Heat transfer from gold nanoparticles to the surrounding medium

HuandHartland[123]investigatedthecoolingdynamicsofAu NPswithvariousdiameters dispersedinwater byfemtosecond pump-probespectroscopy.Fromtheirexperiment,theyconcluded thattheprobe signalversustime profilecouldbeconveniently fittedbyapplyingastretchedexponentialfunction,givenby

F (t)=A·exp

Here isthecoolingtimeconstant,tisthedelaytime,andˇ isthestretching parameter.Byfitting theexperimentalresults, theyfoundanalmostquadraticdependenceofthecoolingtimes onthediameteroftheparticles,whereasthestretching parame-terremainedpracticallyconstantat0.7.Fig.16showstheAuNP size-dependentcoolingtimeconstants.

HuandHartlandgaveanempiricalequationforcalculatingthe NPtemperatureatlowtemperatures;

whereT_iistheinitialtemperatureatthermalequilibriumandT₀is thetemperatureofthesurroundingwater.

Subsequently,Plechet al.[89] measuredthecooling dynam-ics by observing the lattice expansion and contraction of Au NPs dispersed in water. They analytically solved a set of two differentialheat diffusionequations intheLaplace domainand additionally introduced a thermal heat conductance, h, at the NP–water interface. By fitting their analytical results withthe experimentallyobtaineddata,theyestimatedaheatconductance

Fig.16.Characteristictimeconstantforenergydissipationdeterminedusing equa-tion(12)versusdiameter:()experimentaldata;(–)calculatedtemperatureversus timeprofiles.Thedashedlineshowsafittothedataassumingaparabolic depen-denceof ondiameter(i.e., ∝R²).(Reproducedfromref.[123].Reproducedwith permissionfromtheAmericanChemicalSociety.)

of(105±15)MWm⁻²K⁻¹.It wasconsideredthat theNP–water interfacebuildsalayerofafewnanometerssurroundingtheNP surfacewithanaveragetemperaturedefinedbyTm(R).Forasphere withradiusR,thesurfaceareaA_P,andthevolumeV_P,aredefined asA_P=4R² and V_P=4/3R³.Thustheheattransfertermatthe NP–surface–waterboundary,equation(9),reducesto

F= 3h

R (T_L)[TL−Tm(R)] . (14)

Equation(14)showsthattheheatlossthroughtheNP–water interfaceisproportionaltothetemperaturegradientbetweenthe latticeandwater.Furthermore,FisinverselyproportionaltotheNP radius,whichconsequentlyleadstoanincreaseintheheattransfer byreducing theNPsize.Inotherwords,a strongthermal non-equilibriumexistsbetweenTLandTm,andsmallerNPshavegreater heatloss.DependingonthedistancefromtheNP–waterinterface, thesurroundingwatermediumgivesatemperaturedistribution accordingtoequations(14)and(7c)(or(11b)).Fig.17showsan exampleofthecalculatedspatialwatertemperaturecurvesfora 55-nmAuNPexcitedwithdifferentpulsedurations,5nsand150fs, atthreedifferenttimedelays.

InspectionofFig.17revealsthatthenanosecond-laserheatingof AuNPsleadstoagreaterheatdissipationtothesurroundingwater thanthefemtosecond-laserheating.Thelatticeandwater tempera-turesattheNPsurface–waterinterfaceareinquasi-equilibriumfor thenanosecondcase.Incontrast,inthefemtosecond-laserheating, theeffectofheatconductanceisclearlyvisible.Atashortdelayof 100ps,adiscontinuousdecreaseisobservableinthetemperature acrosstheNP–waterinterfacebecauseoftheeffectofthethermal boundaryconductanceh:athermalnon-equilibriumofTLandTm

isestablished.Byincreasingthedelaytime,thedifferencebetween T_LandTmisreducedandvanishesafter600ps.Atthistimedelay, athermalequilibriumisattainedcausedbythecoolingoftheNP latticetemperatureoccurringsimultaneouslywiththeheatingof thesurroundingwater.Thus,theresultsshowthatthefinite ther-malinterfaceconductanceis animportantfactorfor controlling thecoolingofmetalNPsincondensedmedia.Moleculardynamic simulationsoftheheattransferofAuNPsindifferentliquidshave showntemperaturedistributionssimilartothosegiveninFig.17 [124].TheheatconductanceofAgNPsembeddedinaglassmatrix hasbeenestimatedbyJuveetal.[125].

An important consequence of the heat transfer to the sur-rounding liquid of laser-heated metal NPs is the formation of

Fig.17.Calculatedtemperatureprofilesofwatersurroundinga55-nm-diameterAuNPexcitedwithalaserpowerdensityof2.5mJcm⁻²at(a)355nm(pulseduration:5ns) and(b)400nm(pulseduration:150fs)asafunctionoftheradialdistancerfromtheparticlecenter.Thetimedelaysare(a)7.5ns(blackcurve),10ns(redcurve),and15ns (bluecurve);(b)100ps(blackcurve),200ps(redcurve),and600ps(bluecurve).Thestartingpointofthesimulationwas−2 PfromthemaximumoftheGaussiantime profile.Thetemperatureinsidetheparticleisassumedtobeuniform.Temperatures,TcpandTcav,representthetemperaturesofwateratthecriticalpoint(647K)andthe bubbleformationthreshold(573K),whichconsidersthespinodaleffectofwater.

vapornanobubbles.Plechandco-workers[42,126]experimentally investigatedtheformationofvaporbubblesbyprobingthe envi-ronmentalwaterand itsstructuralchangewithX-raysafterthe explosive phase transformation. They estimated that thevapor nanobubblewithaninitialhighpressureexpandsuptoamaximum radius,wheretheinnerpressureisreducedtoanequivalentvalue totheouterwaterpressure,andthenthebubblestartstocollapse.

Themaximumradiusgreatlydependsonthelaserpower.Afterthe firstcycle,thesurroundingliquidsareheatedupagaintothecritical temperaturethroughheattransferfromtheNPsurfaceandanother bubbleformationoccurs.Thisprocessisrepeatedinasequential mannerwithadecreaseinthemaximumbubbleradiiuntilthe metalNPis cooleddownfarbelowthecavitationtemperature.

AnexampleofmeasuredbubbledynamicsbyX-raypump-probe spectroscopytogetherwiththecalculatedbubbleradiiandinner pressuresbyapplyingtheRayleigh–Plessetequationisgiven in Fig.18.

Herethemaximumbubble radiusisreachedafter250ps,at whichtimethepressurewithinthebubbledecreasedtoan equiv-alent value of the outermost water pressure (1atm). The first maximuminpressureat650psdenotesthecollapseofthefirst bubble.Thefollowingcalculatedmodulationsinpressureandsize areonlyexpectedforoscillatorybubblemotion.

Subsequently,Plechandco-workers[43]conductedthe excita-tionofAuNPsdispersedinwaterbyafemtosecond-pulsed-laser andobservedtheirlatticecontractionduringthecoolingprocess.

At a certain laser power density and delay time, the NP lat-ticeremainedconstantwithoutundergoingcontraction,whereno

Fig.18.Bubbleradius(solidcurve)andpressure(dashedcurve)transientsofthe watervaporinsidethebubbleascalculatedfromtheRayleigh–Plessetequation togetherwiththemeasuredradii.(Reprintedwithpermissionfromref.[42]. Copy-right2005,AmericanInstituteofPhysics.)

efficientheat transferoccurred. They assigned this laserpower density as the threshold power density of the formation of a nanobubblethatthermallyinsulatestheAuNPs.Fig.19givesthe relativelatticeexpansiondependentonthelaserpowerdensityfor twoAuNPsizes.

Furthermore,theyestimatedabubbleformationtemperature thatisslightlydependentontheAuNPsizebutisapproximately equaltothecriticalpointofwater(Tcp=647K).

More recently, Plech’s group [127] conducted nanosecond-time-resolvedspectroscopy experimentson60-nmand 80-nm-diameter Au NPs dispersed in water to observe the bubble dynamics. They excited the suspension by single 10-ns pulses at 532nm or 355nm to measure the extinction signals with

Fig.19. LatticeexpansionofAuNPsof52-and94-nmdiametersasdeterminedfrom thepeakshiftofthe(111)powderreflectionat100ps(circles)and1ns(crosses) delaystogetherwithacalculationofthethermalexpansion(lines;dashedline with-outrescaling).Above155Jm⁻²(15.5mJcm⁻²),nopowderreflectionatthe100ps delayisdetectable,indicatingparticlemelting.(Reprintedwithpermissionfromref.

[43].Copyright2006,AmericanInstituteofPhysics.)

Fig.20.Transient-extinctionchangeofan80-nmAuNPsuspensionbyfourcollinearmulticolorlaserprobebeamsexcitedwithasingle10-nslaserpulseat532nmata powerdensityof(A)300Jm⁻²(30mJcm⁻²)and(B)450Jm⁻²(45mJcm⁻²).(Reproducedfromref.[127].ReproducedwithpermissionfromIOPPublishing.)

nanosecond-timeresolutionatseveralwavelengths.Theyobserved anoscillatingsignalofoneperiodfromanegativesignalturningto apositiveone,whichisgiveninFig.20.

Plech’sgroupdiscussedthedetectedextinctionsignalsonthe bases of the calculated Mie extinction spectra of a Au NP and expandingvaporbubbleinwateratdifferenttemperatures.They concludedthattheextinctionsignalatthethresholdlaserpower densityofbubbleformationfirstdecreases,becauseofareduced extinction ofAu NPscaused bya refractive-index changefrom watertovapor(from1.33to1,seeFig.9)andanegligiblysmall extinctionsignalof smallvaporbubbles.Atlongertime delays, thevaporbubblesgrow,leadingtoanextinction(scattering)signal higherthanthatoftheAuNPextinctionatambienttemperature.

The right side of Fig. 21 shows an extinction change

∈(˚)/∈(0)|_t=5nsasafunctionofthelaserpowerdensity.A calcula-tionoftheextinctionasafunctionofthebubblesizeisshownonthe left.Theinterpretationassumesthattheextinctionfirstdecreases onincreasingthepowerdensity(increasingmaximumbubblesize) becauseofthedecouplingoftheLSPRfromthedielectric surround-ing.Atalargerlaserpowerdensity,thebubbleincreasesinsizeand enhancesscatteringasapositivecontributiontotheextinction.It shouldbementionedthatPlechandco-workersappliedratherhigh laserpowerdensities,wellabovethesize-reductionthresholdofAu NPsdispersedinwater[90,93].Sizereductionandreshapingmay leadtoapermanentextinctionreductionandblueshiftofthe plas-monband,andthusshouldbeconsideredintheinterpretationof thesignalobservedhere.

Lapotko’s group [45,128–130] carried out scattering spec-troscopyonasingleAuNPplacedonaglasssubstratesurrounded by water. They excited the NP with a single laser pulse with bothnanosecond-andsubnanosecond-timescaledurations.They detected the time-dependent scattering and transmission sig-nalsusinga CW orpulsed-laserforanalyzinglight. Theyfound thatamuchhigherlaserpowerdensity(100mJcm⁻² fora 250-nm-diameter Au) at an excitation laser wavelength of 532nm is necessary for the threshold of bubble formation than that

expectedtheoreticallybyemployingasimple light-to-heat con-versionmodel.Atthethresholdofbubbleformation,aminimum lifetimeof∼9nswasobserved,whichwasindependentoftheAu NPdiameter.Thisdiscontinuity(thresholdbehavior)couldnotbe detectedbyreplacingtheAuNPswithabsorbingmolecules dis-solvedinwater.Inthiscase,theshortestbubblelifetimemeasured byaphotodetector(transmittance)signalatthevaporbubble gen-erationthresholdwas∼1.6ns,andthelifetimeincreasedgradually with increasing laser power densities with a very low thresh-old.Becauseofthemuchhigherbubblegenerationthresholdand discontinuityin theminimumlifetime, Lapotko’sgroup consid-eredthephenomenonanewcomplexnanosystem,i.e.,plasmonic nanobubble(PNB).Itshouldbepointedoutthattheydidnot con-siderthepossibleeffectsofglasssubstrateoncoolingandbubble nucleation.Fig.22showsthelaserpowerdensitydependentbubble generationprobabilityandlifetimeofa90-nmAuNP.

Lapotko’sgroupdiscussedtheincreasedthresholdforbubble formationonthebasisofthecurvatureeffect.Accordingtothe Young–Laplaceequation,asmallercurvatureradiusofwateryields ahighervaporpressure.Thisadditionalpressurehastobe over-comebeforethebubble,whichisgeneratedbyalaser-heatedAuNP, canexpandintothesurroundingwater.Inaddition,theydetected increasedscatteringorextinctionsignalsbyincreasingthelaser powerdensity.Thus, thelifetimeandmaximum scatteringwas directlycorrelatedwiththebubblesize.

TheexperimentaldataofPlech[127]andLapotko[129] con-tradicttoeachother.Forinstance,onesignificantdifferenceisthe thresholdlaserpowerdensityofbubbleformation.Lapotko’sgroup postulatedmuchhigherlaserpowerdensitiesnecessaryfor bub-bleinitiationthanthetheoreticallyestimatedones.Ontheother hand,Plech’sexperimentaldataagreedwellwiththe thermophys-icalcalculationsandthespinodaleffectofwater.Also,Lapotkodid notobserveanyoscillation-likesignals.

NeumannandBrinkmann[44,131,132]conducted experimen-tal investigations on the nucleation dynamics of bubbles in waterbythe532-nmpulsed-laserexcitation(duration:12ns)of

Fig.21. Left:calculationoftheextinctioncrosssectionofanAuNPwitharadiusof30nm(60nmdiameter)asafunctionofasurroundingvaporbubbleofvariablesize.

Verticallinesdenotetheparticle–vaporinterface.Right:measuredextinctionchangeat532nmthrougha60-nm-diameterAuNPsuspensionintheflowingcapillaryata variablelaserpowerdensityof355-nmpulses.(Reproducedfromref.[127].ReproducedwithpermissionfromIOPPublishing.)

Fig.22.(A)Timeresponsesforsingle80-nmAuNPs(a–c)andamolecular solu-tionofTrypanBlue(d–f)obtainedwithasinglepump-laserpulse(verticaldashed line):(a)subthresholdfluencerepresentingmediumheating,(b)near-threshold powerdensityofbubbleformation,(c)over-thresholdpowerdensityshowing bub-bledynamics,(d)dyeabsorption,(e)bubbleformationpowerdensity,(f)bubble dynamics.Twoarrowsinfshowthedeviationoftheafterbubblesignalfromthe baselinecausedbyresidualheatingofthesurroundingwater.(B)Dependenceofthe PNBlifetime(TPNB,red)andthegenerationprobability(black)aroundsingle90-nm Auspheresonthepowerdensityofasinglepump-laserpulse(0.5ns,532nm); verti-callineseparatesthetypesofcorrespondingtimeresponses(A),andthehorizontal lineseparatestheopticalscatteringeffect.(Reproducedfromref.[129].Reproduced withpermissionfromtheAmericanChemicalSociety.)

microabsorbers, melanosome. They found a rather low bubble thresholdatatemperatureof409K.Theyexplainedthe observa-tionbythespinodaleffectofsuperheatedwater,wherethedensity ofwaterbecomesdramaticallylow(spinodaldensity)and explo-sivelyconvertsintovapor.In otherwords,theaveragedistance betweenthewatermoleculesreachesacriticaldistancebecause ofthedisruptionofhydrogenbondings.Furthermore,Neumann’s groupobtained thebubble radii and expansionvelocities from time-resolvedmicroscopydependentonthelaserpowerdensity.

Thevelocities rangedfrom10to85ms⁻¹ for laser power den-sitiesthat were1.5–8.5timesgreaterthanthethresholdpower density.Ontheotherhand,theshockwavevelocity,generatedby pulse-laser-heatedNPs,isapproximatelythevelocityofsound.

Themechanismsof laser-inducedbubbleformationsassisted bynano-/micro-absorbershavenotbeenfullyunderstood.Clear explanationsfortheexperimentalresultsstillareneeded.