedlines.Inthiscase,thepointsofthecapillarybridgeprofileobtainedfromMDsimulationsclosesttothelowerandupperwallsareremovedforthefittingprocedure.TableS1:Fittingparametersr0andR2ofthetheoreticalprofilesofthecapillarybridgesindicatedbysolidlinesinFig.2ofthemainmanuscript.Errorsinr0andR2are±0.05?;?istheerrorofthetheoreticalprofilerelativetotheMDdata[1].Height[nm]2.53.03.54.04.55.0r0[nm]4.633.883.292.802.452.09R2[nm]5.005.0012.4-15.0-8.61-4.70?0.250.1640.2250.3580.2810.317Height[nm]5.56.06.57.07.58.0r0[nm]1.821.591.391.221.060.94R2[nm]-4.26-4.23-4.29-4.49-4.67-4.98?0.2760.3350.3270.3470.3940.398TableS2:Fittingparametersr0andR2ofthetheoreticalprofilesofthecapillarybridgesindicatedbydashedlinesinFig.2ofthemainmanuscript.Errorsinr0,R2are±0.05?;?istheerrorofthetheoreticalprofilerelativetotheMDdata[1].Height[nm]2.53.03.54.04.55.0r0[nm]4.643.883.292.822.432.09R2[nm]5.015.0915.0-15.0-6.61-4.33S2

2?0.0990.1210.2120.1380.1710.135Height[nm]5.56.06.57.07.58.0r0[nm]1.811.571.371.211.040.92R2[nm]-3.99-3.93-4.05-4.30-4.45-4.76?0.1440.1500.1610.1530.1800.178S3

32.EnergyDensityStoredinCapillaryBridgesbetweenHomogeneousSurfacesInthemainmanuscript,wefindthatthemaximumpotentialenergystoredinawatercapillarybridgebetweenthepatchysurfacedstudiedis9.379nN?nm.Thispotentialenergycorrespondstotheprocessofstretchingthewatercapillarybridgefrom?=2.67nmto?=8.0nm(maximumdistancereachedbeforethecapillarybridgebecomesunstableandbreaks).Ifweassumethattheminimumvolumeassociatedtoourcapillarybridgeis?0=15???10???14??thentheenergydensityofthesystemis9.379nN?nmρ==4470kJ/m3?,??????15?10?14nm3Next,wecomparethisvaluewiththeenergydensityofwatercapillarybridgesexpandingbetweenhomogeneoussurfaces.Weconsiderthreecaseswherethecontactangleofwateris(a)θ=90°,(b)θ=108°,and(c)θ=40°.AsshowninRefs.[1,2],case(a)isfoundwhenSPC/Ewaterisincontactwithour(homogeneous)hydroxylatedsilicawallswithpartialchargesre-scaledbyafactor~0.35.Similarly,case(c)isfoundwhenthehydroxylatedsilicawallshavepartialchargesre-scaledbyafactor~0.6.Case(b),isthecontactangleofSPC/Ewaterwithourhydrophobicwallswheresilicaisnon-hydroxylated.OurresultsaresummarizedinTableS3.TableS3:Energydensityforwatercapillarybridgesbetweenthesurfacesstudied.SurfacePatchyHomogeneousHomogeneousHomogeneousθ=108°θ=90°θ=40°Energydensity4470234837676733(kJ/m3)a)?=??°Thecomponentoftheforceproducedbythecapillarybridgeonthewalls,alongthedirectionpointingawayfromtheconfinedvolume(perpendiculartothewalls),isgivenbybcosθF=?2γw(sinθ+)(S1)hS4

4whereγ=0.053nN/nmistheliquid-vaporsurfacetensionofSPC/EwaterreportedinRefs.[1],bisthethicknessofthewatercapillarybridgeincontactwiththewalls(atitsbase),and?=?=14.0??isthelengthofthepatchalongthewall.Forθ=90°,Eqn.(S1)reducestoF=?2γw,i.e.,thecomponentoftheforcealongthedirectionpointingawayfromtheconfinedvolumeisnegative(attractivewall-wallinteractions;seeFig.S1a)andconstant.Hence,thepotentialenergystoredwhenthewallsaremovedapartfrom?1=2.67nmto?2=8.0nmis:?2?(PE)=PE(?2)?PE(?1)=?∫?(?)??=2γw(?2??1)=7.9097nN?nm?1Thecorrespondingenergydensityis7.9097nN?nmρ==3767kJ/m3?,9015?10?14??3Fig.S1b,showsρ?,90asfunctionof?2=?.FigureS1.(a)Componentoftheforceactingonthewalls(alongthedirectionpointingawayfromtheconfinedvolume,i.e.,wall-wallattractiveforcesarenegative)and(b)potentialenergystoredinthewatercapillarybridgesbetweenhomogenoussurfaceswithdifferentwatercontactanglesθ,asfunctionofthewallsseparation?(duringstretching).Forcomparison,alsoincludedaretheforceandpotentialenergyofthewatercapillarybridgesformedbetweenthepatchysurfacesconsideredinthemainmanuscript.b)?=???°Asketchofthewatercapillarybridgeformedbetweenhydrophobicwalls(e.g.,watercontactangleθ=108°)isincludedinFig.S2.S5

5FigureS2.Sketchofacapillarybridgebetweentwohomogeneoushydrophobicwalls.Inordertocalculatetheworkproducedbytheforceinducedbythewatercapillarybridge,weassumethatthevolumeofthecapillarybridgeisconstant,i.e.,independentofh.Thisimpliesthat?(?)=?0=15???10???14??.Asweshowbelow,thisequationcanbeusedtoextract?=?(?).BycombiningthisexpressionwithEqn.(S1),onecanalsoobtainanexpressionfortheforce?(?).Theexpressionfor?(?)followsfromFig.S2.Specifically,V(h)=wS,whereS=2S1+hbisthecrosssectionareaofthecapillarybridge(seeFig.S2);S1istheareaconfinedbythearcABandthedashedlineAB.ToobtainS1,wenotethatS1canalsobeexpressedasS1=A1-A2,whereA1istheareaoftheregionconfinedbythearcAB,thesegmentAO,andthesegmentBO.A2istheareaofthetriangledefinedbyAOB.Itcanbeshownthat2?1?S=2(πR2?hR2cos)+hb(S2)2π22Where?=∠AOB=2θ?π.Inaddition,onecaneasilyshowthath?R2==?(S3)2cos(π?θ)2???θForhydrophobiccapillarybridges,theconventionisusuallytotakeR2>0,consistentwithEq.S3.UsingEqns.(S3)and(S2),oneobtains?????22S=(+)h+hb(S4)4???2θ2???θSincethevolumeofthebridgeisV(h)=V0=wS,itfollowsfromEqn.(S4)thatS6

6??0????21b(h)=?(+)?=αh+β(S5)wh4???2θ2???θhwherewedefinedtheparameters?????2?0α=?(+)andβ=.4???2θ2???θwFromEqns.(S1)and(S5),wegetbcosθ1F(h)=?2γw(sinθ+h)=ω1+ω2α+ω2βh2(S6)whereω1=?2γwsinθandω2=?2γwcosθForthecaseθ=108°,weobtain:?????2??=?(+)≈?0.1062,?==25nm4???2?2??????1=2??sin?=1.41137??,?2=?2??????=0.45856??Andhence,1F(h)=?1.46+11.464h2AsshowninFig.S1a,thecomponentofforcepointingawayfromtheconfinedvolumeispositive(wall-wallrepulsiveforce)upto?′1=2.80nmanditbecomesnegative(wall-wallattractiveforce)for?>?′1.Therefore,themaximumenergythatcanbestoreinthiscaseisbystretchingthewatercapillarybridgefrom?′1=2.80nmto?2=8nm(weassumethatthebridgedoesnotbecomeunstablefor,atleast,?=8??).Asdoneincase(a),thepotentialenergystoredwhenthewallsaremovedapartfrom?′=12.80nmto?2=8nmis?211?(PE)=PE(?2)?PE(?′1)=?∫?(?)??=?(ω1+ω2α)(?2??′1)?ω2β(?)?′1?2?′1Therefore,weobtain?(PE)=4.93?????.Theenergydensityforthesewallsis:4.93?????ρ==2348??/?3?,10815?10?14??3c)?=??°S7

7Wefollowthesameprocedureofcase(b).Asketchofthewatercapillarybridgeformedbetweenhydrophilicwalls(e.g.,watercontactangleθ=40°)isincludedinFig.S3.FigureS3.Sketchofacapillarybridgebetweentwohomogeneoushydrophilicwalls.hItfollowsfromFig.S3that|R2|=.However,inthecaseofhydrophiliccapillarybridges,2cosθthecommonconventionistodefinetheradiusofcurvaturetobenegative.Thus,hR2=?<02cosθFollowingaproceduresimilartothatshownforcase(b),weobtainfromFig.S3thattheareaofthecapillarybridgecrosssectionisgivenby2?1S=hb?2(πR2+hR2sinθ)(S7)2π2where?=∠AOB=π?2θ.Eqn.(S7)canberewrittenas??2sinθcosθ2S=hb?h(S8)4cos2θCombiningthisexpressionwiththeexpressionforthecapillarybridgevolume,?(?)=?0=Sw,weobtainthefollowingexpressionforb,?0??2sinθcosθ1b(h)=+?=αh+β(S9)wh4cos2θh??2sinθcosθ?0whereα=andβ=.4cos2θwUsingEqns.(S1)and(S9),wefindthatthecomponentoftheforcealongthedirectionpointingawayfromtheconfinedvolumeisgivenbyS8

8bcosθb1F=?2γw(sinθ+)=ω1+ω2=ω1+ω2α+ω2β2hhhwhereω1=?2γwsinθandω2=?2γwcosθ.Forthecaseθ=40°,theseparametersare:??2sinθcosθ?0?=≈0.324?==25nm4cos2θ??1=?2??sin?=?0.953897???2=?2??????=?1.1368??1Therefore,F=?1.32?28.42h2AsshowninFig.S1a,thecomponentofforcepointingawayfromtheconfinedvolumeisalwaysnegativeandhence,thewallseffectivelyattracteachother.Therefore,thepotentialenergycanbestoredonlybystretchingthecapillarybridge(weassumethatthebridgedoesnotbecomeunstablefor,atleast,at?=8??)Asdoneincase(a),thepotentialenergydifferencewhenthewallsaremovedapartfrom?1=2.67nmto?2=8.0nmis:?211?(PE)=PE(?2)?PE(?1)=?∫?(?)??=?(ω1+ω2α)(?2??1)?ω2β(?)?1?2?1i.e,?(PE)=14.14?????.Andhence,theenergydensityforthesewallsis:14.14?????ρ==6733??/?3?,4015?10?14??3S9

9REFERENCES(1)Giovambattista,N.;Almeida,A.B.;Alencar,A.M.;Buldyrev,S.V.ValidationofCapillarityTheoryattheNanometerScalebyAtomisticComputerSimulationsofWaterDropletsandBridgesinContactwithHydrophobicandHydrophilicSurfaces.J.Phys.Chem.C2016,120,1597–1608.(2)Almeida,A.B.;Giovambattista,N.;Buldyrev,S.V.;Alencar,A.M.ValidationofCapillarityTheoryattheNanometerScale.II:StabilityandRuptureofWaterCapillaryBridgesinContactwithHydrophobicandHydrophilicSurfaces.J.Phys.Chem.C2018,122,1556-1569.S10