Forskjell mellom versjoner av «TKP4190 - Fabrikasjon og anvendelse av nanomaterialer»
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* Size dependant crystal growth |
* Size dependant crystal growth |
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===Homogeneous nucleation=== |
===Homogeneous nucleation=== |
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+ | 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. |
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− | <math>f d^{-7}</math> |
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+ | <math>\Delta G = \Delta G_S + \Delta G_V = 4\pi r^2 \gamma + \frac{4}{3}\pi r^3 \Delta G_v</math> |
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+ | 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. |
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+ | 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> |
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+ | <br> |
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+ | 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> |
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+ | <br> |
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+ | This energy originates from random fluctuations. Rate of nucleation can thus be expressed as an Arrhenius equation:<br> |
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+ | <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> |
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===Heterogeneous nucleation=== |
===Heterogeneous nucleation=== |
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===Aggregation vs Crystal Growth=== |
===Aggregation vs Crystal Growth=== |
Revisjonen fra 15. mai 2010 kl. 10:26
Innhold
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>