Base Theory for TMAP8

This page introduces some of the basic theoretical concepts used in TMAP8. However, this list is not exhaustive, and users should refer to the publications listed in the List of Publications supporting SALAMANDER (including Simon et al. (2025)), the List of verification cases, and the SALAMANDER Input File Syntax page for a more comprehensive description of capabilities, theoretical concepts, and available objects.

Tritium transport in solid materials can be divided into two main parts: (1) bulk transport in the interior of the materials and (2) surface reactions. We succinctly describe how TMAP8 models each part below. For more details on fuel cycle modeling, see Fuel Cycle or Fuel Cycle Benchmarking from Meschini et al. (2023).

Bulk Tramsport

Main system of equations (the so-called strong forms)

Bulk transport in TMAP8 can be represented by the generalized equation from McNabb and Foster McNabb and Foster (1963):

(1)

where is the concentrations of the mobile species, is the time, is the flux of the mobile species, is the decay rate constant, and is the generation term. The last term on the right-hand side represents trapping and release. is the trapped species in trap , is a user-defined numerical factor scaling to be closer to for better numerical convergence. and are the release and trapping rate coefficients for trap , is the host material density, and is the concentration of empty trapping sites of type , defined as.

(2)

where is the fraction of host sites that can contribute to trapping. The flux can be defined as

(3)

where is the diffusivity of the mobile species, is the Soret coefficient, and is the temperature. In this instance, the flux accounts for Fickian diffusion and the Soret effect. Note that other tritium transport mechanisms, including the Nernst-Plank effect Masliyah and Bhattacharjee (2006) and stress-induced diffusion Sofronis and McMeeking (1989), can also contribute to the flux of mobile species and be included in such a model in TMAP8.

Trapping is a key mechanism of tritium transport, as hydrogen isotopes tend to be captured by small defects in the solid materials, which slows down diffusion and increases retention. For example, tritium atoms can be trapped in interstitial sites, vacancies, dislocations, grain boundaries, pores, etc. These trapping sites are described with an associated density, which can evolve as a function of space, time, irradiation, etc., as well as rates and energies for trapping and release. In this kind of tritium transport model, the evolution of the concentration of trapped tritium in each trapping site is governed by, for with the number of traps,

(4)

Since TMAP8 uses the finite element method to solve this system of equations, it is necessary to derive the weak form of these equations. The derivation of the weak forms of the equations can be found in Simon et al. (2025) and is reproduced below.

Derivation of the weak forms of the equations

Step 1: Define and rearrange the strong forms of the equations

The strong forms of the governing equations is provided in Eq. (1) and Eq. (4), and can be slightly rearranged as:

(5) and, for : (6) respectively.

Step 2: Multiply every term by a test function

Multiply each term of the equations by an appropriate test function (or for ):

Eq. (5) becomes (7) and, for , Eq. (6) becomes (8)

Step 3: Integrate over the domain

After integration over the domain, we obtain:

(9)

and, for , (10)

Step 4: Integrate by parts and apply the divergence theorem

For the divergence term in Eq. (9), applying integration by parts and the divergence theorem provides where is the boundary of the domain and is the outward-facing normal vector.

This update term is then substituted back into Eq. (9), which leads to: (11)

Since no divergence terms exist in Eq. (10), no integration by parts is needed.

Step 5: Derive the final weak form in inner product notation

Eq. (11) and Eq. (10) can be expressed in inner product notation, where .

Eq. (11) becomes and, for , Eq. (10) becomes These weak forms of the equations can now be used for finite element analysis.

It is important to note that other tritium bulk transport models have been developed. For example, the multi-occupancy trap model enables each trap to capture several hydrogen isotope atoms, knowing that the binding energy of trapping depends on the number and type of other atoms present in the trap. Atomistic level calculations such as density functional theory studies have shown that several atoms can indeed occupy individual traps, especially larger defects such as pores (Fernandez et al., 2015; Heinola et al., 2010). Hodille et al. Hodille et al. (2016), Schmid et al. (2014), and more recently Kaur et al. (2025) have deployed multi-occupancy trapping models. While ongoing efforts are implementing the multi-occupancy model in TMAP8, the current paper uses, verifies, and validates the system of equations described by Eq. (1), Eq. (3), and Eq. (4).

Surface Reactions

At the surface, reactions can be described by several models and assumptions, each with different levels of fidelity or simplicity. One can assume quasi-steady-state equilibrium at the surface if reaction kinetics are much faster than the kinetics of bulk transport or capture the dissociation and recombination reactions taking place at the material's surface. TMAP8 can support these different assumptions, as well as convective transport at the surface (see ver-1ha and ver-1hb). TMAP7 proposed three main surface conditions, namely ratedep", surfdep", and lawdep" (Ambrosek and Longhurst, 2008). TMAP8 reproduces these models in a more general way, which is detailed below.

Dissociation and recombination reaction kinetics - Ratedep conditions

The ratedep condition applies when dissociation and recombination reaction kinetics govern the surface reactions Ambrosek and Longhurst (2008). ratedep assumes that the generation and release rate of molecules is the product of two or more atom concentrations at the surface and a recombination rate coefficient. Once formed, molecules immediately leave the surface.

Note that in TMAP8, just like in TMAP7, atoms in the mesh elements closest to the surface contribute to recombination processes. Atoms deeper in the mesh do not directly contribute to recombinations. However, if the size of the mesh elements is larger than the size of the lattice, which it usually is, atoms from lattice layers deeper in the materials also contribute to recombinations, as opposed to having only surface atoms contribute.

Under the ratedep assumptions, the net flux of atoms of species into the surface is given by

(12)

where denotes different molecules, is the number of atoms of species in a molecule , and is the flux for a given molecule, defined as

(13)

where is the dissociation rate, is the partial pressure, is the recombination rate for molecule , and is the concentration of atomic or complex species . Note that in TMAP8, additional terms can be added if a molecule contains more than two atoms, which was not possible in TMAP7. For example, generating HO was the result of a two-step process, first generating OH and then HO. In TMAP8, however, it is possible to either follow TMAP7's approach or directly account for the reaction between an arbitrary number of atomic or complex species.

The following V&V cases, among others, utilize, verify, and validate the ratedep conditions: ver-1ia, ver-1ib, and val-2e.

For example, ver-1ia considers the following reaction and model:

When two species react on a surface to form a third, it is possible to predict the rate at which equilibration between the species will occur. For example, the reaction between two isotopic species, A and B, is described as

(14)

and the partial pressure of A, B, and AB in equilibrium of the reaction is defined by

(15)

where is the partial pressure of corresponding gas and is the equilibrium constant.

Assuming that the molecular species have the same mass and chemical properties such that there is no enthalpy change associated with this reaction and only configurational entropy is driving the reaction, then

(16)

where is the Gibbs free energy, is the ideal gas constant, and is the temperature.

Therefore, the partial pressure of AB in equilibrium depends on initial partial pressure of A and B:

(17)

At equilibrium, the surface concentrations of A (i.e., ) and B (i.e., ) from Sieverts' law are given by

(18)

and

(19)

where is Sieverts’ solubility. Because we are considering isotopic variants, will be the same for each homonuclear species. Under equilibrium conditions, we also expect

(20)

where is the dissociation coefficient and is the recombination coefficient. That leads to

(21)

Under ratedep conditions, equilibrium is not assumed, but the relationships between the coefficients are maintained. In particular, the recombination and dissociation coefficients are assumed to be independent of the surface species concentrations and gas partial pressures, respectively. If the species molecular masses and solubilities are assumed equal, the dissociation coefficients for AB, A, and B molecules should be identical. Because two different microscopic processes can produce AB (A jumping to find B and B jumping to find A) and only one (A finding A) can form A, and similarly for B, the recombination coefficient for AB should be twice the coefficient for homonuclear molecules. We solve the net current of AB molecules from the surface to the enclosure by

(22)

where is the time, is the surface area, is the Boltzmann’s constant, is the temperature, and is the volume in the enclosure. If diffusion is small, the almost constant numbers of A and B atoms in the gas imply that and should have an almost constant value regardless of the isotopic species composition. The production of A and B in equilibration conditions is given by

(23)

Surface-energy-driven kinetics - Surfdep conditions

The surfdep condition applies when recombination is limited by surface energy. Following the surfdep model, the production rate to form surface species proceeds as the product of random lateral jumps, but release is thermally activated and involves the surface binding energy explicitly. Inversely, the transition from molecules to single atoms is modeled as a two-step process, where molecules are first absorbed by the surface and then dissociate.

When using the surfdep approach, the molecular flux across the surface is then given by (24)

where is the molecular mass, is the barrier energy for molecular entry to the surface (assumed positive), is the surface binding energy of molecule , and is the Debye frequency (10 s for tungsten).

The following V&V cases, among others, utilize and verify the surfdep conditions: ver-1ic and ver-1id.

For example, ver-1ic considers the following reaction and model:

The problem considers the reaction between two isotopic species, A and B, on a surface in surfdep conditions. The reaction between AB, A, and B is the same as in ver-1ia. Therefore, the partial pressure of AB in equilibrium is depends on the initial partial pressures of A and B:

(25)

Under surfdep conditions, there are again no assumptions about equilibrium except in the steady state. Then, the surface concentration of molecules is directly proportional to the gas overpressure. We define the deposition, release, and dissociation coefficients on the surface by

(26)

(27)

and

(28)

where is the mass of species molecules, is the Debye frequency, is the adsorption barrier energy, is the surface binding energy, is the dissociation activation energy, is the Boltzmann constant, and is the temperature.

At steady state, the flux to the surface will be balanced by the flux from the surface, and surface concentration will be related to the gas overpressure by

(29)

where and are the surface concentration and enclosure pressure of gas , respectively.

The conversion of A and B molecules to AB molecules requires several steps. First, homonuclear molecules in the gas must get to the surface. Next, they must dissociate. Then, the individual surface atoms must migrate to sites where they encounter their conjugates and combine. Finally, the AB molecule must leave the surface and return to the gas. These behaviors are described as

(30)

(31)

(32)

(33)

(34)

where is the surface diffusivity or mobility of the atomic species and is the lattice constant.

For the recombination and dissociation steps, we solve

(35)

where is the time, is the surface area, and is the volume in the enclosure. The production of A and B in equilibration conditions is given by

(36)

Surface Equilibrium - Lawdep conditions

Both conditions described capture dissociation and recombination reactions, including their kinetics. However, when the kinetics of the dissociation and recombination processes are much faster than other timescales in the systems (e.g., diffusion), capturing them would likely reduce the required time step and hence increase computational costs without significantly affecting the long-term results. In that case, it is reasonable to assume that the surface reactions are at a quasi-steady state and set the surface concentrations to their equilibrium values.

In TMAP4 and TMAP7, this is coined the lawdep condition. While TMAP8 does not use this terminology, it supports this quasi-steady-state assumption through the ADInterfaceSorption / InterfaceSorption and/or EquilibriumBC capabilities, which enable both Sievert's and Henry's law. Sievert's law applies when diatomic gases dissociate into individual atoms during dissolution and thermodynamic equilibrium is reached. Henry's law applies when no dissociation and recombination reactions take place at the interface and equilibrium is reached.

commentnote:InterfaceKernels vs. Boundary Conditions

Boundary conditions are applied to the boundary of the modeled domain, when the other side of the interface is not being modeled. For example, the TMAP8 boundary condition applying a sorption law at the boundary is EquilibriumBC. Interface kernels, however, are applied at interfaces between two subdomains, such as two different materials with different solubilities. In the case of a sorption law, it also imposes conservation of mass at the interface, as detailed in ADInterfaceSorption / InterfaceSorption. Learn more about InterfaceKernels in the InterfaceKernels System page.

The following V&V cases, among others, use the quasi-steady-state approximation for surface equilibrium:

References

  1. James Ambrosek and GR Longhurst. Verification and Validation of TMAP7. Technical Report INEEL/EXT-04-01657, Idaho National Engineering and Environmental Laboratory, December 2008.[BibTeX]
  2. N. Fernandez, Y. Ferro, and D. Kato. Hydrogen diffusion and vacancies formation in tungsten: Density Functional Theory calculations and statistical models. Acta Materialia, 94:307–318, 2015. URL: https://www.sciencedirect.com/science/article/pii/S1359645415003043, doi:https://doi.org/10.1016/j.actamat.2015.04.052.[BibTeX]
  3. K. Heinola, T. Ahlgren, K. Nordlund, and J. Keinonen. Hydrogen interaction with point defects in tungsten. Phys. Rev. B, 82:094102, Sep 2010. URL: https://link.aps.org/doi/10.1103/PhysRevB.82.094102, doi:10.1103/PhysRevB.82.094102.[BibTeX]
  4. E A Hodille, Y Ferro, N Fernandez, C S Becquart, T Angot, J M Layet, R Bisson, and C Grisolia. Study of hydrogen isotopes behavior in tungsten by a multi trapping macroscopic rate equation model. Physica Scripta, 2016(T167):014011, jan 2016. URL: https://doi.org/10.1088/0031-8949/2016/T167/014011, doi:10.1088/0031-8949/2016/T167/014011.[BibTeX]
  5. Sanjeet Kaur, Daniel R. Mason, Prashanth Srinivasan, Stephen Dixon, Sid Mungale, Teresa Orr, Mikhail Yu. Lavrentiev, and Duc Nguyen-Manh. Effect of multi-occupancy traps on the diffusion and retention of multiple hydrogen isotopes in irradiated tungsten and vanadium. Phys. Rev. Mater., 9:125404, Dec 2025. URL: https://link.aps.org/doi/10.1103/nbwm-bs8m, doi:10.1103/nbwm-bs8m.[BibTeX]
  6. Jacob H Masliyah and Subir Bhattacharjee. Electrokinetic and colloid transport phenomena. John Wiley & Sons, 2006.[BibTeX]
  7. A. McNabb and P. K. Foster. A new analysis of the diffusion of hydrogen in iron and ferritic steels. Transactions of The Metallurgical Society of AIME, 227:618–627, 1963.[BibTeX]
  8. K. Schmid, U. von Toussaint, and T. Schwarz-Selinger. Transport of hydrogen in metals with occupancy dependent trap energies. Journal of Applied Physics, 116(13):134901, 10 2014. URL: https://doi.org/10.1063/1.4896580, doi:10.1063/1.4896580.[BibTeX]
  9. Pierre-Clément A. Simon, Casey T. Icenhour, Gyanender Singh, Alexander D Lindsay, Chaitanya Vivek Bhave, Lin Yang, Adriaan Anthony Riet, Yifeng Che, Paul Humrickhouse, Masashi Shimada, and Pattrick Calderoni. MOOSE-based tritium migration analysis program, version 8 (TMAP8) for advanced open-source tritium transport and fuel cycle modeling. Fusion Engineering and Design, 214:114874, May 2025. doi:10.1016/j.fusengdes.2025.114874.[BibTeX]
  10. P. Sofronis and R.M. McMeeking. Numerical analysis of hydrogen transport near a blunting crack tip. Journal of the Mechanics and Physics of Solids, 37(3):317–350, 1989. URL: https://www.sciencedirect.com/science/article/pii/0022509689900021, doi:https://doi.org/10.1016/0022-5096(89)90002-1.[BibTeX]