Theory and UG for the SurfaceKinetics BC
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@@ -8,14 +8,17 @@ Bulk physics | |
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| H transport | ||
| ^^^^^^^^^^^ | ||
| The model developed by McNabb & Foster :cite:`McNabb1963` is used to model hydrogen transport in materials in FESTIM. | ||
| The principle is to separate mobile hydrogen :math:`c_\mathrm{m}\,[\mathrm{m}^{-3}]` and trapped hydrogen :math:`c_\mathrm{t}\,[\mathrm{m}^{-3}]`. | ||
| The diffusion of mobile particles is governed by Fick’s law of diffusion where the hydrogen flux is | ||
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| .. math:: | ||
| :label: eq_difflux | ||
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| J = -D \nabla c_\mathrm{m} | ||
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| where :math:`D=D(T)\,[\mathrm{m}^{2}\,\mathrm{s}^{-1}]` is the diffusivity. | ||
| Each trap :math:`i` is associated with a trapping and a detrapping rate :math:`k_i\,[\mathrm{m}^{3}\,\mathrm{s}^{-1}]` and :math:`p_i\,[\mathrm{s}^{-1}]`, respectively, as well as a trap density :math:`n_i\,[\mathrm{m}^{-3}]`. | ||
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| The temporal evolution of :math:`c_\mathrm{m}` and :math:`c_{\mathrm{t}, i}` are then given by: | ||
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@@ -29,7 +32,7 @@ The temporal evolution of :math:`c_\mathrm{m}` and :math:`c_{\mathrm{t}, i}` are | |
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| \frac {\partial c_{\mathrm{t}, i}} { \partial t} = k_i c_\mathrm{m} (n_i - c_{\mathrm{t},i}) - p_i c_{\mathrm{t},i} | ||
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| where :math:`S_j=S_j(x,y,z,t)\,[\mathrm{m}^{-3}\,\mathrm{s}^{-1}]` is a source :math:`j` of mobile hydrogen. In FESTIM, source terms can be space and time dependent. These are used to simulate plasma implantation in materials, tritium generation from neutron interactions, etc. | ||
| These equations can be solved in cartesian coordinates but also in cylindrical and spherical coordinates. This is useful, for instance, when simulating hydrogen transport in a pipe or in a pebble. FESTIM can solve steady-state hydrogen transport problems. | ||
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| Soret effect | ||
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@@ -41,18 +44,18 @@ FESTIM can include the Soret effect :cite:`Pendergrass1976,Longhurst1985` (also | |
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| J = -D \nabla c_\mathrm{m} - D\frac{Q^* c_\mathrm{m}}{k_B T^2} \nabla T | ||
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| where :math:`Q^*\,[\mathrm{eV}]` is the Soret coefficient (also called heat of transport) and :math:`k_B` is the Boltzmann constant. | ||
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| Conservation of chemical potential at interfaces | ||
| ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ | ||
| Continuity of local partial pressure :math:`P\,[\mathrm{Pa}]` at interfaces between materials has to be ensured. In the case of a material behaving according to Sievert’s law of solubility (metals), the partial pressure is expressed as: | ||
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| .. math:: | ||
| :label: eq_Sievert | ||
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| P = \left(\frac{c_\mathrm{m}}{K_S}\right)^2 | ||
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| where :math:`K_S\,[\mathrm{m}^{-3}\,\mathrm{Pa}^{-0.5}]` is the material solubility (or Sivert's constant). | ||
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| In the case of a material behaving according to Henry's law of solubility, the partial pressure is expressed as: | ||
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@@ -61,7 +64,7 @@ In the case of a material behaving according to Henry's law of solubility, the p | |
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| P = \frac{c_\mathrm{m}}{K_H} | ||
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| where :math:`K_H\,[\mathrm{m}^{-3}\,\mathrm{Pa}^{-1}]` is the material solubility (or Henry's constant). | ||
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| Two different interface cases can then occur. At the interface between two Sievert or two Henry materials, the continuity of partial pressure yields: | ||
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@@ -85,7 +88,7 @@ At the interface between a Sievert and a Henry material: | |
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| It appears from these equilibrium equations that a difference in solubilities introduces a concentration jump at interfaces. | ||
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| In FESTIM, the conservation of chemical potential is obtained by a change of variables :cite:`Delaporte-Mathurin2021`. The variable :math:`\theta\,[\mathrm{Pa}]` is introduced and: | ||
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| .. math:: | ||
| :label: eq_theta | ||
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@@ -111,7 +114,7 @@ According to the latter, the rate :math:`k(T)` of a thermally activated process | |
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| k(T) = k_0 \exp \left[-\frac{E_k}{k_B T} \right] | ||
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| where :math:`k_0` is the pre-exponential factor, :math:`E_k\,[\mathrm{eV}]` is the process activation energy, and :math:`T\,[\mathrm{K}]` is the temperature. | ||
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| Heat transfer | ||
| ^^^^^^^^^^^^^^ | ||
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@@ -122,7 +125,8 @@ To properly account for the temperature-dependent parameters, an accurate repres | |
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| \rho C_p \frac{\partial T}{\partial t} = \nabla \cdot (\lambda \nabla T) + \sum_i Q_i | ||
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| where :math:`T` is the temperature, :math:`C_p\,[\mathrm{J}\,\mathrm{kg}^{-1}\,\mathrm{K}^{-1}]` is the specific heat capacity, :math:`\rho\,[\mathrm{kg}\,\mathrm{m}^{-3}]` is the material's density, | ||
| :math:`\lambda\,[\mathrm{W}\,\mathrm{m}^{-1}\,\mathrm{K}^{-1}]` is the thermal conductivity and :math:`Q_i\,[\mathrm{W}\,\mathrm{m}^{-3}]` is a volumetric heat source :math:`i`. As for the hydrogen transport problem, the heat equation can be solved in steady state. In FESTIM, the thermal properties of materials can be arbitrary functions of temperature. | ||
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| --------------- | ||
| Surface physics | ||
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@@ -272,9 +276,75 @@ Finally, convective heat fluxes can be applied to boundaries: | |
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| where :math:`h` is the heat transfer coefficient and :math:`T_{\mathrm{ext}}` is the external temperature. | ||
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| Kinetic surface model | ||
| ^^^^^^^^^^^^^^^^^^^^^ | ||
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| Modelling hydrogen retention or outgassing might require considering the kinetics of surface processes. | ||
| A representative example is the hydrogen uptake from a gas phase, when the energy of incident atoms/molecules is not high enough to | ||
| overcome the surface barrier for implantation. The general approach to account for surface kinetics :cite:`Pick1985, Hodille2017, Guterl2019, Schmid2021` consists in | ||
| introducing hydrogen surface species :math:`c_\mathrm{s}\,[\mathrm{m}^{-2}]`. | ||
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| Evolution of hydrogen surface concentration is determined by the atomic flux balance at the surface, as sketched in the simplified energy diagram below. | ||
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| .. thumbnail:: images/potential_diagram.png | ||
| :align: center | ||
| :width: 800 | ||
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| Idealised potential energy diagram for hydrogen near a surface of an endothermic metal. Energy levels are measured from the :math:`\mathrm{H}_2` state | ||
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| The governing equation for surface species is: | ||
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| .. math:: | ||
| :label: eq_surf_conc | ||
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| \dfrac{d c_\mathrm{s}}{d t} = J_\mathrm{bs} - J_\mathrm{sb} + J_\mathrm{vs}~\text{on}~\delta\Omega | ||
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| where :math:`J_\mathrm{bs}\,[\mathrm{m}^{-2}\,\mathrm{s}^{-1}]` is the flux of hydrogen atoms from the subsurface (bulk region just beneath the surface) onto the surface, | ||
| :math:`J_\mathrm{sb}\,[\mathrm{m}^{-2}\,\mathrm{s}^{-1}]` is the flux of hydrogen atoms from the surface into the subsurface, and :math:`J_\mathrm{vs}\,[\mathrm{m}^{-2}\,\mathrm{s}^{-1}]` is the net flux of hydrogen | ||
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There was a problem hiding this comment. does J_vs contains the desorbing flux and all abstraction fluxes (ELy Rideal recombination etc) ? There was a problem hiding this comment. In general, yes, but it should be defined by a user. Here is an example from one validation case on how it can be defined: def J_vs(T, surf_conc, t):
phi_atom = SP * Gamma_atom * (1 - surf_conc / n_surf) # adsorption
phi_exc = Gamma_atom * sigma_exc * surf_conc # abstraction
phi_des = 2 * nu0 * (lambda_des * surf_conc) ** 2 * f.exp(-2 * E_des / F.k_B / T) # desorption
return phi_atom - phi_exc - phi_desThere was a problem hiding this comment. Maybe this could be explicited then? something like There was a problem hiding this comment. In this case, should we somehow clarify that the flux comes onto the surface: There was a problem hiding this comment. Maybe just say J_vs = J_in - J_out where J_in is the sum of all fluxes coming from the vacuum to the surface and J_out from the surface to the vacuum Or something like that and then give a few examples There was a problem hiding this comment. A few examples on different processes that can be included? There was a problem hiding this comment. Yes just like you had in your previous comment There was a problem hiding this comment. I updated this part. Let me know what you think. There was a problem hiding this comment. Looks good to me! |
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| atoms from the vacuum onto the surface. It worth noticing that the current model does not account for possible surface diffusion and is available only for 1D hydrogen transport simulations. | ||
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| The connection condition between surface and bulk domains represents the Robin boundary condition for the hydrogen transport problem. | ||
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| .. math:: | ||
| :label: eq_subsurf_conc | ||
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| -D \nabla c_\mathrm{m} \cdot \mathbf{n} = \lambda_{\mathrm{IS}} \dfrac{\partial c_{\mathrm{m}}}{\partial t} + J_{\mathrm{bs}} - J_{\mathrm{sb}}~\text{on}~\delta\Omega | ||
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| where :math:`\lambda_\mathrm{IS}\,[\mathrm{m}]` is the distance between two interstitial sites in the bulk. | ||
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| .. note:: | ||
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| At the left boundary, the normal vector :math:`\textbf{n}` is :math:`-\vec{x}`. The steady-state approximation of eq. :eq:`eq_subsurf_conc` at the left boundary | ||
| is, therefore, :math:`D\frac{\partial c_\mathrm{m}}{\partial x}=J_\mathrm{bs}-J_\mathrm{sb}` representing eq. (12) in the original work of M.A. Pick & K. Sonnenberg :cite:`Pick1985`. | ||
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| The fluxes for subsurface-to-surface and surface-to-subsurface transitions are defined as follows: | ||
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| .. math:: | ||
| :label: eq_Jbs | ||
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| J_\mathrm{bs} = k_\mathrm{bs} \lambda_\mathrm{abs} c_\mathrm{m} \left(1-\dfrac{c_\mathrm{s}}{n_\mathrm{surf}}\right) | ||
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| .. math:: | ||
| :label: eq_Jsb | ||
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| J_\mathrm{sb} = k_\mathrm{sb} c_\mathrm{s} \left(1-\dfrac{c_\mathrm{m}}{n_\mathrm{IS}}\right) | ||
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| where :math:`n_\mathrm{surf}\,[\mathrm{m}^{-2}]` is the surface concentration of adsorption sites, :math:`n_\mathrm{IS}\,[\mathrm{m}^{-3}]` is the bulk concentration of interstitial sites, | ||
| :math:`\lambda_\mathrm{abs}=n_\mathrm{surf}/n_\mathrm{IS}\,[\mathrm{m}]` is the characteristic distance between surface and subsurface sites, :math:`k_\mathrm{bs}\,[\mathrm{s}^{-1}]` | ||
| and :math:`k_\mathrm{sb}\,[\mathrm{s}^{-1}]` are the rate constants for subsurface-to-surface and surface-to-subsurface transitions, respectively. | ||
| Usually, these rate constants are expressed in the Arrhenius form: :math:`k_i=k_{i,0}\exp(-E_{k,i} / kT)`. Both these processes are assumed to take place | ||
| if there are available sites on the surface (in the subsurface). Possible surface/subsurface saturation is accounted for with terms in brackets. | ||
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| .. note:: | ||
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| In eq. :eq:`eq_Jsb`, the last term in brackets is usually omitted :cite:`Guterl2019, Pick1985, Hodille2017, Schmid2021`, | ||
| since :math:`c_\mathrm{m} \ll n_\mathrm{IS}` is assumed. However, this term is included in some works (e.g. :cite:`Hamamoto2020`) | ||
| to better reproduce the experimental results. | ||
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| ------------ | ||
| References | ||
| ------------ | ||
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| .. bibliography:: bibliography/references.bib | ||
| :style: unsrt | ||
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Vladimir, do you not want to also mention your papers ? I think you implemented and used this kind of model in you laser desorption papers (but it is up to you).
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@ehodille, well, I used, yes. However, I intended to mention here the pioneers, whereas my papers on LID were mainly based on your works.