Page 262 - Academic Press Encyclopedia of Physical Science and Technology 3rd InOrganic Chemistry

P1: GNH Final Pages
Encyclopedia of Physical Science and Technology EN009M-428 July 18, 2001 1:6

Metal Particles and Cluster Compounds 519

physical area of the critical nucleus, there is a much greater
probability for the nucleus to capture atoms by diffusion
than by direct impingement.
The size of a critical nucleus of Ag, for example, can
be roughly estimated by using the capillarity theory with
bulk parameters and a typical deposition rate of 0.1 nm/sec
˚
∗
at 300 K, yielding r = 2.2 A. This suggests that critical
nuclei are of atomic dimension.
Thus, these various models usually give rise to the no-
tion of a critical-sized particle. Particles below this critical
size decay faster than they grow. Of course, this critical
FIGURE 4 Illustration of metal atom accretion to particles size will vary with the type of metal under study, the sur-
(clusters).
face material and temperature, and other factors.
The fate of the heat of condensation has been stud-
theory which shows the relationship between the size of a ied in several theoretical models and is quite interest-
spherical particle and its total free energy. The total free ing. The internal and translational temperatures are not
energy G 0 needed to form a spherical particle of ra- necessarily in thermal equilibrium. Small agglomerates
dius r is taken as the sum of the energy needed to cre- may well be “liquid” because of this extra energy and be-
ate a surface (surface energy) and the volume energy of cause of the fact that melting point decreases with cluster
condensation. Subcritical nuclei (r <r ), formed by col- size.
∗
lisions of thermally equilibrated adatoms jumping around While the kinetics and theory of agglomeration attempt
on the surface, grow with an initial increase in free energy to give size distributions, other theoretical approaches at-
until a critical size is reached, greater than which growth tempt to describe the detailed electronic structure of indi-
continues with a decrease in free energy. Agglomerates vidual particles. The usual approach taken by chemists is
(particles) smaller than r , the radius of the critical nu- to construct the particle from individual atoms and mini-
∗
cleus, are unstable and decompose, while larger clusters mize the electronic energy as a function of shape (bonding
spontaneously grow to form stable particles. It can be seen geometry). In this way the evolution from small agglom-
now that an incident atom that instantaneously thermally erates toward the bulk is clarified. In contrast the physist’s
equilibrates upon striking the surface can hop around on approach is to start with a band-type description of the
the surface for a finite length of time. If it collides with a bulk phase and to investigate how this breaks down as
critical nucleus or larger particle it becomes stabilized on the size gets smaller. In the latter case, a primary goal
the surface of that particle. If this atom collides with a sub- is often the calculation of the density of states function
critical nucleus the resultant particle may become stable of metals. Early in this century there was considerable
if the critical size results or if other adatoms collide with progress made in describing the color of colloids such as
this subcritical nucleus before dissociation occurs. The fi- gold by means of Mie’s theory and the idea of plasmon
nal fate of a migrating atom which fails to become part of responses. More recently, plasmons have been invoked in
a stable nucleus is to desorb. If an incident atom becomes an attempt to explain the surface-enhanced Raman effect.
incorporated into a stable cluster it is considered to be The calculation of the electronic properties of small metal
condensed. The probability that an impinging atom will particles becomes complicated because the spacing be-
condense is called the condensation coefficient. The sec- tween adjacent levels may become large compared to kT ,
ond nucleation theory, the atomistic theory, supports the and classical continuum models break down. It becomes
capillarity theory. The main difference between the two clear that quantum mechanical effects are very important
theories is that the capillarity theory treats a continuously when considering the electronic band structure of small
varying cluster size in terms of a continuously varying free metal clusters.
energy while the atomistic theory considers the change in When comparing theoretical calculations on metal mi-
the number of atoms in a cluster in terms of discontinuous croclusters the location of the d and s bands and their
changes in the binding energy of the adatoms comprising degree of overlap is sometimes used as a guide to their ac-
the cluster. Both theories define the nucleation rate I as curacy. In bulk metals the s band is contained within the
being proportional to the product of the concentrations of d band, and for copper clusters of eight atoms or more the
critical nuclei and the rate at which adatoms join these X∝ − SW theoretical approach predicts s and d overlap.
critical nuclei by diffusion. It is important to note that an However, another method, Hartree–Fock, predicts distant
2
adatom within an area ∼π X − of a critical nucleus will s and d bands for the same eight atom copper ag-
2
diffuse to that nucleus. Since π X − is much larger than the glomerates. Most calculational methods agree with the