Environmental Engineering Reference
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D 0 = 3 h
(2.20-a)
D 0 = 2 h
(2.20-b)
where Eq. (2.20-a) for a particle or a wire with a curved surface and Eq. (2.20-b) for a film
with a plane surface. Now Eq. (2.19) can be rewritten as [77],
γ sl ( D )/γ sl0 = [1- D 0 / D ]/[1-γ sl0 D 0 /( fD )]
(2.21)
with 8κ f 2 /(3γ sl0 ) = D 0 or f = ±[(3γ sl0 D 0 )/(8κ)] 1/2 . The different signs of f correspond to the
tensile (+) and compressive (-) stress on the surface. (In our opinion, the possible physical
background of the positive or negative f could be explained based on the following
mechanism [77]: Atoms at the interface suffer a coordination number reduction, bond
contracts spontaneously, which leads to the enhancement of the atomic binding energy and
hence the tensile stress [82], or f > 0. This is the case of free nanoparticles. When the
interface atoms of different elements are intermixing, CN or the bond strength may increase,
such as alloying or compound formation. The alloying and chemical reaction may alternate
the atomic valences, which may introduce repulsive stress among the ions. An alternative
interpretation is that the element with lower T m has lower bond strength within the layer than
that on the interface, bond expansion of interface atoms is present, which should results in
negative f .)
It is known that for most metals, f is one larger than γ sl0 according to theoretical and
experimental results [77,83-86]. Some molecular dynamics works based on a hard sphere
model show that f has the same magnitude of γ sl0 [87-88]. However, the hard sphere model
itself may lead to this result where strain is absent, which co-exists with f [82]. Thus, Eq.
(2.21) may be simplified as a first order approximation,
γ sl ( D )/γ sl0 ≈ 1- D 0 / D .
(2.22)
Figure 1 shows an agreement between the model prediction of Eq. (2.22) and the
computer simulation results [15] [Because the γ sl ( D )/γ sl0 value of an unknown fcc crystal is
cited for comparison, the simplified form of Eq. (2.22) is employed].
At the same time, γ sl ( D ) values of five organic nanocrystals have also been calculated in
terms of Eq. (2.21) and shown in table 4. The model predictions are consistent with the
experimental observation [13] with the note that there exists a size distribution of
nanocrystals. As the size of crystals decreases, γ sl ( D ) decreases. At D = 4 nm, the decrease in
γ sl ( D ) for different substances is different, which is mainly induced by different D 0 or D 0 / D
values. The drop of γ sl ( D ) values in comparison with the corresponding bulk values reaches
20~40% where the critical diameter of a nucleus may be near 4 nm. Thus, the energetic
resistance for the nucleation procession in liquid may be smaller than what the CNT has
estimated [77].
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