Page 156 - Science at the nanoscale

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June 5, 2009
Formation and Self-Assembly at the Nanoscale
146
7.2 THE BOTTOM-UP APPROACH
As mentioned in Section 6.3, solution-based routes are com-
monly employed in the bottom-up preparation of quantum dots
and nanoparticles. During the preparation, a supersaturation of
This
the product species is often generated in the solution.
can be achieved either by lowering the temperature of the sat-
urated solution at equilibrium, and/or by generating a large
In this sec-
amount of less soluble product species in situ.
tion, we will discuss the homogeneous nucleation and growth
processes for the preparation of nanoparticles.
Although our
discussion will focus on these processes in solution, the funda-
mental concepts should be equally applicable for growth in gas
and solid.
Homogeneous Nucleation
7.2.1
Under the supersaturation condition, the concentration of the
solute (C) exceeds its equilibrium solubility (C o ) and the system
thus possesses a higher chemical potential according to Eq. 7.5.
The system will move toward decreasing G, thus accounting for
the driving force for the nucleation and growth processes:

C

(7.12)
∆G = −RT ln
C o
Thus, when C = C o , ∆G = 0 and the system is at equilibrium.
When C > C o , ∆G is negative and nucleation should occur spon-
taneously.
However, the increase in surface energy needs to be counter-
balanced during crystal growth. Assuming the nucleus is spheri- ch07
cal in shape, there is an increase in chemical potential due to the
surface energy (γ) given by:
2
∆µ s = 4πr γ (7.13)
Since Eq. 7.13 is for a single nucleus of size r, we will rewrite
the change in Gibbs energy (Eq. 7.12) in terms of per unit volume
(∆G v ), and express the reduction in chemical potential for the for-
mation of the new spherical nucleus as:
4 3
∆µ v = πr ∆G v (7.14)
3