In 1972, Okamoto et a/.26 observed strain con­trast around voids in an austenitic stainless steel Fe-18Cr-8Ni-1Si during irradiation in a high — voltage electron microscope. They attributed this con­trast to the segregation strains predicted by Anthony. This is the first reported experimental evidence of RIS. Soon after, a chemical segregation was directly measured by Auger spectroscopy measurements at the surface of a similar alloy irradiated by Ni ions.27

It was then realized that ifthe solute concentration near the point defect sinks reaches the solubility limit, a local precipitation would take place. In 1975, Barbu and Ardell28 observed such a radiation — induced precipitation (RIP) of an ordered Ni3Si phase in an undersaturated Ni-Si alloy.

The analysis of strain contrast and concentration profiles measured by Auger spectroscopy suggested that undersized Ni and Si atoms (which can be more easily accommodated in interstitial sites) were diffus­ing toward point defect sinks, while oversized atoms (such as Cr) were diffusing away. Such a trend, later
confirmed in other austenitic steels and nickel-based alloys,29 led Okamoto and Wiedersich27 to conclude that RIS in austenitic steels was due to the migration of interstitial-solute complexes, and they proposed this new RIS mechanism, in addition to the ones involv­ing vacancies (Figure 2(c)). Then again, Marwick30 explained the same experimental observations by a coupling between fluxes of vacancies and solute atoms, pointing out that thermal diffusion data showed Ni to be a slow diffuser and Cr to be a rapid diffuser in austenitic steels. We will see later that, in spite of many experimental and theoretical studies, the debate on the diffusion mechanisms responsible for RIS in austenitic steels is not over.

Following these debates on RIS mechanisms, it became common to refer to the situation illustrated in Figure 2(a) as segregation by an inverse Kirkendall (IK) effect (the term was coined by Marwick30 in 1977) and to the one in Figure 2(b) as segregation by drag effects, or by migration of vacancy-solute complexes. In the classical Kirkendall effect,31 a gra­dient of chemical species produces a flux of defects. It occurs typically in interdiffusion experiments in A-B diffusion couples, when A and B do not diffuse at the same speed. A vacancy flux must compensate for the difference between the flux of A and B atoms, and this leads to a shift of the initial A/B interface (the Kirkendall plane). The IK effect is due to the same diffusion mechanisms but corresponds to the situa­tion where the gradient of point defects is imposed and generates a flux of solute. The distinction between RIS by IK effect and RIS by migration of defect-solute complexes, initially proposed for the vacancy mechanisms, was soon generalized to inter­stitial fluxes by Okamoto and Rehn.32,33 RIS in dilute alloys, where solute-defect binding energies are clearly defined and often play a key role, is com­monly explained by diffusion of solute-defect complexes, while the IK effect is often more useful to explain RIS in concentrated alloys. This distinction

is reflected in the modeling of RIS (see Section 1.18.3). However, it is clear that RIS can occur in dilute alloys without migration of solute-defect fluxes. Moreover, such a terminology and sharp distinction can be somewhat misleading; the mechanisms are not mutually exclusive. In the case of undersized B atoms, for example, a strong binding between interstitial and B atoms can lead to a rapid diffusion of B by the interstitial (IK effect with DBi > DaO and to the migration of interstitial-solute complexes. More gen­erally, one can always say that RIS results from an IK effect, in the sense that it occurs when a gradient of point defects produces a flux of solute. Nevertheless, because they are widely used, we will refer to these terms at times when they do not create confusion.

The damage function refers to the number of FPs created within the first several picoseconds of the primary recoil event. At longer times, defects migrate from their nascent sites and interact with other defects and microstructural features. As noted earlier, many radiation effects, such as radiation-enhanced diffusion, segregation, and void swelling, depend more strongly on the number of defects that escape their nascent cascades and migrate freely in the lattice before annihilating, trapping, or forming defect clusters. The same general approach used to determine the damage function has been employed to determine the relative fraction of freely migrating defects, that is, e/nNRT, as illustrated by Figure 14. Here, the relative number of Si atoms segregating to the surface during irradiation, per dpa, is plotted versus a characteristic energy of the recoil spectrum, T1/2. It is seen that the fraction decreases rapidly with increasing recoil energy. Simi­lar experiments were performed using radiation — enhanced diffusion, as described in Section 1.07.2.

While ion irradiation has proved extremely useful in illustrating the spectral effects on freely migrating

defects, extracting quantitative information about freely migrating defects from such experiments is difficult. These measurements, unlike the damage function, require very high doses, and several dpa; the buildup of the sink structure must be adequately taken into account. It is also difficult to estimate, for example, how many interstitials are required to transport one Si atom to the surface. We mention in passing that experiments performed using ordering kinetics in order-disorder alloys have provided a more direct measure of the number of freely migrat­ing defects (vacancies in this case), as these experi­ments require doses less than «10~7dpa so that no damage build-up can occur.25 These experiments show similar effects of primary recoil spectrum on the fraction of freely migrating defects, although the fractions of such defects were found to be somewhat higher in these experiments, «5-10%. These frac­tions are in good agreement with radiation-enhanced diffusion experiments using self-ions on Ni, when the effect of sink strength is taken into account.26

From the atomistic and electronic structure point of view, it is legitimate to distinguish between electrically conducting materials on one hand and insulating or semiconducting materials on the other. Indeed, insulating materials exhibit specific behaviors, especially for the point defects. Due to the existence of a gap in the electronic density of states, the point defects may be charged. There is recent evidence that the properties of the point defects, especially their kinetic properties, such as the migration energy, depend a lot on their charge state. The charge of a given point defect depends on the position of the Fermi level within the band gap: a low lying Fermi level (close to the valence band) favors positively charged defects, whereas a Fermi level close to the conduction band favors negatively charged defects. The positions of the Fermi level corresponding to transitions between charge states are called charge transition levels (CTL). The correct determination of these CTL allows the correct prediction of the charge states of the defects, as piloted by Fermi level position, that is, the doping conditions for the material. Standard DFT methods fail to reproduce accurately these CTL, and the research of more accu­rate methods is presently a very active field in the electronic structure community, with major implica­tions for microelectronic research as well as for nuclear materials, especially in view of the aforementioned variation of point defect kinetic properties with their charge state. All these charge aspects of point defects are completely out of range for empirical potentials.

In the last two sections, we exemplify the research on insulating materials by summarizing the available results for two important insulating nuclear materi­als: silicon carbide and uranium dioxide.

Silicon carbide is an important candidate material for fusion and fission applications. Even if it arguably a less crucial material than UO2, we start with this material as its electronic structure is simpler. UO2 is obviously the basic model material for the nuclear fuel of usual reactors.

The two-band approach can be applied to magnetic materials, where the bands spin up and spin down bands have the same capacity (N = nJ = N# = 5). If in addition we assume that the bands have the same width and shape (see, e. g., Figure 11), there is a remarkable collapse of the model onto the single­band EAM form, with a modified embedding function.

The formalism here extends to the two-band model, but the physics is analogous to other magnetic potentials.27 For simplicity, consider a rectangular d-band of full width W centered on E0. The bond energy for a single spin-up band relative to the free atom is given by •E = Z/N —1 w

2 NE/ W dE

— W/2

[16]

where Z" is the occupation of the band and uparrows denote ‘spin up.’

To describe the ferromagnetic case, it is assumed that there are two independent d-bands corresponding to opposite spins, and that these can be projected onto an atom to form a local density of states. For a free atom atomic case, Hund’s rules determine a high-spin case (e. g., S = 2 for iron), and there is an energy Ux associated with transferring an electron to a lower spin state. In the solid, the simplest method is to set Ux to be proportional to the spin with the coefficient of proportionality being an adjustable parameter, E0.

Ux = — EoZ"- Z#I [17]

Defining the spin, S = Z" — Z# and assuming charge neutrality (T = (Z" + Z#)), the two-d-band binding energy on a site i is then as follows:

Ui = u" + U# + ux = + WN (T2 + S2) — TWi/2 — EoSi

[18]

Differentiating this equation about Si gives us the optimal value for the magnetization of a given atom of Si = 2NE0/Wj, and the many-body energy of an atom with T = 6, N = 5 (suppressing the i label) as

U =—6W/5 — 5E2/W E0/Wi < 0.4 = —2W/5 — 4E0 E0/ Wi > 0.4 [19]

where neither band is allowed to have occupancy more than 5 or less than 0. For a material with T d-electrons (where T > 5), transfer of electrons between the spin bands becomes advantageous for W > 10E0/(10 — T). For smaller W, the spin " band is full and the energy is simply proportional to the bandwidth of the # band as in the FS model. Similar cases apply to the T< 5 case when the minority band may be empty.

There has been some controversy about the expression for Ux. In the two-band model, this is a promotion energy from the minority spin band to the majority. In the atomic case, it is the energy to violate Hund’s rule, and the implicit reference state is the high-spin atom. Electron transfer is bound by the number ofelectron, so the function has discontinuous slope at Si = 0.4. By contrast, the approach of Dudarev and coworkers27 uses a Stoner model for the spin energy, which introduces quadratic and quartic terms in Ux(S,). In that case, the implicit reference state is the nonmagnetic solid, and any value of Si is acceptable.

Within the second-moment model, the bandwidth W is given by the square root of p, the sum of the squares of the hopping integral. Applying this, and the usual pairwise repulsion V(r), gives an expres­sion for the two-band energy

и = EV (rj) — ^pj

/

— Б/ yfpjH(2W — 5Eo)

— 4E0H(5E0 — 2 W) [20]

where H is the Heaviside step function, B is a con­stant, and the zero of energy corresponds to the nonmagnetic atom.

Note that this form does not explicitly include S, and that it has the EAM form with an embedding function F(x) = v/x(1 — B/x).

Although this model incorporates magnetism and provides a way to calculate the magnetic moment at each site, it is possible to use it without actually calculating S. The additional many-body repulsive term is similar to, for example, the many-body potential method of Mendelev et al. It is also inter­esting that in the original FS paper, it was not possi­ble to fit the properties of the magnetic elements Fe and Cr; an extra term was added ad hoc. Later para — meterizations of FS potentials for iron with a pure square root for F have not exactly reproduced the elastic constants.28

The implication of this work is that for second — moment type models, there should be a one-to — one relation between the local density p and the magnetic moment. Figure 12 shows this relation for two parameterizations, p from Dudarev-Derlet and Mendelev et al. and the magnetic moment cal­culated with spin-dependent DFT projected onto atoms. It shows that there are two cases. For atoms associated with local defects, the density varies quite sharply with p, while for the crystal under pressure the variation is slower. It is noteworthy that the same broad features are present in both potentials, even though the Mendelev et al. potential was fitted with­out consideration of magnetic properties, albeit with a FS-type embedding function. This suggests that the magnetic effects were unwittingly captured in the fitting process.

1.12.3.1 Why Atomic-Scale Modeling?

First principle ab initio methods for self-consistent calculation of electron-density distribution around moving ions provide the most accurate modeling tech­niques to date. They take into account local chemical and magnetic effects and provide significant potential for predicting material properties. They are used with success in applications where the properties are limited to the nanoscale, for example, microelectronics, cataly­sis, nanoclusters, and so on (Chapter 1.08, Ab Initio Electronic Structure Calculations for Nuclear Materials). A typical scale for this is of the order of a few nm. However, this leaves a significant gap between ab initio methods and those required to model proper­ties of bulk materials arising from radiation damage. These involve phenomena acting over much longer scales, such as interactions between mobile and sessile defects, their thermally activated transport, their response to internal and external stress fields and gra­dients of chemical potential. Models for bulk properties are based on continuum treatments by elasticity, ther­mal conductivity, and rate theories where global defect properties such as formation, annihilation, transport, and interactions are already parameterized at contin­uum level. The only technique that currently bridges the gap in the scales between ab initio and the contin­uum is computer simulation of a large system of atoms, up to 106—108. Atoms move as in classical Newtonian dynamics due to effective forces between them calcu­lated from empirical interatomic potentials and respond to internal and external fields due to tempera­ture, stress, and local imperfections. Atomic-scale mod­eling has provided the results presented in this chapter. In the following section, we present a short description of typical models for simulation of dislocations and their interactions with defects formed by radiation.

The continuous production of SIA clusters in dis­placement cascades is a key process, which makes microstructure evolution under cascade conditions qualitatively different from that during FP producing 1 MeVelectron irradiation. In this case, eqns [10]—[12] should be used for the concentration of mobile defects. The equations for isolated PDs have been considered in detail in the previous section. In order to analyze damage accumulation under cascade irradiation, one needs to define the sink strengths of various defects for the SIA mobile clusters in eqn [12]. We give examples of such calculations for the case when cluster migrates 1D rather than 3D in the following section.

1.13.6.1 Reaction Kinetics of One-Dimensionally Migrating Defects

The 1D migration of the SIA clusters along their Burgers vector direction results in features that dis­tinguish their reaction kinetics from 3D diffusing defects. These were first noticed in and theoreti­cally analyzed for annealing experiments (Lomer and Cottrell,126 Frank et a/.,127 Gosele and Frank,128 Gosele and Seeger,129 and Gosele40) and, then, under irradiation (Trinkaus et a/.19,20 and Borodin130). In this section, we consider the reaction kinetics of 1D migrating clusters with immobile sinks and follow the procedure employed in Barashev et a/.25

Detailed information about the diffusion process of a 1D migrating particle is given by the function u(t, X, x), which is known as Furth’s formula for first passages and has the following probabilistic signifi — cance.1 1 In a diffusion process starting at the point X > 0, the probability that the particle reaches the origin before reaching the point x > X in the time
interval t1 < t < t2 is given by the integral over this interval. For particles undergoing random walk, this function is found to be equal to

1

u(t, X, x) = 2пУ ] i exp

i=1

where D1d is the diffusion coefficient. Using this function, one can write the probability for a particle to survive until time t, that is, not to be absorbed by the barriers placed at the origin and at the point x, as [1

•q(t, X, x) = dt0 [u(t0, X, x) + u(t0, x — X, x)]

_ 4y’exp[—(2i — 1)2р2Р1Р?|x2]sin fnX(2i -1) [111,

n 2i — 1 sin x [ J

i=1

The expected duration of the particle motion until its absorption is given by: 1

v(t, X, x)dt

0

Equation [112] is the classical result of the ‘gambler’s ruin’ problem considered by Feller.131

1.13.6.1.1 Lifetime of a cluster

In order to obtain the lifetime of 1D migrating clusters, one should average truin(X, x) over all possi­ble distances between sinks and initial positions of the clusters, that is, over x and X. For this purpose, the corresponding probability density distribution, ‘(x, X), is required.

Let us assume that all sinks are distributed ran­domly throughout the volume and introduce the 1D density of traps (sinks), L, that is, the number of traps per unit length. In this case, ‘(x, X) can be repre­sented as a product of the probability density for a cluster to find itself between two sinks separated by a distance x, L2x exp(—Lx), and the probability density to find a cluster at a distance X from one of these sinks, 1 =x:

‘(x, X) = L2exp(—Lx), 0 < x < 1,0 < X < x [113]

With this distribution, the cluster lifetime, t1D, and the mean-free path to sinks, l, are:

t1D = <Trum(X, x)>X, x = 1|2D1dL2 [114]

l =<X>X, x = 1|L [115]

where the brackets denote averaging: <>X, x =

o x

dx dX'(x, X)

00

1.13.6.1.2 Reaction rate

It follows from eqn [114] that the reaction rate between 1D migrating clusters and immobile sinks (e. g., Borodin13 ) is given by:

R1D = 2L2D1d C = — г Ad C [116]

l

This equation defines the total reaction rate as a function of L, determined by the concentration and geometry of sinks. If there are different sinks in the system, L is a sum of corresponding contributions Lj from traps of type j In a crystal containing disloca­tions and voids only,

L = Ld + Lc [117]

where subscripts ‘d’ and ‘c’ stand for dislocations and voids, respectively. These partial trap densities are found below.

Consider voids of a particular radius r, randomly distributed over the volume. Without loss of general­ity, the capture radius of a void for a cluster is assumed here to be equal to its geometrical radius, that is, rc, = r,. A void of radius r, is available to react with mobile clusters that lie in a cylinder of this radius around the cluster path. Hence, the partial 1D density of voids of any particular radius, Lci, and the total 1D void density, Lc, are given by

Lc, = prljf (r,) [118]

Lc = Yl, Lc, = Prc2Nc [119]

,

where f (r,) is the SDF of voids (^ f {ri) = Nc is the

,

total void number density) and rc2 is the mean square of the void capture radius. For dislocations

Ld = PrA p* [120]

where pd is the dislocation density defined as the mean number of dislocation lines intersecting a unit area (surface density) and rd is the corresponding capture radius. This can be shown in the following way. The mean number of dislocation lines intersect­ing the cylinder of unit length and radius rd around the cluster path equals the area of the cylinder sur­face, 2шА, times the dislocation density divided by 2. (The factor 2 arises because each dislocation intersects the cylinder twice.) It should be noted that the dislocation sink strength for 3D diffusing defects is usually expressed through the dislocation density, pd, defined as the total length of dislocation lines per unit volume of crystal (volume density). The relationship between p* and Pd depends on the
distribution of the dislocation line directions. For a completely random arrangement, the volume density is twice the surface density, Pd ~ 2p[j (see, e. g., Nabarro132). In this case, eqn [120] is the same as found by Trinkaus et at}9,20

Substituting eqns [117] — [120] into eqn [116], the total reaction rate of the clusters in a crystal contain­ing random distribution of voids and dislocations is found to be130:

R1D = 2(prdfd + Pr2NcJ AdC [121]

For the case, in which immobile vacancy and SIA clusters are also taken into account, the sink strength for 1D diffusing SIA clusters, kg, is equal to

kg = 2 ( 2"^ ^ prc Nc + svclNvcl + ffidAic^ [122]

where svcl and ffid are the interaction cross-sections and Nvcl and ЛА the number densities of the sessile vacancy and SIA clusters, respectively. svcl and ffid are proportional to the product of the loop circum­ference and the corresponding capture radius similar to rd for dislocations.

1.13.6.1.3 Partial reaction rates

A detailed description of the microstructure evolu­tion requires the partial reaction rates, Rj, of the clusters with each particular sink, for example, dis­locations or voids of various sizes.22 According to the definition of the parameters Lj and L, the ratio Lj/L is the probability for a trap to be of type j. Hence, the partial reaction rates are

R = L R [123]

A similar relation between total and partial reaction rates was used in Gosele and Frank.12 Using eqn [116], one can write the partial reaction rate of clus­ters with sinks of type j 2Lj LD1D C where lj = 1/Lj is the mean distance between a cluster and a sink of type j in 1D, cf. eqn [116]. Thus, the partial reaction rate of a specific type of sink depends on the density of that sink and also on the density of all other sinks. This correlation between sinks is characteristic of pure 1D diffusion — reaction kinetics in contrast to 3D diffusion where the leading term of the sink strength of any defect is not correlated with others (see eqn [54]).

Radiation interaction with materials results in the production oflattice defects. The motion and interac­tion between radiation-induced lattice defects among
themselves and the existing microstructure have direct consequences to the macroscopic response of the material to the radiation environment. We present in this section several applications of computational DD to modeling of radiation-induced defects and their effects on mechanical properties.

1.16.3.1 Dislocation Interaction with Radiation-Induced Defects

Using an infinitesimal loop approximation, Kroupa found the stress tensor of a prismatic loop to be of the form s j = kjm bR2/2p, where kjis an orientation factor of order unity, R is the loop radius, and p the distance from the loop center. The total force and its moment on an SIA cluster can be expressed respectively as

Fi = — nj Ojk, ib’k dA

Mi Eijknj bisikdA

where nl, b’k, dA1 refer to the Cartesian components of the normal vector, the Burgers vector, and the habit plane area of the cluster, respectively. As a mobile SIA dislocation loop moves closer to the core of the slip loop, the turning moment on its habit plane increases. When the mechanical work of rotation exceeds a criti­cal value of ~0.1 eV per crowdion, we assume that the cluster changes its Burgers vector and habit plane, and moves to be absorbed into the dislocation core. Thus, the mechanical work for cluster rotation is equated to a critical value (i. e., dW = Jy2 Mddj = A Uerit) and used as a criterion to establish the stand-off distance.35

The two main aspects of dislocation interaction with defect clusters that affect both hardening and ensuing plastic flow localization are (1) dislocation unlocking from defect cluster atmospheres; and (2) destruction of SFTs on nearby slip planes by gliding dislocations. The interaction between grown-in dislo­cations and trapped defect clusters has been shown to lead to unfaulting of vacancy clusters in the form of vacancy loops. It can also result in rotation of the habit plane of mobile SIA clusters. Once either of these possibilities is realized for a vacancy or SIA cluster, it is readily absorbed into the dislocation core. Ghoniem etal3 used these conditions to determine an appropri­ate stand-off distance from the dislocation core, which is free of irradiation-induced defect clusters. It is esti­mated that clusters within a distance of 3-9 nm from the dislocation core in Cu will be absorbed, either by rotation of their Burgers vector or by unfaulting. We will use this estimate as a guide to calculations of long-range interactions of dislocations with sessile
prismatic SIA clusters situated outside the stand-off distance. While the experimentally observed average SFT size is 2.5 nm for oxygen free high conductivity (OFHC) copper, the radius of a sessile interstitial clus­ter, which results from coalescence of smaller mobile clusters, is assumed to be in the range ^4—20 nm. The local density of interstitial defect clusters at the stand­off distance is taken to be in the range 0.6-4 x 1024 m~3, giving an average intercluster spacing of ~18-35й. In subsequent computer simulations, we use the fol­lowing set of material data for Cu: lattice constant a = 0.3615nm, shear modulus m = 45.5 GPa, Poisson’s ratio n = 0.35, and F-R source length L = 1500-2000a.

Consider now the more complex interaction between an expanding Frank-Read (F—R) dislocation source, and the full field of multiple sessile SIA clusters (loops) present in a region of decoration. As the F—R source expands in the elastic field of SIA clusters, each point on the dislocation line will experience a resistive (or attractive) force, which must be overcome for the dislocation to move further. The dislocation line curvature and, hence, the local self-force also change dynamically. To determine the magnitude of collective cluster resistance, systematic calculations for the dynamics of interaction between

SIA clusters are presented (Figure 3). When the SIA clusters are all attractive, the dislocation line is imme­diately pulled into their atmosphere, but as the applied stress is increased, the dislocation remains trapped by the force field of SIA clusters. When the stress is increased to 200 MPa, the line develops an asymmetric configuration as a result of its Burgers vector orienta­tion, and an unzipping instability eventually unlocks the F—R source from the collective cluster atmosphere. This asymmetric unlocking mode is characteristic of a high linear cluster density on smaller sections of the F—R source decoration, where the linear density of SIA clusters is 50 (cluster/lattice constant).

Figure 3 shows the detailed dynamics of the col­lective cluster interaction with an expanding F—R source. A fluctuation in the line shape is amplified by the combined effects of the applied and self-forces on the middle section of the F—R source, and the disloca­tion succeeds in penetrating through the collective cluster field at a critical tensile stress of s11 = 180 MPa (or equivalently at CRSS of t/m = 0.0015). The critical shear stress (in units of the shear modulus) to unlock the F—R source is shown in Figure 4 as a function of the stand-off distance for a fixed intercluster dis­tance of 50a. The results of current calculations are

compared with the analytical estimates of Trinkaus eta/.23 For larger stand-off distances, the current results show a larger critical stress as compared to the analyti­cal estimates, while for stand-offdistances smaller than ^60a, a smaller critical stress is required to unlock the F—R source. When the stand-off distance is large, the applied stress must overcome the self-force, which results from the finite length of the F—R source, in addition to the collective cluster elastic field. At smaller stand-off distances, however, the dislocation easily unlocks by one of the two unzipping instability modes discussed earlier, and the predicted CRSS is smaller than analytical estimates. At intercluster dis­tances smaller than 70a, the dislocation shape instabil­ity results in a CRSS value that is smaller than the corresponding analytical result. It is estimated that the required CRSS is ^0.001m (^50 MPa for copper), for an average intercluster distance of / ~ 50a, and a stand­off distance of ^40a. It is experimentally difficult to determine the local value of / in the decoration region of dislocations, which is likely to vary considerably, depending on the character of the dislocation Burgers vector. However, /~ 50a is an upper bound, while /~ 20-30a is more likely. Since the CRSS is roughly inversely proportional to /, the most likely value of the CRSS to unlock dislocations and start the operation of F—R sources would be TcRss ^100-150 MPa. Depend­ing on the local value of the Schmidt factor, the corresponding uniaxial applied stress is thus likely to be on the order of 200-300 MPa, under conditions of heavy decoration (i. e., at a displacement damage dose of >0.1 dpa).

The current estimates for the unlocking stress are thus consistent with the experimental data, and indi­cate that the operation of F-R sources from deco­rated dislocations can be initiated by one (or both) of the following possibilities:

1. Activated F-R sources are decorated with a statis­tically low linear defect cluster density;

2. Dislocation sources are initiated at stress singula­rities in regions of internal stress concentration.

Anthony set up a thermal diffusion experiment involving vacancies as a driving force18-22,25,63 in aluminum alloys. The gradient of vacancy concentra­tion was produced by a slow decrease of temperature. At the beginning of the experiment, the ratio between solute flux and vacancy flux is the following:

JB _ _ LBB + LAB r-ml

г V V V [20]

LAA ‘ LBB ‘ 2Lab

The volume of the cavity and the amount of solute segregation nearby yield a value for the flux ratio. Note, that secondary fluxes induced by the formation of a segregation profile are neglected in the present analysis.

This experiment, combined with an interdiffusion annealing, could be a way to estimate the complete Onsager matrix. Unfortunately, the same experiment does not seem to be feasible in most alloys, especially in steels. In general, vacancies do not form cavities, and solute segregation induced by quenched vacan­cies is not visible when the vacancy elimination is not concentrated on cavities.

1.18.3.3.1 Diffusion during irradiation

In the 1970s, some diffusion experiments were per­formed under irradiation.64 The main objective was to enhance diffusion by increasing point defect concentrations and thus facilitate diffusion experi­ments at lower temperatures. Another motive was to measure diffusion coefficients of the interstitials cre­ated by irradiation. In general, the point defects reach steady-state concentrations that can be several orders of magnitude higher than the thermal values. In pure metals, some experiments were reliable enough to provide diffusion coefficient values at temperatures that were not accessible in thermal conditions.64

The analysis of the same kind of experiments in alloys happened to be very difficult. A few attempts were made in dilute alloys that led to unrealistic values of solute-interstitial binding energies.65 How­ever, a direct simulation of those experiments using an RIS diffusion model could contribute to a better knowledge of the alloy diffusion properties.

Another technique is to use irradiation to implant point defects at very low temperatures. A slow annealing of the irradiated samples combined with electrical resistivity recovery measurement high­lights several regimes of diffusion; at low tempera­ture, interstitials with low migration energies diffuse alone, while at higher temperatures, vacancies and point defect clusters also diffuse. Temperatures at which a change of slope is observed yield effective migration energies of interstitials, vacancies, and point defect clusters.66 In situ TEM observation of the growth kinetics of interstitial loops in a sample under electron irradiation is another method of determining the effective migration of interstitials.67

Ab initio calculations rely on the use of dedicated codes. Such codes are rather large (a few hundred thousand lines), and their development is a heavy task that usually involves several developers. An easy, though oversimplified, way to categorize codes is to classify them in terms of speed on one hand and accuracy on the other. The optimum speed for the desired accuracy is of course one of the goals of the code developers (together with the addition of new features). Codes can primarily be distinguished by their pseudoization scheme and the type of their basis set. We will not describe many other numerical or programming differences, even though they can influence the accuracy and speed of the codes.

The possible choices in terms of basis sets and pseudoization are discussed in the following para­graphs. Pseudoization scheme and basis set are intri­cate as some bases do not need pseudoization and some pseudoizations presently exist only for specific basis sets. These methodological choices intrinsically lead to accurate but heavy, or conversely fast but approximate, calculations. We also mention some codes, though we have no claim to completeness on that matter. Furthermore, we do not comment on the accuracy and speed of the codes themselves as the developing teams are making continuous efforts to improve their codes, which make such comments inappropriate and rapidly outdated.

A great deal can be said about why MD is a useful simulation technique. Perhaps the most impor­tant statement is that, in this method, one follows the atomic motions according to the principles of classical mechanics as formulated by Newton and Hamilton. Because ofthis, the results are physically as meaningful as the potential U that is used. One does not have to apologize for any approximation in treating the N-body problem. Whatever mechanical, thermodynamic, and statistical mechanical properties that a system of N particles should have, they are all present in the simulation data. Of course, how one extracts these properties from the simulation output — the atomic trajectories — determines how useful the simulation is. We can regard MD simulation as an ‘atomic video’ of the particle motion (which can be displayed as a movie), and how to extract the information in a scien­tifically meaningful way is up to the viewer. It is to be expected that an experienced viewer can get much more useful information than an inexperienced one.

The above comments aside, we present here the general reasons why MD simulation is useful (or unique). These are meant to guide the thinking of the nonexperts and encourage them to discover and appreciate the many significant aspects of this simulation technique.

(a) Unified study of all physical properties. Using MD, one can obtain the thermodynamic, structural, mechanical, dynamic, and transport properties of a system of particles that can be studied in a solid, liquid, or gas. One can even study chemical properties and reactions that are more difficult and will require using quantum MD, or an empirical potential that explicitly models charge transfer.27

(b) Several hundred particles are sufficient to simulate bulk matter. Although this is not always true, it is rather surprising that one can get quite accurate ther­modynamic properties such as equation of state in this way. This is an example that the law of large numbers takes over quickly when one can average over several hundred degrees of freedom.

(c) Direct link between potential model and physical proper­ties. This is useful from the standpoint of funda­mental understanding of physical matter. It is also very relevant to the structure-property correla­tion paradigm in material science. This attribute has been noted in various general discussions of the usefulness of atomistic simulations in material research.28-30

(d) Complete control over input, initial and boundary conditions. This is what provides physical insight into the behavior of complex systems. This is also what makes simulation useful when com­bined with experiment and theory.

(e) Detailed atomic trajectories. This is what one obtains from MD, or other atomistic simulation tech­niques, that experiment often cannot provide. For example, it is possible to directly compute and observe diffusion mechanisms that otherwise may be only inferred indirectly from experi­ments. This point alone makes it compelling for the experimentalist to have access to simulation.

We should not leave this discussion without reminding ourselves that there are significant limitations to MD as well. The two most important ones are as follows:

(a) Need for sufficiently realistic interatomic potential functions U. This is a matter of what we really know fundamentally about the chemical binding of the system we want to study. Progress is being made in quantum and solid-state chemistry and condensed-matter physics; these advances will make MD more and more useful in understand­ing and predicting the properties and behavior of physical systems.

(a) Computational-capability constraints. No computers will ever be big enough and fast enough. On the other hand, things will keep on improving as far as we can tell. Current limits on how big and how long are a billion atoms and about a microsecond in brute force simulation. A billion-atom MD simulation is already at the micrometer length scale, in which direct experimental observations (such as transmission electron microscopy) are available. Hence, the major challenge in MD simulations is in the time scale, because most of the processes of interest and experimental obser­vations are at or longer than the time scale of a millisecond.