Page 109 - Science at the nanoscale

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June 9, 2009
(b)
(a) RPS: PSP0007 - Science-at-Nanoscale 5.1. Surface Energy 99 ch05
Figure 5.6. Schematics showing (a) sintering, and (b) Ostwald ripening.

at the expense of the smaller ones (Fig. 5.6(b)). In this context, the
Gibbs-Thompson equation used in classical crystallisation theory 4
provides us with the correlation:

2σV m
S r = S exp (5.3)
b
rRT
Here, r is the radius of the crystal, σ is the specific surface
energy, V m is the molar volume of the material, S r and S are
b
respectively the solubility of the bulk crystal and a crystal with
radius r. R and T are thermodynamic parameters: R being the gas
constant and T is the absolute temperature.
Equation 5.3 suggests that the solubility of a given crystal is
inversely dependent on its size. When two nanoparticles of dif-
ferent sizes (say r 1 and r 2 , where r 1 ≫ r 2 ) are put together in
solution, each particle will develop an equilibrium solubility with
the surrounding solvent. Thus, the particle with r 2 may dissolve
due to its higher solubility and a solute gradient develops. Con-
sequently, a net diffusion of solute from the vicinity of the smaller
particle to that of the larger particle occurs. In order to maintain
the equilibrium, solute will deposit onto the larger particle while
continuing to dissolve from the smaller particle. Such dissolution
and condensation processes will continue until the complete dis-
solution of the smaller particle. Finally, a larger uniform particle
is obtained as shown in Fig. 5.6(b).
This Ostwald ripening phenomenon is important especially for
solution growth or crystallisation of nanoparticles. In particular,
the mechanism results in the elimination of smaller particles and
thus the size distribution becomes narrower. Ostwald ripening
can be optimised by varying the process temperature and/or by
changing the concentration or the solute supply. The process has
been advantageously used to prepare nanoparticles of narrow size
distributions as discussed in Section 7.2.

4 J. W. Mullin, Crystallization, 3rd Edition, Oxford, 1997.