Principio de Protección al Medio Ambiente

Since the onset of development of ODS steels and NFAs, the prevailing hypothesis has suggested irradiation will influence the long-term stability of the

nanoclusters in these alloys. Several efforts have been made over the past few decades to apply a calculation model describing the evolution of nanoclusters as a result of varying irradiation conditions. Irradiation parameters such as dose rate, temperature, and

irradiation particle will all potentially influence the nanocluster evolution. Ideally, a calculation model would capture the influence of each of these parameters and their relative effect on the long-term stability of the nanoclusters.

One of the earlier models was developed by Nelson, et al. [57]. Within this model, the authors isolate the ballistic effects on nanoparticle dissolution (recoil and

disordering), while overlaying simultaneous growth of nanoclusters via radiation-

enhanced diffusion. First, the volumetric growth rate of nanoclusters due to concentration of solutes in the matrix, c, is written as:

𝑑𝑉 𝑑𝑡 =

3(𝐷+𝐷′)𝑐𝑟

𝑝 (2.16)

in which p is the atomic fraction of solute atoms in the cluster phase, r is the radius of the nanocluster, and (D+D') represents the radiation-enhanced diffusion rate of the solutes. At the same time, total concentration of solute atoms (C) in the system is maintained as:

𝐶 =4

3𝑝𝜋𝑟

3_{𝑛 + 𝑐 (2.17)}

where n is the number density of clusters per unit volume. The net result are equations for the rate of change in the radius of a nanocluster (dr/dt), written as follows:

𝑑𝑟 𝑑𝑡= − 𝜙 𝑁+ 3(𝐷+𝐷′)𝐶 4𝜋𝑝𝑟 − (𝐷 + 𝐷′)𝑟 2_{𝑛 (recoil dissolution) (2.18)} 𝑑𝑟 𝑑𝑡= −𝜓𝐾 + 3(𝐷+𝐷′)𝐶 4𝜋𝑝𝑟 − (𝐷 + 𝐷′)𝑟 2_{𝑛 (disorder dissolution) (2.19)}

In these equations, both the second and third terms are identical and represent the growth rate of the nanoclusters applying Eqs. 2.16 and 2.17. The first terms in Eq. 2.18 and 2.19 each represent the recoil dissolution or disordering dissolution influence, respectively. For recoil dissolution, ϕ is the estimated flux of solute atoms ejected from existing nanoclusters, estimated as 𝜙 = 1014∙ 𝐾 (𝑐𝑚−2𝑠−1) with K as the dose rate (in dpa/s), and N as the density of target atoms per unit volume. For the disordering dissolution term in Eq. 2.19, ψ represents the disordering parameter and is estimated as 𝜓 = 𝑙𝑓, where l is the estimated size of a damage cascade and f is the fraction of solutes that dissolve as a result of disordering. Application of either Eq. 2.18 or 2.19 will result in a solution similar to the one shown in Figure 2.12. These curves each indicate smaller precipitates

will exhibit a positive dr/dt (i.e. growth), while larger precipitates exhibit a negative dr/dt (i.e. shrinkage). The net result is an equilibrium radius, represented by the point at which each curve crosses the horizontal axis in Figure 2.12. Similarly, a plot of the precipitate radius evolution over increasing dose in illustrated in Figure 2.13 for two different dose rates. In both cases, large and small precipitates evolve to converge on an equilibrium size, and this evolution is generally completed within the first 5 dpa of irradiation. Within this model, Nelson et al. [57] acknowledge the least understood variable in Eq. 2.19 is the disordering parameter, ψ. The authors therefore suggest that this parameter may be fitted to existing experimental data to determine the estimated value for different irradiation conditions.

Figure 2.12. Solutions to Eq. 2.19 using values of 𝑲 = 𝟏𝟎−𝟐𝒅𝒑𝒂/𝒔, 𝑫′ =

Figure 2.13. Change in precipitate radius with dose with values of 𝑲 = 𝟏𝟎−𝟐_{𝒅𝒑𝒂/𝒔}

(solid lines) and 𝑲 = 𝟏𝟎−𝟔𝒅𝒑𝒂/𝒔 (dashed lines), from [57].

Another approach to modeling cluster evolution is developed by Martin [60]. The basis of this model is a steady-state solution to the diffusion equations, which produce an equilibrium concentration profile around clustering solid solutions. Details of the

derivation are more complex than the relatively elegant Nelson, et al. solution, and are detailed elsewhere [60]. One of the key conclusions developed through this model is the ballistic effects of irradiation will lead to an increase in configurational entropy of the system. This entropy increase is essentially the same as a rise in temperature of the system. The resulting equivalent temperature (T') is written as [60]:

𝑇′= 𝑇(1 + ∆) (2.20)

where Δ is the temperature dilation factor and is influenced by both the irradiation flux and temperature (T), and can estimated as:

∆= ∆𝑜𝑒

(𝐸∆

𝑘𝐵𝑇) _(2.21)

𝐸_∆ = 𝐸_𝐷 − 𝐸_𝐹 −𝐸𝑚

2 ≅

𝐸𝑚

2 (2.22)

where ED, EF, and Em are the activation energies for solute diffusion, vacancy formation,

and vacancy migration, respectively. An important feature to note in Eq. 2.21, which exhibits arrhenius behavior, is the dilation factor will be reduced at higher temperature. The implication of this temperature dilation is the potential shift in the solubility limits of the solutes in the surrounding matrix, per examples illustrated in Fig. 2.14. Depending on the equilibrium phase diagram of the system, irradiation ballistic effects may potentially induce dissolution of nanoscale phases, or it may facilitate phase separation from solid solution to two (or more) separate phases. In the case of the Martin model, Δ0 is the least

understood parameter, but the opportunity exists to apply this model to existing experimental data and deduce the values of Δ0for various irradiation conditions.

Figure 2.14. Possible alloy behaviors as a result of ballistic effects of irradiation, a) precipitate dissolution, or b) unmixing (at T') or complete disordering (at T''), from [60].

More recently, Chen, et al. [79] has advanced a model originally introduced by Wagner [80] which couples the Gibbs-Thomson model of Ostwald ripening with ballistic dissolution. The model is based on the same competing mechanisms, in which the

diffusion-driven growth evolution of clusters is influenced by the interface coherency between the clusters and the surrounding matrix. They find that fine and coherent clusters (with low interface energy) experience the least dramatic change in size, while incoherent clusters (i.e. high interface energy) more readily dissolve, particularly at lower

temperatures. Similarly to the Nelson, et al. solution [57], this model also predicts an equilibrium size of clusters will be reached over time. This equilibrium state is written as [79]: 𝑑𝑟 𝑑𝑡= 𝐷 𝑟∙ 𝑐−𝑐𝑟 𝑐𝑝−𝑐𝑟− 𝐾𝜓 = 0 (2.23)

in which D is the solute diffusion rate, r is the cluster radius, c is the solute matrix concentration, cp is the solute cluster concentration, and cr is the solute concentration at

the interface with the matrix, given by:

𝑐_𝑟 = 𝑐_∞𝑒𝑥𝑝 (2𝛾𝑖𝑣𝑎𝑡

𝑘𝑇𝑟 ) (2.24)

where 𝑐∞ is the concentration of solutes at a flat interface of the two phases, γi is the

interface energy, vat is the atomic volume within the cluster, T is the temperature, and k is

the Boltzmann constant. The solution space for Eq. 2.23 is illustrated Figure 2.15, in which a finite region of interface energies and cluster radii will lead to cluster growth, while the remaining regions dictate cluster shrinkage.

Figure 2.15. Solution space for Eq. 2.23 under various conditions, from [79]. Finally, a relatively new study by Xu, et al. [35] evaluates a binary Fe-Cu alloy, correlating an atomistically based continuum model to experimental data. The model incorporates thermal and radiation-enhanced diffusion, clustering of Cu precipitates, thermal dissociation and cascade-induced redissolution effects. The model predicts a concentration (i.e. number density) gradient of clusters over time written as [35]:

𝑑𝐶𝑛

𝑑𝑡 = 𝑘𝑛−1

+ _𝐶

1𝐶𝑛−1+ 𝑘𝑛+1− 𝐶𝑛+1− 𝑘𝑛+𝐶1𝐶𝑛− 𝑘𝑛−𝐶𝑛 (2.25)

where C is the concentration (i.e. number density) of n-Cu clusters, and with 𝑘_𝑛+ _{and 𝑘} 𝑛−

as the rate constants for capture and emission of Cu atoms, respectively. The capturing rate is written as:

𝑘_𝑛+ _{= 4𝜋(𝑟}

where r is the radius, either of a Cu atom (r1) or Cu cluster (rn). Likewise, D is the

diffusivity of a Cu atom (D1) or for a Cu cluster (Dn). In this model, Dn is assumed to be

zero. The rate of solute emission is then written as:

𝑘_𝑛− = 4𝜋(𝑟₁+ 𝑟_𝑛−1)(𝐷₁+ 𝐷_𝑛−1)𝐶₀exp (− 𝐸𝑛𝐵 𝑘𝐵𝑇) + 𝑆𝐼𝐶𝑅𝐷 ∙ 4𝜋 3 (𝑟𝑛+ 𝑎0) 3_{∙ 𝜙 ∙} 𝛿2𝑁𝑃𝐾𝐴≥1𝑘𝑒𝑉 𝛿𝑙 𝛿𝑁𝑖𝑜𝑛 (2.27) In this expression, C0 is the matrix atomic number density, 𝐸_𝑛𝐵 is the binding energy of a

Cu atom to the Cu-rich cluster. The SICRD represents the "size-independent cascade re- dissolution parameter" (~1 per PKA), a0 is the lattice parameter of the Fe-matrix, ϕ is the ion flux, and the final derivative term represents the number of PKAs with energy above 1 keV generated per ion per unit depth, which may be calculated using SRIM and the "COLLISION.txt" output file [44]. In this study, Xu et al. initially anneal the sample to induce Cu precipitation as a starting point for all subsequent experiments. Upon

incorporating irradiation at either -20°C or 300°C, the model predicts contrasting evolution of the cluster size distribution as shown in Figure 2.16. In this study, Xu et al. conduct physical experiments corresponding to the simulated irradiation conditions and demonstrate a strong correlation between the model and physical results. It is important to recognize that this calculation is for a binary Fe-Cu system, and may become

incrementally much more cumbersome if attempted on a multi-component system such as ODS or F-M alloys.

Figure 2.16. Model predicted cluster size distributions resulting from irradiation at a) -20°C, and b) 300°C, from [35]

Although each of these calculation models approach the simulation with unique methodology, the foundational theory of multiple active mechanisms persists throughout. Strong evidence is available to suggest the hypothesis of ballistic effects and diffusion- driven growth of clusters are competing upon irradiation. However, a comprehensive and universal solution for predicting the evolution of multi-component solute clusters in b.c.c. Fe-based alloys upon a range of irradiation conditions continues to remain elusive.