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Homogeneous nucleation of supercooled water: Results from a new equation of state Austin, Philip H.; Jeffery, Christopher A. Nov 20, 1997

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JOURNAL OF GEOPHYSICALRESEARCH,VOL. 102,NO. D21, PAGES25,269-25,279,NOVEMBER 20, 1997  Homogeneous nucleation of supercooled waterResults from a new equation of state C. A. Jeffery and P. H. Austin Atmospheric SciencesProgramme, Department of Geography,University of British Columbia Vancouver, British Columbia, Canada  Abstract. A series of laboratory and aircraft measurements have indicated that supercooledliquid water exists to temperatures as low as -70øC. These measurementsalso showthat classicalnucleationtheory, using standard valuesfor the thermodynamic propertiesof supercooledwater, underestimatesthe nucleation rate of ice in liquid water at large supercoolings.New theoretical estimatesfor this •homogeneous nucleation rate are presented, based on a new analytic equation of state for liquid water. The new equation of state, which is accurate over a pressure range of 3000 atmospheresand a temperature range of 1200 K, is used to infer the latent heat of melting, liquid water density, and ice-water surfaceenergy of supercooledwater. Predictionsof the nucleationrate and the homogeneous freezing temperature made by this equation of state are in agreementwith observations at temperatures as cold as -70øC and at pressuresas high as 2000 atmospheres. These results indicate that it is not necessaryto invoke a phasetransition at -45øC to explain aircraft and laboratory observationsof homogeneousice nucleation in supercooledwater clouds. 1. Introduction  Recently,Pruppacher[1995]usedthis ideato explain the failure of classicalnucleationtheory to predict obDuring the last 15 years several new measurements served nucleation rates at low temperatures. He exof the rate of homogeneousnucleation of ice in supertrapolated steepincreases in the density,heat capacity, cooled liquid water have become available. These data latent heat of evaporation, and viscosityand steepdeinclude nucleation rate measurementsin an expansion creases in the density, ice-water surfaceenergy,and latcloudchamberto -45øC [Hagenet al., 1981]and cirrus ent heat of melting consistent with a second-order phase observationsof supercooled water at -35 o --> -40øC transition at -45øC. He then used these new values [Sassenand Dodd, 1988; Heymsfieldand Miloshevich, 1993]. Nucleationrates reportedby theseinvestigators and the observed nucleation rate to derive the activaare substantially higher than those predicted by clas- tion energy required to diffuse water moleculesacross sical nucleation theory using standard estimates of the physicalproperties of water extrapolated to supercooled  temperatures[Pruppacher,1995]. The discrepancy between observed and calculated nucleation rates is one symptom of a larger problem: the absence of a physical model of liquid water that accounts for its unusual properties. These properties  the ice-water  interface.  Although Pruppacher'sapproachwas consistentwith most measurementsof water properties availableprior  to 1995, experimentsconductedby Xie et al. [1993], BartellandHuang[1994],andHuangandBartell [1995] havecast fundamentaldoubt on the idea of a stability limit for liquid water at atmospherictemperaturesand  includethe densitymaximumat 4øC (1 atmosphere) pressures. Xie et al. measured the absolute structure and the rapid changein the specificheat (cp) and factorof liquidwaterto -34øC andfoundno changein other thermodynamic variablesas liquid water is cooled the correlationlengthsof densityfluctuationsas T• was to T•=-45øC. Speedyand Angell [1976]and Speedy approached,in contrastto the strongfluctuationsto be [1982a,b] proposedthe "stabilitylimit conjecture"to expectedbecauseof a stabilitylimit. Huangand Bartell explain the apparently asymptotic behavior of the heat capacity, density, and latent heat, postulating a limit to the mechanical stability of liquid water at T• and atmospheric pressure. Copyright 1997 by the American Geophysical Union. Paper number 97JD02243.  0148-0227/97/97JD- 02243509.00  used electron  diffraction  measurements  of a beam of  water clusterscondensingin supersonicflow to establish the presenceof liquid water at T=-70øC and to infer  a nucleationrate at that temperature.Giventhe large viscosityof water at -70øC, it is possiblethat the liquid dropletsobserved in theseexperiments werenucleating at timescaleslonger than the microsecondobservation timesof the electrondiffractionmeasurement. However, recent computer simulationsindicate that equilibration  25,269  25,270  JEFFERY AND AUSTIN: HOMOGENEOUSNUCLEATION OF SUPERCOOLEDWATER  times are of the order of a few nanoseconds,making it  likely that the liquid phaseis, indeed,still mechanically stable at these very low temperatures[Tanaka, 1996; Speedy,1996]. In this paper we will use a new equationof state for water to derive homogeneous nucleationrates in the temperaturerange -70øC < T < -30øC andthe pressurerangeI • p • 2000 atmospheres.Our approach is similar in spirit to Pruppacher's[1995],in that we reexamine classical nucleation theory in light of new information that has recently becomeavailableon the  ares the molecular jump rate acrossthe interface. Alternative formulationsfor the prefactorchangethe nucleation rate by 1-2 ordersof magnitude at temperatures  above200 K [see,e.g.,HuangandBartell, 1995,Figure 7]. Evaluationof (1) requiresestimatesof the densities Pi and pw, the free energyof germformationAFg, the ac-  tivation energy Ag# , andtheinterface energy (ri/w.The energyof formationAFg, in turn, requiresthe surface  energy(ri/wandanestimateoftheequilibrium germra-  dius [Pruppacherand Klett, 1997, equation7-52, with propertiesof liquidwater. It differsfromhisin that an correction]: analytic equationof state, rather than extrapolations  4tr•øi•raa•  basedon the stability limit conjecture,will providethe  (2)  AFg- 3 ' temperatureand pressuredependence of the density, entropy,latent heat, and surfaceenergyof liquidwater. Determinationof ag is discussed in Section4; it also The activation energyfor diffusionacrossthe ice- requiresknowledgeof the liquid water densityp,oand water interface is also needed for the nucleation rate  the latent heat of mell•ing Lm.  and cannot be obtained directly from the equation of  We usestandardvaluesfor the ice densityPi [Prupstate;this quantitywill be estimatedindependently us- pacherand Klett, 1997,equation3-2] and in section3 ing recentmeasurements of the self-diffusion of water. derive a relation between Lm and the interface energy The resultingpressureand temperaturedependence of (ri/•o The densitypw canbe foundfroman equation the nucleation rate calculated from these parameters of stateof the form p,o= f(p,T), whichalsoyieldsLm  (anda derivedquantitycalledthehomogeneous freezing using Maxwell's relations: temperature)arein agreement with the laboratoryand aircraft measurementsmentioned above, including the new T--70øC nucleationresultsof Huang and Bartell  p -  -  , T  [1995]. We briefly review the classicalnucleationequation and presentthe equationof state in section2. In sections 3, 4, and 5 we estimatethe ice-waterinterfaceenergy,the energyof germformation,andthe activation  =  -  ,  =  energy.Weshowthetemperature andpressure depend- where p is the pressure, a is the specific Helmoltz free ence of the nucleation rate and homogeneous freezing energy, s is the specific entropy, c•o is the specific  temperaturecalculatedfrom theseenergies in section6 volume(1/pw), and the subscriptsw and i denoteliand summarize the results in section 7.  2. Equation of State  quid and ice, respectively. The entropy of ice si is assumed to be independent of pressure; it is determined from an analytic integration of the heat capacity of ice  [Pruppacherand Klett, 1997,equation3-12]:  The classicalrate of homogeneousnucleation of an  si = 1.885log(T) + 0.132T- 5.115  ice germin a liquid water dropletis givenby  (4)  whereT is in Kelvinsand si is in J kg- • K-•. The analytic equation of state used here is described  pih ,,I(tril•o•12 ) exp[-Ag#]} x j - {2Nc(P•økT'• exp[- kT ' (1)  in detail by Jeffery [1996]and will be treated in this  paper as an empirical relation. We include a brief outline of the derivation in Appendix A, with numerical values of the fitted parameters; in this section we show densitiesand heat capacities. The equation of state has where Nc is the number of monomersof water in contact the following form:  with unit area of the ice surface,pw is the liquid water density,pi is the densityof ice, T is the temperature  p- F (pw,T) - po+ 2pub, andNc canbeestimatedaccurately as5.85x 10TMcm-2 [Pruppacher and Klett, 1997,equation7-51] (seethe where F is a function discussedin Appendix A. The notationlist for a completelist of symbols).The term factorof 2 in (5) reflectsthe 2 molesof hydrogenbonds in (1) enclosed in braces(thenucleation prefactor) rep- in eachmoleof water. Equation(5) is a "mixturernodresents the diffusive molecular flux across the liquid-  el"; the hydrogen bonds act as a separate speciesof  solidinterface. Recently,Oxtobyand Harrowell[1992] water, exerting their own pressurePhb. The hydrogen havearguedthat the prefactorusedin (1) underestim- bond fraction is temperature dependent;its form is also  JEFFERY AND AUSTIN: HOMOGENEOUS  NUCLEATION  OF SUPERCOOLED  WATER  25,271  given in Appendix A. Above freezing the bond fraction falls to zero, and the only contribution to the pressure comesfrom the backgroundpressureterm p0. By con-  sideringseparatelythe effect of hydrogenbondson the  freeenergya and (through(3a)) the pressurep, (5)is able to accurately predict the thermodynamic properties of water at supercooledtemperatures. The mixture model describedhere was first employed  by Pooleet al. [1994],whousedthe Van der Waalsequation to calculate p0 and demonstratedqualitatively that  (5) wasableto producethe densitymaximumof water. We have introduced a much more accurate form for p0 and modified the pubformulation of Poole et al. so that its influence is limited to supercooledtemperatures. Figures la and lb showthe agreementbetweensteam  E •  •••  •'  / /  - - - o, /  -'..................  """•-  '•  "•'•  • •  o  • • q  •  •'  '•'•  ••  '•  _  ........ • ,,  •• -...  •  • .  table [Haar et al., 1984]and supercooled measurements [Have and Sorensen,1987] and the equationof state  I  I  I  I  I  I  I  I  in the pressure range 1 bar < p < 2800 bars and -40 -20 0 20 40 60 80 100 the temperature range -40 øC < T < 1000øC. The T (øC) equation predicts a density maximum at pressuresup to 800 bars, as open hydrogenbondsforce a local dens- Figure lb. As in Figure la but magnifiedfor the temity related to their perfect tetrahedral geometry. The peraturerange -40 øC • • • 100øC. bonds are broken at pressuresabove 800 atmospheres, removingthe density maximum. The displacementof  the densitymaximumto lower temperatureswith increasingpressureis in agreementwith measurements the predicted density from the steam table values. The averagerms percentagedensitydeviationwith this bin[Augell,1982,page27]. ning is 0.490, 11.06, 21.7 and 59.7 for the presentequaWe can comparethe steam table valuesfor the denstion of state, Song and Mason [1989], Peng-Robinson ity with the presentequationof state and three other [Melhem et al., 1989],and Van der Waals,respectively. cubicequations of stateby grouping(T, p, p•o triplets  in binsofwidth24 kg m-a in thetemperature andpres- Above supercooledtemperatures the new equation of surerange 0øC < T < 700øC,0.1 bar < p < 1200 bars state is approximately 20 times more accurate than andcalculatingthe averagermspercentage deviationof other equations of state of similar complexity. Adding  supercooleddata to the fit and extendingthe temperature and pressurerange to -34øC < T < 1200øC, and 0.1 bar < p < 3000 bars decreasesthe averagedensity deviation of the presentequation of state from 0.49% to 0.41%.  Figure 2 showsthe heat capacityderivedfrom (5) using(3a), (3b), and the Maxwellrelation:  cp-Tk,oT •,' •  •  --  0 0 • 0  1bar '  At a pressureof 1 bar the equationof state matchesthe  .......... 200 bars  400 bars - 600bars - ß 800 bars• k • • - 1200barsI• • • - - 1600bars • •%•  -  0 -  - -  2200 bars t  observedincreaseof cpasthe temperaturedecreases to  '•'•.• ' '• '••'•._  -45øC. The equation of state does not show a stability limit at supercooledtemperatures.Rather there is a continuouschangefrom the propertiesof supercooled water to those of ice. Not shown here are other water  propertiespredictedby the the equationof state, such as the observedanomalousincreasein cp at the critical point. The equation also predicts a phase trans0 200 400 600 800 1000 ition betweenhigh-densityand low-densityliquid water T (øC) above 1000 bars at supercooledtemperatures and a second critical point for liquid water. These high pres•J•a•e Za, Isobaricdensityof ]Jqu•dw•e•. CJ•c]e by JefferSt pointsare data from Haar et al. [1984]and Hare and sure/lowtemperatureresultsare discussed Sorensen [1987].Boxedregionisexpandedin Figurelb. [1996]. I  2800bars  I  I  I  I  25,272  JEFFERY AND AUSTIN: HOMOGENEOUS  NUCLEATION  OF SUPERCOOLED WATER  Table 1. The kT Values for Different Metalloids  /r-'!  1bar  / I / /"•  •  ..........200bars 400bars  I  -  Metal  k•,  Water  0.32  Antimony  600 bars  i•/'I!ii• 800 bars  Metal  k•,  Bismuth  0.302  0.33  Germanium  0.348  From Turnbull [1950].  [1963,equation13.95]andPruppacher andtflett [1978, equation5-47a]haveproducedtemperature-dependent estimates of rri/wbasedon scalingarguments usingthe latent heat of ice and water. Figure 3 showsthese ex-  trapolations of rri/wtogetherwith the existingexperimental  data  at 0øC.  We will assumea similar temperature dependencefor  (7) usingan empiricallydeterminedconstantk•:  _  I  I  180  200  I  220  I  240  I  260  I  -  (8)  I  280  300  T (K)  where pi is the density of ice, T is in degreesCelsius, and k• = 0.00009 J m -• K -x. The numerical value of k• is independentof temper-  Figure 2. Isobm:icheat,capacitypredictedby [he new ature and pressure;it is chosenusing(1) to yield the observedtemperature dependenceof the nucleationrate equation of' s[a[e at five pressures. Circle poin[s at 1 atmospherein Section6. Figure3 shows(8) evaluated at pressuresof 1 bar, where it has approximately the sameslopeas Pruppacherand tflett's [1978]extra3. Ice-Water Surface Energy polation, and at 550 bars, where the increasein The equation of state provides the density pw and entropy Swof liquid water over a wide range of temperatures and pressure. The evaluation of the nucleation  with pressurehasincreased rri/wat all temperatures.  -F  rate alsorequiresthe ice-watersurfaceenergyeri/was 1 bar  a function of temperature and pressure. We adopt a ..........  pressure-dependent empiricalrelationthat relateseri/w  550 bars  to the latent heat of melting L,•, which is predicted  Pruppacherand Klett [1978]  from the equationof state through(3c). The basisfor this relationis the observationof Turnbull[1950],who  Dufour andDefay [1963] ½h .....  noted that the solid-liquid interface energies erstof a number of metals and metalloids were closelyrelated to L,• through  (• ,•,kTLmps •/31•rl/3,  (7)  wherep• is the densityof the solidphasein molm-a L,• is in J mo1-1andNA is Avogadro's numberThe •  ß  coefficiefft kT is about 0.45 for metals and 0.32 for water  and severalmetalloids. A summary of Turnbull's results for materials with kT •0 0.32 is shown in Table 1. We will use k• - 0.32 below.  Equation (7) cannotbe derivedfrom thermodynamics, but it has consistently been predicted by various moleculartheoriesincluding, most recently,the inhomo-  I  -80  -60  I  -40  I  -20  I  0  T (øC) Figure 3. Thevalueeriiwat supercooled temperatures.  geneouscell model of Liu [1993]and the densityfunc- Valuesat 1 bar and 550 bars are generatedfrom (8) with kT -- 0.32 and k• - 0.00009 J m -2 K -x. Curve tional theoryof McMullen and Oxtoby[1988]. As it stands,(7) doesnot capturethe commonlyas- labeledPruppacherand Klett [1978]is calculatedfrom their equation 5-47a and curve labeled Dufour and Desumedtemperaturedependence of eri/wat atmospheric fay [1963]is from their equation13.95. Squareis from pressures. Equation(7) produces a eri/westimatethat Ketcham and Hobbs [1969],circle is from Woodand decreasessignificantly faster than typical estimatesof Walton[1970],trianglefrom Corielet al. [1971],plus the water-air surface energy or the ice-water surface signis from Jones[1973],crossis from Hardy [1977] energy[Florianoand Angell, 1990]. Dufour and Delay and diamondis from Jones[1973].  JEFFERY AND AUSTIN: HOMOGENEOUS  No measurements are available to directly test the  NUCLEATION  OF SUPERCOOLED WATER  25,273  [Krynickiet al., 1980].ThusAg# is equatedto either  pressuredependence of (8). It can, however,be tested ED [Pruppacher,1972]or to E,[Dufour and Delay, indirectly via predictionsof the homogeneousfreezing 1963].We will determine Ag# fromED(T) usingthe temperature. In section 6 we present homogeneous supercooledself-diffusivity measurementsof Prielmeier were made on bulk freezingtemperaturepredictions madeusing(8) at pres- et al. [1988]. These measurements samplesof supercooleddroplets at pressuresas high as suresas high as 2000 atmospheres. 3000  bars.  There are two different procedures used to calculate activation energiesfrom self-diffusivity measurements; this has led to large discrepanciesin the estimation of  4. Energy of Formation  The freeenergyof germformation(AFg) definedby (2) is the energyrequiredto constructan ice germ of Ag#. The most commonprocedureis to definethe critical radiusag. The criticalradiususedin (2) is effectiveactivationenergyas [Pruppacher,1972]: the equilibrium germ radius found by integrating the equilibrium condition for an ice germ in a supercooled droplet of pure water, which is itself in equilibrium with  Ag# - ED ---Rd(lnD)/d(T-x).  (13)  This relationshipcan be deriveddirectly from (11) humidair. [Pruppacherand Klett, 1997,equations6-52 or from thermodynamicarguments[Wang, 1952] and and 6-10]: has the advantageof removing the unknown coefficient  Do fromthe estimateof Ag#. The implicitassumption in either the algebraic or thermodynamic derivation of  + 2(Pl-w pi x,ad _  T  Pi  •  ai  Here ai is the ice germ radius, and we have includedin  (13) is that the temperaturedependence of ED(T)is weak comparedto the 1IT dependence of the exponential. Although this conditionis satisfiedabove 0øC, it fails in the supercooledregion where the temperature  dependence of Ag# is strong. An alternative is to separately estimate both D and  (9) a correctionto the latent heat of fusionLm for the Do and use (11) directly to determineED. The difpressuredifferencebetweenthe germ and the surround- fusivityD(T) is availablefrom Prielmeieret al. [1988], ing liquid due to germ curvature. The surfaceenergy who have fit their self-diffusivity measurements in the  •ri/•ois estimated fromthelatentheatusing(8).  temperature range 218 K < T < 450 K and the pres-  We integrate(9) fromthefreezingpoint(To,ai = oo) sure range i bar < p < 3000 bars using an empirical to an equilibrium temperature Te and equilibrium ra-  diusag givenvaluesof Lm from (3c), Pi from PruppacherandKlett [1997,equation 3-2]and•ri/•ofrom(8).  Vogel-Tamman-Fulcherequation:  D(T) - D, exp[-B/(T-  T,)].  We showin Appendix B that to a good approximation Using(11),Ag• is givenby (2) can be written as  16•ra•/•  Ag #- ED --RT [T-T, B _in(D•)] '  (14)  (15)  (10)  To estimate Do, we use a separate set of measure-  where the averagesL,•,c and p-[ are definedin Ap-  ments by Harris and Woolf [1980]between 0øC and 50øC in the same pressurerange. We assumethat at thesetemperaturesED (T) is roughlyconstantandwith  AFg-  ^  3[L,•,cPi in(To/Te)]2' ^  pendix B.  this assumption fit (11) to the HarrisandWoolfD(T) values. The fit returns both Do and the averageactivation energy ED. Table 2 summarizesthese results. Theice-wateractivation energyAg# isthefinalphys- The best fit averageactivationenergyED is within the ical quantity needed to evaluate the nucleation rate us- experimentalerror of ED(T = 25øC) determinedby ing(1). Therearenodirectmeasurements of Ag#;it is Woolf [1975]. typically inferred from two measurablequantities: the The activationenergypredictedby (15) at various viscosity ? or the self-diffusivityD. The defining rela- pressuresis shown in Figure 4, which shows a signitionshipsare [Glasstoneet al., 1941] ticantdecrease in g# with increasing pressure.Four  5. Estimating the Activation Energy  estimatesof A## at atmospheric pressureare shown D •-  Doexp(-ED(T)/RT) ?oexp(E,(T)/RT),  (11) in Figure 5. The solid line is a cubicfit [Pruppacher andKlett, 1978,equation7-50a]derivedusing(13) and (12) the self-diffusion data of Pruppacher[1972].The short-  where Do and 70 are approximatelyindependentof temperature. At temperatures near the melting point the  energiesin the Boltzman factor, ED and E,  dashed line labeled  "Corrected"  shows the estimate  of  Ag# obtainedwhenthe dataof Pruppacher [1972]is  agree insteadevaluatedusing(14) and (15). The long-dashed  withinexperimental errorandareabout4.6 Kcal mol-x linelabeled"Preilmeir[1988]"givesthe Ag# estimate  25,274  JEFFERY AND AUSTIN: HOMOGENEOUS  NUCLEATION  OF SUPERCOOLED  Table 2. Self-DiffusivityParametersfor Equation(15) p, bar D,, 10•ø m• s-• B, K -• T,, K Do, 10•ø m• s-•  WATER  C•, kJ mol-•  1  4.14  347  177  349  18.2  100  6.46  455  161  328  18.0  500  8.90  563  143  263  17.5  1000  10.1  622  133  210  16.9  1500  11.2  668  126  175  16.5  2000  8.93  614  131  157  16.3  D,, B and T, are from Prielmeier et al. [1988]. Do and ]•D are from a fit to the results of Harris and Woolf [1980].  using(14) and (15) appliedto the self-diffusion data of 6. Homogeneous Nucleation Rate Prielmeieret al. [1988].The dottedline labeled"PrupWe are now in a positionto evaluate(1) giventhe pacher [1995]"givesthe correctionproposedby Prup-  forL,•, eri/•o, andAg• calculated from(3c), pacher[1995]to theusualA#• estimate obtained using newvalues (8), and (15), respectively, with p•oand Swgivenby the  (13).  Ouranalysis indicates that neglecting thedED/d(T-1 term in the derivationof (13) introducesa substantial  errorin the estimation of ED (andhenceAg•) at supercooledtemperatures.Pruppacher's[1995]correction  equation of state.  Figure6: 0.83'5=4.15in Figure6 showsthe resulting nucleationrates, including the new nucleationmeasure-  ment of Huangand Bartell [1995]usingvery small(3 nm radius)waterdropletsfor the motherphase.These  to (13) (shownby the dottedline in Figure5) assumes droplets have an internal pressureof • 550 bars; nucthe formation of molecular clusters at low temperat-  ures, which can increasetransport of water molecules  (and hencelowerthe ice-wateractivationenergy)even as the activation energy for self-diffusionsharply in-  creases[Pruppacher,1995].Althoughit is possible that molecular clustersdo play a role in germ formation, the reanalysispresentedin this sectionsignificantlyreduces the motivation for introducing a speculativephysical  leation rates at this higher pressure are given by the  dottedline. To evaluatethe energyof formation(AFo) for thesesmall droplets,we use (B5) with ad = 3 nm  anderw/.fromHuangandBartell[1995]: er•o/a - 111.63- 0.13167T,  (16)  process into the formulationof Ag#. We will therefore use (15) and the usual assumptionof equivalence betweenAg• and ED in our analysisof the nucleation  •  Pruppacher and Klett [1978]  Corrected Prielmeir etal.[1988]  equation in section 6. .......  ••  5]  1bar  .......... 100bars ...... 500bars ..... 1000bars ..... 1500bars  • •, .,,. •  .....  2000  bars  i  -6O  -- -- L- ........ i  -60  i  -40  i  -20  '" i  0  i  -40  I  -20  I  0  T (øC) Figure 5. The valueAg• at supercooled temperatures. Line labelsare "Pruppacherand Klett [1978]": Pruppacher'sestimateusing (13) and measurements from  Pruppacher[1972];"Corrected":a reanalysisof Pruppacher[1972]using(14) and (15) insteadof (13); "PriFigure 4. The valueAg• at variouspressures. Val- elmeieret al. [1988]":valuesof Ag• adoptedhere;and uescalculatedfromequation(15) with parametervalues "Pruppacher [1995]",the correctionto Ag• usedby from Table 2. Pruppacher[1995].  T (øC)  JEFFERY AND AUSTIN: HOMOGENEOUS  NUCLEATION  OF SUPERCOOLED  WATER  25,275  melting temperature TM, the cooling rate %, and the droplet volume Vd, TH • T99.99is given by  /o J(T)dT9.21% oo.oo •  (17)  Figure7 showsa comparisonbetween(17) and observations in the pressurerange 0.1 bar < p < 2000 bars. The distance between opposing arrows indicates the range in TH produced by variations in droplet volume V• correspondingto radii between 0.1 y and 1000 ym. The nucleation rates capture the general downward trend in T• at pressuresbelow 1000 bars, although the rate of changeis underestimated.Included in Figure 7 are estimates of TH at 1500 and 2000 bars. These •  i  -70  i  -60  i  -50  i  -40  -30  T (øC)  are connectedby a dotted line on Figure 7, to emphasize their qualitative difference with the size-dependent TH values found at lower pressures. At pressuresof 1500 bars and 2000 bars the equation of state predicts a phase transition at temperatures of -73øC and  Figu,re 6. Comparisonof J at 1 bar and 550 bars. -93øC, respectively.The phasetransition(discussed in The data of Huang and Bartell [1995](diamonds)are detail by Jeffery[1996])betweenhigh-density and lowat 550 bar becauseof the very small size of the droplets.  density liquid water rapidly forces droplet nucleation  The other data (listed below) are at 1 bar. Squares are from Pruppacher[1995],circlesare from Woodand independent of droplet size. Walton [1970],trianglesare from Butorin and $kripov [1972],plus signsare from Demott and Rogers[1990], 7. Summary and Conclusions and crossesare from Hagenet al. [1981].  We have used a new analytic equation of state to  investigate the behavior of the classicalhomogeneous  whereT is in KelvinsandCrwla is in mNm-•. The equilibrium melting temperature, TM was determined  from Wagneret al. [1994];TM is 0 at 1 bar, is-0.8 at 100 bars, is-4.1 at 500 bars, is-8.8 at 1000 bars, is -14.3 at 1500 bars, and is -21.0 at 2000 bars. The ac-  D=0.1 -->1000 !.Lm HDW  tivationenergyA#½wascalculated at 500 bars,which  --> LDW  introducesnegligibleerror. Figure 6 showsgood agreementbetweenobservations at i atmosphere and the theoretical nucleation curve. The data and the theory are not completely independent, however,becausethe surfaceenergyslopeconstant  (ko in (8)) has been selectedto producethe best fit to measurements at i atmosphere and temperatures  warmer than -45øC. The Huang and Bartell [1995] data point at -70øC and 550 bars providesone inde-  pendenttest of (8). Figure6 alsoshowsthe effectof the  strongpressure dependence in A#• on the nucleation rate. At one atmospherethe sharply increasing activation energy forces a decreasein J below T= -60 øC. At 550 atmospheresboth the magnitude and the rate of  I  0.0  I  0.5  I  I  1.0  1.5  I  2.0  I  2.5  increaseof the activationenergyare reduced(seeFig-  ure4), andthe decreasing eri/wincreases the nucleation rate as the temperature is reducedbelow T= -60øC. Additional tests are availablefrom pressure-dependent measurementsof the homogeneous freezingtemperature TH, which can be well-approximatedas the temperature at which 99.99% of a population of droplets of volume Vd will freeze homogeneously.Pruppacher and Klett  p (kbar) Figure 7. Effect of droplet diameter on TH at various pressures. The very short lines at 1500 and 2000 bars are a result of a HDW -+ LDW phase transition. The arrows positioned at the ends of the predicted TH values have been added for emphasis. Squaresare from Kanno  et al. [1975],and diamondsare from Xans andBarnaud [1997,equation7-72] showthat giventhe equilibrium [1975].  25,276  JEFFERY AND AUSTIN: HOMOGENEOUS NUCLEATION  nucleation equation at supercooled temperatures and a range of pressures. The eight fitted coefficientslisted in Appendix A are constrained by several thousand measurementsof the temperature, pressure,density, and entropy of liquid water in the temperature  OF SUPERCOOLED WATER  p - po+ 2pHiS. (A1) For p0 we usea Van der Waals-typeequationthat is a modifiedversionof the SongandMason[1990a,b] equation of state'  range -34øC < T < 1200øC and the pressurerange 1 bar < p < 3000 bars. An additional constraint Po = 1 + bvw RT p•o (A2) pwRT lies in the derivation of the equation of state, which is basedon observationsabout the role playedby strong hydrogenbonds in determining the propertiesof water [Jeffery, 1996; Jeffery and Austin, 1997,manuscriptin Where b(T) is relatedto the excludedvolumeand has preparation]. This foundationin a physicalmodelfor the followingfunctional form: water substantially reduces the parameter space to be searchedfor the best nonlinear least squaresfit to the  -3'a,o,P•o [1-Ab(T)p•o-1 1 ].  b(T)/vB = 0.3exp(-12.15(T/TB + 0.1608) 3) - b•exp(1.066T/T•)+ b2. (A3)  data.  In this paper we have treated the equation of state as an empirical relationship and used it to infer the density and the latent heat of melting and throughthem the surfaceenergy and free energy of germ formation. We have alsopresentedan analysisof the activationenergy for self-diffusionof water, suggestingthat the temperature variation of the ice-water activation energyis less than is conventionally assumedat supercooledtemperatures. In particular, our results indicate that laboratory and aircraft observationsof homogeneousnucleation can be explained without the need to postulate abrupt changesin the activation energyat T • -45øC. The parameter set, inserted into the classical nucleation equation, accurately predicts the recently observed nucleation rate of liquid water at -70øC and atmosphericpressure,and the homogeneous freezingtemperature at high pressures.We have followedmany au-  The secondcomponentof the fitted equationis the  hydrogenbondterm suggested by Pooleet al. [1994]. We follow Poole et al. [1994]and write Helmholtzfree energy as a simple partition function. We assume a  combinationof strongbondswith energye = ½HBand entropyS = SHB and weak bondswith energye = 0 and entropy$ = S0. The numberof configurations of  eachbondtype is then n0 = exp(-So/R) and •HB = exp(--SHB/R), respectively. The total hydrogenbond free energyis assumedto be a mixtureof the two bondtypes[Jeffery,1996]:  AHB =  - f RT ln[no+ nil, exp(-eH,/RT)] -(1 - f)RT ln(n0+ nil,), (A4)  thors[e.g.,HuangandBartell, 1995;Pruppacher,1995] wherethe mixturefractionf is a functionof temperat-  in assumingthat classicalnucleationtheory appliesand ure and density. We separatethe temperatureand densitydependence that, for example, macroscopicparameters such as the so that f (T, Pw) - f* (Pw) f** (T), where solid-liquidsurfaceenergyersthavephysicalmeaningfor small water clustersaway from the equilibriumfreezing point. One example of an alternative approach,using an explicit partition function to calculate the free en-  1PHlS)/er] +C• 2+C•, (A5a) f*(p•o= exp [(p•o-  ergy of formation AFg, was developedby Eadie Eadie [1971]. Future laboratory and aircraft measurements of the nucleation rate at temperatures between -45øC and -70øC and of the homogeneousfreezingtemperature with a variety of cooling rates and droplet sizesat  high pressureswould provide additional detailed tests of both  classical  and statistical-mechanical  f** (T) er -  (A5c)  with T! - 273.15K and 0 < C• < 1. Using(3a), weobtainfor the pressure [Jeffery,1996]  _ p}c9-• a (-fRT[ln no+  PHB  = 2f'p• (P•o -PHB)RT(ln[no + O-2  In this appendix we give the functionalform of the equation of state. For a full discussionof the deriva-  tion, readersare referredto Jeffery [1996]. Programs (in Fortran77)that comparethe density,entropy,and heat capacity of the equationof state againstobservafrom the authors.  As discussed section2, the equationhasthe following  (A6)  nHBexp(--eHB /RT) --In no q-nHB])  Appendix A' Equation of State  basic form:  (A5b)  C2PHB,  nucleation  theory and of the predictions of the equation of state.  tions are available  exp[-O.18(T/Tj)S],  nHBexp(--eHB/RT)]- in[no+ nHB]), where  !  --  (1 + C•) exp[(p•o- PHB)/er] 2X (exp[(pw- PHB)/O.] 2 + C•)2  exp[-0.18(T/Tj)s].  (A7)  JEFFERY AND AUSTIN: HOMOGENEOUS  NUCLEATION  The entropys• can now be obtainedfrom (A1) using (34) and (3b). The determinationof the volume-  OF SUPERCOOLED  t=to t=to  (3b) is discussed in Jeffery[1996]. The following coefficientsare returned from the fit-  25,277  where  dependentfunction that resultsfrom the integration of  ting process(seethe notationlist for definitions):  WATER  (B3)  Note that becauseof the very weak temperature dependenceof pi, it can be separatedout of the average  l/(piT) and represented by a simpletemperatureaverPHB  0.8537g cm-3'  So  -66.156J mo1-1I!'-1 (•0- exp(-$0/R));  age fii.  (B2)gives the Substituting for•2(o•_]_• ai] in(2)using  corrected  SHB  -4.464J tool-1K -1 (•H, -- exp(--$H,/R));  C1  0.3985;  C2  0.15758;  aeos  2.140 vB;  bl  0.24273;  latent  heat'  L,•,c - L,• -ln(To/Te)L,•.  (B4)  Incorporating(B4)into (B2) andusing(2) gives(10) in section  AFt(•)-  16•'o'•/,• 3[Lm,c•iln(To/Te)]2  If we also remove the large droplet approximation and  .0298.  The followingconstantsare passedto the fitting program and are not modified: 1408.4 K;  include (ø-•,•)in (9),(10)becomes  AF• )_  16•a•/• , (B5) -  where  vs  4.1782x 10-• rn• mol-1;  avw  0.5542Pam 6 tool-2'  bvw  4.44 x 10-•;  w' - 2• /a(• _Pi) adpw  1/•w =  0.30532;  ½HB  4'  13.5 kJ mo1-1.  f,T• T=To1/p•dT 1/p• T.  •  (B6) (B7)  and p• is the density of water.  Wehave compared values forAFg (1)andAFg (2)(using  Appendix B' A Closed-Form Approximation to the Free Energy of  a• ----1 •um)with a numericalintegrationof (9) between -70øC • T • -30øC at atmospheric pressure. The rms deviations between the analytic and numeric val-  Formation  The derivationof the criticalradiusag of an icegerm suspendedin a drop of pure water of radius a•, begins  uesare6.72x 10-21 J forAFg (1)and4.41x 10-21 J forAFg (2).Atthese temperatures AFgvaries from0to  with (9), whichrelatesthe differentialsof T, a, a• and ai at thermodynamicequilibrium. Integrationof (9) from equilibriumat the freezingpoint (To, ai -- c•) to (ag, Te) providesthe equilibriumradius. In this ap-  3.5 X 10-19 J, so that the standarddeviationsare 34% of typicalvalues.Therefore(10) and(B5) havebeen  pendix we will do this first by neglectingthe curvature adjustment to the latent heat and the size depend-  stratesthat the effectof dropletcurvatureon AFg is  enceof the liquiddrop(i.e., piLm >> 2(o'i/w/ai)and d(o'i/w/ai)• d(o'w/a/ad) ), thenaddingtheseterms. With both approximationswe have the following from  usedherefor calculationsfor AFg. The smalldifference between the analytic and numeric results also demon-  negligiblefor typical droplet sizes.  Notation 0[•08 •w  -  dT-  T=To  --  ,=c• Ct,l,,,(To)Pi  -  ai  (B1)  InTo _ 2ai/___•_• Te Lm•i ai  •o  (B2)  specificvolumeof water (A2). energy of strong hydrogenbonds in water  O.  Integrating(B1) gives  temperature dependent function.  PHB Pi  (A4). viscosity(12). viscositycoefficient (12). constant(A2). hydrogenbonddensity(A5a). density of ice.  25,278 Pw  rri /w  VB  JEFFERY AND AUSTIN: HOMOGENEOUS  densityof water (1). f functionwidth definedby (A5c). ice-watersurfaceenergy(8). solid-liquidsurfaceenergy(7). water-airsurfaceenergy(16). Boylevolume(A3).  NUCLEATION  T99.99  equilibrium freezing temperature of a supercooled droplet. homogeneous freezingtemperature. equilibrium bulk melting temperature.  number of configurationsof strong hydroa  ad ai avw  AHB b  b•, ba B  molar free energy of open tetrahedral hy-  drogenbonds(A4). temperaturedependentfunction(A3). fitted constants in the expressionfor b  (A3). fitted constantdefinedby (14). Van der Waalsconstant(A2).  bulk freezing temperature of water. temperature at which 99.99% of a population droplets freezes.  number of configurationsof weak hydro-  genbonds(A4). specificHelmholtzfree energy(3b). the radiusof a water droplet(9). the radiusof an ice germ(9). Van der Waalsconstant(A2).  WATER  fitted constantdefinedby (14).  genbonds(A4). •HB  OF SUPERCOOLED  the temperatureof the anomaly. -45øC); dropletvolume(17). functiondefinedby (B6). Acknowledgments. We are grateful to Marcia Baker for introducingus to Poole et al. [1994],to Jon Nelsonfor pointingout the homogeneous freezingmeasurements reported by MacKenzie[1977],to Marcia and BirgerBergersen for many helpful discussions,to two anonymousreviewers whoose comments improved the manuscript. This work is supported by grants from the Atmospheric Environment Serviceand the National Scienceand EngineeringResearch Council  of Canada.  specific heat capacity. Cx, C• D  Do D,  E v, ED  fitted constants intheexpressions forf(T,proReferences (A5a). self-diffusivity(11). self-diffusivity coefficient(11). a fitted constantdefinedby (14). activation energiesfor viscosityand self-  diffusion(11)and (12). fraction of hydrogen bonds that are capable of forming strong bonds.  f** (v)  functiondefinedby (A5a). functiondefinedby (A5b). free energy of formation of an ice germ  (2). ice-wateractivationenergy(13). h  Planck's  J  homogeneous nucleationrate (1).  k  Boltzmann's  ko  constant. constant.  0.00009J m- • K- • slopeconstantin the  (ri/•oexpression (8). kT  0.32,an estimatedconstantdefinedin (7). latent heat of melting(3c). averagelatent (B3). correctedlatent heat includingcurvature.  NA Nc  po  pressure.  background pressurefrom the new equa-  tion of state (A2). PHB  pressureexerted by stronghydrogenbonds  (A6). 8i • 8w  R  So T  Plenum, New York, 1982.  Angell, C. A., M. Oguni, and W. J. 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