Environmental Engineering Reference
In-Depth Information
Equation (15.3) is valid for isotropic solids. For anisotropic solid materials, g is a
tensor. For a crystal comprising terraces, edges, and vertices, the surface energy
may be obtained as a sum over all structural elements. Depending on the strain
resistance of a material, g may be either larger or smaller than s. For various
metals, ghas been calculated using pseudopotentials [Payne et al., 1989]. For (111)
surfaces of Al, Ir, Pt, and Au, the ratio g/swas calculated as 1.3, 1.6, 2.5, and 2.2,
respectively. Thus, for particles of these metals, the chemical potential is expected
to increase with decreasing particle size even more strongly than predicted by the
Gibbs - Thompson relation.
As pointed out by Nagaev, a further complication that has to be considered for solid
particles that are not in equilibrium with their melt or vapor is that material exchange
between the particles and the environment is lacking [Nagaev, 1991, 1992]. He has
demonstrated that in this case of “one-phase thermodynamics,” the Laplace excess
pressure is not a real physical force, but may be considered a formal quantity describ-
ing the surface renormalization of the chemical potential. Nevertheless, the chemical
potential does depend on the size, and, strictly speaking, this dependence is not
determined by the Laplace pressure, but rather by the high contribution of surface
atoms whose energy is different from that of the bulk. Thus, alterations in the chemical
potential are predicted also for thin films, although their curvature is zero. For the par-
ticular case of isotropic spherical solid particles, the predicted dependence of mon the
radius is similar to that given by the Gibbs - Thompson relation [Nagaev, 1992]. This
is a very important conclusion, which lends validity to attempts to utilize thermodyn-
amic notions to predict (at least qualitatively) the properties of particulate materials.
An essential issue concerns the size down to which the laws of classical thermodyn-
amics apply. A simplified answer is that macroscopic thermodynamics is applicable as
long as the splitting d between the electronic energy levels is less than the thermal
energy (see Section 15.2.2):
d
k
B
T
(15
:
4)
It should be pointed out, however, that even when the inequality (15.4) holds, quan-
titative deviations from the laws of classical thermodynamics may be substantial if the
numbers of the surface and volume atoms become comparable. Still, in many cases,
classical thermodynamics provides a reasonable account of the influence of size on
physical properties (e.g., the melting temperature) down to the nanometer range.
For example, using a scanning electron diffraction technique, Buffat and Borel per-
formed an elegant study of the influence of the size of Au nanocrystals supported
on carbon on their melting temperature [Buffat and Borel, 1976]. Their phenomeno-
logical model, based essentially on the equations of classical thermodynamics, is in
excellent agreement with the experimental results in the particle interval from
25 nm down to about 2.5 nm. At smaller sizes, deviations may, however, be signifi-
cant. A number of refinements have been proposed in order to apply thermodynamics
to nano-objects [Hill, 1963, 1964; Rusanov, 2005]. Campbell and co-workers per-
formed microcalorimetric measurements of the heat of adsorption of Pb onto
MgO(100), and showed that the energy of a metal atom in a nanoparticle increases
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