Page 101 - Nanotechnology an introduction

8.2.7. The Addition of Particles to the Solid/Liquid Interface

Many self-assembly processes are based on the addition of nano-objects to a substrate. As noted in Section 8.2.5, a surface can have a
symmetry-breaking effect. Consider a chemically and morphologically unstructured surface (medium 1) brought into contact with a fluid (medium 2)
in which objects (medium 3) are suspended (i.e., their buoyancy is such that they move purely by diffusion (Brownian motion) and are not influenced
by gravity). Suppose that the materials are chosen such that the interfacial free energy is negative (Section 3.2). On occasion, those in the vicinity of
the interface will strike it. The rate of arrival of the particles at the substratum is proportional to the product of particle concentration c in the
b
suspending medium and the diffusion coefficient D of a particle, the constant of proportionality depending on the hydrodynamic regime (e.g.,
convective diffusion); this rate will be reduced by a factor in the presence of an energy barrier; the reduction factor
could easily be a hundred or a thousandfold. Once a particle of radius r adheres to the substratum, evidently the closest the center of a second
particle can be placed to the first one is at a distance 2r from the center of the first; in effect the first particle creates an exclusion zone around it
(Figure 8.5).

Figure 8.5 The concept of exclusion zone. The particles' projected area is hatched. The area enclosed by the dashed lines is the exclusion zone and has twice the radius of the actual particle. The exclusion zone is defined as that area
within which no center of any particle can be placed without violating the condition of no overlap of hard bodies. The cross-hatched area marks the overlap of the exclusion zones of particles numbered 2 and 3. If the particles interact with
each other with longer range forces than the hard body (Born) repulsion, then the radius is increased to an effective radius equal to the distance at which the particle–particle interaction energy ΔG 323 (z) equals the thermal energy k B T.
A corollary of the existence of exclusion zones is that the interface will be jammed (i.e., unable to accept a further particle) at a surface coverage of
substantially less than 100%. The actual value of the jamming limit depends on the shape of the particle; for spheres it is about 54% of complete
surface coverage. This process is known as random sequential addition (RSA). Although a random dispersion of particles in three dimensions is
thereby reduced to a two-dimensional layer, the positions of the particles remain randomml: the radial distribution function is totally unstructured.
Even if the particles can move laterally, allowing the interface to equilibrate in a certain sense, it is still jammed at a surface coverage of well below
100%.

Numerically Simulating RSA
The process is exceptionally easy to simulate: for each addition attempt one selects a point at randomml: if it is further than 2r from the center of
any existing particle a new particle is added (the available area for less symmetrical shapes may have to be computed explicitly) and if it is closer
then the attempt is abandoned. The success of this simple algorithm is due to the fortuitous cancellation of two opposing processes: correlation
and randomization. In reality, if a particle cannot be added at a selected position because of the presence of a previously added one, it will make
another attempt in the vicinity of the first one, because of the Rabinowitch (“cage”) effect [111]; successive attempts are strongly positionally
correlated. One other hand, as a particle approaches the interface through the bulk fluid, it experiences hydrodynamic friction, which exerts a
randomizing effect; the two effects happen to cancel out each other [11].
Functions for Characterizing Nano-Object Addition
We have the fraction of occupied surface θ, equal to the product of the number of objects ν per unit area and the area a occupied by one object;
and the fraction of surface ϕ available for adsorption (sometimes called the available area function). In general, we have for the rate of addition:

(8.12)
where k is the addition rate coefficient (dependent upon the interfacial free energy function ΔG 123 (z), see Section 3.2) and c* is the effective bulk
a
concentration (subsuming hydrodynamic and other factors). Much RSA research concerns the relation of ϕ to θ. An early theory relating them was
Langmuir's: if small objects adsorb to discrete substratum sites larger than the particles,

(8.13)
Substituting this into equation (8.12) and integrating, we see that in Langmuir adsorption the surface is completely filled up (θ → 1) exponentially in
time (for a uniform rate of arrival of particles at the surface). In the absence of discrete sites (or in the presence of such sites that are smaller than
the particles), the particles adsorb wherever they happen to arrive (assumed to be random locations). If a particle arrives such that its center would
fall within the exclusion zone of a previously adsorbed particle (Figure 8.5) its adsorption attempt is rejected. Since the exclusion zone is four times
as large as the particle, we should have

(8.14)
2
but as θ increases, exclusion zones will overlap (Figure 8.5) and compensating terms have to be added, proportional to θ for two overlapping
particles, and so on up to six (the maximum number that can fit; in practice terms up to second or third order will usually be found to be sufficient):

(8.15)
with b = 4 and the coefficients b and b determined by purely geometrical considerations; b = 6√3/π is identical for both irreversible and
3
2
1
3
equilibrium adsorption, whereas the coefficient b varies from about 1.4 for irreversible (random sequential addition, RSA) to about 2.4 for
3
equilibrium (reversible, whether via desorption and readsorption or via lateral movement) adsorption. In this case ϕ → 0 for θ < 1; the “jamming