Classical Nucleation Theory - Aerosols: Measurement, Dosimetry, and Health Effects - page 521

Considering that P α in the nucleation experiments is normally not far from 1 atm and Δ P for the
critical drop is higher essentially we can put P β ≅ Δ P . To know the Debye frequency as a function of
pressure it is enough to know the dependence of sound velocity on pressure (see Equation 19.62) (the
density ρ β is a very weak function of pressure). In Ref. [48] the sound velocity of liquid Cs was mea-
sured in the range of temperatures 300-1500 K and pressure 2 × 10 6 − 6 × 10 7 Pa. The experimental
measurements can be approximated by the following empirical function:

2

)) .

0 1

u ( m s / ≈ ) 1.475 10 ( Pa) × P ( + P T

0

( (19.66)

where P 0 = (7.35, 6.78, and 6.10) × 10 7 Pa for temperatures 508, 530, and 554 K, respectively. For
the irst iteration we set the pressure to be 1 atm (1.013 × 10 5 Pa), then, using the density of liquid
Cs from Table 19.1 and Equations 19.66 and 19.62 we get u = 903, 895, and 886 m/s and ν max =
(1.80, 1.79, and 1.77) × 10 12 s −1 for T = 508, 530, and 554 K, respectively.
The joint solution of Equations 19.61, 19.62, and 19.64 is shown in Figure 19.4 as ln(σ S /σ ∞ ) ver-
sus ln R S . The Tolman length δ can be determined solving the GTKB differential equation (19.5)
which contains d ln (σ S ( R S )/σ ∞ )/ d ln( R S ) in the LHS and the RHS contains only δ/ R S . One can see
from Figure 19.4 that within the experimental accuracy the dependence ln(σ S ( R S )/σ ∞ ) on ln( R S )
can be approximated by a linear function with a constant slope d ln(σ S ( R S )/σ ∞ )/ d ln( R S ) = −0.240,

TABLE 19.1
Important Parameters for Liquid Metals Used in Calculations

Metal T 0 ( K ) ψ (Dyne/cm/K) C D (K) ρ (g/cm 3 ) κ (cm 2 /Dyne)

Li 3300 0.140 8.00 8120 0.47 1.0 × 10 −11

Na 2350 0.100 7.70 5460 0.87 2.3 × 10 −11

Cs 1790 0.048 7.25 3920 1.75 7.0 × 10 −11

Ag 6980 0.161 8.92 14464 9.35 1.4 × 10 −12

Hg 2540 0.210 8.04 3215 12.8 3.8 × 10 −12

Mg 3128 0.254 8.80 7674 1.59 5.1 × 10 −12

Zn 5430 0.167 8.35 6400 6.75 1.9 × 10 −12

0.37

554 K

Cs

0.36

0.35

530 K

0.34

0.33

T = 508 K

0.32

0.31

0.30

0.29

0.28 0.30 0.32 0.34 0.36

In R s (nm)

FIGURE 19.4 ln(σ S /σ ∞ ) versus ln R S as calculated by solution of Equations 19.61 and 19.64 using the experi-
mental measurements of the nucleation rate from the vapor of Cs (Figure 19.3). Open and illed symbols are
results of the irst and second iterations, respectively. Lines are linear ittings.