TKP4190 - Fabrikasjon og anvendelse av nanomaterialer

Pensum

Del I

Crystallization fundamentals

• Supersaturation
• Size dependant crystal growth

Homogeneous nucleation

The free energy associated with nucleation consists of two parts working against each other; the energetically favorable formation of solids and the unfavorable formation of new surfaces. <math>\Delta G = \Delta G_S + \Delta G_V = 4\pi r^2 \gamma + \frac{4}{3}\pi r^3 \Delta G_v</math> Here <math>\Delta G_S</math> is the surface excess free energy, <math>\gamma</math> is the interfacial tension between the phases, <math>\Delta G_V</math> is the volume excess free energy and <math>\Delta G_v</math> is the same per unit volume. At the point where the <math>\Delta G</math>-curve is at its max, we find the critical nucleus size: above this radius the nucleus is stable. Finding this size is straightforward: <math>\frac{\delta \Delta G}{\delta r} = 0 \Rightarrow r_{crit} = \frac{-2\gamma}{\Delta G_v} \Rightarrow \Delta G_{crit} = \frac{16 \pi \gamma^3}{3(\Delta G_v)^2} = \frac{4}{3}\pi r^2_{crit} \gamma</math>
Inserting <math>-\Delta G_v = \frac{k_B T \ln{S}}{\nu}</math> the critical energy for nucleation is <math>\Delta G_{crit} = \frac{16 \pi \gamma^3 \nu^2}{3(k_B T \ln{S})^2}</math>
This energy originates from random fluctuations. Rate of nucleation can thus be expressed as an Arrhenius equation:
<math>J = A \exp(\frac{-\Delta G}{k_B T}) = A \exp(\frac{16 \pi \gamma^3 \nu^2}{3(k_B T \ln{S})^2})</math>