Tritium diffusion in metals is simply the process of atomic tritium moving or hopping through a crystal lattice. Tritium tends to diffuse relatively rapidly through most materials and its diffusion can be measured at relatively low temperatures. Diffusivity, D, is a thermodynamic parameter, and therefore, fol­lows the conventional Arrhenius-type dependence on temperature:

D = Do exp(-ED/RT) [2]

where D0 is a constant and ED is the activation energy of diffusion. Measuring tritium diffusion is nontrivial because of the availability of tritium. Therefore, hydrogen and deuterium are often used as surrogates. From the classic rate theory, it is commonly inferred that the ratio of diffusivities of hydrogen isotopes is equivalent to the inverse ratio of the square root of the masses of the isotopes:

where m is the mass of the respective isotope, and the subscripts tritium and hydrogen refer to tritium and hydrogen, respectively. When this approximation is invoked, the activation energy for diffusion is gener­ally assumed to be independent of the mass of the isotope. Diffusion data at subambient temperatures do not support eqn [3] for a number of metals;2 however, at elevated temperatures, the inverse square root dependence on mass generally provides a rea­sonable approximation (especially for face-centered cubic (fcc) structural metals).3-9 Although eqn [3] provides a good engineering estimate of the relative diffusivity of hydrogen and its isotopes, more advanced theories have been applied to explain experimental data; for example, quantum corrections and anharmo — nic effects can account for experimentally observed differences of diffusivity of isotopes compared to the predictions of eqn [3].3, For the purposes of this

report, we assume that eqn [3] is a good approximation for the diffusion of hydrogen isotopes (as well as for permeation) unless otherwise noted, and we normal­ize reported values and relationships of diffusivity (and permeability) to protium.

4.16.2.2 Solubility

The solubility (K) represents equilibrium between the diatomic tritium molecule and tritium atoms in a metal according to the following reaction:

1 /2T2 $ T [4]

The solubility, like diffusivity, generally follows the classic exponential dependence of thermodynamic parameters:

K = K exp(-AHs/RT) [5]

where K0 is a constant and AHs is the standard enthalpy of dissolution of tritium (also called the heat of solution), which is the enthalpy associated with the reaction expressed in eqn [4]. A word of caution: the enthalpy of dissolution is sometimes reported per mole of gas (i. e., with regard to the reaction T2 $ 2T as in Caskey11), which is twice the value of AHs as defined here. Assuming a dilute solution of dissolved tritium and ideal gas behavior, the chemical equilibrium between the diatomic gas and atomic tritium dissolved in a metal (eqn [4]) is expressed as

1/2 (mTT + RT ln ^TI) = m0 + RTlnо [6]

Ptt

where c0 is the equilibrium concentration of tritium dissolved in the metal lattice in the absence of stress, тіт is the chemical potential of the diatomic gas at a reference partial pressure of pli, and ml is the chemical potential of tritium in the metal at infinite dilution. This relationship is the theoretical origin of Sievert’s law:

co = K (pii)1/2 [7]

where to a first approximation, the solubility is equiv­alent for all isotopes of hydrogen.

It is important to distinguish between solubility and concentration: solubility is a thermodynamic property of the material, while the concentration is a dependent variable that depends on system conditions (including whether equilibrium has been attained). For example, once dissolved in a metal lattice, atomic tritium can interact with elastic stress fields: hydrostatic tension dilates the lattice and increases the concentration of tritium that can dissolve in the metal, while hydrostatic compression decreases the concentration. The relationship that describes this effect in the absence of a tritium flux12-14 is written as

cL = co exp(~RT) [8]

where cL is the concentration of tritium in the lattice subjected to a hydrostatic stress (a = ay-/3), and VT is the partial molar volume of tritium in the lattice. For steels, the partial molar volume of hydrogen is ^2 cm3 mol-1,15 which can be assumed to first order to be the same for tritium. For most systems, the increase of tritium concentration will be relatively small for ordi­nary applied stresses, particularly at elevated tempera­tures; for example, hydrostatic tension near 400 MPa at 673 K results in a ^15% increase in concentration. On the other hand, internal stresses near defects or other stress concentrators can substantially increase the local concentration near the defect. It is unlikely that local concentrations will significantly contribute to elevated tritium inventory in the material, but locally elevated concentrations of hydrogen isotopes become sites for initiating and propagating hydrogen — assisted fracture in structural metals.