Page 19 - Nanotechnology an introduction

(2.2)
where ΔP is the pressure exerted on the crystal compared with zero pressure if the growing phase is an infinite plane. This pressure can do work
on the lattice, compressing the crystal. If β is the isothermal compressibility coefficient, then

(2.3)
where V is the volume of the unperturbed crystal, and l a characteristic lattice parameter of the unperturbed (bulk) crystal. Conversely, the vapor
pressure P of a particle should be inversely dependent on its radius R, as expressed by the Gibbs-Thomson (or Kelvin) law:

(2.4)
where P is the vapor pressure of a surface of infinite radius (i.e., flat), γ and v are, respectively, the surface tension (note that once surface tension
∞
is introduced, it should also be borne in mind that surface tension is itself curvature-dependent [152]) and molecular volume of the material, and k B
and T are, respectively, Boltzmann's constant and the absolute temperature. Equation (2.4) can be used to compare the chemical potentials μ of
two spheres of radii R and R :
1
2

(2.5)
where V is the molar volume.
These equations do not predict an abrupt discontinuity at the nanoscale. To be sure, there are huge differences between the bulk and the
nanoscale (e.g., melting temperature lowered by tens of degrees), but these “unique” properties approach bulk values asymptotically, and the
detectable difference therefore depends on the sensitivity of the measuring apparatus, and hence is not useful for defining a nanoscale. On the
other hand, sometimes there is evidence for qualitative change with diminishing size; as shown, for example, by the appearance of anomalous
crystal structures with no bulk equivalent [137].
2.3. Nucleation
When a vapor is cooled it will ultimately condense, but if it is very pure it may be greatly supercooled because the appearance of a new phase—in
this case, a liquid or a solid—is necessarily a discontinuous process and therefore requires a finite nucleus of the new phase to be formed via
random fluctuations. At first sight this seems to be highly improbable, because the formation of the nucleus requires the energetically unfavorable
creation of an interface between the vapor and the new condensed phase—indeed, were this not energetically unfavorable the vapor could never
be stable. However, if the nucleus is big enough, the energetically favorable contribution of its nonsurface exceeds the surface contribution, and not
only is the nucleus stable, but also grows further, in principle to an unlimited extent (see Figure 2.2). More precisely, when atoms cluster together to
form the new phase, they begin to create an interface between themselves and their surrounding medium, which costs energy A γ, where A is the
1/3
2/3
area of the cluster's surface, equal to (4π) (3nv) , where n is the number of atoms in the cluster. At the same time each atom contributes to the
(negative) cohesive energy of the new phase, equal to nvΔG, where ΔG is the difference in the free energies (per unit volume) between the
uncondensed and condensed phases, the latter obviously having the lower energy. Summing these two contributions, at first the energy will
increase with increasing n, but ultimately the (negative) cohesive energy of the bulk will win. Differentiating with respect to n and looking for maxima
yields

(2.6)
The critical nucleation size can lay reasonable claim to be considered as the boundary of the nanoscale from the mechano-chemical viewpoint. At
least it represents a discontinuity qua crossover point. The critical nucleus size represents the boundary between “surface dominated” (an unstable
state) and “bulk dominated” (a stable state).

Figure 2.2 Sketch of the variation of free energy of a cluster containing n atoms (proportional to the cube of the radius). The maximum corresponds to the critical nucleus size n*. Clusters that have managed through fluctuations to climb
up the free energy slope to reach the critical nucleus size have an equal probability to shrink back and vanish, or to grow up to microscopic size. The shaded zone corresponds to a possible nanoscale (see text).
A corollary of this definition is that nanotechnology is strongly associated with surface technology, since the nanoscale object is necessarily one in
which all or almost all of its constituent atoms have the properties of the surface atoms of a bulk piece of what is atomically the same material.

Note that the nucleus, small as it is, has been characterized above using a thermodynamic continuum approach that is perfectly macroscopic in
spirit; in other words, it is not apparently necessary to explicitly take atomistic behavior into account to provide a good description of the
phenomena. The excellent agreement between the projections of equations based on (2.7) and experimental results on photographic latent images
constituted from only a few silver atoms caused mild surprise when first noticed by Peter Hillson working at the Kodak research laboratories in
Harrow, UK. Molecular dynamics simulations also suggest that the continuum approach seems to be valid down to scales of a few atomic radii.

2.4. Chemical Reactivity