Size Effects in Electrocatalysis of Fuel Cell Reactions on Supported Metal Nanoparticles - Fuel Cell Catalysis: A Surface Science Approach - page 501

spherical metal (e.g., Pt) particle with a size of 1 nm, D 0.9, i.e., 90% of the atoms
are exposed to the surface, and this fact may strongly affect the physical and chemical
properties of the particle. In the so-called scalable interval of particle sizes, physical
properties of materials that are determined by the surface-to-volume ratio scale with
the inverse diameter [Roduner, 2006]. The so-called nonscalable interval provides
the most spectacular manifestations of size effects on material properties, and this
has been widely documented in solid state physics. However, until now, all published
investigations of the electrochemical and electrocatalytic properties of materials have
fallen in the scalable size interval. As discussed below, a variety of physical properties
of nanoparticulate materials in the scalable size interval may be explained within the
thermodynamic approach. After a short introduction to the thermodynamics of dis-
perse systems, we will briefly discuss structural and electronic properties of metal
nanoparticles, which will be essential for further analysis of PSEs in electrocatalysis.

15.2.1 Thermodynamics of Small Particles

The foundations of classical thermodynamics of disperse materials were laid by Gibbs
and are described in nearly every undergraduate course of chemical thermodynamics.
To learn about more recent developments, the reader is referred to Hill's treatise [Hill,
1963, 1964]. The essential quantity in the thermodynamics of disperse materials is the
surface tension g, which for liquids equals the surface free energy s. The excess
surface energy results in the chemical potential of disperse materials m(d ) being
higher than that for the corresponding bulk. For example, for spherical liquid droplets
of diameter d, the chemical potential is given by the Gibbs - Thompson relation:

m(d) ¼ m(d ¼ 1) þ 4y m g
d

(15 : 1)

where y m is the molar volume. An important consequence is that the pressure inside a
droplet exceeds the outer pressure by the so-called Laplace pressure, which for a
sphere is equal to

Dp L ¼ 4g
d

(15 : 2)

Although (15.1) and (15.2) hold strictly only for a liquid in equilibrium with its vapor,
they have been commonly applied also to solid materials, in particular for describing
nanoparticle sintering (see the discussion in [Campbell et al., 2002]). However, a
number of complications must be considered for solid materials. First of all, g= s,
since for a solid a change in the surface area A can be realized either by increasing
the number of surface atoms without changing the interatomic distances between
them (this is related to the first term in (15.3)) or by introducing a strain (this is related
to the second term in (15.3)):

g ¼ s þ A @ s
@ A

(15 : 3)