Biomedical Engineering Reference
In-Depth Information
2 R 03 "
2 #
1 mR 0
w
2
2 3 R 0 m
w
R 0 m
w
1
2
1
2
1
f.m;R 0 / D
C
C
C
2 mR 02 R 0 m
1
3
C
(2.17)
w
and
w D .1 C .R 0 / 2
2R 0 m/ 1=2 :
(2.18)
Here R 0 is actually the dimensionless radius of the curvature of the substrate with
respect to the radius of the critical nuclei r c . In other words, it only makes sense if
the curvature of a foreign body or a substrate refers to the curvature of the critical
nuclei.
Substituting appropriate values of R s , m , cf ,and into ( 2.13 )-( 2.18 ), one
can calculate f ( m , R 0 )andG heter for any nucleation process. Note that the factor
f ( m , R 0 ) varies from 1 to 0. Obviously, this factor plays an important role in the
determination of the heterogeneous nucleation barrier G heter . One can see from
( 2.5 ) that the influence of foreign particles on the nucleation barrier can be fully
characterized by f .
Figure 2.5 ashows f ( m , R 0 ) as a function of R 0 for a given m .When
R 0 ! 0; f .m; R 0 / D 1, implying that the foreign body as a nucleating substrate
will “vanish” completely. In practice, if foreign bodies are too small, i.e., clusters
of several molecules, nucleation on these substrates will not be stable. Then, the
foreign bodies play no role in lowering the nucleation barrier. If R 0 1, the foreign
body can be treated as a flat substrate compared to the critical nuclei. In this case,
f ( m , R 0 ) is solely a function of m (Fig. 2.5 b), and the curvature of the foreign body
has no effect on the nucleation kinetics. Equation ( 2.17 ) is then reduced to
1
4 .2 3m C m 3 /:
f.m;R 0 / D f.m/ D
(2.19)
2.2.2.3
Heterogeneous Nucleation: The Effect of Ice Nucleators
Let us recapture the picture of the heterogeneous nucleation model. On the substrate
surface, some molecular processes occur due to transient visiting molecules that
adsorb, form short-lived unions, break up, desorb, etc. An instantaneous census
would show some distributions of subcritical nuclei (or clusters) with 1, 2, 3, :::
molecules per cluster (cf. Fig. 2.6 ). Taking into account the effect of the substrate
on both the nucleation barrier and the transport process, and the fact that the average
nucleation rate in the fluid phase depends on the density and size of foreign particles
present in the system, the nucleation rate is given by [ 50 ].
D 4aˇ kink Œ.R s / 2 N o f 00 .m/Œf .m/ 1=2 B expŒ f .m/=.T T 2 /
J
(2.20)
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