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 \( \left( {n \, = \,{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 4}}\right.\kern-0pt} \!\lower0.7ex\hbox{$4$}}} \right) \)
|
\( 4\alpha^{3/4} \)
|
\( \alpha^{1/4} \)
|