In the approach described earlier, which considers low concentrations of solutes and defects, the number of independent configurations is rather small, and they can be easily taken into account in kinetics model. The situation is much more complex when considering Fe-Cr with Cr concentration in the range 10-20%. Nevertheless, first results have been obtained by considering the interaction of defects with one or two Cr atoms in the Fe matrix.81 These data could ideally be used to fit an improved empiri­cal potential, but the Fe-Cr system is rather difficult to model because of the strong interplay between magnetic and chemical interactions. This is also clearly one of the challenges in the field.

1.08.4.2.2
Point defects in hcp-Zr

Point defects in hcp-Zr have also been studied via DFT calculations. It was found in particular that the vacancy migration energy is lower by ~0.15 eV within the basal plane than out of the basal plane.82 The situation for the self-interstitial is quite complex, since among the known configurations, at least three configurations are found to have almost the same formation energy (within 0.1 eV): the octahedral (O), split dumbbell (S), and basal octahedral (BO) configurations.83,84

Glue models atoms seek to have as many neighbors as possible; therefore, when a material is cleaved, the surface atoms tend to relax inward toward the bulk to increase cohesion. This effect also arises because of

the longer range of the repulsive part of the potential: at a surface, the further-away atoms are absent. This is in contrast to pair potentials and in agreement with real materials.

1.10.7.2 Cauchy Relations

The functional form of the glue model places fewer restrictions on the elastic constants of materials than pair potentials do; for example, the Cauchy pressure for a cubic metal is as follows20:

-2

Ef,(r 2И2

. j

If the ‘embedding function’ F (minus square root in FS case) has positive curvature, the Cauchy pressure must be positive, as it is for most metals. A minority of metals have negative Cauchy pressure. It is debatable whether this indicates negative curvature ofthe embed­ding function, or a breakdown of the glue model.

There are also some Cauchy-style constraints on the third-order elastic constants. But in general, ‘glue’ type models can fit the full anisotropic linear elastic­ity of a crystal structure.

1.10.7.3 Vacancy Formation

In a near-neighbor second-moment model for fcc, breaking one of twelve bonds reduces the cohesive energy of each atom adjacent to the vacancy by a factor of (1 — J 11/12) = 4.25%. Other glue models give a similar result. Meanwhile, the pairwise (repul­sive) energy is reduced by a full 1/12 = 8.3%. Thus, energy cost to form a vacancy is lower in glue-type models than in pairwise ones. For actual parameterizations, it tends to be less than half the cohesive energy.

1.10.7.4 Alloys

To make alloy potentials in the glue formalism, one needs to consider both repulsive and cohesive terms.

Thinking of the repulsive part as the NFE pair potential, it becomes clear that the long-range behav­ior depends on the Fermi energy. This is composition dependent — the number of valence electrons is criti­cal, so it cannot be directly related to the individual elements. The short-ranged part should reflect the core radii and can be taken from the elements. Despite this obvious flaw, in practice, the pairwise part is usually concentration-independent and is refitted for the ‘cross’ heterospecies interaction.

In the EAM, the function Fj depends on the atom i being embedded, while the charge density JT fj(rij) into which it is embedded depends on the species and position of neighboring atoms. By contrast for FS poten­tials, the function F is a given (square root), while f(rij) is the squared hopping integral, which depends on both atoms. There is no obvious way to relate this heteroatomic hopping integral to the homoatomic ones, but a practical approach is to take a geometric mean21: one might expect this form from considering overlap of exponential tails of wavefunctions.

1.12.2.1 Radiation-Induced Obstacles to Dislocation Glide

Primary damage of structural materials is initiated by the interaction of high-energy atomic particles with material atoms to cause the energetic recoil and displacement of primary knock-on atoms (PKAs). PKA energy can vary from a few tens to tens of thousands of electron volt and the PKA spectrum can be calculated for a particular position in a particular installation.5 A PKA with energy >~1 keV gives rise to a displacement cascade that produces a localized distribution of point defects (vacancies and self-interstitial atoms, SIAs) and their clusters (see Chapter 1.11, Primary Radiation Damage Forma­tion). Further evolution of these defects produces specific microstructures that depend on the irradia­tion type, ambient temperature, and the material and its initial structure (see Chapter 1.13, Radiation Damage Theory). This radiation-induced micro­structure consists typically of voids, gas-filled bubbles, DLs (that can evolve into a dislocation network), secondary-phase precipitates, and other extended defects specific to the material, for example, SFTs in face-centered cubic (fcc) metals. These features are generally obstacles to the dislocation motion. Their size is typically6 in the range of nanometers to tens of nanometers and their number density may reach ^1024m~3. At this density, the mean distance between obstacles can be as short as ~ 10 nm, and such a high density of small defects, particularly those with a dislocation character, makes the mechanisms of radi­ation effects on mechanical properties very different from those due to other treatments.

The diffusion coefficient of vacancies is an important parameter for microstructural evolution, for it deter­mines the rate of mutual recombination of PDs. Migrating vacancies can also meet solute or impurity atoms and form immobile complexes, which can then dissociate. In quasi-equilibrium, when the rates of complex formation and dissociation events are equal to each other:

zv+C’Y C’s = n-Cvs [103]

Here, Cvs and Cs are the concentrations of com­plexes and solute atoms, respectively, C and Cv are the concentrations of free (unpaired) solute atoms and vacancies, respectively, v+ and v- are the frequencies of complex formation and dissociation events, respectively, and z is a geometrical factor, which is of the order of the coordination number for complexes with a short-range (first-nearest neigh­bor) interaction and unity for long-range interac­tions. The binding energy of the complex, £{(,, is defined from v-/v+ = exp(b£jbs). The solute con­centration is generally much higher than that of vacancies, hence

C0 « Cs

cv = Cv — Cvs [104]

Substituting these into eqn [103], one obtains aCyCs exp(bEys)

1 + aCs exp(bEjbs)

The total vacancy concentration is, therefore,

Cv = Cv + Cvs = Cv [1 + aCsexp(bEbs)] [106]

The effective diffusion coefficient of vacancies may be defined as

Dv

1 + a Cs exp(b£[bs) exp [-b(Evm + Ebs)]

While the vacancy concentration is approximately equal to

Cv « CvaCsexp^) [108]

The vacancy flux is, thus, equal to that in the absence of impurities,

Df Cv = DvCv [109]

which is supported by the measurements of the self-diffusion energy, which is almost independent of the presence of impurities. The main conclusion is that the total vacancy flux does not depend on the presence of impurity atoms. However, impurity trapping may affect the recombination rate and hence Cv may be increased.

In this section, we develop the basic ingredients of computational DD, following closely the parametric method. The key idea here is that dislocations are divided into segments of smooth parametric curves in three dimensions. Each segment has two nodes, one on each end of the segment. Forces are computed on the segments, and effectively calculated at the nodes. An equation of motion (EOM) is then derived for each node, and the whole system is assembled in a matrix form, very similar to the finite element method (FEM). The matrix system is solved with time increments by either explicit or implicit integra­tion methods. We will now go into some of the computational details.

1.16.2.1 Isotropic Crystals

When the Green’s functions are known, the elastic field of a dislocation loop can be constructed by a surface integration. The starting point in this

calculation is the displacement field in a crystal con­taining a dislocation loop, which can be expressed	Consider now the virtual motion of the disloca­tion loop. The mechanical power during the virtual motion is composed of two parts: (1) change in the elastic energy stored in the medium upon loop motion under the influence of its own stress (i. e., the change in the loop self-energy); (2) the work done on moving the loop as a result of the action of external and internal stresses, excluding the stress contribution ofthe loop itself. These two components constitute the work of Peach-Koehler.27 The main idea of DD is to derive approximate equations of motion from the principle of virtual power dissipa­tion of the second law of thermodynamics,15 by finding virtual Peach-Koehler forces that would result in the simultaneous displacement of all dislo­cation loops in the crystal. A major simplification is that this many-body problem is reduced to the single loop problem. In this simplification, instead of moving all the loops simultaneously, they are moved sequentially, with the motion of each one against the collective field of all other loops. The approach is reminiscent of the single-electron simplification of the many-electron problem in quantum mechanics. If the material is assumed to be elastically isotro­pic and infinite, a great reduction in the level of required computations ensues. First, surface integrals can be replaced by line integrals along the disloca­tion. Second, Green’s functions and their derivatives have analytical solutions. Thus, the starting point in most DD simulations so far is a description of the elastic field of dislocation loops of arbitrary shapes by line integrals of the form proposed in de Wit26 as
dk [2]
where m and n are the shear modulus and Poisson’s ratio, respectively, b is the Burgers vector with Car­tesian components b,, and the vector potential Ak(R) = EijkX, Sj/[R(R T R • s)] satisfies the differen­tial equation CpikA^p(R) = XjRT3, where s is an arbi­trary unit vector. The radius vector R connects a source point on the loop to a field point, as shown in Figure 1, with Cartesian components R,, succes­sive partial derivatives R, jk. .., and magnitude R. The line integrals are carried out along a closed contour C defining the dislocation loop of differential arc length dl of components dlk.
41 nEkmn(R, ijm dijR, ppm)
[3]
segments. Reproduced from Ghoniem, N. M.; Huang, J.; Wang, Z. Philos. Mag. Lett. 2001, 82(2), 55-63.
where m and n are the shear modulus and Poisson’s ratio, respectively. The line integral is discretized, and the stress field of dislocation ensembles is obtained by a summation process over line segments, as shown in Figure 1. Once the parametric curve for the dislocation segment is mapped onto the scalar interval {o 2 [0,1]}, the stress field everywhere is obtained as a fast numerical quadrature sum.13 The Peach-Koehler force exerted on any other dislocation segment can be obtained from the total stress field (external and internal) at the segment as27 FPK = s • b x t [4]

The total self-energy of the dislocation loop is deter­mined by double line integrals. However, Gavazza and Barnett28 have shown that the first variation in the self-energy of the loop can be written as a single line integral, and that the majority ofthe contribution is governed by the local line curvature. Based on these methods for evaluation of the interaction and self-forces, the weak variational form of the govern­ing EOM of a single dislocation loop was developed by Ghoniem et al.14 as

(Ft — BakVa)drkds = 0 [5]

г

Here, Fk are the components of the resultant force, consisting of the Peach-Koehler force FPK (generated by the sum of the external and internal stress fields), the self-force Fs, and the osmotic force Fo (in case climb is also considered14). The resistivity matrix (inverse mobility) is Bak, Va are the velocity vector components, and the line integral is carried out along the arc length of the dislocation ds. To simplify the problem, let us define the following dimensionless parameters:

r.= £ f*= £ t-=m

a ma B

Here, a is lattice constant, m the shear modulus, and t is time. Hence eqn [5] can be rewritten in dimen­sionless matrix form as

r — 8r*T(f*- d?)dS- = 0 [6]

Here, f — = fi-,/2*,f-]T and r — = [r-, r*, r-]T, are all dependent on the dimensionless time t*. Following Ghoniem et a/.,14 a closed dislocation loop can be divided into Ns segments. In each segment j, we can choose a set of generalized coordinates qm at the two ends, thus allowing parameterization of the form

r- = CQ [7]

Here, C = [Ci(ffl), C2(m),, Cm(m)], Cj(m), (i = 1, 2,…,m) are shape functions dependent on the param­eter (0 < m < 1), and Q = [q1, q2,…, qm]T, with q2 a set of generalized coordinates. Substituting eqn [7] into eqn [6], we obtain

Let,

CTf-ds, ky

г

Following a similar procedure to the FEM, we assem­ble the EOM for all contiguous segments in global matrices and vectors, as

Ns N,

F = fj, К = k

j=i Z=i

Then, from eqn [8] we get

,dQ

The L-coefficients characterize the kinetic response of an alloy to a gradient of chemical potential. In practice, what is imposed is a composition gradient.

Chemical potential gradients, and therefore the fluxes, are assumed to be proportional to concentra­tion gradients, (eqn [9]) leading to the generalized Fick’s laws

J = — X D t1?]

A diffusion experiment consists of measurement of some of the terms of the diffusivity matrix D, y. These terms cannot be determined one by one because at least two concentration gradients are involved in a diffusion experiment. Note, that the L-coefficients can be traced back only ifthe whole diffusivity matrix and the thermodynamic factors are known. Further­more, most of the diffusion experiments are per­formed in thermal conditions and do not involve the interstitial diffusion mechanism.

In the following section, two examples of thermal diffusion experiments are introduced. Then, a few irradiation diffusion experiments are reviewed. The difficulty of measuring the whole diffusivity matrix is emphasized.

1.08.1 Introduction

Electronic structure calculations did not start with the so-called ab initio calculations or in recent years. The underlying basics date back to the 1930s with an understanding of the quantum nature of bonding in solids, the Hartree and Fock approximations, and the Bloch theorem. A lot was understood of the elec­tronic structure and bonding in nuclear materials using semiempirical electronic structure calculations, for example, tight binding calculations.1 The impor­tance ofthese somewhat historical calculations should not be overlooked. However, in the following sections, we focus on ‘ab initio’ calculations, that is, density functional theory (DFT) calculations. One must acknowledge that ‘ab initio calculations’ is a rather vague expression that may have different meanings depending on the community. In the present chapter we use it, as most people in the materials science community do, as a synonym for DFT calculations.

The popularity of these methods stems from the fact that, as we shall see, they provide quantitative results on many properties of solids without any adjustable parameters, though conceptual and technical difficul­ties subsist that should be kept in mind. The presen­tation is divided as follows. Methodologies and tools are briefly presented in the first section. The next two sections focus on some examples of ab initio results on metals and alloys on one hand and insulating materials on the other.

A widely adopted model used in many early MD simulations in statistical mechanics is the Lennard — Jones (6-12) potential, which is considered a reason­able description of van der Waals interactions between closed-shell atoms (noble gas elements, Ne, Ar, Kr, and Xe). This model has two parameters that are fixed by fitting to selected experimental data. One should recognize that there is no one single physi­cal property that can determine the entire potential function. Thus, using different data to fix the model parameters of the same potential form can lead to different simulations, making quantitative compar­isons ambiguous. To validate a model, it is best to calculate an observable property not used in the fitting and compare with experiment. This would provide a test of the transferability of the potential, a measure of robustness of the model. In fitting model parameters, one should use different kinds of properties, for example, an equilibrium or ther­modynamic property and a vibrational property to capture the low — and high-frequency responses (the hope is that this would allow a reasonable inter­polation over all frequencies). Since there is consid­erable ambiguity in what is the correct method of fitting potential models, one often has to rely on agreement with experiment as a measure of the goodness of potential. However, this could be mis­leading unless the relevant physics is built into the model.

For a qualitative understanding of MD essentials, it is sufficient to assume that the interatomic

potential U can be represented as the sum of two — body interactions

U (r1;… rN) V (rj) [7]

><J

where ry = |r, — ry I is the separation distance between particles i and j. V is the pairwise additive interaction, a central force potential that is a func­tion of only the scalar separation distance between the two particles, ry. A two-body interaction energy commonly used in atomistic simulations is the Lennard-Jones potential

V (r) = 4e[(s/r)12 — (s/r)6 ] [8]

where e and s are the potential parameters that set the scales for energy and separation distance, respectively. Figure 4 shows the interaction energy rising sharply when the particles are close to each other, showing a minimum at intermediate separation and decaying to zero at large distances. The interatomic force is also sketched in Figure 4. The particles repel each other when they are too close, whereas at large separa­tions they attract. The repulsion can be understood as arising from overlap of the electron clouds, whereas the attraction is due to the interaction between the induced dipole in each atom. The value of 12 for the first exponent in V(r) has no special significance, as the repulsive term could just as well be replaced by an exponential. The value of 6 for the second expo­nent comes from quantum mechanical calculations (the so-called London dispersion force) and therefore

is not arbitrary. Regardless of whether one uses eqn [8] or some other interaction potential, a short-range repulsion is necessary to give the system a certain size or volume (density), without which the particles will collapse onto each other. A long-range attraction is also necessary for cohesion of the system, without which the particles will not stay together as they must in all condensed states of matter. Both are necessary for describing the physical properties of the solids and liquids that we know from everyday experience.

Pair potentials are simple models that capture the repulsive and attractive interactions between atoms. Unfortunately, relatively few materials, among them the noble gases (He, Ne, Ar, etc.) and ionic crystals (e. g., NaCl), can be well described by pair potentials with reasonable accuracy. For most solid engineering materials, pair potentials do a poor job. For example, all pair potentials predict that the two elastic con­stants for cubic crystals, C12 and C44, must be equal to each other, which is certainly not true for most cubic crystals. Therefore, most potential models for engi­neering materials include many-body terms for an improved description of the interatomic interaction. For example, the Stillinger-Weber potential16 for silicon includes a three-body term to stabilize the tetrahedral bond angle in the diamond-cubic struc­ture. A widely used typical potential for metals is the embedded-atom method17 (EAM), in which the many-body effect is introduced in a so-called embed­ding function.

1.10.12.2.1 FeCr

Steel is of particular importance to radiation damage. Stainless steel is based on FeCr alloys, which have been observed by first principles calculation to exhibit unusual energy of solution. For small Cr concentrations, the energy of solution is negative; however, once the concentration exceeds about 10%, it changes sign. Thus, the FeCr system has a miscibility gap, but even at 0 K, there is a finite Cr concentration in the Fe-rich region. The underlying physics of this is that it is favorable for a Cr atom to dissolve in ferromagnetic Fe, provided the Cr spin is opposite to the Fe. Two adjacent Cr cannot be anti­parallel to each other and to the Fe matrix. Thus, nearby Cr atoms suffer magnetic frustration, which leads to repulsion between Cr atoms in FeCr not seen in pure Fe or pure Cr. Reproducing this effect in a potential is a challenging problem.

In early work, EAM was regarded as being inap­propriate for bcc metals (this turned out to be due to the use of rapidly decaying functions). The original FS functional form stabilized bcc elements, but they were unable to obtain a good fit for the elastic constants in Fe and Cr without introducing further parameters.

The two-band model can be applied to the FeCr system46 by assuming that the material can be treated as ferromagnetic, and using s and d as the two bands. They adopted the functional form of the interactions from the iron potential by Ackland and Mendelev, scaling the Cr electron density by the ratio of the atomic numbers 24/26. The CrCr potential was refitted to elastic and point defect properties. As the previous Fe parameterization incorporated with effect of s-electrons in a single embedding function, the so — called s-band density ofthis model in fact depends only on the FeCr cross potential. It described the excess energy of alloying by a many-body rather than pairwise additive effect. By choosing values which favor Fe atoms with a single Cr neighbor, this potential gives the skew solubility. This is an ingenious solution: magnetic frustration is essentially a 2 + N-body effect. Cr atoms repel when in an Fe rich ferromagnetic environment; this is neatly captured by the long — ranged Slater orbital used for the s-electron. It is
debatable whether this term is really capturing phys­ics associated with the s-band.

A related approach47 created a potential in which the embedding function depends directly on the local Cr concentration. The skew embedding function readily reproduced the phase diagram, which was the intention of the work. However, the short-range ordering and the Cr-Cr repulsion which appears to underlie the physics of radiation damage are less well reproduced.

1.10.12.2.2 FeC

Carbon dissolves readily in iron, producing a strengthening effect that underlies all steel. The physics of this is rather complex: the solution energy is very high (6 eV), and carbon adopts an interstitial position in bcc Fe with a barrier of 0.9 eV to migration. It is attracted to tensile regions of the crystal and to vacancies. It is repelled from compressive regions, including interstitial atoms, although the asymmetry of the interstitial means there are some tensile sites at larger distances which are favorable. First principles calculation also shows that the carbon forms covalently bonded pairs in a vacancy site, and the energy gained from the bond more than compensated for the reduced space available to the second carbon atom. These criteria prove rather demanding for parameterizing FeC potentials, even though they only cover composi­tions with vey low carbon concentrations.

An early pair potential by Johnson48 proved extremely successful, and it was only once first prin­ciples calculation revealed the repulsion between C and interstitials that a major problem was revealed. Although interstitial atoms are specific to radiation damage applications, there is a strong implication that the binding to other overcoordinated regions such as dislocation cores may be wrong.

It appears to be very difficult to obtain the correct bonding of carbon in all the cases above with smooth EAM-type functions. Even in recent potentials,49 like those by Johnson, carbon binds chemically to the interstitial.

There is a qualitative explanation for this. Elec­tronic structure calculation50 shows that the electrons pile up between the two nearest neighbors in the octa­hedral configuration, essentially forming two FeC bonds. However, all the simple potentials described above obtain similar bonding from all six neighbors, stabilizing the octahedral site because the tetrahedral site has only four neighbors. This approach favors carbon bonding to highly coordinated defects, and underlies the bonding to interstitials. An EAM potential with a Tersoff-Brenner style saturation in the C cohesion has addressed this problem.51 This is tuned to saturate at two near neighbors, and so favors the octahedral site but not overcoordination. As a con­sequence, it does not bind carbon to the interstitial.

1.10.12.1 Austenitic Steel

Few potentials exist for fcc iron. Calculating high — temperature phase transitions is a subtle process invol­ving careful calculation of free energy differences, which makes it difficult to incorporate in the fitting process. Although the bcc-fcc transition has been reported for one EAM iron potential,52 it is probably fortuitous and has been disputed.53 In any case, it is at far higher temperature than experimentally observed. Worse, it is likely that magnetic entropy plays a significant role,54 and the magnetic degrees of free­dom are seldom included in potentials.

Some very recent progress has been made; an analytic bond order potential53 shows bcc-fcc-bcc transitions for iron and an MEAM parameterization by Baskes successfully reproduces the bcc-fcc-bcc phase transitions in iron on heating by using temper­ature-dependent parameters. It seems certain that the challenge of austenitic steel will be receiving more attention in the next few years.

Atomic-scale simulation by computer has become a powerful tool for investigation of material properties and processes that cannot be achieved by experimen­tal techniques or other theoretical methods. This is due to the increasing power of computers, the devel­opment of new efficient modeling codes, and the extensive usage of ab initio calculations for probing atomic mechanisms and generating data for design of new IAPs. Moreover, increasing length and time scales attainable by atomistic modeling provides overlap with experimental scales in some cases, thereby allowing direct verification of modeling results.26 In this chapter, we have described only a selection of the results obtained within the last decade by atomic-scale modeling of DD in irradiated bcc and fcc metals. The examples presented and references cited demonstrate how detailed insight into mechanisms can be gained by such modeling. For some obstacles to dislocation motion, for exam­ple, many inclusion-like obstacles, strengthening is controlled by the dislocation line shape at breakaway and can be parameterized using the existing elastic­ity theory models. In other cases, for example, dislocation-like obstacles, reactions and their results, including obstacle strength, depend very much on the material, and dislocation character and core struc­ture; dislocation behavior is also sensitive to condi­tions such as interaction geometry, temperature, and strain rate. The variety of outcomes for dislocation­like obstacles is complicated and wide, and although features of these reactions can be understood within the framework of the continuum theory of disloca­tions, for example, Frank’s rule, a general formaliza­tion of the reactions in terms of obstacle strength and reaction product does not exist. Nevertheless, it has been seen that the insight gained by simulation has allowed the outcome of reactions to be classified in a meaningful way (Table 1). This will allow for validation of higher-level DD modeling of these reac­tions using the continuum approximation. An excel­lent example of the way in which this can be done has been provided by Martinez and coworkers,96,97 who used MD and DD to simulate the same dislocation — SFT interactions. The continuum modeling of these nanometer-scale obstacles was verified by the atomic simulation, and this enabled a large number of inter­action geometries and conditions to be investigated successfully by DD. Unfortunately, successful over­laps in scale of atomistic modeling and experiment or/and continuum modeling are still rare. Efforts in all techniques are necessary to progress understand­ing of mechanisms and their parameterization for predictive modeling tools that can be applied to irradiated materials.

Investigations such as those just discussed bring assurance that atomic-scale modeling is correct at least qualitatively and is invaluable in cases where scale overlap of techniques is not yet achieved. Two main problems exist with regard to the quality of its quantitative outcomes. One is concerned with time scale. As already mentioned, the limit on time scale is the main disadvantage of current atomic-scale mod­eling. The maximum simulation time achieved so far is of the order of a few hundred nanoseconds. For dislocation studies, this allows dislocations with velocity as low as 0.5 ms-1 to be modeled — and it may be that some processes are insensitive to veloc­ity at this level27 — but the overall strain rate (105 s-1) is fast compared with experiment, and the interaction time (~100 ns) with an obstacle is too short for thermally-activated processes to be sampled. Devel­opment of new methods for keeping atomic-level accuracy over at least microsecond to second time scales is necessary to progress to the next step toward predictive modeling for engineering applications. An example of a new generation technique for simulating realistic strain rates whilst retaining atomic-scale detail was published recently.98 The new technique combines atomic-scale modeling for estimating vacancy migration barriers in the vicinity of an edge dislocation and kinetic Monte Carlo (MC) for simulating vacancy kinetics in a crystal with a specified dislocation density. The technique was successfully applied to simulate the process of power-law creep over a macroscopic time scale with microscopic fidelity. The other problem is concerned with accuracy in describing interatomic interactions. Much of the research described in this chapter, based as it is on empirical EAM IAPs, is more related to the behavior of model metals with bcc or fcc crystal structure in general than to the elements Fe or Cu in particular. This difficulty will become more acute with the demand for more sophisticated, radiation — resistant alloys, and future investigations of chemical effects on plasticity will require IAPs that incorpo­rate chemistry in a meaningful way. More on this is presented in Chapter 1.10, Interatomic Potential Development.

We would like to conclude on an optimistic note. It is clear that significant progress has been achieved over the last decade in understanding the details of the atomic-scale mechanisms involved in dislocations dynamics in structural metals in a reactor environ­ment. The small, nanoscale nature of the obstacles created by radiation damage is such that the tech­niques described here provide uniquely valuable

information, despite the limitations they currently experience.