As shown earlier, steady-state permeation of hydro­gen through materials is normally governed only by solubility and diffusivity. It has been shown23 that at low pressures, permeation can also be limited by dissociation at the surface. Due to limited data in the literature on this effect (and questions about whether this condition ever really exists), we do not consider this effect in this chapter. It is also possible for permeation to be limited by the rate at which atoms can recombine back into molecules. With the exception of extremely high temperatures, this recom­bination is necessary for hydrogen to be released from a material. Wherever the release rate from a surface is limited by recombination, the boundary condition at that boundary is given by:

Jr = krc2 [19]

where kr is the recombination-rate constant and c is the concentration of hydrogen near the surface (for this discussion, we assume that there is no surface rough­ness). The units for k and c are m4 s-1 per mol of H2 and mol H2 m~3, respectively.

There are two specific types of conditions that can lead to the hydrogen release being rate limited by recombination. One of them occurs for plasma­facing materials in which the recombination-rate coefficient is relatively low, and the implantation rate is high. With this condition, the concentration of hydrogen in the very-near plasma-exposed surface will increase to the point at which Jr is effectively equal to the implantation rate. It is not exactly equal to the implantation rate because there is permeation away from that surface to the downstream surface. The other condition that can lead the hydrogen release being controlled by recombination is when the rate of ingress at the upstream boundary is very low. This condition can occur either when the upstream pressure is extremely low or a barrier is placed on the upstream surface, and the downstream surface has a relatively low recombination-rate con­stant. In the extreme case, the release rate at the downstream side is so slow that the hydrogen con­centration becomes uniform throughout the material. The release rate from the downstream surface will be krc2, where c now represents the uniform concentra­tion. From c = KpTT and eqn [19], it can easily be shown that the recombination-limited permeation is linearly dependent on pressure, rather than having the square root of pressure dependence of diffusion — limited permeation.

There are various derivations and definitions of the recombination-rate constant. In the case of intense plasma exposure in which extreme near-surface concentrations are generated, Baskes24 derived the recombination-rate constant with the assumption that the rate was controlled by the process of bulk atoms jumping to the surface, combining with surface atoms, and then desorbing. His expression for the recombination-rate constant is

s

-t Кexp where C is a constant, s is the sticking constant, which depends on the cleanliness of the material surface, and EX = DHs + Ed > 0, otherwise EX = 0. The sticking constant can be anywhere from 1 for clean surfaces to 10-4 or smaller for oxidized surfaces.

Pick and Sonnenberg25 solved the recombination — rate constant for the case where the near-surface con­centration of hydrogen is small. In the limit of low surface concentration, the rate of atom jump to the surface does not play an important role in the recom­bination rate, thus eliminating EX from the exponen­tial. The sticking constant in the Pick and Sonnenberg model is thermally activated: s = s0 exp(-2Ec/RT), where s0 is the sticking coefficient and EC is the activation energy for hydrogen adsorption.

Wampler26 also studied the case of low near­surface concentration to arrive at an expression for the recombination-rate constant. He assumed equi­librium between hydrogen atoms in surface chemi­sorption sites and atoms in solution, deriving the recombination-rate constant as

nsn 2D Hs

(bnL)2 exp RT" where ns is the area density of surface chemisorption sites, and n is the jump frequency.

These expressions differ, but also display many similarities. Unfortunately, the surface cleanliness dominates the rate of recombination and these theo­retical relationships are relevant only for sputter — cleaned surfaces and very low pressures. For example, Causey and Baskes27 showed that the Baskes24 model predicts fairly accurate results for plasma-driven per­meation of deuterium in nickel. Comparison with values in the literature for nickel showed other results to differ by as much as four orders of magnitude and to have significantly different activation energies.

4.16.2.3 Irradiation and Implantation

Irradiation and implantation can affect the transport of hydrogen isotopes in materials. Since these effects can be complex and depend on the conditions of the mate­rials and the environment, it can be difficult to draw broad conclusions from the literature. Nevertheless, changes in apparent transport properties are generally attributed to damage and the creation of hydrogen traps28-31 (see also Chapter 1.03, Radiation-Induced Effects on Microstructure). Therefore, the effects of
irradiation and implantation will depend sensitively on the characteristics of the traps that are created by these processes. The density of damage is an important con­sideration: for example, it has been shown that helium bubbles are not effective trapping sites for steels,32 likely because in these experiments, the density of helium bubbles was relatively low. The energy of the trap will determine the coverage as a function of tem­perature (eqn [9]): generally, the effect of trapping will be stronger at low temperatures, especially in materials with a low solubility (Figure 2), which can result in substantial increases in hydrogen isotope inventory compared to hydrogen content predicted from lattice solubility. Additionally, irradiation may increase ioni­zation of hydrogen isotopes, thus enhancing apparent permeation.29 Reactor environments can defeat per­meation barriers, for example, by damaging the integ­rity of oxide layers; this is discussed at the end of this chapter.