REEXAMINATION OF THE SCALING OF HOMOGENEOUS
NUCLEATION RATES FOR WATER
B. N. HALE1
Physics Department and Cloud & Aerosol Science Laboratory, Missouri University of Science &
Technology, Rolla, MO 65409 U. S. A
1
Keywords: NUCLEATION, SCALING, WATER, EXPERIMENT, THEORY
INTRODUCTION
The temperature dependent scaling of the homogeneous nucleation rate, J, offers a comprehensive
comparison of rates over a range of temperature ratios, T/Tc (where Tc is the critical temperature) and
supersaturation ratios, S = P/P₀ (where P and P0 are the ambient and equilibrium vapor pressures).
Experimental rates are often obtained via physically different background processes while the theoretical
predictions range from classical analyses to Monte Carlo (MC) and Molecular Dynamics (MD) computer
simulations with several choices of effective pair potentials. The scaling plot enables one to examine all
these data simultaneously using two parameters: , the excess surface entropy per molecule (in units of k,
the Boltzmann constant) and the critical temperature for the substance or the model potential in the case of
the theoretical predictions. We examine here the scaling plot for water (Hale, 2004) and (Wyslouzil and
Wölk, 2016) together with recent MD (Tanaka, 2014), (Angelil, 2015) and MC predictions at higher rates
(Hale and DiMattio, 2017). The latter higher rates place the physical process in the realm of smaller
critical cluster sizes where the discreteness of the cluster size can play a significant role and affect the
scaling exponents.
THE SCALING PLOT
In the scaling plot (Hale, 2004) -log [J/JS] is plotted vs. the scaled free energy of formation (Hale,
1986) [16π/3] Ω³ [Tc/T-1]³/(lnS)² where JS is a typical monomer flux factor times the monomer number
density -- often taken to be 10²⁶ in the cgs system of units. The excess surface entropy per molecule, Ω, is
approximately 1.5 for substances with a dipole moment and 2.0 for normal substances. The scaling plot
for the experimental and theoretical water data is shown in Fig. 1.
MC NUCLEATION RATES FOR THE TIP4P POTENTIAL MODEL
AT LARGE SUPERSATURATION RATIOS
In the interest of generating Monte Carlo simulation based nucleation rates near 1023 cm-3s-1 the free
energy differences for small TIP4P clusters (Hale and DiMattio, 2004) at 260K, 280K and 300K (see Fig.
2) are used to predict nucleation rates at high supersaturation ratios. The results are added to Fig. 1.
Details of this process are in (Hale, 2010). The predicted nucleation rates are given in Table I.
Table I
T/K
S
J/cm-3s-1
260
260
260
260
260
260
260
260
260
6
6.2
6.4
6.6
6.8
18.7
46
10.4
150
2.70E+09
1.52E+10
6.71E+10
2.44E+11
2.44E+11
1.45E+20
3.32E+23
2.77E+16
1.40E+26
T/K
S
J/cm-3s-1
T/K
S
J/cm-3s-1
280
280
280
280
280
280
280
280
4.3
4.5
4.7
4.9
5.1
9.7
19.2
6.3
1.85E+09
4.83E+10
7.97E+11
8.24E+12
5.57E+13
1.86E+20
3.74E+23
5.26E+16
300
300
300
300
300
300
300
300
3.5
3.6
3.7
3.8
3.9
5.9
9.9
4.4
2.45E+11
2.11E+12
1.53E+13
9.31E+13
4.70E+14
2.54E+20
4.33E+23
1.68E+17
Fig.1. Homogeneous nucleation rates, J, for water vs. the scaled free energy of formation. Much of the data in
this plot was presented in Fig. 12 of (Wyslouzil and Wölk, 2016) who generously shared data files for creating the
present figure which now includes data for the MD generated nucleation rates, (Tanaka et al, 2014), (Angelil et al.
2015), (Dumitrescu et al. 2017) and MC generated nucleation rates (Hale and DiMattio, 2017). See Fig. 10 of
(Wyslouzil and Wölk, 2016) for remaining data sources identified at the side.
Fig 2. Helmholtz free energy differences for small TIP4P clusters scaled with [Tc/T – 1] (Hale and DiMattio, 2004).
The -f = ln[Q(n)/(Q(n-1)Q(1)vn/V)] where Q(n) are the n- cluster canonical configuration integrals and vn and V
are the small n-cluster volume and system volumes respectively. The free energy differences are calculated via the
Bennett technique. See (Hale, 2010) for additional formalism details.
COMMENTS AND CONCLUSIONS
For inclusion in Fig. 1 the nucleation rates in Table I were plotted vs. the scaled energy of formation
using the literature reported value of the critical temperature for TIP4P, Tc = 588K (Vega, 2011). There
was no convincing value of Tc with which the free energy differences scaled in the Monte Carlo
simulations. For purposes of demonstrating the scaling analysis Tc = 588K is used in Fig 2. It is
essential to note the importance of the intercept at infinite n cluster size in Fig. 2. This intercept is equal
to ln [liquid/vapor monomer] and enters crucially into the determination of the n-cluster number density and
subsequently the nucleation rate. Scaling does not occur unless this intercept reliably reproduces the ratio
of liquid to vapor monomer number densities, with the correct temperature dependence. See (Hale, 2010)
for further details.
The TIP4P data points in Fig. 1 from the present MC analysis appear to agree reasonably with the
results of (Merikanto et al.2004) and surprisingly with a range of experimental data points. A feature of
note is that the present TIP4P MC nucleation rates appear to have an  slightly larger than 1.47. At high
nucleation rates the TIP4P MC data points are consistent with the MD results of (Dumitrescu et al. 2017)
and not far from the MD results for the SPC/E potential (Tanaka et al, 2014) and (Angelil et al., 2015).
Angelil et al. have found an exponent of 1.7 at high nucleation rates whereas the experimental data at
lower rates are more aligned with the 3/2 value. This very interesting result will no doubt have important
consequences in understanding the role which the effective potential model and the critical cluster size
play in the scaling analysis.
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