Introduction to Nanotechnology

330 BIOLOGICAL MATERIALS
hardened form. Bone contains, for example, many rod-shaped inorganic mineral
crystals with typical 5 nm diameters, and lengths ranging from 20 to 200 nm.
The kinetics for the self-assembly of many of these films involved in
biomineralization can be approximately modeled as the initial joining together or
dimerization of two monomers
R+R+R, (12.7)
with a low equilibrium constant KD = CD/(cF)’, followed by the stepwise or
sequential addition of more monomers
(12.8)
with a much larger equilibrium constant K = Cn+l/CFC,,. These two constants
exercise control over the rate at which the reaction proceeds. For the case under
consideration, KD < K, the concentration of free or unbound monomers C, always
remains below a critical concentration Co = 1 /K, specifically, C, < Co. When the
total concentration of free and clustered (Le., bound) monomers CT satisfies the
condition CT < Co , then the free monomer concentration C, increases with increases
in CT. When a high enough concentration is provided so that CT becomes larger than
the critical value (i.e., CT > Co), then the aggregate forms and grows for further
increases in CT. In analogy with this model, self-assembly kinetics often involves a
slow dimer formation step followed by faster propagation steps, with KD