Homogeneous Nucleation of H2O and D2O in Comparison: The

J. Phys. Chem. B 2001, 105, 11683-11701

11683

Homogeneous Nucleation of H2O and D2O in Comparison: The Isotope Effect† Judith Wo1 lk and Reinhard Strey* Institut fu¨ r Physikalische Chemie, UniVersita¨ t zu Ko¨ ln, Luxemburger Str. 116, D-50939 Ko¨ ln, Germany ReceiVed: April 26, 2001; In Final Form: June 25, 2001

In the past century, the homogeneous nucleation of light water (H2O) has repeatedly been studied using various experimental techniques. Generally, the onset of nucleation was recorded, while less frequently, the actual nucleation rates were determined. In contrast, the nucleation of heavy water (D2O) has been examined only in a single instance with no nucleation rates measured. Here, we report the first nucleation rate study of D2O along with nucleation rate measurements for H2O, which we repeated for comparison under identical conditions. We find that the nucleation rates for H2O and D2O differ by a factor of 2500, if compared at the same respective vapor pressure pv and temperature T, whereas the comparison at the same supersaturation S shows an agreement within experimental scatter. Also, the numbers of molecules in the critical clusters, which are determined from the slopes of the ln J versus ln S curves, are nearly the same for both isotopic waters. A satisfactory agreement with previous nucleation rate measurements of H2O made by Viisanen et al. (Viisanen, Y.; Strey, R.; Reiss, H. J. Chem. Phys. 1993, 99, 4680; 2000, 112, 8205) is observed, if the onset supersaturations S0 at nucleation rates of J0 ) 107 cm-3 s-1 are compared. Using the most recent expressions for temperature-dependent vapor pressures, we calculated surface tensions and densities predictions by the classical Becker-Do¨ring nucleation theory. Around T ) 240 K, the predictions quantitatively agree with the experimental data. However, as in the case of other systems (e.g., alcohols and alkanes), classical theory shows a stronger temperature dependence than experimentally observed. A temperature-dependent correction of the classical theory is developed which permits analytical calculation of nucleation rates as function of supersaturation and temperature over extended ranges.

I. Introduction Water is the most important liquid for life. Accordingly, considerable research efforts are undertaken to clarify the physicochemical properties of water. These are of interest for understanding nucleation and condensation of atmospherically relevant mixtures, e.g., in cloud formation. In nature, water is mainly encountered as H2O, but other isotopic waters, prominently D2O, are also found in seawater to a substantial fraction.1 For scientific purposes D2O has become an indispensable substance, e.g., in small-angle neutron scattering (SANS) or nuclear magnetic resonance (NMR) applications. Recently, the first successful experiments on detecting D2O condensation in supersonic nozzle flow by SANS have been conducted.2,3 The condensing material for contrast reasons had to be D2O. In nozzle flow, as well as in many other technological and environmental applications, droplet formation passes through nucleation, growth, and aging stages. The size distribution and polydispersity of the precipitate depend on the initial nucleation process. The validity of quantitative theoretical modeling of the nucleation processes has to be checked against experiments. Accordingly, there is considerable fundamental interest in experimental studies of homogeneous nucleation of D2O as a model system. Although H2O and D2O are electronically identical, the mass, the vibrational properties, the dipole moment, the moment of inertia, and other properties of the two isotopic waters differ somewhat. They express themselves also in slight but significant differences in the thermo-physical properties. This may be †

Part of the special issue “Howard Reiss Festschrift”. * To whom correspondence should be addressed. E-mail: [email protected]. FAX: 49 221 470 5104.

judged, if one recalls that, e.g., the critical temperatures are Tc(H2O) ) 647.15 K and Tc(D2O) ) 643.89 K, the boiling points are Tb(H2O) ) 373.15 K and Tb(D2O) ) 374.57 K, and the triple points are Tp(H2O) ) 273.16 K and Tp(D2O) ) 276.97 K.4 The anomalous density maximum occurs at Tmax(H2O) ) 3.984 °C versus Tmax(D2O) ) 11.185 °C.5 In quantitative examinations of nucleation, D2O was considered almost as early as H2O. Since these early investigations by Flood and Tronstad6 in 1935, the nucleation properties of D2O have apparently not been studied again. Due to further development of the measuring techniques, we are able to report here the first nucleation rate study of D2O. To see most sensitively which differences might exist, if compared to H2O, we repeated nucleation rate measurements for H2O, which had previously been performed by Viisanen et al.7 with the same chamber, and conducted them under conditions identical to those for D2O. The nucleation of water has repeatedly been examined in history. The first quantitative study dates back to Wilson,8 who over 100 years ago used an expansion chamber for generating the supersaturations necessary. Later, Volmer and Flood9 in 1934 and Sander and Damko¨hler10 in 1943 investigated water nucleation using different types of expansion chambers. In 1951 Wegener11 and co-workers used a shock tube and in 1969 a nozzle to study water nucleation. Also, Courtney12 in 1961 studied water nucleation. Katz and Ostermier13 in 1967 and Heist and Reiss14 in 1973 applied diffusion cloud chambers only at higher temperatures because of the comparatively high triple point of water. In all these works, the onset of nucleation, but no nucleation rates were determined. To obtain nucleation rates, it has proven useful to decouple nucleation and growth. This is most effectively done in so-called

10.1021/jp0115805 CCC: $20.00 © 2001 American Chemical Society Published on Web 09/07/2001

11684 J. Phys. Chem. B, Vol. 105, No. 47, 2001 “nucleation pulse chambers“. Kassner and co-workers15 developed in 1965 a single-piston expansion chamber for producing short pressure pulses. Using this device, Miller16,17 measured water nucleation rates in 1976. Starting with a well-defined saturated vapor, increasing supersaturations were achieved by increasing the expansion ratios and accordingly lowering the final temperatures. The droplets nucleated were photographed and countedsa rather time-consuming process. The pressure pulse was numerically integrated, and an effective time interval during which nucleation occurred was determined. With the sensitive time interval known, nucleation rates between 102 and 105 cm-3 s-1 could be determinedsfor the first time in the gas phase. The tedious counting procedure was circumvented in a new development by Wagner and Strey in 1981.18 On the basis of the nucleation pulse idea, short pressure pulses on the order of 1 ms were produced in a small chamber, much smaller in dimensions than those in Kassner’s original, by two sequentially released pistons. The real improvement was that the number density of droplets was measured instantaneously by constant angle Mie-scattering19 utilizing the characteristic light scattering patterns for growing droplets. Nucleation rates between 105 and 109 cm-3 s-1 for water were obtained. The way the chambers at that time operated only nonisothermal J-S curves could be measured. A detailed description of the two-piston expansion chamber was given in 1984.20 In 1986 the two-piston chamber was reconstructed into a twovalve expansion chamber, which is also used for the data presented in the present paper. A detailed description has been given by Strey et al.21 in 1994. The two-valve expansion chamber differs from the previous two-piston chamber by two essential features. First, the two pistons have been replaced by two valves connecting sequentially additional pressure-controlled volumes to the main chamber volume. By setting the respective pressures in the additional volumes, smooth pressure pulses with high reproducibility are obtained because the accelerated motion and sudden stops of the pistons are avoided. Second, vapors and carrier gas are premixed in a separate receptacle before the desired amount of the vapor/carrier gas mixture is transferred to the expansion chamber. Previously, saturation of the carrier gas with the vapor was performed inside the chamber and had to be awaited. Since the ratio of vapor to carrier gas pressure is known from preparing the mixture and kept constant for a series of fillings, one can set the initial vapor pressure by selecting the total pressure to any desired value, while the magnitude of the expansion controls the final temperature. In this fashion, isothermal J-S curves can be and have been measured for a series of unary, binary, and even ternary vapor/carrier gas mixtures. Homogeneous nucleation rates for water have been measured by Viisanen et al.7 in 1993. An important observation made in that paper was that the homogeneous nucleation process of water is not dependent on the nature of the carrier gas. Preliminary measurements performed in the context of the present study in 199822 lead to a reanalysis of the previous experimental procedure and a recalculation of the previous data from Viisanen et al.7 A few (minor) inaccuracies were disclosed as detailed in the erratum.7,23 Since then, further water nucleation rate curves have been measured when examining the nonideal mixtures of water-n-alcohols.24,25 One more academic reason to study H2O and D2O is that mixing the two isotopic waters a nearly ideal binary system can be obtained, useful for testing binary nucleation theories.26 In this paper, we confine ourselves to the pure unary systems H2O and D2O, both with argon as carrier gas.

Wo¨lk and Strey II. Theory The theoretical treatment of homogeneous nucleation dates back to 1925, when the classical theory was first formulated by Volmer and Weber.27 Farkas28 and Becker and Do¨ring29 perfected the description, making it the most successful model for quantitative prediction of nucleation phenomena. Comprehensive treatments of nucleation are found in the books edited by Zettelmoyer.30,31 Various theoretical developments have been reported in recent years.32-54 A useful review was provided by Oxtoby55 describing some advanced treatments which try to avoid some of the idealizing assumptions of the classical theory. Here the treatment by Girshick and Chiu40 and the latest calculations by Reiss et al.56 are particularly successful modifications. A. Nucleation Theory. All theoretical treatments arrive at an essentially common expression for the nucleation rate J

J ) K exp(-∆G*/kT)

(1)

In this expression, the kinetic prefactor K and ∆G*, the formation free energy of a critical cluster, depend on the theoretical approach. Some of the theories will be described shortly in the following. A.1. Becker and Do¨ ring. The first step in building a new phase is the so-called nucleation process. A nucleus is defined as a critical cluster in unstable equilibrium with the surrounding vapor. Gibbs57 assumed that the nucleus has the same properties as a macroscopic liquid drop. In the nucleation literature, this assumption is called the “capillarity approximation”. In classical nucleation theory, one calculates the radius of the cluster from the ratio of the vapor pressure pv of the nucleus and the equilibrium vapor pressure of the macroscopic phase pve

ln

pv 2σVm ) pve rkT

(2)

In this so-called Gibbs-Thomson or Kelvin equation, σ is the surface tension of the macroscopic fluid-vapor interface. The molecular volume Vm is calculated from the macroscopic density and the corresponding molecular mass m. r is the radius of the spherical liquid drop, while k is the Boltzmann constant and T the temperature. It is obvious that the classical theory has the important advantage that it allows the calculation of nucleation rates for arbitrary systems by only using macroscopic substance properties. The ratio of the vapor pressures

S)

pv pve

(3)

is called the supersaturation S. With

n*Vm )

4πr*3 3

(4)

where r* is the radius of the critical cluster and with the GibbsThomson equation, the number of particles in the critical cluster n* is readily found, and it follows that

n* )

32πVm2σ3 3(kT ln S)3

(5)

The work of nucleus formation or the Gibbs free energy is given by

∆G* ) -n*∆µ + 4πr*2σ

(6)

Homogeneous Nucleation of H2O and D2O

J. Phys. Chem. B, Vol. 105, No. 47, 2001 11685

where

∆µ ) µg - µl ) kT ln S

(7)

is the chemical potential difference of the molecules in the supersaturated vapor and in the liquid phase at the same pressure. The preexponential factor in eq 1 was first calculated by Becker and Do¨ring 1935.29 With eqs 1-7, the classical nucleation theory gives the following expression for the nucleation rate:

JBD )

() {

}

pv 2 -16πVm2σ3 2σ exp V πm m kT 3(kT)3(ln S)2

x

(8)

A.2. Girshick-Chiu. In the classical Becker-Do¨ring theory, the formation-free energy for the cluster of the size n ) 1 is nonzero. Girshick and Chiu40 attempted to achieve a remedy of this inconsistency by correcting the formation free energy by subtracting ∆G of the monomer. The result was that ∆G is zero in case of the monomer

∆G*GC ∆G*BD ) - (-ln S + Θ) kT kT with Θ )

(36π)1/3Vm2/3σ (9) kT

where Θ is the surface energy of a monomer multiplied with its surface area in kT. The nucleation rate is then given as

{

}

∆G*GC exp{Θ} ) JBD kT S

JGC ) K exp -

(10)

Importantly, the nucleation rates according to this so-called selfconsistent treatment have a different dependence on temperature than the nucleation rates calculated by the classical theory. A.3. Reiss-Kegel-Katz. One of the latest further developments based on the classical theory is the Reiss, Kegel, and Katz approach.56 Here the number density of the critical clusters, which is normally defined by a Boltzmann-type expression Nn ) N1 exp{-∆G*/kT}, is modified into

R Nn ) N1 exp{-∆G*/kT} S

(11)

where S is again the supersaturation and R is the so-called “replacement free energy”. With this definition, the nucleation rate is given by

JRKK )

(x

)

n*NAkTFκ m

-1

exp{Θ} JBD S

(12)

where κ is the isothermal compressibility. As can be seen very easily, the new theory developed by Reiss et al. differs just by another prefactor from the classical theory and is similar to the one by Girshick and Chiu. The prefactor is a slightly different function of temperature T and supersaturation S through n* in the square root. A quantitative comparison of the three theories shows that the newest development by Reiss, Kegel and Katz56 yields very minor deviations from Girshick and Chiu,40 whereas the classical theory differs on several orders of magnitude from both of them. B. Measurement of the Number of Molecules in Critical Clusters. It should be mentioned that theoretical work of Kashchiev58 in 1982 opened a route for obtaining additional, independent information from experiment. He proved that the

slope of isothermal J-S curves is roughly equal to the number of molecules n* in the critical cluster, an observation that had independently been made and used by a Russian group already in 1979,59 who, however, did not write down the explicit derivation. In any case

(∂∂ lnln SJ) = n*

(13)

T

More general arguments given by Kashchiev58 and an analysis based on the statistical mechanics of fluctuations60 by Viisanen et al.7 lead to the conclusion that eq 13 approximately yields the excess number ∆n* ) n* - nj of molecules in the critical cluster over that present in the same volume before the fluctuation occurred, rather than the total number n* of molecules itself. This result has been confirmed using a generalized thermodynamic approach by Oxtoby and Kashchiev.61 While this is theoretically correct, for the practical case of formation of liquid clusters in dilute vapors, ∆n* is to a very good approximation equal to n*. Thus, isothermal nucleation rate curves allow the determination of the number of molecules in the critical cluster n*. The interest in this observation is that a model-independent experimental check of the Gibbs-Thomson equation is possible. Needless to say, the direct measurement of the molecular content of the nuclei, obtained without recourse to any specific nucleation theory, is a useful additional information when formulating improved nucleation theories. For comparison with theories, one may use the nucleation rate J in eq 13, as most generally given by eq 1 and

( ) ( ∂ ln J ∂ ln S

)-

T

) ( )

∂(∆G*/kT) ∂ ln S

+

T

∂ ln K ∂ ln S

T

) n* + x

(14)

The difference x depends on the respective theory or rather on the kinetic prefactor of the theory considered. For example, in the classical theory, the (∂ ln K)/(∂ ln S) term or x is equal to 2, and in the self-consistent theory, it is equal to 1. Such differences are clearly within the uncertainty of our present experiments. III. Experiment As mentioned in the Introduction, in recent years, an advanced version of an expansion chamber for the study of homogeneous nucleation by means of the nucleation pulse method has been developed.21 In the course of the development, the in- and outletvalves of the chamber have also been modified, as can be seen in Figure 1. These new valves allow to close the chamber without compressing the gas inside the chamber volume, causing a smaller error in the pressure adjustment and reading. The other details of the chamber are as described in ref 21. Accordingly, only a brief summary is necessary here to describe the basics of the experiments and the actual nucleation rate data and measurables. A. Nucleation Pulse Measurements. The vapor and the carrier gas are premixed in a separate receptacle before the desired amount of the vapor/carrier gas mixture is transferred to the expansion chamber. A vaporizer separately thermostated to a temperature above the chamber temperature supplies sufficiently high vapor pressures. The vaporizer is partly filled with the liquid to be studied, dissolved gases being removed by vacuum pumps. Next, the vapor from the vaporizer is admitted to the receptacle and then filtered carrier gas (usually a noble gas, e.g., argon) is added in excess. The pressures of the gases admitted are monitored by pressure transducers (MKS Baratron). This procedure allows a precise setting of the vapor fraction of the vapor considered. The vapor and the carrier gas

11686 J. Phys. Chem. B, Vol. 105, No. 47, 2001

Wo¨lk and Strey

Figure 1. Axial section of the chamber, illustrating the light-scattering arrangement and the position of the piezo-electric pressure transducer.

TABLE 1: Thermo-Physical Parameters H2O 93.6635 + 0.009133T - 0.000275T2 0.08 tanh(x) + 0.7415tr0.33 + 0.32 x ) (T - 225)/46.2 exp(77.34491-7235.42465/T - 8.2 ln T + 0.0057113T) 10-11(a - bt + ct2 - dt3 + et4 - ft5) a ) 50.9804 b ) 0.374957 c ) 7.21324‚10-3

σH2O (mN/m)4,7 FH2O (g/cm3) pveH2O (Pa)66 κH2O (Pa-1)68

Tc(H2O) ) 647.15 K4

D2O 93.6635 + 0.009133T′ - 0.000275T′2 + 0.338 0.09 tanh(x) + 0.847t0.33 r x ) T - 231/51.5 ) pc exp{Tc/T(R1τ + R2τ1.9 + R3τ2 + R4τ5.5 + R5τ10)} R1 ) -7.81583 R2 ) 17.6012 R3 ) -18.1747 10-11(a - bt + ct2 - dt3 + et4 - ft5) a ) 53.5216 b ) 0.4536 c ) 8.7212 × 10-3

σD2O (mN/m)67 FD2O (g/cm3) pve D2O (Pa)64,65

κD2O (Pa-1)69

Tc(D2O) ) 643.89 K64 pc(D2O) ) 21.66 MPa64 T temperature in [K] t temperature in [°C]

are allowed to mix in the receptacle. The vapor fraction ω in the vapor-carrier gas mixture is fixed for a whole set of experiments. The expansion chamber is flushed with the vapor/ carrier gas mixture through the reconstructed in- and outletvalves (cf. Figure 1). The actual partial vapor pressure pv in the measuring chamber before expansion is determined by the initial total pressure p0 by

pv ) ωp0

(15)

The actual initial vapor pressure pv, and thus the supersaturation ratio achieved, is obtained by variation of the initial total pressure p0. The degree of expansion and recompression define a pressure pulse of about 1 ms. The recompression (usually between 2 and 6% of ∆pmax) is chosen to reduce the nucleation

tr ) (Tc - T)/Tc d ) 6.41785 × 10-5 e ) 0.343024 × 10-6 f ) 0.684212 × 10-9 T′ ) T‚1.022 tr ) (Tc - T)/Tc R4 ) -3.92488 R5 ) 4.19174 τ) 1 - T/Tc d ) 8.5541 × 10-5 e ) 5.4089 × 10-7 f ) 1.3478 × 10-9

rate by at least 2 orders of magnitude. The nuclei formed during the nucleation pulse can still grow and develop into droplets of micron size. The pressure in the expansion chamber is measured as a function of time during the pressure pulse by means of a fast piezo-electric pressure transducer. This pressure transducer (c.f Figure 1) is calibrated frequently. The expanded, supersaturated state, during which nucleation occurs, is maintained for a time interval ∆texp of about 1 ms. Pulses of variable depth and length can be produced. Due to the steep decrease of the nucleation rate with decreasing supersaturation, significant nucleation is only initiated during the pulse plateau, which corresponds to practically constant temperature and supersaturation. The difference ∆p ) p0 - p averaged over the pulse divided by p0 is given in column 2 of Table 2. The ∆p/p0 allow

Homogeneous Nucleation of H2O and D2O

J. Phys. Chem. B, Vol. 105, No. 47, 2001 11687

TABLE 2: Experimental Results of the Nucleation Measurements of Light and Heavy Watera IIa H2O, Argon 260 K, T0 ) 30 °C, ω j v ) 0.02641 p0

[∆p/p0]exp

[∆p/p0]

J

T

88.24 88.24 88.24 92.88 92.88

0.3230 0.3228 0.3230 0.3230 0.3232

0.3240 0.3238 0.3240 0.3240 0.3242

2.17 × 2.67 × 107 2.90 × 107 2.20 × 108 2.09 × 108

p0

[∆p/p0]exp

[∆p/p0]

J

0.3235 0.3235 0.3233 0.3239 0.3239 0.3243

5.67 × 6.20 × 108 6.30 × 108 1.32 × 108 1.34 × 108 1.24 × 108

107

259.84 259.87 259.84 259.84 259.81

S 7.19 7.18 7.19 7.57 7.58

p0 92.88 92.88 97.77 97.77 97.77

[∆p/p0]exp 0.3231 0.3231 0.3226 0.3224 0.3230

[∆p/p0]

J

T

S

0.3241 0.3241 0.3236 0.3234 0.3240

1.94 × 2.27 × 108 1.08 × 109 1.09 × 109 1.08 × 109

259.82 259.82 259.90 259.93 259.84

7.58 7.58 7.94 7.92 7.97

[∆p/p0]

J

0.3242 0.3238 0.3242 0.3234 0.3235

2.15 × 10 1.94 × 107 2.39 × 107 4.78 × 106 3.63 × 106

108

H2O, Argon 260 K, T0 ) 30 °C, ω j v ) 0.02646 97.77 97.77 97.77 92.88 92.88 92.88

0.3225 0.3225 0.3223 0.3229 0.3229 0.3233

T 108

259.92 259.92 259.95 259.86 259.86 259.79

S 7.94 7.94 7.93 7.58 7.58 7.61

p0 88.24 88.24 88.24 83.83 83.83

[∆p/p0]exp 0.3232 0.3228 0.3232 0.3224 0.3225

T

S

7

259.81 259.87 259.81 259.93 259.92

7.23 7.19 7.22 6.80 6.81

H2O, Argon 260 K, T0 ) 30 °C, ω j v ) 0.02623 p0

[∆p/p0]exp

[∆p/p0]

J

T

S

p0

[∆p/p0]exp

[∆p/p0]

J

T

S

97.77 97.77 97.77 97.77 92.88 92.88 92.88

0.3229 0.3225 0.3227 0.3225 0.3234 0.3230 0.3230

0.3239 0.3235 0.3237 0.3235 0.3244 0.3240 0.3240

2.85 × 108 2.89 × 108 2.75 × 108 2.62 × 108 5.53 × 107 5.30 × 107 5.74 × 107

259.85 259.91 259.88 259.91 259.77 259.83 259.83

7.92 7.87 7.90 7.87 7.56 7.52 7.52

92.88 88.24 88.24 88.24 88.24 83.83

0.3226 0.3223 0.3229 0.3229 0.3227 0.3223

0.3236 0.3233 0.3239 0.3239 0.3237 0.3233

4.12 × 107 7.55 × 106 9.04 × 106 9.54 × 106 1.15 × 107 1.98 × 106

259.90 259.94 259.85 259.85 259.88 259.94

7.49 7.08 7.14 7.14 7.13 6.74

H2O, Argon 260 K, T0 ) 30 °C, ω j v ) 0.02653 p0

[∆p/p0]exp

[∆p/p0]

J

T

S

p0

[∆p/p0]exp

[∆p/p0]

J

T

S

97.77 97.77 97.77 97.77 97.77 92.88

0.3225 0.3234 0.3235 0.3235 0.3231 0.3231

0.3235 0.3244 0.3245 0.3245 0.3241 0.3241

5.85 × 108 9.15 × 108 9.18 × 108 8.99 × 108 7.12 × 108 1.35 × 108

259.92 259.78 259.76 259.76 259.83 259.83

7.96 8.04 8.05 8.05 8.02 7.61

92.88 92.88 92.88 88.24 88.24

0.3229 0.3225 0.3232 0.3226 0.3224

0.3239 0.3235 0.3242 0.3236 0.3234

1.24 × 108 8.15 × 107 9.95 × 107 1.33 × 107 1.37 × 107

259.86 259.92 259.81 259.90 259.93

7.59 7.56 7.62 7.20 7.18

p0

[∆p/p0]exp

[∆p/p0]

J

T

S

p0

[∆p/p0]exp

[∆p/p0]

J

T

S

97.77 97.77 97.77 97.77 92.88

0.3227 0.3228 0.3228 0.3229 0.3226

0.3237 0.3238 0.3238 0.3239 0.3236

1.52 × 108 1.99 × 108 2.06 × 108 2.15 × 108 3.57 × 107

259.87 259.86 259.86 259.84 259.89

7.78 7.80 7.79 7.80 7.40

92.88 92.88 92.88 88.24

0.3233 0.3223 0.3228 0.3229

0.3243 0.3233 0.3238 0.3239

5.11 × 107 4.46 × 107 5.89 × 107 7.62 × 106

259.78 259.93 259.86 259.84

7.45 7.37 7.41 7.04

p0

[∆p/p0]exp

[∆p/p0]

J

[∆p/p0]

J

0.3244 0.3238 0.3237 0.3242 0.3239 0.3237

3.19 × 2.61 × 108 2.91 × 108 4.66 × 107 5.29 × 107 5.00 × 107

0.3235 0.3240 0.3238 0.3237 0.3237

7.95 × 8.21 × 106 6.70 × 106 2.11 × 106 1.75 × 106

H2O, Argon 260 K, T0 ) 30 °C, ω j v ) 0.02587

H2O, Argon 260 K, T0 ) 30 °C, ω j v ) 0.02640 97.77 97.77 97.77 92.88 92.88 92.88

0.3234 0.3228 0.3227 0.3232 0.3229 0.3227

T 108

259.78 259.87 259.88 259.81 259.85 259.88

S 8.01 7.95 7.94 7.58 7.56 7.54

p0 88.24 88.24 88.24 83.83 83.83

[∆p/p0]exp 0.3225 0.3230 0.3228 0.3227 0.3227

T

S

106

259.92 259.84 259.87 259.88 259.88

7.15 7.19 7.18 6.81 6.82

H2O, Argon 260 K, T0 ) 30 °C, ω j v ) 0.02624 p0

[∆p/p0]exp

[∆p/p0]

J

T

S

p0

[∆p/p0]exp

[∆p/p0]

J

T

S

92.88 92.88 92.88 92.88 92.88

0.3227 0.3225 0.3232 0.3225 0.3231

0.3237 0.3235 0.3242 0.3235 0.3241

2.12 × 107 2.83 × 107 2.68 × 107 2.62 × 107 3.20 × 107

259.88 259.91 259.80 259.91 259.82

7.50 7.49 7.54 7.48 7.53

88.24 88.24 88.24 88.24

0.3223 0.3224 0.3231 0.3222

0.3233 0.3234 0.3241 0.3232

4.12 × 106 3.83 × 106 4.22 × 106 3.77 × 106

259.94 259.93 259.82 259.96

7.09 7.10 7.16 7.08

H2O, Argon 260 K, T0 ) 30 °C, ω j v ) 0.02651 p0

[∆p/p0]exp

[∆p/p0]

J

T

S

p0

[∆p/p0]exp

[∆p/p0]

J

T

S

97.77 97.77 97.77 102.92 102.92

0.3229 0.3229 0.3227 0.3227 0.3232

0.3239 0.3239 0.3237 0.3237 0.3242

2.90 × 108 3.15 × 108 3.50 × 108 1.51 × 109 1.81 × 109

259.86 259.86 259.89 259.89 259.81

7.99 7.99 7.98 8.39 8.44

102.92 102.92 102.92 100.34

0.3223 0.3225 0.3233 0.3229

0.3233 0.3235 0.3243 0.3239

1.40 × 109 1.47 × 109 1.62 × 109 7.51 × 108

259.95 259.92 259.79 259.86

8.36 8.37 8.45 8.20

11688 J. Phys. Chem. B, Vol. 105, No. 47, 2001

Wo¨lk and Strey

TABLE 2. (Continued) IIa (Continued) H2O, Argon 250 K, T0 ) 25 °C, ω j v ) 0.01677 p0

[∆p/p0]exp

[∆p/p0]

J

T

S

p0

[∆p/p0]exp

[∆p/p0]

J

T

S

91.11 91.11 91.11 86.55 86.55 86.55 82.23 82.23 82.23 82.23

0.3584 0.3592 0.3587 0.3593 0.3596 0.3590 0.3591 0.3589 0.3592 0.3592

0.3601 0.3609 0.3604 0.3610 0.3613 0.3607 0.3608 0.3606 0.3609 0.3609

2.42×109 2.34×109 2.69×109 9.08×108 7.03×108 7.12×108 1.56×108 1.36×108 1.14×108 9.83×107

249.84 249.71 249.79 249.70 249.65 249.75 249.73 249.76 249.71 249.71

10.45 10.56 10.49 10.04 10.08 10.00 9.52 9.49 9.53 9.53

78.12 78.12 78.12 74.21 74.21 74.21 70.50 70.50 70.50

0.3590 0.3587 0.3585 0.3586 0.3588 0.3586 0.3584 0.3587 0.3596

0.3607 0.3604 0.3602 0.3603 0.3605 0.3603 0.3601 0.3604 0.3613

2.35×107 2.00×107 2.41×107 4.00×106 3.25×106 2.99×106 7.41×105 5.96×105 7.47×105

249.75 249.79 249.83 249.81 249.78 249.81 249.84 249.79 249.65

9.03 9.00 8.98 8.54 8.56 8.53 8.09 8.12 8.21

p0

[∆p/p0]exp

[∆p/p0]

J

[∆p/p0]

J

T

S

0.3603 0.3609 0.3604 0.3605 0.3605 0.3608 0.3605 0.3608 0.3612 0.3606 0.3601 0.3612

1.06×107

249.80 249.71 249.79 249.77 249.77 249.72 249.77 249.72 249.66 249.76 249.83 249.66

8.86 8.93 8.87 8.87 8.44 8.47 8.43 8.46 8.51 8.03 7.98 8.08

H2O, Argon 250 K, T0 ) 25 °C, ω j v ) 0.01652 T

S

p0

[∆p/p0]exp

91.11 91.11 91.11 91.11 86.55 86.55 86.55 86.55 82.23 82.23 82.23 82.23

0.3595 0.3588 0.3586 0.3590 0.3585 0.3589 0.3587 0.3593 0.3585 0.3596 0.3590 0.3594

0.3612 0.3605 0.3603 0.3607 0.3602 0.3606 0.3604 0.3610 0.3602 0.3613 0.3607 0.3611

1.37×109

p0

[∆p/p0]exp

[∆p/p0]

J

T

S

p0

[∆p/p0]exp

[∆p/p0]

J

T

S

91.11 91.11 91.11 86.55 86.55 86.55 86.55 82.23 82.23 82.23 80.17

0.3594 0.3591 0.3595 0.3592 0.3595 0.3587 0.3590 0.3586 0.3588 0.3585 0.3587

0.3611 0.3608 0.3612 0.3609 0.3612 0.3604 0.3607 0.3603 0.3605 0.3602 0.3604

3.49×109 3.06×109 2.76×109 8.95×108 7.55×108 7.64×108 6.79×108 1.23×108 1.33×108 1.33×108 4.30×107

249.68 249.73 249.67 249.71 249.67 249.79 249.75 249.81 249.78 249.83 249.79

10.59 10.55 10.60 10.03 10.06 9.96 10.01 9.47 9.49 9.45 9.24

80.17 80.17 80.17 76.16 76.16 76.16 76.16 76.16 72.35 72.35

0.3589 0.3588 0.3596 0.3592 0.3585 0.3584 0.3588 0.3594 0.3596 0.3594

0.3606 0.3605 0.3613 0.3609 0.3602 0.3601 0.3605 0.3611 0.3613 0.3611

5.42×107 4.85×107 5.72×107 9.28×106 6.38×106 8.53×106 7.16×106 6.98×106 1.63×106 1.62×106

249.76 249.78 249.65 249.71 249.83 249.84 249.78 249.68 249.65 249.68

9.26 9.25 9.33 8.83 8.76 8.74 8.78 8.86 8.43 8.41

p0

[∆p/p0]exp

[∆p/p0]

J

0.3947 0.3941 0.3950 0.3953 0.3950

0.3974 0.3968 0.3977 0.3980 0.3977

2.38×107

p0

[∆p/p0]exp

[∆p/p0]

J

T

S

p0

[∆p/p0]exp

[∆p/p0]

J

T

S

99.45 99.45 99.45 104.68

0.3950 0.3948 0.3950 0.3945

0.3977 0.3975 0.3977 0.3972

2.97×107 2.36×107 2.50×107 9.64×107

239.54 239.57 239.54 239.62

11.40 11.37 11.40 11.92

104.68 104.68 110.19 110.19

0.3946 0.3942 0.3946 0.3946

0.3973 0.3969 0.3973 0.3973

1.06×108 1.09×108 4.67×108 4.33×108

239.60 239.67 239.60 239.60

11.93 11.86 12.55 12.56

p0

[∆p/p0]exp

[∆p/p0]

J

[∆p/p0]

J

T

S

0.3973 0.3971 0.3978 0.3973 0.3979

2.94×108

0.3970 0.3972 0.3976 0.3976 0.3976

4.49×108

239.65 239.61 239.55 239.55 239.55

12.52 12.56 11.38 11.38 11.38

1.31×109 1.38×109 1.37×109 3.84×108 3.99×108 3.38×108 3.95×108 6.92×107 8.22×107 8.29×107 8.23×107

249.66 249.77 249.80 249.74 249.82 249.76 249.79 249.69 249.82 249.64 249.74 249.68

10.45 10.35 10.33 10.38 9.80 9.85 9.83 9.90 9.31 9.44 9.37 9.41

78.12 78.12 78.12 78.12 74.21 74.21 74.21 74.21 74.21 70.50 70.50 70.50

0.3586 0.3592 0.3587 0.3588 0.3588 0.3591 0.3588 0.3591 0.3595 0.3589 0.3584 0.3595

1.23×107 1.29×107 1.36×107 1.79×106 1.99×106 1.42×106 3.35×106 2.34×106 4.48×105 4.41×105 7.37×105

H2O, Argon 250 K, T0 ) 25 °C, ω j v ) 0.01677

H2O, Argon 240 K, T0 ) 20 °C, ω j v ) 0.006797 99.45 99.45 99.45 99.45 99.45

2.15×107 2.12×107 3.02×107 2.62×107

T 239.58 239.68 239.54 239.49 239.54

S 11.34 11.24 11.38 11.43 11.38

p0 94.48 94.48 94.48 94.48 89.75

[∆p/p0]exp 0.3944 0.3944 0.3946 0.3948 0.3947

[∆p/p0]

J

T

S

0.3971 0.3971 0.3973 0.3975 0.3974

4.51×106

239.63 239.63 239.60 239.57 239.58

10.73 10.71 10.75 10.78 10.22

3.74×106 4.35×106 6.18×106 1.31×106

H2O, Argon 240 K, T0 ) 20 °C, ω j v ) 0.006808

H2O, Argon 240 K, T0 ) 20 °C, ω j v ) 0.006650 110.19 110.19 110.19 110.19 113.02

0.3946 0.3944 0.3951 0.3946 0.3952

2.42×108 2.81×108 2.39×108 5.86×108

T 239.60 239.63 239.52 239.60 239.50

S 12.28 12.24 12.36 12.26 12.69

p0 113.02 113.02 101.71 101.71 101.71

[∆p/p0]exp 0.3943 0.3945 0.3949 0.3949 0.3949

5.35×108 2.84×107 2.24×107 3.56×107

H2O, Argon 240 K, T0 ) 20 °C, ω j v ) 0.006862 p0

[∆p/p0]exp

[∆p/p0]

J

T

S

p0

[∆p/p0]exp

[∆p/p0]

J

T

S

113.02 113.02 113.02 107.37 107.37

0.3948 0.3952 0.3947 0.3946 0.3942

0.3975 0.3979 0.3974 0.3973 0.3969

1.23×109 1.27×109 1.38×109 3.53×108 3.18×108

239.57 239.50 239.59 239.60 239.67

13.01 13.09 13.01 12.34 12.26

107.37 102.00 102.00 102.00 91.80

0.3945 0.3951 0.3946 0.3943 0.3942

0.3972 0.3978 0.3973 0.3970 0.3969

3.20×108 8.31×107 7.63×107 8.12×107 3.74×106

239.62 239.52 239.60 239.65 239.67

12.32 11.80 11.71 11.67 10.48

Homogeneous Nucleation of H2O and D2O

J. Phys. Chem. B, Vol. 105, No. 47, 2001 11689

TABLE 2. (Continued) IIa (Continued) H2O, Argon 240 K, T0 ) 20 °C, ω j v ) 0.006917 p0 107.37 107.37 107.37 103.08 103.08 103.08 97.92

[∆p/p0]exp 0.3949 0.3951 0.3946 0.3947 0.3946 0.3945 0.3943

[∆p/p0]

J

0.3976 0.3978 0.3973 0.3974 0.3973 0.3972 0.3970

3.43×108 4.28×108 3.71×108 1.33×108 1.17×108 1.21×108 2.59×107

T 239.56 239.52 239.60 239.59 239.60 239.62 239.65

S 12.49 12.52 12.42 11.94 11.93 11.92 11.28

p0 97.92 97.92 93.03 93.03 93.03 88.37

[∆p/p0]exp 0.3943 0.3949 0.3947 0.3953 0.3951 0.3951

[∆p/p0]

J

T

S

0.3970 0.3976 0.3974 0.3980 0.3978 0.3978

2.46×107

239.65 239.56 239.59 239.49 239.52 239.52

11.29 11.39 10.78 10.89 10.85 10.31

3.66×107 6.24×106 9.10×106 5.38×106 9.63×105

H2O, Argon 240 K, T0 ) 20 °C, ω j v ) 0.006827 p0

[∆p/p0]exp

[∆p/p0]

J

T

S

p0

[∆p/p0]exp

[∆p/p0]

J

T

S

103.08 103.08 103.08 96.90 96.90 96.90

0.3948 0.3947 0.3954 0.3949 0.3950 0.3946

0.3975 0.3974 0.3981 0.3976 0.3977 0.3973

6.15×107 7.03×107 7.24×107 1.38×107 1.12×107 1.48×107

239.57 239.59 239.47 239.55 239.54 239.60

11.82 11.80 11.91 11.12 11.13 11.08

92.05 92.05 92.05 87.45 87.45

0.3949 0.3948 0.3951 0.3943 0.3946

0.3976 0.3975 0.3978 0.3970 0.3973

2.78×106 2.06×106 2.05×106 3.75×105 3.31×105

239.55 239.57 239.52 239.65 239.60

10.57 10.55 10.59 9.95 10.00

H2O, Argon 230 K, T0 ) 10 °C, ω j v ) 0.003404 p0

[∆p/p0]exp

[∆p/p0]

J

T

S

p0

[∆p/p0]exp

[∆p/p0]

J

T

S

103.07 103.07 103.07 97.92 97.92 97.92 93.02 93.02

0.4056 0.4056 0.4064 0.4060 0.4061 0.4057 0.4059 0.4056

0.4087 0.4087 0.4095 0.4091 0.4092 0.4088 0.4090 0.4087

2.11×108 1.91×108 1.70×108 5.72×107 6.13×107 5.92×107 1.42×107 1.18×107

229.58 229.58 229.45 229.51 229.50 229.56 229.53 229.58

16.06 16.06 16.28 15.37 15.39 15.31 14.58 14.51

93.02 93.02 88.37 88.37 88.37 83.95 83.95 83.95

0.4064 0.4055 0.4061 0.4057 0.4063 0.4055 0.4060 0.4055

0.4095 0.4086 0.4092 0.4088 0.4094 0.4086 0.4091 0.4086

1.43×107 1.30×107 3.37×106 3.78×106 4.28×106 6.52×105 6.61×105 9.33×105

229.45 229.59 229.50 229.56 229.46 229.59 229.51 229.59

14.70 14.49 13.89 13.80 13.93 13.07 13.18 13.07

H2O, Argon 230 K, T0 ) 10 °C, ω j v ) 0.003433 p0

[∆p/p0]exp

[∆p/p0]

J

T

S

p0

[∆p/p0]exp

[∆p/p0]

J

T

S

103.07 103.07 103.07 97.92 97.92 97.92 93.02 93.02

0.4063 0.4062 0.4065 0.4064 0.4059 0.4055 0.4061 0.4058

0.4094 0.4093 0.4096 0.4095 0.4090 0.4086 0.4092 0.4089

1.23×108 1.36×108 1.55×108 3.56×107 4.11×107 3.73×107 1.08×107 1.10×107

229.46 229.48 229.43 229.45 229.53 229.59 229.49 229.54

15.97 15.94 16.02 15.19 15.08 14.97 14.37 14.29

93.02 93.02 88.37 88.37 88.37 84.83 84.83 84.83

0.4058 0.4057 0.4058 0.4059 0.4057 0.4059 0.4062 0.4065

0.4089 0.4088 0.4089 0.4090 0.4088 0.4090 0.4093 0.4096

1.46×107 1.52×107 3.38×106 3.42×106 2.46×106 5.84×105 9.40×105 6.42×105

229.54 229.56 229.54 229.53 229.56 229.53 229.48 229.43

14.30 14.28 13.58 13.60 13.57 13.06 13.12 13.18

H2O, Argon 230 K, T0 ) 10 °C, ω j v ) 0.003340 p0

[∆p/p0]exp

[∆p/p0]

J

T

S

p0

[∆p/p0]exp

[∆p/p0]

J

T

S

103.07 103.07 103.07 108.49 108.49 108.49 108.49 97.65 97.65

0.4065 0.4061 0.4061 0.4061 0.4066 0.4057 0.4063 0.4063 0.4061

0.4096 0.4092 0.4092 0.4092 0.4097 0.4088 0.4094 0.4094 0.4092

1.25×108 1.11×108 1.56×108 4.92×108 5.51×108 4.72×108 5.95×108 4.83×107 3.83×107

229.43 229.49 229.49 229.49 229.41 229.56 229.46 229.46 229.49

16.00 15.90 15.92 16.75 16.87 16.63 16.79 15.12 15.07

97.65 95.20 95.20 95.20 90.44 90.44 90.44 85.92 85.92

0.4061 0.4061 0.4055 0.4057 0.4064 0.4056 0.4054 0.4063 0.4053

0.4092 0.4092 0.4086 0.4088 0.4095 0.4087 0.4085 0.4094 0.4084

3.40×107 2.09×107 1.92×107 1.89×107 5.14×106 4.70×106 5.96×106 1.43×106 8.61×105

229.49 229.49 229.59 229.56 229.45 229.57 229.61 229.46 229.62

15.06 14.69 14.56 14.60 14.02 13.85 13.67 13.24 13.09

[∆p/p0]

J

T

S

0.4086 0.4089 0.4091 0.4095 0.4091 0.4091 0.4090 0.4085

2.49×107

229.59 229.54 229.51 229.45 229.51 229.51 229.53 229.61

14.55 14.62 14.66 14.02 13.93 13.94 13.21 13.11

H2O, Argon 230 K, T0 ) 10 °C, ω j v ) 0.003331 p0 108.49 108.49 108.49 105.78 105.78 105.78 100.49 100.49 100.49

[∆p/p0]exp 0.4059 0.4059 0.4060 0.4061 0.4058 0.4061 0.4055 0.4058 0.4060

[∆p/p0]

J

0.4090 0.4090 0.4091 0.4092 0.4089 0.4092 0.4086 0.4089 0.4091

4.85×108 3.94×108 4.41×108 3.01×108 2.49×108 3.05×108 7.92×107 8.54×107 8.32×107

T 229.53 229.53 229.51 229.49 229.54 229.49 229.59 229.54 229.51

S 16.64 16.65 16.67 16.28 16.21 16.29 15.31 15.40 15.44

p0 95.46 95.46 95.46 90.69 90.69 90.69 86.16 86.16

[∆p/p0]exp 0.4055 0.4058 0.4060 0.4064 0.4060 0.4060 0.4059 0.4054

2.10×107 2.95×107 7.54×106 5.96×106 6.97×106 1.11×106 7.08×105

11690 J. Phys. Chem. B, Vol. 105, No. 47, 2001

Wo¨lk and Strey

TABLE 2. (Continued) IIa (Continued) H2O, Argon 220 K, T0 ) 10 °C, ω j v ) 0.001520 p0

[∆p/p0]exp

[∆p/p0]

J

T

S

p0

[∆p/p0]exp

[∆p/p0]

J

T

S

103.38 103.38 103.38 103.38 98.21 98.21 98.21

0.4680 0.4686 0.4681 0.4687 0.4679 0.4678 0.4684

0.4744 0.4750 0.4745 0.4751 0.4743 0.4742 0.4748

1.73×108 1.70×108 1.16×108 2.26×108 4.31×107 5.63×107 5.45×107

218.97 218.87 218.96 218.85 218.99 219.01 218.90

21.38 21.64 21.42 21.68 20.27 20.22 20.47

93.30 93.30 93.30 93.30 88.64 88.64

0.4685 0.4681 0.4680 0.4684 0.4684 0.4684

0.4749 0.4745 0.4744 0.4748 0.4748 0.4748

2.70×107 1.78×107 1.92×107 1.59×107 8.88×106 6.32×106

218.88 218.96 218.97 218.90 218.90 218.90

19.47 19.33 19.30 19.44 18.47 18.47

p0

[∆p/p0]exp

[∆p/p0]

J

[∆p/p0]

J

T

S

H2O, Argon 220 K, T0 ) 10 °C, ω j v ) 0.001567 103.38 103.38 103.38 103.38 98.21 98.21 98.21

0.4688 0.4686 0.4680 0.4684 0.4680 0.4685 0.4680

0.4752 0.4750 0.4744 0.4748 0.4744 0.4749 0.4744

1.99×108

p0

[∆p/p0]exp

[∆p/p0]

J

0.4751 0.4748 0.4752 0.4744 0.4746 0.4744 0.4742

3.36×108

2.04×108 1.52×108 1.92×108 8.82×107 1.04×108 9.65×107

T 218.83 218.87 218.98 218.90 218.98 218.89 218.98

S

p0

22.38 22.30 22.03 22.19 20.92 21.15 20.94

98.21 93.30 93.30 93.30 88.64 88.64

[∆p/p0]exp 0.4679 0.4683 0.4685 0.4681 0.4676 0.4679

7

0.4743 0.4747 0.4749 0.4745 0.4739 0.4743

7.84×10 3.43×107 3.74×107 2.65×107 1.51×107 6.95×106

218.99 218.92 218.89 218.96 219.05 218.99

20.90 20.00 20.09 19.92 18.74 18.85

[∆p/p0]

J

T

S

0.4744 0.4744 0.4747 0.4742 0.4743 0.4746

1.56×108

218.98 218.98 218.92 219.01 219.00 218.94

21.83 20.75 20.86 20.66 19.67 19.77

H2O, Argon 220 K, T0 ) 10 °C, ω j v ) 0.001635 103.38 103.38 103.38 103.38 98.21 98.21 98.21

0.4687 0.4684 0.4688 0.4680 0.4682 0.4680 0.4678

3.54×108 4.40×108 3.46×108 2.26×108 1.63×108 1.40×108

T 218.85 218.91 218.83 218.98 218.94 218.98 219.01

S

p0

23.29 23.15 23.35 22.98 21.91 21.83 21.74

98.21 93.30 93.30 93.30 88.64 88.64

[∆p/p0]exp 0.4680 0.4680 0.4683 0.4678 0.4679 0.4682

5.79×107 6.47×107 5.48×107 1.74×107 2.53×107

H2O, Argon 220 K, T0 ) 10 °C, ω j v ) 0.001708 p0

[∆p/p0]exp

[∆p/p0]

J

T

S

p0

[∆p/p0]exp

[∆p/p0]

J

T

S

93.30 93.30 93.30 88.64 88.64 88.64 84.20 84.20 84.20 84.20 84.20 79.99

0.4684 0.4684 0.4688 0.4688 0.4679 0.4679 0.4676 0.4681 0.4681 0.4687 0.4685 0.4679

0.4748 0.4748 0.4752 0.4752 0.4743 0.4743 0.4739 0.4745 0.4745 0.4751 0.4749 0.4743

1.90×108 1.69×108 1.78×108 5.72×107 4.63×107 4.34×107 1.78×107 1.27×107 1.36×107 2.83×107 1.43×107 3.99×106

218.91 218.91 218.84 218.84 219.00 219.00 219.05 218.96 218.96 218.86 218.89 219.00

21.83 21.84 22.00 20.90 20.54 20.54 19.40 19.59 19.58 19.82 19.74 18.53

79.99 79.99 79.99 79.99 75.99 75.99 75.99 75.99 74.09 74.09 74.09

0.4677 0.4680 0.4682 0.4680 0.4681 0.4680 0.4683 0.4679 0.4681 0.4676 0.4688

0.4740 0.4744 0.4746 0.4744 0.4745 0.4744 0.4747 0.4743 0.4745 0.4739 0.4752

3.13×106 5.25×106 3.39×106 5.61×106 1.95×106 1.87×106 1.46×106 1.02×106 1.14×106 8.07×105 6.89×105

219.03 218.98 218.94 218.98 218.96 218.98 218.93 219.00 218.96 219.05 218.84

18.47 18.57 18.63 18.57 17.66 17.63 17.74 17.62 17.22 17.08 17.47

H2O, Argon 220 K, T0 ) 10 °C, ω j v ) 0.001645 p0

[∆p/p0]exp

[∆p/p0]

J

T

S

p0

[∆p/p0]exp

[∆p/p0]

J

T

S

93.30 93.30 93.30

0.4688 0.4683 0.4684

0.4752 0.4747 0.4748

8.61×107 4.87×107 6.78×107

218.84 218.92 218.91

21.20 21.00 21.04

93.30 88.64

0.4688 0.4687

0.4752 0.4751

5.62×107 3.05×107

218.84 218.85

21.20 20.09

[∆p/p0]

J

T

S

0.3237 0.3234 0.3227 0.3226

9.49×107

259.78 259.83 259.94 259.95

7.70 7.68 7.62 7.22

IIb D2O, Argon, 260 K, T0 ) 30 °C, ω j v ) 0.02214 p0 92.60 92.60 92.60 87.97 87.97

[∆p/p0]exp 0.3229 0.3222 0.3216 0.3222 0.3219

[∆p/p0]

J

0.3239 0.3232 0.3226 0.3232 0.3229

5.84×108 4.54×108 4.54×108 1.31×108 8.65×107

T 259.75 259.86 259.95 259.86 259.91

S 8.12 8.05 8.00 7.65 7.63

p0 87.97 87.97 87.97 83.57

[∆p/p0]exp 0.3227 0.3224 0.3217 0.3216

9.01×107 9.81×107 1.71×107

D2O, Argon, 260 K, T0 ) 30 °C, ω j v ) 0.02153 p0

[∆p/p0]exp

[∆p/p0]

J

T

S

p0

[∆p/p0]exp

[∆p/p0]

J

T

S

92.60 92.60 92.60 92.60 87.97 87.97

0.3226 0.3221 0.3222 0.3223 0.3217 0.3224

0.3236 0.3231 0.3232 0.3233 0.3227 0.3234

1.26×108 1.42×108 1.29×108 1.47×108 1.96×107 2.27×107

259.78 259.86 259.85 259.83 259.92 259.81

7.88 7.83 7.85 7.85 7.41 7.47

87.97 87.97 87.97 83.57 83.57

0.3218 0.3215 0.3218 0.3217 0.3223

0.3228 0.3225 0.3228 0.3227 0.3233

1.84×107 3.23×107 3.13×107 3.72×106 7.37×106

259.91 259.95 259.91 259.92 259.83

7.42 7.39 7.41 7.04 7.09

Homogeneous Nucleation of H2O and D2O

J. Phys. Chem. B, Vol. 105, No. 47, 2001 11691

TABLE 2. (Continued) IIb (Continued) D2O, Argon, 260 K, T0 ) 30 °C, ω j v ) 0.02220 p0

[∆p/p0]exp

[∆p/p0]

J

T

92.60 92.60 92.60 87.97 87.97 87.97 88.64 88.64 88.64 85.09 85.09 85.09 85.09

0.3218 0.3222 0.3221 0.3220 0.3226 0.3216 0.4679 0.4681 0.4686 0.4677 0.4676 0.4683 0.4679

0.3228 0.3232 0.3231 0.3230 0.3236 0.3226 0.4743 0.4745 0.4750 0.4740 0.4739 0.4747 0.4743

2.43×108

p0

[∆p/p0]exp

[∆p/p0]

J

0.3233 0.3231 0.3230 0.3234 0.3234 0.3235 0.3230 0.3239

2.90×108

3.41×108 2.95×108 6.81×107 6.64×107 5.46×107 1.77×107 2.50×107 2.56×107 1.01×107 9.29×106 7.70×106 6.98×106

259.92 259.86 259.88 259.89 259.80 259.95 219.00 218.96 218.87 219.03 219.05 218.92 219.00

S 8.05 8.08 8.07 7.65 7.71 7.62 19.80 19.88 20.07 18.93 18.88 19.14 18.99

p0 85.09 85.09 80.84 80.84 80.84 80.84 76.79 76.79 76.79 83.57 83.57 83.57 83.57

[∆p/p0]exp 0.4681 0.4685 0.4682 0.4685 0.4684 0.4676 0.4679 0.4680 0.4677 0.3222 0.3221 0.3217 0.3217

[∆p/p0]

J

T

S

0.4745 0.4749 0.4746 0.4749 0.4748 0.4739 0.4743 0.4744 0.4740 0.3232 0.3231 0.3227 0.3227

6.97×106

218.96 218.89 218.94 218.89 218.91 219.05 219.00 218.98 219.03 259.86 259.88 259.94 259.94

19.06 19.22 18.15 18.27 18.22 17.94 17.14 17.17 17.09 7.29 7.28 7.25 7.25

[∆p/p0]

J

T

S

0.3227 0.3228 0.3235 0.3239 0.3236 0.3236 0.3232

1.33×107

259.94 259.93 259.82 259.76 259.80 259.80 259.86

7.28 7.29 7.35 7.01 6.99 6.99 6.61

1.15×107 4.08×106 3.85×106 3.91×106 3.55×106 1.11×106 1.10×106 9.96×105 9.17×106 7.43×106 5.99×106 4.96×106

D2O, Argon, 260 K, T0 ) 30 °C, ω j v ) 0.02230 92.60 92.60 92.60 87.97 87.97 87.97 83.57 75.42

0.3223 0.3221 0.3220 0.3224 0.3224 0.3225 0.3220 0.3229

T

2.93×108 3.18×108 1.00×108 9.45×107 9.36×107 1.78×107 4.06×105

259.85 259.88 259.89 259.83 259.83 259.82 259.89 259.76

S 8.12 8.10 8.10 7.73 7.73 7.74 7.31 6.66

p0 83.57 83.57 83.57 79.39 79.39 79.39 75.42

[∆p/p0]exp 0.3217 0.3218 0.3225 0.3229 0.3226 0.3226 0.3222

1.29×107 2.22×107 3.96×106 3.39×106 3.51×106 4.38×105

D2O, Argon, 260 K, T0 ) 30 °C, ω j v ) 0.02196 p0

[∆p/p0]exp

[∆p/p0]

J

T

S

p0

[∆p/p0]exp

[∆p/p0]

J

T

S

92.60 92.60 92.60 95.46 95.46

0.3226 0.3228 0.3224 0.3223 0.3222

0.3236 0.3238 0.3234 0.3233 0.3232

1.68×108 1.71×108 1.44×108 4.40×108 4.42×108

259.79 259.76 259.83 259.84 259.86

8.03 8.05 8.01 8.25 8.24

95.46 98.42 98.42 98.42 88.82

0.3218 0.3218 0.3217 0.3216 0.3228

0.3228 0.3228 0.3227 0.3226 0.3238

3.86×108 9.44×108 1.04×109 1.33×109 1.01×108

259.92 259.92 259.93 259.95 259.76

8.20 8.46 8.44 8.44 7.72

p0

[∆p/p0]exp

[∆p/p0]

J

[∆p/p0]

J

T

S

0.3237 0.3230 0.3229 0.3235 0.3227 0.3238 0.3230 0.3231 0.3236

1.57×106 1.61×106 1.10×106 5.70×105 3.29×105 4.45×105 2.12×105 4.08×105 7.09×105

259.78 259.89 259.91 259.81 259.94 259.77 259.89 259.88 259.80

7.02 6.97 6.96 6.83 6.77 6.86 6.79 6.80 6.84

[∆p/p0]

J

T

S

0.3601 0.3604 0.3602 0.3594 0.3600 0.3600

7.73×107 1.60×107 1.13×107 1.06×107 1.47×107 3.61×106

249.73 249.68 249.71 249.82 249.74 249.74

9.76 9.31 9.29 9.20 9.27 8.80

[∆p/p0]

J

T

S

0.3606 0.3603 0.3598 0.3601 0.3599 0.3603 0.3604 0.3600

1.84×107

249.64 249.69 249.77 249.72 249.75 249.69 249.67 249.73

9.12 9.08 9.03 9.07 8.58 8.63 8.66 8.59

D2O, Argon, 260 K, T0 ) 30 °C, ω j v ) 0.02216 T 7

88.82 88.82 88.82 88.82 84.38 84.38 84.38 84.38 80.16 80.16

0.3225 0.3223 0.3221 0.3226 0.3219 0.3225 0.3218 0.3227 0.3219 0.3223

0.3235 0.3233 0.3231 0.3236 0.3229 0.3235 0.3228 0.3237 0.3229 0.3233

6.29×10 4.68×107 5.13×107 6.08×107 7.86×106 8.85×106 6.88×106 8.83×106 1.08×106 1.42×106

p0

[∆p/p0]exp

[∆p/p0]

J

259.81 259.85 259.88 259.80 259.91 259.81 259.92 259.78 259.91 259.85

S 7.76 7.74 7.72 7.77 7.32 7.37 7.32 7.40 6.96 6.98

p0 80.16 80.16 80.16 78.16 78.16 78.16 78.16 78.16 78.16

[∆p/p0]exp 0.3227 0.3220 0.3219 0.3225 0.3217 0.3228 0.3220 0.3221 0.3226

D2O, Argon, 250 K, T0 ) 25 °C, ω j v ) 0.01239 91.11 91.11 91.11 91.11 86.55 86.55

0.3581 0.3585 0.3582 0.3586 0.3580 0.3581

0.3598 0.3602 0.3599 0.3603 0.3597 0.3598

5.46×108

p0

[∆p/p0]exp

[∆p/p0]

J

0.3604 0.3602 0.3604 0.3600 0.3597 0.3595 0.3600 0.3600 0.3594

3.76×108

5.47×108 5.04×108 4.44×108 6.97×107 6.71×107

T 249.77 249.71 249.76 249.69 249.79 249.77

S 10.24 10.28 10.26 10.30 9.71 9.73

p0 86.55 82.22 82.22 82.22 82.22 78.11

[∆p/p0]exp 0.3584 0.3587 0.3585 0.3578 0.3583 0.3583

D2O, Argon, 250 K, T0 ) 25 °C, ω j v ) 0.01209 91.11 91.11 91.11 91.11 86.55 86.55 86.55 86.55 86.55

0.3587 0.3585 0.3587 0.3583 0.3580 0.3579 0.3583 0.3583 0.3578

4.71×108 4.94×108 4.50×108 1.15×108 9.33×107 9.26×107 8.88×107 9.68×107

T 249.67 249.70 249.67 249.73 249.78 249.80 249.73 249.73 249.81

S 10.08 10.06 10.08 10.02 9.48 9.47 9.52 9.52 9.46

p0 82.23 82.23 82.23 82.23 78.12 78.12 78.12 78.12

[∆p/p0]exp 0.3589 0.3586 0.3581 0.3584 0.3582 0.3586 0.3587 0.3583

1.64×107 1.50×107 1.97×107 2.46×106 3.49×106 1.55×106 2.75×106

11692 J. Phys. Chem. B, Vol. 105, No. 47, 2001

Wo¨lk and Strey

TABLE 2. (Continued) IIb (Continued) D2O, Argon, 250 K, T0 ) 25 °C, ω j v ) 0.01244 p0

[∆p/p0]exp

[∆p/p0]

J

T

S

p0

[∆p/p0]exp

[∆p/p0]

J

T

S

91.11 91.11 91.11 86.55 86.55 86.55 86.55 82.23 82.23 82.23

0.3585 0.3581 0.3579 0.3579 0.3590 0.3584 0.3583 0.3577 0.3577 0.3583

0.3602 0.3598 0.3595 0.3595 0.3607 0.3601 0.3600 0.3593 0.3593 0.3600

8.43×108 8.68×108 9.23×108 1.92×108 2.35×108 1.97×108 1.93×108 3.31×107 3.95×107 3.16×107

249.71 249.77 249.81 249.81 249.63 249.73 249.74 249.84 249.84 249.74

10.32 10.28 10.26 9.74 9.88 9.80 9.79 9.22 9.22 9.30

82.23 78.12 78.12 78.12 78.12 74.21 74.21 74.21 74.21

0.3584 0.3583 0.3580 0.3585 0.3584 0.3582 0.3582 0.3581 0.3588

0.3601 0.3600 0.3597 0.3602 0.3601 0.3599 0.3599 0.3598 0.3605

3.14×107 5.07×106 6.69×106 6.77×106 4.40×106 5.50×105 1.06×106 8.02×105 1.90×106

249.73 249.74 249.79 249.71 249.73 249.76 249.76 249.77 249.66

9.31 8.84 8.80 8.86 8.85 8.39 8.38 8.36 8.45

D2O, Argon, 250 K, T0 ) 25 °C, ω j v ) 0.01231 p0

[∆p/p0]exp

[∆p/p0]

J

T

S

p0

[∆p/p0]exp

[∆p/p0]

J

T

S

91.11 91.11 91.11 91.11 86.55 86.55 86.55 82.23 82.23

0.3586 0.3586 0.3580 0.3588 0.3579 0.3584 0.3583 0.3587 0.3582

0.3603 0.3603 0.3597 0.3605 0.3595 0.3601 0.3600 0.3604 0.3599

4.83×108 5.52×108 5.02×108 5.37×108 1.32×108 1.62×108 1.62×108 3.58×107 2.69×107

249.69 249.69 249.79 249.66 249.80 249.72 249.74 249.68 249.76

10.25 10.24 10.16 10.27 9.64 9.71 9.69 9.25 9.20

82.23 82.23 79.76 79.76 79.76 75.77 75.77 75.77

0.3580 0.3576 0.3582 0.3586 0.3580 0.3576 0.3579 0.3584

0.3597 0.3592 0.3599 0.3603 0.3597 0.3592 0.3595 0.3601

3.12×107 3.03×107 1.12×107 1.04×107 9.02×106 1.74×106 1.45×106 1.52×106

249.79 249.85 249.76 249.69 249.79 249.85 249.80 249.72

9.17 9.12 8.93 8.97 8.90 8.41 8.44 8.50

D2O, Argon, 240 K, T0 ) 20 °C, ω j v ) 0.006649 p0

[∆p/p0]exp

[∆p/p0]

J

T

S

p0

[∆p/p0]exp

[∆p/p0]

J

T

S

77.98 77.98 77.98 77.98 74.08 74.08 74.08 66.86

0.3947 0.3953 0.3945 0.3941 0.3950 0.3941 0.3945 0.3951

0.3974 0.3980 0.3972 0.3968 0.3977 0.3968 0.3972 0.3978

4.95×107 4.50×107 4.30×107 4.34×107 1.46×107 1.38×107 1.47×107 1.44×106

239.58 239.48 239.61 239.68 239.53 239.68 239.61 239.52

11.90 12.00 11.85 11.78 11.34 11.19 11.27 10.25

74.08 70.38 70.38 70.38 70.38 70.38 66.86

0.3943 0.3945 0.3951 0.3944 0.3954 0.3946 0.3949

0.3970 0.3972 0.3978 0.3971 0.3981 0.3973 0.3976

1.21×107 5.44×106 4.31×106 3.40×106 2.59×106 2.83×106 1.40×106

239.65 239.61 239.52 239.63 239.47 239.60 239.55

11.23 10.69 10.79 10.68 10.84 10.71 10.22

p0

[∆p/p0]exp

[∆p/p0]

J

[∆p/p0]

J

T

S

0.3969 0.3971 0.3971 0.3970 0.3976 0.3975 0.3974

1.33×108 1.25×108 2.78×107 3.27×107 3.31×107 3.48×107 9.60×106

239.66 239.63 239.63 239.65 239.55 239.57 239.58

11.91 11.96 11.35 11.33 11.45 11.43 10.84

[∆p/p0]

J

T

S

0.3968 0.3970 0.3974 0.3968 0.3971 0.3968 0.3972 0.3977

5.43×107

239.68 239.65 239.58 239.68 239.63 239.68 239.61 239.53

11.81 11.86 11.91 11.23 11.28 11.23 11.28 10.81

D2O, Argon, 240 K, T0 ) 20 °C, ω j v ) 0.006716 82.08 82.08 82.08 82.08 82.08 77.98 77.98

0.3947 0.3950 0.3941 0.3948 0.3946 0.3947 0.3950

0.3974 0.3977 0.3968 0.3975 0.3973 0.3974 0.3977

2.86×108

p0

[∆p/p0]exp

[∆p/p0]

J

0.3980 0.3968 0.3969 0.3968 0.3977 0.3974 0.3978 0.3972

1.83×108

2.96×108 3.47×108 4.53×108 4.91×108 1.59×108 1.25×108

T 239.58 239.53 239.68 239.57 239.60 239.58 239.53

S 12.65 12.70 12.53 12.66 12.62 12.01 12.06

p0 77.98 77.98 74.08 74.08 74.08 74.08 70.38

[∆p/p0]exp 0.3942 0.3944 0.3944 0.3943 0.3949 0.3948 0.3947

D2O, Argon, 240 K, T0 ) 20 °C, ω j v ) 0.006670 82.08 82.08 82.08 82.08 82.08 82.08 77.98 77.98

0.3953 0.3941 0.3942 0.3941 0.3950 0.3947 0.3951 0.3945

1.48×108 1.91×108 2.34×108 2.83×108 3.13×108 9.11×107 5.92×107

T 239.48 239.68 239.66 239.68 239.53 239.58 239.52 239.61

S 12.67 12.44 12.46 12.45 12.60 12.55 12.00 11.89

p0 77.98 77.98 77.98 74.08 74.08 74.08 74.08 70.37

[∆p/p0]exp 0.3941 0.3943 0.3947 0.3941 0.3944 0.3941 0.3945 0.3950

7.01×107 8.32×107 1.60×107 2.24×107 1.60×107 1.92×107 5.08×106

D2O, Argon, 240 K, T0 ) 20 °C, ω j v ) 0.006669 p0

[∆p/p0]exp

[∆p/p0]

J

T

S

p0

[∆p/p0]exp

[∆p/p0]

J

T

S

82.08 82.08 82.08 77.98 77.98 77.98 74.08 74.08 74.08

0.3950 0.3950 0.3942 0.3945 0.3943 0.3941 0.3947 0.3944 0.3945

0.3977 0.3977 0.3969 0.3972 0.3970 0.3968 0.3974 0.3971 0.3972

3.59×108 4.30×108 3.80×108 1.06×108 9.46×107 1.15×108 2.90×107 3.13×107 3.07×107

239.53 239.53 239.66 239.61 239.65 239.68 239.58 239.63 239.61

12.60 12.60 12.46 11.89 11.86 11.82 11.33 11.28 11.29

70.37 70.37 70.37 70.37 66.85 66.85 63.51 63.51 63.51

0.3941 0.3950 0.3952 0.3953 0.3953 0.3946 0.3947 0.3954 0.3946

0.3968 0.3977 0.3979 0.3980 0.3980 0.3973 0.3974 0.3981 0.3973

6.20×106 8.65×106 7.68×106 6.57×106 1.91×106 1.39×106 5.06×105 5.14×105 5.32×105

239.68 239.53 239.50 239.48 239.48 239.60 239.58 239.47 239.60

10.66 10.81 10.84 10.86 10.32 10.21 9.71 9.81 9.69

Homogeneous Nucleation of H2O and D2O

J. Phys. Chem. B, Vol. 105, No. 47, 2001 11693

TABLE 2. (Continued) IIb (Continued) D2O, Argon, 240 K, T0 ) 20 °C, ω j v ) 0.006772 p0

[∆p/p0]exp

[∆p/p0]

J

T

S

p0

[∆p/p0]exp

[∆p/p0]

J

T

S

82.08 82.08 82.08 82.08 82.08 82.08 82.08 84.62

0.3954 0.3942 0.3945 0.3943 0.3952 0.3946 0.3952 0.3948

0.3981 0.3969 0.3972 0.3970 0.3979 0.3973 0.3979 0.3975

3.33×108 3.87×108 5.07×108 4.47×108 3.27×108 3.96×108 5.28×108 1.08×109

239.47 239.67 239.62 239.65 239.50 239.60 239.50 239.57

12.87 12.64 12.70 12.67 12.84 12.72 12.84 13.15

84.62 84.62 84.62 77.98 77.98 77.98 74.08

0.3943 0.3948 0.3945 0.3946 0.3944 0.3947 0.3942

0.3970 0.3975 0.3972 0.3973 0.3971 0.3974 0.3969

1.01×109 9.63×108 1.07×109 1.41×108 1.75×108 1.57×108 4.61×107

239.65 239.57 239.62 239.60 239.63 239.58 239.67

13.05 13.15 13.09 12.08 12.05 12.10 11.41

D2O, Argon, 240 K, T0 ) 20 °C, ω j v ) 0.006854 p0

[∆p/p0]exp

[∆p/p0]

J

T

S

p0

[∆p/p0]exp

[∆p/p0]

J

T

S

74.08 74.08 74.08 74.08 70.38 70.38 70.38

0.3950 0.3946 0.3943 0.3954 0.3950 0.3950 0.3950

0.3977 0.3973 0.3970 0.3981 0.3977 0.3977 0.3977

3.03×107 2.97×107 3.70×107 4.46×107 1.58×107 1.86×107 1.73×107

239.54 239.60 239.65 239.47 239.54 239.54 239.54

11.69 11.62 11.57 11.76 11.11 11.10 11.10

66.86 66.86 66.86 63.51 63.51 63.51

0.3942 0.3945 0.3953 0.3948 0.3941 0.3941

0.3969 0.3972 0.3980 0.3975 0.3968 0.3968

4.00×106 3.65×106 4.50×106 1.29×106 7.92×105 8.46×105

239.67 239.62 239.49 239.57 239.68 239.68

10.41 10.47 10.60 9.99 9.89 9.89

p0

[∆p/p0]exp

[∆p/p0]

J

[∆p/p0]

J

T

S

0.4086 0.4083 0.4087 0.4083 0.4091 0.4093 0.4090 0.4086 0.4089 0.4084 0.4082

4.09×107 3.38×107 4.11×107 2.55×107 1.24×107 1.22×107 1.25×107 1.08×107 2.89×106 2.74×106 3.36×106

229.57 229.62 229.55 229.62 229.49 229.46 229.50 229.57 229.52 229.60 229.63

14.61 14.54 14.62 14.54 13.99 14.05 13.96 13.88 13.26 13.14 13.09

[∆p/p0]

J

T

S

0.4085 0.4090 0.4086 0.4089 0.4085 0.4089 0.4083 0.4086

2.05×107 2.16×107 1.94×107 6.03×106 4.76×106 5.11×106 1.41×106 2.15×106

229.59 229.51 229.57 229.52 229.59 229.52 229.62 229.57

14.15 14.25 14.17 13.52 13.43 13.52 12.72 12.78

[∆p/p0]

J

T

S

0.4086 0.4087 0.4087 0.4086 0.4090 0.4088 0.4087 0.4089

3.81×107

229.57 229.56 229.56 229.57 229.51 229.54 229.56 229.52

14.66 14.69 13.95 13.93 14.03 13.28 13.25 13.30

D2O, Argon, 230 K, T0 ) 10 °C, ω j v ) 0.002587 88.36 88.36 88.36 88.36 93.01 93.01 93.01 93.01 95.89 95.89 95.89

0.4060 0.4056 0.4063 0.4057 0.4060 0.4054 0.4058 0.4054 0.4062 0.4057 0.4060

0.4091 0.4087 0.4094 0.4088 0.4091 0.4085 0.4089 0.4085 0.4093 0.4088 0.4091

3.77×107

p0

[∆p/p0]exp

[∆p/p0]

J

4.24×107 4.44×107 4.77×107 1.75×108 1.66×108 1.95×108 1.77×108 3.83×108 3.69×108 3.77×108

T 229.49 229.55 229.44 229.54 229.49 229.58 229.52 229.58 229.46 229.54 229.49

S 15.03 14.95 15.11 14.96 15.84 15.68 15.78 15.68 16.38 16.25 16.33

p0 86.54 86.54 86.54 86.54 82.21 82.21 82.21 82.21 78.10 78.10 78.10

[∆p/p0]exp 0.4055 0.4052 0.4056 0.4052 0.4060 0.4062 0.4059 0.4055 0.4058 0.4053 0.4051

D2O, Argon, 230 K, T0 ) 10 °C, ω j v ) 0.002641 95.89 95.89 95.89 91.10 91.10 91.10 86.54 86.54 86.54

0.4058 0.4061 0.4056 0.4052 0.4059 0.4056 0.4056 0.4056 0.4060

0.4089 0.4092 0.4087 0.4083 0.4090 0.4087 0.4087 0.4087 0.4091

4.68×108

p0

[∆p/p0]exp

[∆p/p0]

J

0.4094 0.4090 0.4088 0.4086 0.4086 0.4084 0.4085 0.4091 0.4086

4.12×108

5.63×108 5.17×108 1.76×108 1.97×108 1.75×108 6.44×107 6.16×107 6.32×107

T 229.52 229.47 229.55 229.62 229.51 229.55 229.55 229.55 229.49

S 16.60 16.68 16.55 15.62 15.80 15.72 14.93 14.94 15.03

p0 82.21 82.21 82.21 78.10 78.10 78.10 74.20 74.20

[∆p/p0]exp 0.4054 0.4059 0.4055 0.4058 0.4054 0.4058 0.4052 0.4055

D2O, Argon, 230 K, T0 ) 10 °C, ω j v ) 0.002678 93.01 93.01 93.01 88.36 88.36 88.36 88.36 83.94 83.94

0.4063 0.4059 0.4057 0.4055 0.4055 0.4053 0.4054 0.4060 0.4055

3.45×108 3.33×108 1.01×108 1.08×108 1.26×108 1.08×108 4.50×107 3.77×107

T 229.44 229.51 229.54 229.57 229.57 229.60 229.59 229.49 229.57

S 16.46 16.35 16.30 15.44 15.44 15.39 15.41 14.78 14.66

p0 83.94 83.94 79.74 79.74 79.74 75.76 75.76 75.76

[∆p/p0]exp 0.4055 0.4056 0.4056 0.4055 0.4059 0.4057 0.4056 0.4058

3.94×107 1.10×107 1.29×107 1.27×107 3.06×106 2.28×106 3.37×106

D2O, Argon, 230 K, T0 ) 10 °C, ω j v ) 0.002663 p0

[∆p/p0]exp

[∆p/p0]

J

T

S

p0

[∆p/p0]exp

[∆p/p0]

J

T

S

93.01 93.01 93.01 93.01 93.01 88.36 88.36 88.36

0.4059 0.4059 0.4054 0.4062 0.4061 0.4053 0.4057 0.4055

0.4090 0.4090 0.4085 0.4093 0.4092 0.4084 0.4088 0.4086

2.71×108 2.19×108 2.18×108 2.69×108 2.60×108 6.95×107 7.24×107 8.13×107

229.51 229.51 229.59 229.46 229.47 229.60 229.54 229.57

16.26 16.27 16.13 16.33 16.31 15.31 15.40 15.35

88.36 83.94 83.94 83.94 83.94 79.74 79.74

0.4056 0.4054 0.4054 0.4061 0.4057 0.4055 0.4064

0.4087 0.4085 0.4085 0.4092 0.4088 0.4086 0.4095

6.65×107 1.96×107 2.03×107 2.19×107 2.84×107 6.35×106 1.08×107

229.55 229.59 229.59 229.47 229.54 229.57 229.43

15.38 14.56 14.56 14.72 14.63 13.85 14.06

11694 J. Phys. Chem. B, Vol. 105, No. 47, 2001

Wo¨lk and Strey

TABLE 2. (Continued) IIb (Continued) D2O, Argon, 230 K, T0 ) 10 °C, ωv ) 0.002661 p0

[∆p/p0]exp

[∆p/p0]

J

T

S

p0

[∆p/p0]exp

[∆p/p0]

J

T

S

93.01 93.01 93.01 88.36 88.36 88.36 83.94 83.94 83.94

0.4060 0.4059 0.4054 0.4057 0.4056 0.4058 0.4058 0.4056 0.4052

0.4091 0.4090 0.4085 0.4088 0.4087 0.4089 0.4089 0.4087 0.4083

3.29×108 3.02×108 2.93×108 1.04×108 1.13×108 1.17×108 4.05×107 3.88×107 3.88×107

229.49 229.51 229.59 229.54 229.55 229.52 229.52 229.55 229.62

16.28 16.25 16.12 15.39 15.37 15.42 14.65 14.60 14.50

83.94 79.74 79.74 79.74 79.74 75.76 75.76 75.76 75.76

0.4058 0.4053 0.4059 0.4052 0.4061 0.4055 0.4064 0.4054 0.4052

0.4089 0.4084 0.4090 0.4083 0.4092 0.4086 0.4095 0.4085 0.4083

4.74×107 1.38×107 1.36×107 1.20×107 1.36×107 2.51×106 2.77×106 2.60×106 3.94×106

229.52 229.60 229.51 229.62 229.47 229.57 229.43 229.59 229.62

14.64 13.81 13.93 13.79 13.99 13.15 13.34 13.14 13.09

p0

[∆p/p0]exp

[∆p/p0]

J

T

S

p0

[∆p/p0]exp

[∆p/p0]

J

T

S

86.57 86.57 86.57 86.57 86.57 82.24 82.24

0.4679 0.4682 0.4684 0.4683 0.4676 0.4677 0.4687

0.4743 0.4746 0.4748 0.4747 0.4739 0.4740 0.4751

1.46×108 1.30×108 1.29×108 1.34×108 1.30×108 5.41×107 5.06×107

218.98 218.93 218.89 218.91 219.03 219.02 218.84

21.62 21.77 21.84 21.79 21.49 20.46 20.89

82.24 82.24 78.13 78.13 78.13 78.13 74.22

0.4679 0.4678 0.4681 0.4684 0.4676 0.4683 0.4682

0.4743 0.4742 0.4745 0.4748 0.4739 0.4747 0.4746

4.52×107 4.88×107 2.19×107 1.66×107 1.99×107 2.09×107 5.85×106

218.98 219.00 218.95 218.89 219.03 218.91 218.93

20.55 20.51 19.60 19.71 19.40 19.66 18.64

p0

[∆p/p0]exp

[∆p/p0]

J

T

S

p0

[∆p/p0]exp

[∆p/p0]

J

T

S

86.57 86.57 86.57 86.57 82.24 82.24 82.24 82.24 78.13 78.13 78.13 78.13

0.4685 0.4688 0.4686 0.4682 0.4684 0.4679 0.4677 0.4681 0.4679 0.4682 0.4684 0.4680

0.4749 0.4752 0.4750 0.4746 0.4748 0.4743 0.4740 0.4745 0.4743 0.4746 0.4748 0.4744

2.02×108 2.91×108 2.71×108 2.57×108 1.66×108 1.38×108 1.51×108 1.61×108 6.04×107 5.79×107 4.76×107 6.62×107

218.88 218.82 218.86 218.93 218.90 218.98 219.02 218.95 218.98 218.93 218.90 218.97

23.45 23.61 23.51 23.33 22.25 22.02 21.94 22.10 20.91 21.05 21.13 20.95

74.22 74.22 74.22 74.22 70.51 70.51 70.51 70.51 66.99 66.99 66.99 66.99

0.4677 0.4676 0.4679 0.4679 0.4681 0.4682 0.4676 0.4679 0.4677 0.4677 0.4678 0.4677

0.4740 0.4739 0.4743 0.4743 0.4745 0.4746 0.4739 0.4743 0.4740 0.4740 0.4742 0.4740

1.85×107 1.80×107 2.22×107 2.25×107 9.48×106 9.05×106 8.13×106 8.05×106 3.83×106 2.60×106 3.43×106 1.96×106

219.02 219.04 218.98 218.98 218.95 218.93 219.04 218.98 219.02 219.02 219.00 219.02

19.78 19.75 19.87 19.87 18.95 19.00 18.76 18.86 17.87 17.86 17.90 17.86

p0

[∆p/p0]exp

[∆p/p0]

J

T

S

p0

[∆p/p0]exp

[∆p/p0]

J

T

S

86.57 86.57 86.57 86.57 82.24 82.24 82.24

0.4687 0.4685 0.4680 0.4682 0.4684 0.4683 0.4676

0.4751 0.4749 0.4744 0.4746 0.4748 0.4747 0.4739

2.79×108 3.07×108 2.50×108 2.46×108 1.15×108 1.39×108 9.70×107

218.84 218.88 218.97 218.93 218.89 218.91 219.04

22.76 22.67 22.43 22.54 21.49 21.45 21.13

82.24 78.13 78.13 78.13 74.22 74.22

0.4683 0.4679 0.4682 0.4679 0.4681 0.4682

0.4747 0.4743 0.4746 0.4743 0.4745 0.4746

1.44×108 4.12×107 5.12×107 4.37×107 2.06×107 1.71×107

218.91 218.98 218.93 218.98 218.95 218.93

21.45 20.21 20.32 20.20 19.28 19.32

p0

[∆p/p0]exp

[∆p/p0]

J

T

S

p0

[∆p/p0]exp

[∆p/p0]

J

T

S

86.57 86.57 86.57 86.57 82.24 82.24 82.24 82.24 78.13

0.4676 0.4677 0.4676 0.4684 0.4677 0.4679 0.4676 0.4683 0.4675

0.4739 0.4740 0.4739 0.4748 0.4740 0.4743 0.4739 0.4747 0.4738

1.49×108 9.22×107 1.13×108 1.34×108 4.78×107 4.08×107 5.12×107 5.49×107 1.67×107

219.03 219.02 219.03 218.89 219.02 218.98 219.03 218.91 219.05

21.34 21.38 21.33 21.69 20.33 20.40 20.27 20.59 19.22

78.13 78.13 78.13 74.22 74.22 74.22 74.22 70.51 70.51

0.4677 0.4680 0.4677 0.4678 0.4680 0.4680 0.4675 0.4684 0.4681

0.4740 0.4744 0.4740 0.4742 0.4744 0.4744 0.4738 0.4748 0.4745

1.93×107 2.46×107 1.61×107 8.46×106 9.58×106 8.06×106 8.41×106 3.08×106 1.58×106

219.02 218.96 219.02 219.00 218.96 218.96 219.05 218.89 218.95

19.31 19.43 19.31 18.38 18.45 18.46 18.27 17.66 17.57

p0

[∆p/p0]exp

[∆p/p0]

J

[∆p/p0]

J

T

S

0.4740 0.4739 0.4743 0.4739 0.4740 0.4748 0.4750 0.4743 0.4742 0.4746

1.27×108

0.4742 0.4750 0.4743 0.4739 0.4750 0.4742 0.4746 0.4746 0.4747

1.17×107

219.00 218.86 218.98 219.04 218.86 219.00 218.93 218.93 218.91

19.17 19.47 18.24 18.14 18.52 18.21 17.44 17.43 17.47

D2O, Argon, 220 K, T0 ) 10 °C, ω j v ) 0.001235

D2O, Argon, 220 K, T0 ) 10 °C, ω j v ) 0.001324

D2O, Argon, 220 K, T0 ) 10 °C, ω j v ) 0.001278

D2O, Argon, 220 K, T0 ) 10 °C, ω j v ) 0.001226

D2O, Argon, 220 K, T0 ) 10 °C, ω j v ) 0.001279 86.57 86.57 82.24 82.24 82.24 82.24 78.13 78.13 78.13 74.22

0.4677 0.4676 0.4679 0.4676 0.4677 0.4684 0.4686 0.4679 0.4678 0.4682

1.72×108 6.47×107 8.59×107 8.44×107 8.84×107 3.21×107 3.53×107 3.56×107 1.65×107

T 219.02 219.04 218.98 219.04 219.02 218.89 218.86 218.98 219.00 218.93

S 22.30 22.27 21.28 21.16 21.20 21.50 20.50 20.21 20.18 19.31

p0 74.22 74.22 70.51 70.51 70.51 70.51 66.99 66.99 66.99

[∆p/p0]exp 0.4678 0.4686 0.4679 0.4676 0.4686 0.4678 0.4682 0.4682 0.4683

1.34×107 4.68×106 6.49×106 5.10×106 2.24×106 1.26×106 1.41×106 1.64×106

Homogeneous Nucleation of H2O and D2O

J. Phys. Chem. B, Vol. 105, No. 47, 2001 11695

TABLE 2. (Continued) IIb (Continued) D2O, Argon, 220 K, T0 ) 10 °C, ω j v ) 0.001283 p0

[∆p/p0]exp

[∆p/p0]

J

T

S

p0

[∆p/p0]exp

[∆p/p0]

J

T

S

86.57 86.57 86.57 86.57 82.24 82.24 78.13 78.13 78.13

0.4680 0.4678 0.4681 0.4679 0.4675 0.4677 0.4680 0.4679 0.4676

0.4744 0.4742 0.4745 0.4743 0.4738 0.4740 0.4744 0.4743 0.4739

4.53×108 2.04×108 2.08×108 1.93×108 1.06×108 9.51×107 3.61×107 3.13×107 2.91×107

218.97 219.00 218.95 218.98 219.05 219.02 218.97 218.98 219.04

22.51 22.43 22.55 22.46 21.18 21.27 20.32 20.28 20.14

74.22 74.22 74.22 74.22 74.22 70.51 70.51 70.51

0.4680 0.4682 0.4683 0.4676 0.4679 0.4676 0.4676 0.4678

0.4744 0.4746 0.4747 0.4739 0.4743 0.4739 0.4739 0.4742

3.19×107 1.06×107 8.68×106 7.64×106 9.04×106 2.31×106 3.02×106 2.33×106

218.97 218.93 218.91 219.04 218.98 219.04 219.04 219.00

19.31 19.39 19.43 19.14 19.26 18.19 18.20 18.27

D2O, Argon, 220 K, T0 ) 10 °C, ω j v ) 0.001283 p0

[∆p/p0]exp

[∆p/p0]

J

T

S

p0

[∆p/p0]exp

[∆p/p0]

J

T

S

78.13 78.13 78.13 78.13 78.13 78.13 74.22 74.22 74.22 74.22 74.22

0.4681 0.4684 0.4681 0.4679 0.4683 0.4680 0.4678 0.4687 0.4686 0.4684 0.4682

0.4745 0.4748 0.4745 0.4743 0.4747 0.4744 0.4742 0.4751 0.4750 0.4748 0.4746

3.56×107 1.86×107 2.08×107 1.86×107 1.87×107 2.38×107 7.86×106 8.76×106 1.14×107 8.41×106 8.43×106

218.95 218.89 218.95 218.98 218.91 218.97 219.00 218.84 218.86 218.89 218.93

20.35 20.48 20.35 20.27 20.43 20.31 19.22 19.57 19.54 19.45 19.38

70.51 70.51 70.51 70.51 66.99 66.99 66.99 63.64 63.64 63.64 63.64

0.4686 0.4678 0.4681 0.4687 0.4684 0.4675 0.4682 0.4678 0.4687 0.4678 0.4677

0.4750 0.4742 0.4745 0.4751 0.4748 0.4738 0.4746 0.4742 0.4751 0.4742 0.4740

4.15×106 2.89×106 3.08×106 2.36×106 8.73×105 1.19×106 1.18×106 3.12×105 7.33×105 8.86×105 4.12×105

218.86 219.00 218.95 218.84 218.89 219.05 218.93 219.00 218.84 219.00 219.02

18.56 18.27 18.38 18.60 17.56 17.24 17.48 16.49 16.79 16.48 16.44

a T /°C chamber temperature, ω j v vapor fraction in the mixture, p0/kPa initial total pressure, ∆p difference between p0 and the actual pressure pexp 0 during the nucleation pulse, J/cm-3 s-1 experimental nucleation rate, T/K experimental temperature, S supersaturation.

calculating the nucleation temperature T according to Poisson’s law

T ) T0(1 - (∆p/p0))(γ-1)/γ

(16)

where T0 is the (starting) chamber temperature. The ratio γ of the specific heats is obtained using the Richarz formula,62 accounting for the influence of the vapor on the value of γ. During one series of measurements, the vapor fraction ω and thus γ remain constant. Furthermore, T0 and ∆p are kept constant, and thus, a constant nucleation temperature T is obtained. In the actual experiments, a reproducibility of T within a few hundredth of 1 °C is achieved (cf. column 5 in Table 2). The vapor fraction ω remains unchanged during expansion. Therefore, one can easily calculate the actual vapor pressure in the expanded state. Then, from the known equilibrium vapor pressure pve63 (Table 1) at nucleation temperature T, the supersaturation S is calculated from

S)

ωp0 pve(T)

(17)

S is adjusted by setting p0, given as column 6 in Table 2. B. Nucleation Rates. The number concentration Cexp of droplets is observed by light scattering, as described by Wagner.19 Since the nucleation process is stationary, the nucleation rate is then given by

J)

C exp ∆t exp

(18)

Nucleation rates ranging from about 105 to 109 cm-3 s-1 can be measured with the present experimental system. The nucleation rates J are given in column 4 of Table 2. C. Physicochemical Properties of H2O and D2O. C.1. Materials. H2O was twice distilled in a quartz column, and D2O

was purchased from Cambrige Isotope Laboratories and had an isotopic purity of 99.9%. To be able to calculate the supersaturation S in the experiment, we needed the equilibrium vapor pressure pve at the experimental temperature T. To calculate theoretical nucleation rates, we needed in addition the surface tensions and densities for the two isotopic waters in the supercooled state. For the calculation of the revised theory by Reiss et al. also the compressibility κ is needed. There have been considerable efforts over the years to measure these quantities at low temperatures. The best correlations we have been able to locate or generate are given in Table 1. The graphic representation of surface tensions, equilibrium vapor pressures, and densities for the temperature range of interest (220-320 K) is given in Figure 2. C.2. Density. The density of water is the most well-known property. Although in the supercooled state the data for D2O are less reliable, it is known to the fourth significant digit, as demonstrated by Vedamuthu et al.5 These authors also show and argue that many properties of D2O can be judged much more precisely if the generally better known quantities of H2O are used as a reference. The properties of D2O are then obtained by rescaling the temperature axis. This procedure has been applied to the density. The density maximum for H2O is located at 3.9 °C, while for D2O, the maximum is at 11.1. Accordingly, a shift by 7.2 °C brings them together, leaving a ratio of F(D2O)/ F(H2O) ) 1.106.5 We follow this reasoning. However, we mention that extrapolation of Kell’s fit equation to 220 K or so leads to unreasonably low densities.63 Therefore, the fit equation for density has to account for the fact that the supercooled water will eventually approach a density of amorphous water ice. For that reason, we have developed a fit function based on the following arguments. At Tc ) 647.14 K, the density is Fc ) 0.322 g/cm3. It increases as T decreases according to critical scaling as (F - Fc) ) ctβ, where c is substance-specific but β ) 0.33 is a universal exponent. Cooperative hydrogen bonding and ordering lead to steplike change from the monotonically

11696 J. Phys. Chem. B, Vol. 105, No. 47, 2001

Wo¨lk and Strey

Figure 3. Experimental nucleation rates J of heavy water (rectangles) and light water (circles) as a function of the actual vapor pressure pv for various constant nucleation temperatures (220< T/K < 260). Note that the nucleation of D2O requires much smaller actual vapor pressures pv than H2O. Figure 2. Density, equilibrium vapor pressure, and surface tension of H2O and D2O as a function of temperature. The circles belong to light water and the rectangles to heavy water.

increasing density of liquid water to the 8% lower value of supercooled water or amorphous ice. To account for this feature, we have added an a tanh((T - T0)/b) term, where a takes care of the overall density difference of water and ice, T0 is the 50% temperature of the transition, and b accounts for the sharpness of the transition (error < 0.5%). C.3. Equilibrium Vapor Pressure. The equilibrium vapor pressure of H2O is about 10% higher than that of D2O. To check the equilibrium vapor pressure expressions given in the literature, we have measured the vapor pressure for 5, 15, and 25 °C for both H2O and D2O. The agreement with the correlations given by Hill64,65 for D2O and by Dillmann21,49 for H2O was excellent. Following the physical argument discussed above for the densities, the analogous rescaling of the temperature results in a dividing factor. Using the equilibrium vapor pressure correlation for H2O, we generated the D2O curve by defining a rescaled temperature T′ ) T / 1.008. The rescaled temperature T′ renders the fit parameters in a well-defined way. We have compared the calculated vapor pressure using Hill’s correlation and the one generated by Wagner.66 Only insignificant differences are observed (3% at low temperatures and 1% at high temperatures). Therefore, either one can be used in order to calculate equilibrium vapor pressures. C.4. Surface Tension. The surface tension of light water is taken from the Handbook of Chemistry and Physics4 and that of heavy water from “The Surface Tension of Pure Liquid Compounds” by Jasper.67 The data for heavy water cover a temperature range between 5 and 75 °C and those for light water between 0 and 100 °C. The light water data points are described to within 0.3% by a correlation, which was already employed by Viisanen.7 A correlation for D2O can be obtained when the temperature of the H2O function is rescaled by T′ ) T‚1.022, providing an even better agreement (error 0.05%) with the D2O data points. C.5. Compressibility. The compressibility for light water and the corresponding correlation are taken from Kell and Whalley.68 For heavy water, the compressibilities are given by Millero and Lepple69 in a temperature range between 5 and 65 °C. Using these data and the same kind of function as those for light water, we obtained the fit parameters for heavy water. This fit function agrees very well with the measured compressibilities (error 0.05

%). We note that the compressibility occurs only in the prefactor and has accordingly an insignificant effect on the quantitative analyses. D. Accuracy of the Results. The experimental limitations and accuracy obtainable with the present version of the nucleation pulse chamber have been described in detail in two recent papers.21,7 The accuracy of the light-scattering method was determined by comparison to a different technique for absolute number concentration measurements.70 The number density is accurate to about a factor of 2, i.e., ( 200%, while the reproducibility is much better, only about ( 15%. We noted that the experimentally observed nucleation rates differ if we start the measurements from different initial chamber temperatures. This effect was also noticed by Wagner, Strey, and Viisanen in 1992 and was first examined by Viisanen et al. 1993.7 They observed that a correction function of the form

( ) ( ) [ ( )] ∆p ∆p ) p0 p0

1+a

exp

∆p p0

b

(19)

exp

with a ) 0.2915 and b ) 4.036 was able to bring nucleation rates from different initial temperatures into mutual agreement. We have corrected our measurements with eq 19. The results ∆p/p0 are given in column 3 of Table 2. With this new ∆p/p0, we calculated using Poisson’s law (eq 17) the new nucleation temperature, which is given in column 5 of Table 2. We have kept the expansion ratios as small as possible in order to minimize any effect of inaccuracies induced by the use of eq 19. In Figure 3, the actual vapor pressure is employed in place of supersaturation. The vapor pressure pv is known from preparing the mixture and free of all thermo-physical parameters and, therefore, accurate to better than 0.5%. The supersaturations are calculated using the equilibrium vapor pressure pve, accurate to 1% at the highest and 3% at the lowest temperatures. These errors are within symbol width in Figure 4 and are consequently not shown. The onset supersaturations S0 carry essentially the error from the equilibrium vapor pressure determination of up to 3%. The statistical temperature errors amounts to a small fraction of a percent, as can be seen from Table 2 and is therefore within the symbol width in Figure 3. The errors of the critical cluster sizes n* from the error in the slopes of the ln J - ln pv, or ln S curves are about 10%. The improved experimental accuracy required, furthermore, consideration of the validity of the assumption of ideal adiabatic

Homogeneous Nucleation of H2O and D2O

Figure 4. Experimental nucleation rates J of light water (circles) in comparison to the nucleation rates of heavy water (rectangles) as a function of supersaturation S for five constant temperatures ranging from 220 to 260 K. The numerical data are compiled in Table 2 in the appendix. The dashed line belongs to a constant nucleation rate J0 ) 107 cm-3 s-1 and allows to determine the onset supersturation S0.

behavior of the vapor carrier gas mixtures. The effect is found to be so small that it does introduce only negligible systematic errors. IV. Results and Analysis In the present paper, we report homogeneous nucleation rates of heavy (D2O) and light water (H2O) with argon as carrier gas. We confine ourselves to the use of argon because, as Viisanen et al.7 showed, the homogeneous nucleation of water is independent of whether helium, neon, argon, krypton, or xenon was used. After the model-free presentation of the new data, a comparison between previous H2O measurements by Viisanen et al.,7,23 and the new data are made to check the reproducibility of the experiment. In a first analysis, we use the fact that measurements of isothermal nucleation rate curves permit direct determination of the number of molecules in the critical cluster, as elaborated earlier.21,7 The molecular contents of the nuclei obtained for both isotopic waters are compared for various temperatures. In the third part of this section, we compare our experimental results to predictions of two different nucleation theories. Toward end, a test of the Gibbs-Thomson equation is performed, useful because the Gibbs-Thomson equation is used in many theoretical works to calculate the number of particles in the critical cluster. A. Homogeneous Nucleation Rates of Light and Heavy Water. Homogeneous nucleation rates J ranging from about 105 to 109 cm-3 s-1 were measured quantitatively as functions of vapor pressure pv at constant nucleation temperature T, where T ranged between 220 and 260 K in steps of 10 K. In Figure 3, the nucleation rates of H2O (circles) and D2O (squares) are shown as function of the actual vapor pressure pv at nucleation conditions. This way of comparing the experimental data is as modelfree as possible. As one can see, the experimental nucleation rates of heavy water at the same absolute vapor pressure are higher than the nucleation rates of light water by a factor of 2500 under otherwise identical conditions (temperature, expansion ratio and chamber temperature). On the other hand, the same nucleation rate J at the same temperature is observed for lower vapor pressure for D2O than for H2O. It can be seen that the data do not show any significant deviations from the indicated linear fits. Also, it is obvious that the slope at the same temperatures for H2O and D2O is nearly the same. Since

J. Phys. Chem. B, Vol. 105, No. 47, 2001 11697

Figure 5. Measured onset supersaturations S0 for H2O, corresponding to an onset nucleation rate of J0 ) 107 cm-3 s-1, are compared to the onset supersaturations of previous measurements (triangles up, uncorrected data of Viisanen et al.;7 triangles down, corrected data of Viisanen et al.23 based on Viisanen et al.;7 open circles, preliminary data by Wo¨lk and Strey;22 full circles, this work).

nucleation and condensation requires supersaturation, the next step is to compare the experimental results as a function of vapor supersaturation S at constant temperature T. The comparison is shown in Figure 4. In contrast to Figure 3, now the experimental nucleation rate of H2O and of D2O superimpose at the same temperature and supersaturation within experimental scatter. While one might have expected this in view of the similarity of H2O and D2O, the thermo-physical parameters of the two isotopic waters do differ quite substantially (cf. Figure 2). B. Comparison with Previous Data. As mentioned before, preliminary measurements performed in 199822 lead to a reanalysis of the experimental procedure and a repeated analysis of the previous data by Viisanen et al.7 To check the reproducibility of the experiment, we present a comparison between these corrected H2O measurements by Viisanen et al.23 and the new data in Figure 5. Since the actual experimental temperatures differ somewhat, the comparison is best performed by comparing the onset supersaturations S0 corresponding to a nucleation rate of J0 ) 107 cm-3 s-1 as function of temperature. The agreement between the two most recent series is quite good (open circles, ref 22; full circles, this work). Only at 220 K is there a small, but systematic deviation. It should be noted, however, that at the low temperatures the vapor pressures are already quite low so that minor adsorption losses, e.g., to the walls have a comparatively large effect. The older and recalculated Viisanen, Strey, and Reiss 7,23 data (open triangles down) differ somewhat from the present data, but actually agree within experimental error. For comparison, the original, noncorrected data from Viisanen et al.7 are shown as open triangles up. These clearly differ from the results of our preliminary series in 1998 (open circles), which initiated the repeated analysis by Viisanen et al..23 C. Molecular Content of the Critical Clusters. From the slopes of the experimental data in Figure 4, the number of molecules in the critical clusters is determined for both isotopic waters. Figure 6 shows the molecular content of the H2O and D2O nuclei as a function of temperature. As can be seen, the number of molecules in the critical cluster

11698 J. Phys. Chem. B, Vol. 105, No. 47, 2001

Wo¨lk and Strey

Figure 8. Same as the caption of Figure 7 for D2O.

Figure 6. Molecular content of nuclei n* for light (circles) and heavy water (rectangles) as function of nucleation temperature T.

Figure 9. Predictions of the classical theory (BD) for light water (full lines) and heavy water (dashed lines) at five constant temperatures T.

Figure 7. Comparison of experimental nucleation rates of H2O (circles) with the predictions of the classical theory. The full lines belong to the classical Becker-Do¨ring theory (BD).

varies from 20 to 36 molecules both for H2O and D2O in the temperature range from 220 to 260 K. The actual number of molecules show a quite large uncertainty arising from the sensitivity of the slopes on various influences. At first, one should realize that the ln J versus ln S curves are actually curved and not straight lines. Second, depending on the number of data points obtained, the slope tends to be flatter if more data points at high rates are included. On the other hand, the accuracy of the data points at lower rates becomes increasingly poorer. We therefore have decided to base the [(∂ ln J)/(∂ ln S)]T determination on all data points for each temperature and live with the relatively large error of 10%. This is the error indicated by the error bars in Figure 6. D. Comparison with Theory. Of the numerous nucleation theories, we have selected two. The classical nucleation theory of Becker and Do¨ring29 is the most commonly used model for quantitative prediction of nucleation phenomena. Despite some shortcomings, it has dominated our perception of the nucleation process. Various theoretical improvements have been reported in recent years. Here we compare the experimental nucleation rates only with the classical theory and one of the most recent modifications by Reiss et al.56 In Figure 7, the experimental nucleation rate curves for light water are compared with the classical theory. In Figure 8, the same comparison is shown for heavy water.

The calculations have been performed using the thermophysical parameters compiled in Table 1. The curves calculated from the classical theory are in quite close agreement with the measured data sets, as can nicely be seen by comparing the full lines to the respective data points. However, it is also obvious that the classical theory predicts a disparate temperature dependence. A nearly perfect agreement of the classical theory and experiment is found near 240 K. There, both absolute rate and slope of the curves agree. These results and the fact that the nucleation rates as a function of supersaturation are the same for both isotopic waters leads to the conclusions that the theoretical predictions for both should be the same, too. To support this statement, the predictions of the classical theory for both isotopic waters are shown as a function of supersaturation in Figure 9. At a temperature of 260 K, we find absolute agreement for both isotopic waters. For the lower temperatures, a small deviation is found, which tends become bigger with decreasing temperature. The maximal difference between the predictions of classical theory for both isotopic waters is found at 220 K, where it is about a factor of 2. If we consider that the parameters surface tension and density, which are used to calculate the theoretical predictions, are all extrapolated to low temperatures (cf. Figure 2), it is actually quite surprising that the agreement is that good. In contrast to the classical theory, the self-consistent theory by Reiss et al.56 predicts too high a nucleation rate over the whole temperature range investigated. Dividing the predictions of this theory by a constant factor of 6000, however, brings the classical and self-consistent theory into absolute agreement at T ) 240 K, as shown in Figure 10.

Homogeneous Nucleation of H2O and D2O

J. Phys. Chem. B, Vol. 105, No. 47, 2001 11699

Figure 10. Experimental nucleation rates of H2O (circles) as shown in Figure 4 compared to the predictions of the Reiss-Kegel-Katz theory (RKK; dashed lines) (divided by a factor of 6000) along with the classical theory (BD; solid lines). Note the improved temperature dependence of the RKK theory.

Figure 12. Comparison of experimental and theoretical nucleation rates for H2O Jexp/Jtheo vs 1/T. The left scale ln(Jexp/Jtheo) can be read as the difference (in kT) between the experimental and theoretical work of formation of a nucleus.

Figure 11. Same as the caption of Figure 10 for D2O.

As can be seen, the prediction of the self-consistent theory (dashed line) (divided by a factor of 6000) shows not only the correct slope but, in contrast to the classical theory, also an improved temperature dependence. The latter two observations can also be made for heavy water which is shown in Figure 11. E. Empirical Correlation for Water Nucleation. A useful way of comparing experiment and theory is presented in Figure 12, where we divided experimental by theoretical nucleation rates for H2O. In this kind of comparison, whole nucleation rate curves collapse into a cloud of points, one for each experimental temperature. The mean values of the logarithms of Jexp/Jtheo values are plotted versus the mean inverse experimental temperature (1/T). As can be seen, the data behave as

ln

Jexp 1 )A+B Jtheo T

(20)

with A ) ln(Kexp/Ktheo) and B ) (∆G/theo - ∆G/exp)/k, if we assume the general validity of eq 1. So the left scale ln(Jexp/ Jtheo) in Figure 12 can be read as the difference (in kT) between the theoretical and experimental work of nucleus formation.72 The incorrect temperature dependence of the classical theory is obvious, while the temperature of perfect agreement is found from 1/T ) 0.00421 [1/K] for light water, which corresponds to a temperature of T ) 238 K. In Figure 13, the same analysis is performed for heavy water. For heavy water, the perfect agreement between theory and experiment is found at 1/T ) 0.00418 [1/K], which is T ) 239

Figure 13. Same as the caption of Figure 12 for D2O.

K. Thus, the classical theory predicts the correct nucleation rates for the two isotopic waters around 240 K, which might be of interest in cloud models working in that temperature range. Furthermore, the slopes of nucleation rate curves as well as the classical theory coincide, which might be an even more important observation. To compare also the self-consistent theory by Reiss et al.56 to the experimental data, we applied the same procedure plotted it in Figures 12 and 13 together with the linear fit (eq 20) to compare the temperature dependence of the both theories more easily. For H2O, we find B ) 6.5 × 103 K for the classical theory and B ) 3.7 × 103 K for the self-consistent theory with intercepts A ) -27.56 and -24.63, respectively. For D2O, the slope for the theories are somewhat steeper compared to H2O, B ) 8.6 × 103 and 5.8 × 103 K, respectively, and the intercepts are A ) -35.98 for the classical theory and A ) -30.74 for the Reiss-Kegel-Katz theory. Clearly, the self- consistent theory shows an improved temperature dependence. Turning eq 20 around, an empirical correlation for water nucleation rates over extended ranges of supersaturation and temperature can be given

11700 J. Phys. Chem. B, Vol. 105, No. 47, 2001

(

J ) JBD exp A +

B T

)

Wo¨lk and Strey

(21)

We expect that this equation can safely be used to calculate homogeneous nucleation rates for water from 100 up to 1020 cm-3 s-1 for a supersaturation range of 5 < S < 200 and a temperature range of about 200 < T/K < 270. As a critical test, we calculated the onset supersaturation S0 for a rate of 100 cm-3 s-1 as they are measured in diffusion cloud chambers. The data by Heist and Reiss14 for H2O are compared with these supersaturations in Figure 14. For this comparison, we have scanned Figure 3 from reference.14 We note that our predictions of classical theory (crosshair) and the calculations by Heist and Reiss14 agree quantitatively. This shows that our as well as their thermophysical parameters were properly chosen. The onset supersaturation S0 calculated from eq 21 using the A and B values for H2O given above are shown as thick line in Figure 14. A rather good agreement between their results, and our calculated supersaturations is found for the examined temperature range. The agreement is noteworthy insofar as our actual H2O measurements were performed in the 105 to 109 cm-3 s-1 range but predict rates around 100 cm-3 s-1 properly. Also, our measurements are based on the temperature range between 220 and 260 K but correctly predict onset in the 280-320 K range. In our opinion, these observations lend strong support for the correct predictions of the supersaturation dependence of the classical theory (steepness of the J-S curves) and the sufficient accounting of eq 20 of the disparate temperature dependence the classical theory bears. With the constants A and B for D2O, nucleation rate calculations for heavy water can be performed in order to predict nucleation rates for the above-mentioned aerosol-SANS measurements. There nucleation rates extending into the range of 1018 cm-3 s-1 are needed. Such critical test of this equation will be presented in a forthcoming paper. F. Test of the Gibbs Thomson Equation. As emphasized by Strey et al.,21 one of the major advances of nucleation rate measurements as opposed to mere onset determinations, the nucleation rate curves permit a direct test of the GibbsThomson equation, which is used in theories to account for the curvature dependence of the equilibrium vapor pressure. We test here the Gibbs-Thomson equation comparing the critical cluster size obtained from the slopes of the experimental ln J versus ln S curves (eq 13) to the n* calculated from eq 5. For this calculation, the actual experimental supersaturations and the corresponding experimental temperatures are used. In Figure 15, the mean value of the n* calculated from the GibbsThomson equation is plotted against the n* obtained from the slopes of the experimental curves. As can be seen, the prediction of Gibbs-Thomson is quite good both for light and heavy water, which is surprising if one considers that the calculations use the macroscopic surface tension and density. Actually, this aspect in retrospect justifies comparison of our results with the predictions of the classical nucleation theory, which includes the so-called “capillarity approximation” and which is so often drawn into doubt. Conclusions Measurements of homogeneous nucleation rates of D2O and H2O under identical conditions permit a number of interesting observations. Because we have two very similar but distinct substances, all experimental uncertainties sensitively show up by comparison. At first sight, surprisingly, the nucleation rates

Figure 14. Variation of the onset supersaturation of water vapor referring to a nucleation rate of J ) 100 cm-3 s-1 as a function of temperature as given by Heist and Reiss [14]. These results were scanned from ref 14 (page 670), and the onset supersaturations S0 calculated using eq 21 are included (thick line). Note that the predictions of the classical theory calculated by Heist and Reiss14 and those calculated in this work (crosshair) agree quantitatively.

Figure 15. Experimental n*’s (nucleation theorem) are presented together with the corresponding prediction by the Gibbs-Thomson (GT) equation (eq 5) for light (circles) and heavy water (rectangles). The dashed line stands for total agreement between the experimental n* and the Gibbs-Thomson equation.

of D2O are found to be larger than those of H2O by a factor of 2500 at the same vapor pressure and temperature. However, if one uses the corresponding equilibrium vapor pressure expressions to calculate supersaturations, one finds the D2O rates to superimpose with those of H2O at the same respective supersaturation S and temperature T. This observation holds for all temperatures 220 < T/K < 260 we examined. Such superposition has to be accounted for by nucleation theories. Given the superposition, it comes as no surprise that experimentally we find the number of water molecules in the critical clusters to be the same for the two isotopic waters within experimental error. Using the most recent and best empirical expressions for equilibrium vapor pressures, surface tensions, and densities as functions of temperature, we have calculated predictions using the classical nucleation theory to perform a comparison on absolute scale. The classical theory is found to correctly predict

Homogeneous Nucleation of H2O and D2O the nucleation rate for one particular temperature for each substance. The temperatures, for which classical theory and experimental rate quantitatively agree, are 238 K for H2O and 239 K for D2O. However, the classical theory predicts an incorrect temperature dependence. It is stronger in theory than in experiment. This finding is consistent with observations by a number of authors on a variety of vapors.20,71-73 The revised nucleation theory by Reiss et al.,56 in contrast, provides an improved temperature dependence, but is off by a large constant factor. This factor is 6 × 103 both for light and heavy water. Both the classical and the revised RKK theory predict for the ratio of the nucleation rates of the two isotopic waters at most a factor of 2 at the low-temperature end of our measuring range. Such difference is within our present experimental uncertainty. Toward the end of our analysis, we developed an empirical correction function which, if applied to the classical theory, permits calculating nucleation rates for water over wide ranges of supersaturation and temperature. We tested this empirical correlation to predict the onset conditions in a diffusion cloud chamber. Such measurements have been published by Heist and Reiss in 1973.14 A quantitative agreement is found (cf. Figure 14), although our measurements were performed in a much lower temperature and a much higher nucleation rate range. Such agreement requires that the slope of the nucleation rate versus supersaturation curves is properly predicted by classical theory. This may be directly tested if the Gibbs-Thomson equation is used to calculate the molecular content of the critical clusters and compared to those numbers of molecules in the critical cluster obtained from the nucleation theorem. The agreement we find (cf. Figure 15) is sufficient. Acknowledgment. The nucleation pulse chamber was developed in the laboratory of M. Kahlweit at the Max-PlanckInstitut fu¨r Biophysikalische Chemie in Go¨ttingen. We are indebted to him for supporting the research over so many years. We also gratefully acknowledge the permission by the MaxPlanck-Gesellschaft to use and further develop the chamber in Cologne. The present state of the chamber is the result of longlasting and pleasant cooperations mainly with P.E. Wagner and Y. Viisanen and also with J. Hru´by and P. van Remoortere. To the latter we are indebted for instructions and help with the experiment in the early stages of the present measurements. We also like to thank J. Cheung for helping us find some of the thermo-physical parameters of water, which are used in this paper. References and Notes (1) Due to the vast excess and due to fast isotope exchange, mainly HDO is originally present. (2) Wyslouzil, B. E.; Cheung, J. L.; Wilemski, G.; Strey, R. Phys. ReV. Lett. 1997, 79, 431. (3) Heath, C. H.; Streletzky, K. A.; Wo¨lk, J.; Wyslouzil, B. E.; Strey, R. D2O-H2O Condensation in Supersonic Nozzles: I. Experiment. Nucleation and Atmosperic Aerosols, 15th International Conference, Rolla, MI, Aug 6-11, 2000; AIP: Melville, NY, 2000. (4) Handbook of Chemistry and Physics, 72nd ed.; CRC Press: Boston, 1991-1992. (5) Vedamuthu, M.; Singh, S.; Robinson, G. W. J. Chem. Phys. 1996, 100, 3825. (6) Flood, H.; Tronstad, L. Z. Phys. Chem. A 1936, 175, 347. (7) Viisanen, Y.; Strey, R.; Reiss, H. J. Chem. Phys. 1993, 99, 4680. (8) Wilson, C. R. T. Philos. Trans. R. Soc. London, Ser. A 1897, 189, 265. (9) Volmer, M.; Flood, H. Z. Phys. Chem. A 1934, 190, 273. (10) Sander, A.; Damko¨hler, G. Naturwissenschaften 1943, 31, 460. (11) Wegener, P.; Lundquist, G. J. Appl. Phys. 1951, 22, 233. (12) Courtney, W. G. J. Chem. Phys. 1961, 35, 2249. (13) Katz, J. L.; Ostermier, B. J. J. Chem. Phys. 1967, 47, 478.

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