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 using TMAP8 (including Simon et al. (2025)), the List of verification cases, and the Complete TMAP8 Input File Syntax page for a more comprehensive description of capabilities, theoretical concepts, and available objects.
Surface Reactions
At surfaces, atoms and molecules go through dissociations and recombinations, which are captured in TMAP8. There are two main dissociation/recombination conditions that TMAP7 offered, namely ratedep and surfdep. These cases are both reproduced in TMAP8 in a more general way, and are detailed below:
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 nodes 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 nodes in the mesh 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
(1)
where denotes different molecules, is the number of atoms of species in a molecule , and is the flux for a given molecule, defined as
(2)
where is the dissociation rate, is the partial pressure, and is the recombination rate for molecule . is the concentration of atomic or complex species “i”. 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 of complex species.
The following V&V cases, among others, utilize 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
(3)
and the partial pressure of A, B, and AB in equilibrium of the reaction is defined by
(4)
where is the partial pressure of corresponding gas , 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
(5)
where is the Gibbs free energy, is the ideal gas constant, is the temperature.
Therefore, the partial pressure of AB in equilibrium depends on initial partial pressure of A and B:
(6)
At equilibrium, the surface concentrations of A (i.e., ) and B (i.e., ) from Sieverts' law are given by
(7)
and
(8)
where is Sieverts’ solubility. Because we are considering isotopic variants, will be the same for each homonuclear species. Under equilibrium conditions, we also expect
(9)
where is the dissociation coefficient and is the recombination coefficient. That leads to
(10)
Under the ratedep condition, 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 of the coefficient for homonuclear molecules. We solve the net current of AB molecules from the surface to the enclosure by
(11)
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
(12)
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 dissociates.
When using the surfdep approach, the molecular flux across the surface is then given by (13)
where is the molecular mass, is the barrier energy for molecular entry to the surface (assumed positive), and is the surface binding energy of molecule , and is the Debye frequency ( 10 s for tungsten).
The following V&V cases, among others, utilize 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 condition. The reaction between AB, A, and B is the same as in ver-1ia. Therefore, the partial pressure of AB in equilibrium is a constant value depends on the initial partial pressures of A and B:
(14)
Under surfdep condition, there are again no assumptions about equilibrium except in the steady state. Then, the surface concentration of molecules is directly proportional to the gas over-pressure. We define the deposition, release, and dissociation coefficients on the surface by
(15)
(16)
and
(17)
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 over-pressure by
(18)
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. Here we assume there is a probability of unity of their combination once they find each other. Finally, the AB molecule must leave the surface and return to the gas. These behaviors are described as
(19)
(20)
(21)
(22)
(23)
where is the surface diffusivity or mobility of the atomic species and is the lattice constant.
For the recombination step and dissociation step, we solve
(24)
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
(25)
Surface Equilibrium
Both conditions described capture dissociation and recombination reactions, including their kinetics. However, when the kinetics of the dissociation/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 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/recombination reactions take place at the interface, and equilibrium is reached.
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
- 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]
- 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]