Section 'Trapping kinetics' is still under construction!!

# TechPaper #2016002: Trapping kinetics

## Compatibility

MatCalc version: 6.0100 - …

Author: Y. Shan

Created: 2016-08-09

Revisions:

## Objectives

This paper describes how trapping for kinetics is evaluated in *MatCalc*. It serves as a basis for the discussion for the trapping model.

# Main document

## Thermodynamics

The kinetics for trapped elements are obtained by the thermodynamic extremal principle [1]

\[
Q = -\dot{G} .
\]

The dissipation $Q$ at dislocations of an element $i$ due to diffusion can be approximated by [2]

\[
Q \approx \frac{ R_g T a^2 V_L }{2 D_i \Omega y_{Li} V_{T,disl}} \dot{y}_T^2 \left( \ln \left( \frac{R}{a} \right) - \frac{3}{4} \right) ,
\]

with $a$ being the acting radius of a dislocation (sites within this radius are considered as trap positions) in a cylindrical unit cell of radius $R$. $D_i$ the diffusion coefficient of element $i$. $V_{T,disl}$ is the molar volume for trap sites in dislocations and is given by

\[
V_{T,disl} = \frac{1}{\rho \pi a^2},
\]

with $\rho$ being the dislocation density.

Using the thermodynamic extremal principle, this leads to the following evolution of trapped elements

\[
\dot{y_T} = \frac{2D_i}{a^2\left(\ln\frac{R}{a}-\frac{3}{4}\right)} y_L \left( \ln\frac{y_L(1-y_T)}{y_T(1-y_L)} + \frac{\Delta E}{R_gT} \right) .
\]

The dissipation $Q$ at grain boundaries due to diffusion is given by [2]

\[
Q = \frac{ R_g T \delta V_L}{10 D_i \Omega y_{Li} V_{T,gb}} \dot{y}_T^2 ,
\]

and the molar volume of trap positions for grain boundaries

\[
V_{T,gb} = \frac{\left(2R_G + \delta \right)^3}{12 R_G^2 \delta} \approx \frac{2 R_G}{3 \delta} ,
\]

with $R_G$ being the grain radius in a unit cell and $\delta$ the grain boundary thickness, which is considered to be the positions for trapping.

The evolution of trapped elements at grain boundaries is finally given by
\[
\dot{y_T} = \frac{10 D_i}{R_G \delta} y_L \left( \ln\frac{y_L(1-y_T)}{y_T(1-y_L)} + \frac{\Delta E}{R_gT} \right) .
\]

## References

[1] J. Svoboda, I. Turek, F.D. Fischer, Application of the thermodynamic extremal principle to modelling of thermodynamic processes in material science, Phil. Mag. 85 (2005) 3699-3707.

[2] J. Svoboda, G.A. Zickler, E. Kozeschnik, F.D. Fischer, Kinetics of interstitial segregation in Cottrell atmospheres and grain boundaries, Phil. Mag. Lett. 95 (2015) 458-465.