MatCalc version: 6.0100 - …

Author: Y. Shan

Created: 2016-08-01

Revisions:

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

Trapping of an element $i$ on a trap $k$ reduces the system energy by an amount of $\Delta E$. Traps are usually system defects such as solute elements, dislocations or grain boundaries. The molar concentration $c_i$ of the trapped element $i$ can be divided into a lattice (free) part, denoted by indices $Li$, and into trapped parts by traps $k$, denoted by indices $Tki$

\[ c_i = c_{Li} + \sum_{k}c_{Tki} . \]

The occupancy of all lattice positions by an element $i$ is given by $y_{Li}$ and vice versa for trap positions of a trap $k$ by $y_{Tki}$. These are the product of the molar concentrations $c_{Li}$, $c_{Tki}$ and the molar volumes $V_{L}$, $V_{Tk}$

\[ y_{Li} = c_{Li} V_{L}, \quad y_{Tki} = c_{Tki} V_{Tk} . \]

The molar volume $V_{L}$ corresponds to the volume of 1 mole of lattice positions, whereas the molar volume $V_{Tk}$ corresponds to the volume of 1 mole of trap positions of a trap $k$. The sum of the inverse molar volumes gives the inverse molar volume of the system $\Omega$

\[ \frac{1}{\Omega} = \frac{1}{V_L}+\sum_{k}\frac{1}{V_{Tk}} . \]

As for the molar volume of trapping sites $V_{Tk}$, the value is given by the amount of trapping sources, expressed by a molar concentration $c_k$, and their range, given by the coordination number $Z_k$

\[ V_{Tk} = \frac{1}{Z_k c_k} . \]

Now the Gibbs Energy can be expressed by

\[G = G_0 + R_gT \left[ \frac{\Omega}{V_L}\left( \sum\limits_{i} y_{Li} \ln y_{Li} + (1- \sum\limits_{i} y_{Li}) \ln (1-\sum\limits_{i} y_{Li})\right) + \\ \sum\limits_{k} \frac{\Omega}{V_{Tk}} \left(\sum\limits_{i} y_{Tki} \ln y_{Tki} + (1- \sum\limits_{i} y_{Tki}) \ln (1-\sum\limits_{i} y_{Tki} \right)\right] - \sum\limits_{k}\sum\limits_{i}\frac{\Omega}{V_{Tk}}y_{Tki}\Delta E_{ki}. \]

The equilibrium case for element $j$ trapped on a trap $l$ can be calculated by $\dot{G}=0$

\[ \frac{y_{Lj}}{1-\sum\limits_{i}y_{Li}} \frac{1-\sum\limits_{k}\sum\limits_{i}y_{Tki}}{y_{Tlj}} = \exp\left(\frac{\Delta E_{lj}}{R_gT}\right) . \]

[1] J. Svoboda, F. D. Fischer, Modelling for hydrogen diffusion in metals with traps revisited, Acta Mater. 60 (2012) 1211-1220.

[2] F. D. Fischer, J. Svoboda, E. Kozeschnik, Interstitial diffusion in systems with multiple sorts of traps, Model. Simul. Mater. Sci. Eng. 21 (2013) 025008/1-13.