In real metals, the vacancy formation energy is typically one-third of the cohesive energy, the discrepancy coming yet again from the strengthening of bonds to undercoordinated atoms.Как выбрать гостиницу для кошек
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Category Archives: Comprehensive nuclear materials
MD Case Studies
In the following section, we present a set of case studies that illustrate the fundamental concepts discussed earlier. The examples are chosen to reflect the application of MD to mechanical properties of crystalline solids and the behavior of defects in them. More detailed discussions of these topics, especially in irradiated materials, can be found in Chapter 1.11, Primary Radiation Damage Formation and Chapter 1.12, Atomic-Level Level Dislocation Dynamics in Irradiated Metals.
Perhaps the most widely used test case for an atomistic simulation program, or for a newly implemented potential model, is the calculation of equilibrium lattice constant a0, cohesive energy Ecoh, and bulk modulus B. Because this calculation can be performed using a very small number of atoms, it is also a widely used test case for first-principle simulations (see Chapter 1.08, Ab Initio Electronic Structure Calculations for Nuclear Materials). Once the equilibrium lattice constants have been determined, we can obtain other elastic constants ofthe crystal in addition to the bulk modulus. Even though these calculations are not MD perse, they are important benchmarks that practitioners usually perform, before embarking on MD simulations ofsolids. This case study is discussed in Section 1.09.6.1.1.
Following the test case at zero temperature, MD simulations can be used to compute the mechanical properties of crystals at finite temperature. Before computing other properties, the equilibrium lattice constant at finite temperature usually needs to be determined first, to account for the thermal expansion effect. This case study is discussed in Section 1.09.6.1.2.
Description of Displacement Cascades
In a crystalline material, a displacement cascade can be visualized as a series of elastic collisions that is initiated when a given atom is struck by a high-energy neutron (or incident ion in the case of ion irradiation). The initial atom, which is called the primary knock-on atom (PKA), will recoil with a given amount of kinetic energy that it dissipates in a sequence of collisions with other atoms. The first of these are termed secondary knock-on atoms and they will in turn lose energy to a third and subsequently higher ordered knock-ons until all of the energy initially imparted to the PKA has been dissipated. Although the physics is slightly different, a similar event has been observed on billiard tables for many years.
Perhaps the most important difference between billiards and atomic displacement cascades is that an atom in a crystalline solid experiences the binding forces that arise from the presence ofthe other atoms. This binding leads to the formation of the crystalline lattice and the requirement that a certain minimum kinetic energy must be transferred to an atom before it can be displaced from its lattice site. This minimum energy is called the displacement threshold energy (Ed) and is typically 20 to 40 eV for most metals and alloys used in structural applications.16
If an atom receives kinetic energy in excess of Ed, it can be transported from its original lattice site and come to rest within the interstices of the lattice. Such an atom constitutes a point defect in the lattice and is called an interstitial or interstitial atom. In the case of an alloy, the interstitial atom may be referred to as a self-interstitial atom (SIA) if the atom is the primary alloy component (e. g., iron in steel) to distinguish it from impurity or solute interstitials. The SIA nomenclature is also used for pure metals, although it is somewhat redundant in that case. The complementary point defect is formed if the original lattice site remains vacant; such a site is called a vacancy (see Chapter 1.01, Fundamental Properties of Defects in Metals for a discussion of these defects and their properties). Vacancies and interstitials are created in equal numbers by this process and the name Frenkel pair is used to describe a single, stable interstitial and its related vacancy. Small clusters of both point defect types can also be formed within a displacement cascade.
The kinematics of the displacement cascade can be described as follows, where for simplicity we consider the case of nonrelativistic particle energies with one particle initially in motion with kinetic energy E0 and the other at rest. In an elastic collision between two such particles, the maximum energy transfer (Em) from particle (1) to particle (2) is given by
Em = 4EoA1A2/ (A1 + A2)2 I1]
where Ax and A2 are the atomic masses of the two particles. Two limiting cases are of interest. If particle 1 is a neutron and particle 2 is a relatively heavy element such as iron, Em ~ 4E0/A. Alternately, if A1 = A2, any energy up to E0 can be transferred. The former case corresponds to the initial collision between a neutron and the PKA, while the latter corresponds to the collisions between lattice atoms ofthe same mass.
Beginning with the work of Brinkman mentioned above, various models were proposed to compute the total number of atoms displaced by a given PKA as a function of energy. The most widely cited model was that of Kinchin and Pease.17 Their model assumed that between a specified threshold energy and an upper energy cut-off, there was a linear relationship between the number of Frenkel pair produced and the PKA energy. Below the threshold, no new displacements would be produced. Above the high-energy cut-off, it was assumed that the additional energy was dissipated in electronic excitation and ionization. Later, Lindhard and coworkers developed a detailed theory for energy partitioning that could be used to compute the fraction of the PKA energy that was dissipated in the nuclear system in elastic collisions and in electronic losses.18 This work was used by Norgett, Robinson, and Torrens (NRT) to develop a secondary displacement model that is still used as a standard in the nuclear industry and elsewhere to compute atomic displacement rates.19
The NRT model gives the total number of displaced atoms produced by a PKA with kinetic energy epka as
Vnrt = 0.8Td(EPKA)/2Ed [2]
where Ed is an average displacement threshold energy.16 The determination of an appropriate average displacement threshold energy is somewhat ambiguous because the displacement threshold is strongly dependent on crystallographic direction, and details of the threshold surface vary from one potential to another. An example of the angular dependence is shown in Figure 1,20 for MD simulations in iron obtained using the Finnis-Sinclair potential.21 Moreover, it is not obvious how to obtain a unique definition for the angular average. Nordlund and coworkers22 provide a comparison of threshold behavior obtained with 11 different iron potentials and discusses several different possible definitions of the displacement threshold energy. The factor Td in eqn [2] is called the damage energy and is a function of EPKA. The damage energy is the amount of the initial PKA energy available to cause atomic displacements, with the fraction of the PKA’s initial kinetic energy lost to electronic excitation being responsible for the difference between EPKA and Td. The ratio of Td to EPKA for iron is shown in Figure 2 as a function of PKA energy, where the analytical fit to Lindhard’s theory described by Norgett and coworkers19 has been used to obtain Td.
Note that a significant fraction of the PKA energy is dissipated in electronic processes even for energies
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[210] [221] [211] |
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Knock-on direction |
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Figure 1 Angular dependence of displacement threshold energy for iron at 0 K. Reproduced from Bacon, D. J.; Calder, A. F.; Harder, J. M.; Wooding, S. J. J. Nucl. Mater. 1993, 205, 52-58. |
as low as a few kiloelectronvolts. The factor of 0.8 in eqn [2] accounts for the effects of realistic (i. e., other than hard sphere) atomic scattering; the value was obtained from an extensive cascade study using the binary collision approximation (BCA).23,2
The number of stable displacements (Frenkel pair) predicted by both the original Kinchin-Pease model and the NRT model is shown in Figure 3 as a function of the PKA energy. The third curve in the figure will be discussed below in Section 1.11.3. The MD results presented in Section 1.11.4.2 indicate that vNRT overestimates the total number of
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Figure 2 Ratio of damage energy (Td) to PKA energy (Epka) as a function of PKA energy. |
Frenkel pair that remain after the excess kinetic energy in a displacement cascade has been dissipated at about 10 ps. Many more defects than this are formed during the collisional phase of the cascade; however, most of these disappear as vacancies and interstitials annihilate one another in spontaneous recombination reactions.
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One valuable aspect of the NRT model is that it enabled the use of atomic displacements per atom (dpa) as an exposure parameter, which provides a common basis of comparison for data obtained in different types of irradiation sources, for example, different neutron energy spectra, ion irradiation, or electron irradiation. The neutron energy spectrum can vary significantly from one reactor to another depending on the reactor coolant and/or moderator (water, heavy water, sodium, graphite), which leads to differences in the PKA energy spectrum as will be discussed below. This can confound attempts to correlate irradiation effects data on the basis of parameters such as total neutron fluence or the fluence above some threshold energy, commonly 0.1 or 1.0 MeV. More importantly, it is impossible to correlate any given neutron fluence with a charged particle fluence. However, in any of these cases, the PKA energy spectrum and corresponding damage energies can be calculated and the total number of displacements obtained using eqn [2] in an integral calculation. Thus, dpa provides an environment — independent radiation exposure parameter that in
many cases can be successfully used as a radiation damage correlation parameter.25 As discussed below, aspects of primary damage production other than simply the total number of displacements must be considered in some cases.
Characterization of Cascade-Produced Primary Damage
The NRT displacement model is most correct for irradiation such as 1 MeV electrons, which produce only low-energy recoils and, therefore, the FPs. At higher recoil energies, the damage is generated in the form of displacement cascades, which change both the production rate and the nature ofthe defects produced. Over the last two decades, the cascade process has been investigated extensively by molecular dynamics (MD) and the relevant phenomenology is described in Chapter 1.11, Primary Radiation Damage Formation and recent publications.43,44 For the purpose of this chapter the most important findings are (see discussion in the Chapter 1.11, Primary Radiation Damage Formation):
• For energy above ^0.5 keV, the displacements are produced in cascades, which consist of a collision and recovery or cooling-down stage.
• A large fraction of defects generated during the collision stage of a cascade recombine during the cooling-down stage. The surviving fraction of defects decreases with increasing PKA energy up to ~10keV, when it saturates at a value of ^30% of the NRT value, which is similar in several metals and depends only slightly on the temperature.
• By the end of the cooling-down stage, both SIA and vacancy clusters can be formed. The fraction of defects in clusters increases when the PKA energy is increased and is somewhat higher in face-centered cubic (fcc) copper than in bcc iron.
• The SIA clusters produced may be either glissile or sessile. The glissile clusters of large enough size (e. g., >4 SIAs in iron) migrate 1D along close — packed crystallographic directions with a very low activation energy, practically a thermally, similar to the single crowdion.45,46 The SIA clusters produced in iron are mostly glissile, while in copper they are both sessile and glissile.
• The vacancy clusters produced may be either mobile or immobile vacancy loops, stacking-fault tetrahedra (SFTs) in fcc metals, or loosely correlated 3D arrays in bcc materials such as iron.
As compared to the FP production, the cascade damage has the following features.
• The generation rates of single vacancies and SIAs are not equal: Gv = Gi and both smaller than that given by the NRT standard, eqn. [2]: Gv, Gi <Gnrt.
• Mobile species consist of 3D migrating single vacancies and SIAs, and 1D migrating SIA and vacancy clusters.
• Sessile vacancy and SIA clusters, which can be sources/sinks for mobile defects, can be formed.
The rates of PD production in cascades are given by
Gv = Gnrt(1 — er)(1 — ev) [4]
Gi = Gnrt(1 — er)(1 — Єі) [5]
where er is the fraction of defects recombined in cascades relative to the NRT standard value, and ev and ei are the fractions of clustered vacancies and SIAs, respectively.
One also needs to introduce parameters describing mobile and immobile vacancy and SIA-type clusters of different size. The production rate of the clusters containing x defects, G(x), depends on cluster type, PKA energy and material, and is connected with the fractions e as
^xGa(x)=eaGNRT(1 — er) [6]
x = 2
where a = v, i for the vacancy and SIA-type clusters, respectively. The total fractions ev and Єі of defects in clusters are given by the sums of those for mobile and immobile clusters,
ea = ea+ea [7]
where the superscripts ‘s’ and ‘g’ indicate sessile and glissile clusters, respectively. In the mean-size approximation
(x) = d(x — <x4» [8]
where j = s, g; 8(x) is the Kronecker delta and <xoj) is the mean cluster size and
G = <xa)-1GNRT(1 — er)ei [9]
Also note that although MD simulations46 show that small vacancy loops can be mobile, this has not been incorporated into the theory yet and we assume that they are sessile: evg = 0 and esv = ev.
KMC Modeling of Microstructure Evolution Under Radiation Conditions
KMC models are now widely used for simulating radiation effects on materials.21- 0 Advantages of KMC models include the ability to capture spatial correlations in a full 3D simulation with atomic resolution, while ignoring the atomic vibration time scales captured by MD models. In KMC, individual point defects, point defect clusters, solutes, and impurities are treated as objects, either on or off an underlying crystallographic lattice, and the evolution of these objects is modeled over time. Two general approaches have been used in KMC simulations, object KMC (OKMC) and event KMC (EKMC),35,36 which differ in the treatment of time scales or step between individual events. Within the OKMC designation, it is also possible to further subdivide the techniques into those that explicitly treat atoms and atomic interactions, which are often denoted as atomic KMC (AKMC), or lattice KMC (LKMC), and which were recently reviewed by Becquart and Domain,45 and those that track the defects on a lattice, but without complete resolution of the atomic arrangement. This later technique is predominately referred to as object Monte Carlo and used in such codes as BIGMAC27 or LAKIMOCA.28 More recently, several algorithmic ideas have been identified that, in combination, promise to deliver breakthrough KMC simulations for materials computations by making their performance essentially independent of the particle density and the diffusion rate disparity, and these will be further discussed as outstanding areas for future research at the end of the chapter.
KMC modeling of radiation damage involves tracking the location and fate of all defects, impurities, and solutes as a function oftime to predict microstructural evolution. The starting point in these simulations is often the primary damage state, that is, the spatially correlated locations of vacancy, self-interstitials, and transmutants produced in displacement cascades resulting from irradiation and obtained from MD simulations, along with the displacement or damage rate which sets the time scale for defect introduction. The rates of all reaction-diffusion events then control the subsequent evolution or progression in time and are determined from appropriate activation energies for diffusion and dissociation; moreover, the reactions and rates of these reactions that occur between species are key inputs, which are assumed to be known. The defects execute random diffusion jumps (in one, two, or three dimensions depending on the nature of the defect) with a probability (rate) proportional to their diffusivity. Similarly, cluster dissociation rates are governed by a dissociation probability that is proportional to the binding energy of a particle to the cluster. The events to be performed and the associated time-step of each Monte Carlo sweep are chosen from the RTA.17,18 In these simulations, the events which are considered to take place are thus diffusion, emission, irradiation, and possibly transmutation, and their corresponding occurrence rates are described below.
Variable Stoichiometry/Associate Species Models
As noted 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, modeling of complex systems such as U-Pu-Zr and (U, Pu)O2±x has been exceptionally difficult. For example, actinide oxide fuel is understood to be nonstoichiometric almost exclusively due to oxygen site vacancies and interstitials. As a result, the fluorite — structure phase has been treated as being composed of various metal-oxygen species with no vacancies on the metal lattice.
An early and successful modeling approach has employed a largely empirical use of variable stoichiometry species that are mixed as subregular solutions to fit experimental information.17-20 This technique can be viewed as a variant of the associate species method.21 In the approach, thermochemical values were determined from the phase equilibria, that is, the phase boundaries, and data for the temperature — composition-oxygen potential [mO2 = RT ln(pO*)] where pO2 is a dimensionless quantity defined by the oxygen pressure divided by the standard state pressure of 1 bar. The UO2±x phase, for example, was treated as a solid solution of UO2 and UaOb where the values of a and b were determined by a fit to experimental data. Figure 3 illustrates the trial and error process using a limited data set to obtain the species stoichiometry which results in the best fit to the data. As can be seen, a variety of stoichiometries for the constituent species yield differing curves ofln(pO*) versus fx), with the most appropriate matching the slope of 2. Thus, for this example U10/3O23/3 provides for an optimum fit between U3O7 and U4O9, and its solution with UO2 best reproduces the observed oxygen potential behavior. Utilizing a much more extensive data set from a variety of sources resulted in a set of best fits to the data, yet they required three solid solutions to adequately represent the entire compositional range for UO2±x These are
-4
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Figure 3 The іп(рО|) dependence as a function of x for UO2+x and of f(x) for several solid-solution species’ stoichiometries for an illustrative oxygen pressure — temperature-composition data set. Coincidence with the theoretical slope of 2 indicates the proper solution model. Reproduced from Lindemer, T. B.; Besmann, T. M. J. Nucl. Mater. 1985, 130, 473-488.
UO2+x (high hyperstoichiometry, i. e., large values of x): UO2 + U3O7,
UO2+x (low hyperstoichiometry, i. e., smaller
values of x): UO2 + U2O45, and
UO2-x (hypostoichiometric): UO2 + U1/3.
The results of the models for UO2±x are plotted in Figure 4 together with the entire data set used for optimizing the system.
The above models for UO2±x have been widely adopted, as has been a similar model of PuO2-x.16 These have also been combined to construct a successful model for (U, Pu)O2±x.16 Lewis eta/.22 used an analogous technique for UO2±x. Lindemer23 and Runevall et a/.24 have generated successful models of CeO2-x. Runevall et a/.24 also used the method for NpO2-x, AmO2-x (with the work of Thiriet and Konings25), (U, Am)O2±x, (Th, U)O2±x, (U, Ce)O2_* (Pu, Am)O2±x, and (U, Pu, Am)O2±x. They noted that results for the (Th, U)O2+x were less successful perhaps because of the difficulty in the measurements
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1000 1500 2000 2500 3000
Temperature (K)
Figure 4 Oxygen potential plotted versus x for the models of UO2±x of Lindemer and Besmann17 overlaid with the entire data set used for the optimization.
made near stoichiometry. Osaka et a/.26-28 used the approach to successfully represent the (U, Am)O2_x, (U, Pu, Am)O2_x and (Am, Th)O2_x phases.
Creation and Elimination of Point Defects
Because RIS is due to fluxes of excess point defects, modeling must take into account their creation and elimination mechanisms. It must, for example, reproduce the ratio between vacancy and interstitial concentrations that controls the respective weights of annihilation by recombination or elimination at sinks. The situation from this point of view is very different from phase transformations during thermal aging, where usually only vacancy diffusion occurs, and simulations can be performed with nonphysical point defect concentrations and a correction of the timescale (see, e. g., Le Bouar and Soisson78 and Soisson and Fu12 ).
During electron or light ion irradiation, defects are homogeneously created in the material, with a frequency directly given by the radiation flux (in dpa s~ ), a condition that is easily modeled in continuous models, mean-field models,12,14 or Monte Carlo simulations.16,118 The formation of defects in displacement cascades during irradiation by heavy particles can also be introduced in kinetic models, using the point defect distributions computed by molecular dynamics.126 The annihilation mechanisms at sinks such as surfaces or grain boundaries are, for the time being, simulated using very simple approximations (perfects sinks, no formation/annihilation of kinks on dislocations, or steps on surfaces). This should not affect the basic coupling between diffusion fluxes, but the long-term evolution of the sink microstructure — which will eventually have an impact on the chemical distribution — cannot be taken into account.
Finally, thermally activated point defect formation mechanisms that operate during thermal aging, are taken into account in some simulations.11-14 Simulations that do not include the thermal production are then valid only at sufficiently low temperatures, when defect concentrations under irradiation are much larger than equilibrium ones.
Primary and Weighted Recoil Spectra
A description of irradiation damage must also consider the distribution of recoils in energy and space. The primary recoil spectrum describes the relative number of collisions in which the amount of energy between Tand T + dTis transferred from the primary
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