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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±0.05Å;?istheerrorofthetheoreticalprofilerelativetotheMDdata[1].Height[nm]2.53.03.54.04.55.0r0[nm]4.633.883.292.802.452.09R2[nm]5.005.0012.4-15.0-8.61-4.70ϵ0.250.1640.2250.3580.2810.317Height[nm]5.56.06.57.07.58.0r0[nm]1.821.591.391.221.060.94R2[nm]-4.26-4.23-4.29-4.49-4.67-4.98ϵ0.2760.3350.3270.3470.3940.398TableS2:Fittingparametersr0andR2ofthetheoreticalprofilesofthecapillarybridgesindicatedbydashedlinesinFig.2ofthemainmanuscript.Errorsinr0,R2are±0.05Å;?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
2ϵ0.0990.1210.2120.1380.1710.135Height[nm]5.56.06.57.07.58.0r0[nm]1.811.571.371.211.040.92R2[nm]-3.99-3.93-4.05-4.30-4.45-4.76ϵ0.1440.1500.1610.1530.1800.178S3
32.EnergyDensityStoredinCapillaryBridgesbetweenHomogeneousSurfacesInthemainmanuscript,wefindthatthemaximumpotentialenergystoredinawatercapillarybridgebetweenthepatchysurfacedstudiedis9.379nN∗nm.Thispotentialenergycorrespondstotheprocessofstretchingthewatercapillarybridgefromℎ=2.67nmtoℎ=8.0nm(maximumdistancereachedbeforethecapillarybridgebecomesunstableandbreaks).Ifweassumethattheminimumvolumeassociatedtoourcapillarybridgeis?0=15??∗10??∗14??thentheenergydensityofthesystemis9.379nN∗nmρ==4470kJ/m3?,????ℎ?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,thepotentialenergystoredwhenthewallsaremovedapartfromℎ1=2.67nmtoℎ2=8.0nmis:ℎ2∆(PE)=PE(ℎ2)−PE(ℎ1)=−∫?(ℎ)?ℎ=2γw(ℎ2−ℎ1)=7.9097nN∗nmℎ1Thecorrespondingenergydensityis7.9097nN∗nmρ==3767kJ/m3?,9015∗10∗14??3Fig.S1b,showsρ?,90asfunctionofℎ2=ℎ.FigureS1.(a)Componentoftheforceactingonthewalls(alongthedirectionpointingawayfromtheconfinedvolume,i.e.,wall-wallattractiveforcesarenegative)and(b)potentialenergystoredinthewatercapillarybridgesbetweenhomogenoussurfaceswithdifferentwatercontactanglesθ,asfunctionofthewallsseparationℎ(duringstretching).Forcomparison,alsoincludedaretheforceandpotentialenergyofthewatercapillarybridgesformedbetweenthepatchysurfacesconsideredinthemainmanuscript.b)?=???°Asketchofthewatercapillarybridgeformedbetweenhydrophobicwalls(e.g.,watercontactangleθ=108°)isincludedinFig.S2.S5
5FigureS2.Sketchofacapillarybridgebetweentwohomogeneoushydrophobicwalls.Inordertocalculatetheworkproducedbytheforceinducedbythewatercapillarybridge,weassumethatthevolumeofthecapillarybridgeisconstant,i.e.,independentofh.Thisimpliesthat?(ℎ)=?0=15??∗10??∗14??.Asweshowbelow,thisequationcanbeusedtoextract?=?(ℎ).BycombiningthisexpressionwithEqn.(S1),onecanalsoobtainanexpressionfortheforce?(ℎ).Theexpressionfor?(ℎ)followsfromFig.S2.Specifically,V(h)=wS,whereS=2S1+hbisthecrosssectionareaofthecapillarybridge(seeFig.S2);S1istheareaconfinedbythearcABandthedashedlineAB.ToobtainS1,wenotethatS1canalsobeexpressedasS1=A1-A2,whereA1istheareaoftheregionconfinedbythearcAB,thesegmentAO,andthesegmentBO.A2istheareaofthetriangledefinedbyAOB.Itcanbeshownthat2ϕ1ϕS=2(πR2−hR2cos)+hb(S2)2π22Whereϕ=∠AOB=2θ−π.Inaddition,onecaneasilyshowthathℎR2==−(S3)2cos(π−θ)2???θForhydrophobiccapillarybridges,theconventionisusuallytotakeR2>0,consistentwithEq.S3.UsingEqns.(S3)and(S2),oneobtainsϕϕ???22S=(+)h+hb(S4)4???2θ2???θSincethevolumeofthebridgeisV(h)=V0=wS,itfollowsfromEqn.(S4)thatS6
6ϕ?0ϕ???21b(h)=−(+)ℎ=αh+β(S5)wh4???2θ2???θhwherewedefinedtheparametersϕϕ???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)uptoℎ′1=2.80nmanditbecomesnegative(wall-wallattractiveforce)forℎ>ℎ′1.Therefore,themaximumenergythatcanbestoreinthiscaseisbystretchingthewatercapillarybridgefromℎ′1=2.80nmtoℎ2=8nm(weassumethatthebridgedoesnotbecomeunstablefor,atleast,ℎ=8??).Asdoneincase(a),thepotentialenergystoredwhenthewallsaremovedapartfromℎ′=12.80nmtoℎ2=8nmisℎ211∆(PE)=PE(ℎ2)−PE(ℎ′1)=−∫?(ℎ)?ℎ=−(ω1+ω2α)(ℎ2−ℎ′1)−ω2β(−)ℎ′1ℎ2ℎ′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.S3thattheareaofthecapillarybridgecrosssectionisgivenby2ϕ1S=hb−2(πR2+hR2sinθ)(S7)2π2whereϕ=∠AOB=π−2θ.Eqn.(S7)canberewrittenasϕ−2sinθcosθ2S=hb−h(S8)4cos2θCombiningthisexpressionwiththeexpressionforthecapillarybridgevolume,?(ℎ)=?0=Sw,weobtainthefollowingexpressionforb,?0ϕ−2sinθcosθ1b(h)=+ℎ=αh+β(S9)wh4cos2θhϕ−2sinθcosθ?0whereα=andβ=.4cos2θwUsingEqns.(S1)and(S9),wefindthatthecomponentoftheforcealongthedirectionpointingawayfromtheconfinedvolumeisgivenbyS8
8bcosθb1F=−2γw(sinθ+)=ω1+ω2=ω1+ω2α+ω2β2hhhwhereω1=−2γwsinθandω2=−2γwcosθ.Forthecaseθ=40°,theseparametersare:ϕ−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,atℎ=8??)Asdoneincase(a),thepotentialenergydifferencewhenthewallsaremovedapartfromℎ1=2.67nmtoℎ2=8.0nmis:ℎ211∆(PE)=PE(ℎ2)−PE(ℎ1)=−∫?(ℎ)?ℎ=−(ω1+ω2α)(ℎ2−ℎ1)−ω2β(−)ℎ1ℎ2ℎ1i.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