Reaction mechanisms | Differential form f(α) | Integral form g(α) |
---|---|---|
A2—Nucleation and nuclei growth (Avrami Eq. 1) | \( 2(1 - \alpha )[ - \ln (1 - \alpha )]^{1/2} \) | \( [ - \ln (1 - \alpha )]^{1/2} \) |
A3—Nucleation and nuclei growth (Avrami Eq. 2) | \( 3(1 - \alpha )[ - \ln (1 - \alpha )]^{3/2} \) | \( [ - \ln (1 - \alpha )]^{1/3} \) |
A4—Nucleation and nuclei growth (Avrami Eq. 3) | \( 4(1 - \alpha )[ - \ln (1 - \alpha )]^{3/4} \) | \( [ - \ln (1 - \alpha )]^{1/4} \) |
R2—Phase boundary controlled reaction (contracting area) | \( 2(1 - \alpha )^{1/2} \) | \( [1 - (1 - \alpha )]^{1/2} \) |
R3—Phase boundary controlled reaction (contracting volume) | \( 3(1 - \alpha )^{2/3} \) | \( [1 - (1 - \alpha )]^{1/3} \) |
D1—One-dimensional diffusion | \( (1/2)\alpha \) | \( \alpha^{2} \) |
D2—Two-dimensional diffusion (Valensi equation) | \( [ - \ln (1 - \alpha )]^{ - 1} \) | \( (1 - \alpha )\ln (1 - \alpha ) + \alpha \) |
D3—Three-dimensional diffusion (Jander equation) | \( (3/2)[1 - (1 - \alpha )^{1/3} ]^{ - 1} (1 - \alpha )^{2/3} \) | \( [1 - (1 - \alpha )^{1/3} ]^{2} \) |
D4—Three-dimensional diffusion (Ginstling–Brounshtein equation) | \( (3/2)[1 - (1 - \alpha )^{1/3} ]^{ - 1} \) | \( [1 - (2/3)\alpha )] - (1 - \alpha )^{2/3} \) |
F1—Random nucleation with one nucleus on the individual particle | \( 1 - \alpha \) | \( - \ln (1 - \alpha ) \) |
F2—Random nucleation with two nuclei on the individual particle | \( (1 - \alpha )^{2} \) | \( 1/(1 - \alpha ) \) |
F3—Random nucleation with three nuclei on the individual particle | \( (1/2)(1 - \alpha )^{3} \) | \( 1/(1 - \alpha )^{2} \) |
P1—Mampel power law \( \left( {n \, = \,{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \right) \) | \( 2\alpha^{1/2} \) | \( \alpha^{1/2} \) |
P2—Mampel power law \( \left( {n \, = \,{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}} \right) \) | \( 3\alpha^{2/3} \) | \( \alpha^{1/3} \) |
P3—Mampel power law \( \left( {n \, = \,{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 4}}\right.\kern-0pt} \!\lower0.7ex\hbox{$4$}}} \right) \) | \( 4\alpha^{3/4} \) | \( \alpha^{1/4} \) |