Page 102 - Nanotechnology an introduction

limit” at which ϕ = 0 is θ ≈ 0.55 for spheres adsorbing irreversibly.
J
The RSA formalism was developed in the context of particles interacting predominantly via hard body repulsion. The particle radius r is implicitly
considered to be the hard body radius r. “Soluble” (stably suspended) particles must have repulsive particle–particle interactions and cannot in fact
approach each other to a center-to-center distance of 2r but will behave as particles of an effective radius r′, where r′ is that value of z at which the
total interfacial (IF) interaction energy (see Section 3.2) ΔG (323) (z) ~ k T, the thermal energy.
B
Generalized Ballistic Deposition
The ballistic deposition (BD) model describes objects falling onto a surface under the influence of gravity. Whereas in RSA if a particle attempts to
land with its center within the exclusion zone around a previously adsorbed particle it is rejected, in BD the particle is not eliminated, since it is not
buoyant enough to diffuse away, but rolls along on top of previously adsorbed particles until it finds space to adsorb. The coefficients of equation
(8.15) are then different, namely b = b = 0 and b ≈ −9.95. BD and RSA can be combined linearly to give generalized ballistic deposition (GBD),
3
1
2
where
(8.16)
with the parameter j defined as

(8.17)
where p′ is the probability that a particle arriving via correlated diffusion (“rolling”) at a space large enough to accomodate it will remain (i.e., will
surmount any energy barrier), and p is the probability that a particle arriving directly at a space large enough to accomodate it will remain. p is
clearly related to the lateral interaction (“stickiness”) of particles for each other, and as j → ∞ the model describes nanoparticle aggregation at a
surface. Essentially, the exclusion zones are thereby annihilated, and ϕ can be simplified to eqn (8.13).

Two-Dimensional Crystallization
If, however, the particles can adhere to each other on the interface, the possibility for organizing arises. This has been very clearly demonstrated
when lateral mobility was expressly conferred on the particles by covering the substratum with a liquid-crystalline lipid bilayer and anchoring the
particles (large spherical proteins) in the bilayer through a hydrophobic “tail”[142]. The particles structure themselves to form a two-dimensional
ordered array (crystal). When such an affinity exists between the particles trapped at the interface, the exclusion zones are annihilated. From this
fact alone (which can be very easily deduced from the kinetics of addition [142]) one cannot distinguish between random aggregation forming a
diffusion-limited aggregate (DLA) (cf. reaction-limited aggregation, RLA) and two-dimensional crystallization; they can generally be distinguished,
however, by the fact that in the latter the crystal unit cell size is significantly bigger than the projected area of the particle (cf. a three-dimensional
protein crystal: typically about 70% of the volume of such a crystal is occupied by solvent). The process of two-dimensional crystallization has two
characteristic timescales: the interval τ between the addition of successive particles to the interface
a

(8.18)
where a is the area per particle, F is the flux of particles to an empty surface (proportional to the bulk particle concentration and some power < 1 of
the coefficient of diffusion in three dimensions), and ϕ is the fraction of the surface available for addition, which is some function of θ, the fractional
surface coverage of the particles at the interface; and the characteristic time τ for rearranging the surface by lateral diffusion (with a diffusion
D
coefficient D )
2

(8.19)
If τ ≫ τ then lateral diffusion is encumbered by the rapid addition of fresh particles before self-organization can occur and the resulting structure is
D
a
indistinguishable from that of random sequential addition. Conversely, if τ ≫ τ there is time for two-dimensional crystallization to occur. Note that
D
a
some affinity-changing conformational change needs to be induced by the interface, otherwise the particles would already aggregate in the bulk
suspension. In the example of the protein with the hydrophobic tail, when the protein is dissolved in water the tail is buried in the interior of the
protein, but partitions into the lipid bilayer when the protein arrives at its surface.
Another intriguing example of interfacial organization is the heaping into cones of the antifreeze glycoprotein (AFGP), consisting of repeated
alanine–alanine–glycosylated threonine triplets, added to the surface of a solid solution of nanocrystalline Si Ti O [102]. Under otherwise
0.6 0.4 2
identical conditions, on mica the glycoprotein adsorbs randomly sequentially. Despite the simplicity of the structure of AFGP, its organization at the
silica–titania surface appears to be a primitive example of programmable self-assembly (Section 8.2.8).

8.2.8. Programmable Self-Assembly
The addition of monomers to a growing crystal is self-assembly (Section 6.4), but the result is not useful as a nanofacturing process because there
is nothing to limit growth, except when the imposed conditions are carefully chosen (Section 6.1.2). Crystallization is an example of passive or
nonprogrammable self-assembly. Its ubiquity might well engender a certain pessimism regarding the ultimate possibility of realizing true self-
assembly, the goal of which is to produce vast numbers of identical, prespecified arbitrary structures. Yet, in biology, numerous examples are
known (see also Section 1.4): the final stages of assembly of bacteriophage viruses, of ribosomes and of microtubules; they occur not only in vivo
but can also be demonstrated in vitro by simply mixing the components together in a test-tube. As apparent examples of what might be called
“passive” self-assembly, in which objects possessing certain asymmetric arrangements of surface affinities are randomly mixed and expected to
produce ordered structures [57], they seem to contradict the predictions of 8.2.2 and 8.2.3.
It has long been known that biomolecules are constructions: that is, they have a small number of macroscopic (relative to atomic vibrations)
degrees of freedom, and can exist in a small number (≥ 2) of stable conformations. Without these properties, the actions of enzymes, active
carriers such as hemoglobin, and the motors that power muscle, etc., are not understandable. Switching from one conformation to another is