Size Effects in Electrocatalysis of Fuel Cell Reactions on Supported Metal Nanoparticles - Fuel Cell Catalysis: A Surface Science Approach - page 503

with decreasing size below about 2.5 nm much more dramatically than predicted by
the Gibbs - Thompson relation [Campbell et al., 2002]. They proposed a simple pair-
wise bond-additivity model, which reproduces the dramatic dependence of the energy
on the cluster size very well in the size interval from 1 to 4 nm. Sun developed a bond
order - length - strength correlation model based on the following principles: (i) relax-
ation of bond length in nanoparticles because of the reduced atomic coordination num-
bers (CN) at the surface; (ii) increased binding energy of the relaxed bonds; and (iii)
concomitant variation of cohesive energy of the surface atoms from the bulk value
[Sun, 2007]. The model is capable of explaining a variety of nanoscale phenomena,
including phase transitions, sintering, chemical reactivity, and thermal stability.
Although, for obvious reasons, classical thermodynamics cannot provide a quanti-
tative account of the vast variety of phenomena occurring on the nanoscale, it does
make some useful semiquantitative predictions in the scalable size interval. For
example, based on the Gibbs - Thompson relation, Plieth predicted that for a metal
composed of small particles, the redox potential E(d ) of the M z þ /M 0 transition
must scale with the inverse particle diameter and shift negatively with respect to the
value characteristic of the bulk metal [Plieth, 1982]:

E(d) ¼ E(d ¼ 1) 4y m g
zFd

(15 : 5)

Despite the limitations of (15.5) [Horanyi, 1985], it explains reasonably well the
Ostwald ripening of Ag particles by the dissolution of small particles and the
growth of large particles [Redmond et al., 2005; Schroeder et al., 2006]. The negative
shift of the Cu 2 þ /Cu 0 redox potential for 3 nm colloidal Cu particles with respect to
the value for bulk Cu has been confirmed, for example, in Savinova et al. [1988].
It has recently been suggested that the thermodynamic approach may also be help-
ful in understanding PSEs in catalysis and adsorption [Parmon, 2007; Savinova,
2006]. The idea is essentially based on the assumption that the activation energy
for adsorption (desorption) occurring on the surface of heterogeneous catalysts must
depend on the chemical potential of the latter through the Brønsted - Evans -
Polanyi rule establishing a linear relationship between the activation energy and
the heat of reaction. Given that the chemical potential of a catalyst in a dispersed
state depends on the particle size, this simple reasoning predicts a particle size depen-
dence of adsorption/desorption and heterogeneous catalytic reactions. In other words,
the adsorption/desorption energetics must be influenced by particle size. An influence
of the chemical potential of metal nanoparticles on the rates of electrochemical
reactions on their surfaces has been proposed [Nagaev, 1992]. It would, therefore,
be very interesting to use more rigorous computational approaches to estimate chemi-
cal potentials of metal nanoparticles relevant to fuel cell electrocatalysis and consider
their possible influence on the activation barriers for electrocatalytic reactions.
Although one may hardly expect quantitative agreement with experiment, since the
surface energy must depend on the chemical environment and on the electrode poten-
tial, the thermodynamic approach may help in understanding and predicting some
essential trends.