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>

Heterogeneous nucleation

Critical energy changed due to availability of a solid surface. <math>\Delta G_{crit,hetr} = \phi\Delta G_{crit,hom}, \phi = \frac{1}{4}(2+\cos{\theta})(1-\cos{\theta})</math>

Growth rate limits

Diffusion controlled growth

Growth as change of particle radius per time is given as <math>\frac{dr}{dt} = D(C-C_S)\frac{V_m}{r}</math> where r is the radius, D is the diffusion coefficient of the growth species, C is the bulk concentration, <math>C_S</math> is the solubility concentration and <math>V_m</math> is the molecular volume. Solving gives <math>r^2 = 2D(C-C_S)V_mt + r_0^2</math>

• Diffusion controlled growth promotes unisized particles
• Can be obtained by increasing viscosity or introducing a diffusion barrier

Radius difference between particles decreases with time: <math>\delta r = \frac{r_0\delta r_0}{\sqrt{k_Dt + r_0^2}}</math>

Surface integration controlled growth

Growth given by <math> G = k_g(S-1)^g</math>

• Spiral growth (most common): g = 2 at very low supersaturation and g = 1 at large supersaturation
• 2D Nucleation: g > 2
• Rough growth: g=1

Mononuclear growth (layer by layer): <math>\frac{dr}{dt} = k_mr^2 \Rightarrow \frac{1}{r}=\frac{1}{r_0} - k_mt</math> and radius difference increases with time <math>\delta r = \frac{\delta r_0}{(1-k_mr_0t)^2}</math>
Polynuclear growth (multiple layers growing at once): <math>\frac{dr}{dt} = k_p \Rightarrow r=k_pt+r_0</math> and radius difference remains unchanged <math>\delta r = \delta r_0</math>