Achlys Tritium Diffusion

The Achlys application aims to model macroscopic tritium transport processes through fusion materials using the MOOSE Finite Element Framework Permann et al. (2020), Gaston et al. (2015), Lindsay et al. (2021).

The Achlys source code is hosted on Github, a dockerfile is available to easily obtain the software.

Governing Equations

Achlys implements a formulation of the Foster-McNabb equations presented byHodille et al. (2015) and Delaporte-Mathurin et al. (2019) as shown in Eq. (1), Eq. (2), and Eq. (3).

$(1)var element = document.getElementById("moose-equation-0f87a961-e5d4-4664-a4ff-cd8e01e0962e");katex.render("\\frac{\\partial C_{m}}{\\partial t} = \\nabla \\cdot \\left( D \\left(T \\right) \\nabla C_{m} \\right) - \\sum \\frac{\\partial C_{t,i}}{\\partial t} + S_{ext} ", element, {displayMode:true,throwOnError:false});$

$(2)var element = document.getElementById("moose-equation-297340b2-16bf-4406-9a93-5bd9d7fdc627");katex.render("\\frac{\\partial C_{t,i}}{\\partial t} = \\nu_m \\left(T\\right) C_m \\left(n_i - C_{t,i} \\right) - \\nu_i\\left(T\\right) C_{t,i} ", element, {displayMode:true,throwOnError:false});$

$(3)var element = document.getElementById("moose-equation-ee103be4-9007-4d7e-a74f-99ae481cd16d");katex.render(" \\rho_m C_p \\frac{\\partial T}{\\partial t} = \\nabla \\cdot \\left(k \\nabla T \\right)", element, {displayMode:true,throwOnError:false});$

$var element = document.getElementById("moose-equation-c2f2739b-be22-4bfa-8473-27c79e3b85f2");katex.render("D", element, {displayMode:false,throwOnError:false});$ is the diffusivity of the species through some material in units of $var element = document.getElementById("moose-equation-3270835a-098e-4f45-84c7-8eba7ec4f1e4");katex.render("m^{2}s^{-1}", element, {displayMode:false,throwOnError:false});$ . This is modelled by an Arhhenius relation as given by Eq. (4). The D0 and energy value parameters are determined experimentally for any pair of solute atom and bulk material lattice. $(4)var element = document.getElementById("moose-equation-8ec7677f-5547-407e-ba71-f2c4fb3e276f");katex.render(" D(T) = D_0 \\exp \\left( -E_{diff} / k_{B} T \\right)", element, {displayMode:true,throwOnError:false});$

$var element = document.getElementById("moose-equation-53978c04-ab6d-4dc3-afb2-6f18a68c7a40");katex.render("\\nu_{m}", element, {displayMode:false,throwOnError:false});$ , in units of $var element = document.getElementById("moose-equation-72404bb4-9ae3-45a7-ba2c-169c9bdbfd29");katex.render("m^{3}s^{-1}", element, {displayMode:false,throwOnError:false});$ , is the reaction rate for the trapping process and is modelled by Eq. (5) where $var element = document.getElementById("moose-equation-2f86be53-ff8a-429b-a431-5f46a2770e55");katex.render("\\lambda", element, {displayMode:false,throwOnError:false});$ is the lattice constant in $var element = document.getElementById("moose-equation-7593f3ed-1788-494c-8b56-f56d5726c95a");katex.render("m", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-1f7de1d7-4108-43aa-a2c6-78d096f03384");katex.render("N_{sol}", element, {displayMode:false,throwOnError:false});$ is the number density of solute sites in the material. $(5)var element = document.getElementById("moose-equation-c2e2d2ec-ae2d-42ec-a85d-15833bc5ef88");katex.render(" \\nu_{m} = \\frac{D(T)}{\\lambda^2 N_{sol}}", element, {displayMode:true,throwOnError:false});$

$var element = document.getElementById("moose-equation-79776890-f08c-432c-b794-4d34cc66a73e");katex.render("\\nu_{i}", element, {displayMode:false,throwOnError:false});$ , in units of $var element = document.getElementById("moose-equation-fdd9e06f-ead0-488f-9f75-1797b9a0f29d");katex.render("s^{-1}", element, {displayMode:false,throwOnError:false});$ , is the reaction rate for the de-trapping process from the $var element = document.getElementById("moose-equation-24de7790-cab9-45c9-b4f8-ca4caa60d6c8");katex.render("i", element, {displayMode:false,throwOnError:false});$ -th trapping site. This is modelled by the Arrhenius type equation as given by Equation Eq. (6) where E is the energy barrier a trapped atom must overcome to leave the site and $var element = document.getElementById("moose-equation-5a5f849d-494c-4395-ae63-3644a8eed182");katex.render("\\nu_{0}", element, {displayMode:false,throwOnError:false});$ is referred to as the attempt frequency. $(6)var element = document.getElementById("moose-equation-aa96be03-8d30-461c-80d6-dbc462f2be3a");katex.render(" \\nu_{i} = \\nu_0 \\exp \\left( -E_{i} / k_{B} T \\right)", element, {displayMode:true,throwOnError:false});$

Kernel selection

The kernels selected for each of the governing equations given in Eq. (1), Eq. (2), and Eq. (3) are shown below. Note that all kernels are provided by the MOOSE framework without modification except in a single case: the ADTrappingEquilibriumEquation is a custom kernel implemented within Achlys.

Note also that Automatic Differentiation Lindsay et al. (2021) has been employed within Achlys so AD kernels, BCs, and objects should be used wherever available. More information on this system is available through the MOOSE documentation.

Mobile Concentration

$var element = document.getElementById("moose-equation-5c34de02-810c-4119-8a8b-3f6acc71653c");katex.render("\\underbrace{\\left(\\psi, \\frac{\\partial C_{m}}{\\partial t}\\right) }_{\\text{(i) \\:ADTimeDerivative}} - \\underbrace{\\bigg(\\nabla\\psi, D\\nabla C_m \\bigg)} _{\\text{(ii) \\:ADMatDiffusion}} + \\underbrace{\\left(\\psi, \\sum \\frac{\\partial C_{t,i}}{\\partial t} \\right) }_{\\text{(iii) \\:ADCoupledTimeDerivative}} + \\underbrace{\\bigg( \\psi,-S_{ext} \\bigg)}_{\\text{(iv) \\:ADBodyForce}} = 0", element, {displayMode:true,throwOnError:false});$

Trapped Concentration, site $var element = document.getElementById("moose-equation-eed130cd-4a38-4596-89f1-37509444855f");katex.render("i", element, {displayMode:false,throwOnError:false});$

$var element = document.getElementById("moose-equation-eb9ae2e7-14c2-4bd9-9e53-c5c2407150c2");katex.render("\\underbrace{\\left(\\psi, \\frac{\\partial C_{t,i}}{\\partial t}\\right) }_{\\text{(i) \\:ADTimeDerivative}} - \\underbrace{\\bigg(\\psi, \\nu_m C_m(n_i - C_{t,i}) \\bigg) + \\bigg(\\psi,\\nu_i C_{t,i} \\bigg) }_{\\text{( iii ) \\: ADTrappingEquilibriumEquation}} = 0", element, {displayMode:true,throwOnError:false});$

Temperature Evolution

$var element = document.getElementById("moose-equation-5ba9745d-5280-4ebe-8825-84d49856ca0b");katex.render("\\underbrace{\\left(\\psi, \\rho C_p \\frac{\\partial T}{\\partial t}\\right) }_{\\text{ADHeatConductionTimeDerivative}} - \\underbrace{\\bigg(\\nabla\\psi, k\\nabla T \\bigg) }_{\\text{ADHeatConduction}} = 0", element, {displayMode:true,throwOnError:false});$

Symbols, units, and meanings

Here, $var element = document.getElementById("moose-equation-53e33ccc-80cb-46f4-82ce-4672d84b1968");katex.render("C", element, {displayMode:false,throwOnError:false});$ symbols denote concentrations of the hydrogen isotope either in the trapped or mobile phase as indicated by the relevant subscript. $var element = document.getElementById("moose-equation-357a6d0a-cd2d-4889-9e63-ed1d9d823887");katex.render("n_i", element, {displayMode:false,throwOnError:false});$ is the concentration of a given trap type and other symbols are defined in Table 1. Mathematically, the units of concentration can either be expressed in terms of $var element = document.getElementById("moose-equation-a224e928-a3c0-4f95-ba88-8e4f97053fda");katex.render("m^{-3}", element, {displayMode:false,throwOnError:false});$ or atomic fraction; the Achlys application uses atomic fraction in its calculations.

Table 1: Nomenclature and symbol meanings

A Note on Units and Scaling

Achlys solves for the concentrations of the mobile and each trapped phase in the units of atomic fraction, this is the concentration in $var element = document.getElementById("moose-equation-7c82ee3f-89e0-4e00-8d34-8fadcba91c56");katex.render("\\text{m}^{-3}", element, {displayMode:false,throwOnError:false});$ divided by the number density of solute atoms. Most physical processes are unaffected by this conversion and the SI units can be retrieved by applying a simple scale factor to the outputs, an exception being recombination boundary conditions which evaluate the square of the mobile concentration. In this case the scale factor must be re-applied within the boundary condition (see ADSurfaceRecombination for example).

The concentrations are scaled down in this way as a bug was experienced in early development where diffusion kernels would cease to function once the magnitude of the scalar field surpassed some threshold. This issue is yet to be further investigated.

Energies are expressed in eV (though only appear in non-dimensional ratios - so transforms could be possible) and all other units are SI.

source

Complete source definitions for all objects implemented within Achlys are detailed below.

Achlys kernels

ADTrappingEquilibriumEquationCompute the local movement of particles between the mobile phaseand the trapped phase for a specific trap type

Achlys BCs

ADSurfaceRecombinationImposes the integrated boundary condition $var element = document.getElementById("moose-equation-c8eaa863-ac56-4eb3-a1b4-8380569e39c8");katex.render("\\frac{\\partial u}{\\partial n}= kv^2", element, {displayMode:false,throwOnError:false});$ , where $var element = document.getElementById("moose-equation-be635440-9ff9-44a9-8af8-1d7822112770");katex.render("v", element, {displayMode:false,throwOnError:false});$ is a variable and the constant, k, is calculated from an Arhhenious expression based on provided (constant) values of E and T
ADSurfaceRecombinationCoupledTImposes the integrated boundary condition $var element = document.getElementById("moose-equation-9dcd052d-bdd3-4d7d-9062-9416566b8f8c");katex.render("\\frac{\\partial u}{\\partial n}= kv^2", element, {displayMode:false,throwOnError:false});$ , where $var element = document.getElementById("moose-equation-ae24f0ab-3ba1-40f0-9fed-6f1d85cdab06");katex.render("v", element, {displayMode:false,throwOnError:false});$ is a variable and the constant, k, is calculated from an Arhhenious expression based on an energy barrier, E, and a coupled Temperature.
DirichletEquivalentPfcImplantBCImposes the essential boundary condition $var element = document.getElementById("moose-equation-fab6db50-c76b-439c-94f6-82698b362632");katex.render("u=g", element, {displayMode:false,throwOnError:false});$ , where $var element = document.getElementById("moose-equation-053ee72a-89ab-4b4c-bbde-1f35ec709f53");katex.render("g", element, {displayMode:false,throwOnError:false});$ is a constant calculated assuming the recombination fluxquickly becomes equal to the incident flux. The recombination term is calculated using the simulation temperature and the pre-exponent and Energy values provided
DirichletPfcFunctionFluxRampBCImposes the essential boundary condition $var element = document.getElementById("moose-equation-074c5e96-fd5d-4e88-b0b0-bacfdc928b1d");katex.render("u=g", element, {displayMode:false,throwOnError:false});$ , where $var element = document.getElementById("moose-equation-a8e71907-da93-40c0-9a87-90a5390f9ea6");katex.render("g", element, {displayMode:false,throwOnError:false});$ is a constant calculated assuming the recombination fluxquickly becomes equal to the incident flux. The recombination term is calculated using the simulation temperature and the pre-exponent and Energy values provided
PfcFunctionFluxRampBCImposes the essential boundary condition $var element = document.getElementById("moose-equation-9dd39981-fb20-4596-a93e-2e6c43fbdefd");katex.render("u=g", element, {displayMode:false,throwOnError:false});$ , where $var element = document.getElementById("moose-equation-61e23337-5e8e-4cca-b00b-71253133c0f2");katex.render("g", element, {displayMode:false,throwOnError:false});$ is a constant calculated assuming the recombination fluxquickly becomes equal to the incident flux. The recombination term is calculated using the simulation temperature and the pre-exponent and Energy values provided

Achlys Materials

Achlys Postprocessors

SideDesorptionFluxComputes the integral of the flux over the specified boundary
SideMaxValueFind a specific target value along a sampling line. The variable values along the line should change monotonically. The target value is searched using a bisection algorithm.
SideMinValueFind a specific target value along a sampling line. The variable values along the line should change monotonically. The target value is searched using a bisection algorithm.
ValueAtPointFind a specific target value along a sampling line. The variable values along the line should change monotonically. The target value is searched using a bisection algorithm.
VariableIntegralComputes a volume integral of the specified variable

References

Rémi Delaporte-Mathurin, Etienne A. Hodille, Jonathan Mougenot, Yann Charles, and Christian Grisolia. Finite element analysis of hydrogen retention in iter plasma facing components using festim. Nuclear Materials and Energy, 21:100709, 2019. URL: https://www.sciencedirect.com/science/article/pii/S2352179119300547, doi:https://doi.org/10.1016/j.nme.2019.100709.

@article{DELAPORTE2019,
    author = "Delaporte-Mathurin, Rémi and Hodille, Etienne A. and Mougenot, Jonathan and Charles, Yann and Grisolia, Christian",
    title = "Finite element analysis of hydrogen retention in ITER plasma facing components using FESTIM",
    journal = "Nuclear Materials and Energy",
    volume = "21",
    pages = "100709",
    year = "2019",
    issn = "2352-1791",
    doi = "https://doi.org/10.1016/j.nme.2019.100709",
    url = "https://www.sciencedirect.com/science/article/pii/S2352179119300547",
    keywords = "Fusion, Tritium, Finite elements, Plasma facing components, Hydrogen isotopes",
    abstract = "The behaviour of hydrogen isotopes in ITER monoblocks was studied using the code FESTIM (Finite Element Simulation of Tritium In Materials) which is introduced in this publication. FESTIM has been validated by reproducing experimental data and the Method of Manufactured Solutions was used for analytical verification. Following relevant plasma scenarios, both transient heat transfer and hydrogen isotopes (HIs) diffusion have been simulated in order to assess HIs retention in monoblocks. Relevant materials properties have been used. Each plasma cycle is composed of a current ramp up, a current plateau, a current ramp down and a resting phase before the following shot. 100 cycles are simulated. The total HIs inventory in the tokamak during resting phases reaches 1.8×10−3mg whereas during the implantation phases it keeps increasing as a power law of time. Particle flux on the cooling channel of the monoblock is also computed. The breakthrough time is estimated to be t=1×105s which corresponds to 24 cycles. Relevance of 2D modelling has been demonstrated by comparing the total HIs inventory obtained by 2D and 1D simulations. Using 1D simulations, a relative error is observed compared to 2D simulations which can reach -25\% during the resting phase. The error during implantation phases keeps increasing."
}

Derek R. Gaston, Cody J. Permann, John W. Peterson, Andrew E. Slaughter, David Andrš, Yaqi Wang, Michael P. Short, Danielle M. Perez, Michael R. Tonks, Javier Ortensi, Ling Zou, and Richard C. Martineau. Physics-based multiscale coupling for full core nuclear reactor simulation. Annals of Nuclear Energy, 84:45–54, 2015.

@article{moose_multi_gaston2015,
    author = "Gaston, Derek R. and Permann, Cody J. and Peterson, John W. and Slaughter, Andrew E. and Andr{\v{s}}, David and Wang, Yaqi and Short, Michael P. and Perez, Danielle M. and Tonks, Michael R. and Ortensi, Javier and Zou, Ling and Martineau, Richard C.",
    title = "Physics-based multiscale coupling for full core nuclear reactor simulation",
    year = "2015",
    journal = "Annals of Nuclear Energy",
    volume = "84",
    pages = "45--54",
    publisher = "Elsevier"
}

E.A. Hodille, X. Bonnin, R. Bisson, T. Angot, C.S. Becquart, J.M. Layet, and C. Grisolia. Macroscopic rate equation modeling of trapping/detrapping of hydrogen isotopes in tungsten materials. Journal of Nuclear Materials, 467:424–431, 2015. URL: https://www.sciencedirect.com/science/article/pii/S0022311515300660, doi:https://doi.org/10.1016/j.jnucmat.2015.06.041.

@article{HODILLE2015,
    author = "Hodille, E.A. and Bonnin, X. and Bisson, R. and Angot, T. and Becquart, C.S. and Layet, J.M. and Grisolia, C.",
    title = "Macroscopic rate equation modeling of trapping/detrapping of hydrogen isotopes in tungsten materials",
    journal = "Journal of Nuclear Materials",
    volume = "467",
    pages = "424-431",
    year = "2015",
    issn = "0022-3115",
    doi = "https://doi.org/10.1016/j.jnucmat.2015.06.041",
    url = "https://www.sciencedirect.com/science/article/pii/S0022311515300660",
    keywords = "fuel retention, deuterium, tungsten, modeling",
    abstract = "Relevant parameters for trapping of Hydrogen Isotopes (HIs) in polycrystalline tungsten are determined with the MHIMS code (Migration of Hydrogen Isotopes in MaterialS) which is used to reproduce Thermal Desorption Spectrometry experiments. Three types of traps are found: two intrinsic traps (detrapping energy of 0.87 eV and 1.00 eV) and one extrinsic trap created by ion irradiation (detrapping energy of 1.50 eV). Then MHIMS is used to simulate HIs retention at different fluences and different implantation temperatures. Simulation results agree well with experimental data. It is shown that at 300 K the retention is limited by diffusion in the bulk. For implantation temperatures above 500 K, the retention is limited by trap creation processes. Above 600 K, the retention drops by two orders of magnitude as compared to the retention at 300 K. With the determined detrapping energies, HIs outgassing at room temperature is predicted. After ions implantation at 300 K, 45\% of the initial retention is lost to vacuum in 300 000 s while during this time the remaining trapped HIs diffuse twice as deep into the bulk."
}

Alexander Lindsay, Roy Stogner, Derek Gaston, Daniel Schwen, Christopher Matthews, Wen Jiang, Larry K Aagesen, Robert Carlsen, Fande Kong, Andrew Slaughter, and others. Automatic differentiation in metaphysicl and its applications in moose. Nuclear Technology, pages 1–18, 2021.

@article{moose_ad_lindsay2021,
    author = "Lindsay, Alexander and Stogner, Roy and Gaston, Derek and Schwen, Daniel and Matthews, Christopher and Jiang, Wen and Aagesen, Larry K and Carlsen, Robert and Kong, Fande and Slaughter, Andrew and others",
    title = "Automatic Differentiation in MetaPhysicL and Its Applications in MOOSE",
    journal = "Nuclear Technology",
    pages = "1--18",
    year = "2021",
    publisher = "Taylor \\& Francis"
}

Cody J. Permann, Derek R. Gaston, David Andrš, Robert W. Carlsen, Fande Kong, Alexander D. Lindsay, Jason M. Miller, John W. Peterson, Andrew E. Slaughter, Roy H. Stogner, and Richard C. Martineau. MOOSE: enabling massively parallel multiphysics simulation. SoftwareX, 11:100430, 2020. URL: http://www.sciencedirect.com/science/article/pii/S2352711019302973, doi:https://doi.org/10.1016/j.softx.2020.100430.

@article{moose_permann2020,
    author = "Permann, Cody J. and Gaston, Derek R. and Andr{\v{s}}, David and Carlsen, Robert W. and Kong, Fande and Lindsay, Alexander D. and Miller, Jason M. and Peterson, John W. and Slaughter, Andrew E. and Stogner, Roy H. and Martineau, Richard C.",
    title = "{MOOSE}: Enabling massively parallel multiphysics simulation",
    year = "2020",
    journal = "{SoftwareX}",
    volume = "11",
    pages = "100430",
    issn = "2352-7110",
    doi = "https://doi.org/10.1016/j.softx.2020.100430",
    url = "http://www.sciencedirect.com/science/article/pii/S2352711019302973",
    keywords = "Framework, Finite-element, Parallel, Multiphysics, Multiscale"
}

Governing Equations
Kernel selection
Symbols , units , and meanings
A Note on Units and Scaling
Achlys kernels
Achlys BCs
Achlys Materials
Achlys Postprocessors
References

Module

Examples

Source Code

Comparison to Analytical Solution

Thermal Desorption Spectrum

Kernels

BCs

Materials

Postprocessors

Achlys Tritium Diffusion

Governing Equations

Kernel selection

Mobile Concentration

Trapped Concentration, site $var element = document.getElementById("moose-equation-eed130cd-4a38-4596-89f1-37509444855f");katex.render("i", element, {displayMode:false,throwOnError:false});$

Temperature Evolution

Symbols, units, and meanings

A Note on Units and Scaling

source

Achlys kernels

Achlys BCs

Achlys Materials

Achlys Postprocessors

References

Parameter Name	Symbol(s)	Unit
Species Concentration	$var element = document.getElementById("moose-equation-f1f698a0-97c2-483d-85e7-0d67ad902c7c");katex.render("C, C_i, C_m", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-aa2fd185-e486-402f-9ed0-87e69ed19698");katex.render("m^{-3}", element, {displayMode:false,throwOnError:false});$
Trap Concentration	$var element = document.getElementById("moose-equation-08438bdb-d318-4982-a033-5ce0a74a86a2");katex.render("n_i", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-b2866241-03d9-4b39-b493-98be63ddf0db");katex.render("m^{-3}", element, {displayMode:false,throwOnError:false});$
time	$var element = document.getElementById("moose-equation-8c9d1bf5-0158-43ac-93c4-70d81d4565b6");katex.render("t", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-a59c3ab8-f329-4519-a5e5-b72c81f22239");katex.render("s", element, {displayMode:false,throwOnError:false});$
External sources	$var element = document.getElementById("moose-equation-352251a8-9545-46e0-a0aa-374d77ef6dcd");katex.render("S_{ext}", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-f6be25f6-132e-4f80-8235-31fad5cc77f3");katex.render("m^{-3}s^{-1}", element, {displayMode:false,throwOnError:false});$
Trapping reaction rate	$var element = document.getElementById("moose-equation-ee9dac36-0134-43fe-be7b-b164238a6f35");katex.render("\\nu_{m}", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-f1377c49-50ba-483d-ac9e-09d0fcd2aecf");katex.render("m^{3}s^{-1}", element, {displayMode:false,throwOnError:false});$
Lattice constant	$var element = document.getElementById("moose-equation-530a3931-4125-4ae2-a5e2-06c35e3a64f8");katex.render("\\lambda", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-29b89522-f91f-4eed-9541-16b5aa6915ce");katex.render("m", element, {displayMode:false,throwOnError:false});$
Lattice sites	$var element = document.getElementById("moose-equation-d7e3e5d0-05b6-45c4-b404-185a5b20d27e");katex.render("N_{sol}", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-1273a325-be69-4f3a-915d-db0afb5257da");katex.render("m^{-3}", element, {displayMode:false,throwOnError:false});$
De-trapping reaction rate	$var element = document.getElementById("moose-equation-2ad3acb9-34c5-441a-b4d0-da8074f063f0");katex.render("\\nu_{i}", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-5a129645-49e4-474f-977f-33003f43db0f");katex.render("s^{-1}", element, {displayMode:false,throwOnError:false});$
De-trapping pre-exponential factor	$var element = document.getElementById("moose-equation-87da2dd4-caa9-47fc-9696-2697aeb60c6d");katex.render("\\nu_0", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-1bbc5299-bcbd-463f-be5e-5848343cf23b");katex.render("s^{-1}", element, {displayMode:false,throwOnError:false});$
Energy barrier	$var element = document.getElementById("moose-equation-10981541-8014-47e5-a4ca-d9b4478dfa31");katex.render("E", element, {displayMode:false,throwOnError:false});$	eV
Boltzman Constant	$var element = document.getElementById("moose-equation-5995f5ef-6410-4935-8409-b574bdb0363d");katex.render("k_b", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-cc74ea2f-79ac-47db-864a-035d50d01b53");katex.render("eV K^{-1}", element, {displayMode:false,throwOnError:false});$
Diffusion Coefficient	$var element = document.getElementById("moose-equation-25877d95-5962-4e53-aa9f-81141259885b");katex.render("D", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-880a33c8-32dd-44ab-9b2e-d6e37984d8f9");katex.render("m^2\\:s^{-1}", element, {displayMode:false,throwOnError:false});$
Diffusion pre-exponential factor	$var element = document.getElementById("moose-equation-838e306d-5889-49a3-86cf-b012d07eae7c");katex.render("D_0", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-d7035aaa-f8ab-467d-ab9c-26dc7348e4a8");katex.render("m^2\\:s^{-1}", element, {displayMode:false,throwOnError:false});$
Lattice Density	$var element = document.getElementById("moose-equation-8ee0158a-9aae-4379-bb3c-1278e5d72262");katex.render("\\rho", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-e06c3fdc-f797-4d20-85c0-18db6153ae15");katex.render("m^{-3}", element, {displayMode:false,throwOnError:false});$
Temperature	$var element = document.getElementById("moose-equation-fa5e21a3-7776-4f67-bcdd-99f02fa17851");katex.render("T", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-8c4c340b-f6c9-46b5-affb-c70aa4519080");katex.render("K", element, {displayMode:false,throwOnError:false});$
Thermal Conductivity	$var element = document.getElementById("moose-equation-07eeb944-d90a-46eb-8d50-e556fbbb5928");katex.render("k", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-58551ae1-a4e9-47c4-9069-e79b459e51b4");katex.render("W\\:k^{-1}", element, {displayMode:false,throwOnError:false});$
Specific Heat Capacity	$var element = document.getElementById("moose-equation-df01270e-f4f3-4ccc-addb-50c79edf4001");katex.render("C_p", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-4d79c3be-051e-4418-b247-c8c822d100c3");katex.render("J \\:kg^{-1} K^{-1}", element, {displayMode:false,throwOnError:false});$
Mass Density	$var element = document.getElementById("moose-equation-605ae47d-945a-4d8e-a98e-1aae5a7acf56");katex.render("\\rho_m", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-456a49e8-728a-4925-b48f-977c1ebe326a");katex.render("kg\\: m^{-3}", element, {displayMode:false,throwOnError:false});$

Achlys Tritium Diffusion

Governing Equations

Kernel selection

Mobile Concentration

Trapped Concentration, site var element = document.getElementById("moose-equation-eed130cd-4a38-4596-89f1-37509444855f");katex.render("i", element, {displayMode:false,throwOnError:false});

Temperature Evolution

Symbols, units, and meanings

A Note on Units and Scaling

References

Trapped Concentration, site $var element = document.getElementById("moose-equation-eed130cd-4a38-4596-89f1-37509444855f");katex.render("i", element, {displayMode:false,throwOnError:false});$