Theory and UG for the SurfaceKinetics BC

docs/source/bibliography/references.bib

@@ -4,7 +4,7 @@ @article{McNabb1963
 	year         = 1963,
 	journal      = {Trans. Metall. Soc. AIME},
 	volume       = 227,
-	pages        = 618,
+	pages        = 618
 }
 @article{Longhurst1985,
 	title        = {{The soret effect and its implications for fusion reactors}},
@@ -33,7 +33,7 @@ @article{Delaporte-Mathurin2021
 	number       = 3,
 	pages        = {036038},
 	issn         = {0029-5515},
-	url          = {https://iopscience.iop.org/article/10.1088/1741-4326/abd95f},
+	url          = {https://iopscience.iop.org/article/10.1088/1741-4326/abd95f}
 }
 @phdthesis{Delaporte-Mathurin2022,
 	title        = {{Hydrogen transport in tokamaks : Estimation of the ITER divertor tritium inventory and influence of helium exposure}},
@@ -54,4 +54,58 @@ @article{Schmid2016
 	pages        = {014025},
 	issn         = {0031-8949},
 	url          = {https://iopscience.iop.org/article/10.1088/0031-8949/T167/1/014025}
+}
+@article{Guterl2019,
+	title        = {Effects of surface processes on hydrogen outgassing from metal in desorption experiments},
+	author       = {Guterl, Jerome and Smirnov, RD and Snyder, P},
+	year         = 2019,
+	journal      = {Nuclear Fusion},
+	publisher    = {IOP Publishing},
+	volume       = 59,
+	number       = 9,
+	pages        = {096042},
+	url          = {https://iopscience.iop.org/article/10.1088/1741-4326/ab280a/meta}
+}
+@article{Pick1985,
+	title        = {A model for atomic hydrogen-metal interactions—application to recycling, recombination and permeation},
+	author       = {Pick, MA and Sonnenberg, K},
+	year         = 1985,
+	journal      = {Journal of Nuclear Materials},
+	publisher    = {Elsevier},
+	volume       = 131,
+	number       = {2-3},
+	pages        = {208--220},
+	url          = {https://www.sciencedirect.com/science/article/abs/pii/0022311585904593}
+}
+@article{Hodille2017,
+	title        = {Simulations of atomic deuterium exposure in self-damaged tungsten},
+	author       = {Hodille, EA and Zalo{\v{z}}nik, A and Markelj, S and Schwarz-Selinger, T and Becquart, CS and Bisson, R{\'e}gis and Grisolia, Christian},
+	year         = 2017,
+	journal      = {Nuclear Fusion},
+	publisher    = {IOP Publishing},
+	volume       = 57,
+	number       = 5,
+	pages        = {056002},
+	url          = {https://iopscience.iop.org/article/10.1088/1741-4326/aa5aa5/meta}
+}
+@article{Schmid2021,
+	title        = {On the use of recombination rate coefficients in hydrogen transport calculations},
+	author       = {Schmid, K and Zibrov, M},
+	year         = 2021,
+	journal      = {Nuclear Fusion},
+	publisher    = {IOP Publishing},
+	volume       = 61,
+	number       = 8,
+	pages        = {086008},
+	url          = {https://iopscience.iop.org/article/10.1088/1741-4326/ac07b2/meta}
+}
+@article{Hamamoto2020,
+	title        = {Comprehensive modeling of hydrogen transport and accumulation in titanium and zirconium},
+	author       = {Hamamoto, Yoshiki and Uchikoshi, Takeru and Tanabe, Katsuaki},
+	year         = 2020,
+	journal      = {Nuclear Materials and Energy},
+	publisher    = {Elsevier},
+	volume       = 23,
+	pages        = 100751,
+	url          = {https://www.sciencedirect.com/science/article/pii/S2352179120300272}
 }

docs/source/theory.rst

@@ -272,9 +272,76 @@ Finally, convective heat fluxes can be applied to boundaries:
 
 where :math:`h` is the heat transfer coefficient and :math:`T_{\mathrm{ext}}` is the external temperature.
 
----------------
+Kinetic surface model
+^^^^^^^^^^^^^^^^^^^^^
+
+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}`. 
+
+Evolution of hydrogen surface concentration is determined by the atomic flux balance at the surface, as sketched in the simplified energy diagram below.
+
+.. figure:: images/potential_diagram.png
+    :align: center
+    :width: 800
+    :alt: Idealised potential energy diagram for hydrogen near a surface of an endothermic metal. Energy levels are measured from the :math:`\mathrm{H}_2` state
+
+    Idealised potential energy diagram for hydrogen near a surface of an endothermic metal. Energy levels are measured from the :math:`\mathrm{H}_2` state
+
+The governing equation for surface species is:
+
+.. math::
+    :label: eq_surf_conc
+
+    \dfrac{d c_\mathrm{s}}{d t} = J_\mathrm{bs} - J_\mathrm{sb} + J_\mathrm{vs}
+
+where :math:`J_\mathrm{bs}` is the flux of hydrogen atoms from the subsurface (bulk region just beneath the surface) onto the surface, 
+:math:`J_\mathrm{sb}` is the flux of hydrogen atoms from the surface into the subsurface, and :math:`J_\mathrm{vs}` is the net flux of hydrogen 
+atoms from the vacuum onto the surface. The current model does not account for possible surface diffusion and, therefore, is limited to
+one-dimensional hydrogen transport simulations.
+
+The connection condition between surface and bulk domains represents the Robin boundary condition for the hydrogen transport problem.
+
+.. math::
+    :label: eq_subsurf_conc
+
+    -D \nabla c_\mathrm{m} \cdot \mathbf{n} = \lambda_{\mathrm{IS}} \dfrac{\partial c_{\mathrm{m}}}{\partial t} + J_{\mathrm{bs}} - J_{\mathrm{sb}}
+
+where :math:`\lambda_\mathrm{IS}` is the distance between two interstitial sites in the bulk. 
+
+.. note::
+
+    At steady state and :math:`x=0`, :eq:`eq_subsurf_conc` is reduced to :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`.
+
+The fluxes for subsurface-to-surface and surface-to-subsurface transitions are defined as follows:
+
+.. math::
+    :label: eq_Jbs
+
+    J_\mathrm{bs} = k_\mathrm{bs} \lambda_\mathrm{abs} c_\mathrm{m} \left(1-\dfrac{c_\mathrm{s}}{n_\mathrm{surf}}\right)
+
+.. math::
+    :label: eq_Jsb
+
+    J_\mathrm{sb} = k_\mathrm{sb} c_\mathrm{s} \left(1-\dfrac{c_\mathrm{m}}{n_\mathrm{IS}}\right)
+
+where :math:`n_\mathrm{surf}` is the surface concentration of adsorption sites, :math:`n_\mathrm{IS}` is the bulk concentration of interstitial sites,
+:math:`\lambda_\mathrm{abs}=n_\mathrm{surf}/n_\mathrm{IS}` is the characteristic distance between surface and subsurface sites, :math:`k_\mathrm{bs}` 
+and :math:`k_\mathrm{sb}` 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_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.
+
+.. note::
+
+    In eq. :eq:`eq_Jsb`, the last term in brackets is usually omitted, 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.   
+
+
+------------
 References
----------------
+------------
 
 .. bibliography:: bibliography/references.bib
     :style: unsrt

docs/source/userguide/boundary_conditions.rst

@@ -105,6 +105,31 @@ Where :math:`Kd` is the dissociation coefficient, :math:`P` is the partial press
 
     my_bc = DissociationFlux(surfaces=2, Kd_0=2, E_Kd=0.1, P=1e05)
 
+Kinetic surface model (1D)
+^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+Kinetic surface model can be included to account for the evolution of adsorbed hydrogen on a surface with the :class:`festim.SurfaceKinetics` class.
+Refer to the :ref:`theory` section for more details.
+
+.. testcode:: BCs
+
+    from festim import t
+    import fenics as f
+
+    def k_bs(T, surf_conc, t):
+        return 1e13*f.exp(-0.2/k_b/T)
+
+    def k_sb(T, surf_conc, t):
+        return 1e13*f.exp(-1.0/k_b/T)
+
+    def J_vs(T, surf_conc, t):
+
+        J_des = 2e5*surf_conc**2*f.exp(-1.2/k_b/T)
+        J_ads = 1e17*(1-surf_conc/1e17)**2*f.conditional(t<10, 1, 0)
+
+        return J_ads - J_des
+
+    my_bc = SurfaceKinetics(k_bs=k_bs, k_sb=k_sb, lambda_IS=1.1e-10, n_surf=1e17, n_IS=6.3e28, J_vs=J_vs, surfaces=3, initial_condition=0, t=t)
 
 Sievert's law of solubility
 ^^^^^^^^^^^^^^^^^^^^^^^^^^^