|
reacting against the formation of chemical concentration gradients at sinks produced by the IK effect (last term of LHS of eqn [24]). Equation [24] can be used in both dilute and concentrated alloys. Differences between models arise when one evaluates specific partial diffusion coefficients.
The first RIS model in dilute fcc alloys, designed by Johnson and Lam,105 introduced an explicit variable for solute-point defect complexes. The same kind of approach has been used by Faulkner et a/.,106 although it has been shown to be incorrect in specific cases.86,107 A more rigorous treatment relies on the linear response theory, with a clear correspondence between the atomic jump frequencies and the L-coefficients. The first RIS model derived from a rigorous estimation of fluxes was devoted to fcc dilute alloys,108 and then to bcc dilute alloys.87
In concentrated alloys, due to the greater complexity and the lack of experimental data, further simplifications and more approximate diffusion models are used to simulate RIS.
Heavy ions enjoy the benefit of high dose rates resulting in the accumulation of high doses in short times. Also, because they are typically produced in the energy range of a few MeV, they are very efficient at producing dense cascades, similar to those produced by neutrons. The disadvantage is that as with electrons, the high dose rates require large
|
|
|
1.2
1.0
0.8
0.6
0.4
0.2
0
|
|
 |
|
Figure 33 (a) Subsurface swelling resulting from 5 MeV Ni+ ion irradiation of Fe-15Cr-35Ni at 625 °C and (b) displacement rate and ion deposition rate calculated for 5 MeV Ni2+ on nickel. Adapted from Garner, F. A. J. Nucl. Mater. 1983, 117, 177-197; Lee, E. H.; Mansur, L. K.; Yoo, M. H. J. Nucl. Mater. 1979, 85&86, 577-581.
|
|
temperature shifts so that irradiations must be conducted at temperatures of ^500 °C in order to create similar effects as neutron irradiation at ^300 °C. Clearly, there is not much margin for studying neutron irradiations at higher reactor temperature as higher ion irradiation temperatures will cause annealing. Another drawback is the short penetration depth and the continuously varying dose rate over the penetration depth. Figure 32 shows the damage profile for several heavy ions incident on nickel. Note that the damage rate varies continuously and
peaks sharply at only 2 pm below the surface. As a result, regions at a very well-defined depth from the surface must be isolated and sampled in order to avoid dose or dose rate variation effects from sample to sample. Small errors (500 nm) made in locating the volume to be characterized can result in a dose that varies by a factor of 2 from the target value.
A problem that is rather unique to nickel ion irradiation of stainless steel or nickel-base alloys is that in addition to the damage they create, each bombarding Ni ion constitutes an interstitial. Figure 33(a) shows
that 5 MeV Ni2+ irradiation of a Fe-15Cr-35Ni alloy resulted in high swelling in the immediate subsurface region compared to that near the damage peak. As shown in Figure 33(b), the Ni2+ ions come to rest at a position just beyond the peak damage range. So even though the peak damage rate is about 3 x that at the surface, swelling at that location is suppressed by about a factor of 5 compared to that at the surface.46 The reason is that the bombarding Ni2+ ions constitute interstitials and the surplus of interstitials near the damage peak results in a reduction of the void growth rate.47,48 In the dose rate-temperature regime where recombination is the dominant point defect loss mechanism, interstitials injected by Ni2+ ion bombardment may never recombine as there is no corresponding vacancy production.
The relaxed atomic structure from Section 1.09.6.2.1 at zero stress can be used to construct initial conditions for MD simulations for computing dislocation mobility at finite temperature. The dislocation in Section 1.09.6.2.1 is periodic along its length (z-axis) with a relatively short repeat distance (2 [Г12]). In a real crystal, the fluctuation of the dislocation line can be important for its mobility. Therefore, we extend the simulation box length by five times along z-axis by replicating the atomic structure before starting the MD simulation. Thus, the MD simulation cell has dimensions 30[111], 40[T10], 10 [T 12] along the x, y z axes, respectively, and contains 10 7070 atoms.
In the following section, we compute the dislocation velocity at several shear stresses at T = 300 K. For simplicity, the simulation in which the shear stress is applied is performed under the NVT ensemble. However, the volume of the simulation cell needs to be adjusted from the zero-temperature value to accommodate the thermal expansion effect. The cell dimensions are adjusted by a series of NVT simulations using an approach similar to that used in Section 1.09.6.1.2, except that exx, £yy, ezz are allowed to adjust independently. As we have found in Section
1.09.6.1.2 that for a perfect crystal, the thermal strain at 300 K is e = 0.00191, exx, £yy, ezz are initialized to this value at the beginning of the equilibration.
After the equilibration for 10 ps, we perform MD simulation under different shear stresses axy up to 100 MPa. The simulations are performed under the NVT chain method using the Velocity Verlet algorithm with At = 1 fs. The shear stress is applied by adding external forces on surface atoms, in the same way as in Section 1.09.6.2.1 . The atomic configurations are saved periodically every 1 ps. For each saved configuration, the CSD parameter45 of each atom is computed. Due to thermal fluctuation, certain atoms in the bulk can also have CSD values exceeding 0.6 A2. Therefore, only the atoms whose CSD value is between 4.5 and 10.0 A2 are classified as dislocation core atoms.
Figure 9(a) plots the average position (x) of dislocation core atoms as a function of time at different applied stresses. Due to PBC in x-direction, it is possible to have certain core atoms at the left edge of the cell with other core atoms at the right edge of the cell, when the dislocation core moves to the cell border. In this case, we need to ensure that all atoms are within the nearest image of one another, when computing their average position in x-direction. When the configurations are saved frequently enough, it is impossible for the dislocation to move by more than the box length in the x-direction since the last time the configuration was saved. Therefore, the average dislocation position (x) at a given snapshot is taken to be the nearest image of the average dislocation position at the previous snapshot so that the (x)(t) plots in Figure 9(a) appear as smooth curves.
Figure 9(a) shows that all the (x)(t) curves at t= 0 have zero slope and nonzero curvature,
indicating that the dislocation is accelerating. Eventually, (x) becomes a linear function of t, indicating that the dislocation has settled down into steady-state motion. The dislocation velocity is computed from the slope of the (x){t) in the second half of the time period. Figure 9(b) plots the dislocation velocity obtained in this way as a function of the applied shear stress. The dislocation velocity appears to be a linear function of stress in the low stress limit, with mobility M = v/(exy • b) = 2.6 x 104Pa-1 s_1. Dislocation mobility is one of the important material input parameters to dislocation dynamics (DD) simulations.46-48
For accurate predictions of the dislocation velocity and mobility, MD simulations must be performed for a long enough time to ensure that steady-state dislocation motion is observed. The simulation cell size also needs to be varied to ensure that the results have converged to the large cell limit. For large simulation cells, parallel computing is usually necessary to speed up the simulation. The LAMMPS program49 (http://lammps. sandia. gov) developed at Sandia National Labs is a parallel simulation program that has been widely used for MD simulations of solids.
Among the features visible in the two cascades shown in Figure 6 are a number of small interstitial clusters. For example, the cascade debris from the 1 keV cascade in Figure 6(c) contains only seven stable interstitials, but five of them (71%) are in clusters: one di-interstitial and one tri-interstitial. This tendency for point defects to cluster is characteristic of energetic displacement cascades, and it differentiates neutron and ion irradiation from typical 0.5 to 1 MeV electron irradiation, which primarily produces only isolated Frenkel pair defects. The differences between in-cascade vacancy and interstitial clustering discussed below, and the fact that their migration behavior is also quite different, have a profound influence on radiation-induced microstructural evolution at longer times. This impact of point defect clusters on microstructural evolution is discussed in detail in Chapter 1.13, Radiation Damage Theory.
The ME [18] is a continuity equation (with the source term) for the SDF of defect clusters in a discreet space of their size. This equation provides the most accurate description of cluster evolution in the framework ofthe mean-field approach describing all possible stages, that is, nucleation, growth, and coarsening of the clusters due to reactions with mobile defects (or solutes) and thermal emission of these same species. The ME is a set of coupled differential equations describing evolution of the clusters ofeach particular size. It can be used in several ways. For short times, that is, a small number of cluster sizes, the set of equations can be solved numerically.74 For longer times the relevant physical processes require accounting for clusters containing a very large number of PDs or atoms (^106 in the case of one-component clusters like voids or dislocation loops and ^1012 in the case of two-component particles like gas bubbles). Numerical integration of such a system is feasible on modern computers, but such calculations are overly time consuming. Two types of procedures have been developed to deal with this situation: grouping techniques (see, e. g., Feder et a/.,69
Wagner and Kampmann,70 and Kiritani75) and differential equation approximations in continuous space of sizes (see, e. g., Goodrich67,68, Bondarenko and Konobeev,76 Ghoniem and Sharafat,77 Stoller and Odette,78 Hardouin Duparc eta/.,79 Wehner and Wolfer,80 Ghoniem,81 and Surh eta/.82). The correspondence between discrete microscopic equations and their continuous limits has been the subject of an enormous amount of theoretical work. The equations of thermodynamics, hydrodynamics, and transport equations, such as the diffusion equation, are all examples of statistically averaged or continuous limits of discrete equations for a large number of particles. The extent to which the two descriptions give equivalent mathematical and physical results has been considered by Clement and Wood.83 In the following two sections, we briefly discuss these methods.
AKMC is a versatile method that can be used to simulate the evolution of materials with complex microstructure at the atomic scale by modeling the elementary atomic mechanisms. It has been used extensively to study phase transformations such as precipitation, phase separation, and/or ordering in many systems, as discussed in a recent review.92 Despite the fact that the algorithm is fairly simple, the method is most of the time nontrivial to implement in the case of realistic materials (as opposed to AB alloys for instance). Indeed, the determination of the total potential energy of the system, that is, the construction of the cohesive model when the chemistry ofthe system under study is complex and involves many species or a complex crystallographic structure, is difficult to obtain. Furthermore, the knowledge of all the possible events and the rates at which they occur, that is, the possible migration paths as well as their energies is nontrivial. On rigid lattices, the migration paths are more obvious to determine and cluster expansion type methods may be extended to determine the saddle point energies as a function of the local chemical environment. This can, however, take a very large amount of calculation time when there is a drastic difference in the local environment. Furthermore, complicated correlated motions such as the adatom diffusion on the (100) surfaces of fcc metals which occurs by a two-atom concerted displacement, in which the adatom replaces a surface atom, which in turn becomes an adatom, cannot be modeled within the simple scheme usually followed in AKMC of jumps to 1nn neighbor sites.
Another drawback is that to be efficient, it is tempting to use rigid lattices as a large number of KMC steps have to be performed. This can lead to an approximate (or even completely unrealistic) treatment of microstructure elements such as incoherent carbide precipitates, SIA clusters, or interstitial dislocation loops. Note, however, that it is possible to perform off-lattice AKMC, which will of course require more time consuming simulations, as proposed recently by Mason et a/.93 to investigate phase transformation in Al-Cu-Mg alloys. The authors noticed that the use of flexible lattices instead of rigid ones affected the mobility of the vacancies as well as the driving force of the reaction and therefore the rate at which phase separation took place. Furthermore, note that off-lattice AKMC also requires an equilibrium continuous cohesive model, which is difficult to build for multicomponent alloys.
At the moment, OKMC methods have been mostly used to investigate the annealing of the primary damage as in Heinisch and Singh14 or Domain eta/.28 or the effect of temperature change in the damage accumula- tion,94 but its strongest contribution in the field seems to be the study of parameters or assumptions such as the motion, 3D versus 1D motion, mobility of the SIA clusters,95-98 or corroboration of theoretical assumptions such as the analytical description of the sink strength.99 They have been used also to model as well as reexamine simple experiments such as He desorption in W1 0 or in Fe101 as well as the influence of C in isochronal annealing experiments. It can also be used to determine the production rate or source term (i. e., the ‘irradiation flux’) in mean field rate theory (MFRT) models, as discussed in the chapter on MFRT. As no spatial correlation is explicitly considered in these techniques, the source term has to take into account intracascade agglomeration and recombination. The amount of agglomeration can be obtained by annealing the cascade debris using OKMC.1
In the OKMC, the evolution of individual objects is simulated on the basis of time scales that encompass individual atomic diffusive jumps, dominated by the very fast events. This method is not efficient at high temperatures and/or high doses. The difficulty is the inability to model sufficiently high doses necessary for macroscopic materials behavior due to the focus on fast dynamics.
The time-step between events is much longer in EKMC models, which require that a reaction (e. g., clustering among like defects, annihilation among opposite defects, cluster dissolution, or new cascade introduction) occur within each Monte Carlo sweep. EKMC can therefore simulate much longer times and therefore simulate materials evolution over higher doses. It is most efficient when few objects are present in the simulation box. But questions relate to whether the time-steps are too large to reliably capture the underlying fast dynamics and whether the assumed binary interactions are sufficient to reliably calculate interaction probabilities. Further, EKMC models developed to date have not included all of the relevant microstructural evolution mechanisms, but they do represent an interesting approach in the limit of long time-step Monte Carlo simulations.
Tabulated thermochemical data have been available from a number of sources for several decades. For general substances, the most familiar have been the NIST-JANAF Thermochemical Tables54 and Thermochemical Data of Pure Substances55 The data are generally given as 298.15 K values, and columns of values such as Gibbs free energy, heat, entropy, and heat capacity are listed incrementally with temperature. The NIST-JANAF Thermochemical Tables are also available online through the National Institute for Standards and Technology (NIST). One of the key issues in using thermochemical data is the consistency of the standard states. The current commonplace usage is that the standard state is defined as 298.15 K and 1 bar (100 kPa) pressure. Small, but potentially important, errors can arise if data with different standard states are combined, for example, values at standard state pressure of 1 atm and of 1 bar are used together.
Much of the thermochemical data compilations are currently available as computer databases. In addition to the NIST-JANAF Thermochemical Tables54 is that of the Scientific Group Thermodata Europe (SGTE),56 which is well-established and has an ongoing program to assess data and add new species and phases. The same is true for the databases provided by THERMODATA57 in Grenoble, France, which has compound and solution values. Another source is MALT,46 supplied by Kagaku Gijutsu-Sha in Japan, which is more limited than SGTE,56 focusing on data that directly support industry issues. There have also been databases developed specifically for nuclear applications including THERMODATA,57 which has databases for both ex-vessel applications, NUCLEA, and for mixed oxide fuel (MOX). Kurata58 has developed a limited thermochemical database focused on metallic fuels. A database dedicated to zirconium alloys of interest for nuclear applications called ZIRCOBASE59,60 is available with fully developed representations of a number of zirconium- containing binary systems and some ternaries. The binaries and ternaries can be combined in generating higher order systems often with reasonably good accuracy. An SGTE56 nuclear materials database is also available containing most ofthe gaseous species and simple compounds of interest. An advanced nuclear fuel-specific database initiated by the Commissariat a l’Energie Atomique, FUELBASE,31 and which is expected to be moved under the auspices of the Nuclear Energy Agency with the Organization for Economic Cooperation and Development, is described in more detail in Chapter 2.02, Thermodynamic and Thermophysical Properties of the Actinide Oxides.
Information on the most common compounds and, in recent years, solution phases for many important systems has become available in the literature and is included in databases such as those noted above. However, much important data and models are not available for nuclear systems, which have not received the same attention as, for example, commercial steels. With advances in first principles modeling, some stoichiometric compounds for which there is limited or no experimental information can have values computationally determined. This is more likely for gaseous species than for condensed phases because of the greater ease in modeling the vapor. Another approach to filling in needed data is to use simple estimation techniques. The heat capacity of a complex oxide can be fairly accurately represented by the linear summation of the values of the constituent oxides. A linear relationship with atomic number is often seen in the enthalpy of formation of analogous compounds. These and other methods are discussed extensively in Kubaschewski et al.14
Equilibrium computational software packages typically will automatically acquire the needed data from accompanying selected databases. The published and commercial databases are generally assessed, meaning that they are compatible with broadly accepted values for the systems and when used with other standard values in the database thus yield correct thermochemical and phase relations. However, caution is needed when using those data with additional values obtained from other sources such as published experimental or computed values so that fundamental relationships such as phase equilibria are preserved. Another very significant issue is the completeness ofthe information. A simple example is UO2 where calculations can be performed using database values for the phase, whereas in reality the phase varies in stoichiometry as UO2±x and without including a representation for the nonstoichiometry any conclusions will be in doubt. Given the great complexity ofthe fuel and fission product phases described in Chapter 2.01, The Actinides Elements: Properties and Characteristics; Chapter 2.02, Thermodynamic and Thermophysical Properties of the Actinide Oxides; and Chapter 2.20, Fission
Product Chemistry in Oxide Fuels, it is apparent that a thermochemical model of fuel undergoing burnup is far from complete. The metallic fuel composition U-Pu-Zr is reasonably well represented,61 largely from the constituent binaries, yet the fuel after significant burnup will also contain bred actinides and fission products. Similarly, the oxide fluorite fuel phase with uranium and plutonium has perhaps been completely represented (see Chapter 2.02, Thermodynamic and Thermophysical Properties of the Actinide Oxides), but it too has yet to be modeled containing other TRU elements and fission products. High burnup fuels will also generate other phases, as noted in Chapter 2.20, Fission Product Chemistry in Oxide Fuels, and these too are often complex solid solutions with numerous components. Thus, the critical question in thermochemical modeling is, does the database contain values and representations for all the species and phases of interest? Without inclusion of all important phases, the accuracy of any conclusions from calculations will be in question.
As noted above, most databases are assessed, which implies that the included data have been evaluated with regard to the sources and methodologies used to obtain the data. It also implies that the data are consistent with information for other phases and species containing one or more of the same compo — nents/elements. Calculations of properties must return the appropriate relationships between phases and species (e. g., activities and phase equilibria). Thus, the use of data from multiple sources raises the specter of inconsistent values being used, leading to inaccurate representations. Assuring that the data are consistent between sources through checks of relationships such as known phase equilibria is important to maintaining confidence in the information providing accurate results.
Ab initio calculations can be applied to almost any solid once the limitations in cell sizes and number of atoms are taken into account. Among the materials of nuclear interest that have been studied one can cite the following: metals, particularly iron, tungsten, zirconium, and plutonium; alloys, especially iron alloys (FeCr, FeC to tackle steel, etc.); models of fuel materials, UO2, U-PuO2, and uranium carbides; structural carbides (SiC, TiC, B4C, etc.); waste materials (zircon, pyrochlores, apatites, etc.).
In this section, we rapidly expose the types of studies that can be done with ab initio calculations. The last two sections on metallic alloys and insulating materials will allow us to go into detail for some specific cases.
1.08.3.1 Perfect Crystal
1.08.3.1.1 Bulk properties
Dealing with perfect crystals, ab initio calculations provide information about the crystallographic and electronic structure of the perfect material. The properties of usual materials, such as standard metals, band insulators, or semi-conductors, are basically well reproduced, though some problems remain, especially for nonconductors (see Section 1.08.5.1 on SiC). However, difficulties arise when one wishes to tackle the properties of highly correlated materials such as uranium oxide (Section 1.08.5.2). For instance, no ab initio code, whatever the complexity and refinements, is able to correctly predict the fact that plutonium is nonmagnetic. In such situations, the nature of the chemical bonding is still poorly understood, so the correct physical ingredients are probably not present in today’s codes. These especially difficult cases should not mask the very impressive precision of the results obtained for the crystal structure, cohesive energy, atomic vibrations, and so on of less difficult materials.
1.08.3.1.2 Input for thermodynamic models
The information on bulk materials can be gathered in thermodynamical models. Most ab initio calculations are performed at zero temperature. Even with this restriction, they can be used for thermodynamical studies. First, ab initio calculations enable one to consider phases that are not accessible to experiments. It is thus possible to compare the relative stability of various (real or fictitious) structures for a given composition and pressure.
Considering alloys, it is possible to calculate the cohesive energy of various crystallographic arrangements. Solid solutions can also be modeled by so-called special quasi-random structures (SQS). Beyond a simple comparison of the energies of the various structures, when a common underlying crystalline network exists for all the considered phases, the information about the cohesive energies can be used to parameterize rigid lattice inter-atomic interaction models (i. e., pair, triplet, etc., interactions) that can be used to perform computational thermodynamics (see Chapter
1. 17, Computational Thermodynamics: Application to Nuclear Materials). These interactions can then be used in mean field or Monte-Carlo simulations to predict phase stabilities at nonzero temperature.2
As examples of this kind of studies one can cite the determination of solubility limits (e. g., Zr and Sc in aluminum22) and the exploration of details of the phase diagrams (e. g., the inversion of stability in the iron-rich side of the Fe-Cr diagram2 ).
Directly considering nonzero temperature in ab initio simulation is also possible, though more difficult. First, one can calculate for a given composition and structure the electronic and vibrational entropy (through the phonon spectrum), which leads to the variation in heat capacity with temperature. Nontrivial thermodynamic integrations can then be used to calculate the relative stability of various structures at nonzero temperature. Second, one can perform ab initio molecular dynamics simulations to model finite temperature properties (e. g., thermal expansion).
In steels, the presence of carbon, even though its concentration is very low, considerably affects defect properties because of the strong carbon-defect
interaction. DFT calculations reproduce the well — known fact that carbon is located in octahedral sites, and they also confirm the strong attraction between interstitial carbon and a monovacancy, with a binding energy of about 0.5 eV.72-74 This strong attraction is the origin of the confusing discrepancy between the vacancy migration energy in ultrapure iron, ^0.6 eV, and the effective vacancy migration energy in iron with carbon or in steels, that is, ~1.1 eV, which corresponds to first order to the sum of the vacancy migration energy and the carbon-vacancy binding energy.74 More interestingly, DFT calculations predict that the complex formed by a vacancy and two carbon atoms, VC2, is extremely stable, due to the formation of a strong covalent bond between the carbon atoms. The VC-C binding energy is indeed close to 1 eV,72-74 and VC2 complexes are expected to play a very important role.
The interaction between carbon and selfinterstitials is also attractive but weaker. In agreement with experiments,75 DFT calculations confirmed a binding energy of ^0.2 eV76 and predict, at variance with initial empirical potential results, that the nearest-neighbor configurations are repulsive and that the most attractive configuration is that shown in Figure 7. This shortcoming of empirical potentials was overcome recently with an improved potential derived taking into account information from the electronic structure.77
The strong interaction of carbon with vacancies also affects the energetics of helium-vacancy clusters, and it is important to take this into account to reproduce, for example, thermal helium desorption experiments performed in iron.78
Similar calculations have been performed with
72
nitrogen.
|
