Bulk Diffusion-Controlled Thermal Desorption Spectroscopy with Examples for Hydrogen in Iron

Abstract

Bulk diffusion-controlled thermal desorption spectroscopy (TDS) is studied by solving the corresponding transport equations numerically as well as analytically with appropriate approximations. The two solutions are compared in order to validate the derived equations including the Kissinger equation. Besides the diffusion of the desorbed species through the sample, trapping of the species at special lattice sites within the sample is included in the numerical and approximate analytical solutions. Trapping energies are mono-energetic, multi-energetic, or are described by a box-type distribution. TDS-peaks were simulated for different heating rates, sample thicknesses, trap concentrations, and initial degrees of trap saturation. It is shown that for the case of mono-energetic traps, Kissinger’s equation is obeyed for both numerical and analytical results. This widely used equation for reaction rate-controlled studies is derived in an explicit form for diffusion-controlled processes. Together with a newly derived relation between maximum desorption rate and temperature, TDS-spectra yield information about diffusion coefficient, trap energies, and trap concentration as well as trap saturation. This is exemplified using data of two experimental studies. Although the numerical and analytical treatment is in general applicable to all diffusion species, hydrogen in iron alloys is used as a model system because of its technological importance and the increasing number of experimental work with this material.

Appendices

Appendix A

1.1 Approximate Solutions for the Desorption Rate J _m/c _o of Non-saturated Mono-energetic Traps

The maximum desorption rate is derived from Eq. [37]

$$ \left[ {\frac{\partial \ln J}{\partial T} - \frac{Q}{{RT^{2} }}} \right] = - \frac{{\pi^{2} }}{{l^{2} }}\frac{{D_{\text{f}} }}{{\theta \left( {1 + \frac{{c_{{{\text{t}}0}} }}{{c_{0} y_{\text{t}} }}} \right)}} - \frac{{E_{\text{t}} }}{{RT^{2} \left( {1 + \frac{{c_{0} y_{\text{t}} }}{{c_{{{\text{t}}0}} }}} \right)}}. $$

(A1)

For the limiting case of weak trapping or

$$ \frac{{c_{\text{to}} }}{{c_{\text{o}} y_{1} }} \ll 1 $$

(A2)

Equation [A1] reduces to

$$ \left[ {\frac{\partial \ln J}{\partial T} - \frac{Q}{{RT^{2} }}} \right] = - \frac{{\pi^{2} }}{{l^{2} }}\frac{{D_{\text{f}} }}{\theta } $$

(A3)

with the solution

$$ J(T) = J(T_{\text{o}} )\exp \left[ { - \frac{{\pi^{2} }}{{\theta l^{2} }}\int\limits_{{T_{\text{o}} }}^{T} {D_{\text{f}} {\text{d}}T} - \frac{Q}{RT}} \right] = \frac{{J(T_{\text{o}} )}}{{D_{\text{o}} }}D_{\text{f}} \exp \left[ { - \frac{{\pi^{2} }}{{\theta l^{2} }}\int\limits_{{T_{\text{o}} }}^{T} {D_{\text{f}} {\text{d}}T} } \right]. $$

(A4)

1.2 Large Trap Concentration and/or Large Trap Energy

Defined as

$$ \frac{{c_{\text{to}} }}{{c_{\text{o}} y_{1} }} \gg 1 $$

(A5)

Equation [A1] reduces to

$$ \frac{\partial \ln J}{\partial T} = - \frac{{E_{\text{t}} - Q}}{{RT^{2} }} $$

(A6)

with the solution

$$ \ln \frac{J}{{J(T_{\text{o}} )}} = + \frac{{E_{\text{t}} - Q}}{R}\left( {\frac{1}{T} - \frac{1}{{T_{\text{o}} }}} \right). $$

(A7)

Equation [25] and the definition of the filling factor f in Section III–E yields

$$ J(T_{\text{o}} ) = \frac{\pi }{l}c_{\text{f}} (T_{\text{o}} )D_{\text{f}} (T_{\text{o}} ) = \frac{\pi }{l}c_{\text{o}} fy_{\text{t}} (T_{\text{o}} )D_{\text{f}} (T_{\text{o}} ), $$

(A8)

which is inserted in Eq. [A7] leading to the final result for T = T _m

$$ J_{\text{m}} = f\frac{{\pi c_{\text{o}} }}{l}D_{\text{o}} \exp \left( {\frac{{E_{\text{t}} - Q}}{{RT_{\text{m}} }}} \right) = f\frac{{\pi c_{\text{o}} }}{l}D_{\text{f}} (T_{\text{m}} )y_{\text{t}} (T_{\text{m}} ). $$

(A9)

In the following, a box-type distribution is assumed, where for the case of non-saturated traps Eq. [45] was derived as

$$ \begin{aligned} \frac{\partial \ln J}{\partial T}\left[ {1 + \int\limits_{E1}^{E2} {\exp \left( {\frac{ - E}{RT}} \right)\frac{{c_{\text{to}} }}{{\left( {E_{1} - E_{2} } \right)c_{\text{o}} }}{\text{d}}E} } \right] \hfill \\ = \frac{Q}{{RT^{2} }} - \frac{{\pi^{2} D_{\text{f}} }}{{\theta l^{2} }} - \int\limits_{E1}^{E2} {\frac{E - Q}{{RT^{2} }}\exp \left( {\frac{ - E}{RT}} \right)\frac{{c_{\text{to}} }}{{\left( {E_{1} - E_{2} } \right)c_{\text{o}} }}{\text{d}}E} \hfill \\ \end{aligned}. $$

(A10)

Strong trapping is defined as

$$ \int\limits_{E1}^{E2} {\exp \left( {\frac{ - E}{RT}} \right)\frac{{c_{\text{to}} }}{{\left( {E_{1} - E_{2} } \right)c_{\text{o}} }}{\text{d}}E} \gg 1 $$

(A11)

leading to

$$ \begin{aligned} \frac{\partial \ln J}{\partial T} & \approx \frac{{ - \int\limits_{E1}^{E2} {\frac{E - Q}{{RT^{2} }}\exp \left( {\frac{ - E}{RT}} \right){\text{d}}E} }}{{\int\limits_{E1}^{E2} {\exp \left( {\frac{ - E}{RT}} \right){\text{d}}E} }} \\ & = \frac{{\left( {E_{2} - Q - RT} \right)\exp \left( {\frac{{ - E_{2} }}{RT}} \right) - \left( {E_{1} - Q - RT} \right)\exp \left( {\frac{{ - E_{1} }}{RT}} \right)}}{{RT^{2} \left[ {\exp \left( {\frac{{ - E_{2} }}{RT}} \right) - \exp \left( {\frac{{ - E_{1} }}{RT}} \right)} \right]}}. \\ \end{aligned} . $$

(A12)

For broad distributions defined as $\exp \left( {\frac{{ - E_{2} }}{RT}} \right) \gg \exp \left( {\frac{{ - E_{1} }}{RT}} \right)$ and Eq. [A12] yields

$$ \frac{\partial \ln J}{\partial T} = \frac{{\left( {E_{2} - Q - RT} \right)}}{{RT^{2} }}. $$

(A13)

Integration of Eq. [A13] gives

$$ \frac{J(T)}{{J(T_{\text{o}} )}} = \frac{{T_{\text{o}} }}{T}\exp \left[ {\left( {\frac{{E_{2} - Q}}{R}} \right)\left( {\frac{1}{T} - \frac{1}{{T_{0} }}} \right)} \right]. $$

(A14)

As for mono-energetic traps (Eq. [A8]), the box distribution gives for low coverage of traps (cf. Eq. [20] for f ≪ 1)

$$ J(T_{\text{o}} ) = \frac{\pi }{l}c_{\text{f}} (T_{\text{o}} )D_{\text{f}} (T_{\text{o}} ) = \frac{\pi }{l}c_{\text{o}} \frac{{f\left( {E_{2} - E_{1} } \right)}}{{RT_{\text{o}} }}\exp \left[ {\frac{{E_{2} }}{{RT_{\text{o}} }}} \right]D_{\text{f}} (T_{\text{o}} ). $$

(A15)

Inserting Eq. [A15] in Eq. [A16] yields

$$ J(T) = \frac{{T_{\text{o}} }}{T}\exp \left( {\frac{{E_{2} - Q}}{RT}} \right)\frac{\pi }{l}c_{\text{o}} \frac{{f\left( {E_{2} - E_{1} } \right)}}{{RT_{\text{o}} }}D_{\text{fo}} $$

(A16)

or

$$ \ln J(T) = \frac{{E_{2} - Q}}{RT} + \ln \frac{{f\pi c_{\text{o}} D_{\text{fo}} \left( {E_{2} - E_{1} } \right)}}{lRT}. $$

(A17)

Appendix B

2.1 Approximate Solutions for Minima of TDS-Data

In the following, it will be assumed that hydrogen could be in two traps besides being free. Thus, three peaks are expected with two minima in between and Eq. [13] applies

$$ \frac{{D_{\text{f}} }}{\theta }\frac{{\partial^{2} c_{\text{f}} }}{{\partial x^{2} }} - \frac{{c_{{{\text{t}}10}} c_{\text{f}} y_{1} E_{{1{\text{t}}}} }}{{RT_{\hbox{min} }^{2} c_{0} \left( {y_{1} + c_{\text{f}} } \right)^{2} }} - \frac{{c_{{{\text{t}}20}} c_{\text{f}} y_{2} E_{{2{\text{t}}}} }}{{RT_{\hbox{min} }^{2} c_{0} \left( {y_{2} + c_{\text{f}} } \right)^{2} }} = 0. $$

(B1)

Applying the cosinusoidal profile for c _f as given by Eq. [24] yields

$$ - \frac{{D_{\text{f}} }}{\theta }\frac{{\pi^{2} c_{\text{fm}} }}{{l^{2} }} - \frac{{c_{{{\text{t}}10}} c_{\text{fm}} y_{1} E_{{1{\text{t}}}} }}{{RT_{\hbox{min} }^{2} c_{0} \left( {y_{1} + c_{\text{fm}} } \right)^{2} }} - \frac{{c_{{{\text{t}}20}} c_{\text{fm}} y_{2} E_{{2{\text{t}}}} }}{{RT_{\hbox{min} }^{2} c_{0} \left( {y_{2} + c_{\text{fm}} } \right)^{2} }} = 0. $$

(B2)

At the first minimum between the first and second TDS-peak, the condition c _fm ≫ y ₁ ≫ y ₂ holds and the last equation becomes

$$ \frac{{D_{\text{f}} }}{\theta }\frac{{\pi^{2} c_{\text{fm}} }}{{l^{2} }} = - \frac{{c_{{{\text{t}}10}} y_{1} E_{{1{\text{t}}}} }}{{RT^{2} c_{0} c_{\text{fm}} }}. $$

(B3)

Using J _min for c _fm as defined by Eq. [25] gives

$$ \frac{{J_{\hbox{min} }^{2} T_{\hbox{min} }^{2} }}{\theta } = \frac{{D_{\text{f}} c_{{{\text{t}}10}} y_{1} E_{{1{\text{t}}}} }}{{Rc_{0} }} $$

(B4)

or

$$ \ln \frac{{J_{\hbox{min} }^{2} T_{\hbox{min} }^{2} }}{\theta } = \frac{{E_{{1{\text{t}}}} - Q}}{{RT_{\hbox{min} } }} + \ln \frac{{D_{\text{o}} y_{{{\text{o}}1}} c_{{{\text{t}}10}} E_{{1{\text{t}}}} }}{{Rc_{0} }}. $$

(B5)

At the second minimum between the second and third peak, the condition y ₁ ≫ c _fm ≫ y ₂ applies and Eq. [B2] gives

$$ - \frac{{D_{\text{f}} }}{\theta }\frac{{\pi^{2} c_{\text{fm}} }}{{l^{2} }} - \frac{{c_{{{\text{t}}20}} y_{2} E_{{2{\text{t}}}} }}{{RT^{2} c_{0} c_{\text{fm}} }} = 0. $$

(B6)

Thus analogously to Eq. [B5], the following relation holds

$$\ln \frac{J_{\hbox{min}}^{2} T_{\hbox{min}}^{2}}{\theta} = \frac{E_{{2{\hbox{t}}}} - Q} {RT_{\hbox{min}}} + \ln \frac{D_{\hbox{o}} y_{{\hbox{o}}2} c_{{\hbox{t}}20} E_{2{\hbox{t}}}}{Rc_{0}}. $$

(B7)