Page 20 - Nanotechnology an introduction

Consider a heterogeneous reaction A + B → C, where A is a gas or a substance dissolved in a liquid and B is a solid. Only the surface atoms are
able to come into contact with the environment, hence for a given mass of material B the more finely it is divided the more reactive it will be, in
terms of numbers of C produced per unit time.
The above considerations do not imply any discontinuous change upon reaching the nanoscale. Granted, however, that matter is made up of
atoms, the atoms situated at the boundary of an object are qualitatively different from those in the bulk (Figure 2.3). A cluster of six atoms (in two-
dimensional Flatland) has only one bulk atom (see Figure 2.1), and any smaller cluster is “all surface”. This may have a direct impact on chemical
reactivity: it is to be expected that the surface atoms are individually more reactive than their bulk neighbors, since they have some free valences
(i.e., bonding possibilities). Consideration of chemical reactivity (its enhancement for a given mass, by dividing matter into nanoscale-sized pieces)
therefore suggests a discontinuous change when matter becomes “all surface”.

Figure 2.3 The boundary of an object shown as a cross-section in two dimensions. The surface atoms (white) are qualitatively different from the bulk atoms (gray), since the latter have six nearest neighbors (in two-dimensional cross-
section) of their own kind, whereas the former only have four.
Another implication concerns solubility: the vapor pressure P of a droplet, and by extension the solubility of a nanoparticle, increases with
diminishing radius r according to the Kelvin equation (cf. equation 2.4)

(2.7)
In practice, however, the surface atoms may have already satisfied their bonding requirements by picking up reaction partners from the
environment. For example, many metals become spontaneously coated with a film of their oxide when left standing in air, and as a result are
chemically more inert than pure material. These films are typically thicker than one atomic layer, implying complex growth processes involving
internal transport or diffusion. For example, on silicon the native oxide layer is about 4 nm thick. This implies that a piece of freshly cleaved silicon
undergoes some lattice disruption enabling oxygen atoms to effectively penetrate deeper than the topmost layer. Thus it may happen that if the
object is placed in the “wrong” environment, the surface compound may be so stable that the nanoparticles coated with it are actually less reactive
than the same mass of bulk matter. A one centimeter cube of sodium taken from its protective fluid (naphtha) and thrown into a pool of water will act
in a lively fashion for some time, but if the sodium were first cut up into one micrometer cubes, most of the metallic sodium would have already
reacted with moist air before reaching the water.
Consider the prototypical homogeneous reaction

(2.8)
and suppose that the reaction rate coefficient k is much less than the diffusion-limited rate, that is, , where d and D are
f
the molecular radii and diffusivities respectively. Then [147]:

(2.9)

where a and b are the numbers (concentrations) of A and B, the angular brackets denote expected numbers, and γ is the number of C molecules
t
2
created up to time t. The term Δ (γ) expresses the fluctuations in : supposing that γ approximates to a Poisson distribution,
t
t
2
then Δ (γ) will be of the same order of magnitude as . The kinetic mass action law (KMAL) putting etc., the subscript 0 denoting
t
2
initial concentration at t = 0, is a first approximation in which Δ (γ) is supposed negligibly small compared to 〈a〉 and 〈b〉, implying that 〈a〉〈b〉 =
t
2
〈ab〉, whereas strictly speaking it is not since a and b are not independent. Nevertheless, the neglect of Δ (γ) is justified for molar quantities of
t
starting reagents (except near the end of the process, when 〈a〉 and 〈b〉 become very small), but not for reactions in ultrasmall volumes
(nanomixers).
2
These number fluctuations, i.e. the Δ (γ) term, constantly tend to be eliminated by diffusion. On the other hand, because of the correlation between
t
a and b, initial inhomogeneities in their spacial densities may (depending on the relative rates of diffusion and reaction) lead to the development of
−1
zones enriched in either one or the other faster than the enrichment can be eliminated by diffusion. Hence instead of A disappearing as t (when
a = b ), it is consumed as t −3/4 , and in the case of a reversible reaction, equilibrium is approached as t −3/2 . Deviations from perfect mixing are
0
0
more pronounced in dimensions lower than three.
As the reaction volume decreases, the relative importance of fluctuations in the concentrations of the various chemical species increases, but this
does not in itself constitute a qualitative change. In cases more complicated than the single reaction A + B → C, however, in which additional
parallel or sequential reactions are possible, the outcome may actually change. For example, consider what happens if an additional reaction A +
C → D is possible. In the usual bulk situation A-rich regions will rapidly become coated by C-rich regions, which in turn will further react with A to
form D. If, however, C gets diluted in the ocean of B before it can undergo further reaction with A, D will be formed in very poor yield (see Figure
2.4). Ways of achieving such rapid dilution could be to somehow divide the reaction space into very small volumes, or to impose a drastically
violent mixing regime on the bulk, although friction with the walls of the containing vessel prevent turbulence from filling the entire volume.