Comprehensive nuclear materials 1 01 fundamental properties of defects in metals

Comprehensive nuclear materials 1 01 fundamental properties of defects in metals Comprehensive nuclear materials 1 01 fundamental properties of defects in metals Comprehensive nuclear materials 1 01 fundamental properties of defects in metals Comprehensive nuclear materials 1 01 fundamental properties of defects in metals Comprehensive nuclear materials 1 01 fundamental properties of defects in metals Comprehensive nuclear materials 1 01 fundamental properties of defects in metals Comprehensive nuclear materials 1 01 fundamental properties of defects in metals

1.01 Fundamental Properties of Defects in Metals W G Wolfer Ktech Corporation, Albuquerque, NM, USA; Sandia National Laboratories, Livermore, CA, USA Published by Elsevier Ltd 1.01.1 1.01.2 1.01.3 1.01.3.1 1.01.3.2 1.01.3.3 1.01.4 1.01.4.1 1.01.4.2 1.01.4.3 1.01.4.4 1.01.5 1.01.5.1 1.01.5.2 1.01.5.3 1.01.6 1.01.6.1 1.01.6.2 1.01.6.3 1.01.7 1.01.7.1 1.01.7.2 1.01.7.3 1.01.7.4 1.01.7.4.1 1.01.7.4.2 1.01.7.4.3 1.01.7.4.4 1.01.8 1.01.8.1 1.01.8.2 1.01.8.2.1 1.01.8.2.2 1.01.8.3 1.01.9 Appendix A A1 A2 Appendix B B1 B2 B3 B4 B5 References Introduction The Displacement Energy Properties of Vacancies Vacancy Formation Vacancy Migration Activation Volume for Self-Diffusion Properties of Self-Interstitials Atomic Structure of Self-Interstitials Formation Energy of Self-Interstitials Relaxation Volume of Self-Interstitials Self-Interstitial Migration Interaction of Point Defects with Other Strain Fields The Misfit or Size Interaction The Diaelastic or Modulus Interaction The Image Interaction Anisotropic Diffusion in Strained Crystals of Cubic Symmetry Transition from Atomic to Continuum Diffusion Stress-Induced Anisotropic Diffusion in fcc Metals Diffusion in Nonuniform Stress Fields Local Thermodynamic Equilibrium at Sinks Introduction Edge Dislocations Dislocation Loops Voids and Bubbles Capillary approximation The mechanical concept of surface stress Surface stresses and bulk stresses for spherical cavities Chemical potential of vacancies at cavities Sink Strengths and Biases Effective Medium Approach Dislocation Sink Strength and Bias The solution of Ham Dislocation bias with size and modulus interactions Bias of Voids and Bubbles Conclusions and Outlook Elasticity Models: Defects at the Center of a Spherical Body An Effective Medium Approximation The Isotropic, Elastic Sphere with a Defect at Its Center Representation of Defects by Atomic Forces and by Multipole Tensors Kanzaki Forces Volume Change from Kanzaki Forces Connection of Kanzaki Forces with Transformation Strains Multipole Tensors for a Spherical Inclusion Multipole Tensors for a Plate-Like Inclusion 5 10 12 12 13 15 15 16 16 17 20 21 22 23 25 26 26 26 27 29 29 29 30 31 32 32 33 33 35 35 37 38 38 38 41 41 42 43 43 43 44 Fundamental Properties of Defects in Metals Abbreviations bcc CA CD dpa EAM fcc hcp INC IHG SIA Vac Body-centered cubic Cavity model Center of dilatation model Displacements per atom Embedded atom method Face-centered cubic Hexagonal closed packed Inclusion model Inhomogeneity model Self-interstitial atom Vacancy 1.01.1 Introduction Several fundamental attributes and properties of crystal defects in metals play a crucial role in radiation effects and lead to continuous macroscopic changes of metals with radiation exposure These attributes and properties will be the focus of this chapter However, there are other fundamental properties of defects that are useful for diagnostic purposes to quantify their concentrations, characteristics, and interactions with each other For example, crystal defects contribute to the electrical resistivity of metals, but electrical resistivity and its changes are of little interest in the design and operation of conventional nuclear reactors What determines the selection of relevant properties can best be explained by following the fate of the two most important crystal defects created during the primary event of radiation damage, namely vacancies and self-interstitials The primary event begins with an energetic particle, a neutron, a high-energy photon, or an energetic ion, colliding with a nucleus of a metal atom When sufficient kinetic energy is transferred to this nucleus or metal atom, it is displaced from its crystal lattice site, leaving behind a vacant site or a vacancy The recoiling metal atom may have acquired sufficient energy to displace other metal atoms, and they in turn can repeat such events, leading to a collision cascade Every displaced metal atom leaves behind a vacancy, and every displaced atom will eventually dissipate its kinetic energy and come to rest within the crystal lattice as a self-interstitial defect It is immediately obvious that the number of self-interstitials is exactly equal to the number of vacancies produced, and they form Frenkel pairs The number of Frenkel pairs created is also referred to as the number of displacements, and their accumulated density is expressed as the number of displacements per atom (dpa) When this number becomes one, then on average, each atom has been displaced once At the elevated temperatures that exist in nuclear reactors, vacancies and self-interstitials diffuse through the crystal As a result, they will encounter each other, either annihilating each other or forming vacancy and interstitial clusters These events occur already in their nascent collision cascade, but if defects escape their collision cascade, they may encounter the defects created in other cascades In addition, migrating vacancy defects and interstitial defects may also be captured at other extended defects, such as dislocations, cavities, grain boundaries and interface boundaries of precipitates and nonmetallic inclusions, such as oxide and carbide particles The capture events at these defect sinks may be permanent, and the migrating defects are incorporated into the extended defects, or they may also be released again However, regardless of the complex fate of each individual defect, one would expect that eventually the numbers of interstitials and vacancies that arrive at each sink would become equal, as they are produced in equal numbers as Frenkel pairs Therefore, apart from statistical fluctuations of the sizes and positions of the extended defects, or the sinks, the microstructure of sinks should approach a steady state, and continuous irradiation should change the properties of metals no further It came as a big surprise when radiation-induced void swelling was discovered with no indication of a saturation In the meantime, it has become clear that the microstructure evolution of extended defects and the associated changes in macroscopic properties of metals in general is a continuing process with displacement damage The fundamental reason is that the migration of defects, in particular that of self-interstitials and their clusters, is not entirely a random walk but is in subtle ways guided by the internal stress fields of extended defects, leading to a partial segregation of self-interstitials and vacancies to different types of sinks Guided then by this fate of radiation-produced atomic defects in metals, the following topics are presented in this chapter: The displacement energy required to create a Frenkel pair The energy stored within a Frenkel pair that consists of the formation enthalpies of the selfinterstitial and the vacancy The dimensional changes that a solid suffers when self-interstitial and vacancy defects are created, Fundamental Properties of Defects in Metals and how these changes manifest themselves either externally or internally as changes in lattice parameter These changes then define the formation and relaxation volumes of these defects and their dipole tensors The regions occupied by the atomic defects within the crystal lattice possess a distorted, if not totally different, arrangement of atoms As a result, these regions are endowed with different elastic properties, thereby changing the overall elastic constants of the defect-containing solid This leads to the concept of elastic polarizability parameters for the atomic defects Both the dipole tensors and elastic polarizabilities determine the strengths of interactions with both internal and external stress fields as well as their mutual interactions When the stress fields vary, the gradients of the interactions impose drift forces on the diffusion migration of the atomic defects that influences their reaction rates with each other and with the sinks At these sinks, vacancies can also be generated by thermal fluctuations and be released via diffusion to the crystal lattice Each sink therefore possesses a vacancy chemical potential, and this potential determines both the nucleation of vacancy defect clusters and their subsequent growth to become another defect sink and part of the changing microstructure of extended defects The last two topics, and 7, as well as topic 1, will be further elaborated in other chapters where mc2 is the rest energy of an electron and L ¼ mM/(m þ M)2 The approximation on the right is adequate because the electron mass, m, is much smaller than the mass, M, of the recoiling atom Changing the direction of the electron beam in relation to the orientation of single crystal film specimens, one finds that the threshold energy varies significantly However, for polycrystalline samples, values averaged over all orientations are obtained, and these values are shown in Figure for different metals as a function of their melting temperatures.1 First, we notice a trend that Td increases with the melting temperature, reflecting the fact that larger energies of cohesion or of bond strengths between atoms also lead to higher melting temperatures We also display values of the formation energy of a Frenkel pair Each value is the sum of the corresponding formation energies of a self-interstitial and a vacancy for a given metal These energies are presented and further discussed below The important point to be made here is that the displacement energy required to create a Frenkel pair is invariably larger than its formation energy Clearly, an energy barrier exists for the recoil process, indicating that atoms adjacent to the one that is being displaced also receive some additional kinetic energy that is, however, below the displacement energy Td and is subsequently dissipated as heat The displacement energies listed in Table and shown in Figure are averaged not only over crystal orientation but also over temperature for those metals 50 Displacement energy Td(eV) fcc Frenkel pair (eV) bcc Frenkel pair (eV) Scattering of energetic particles from external sources, be they neutrons, electrons, ions, or photons, or emission of such particles from an atomic nucleus, imparts a recoil energy When this recoil energy exceeds a critical value, called the threshold displacement energy, Td, Frenkel pairs can be formed To measure this displacement energy, an electron beam is employed to produce the radiation damage in a thin film of the material, and its rise in electrical resistivity due to the Frenkel pairs is monitored By reducing the energy of the electron beam, the resistivity rise is also reduced, and a threshold electron energy, Emin, can be found below which no Frenkel pairs are produced The corresponding recoil energy is given by relativistic kinematics as   2mc ỵ Emin m Emin ẵ1 E % ỵ Td ẳ LEmin 2mc ỵ LEmin 2mc M Displacement and Frenkel pair energies (eV) 1.01.2 The Displacement Energy 40 30 20 10 0 500 1000 1500 2000 2500 3000 3500 Melting temperature (K) 4000 Figure Energies of displacement and energies of Frenkel pairs for elemental metals as a function of their melting temperatures Fundamental Properties of Defects in Metals Table Element Displacement and Frenkel pair energies of elemental metals Symbol Z Melt temp ( K) M Td (eV) Frenkel pair energy (eV) fcc Silver Aluminum Gold Cadmium Cobalt Chromium Copper Iron Indium Iridium Magnesium Molybdinum Niobium Neodymium Nickel Lead Palladium Platinum Rhenium Tantalum Titanium Vanadium Tungsten Zinc Zirconium Ag Al Au Cd Co Cr Cu Fe In Ir Mg Mo Nb Nd Ni Pb Pd Pt Re Ta Ti V W Zn Zr 47 13 79 48 27 24 29 26 49 77 12 42 41 60 28 82 46 78 75 73 22 23 74 30 40 107.9 26.98 197.0 112.4 58.94 52.01 63.54 55.85 114.8 192.2 24.32 95.95 92.91 144.3 58.71 207.2 106.4 195.1 186.2 181.0 47.90 50.95 183.9 65.38 91.22 fcc fcc fcc hcp hcp bcc fcc bcc tetragonal fcc hcp bcc bcc hcp fcc fcc fcc fcc hcp bcc hcp bcc bcc hcp hcp 1235 933.5 1337 594.2 1768 2180 1358 1811 429.8 2719 923.2 2896 2750 1289 1728 600.6 1828 2041 3458 3290 1941 2183 3695 692.7 2128 26.0 15.3 34.0 19.0 23.0 28.0 18.3 17.4 10.5 46.0 10.0 32.4 28.2 9.30 22.0 11.8 34.0 34.0 44.0 26.7 20.8 28.0 44.0 12.0 22.5 bcc 6.52 4.96 5.98 4.40 6.66 4.03 6.69 4.01 10.5 3.24 6.51 12.7 5.77 3.26 8.90 Source: Displacement energies from Jung, P In Landolt-Boărnstein; Springer-Verlag: Berlin, 1991; Vol III/25, pp 8–11 50 Td fcc Td bcc Td hcp and others Td, eV 40 Displacement energy (eV) where the displacement energy has been measured as a function of irradiation temperature For some materials, such as Cu, a significant decrease of the displacement energy with temperature has been found However, a definitive explanation is still lacking Close to the minimum electron energy for Frenkel pair production, the separation distance between the self-interstitial and its vacancy is small Therefore, their mutual interaction will lead to their recombination With increasing irradiation temperature, however, the self-interstitial may escape, and this would manifest itself as an apparent reduction in the displacement energy with increasing temperature On the other hand, Jung2 has argued that the energy barrier involved in the creation of Frenkel pairs is directly dependent on the temperature in the following way This energy barrier increases with the stiffness of the repulsive part of the interatomic potential; a measure for this stiffness is the bulk modulus Indeed, as Figure demonstrates, the displacement energy increases with the bulk modulus Since the bulk modulus decreases with temperature, so will the displacement energy The correlation of the displacement energy with the bulk modulus appears to be a somewhat better 30 20 10 0 50 100 150 200 250 300 350 400 Bulk modulus (GPa) Figure Displacement energies for elemental metals as a function of their bulk modulus empirical relationship than the correlation with the melt temperature However, one should not read too much into this, as the bulk modulus B, atomic Fundamental Properties of Defects in Metals volume O, and melt temperature of elemental metals approximately satisfy the rule BO % 100kB Tm discovered by Leibfried3 and shown in Figure 1.01.3 Properties of Vacancies 1.01.3.1 Vacancy Formation The thermal vibration of atoms next to free surfaces, to grain boundaries, to the cores of dislocations, etc., make it possible for vacancies to be created and then diffuse into the crystal interior and establish an equilibrium thermal vacancy concentration of  f  EV TSVf eq ẵ2 CV T ị ¼ exp À kB T given in atomic fractions Here, EVf is the vacancy formation enthalpy, and SVf is the vacancy formation entropy The thermal vacancy concentration can be measured by several techniques as discussed in Damask and Dienes,4 Seeger and Mehrer,5 and Siegel,6 and values for EVf have been reviewed and tabulated by Ehrhart and Schultz;7 they are listed in Table When these values for the metallic elements are plotted versus the melt temperature in Figure 4, an approximate correlation is obtained, namely EVf % Tm =1067 ½3Š 4000 Bulk mod.*atom vol./(100 k) 3500 3000 2500 1500 1000 500 Using the Leibfried rule, a new approximate correlation emerges for the vacancy formation enthalpy that has become known as the cBO model8; the constant c is assumed to be independent of temperature and pressure As seen from Figure 5, however, the experimental values for EVf correlate no better with BO than with the melting temperature It is tempting to assume that a vacancy is just a void and its energy is simply equal to the surface area 4pR2 times the specific surface energy g0 Taking the atomic volume as the vacancy volume, that is, O ¼ 4pR3/3, we show in Figure the measured vacancy formation enthalpies as a function of 4pR2g0, using for g0 the values9 at half the melting temperatures It is seen that EVf is significantly less, by about a factor of two, compared to the surface energy of the vacancy void so obtained Evidently, this simple approach does not take into account the fact that the atoms surrounding the vacancy void relax into new positions so as to reduce the vacancy volume VVf to something less than O The difference VVrel ¼ VVf À O 500 1000 1500 2000 2500 3000 3500 4000 Melt temperature (K) Figure Leibfried’s empirical rule between melting temperature and the product of bulk modulus and atomic volume ½4Š is referred to as the vacancy relaxation volume The experimental value7 for the vacancy relaxation of Cu is À0.25O, which reduces the surface area of the vacancy void by a factor of only 0.825, but not by a factor of two The difference between the observed vacancy formation enthalpy and the value from the simplistic surface model has recently been resolved It will be shown in Section 1.01.7 that the specific surface energy is in fact a function of the elastic strain tangential to the surface, and when this surface strain relaxes, the surface energy is thereby reduced At the same time, however, the surface relaxation creates a stress field in the surrounding crystal, and hence a strain energy As a result, the energy of a void after relaxation is given by FC ẵeRị; e ẳ 4pR2 gẵeRị; e ỵ 8pR3 me2 Rị 2000 ẵ5 The first term is the surface free energy of a void with radius R, and it depends now on a specific surface energy that itself is a function of the surface strain e(R) and the intrinsic residual surface strain e* for a surface that is not relaxed The second term is the strain energy of the surrounding crystal that depends on its shear modulus m The strain dependence of the specific surface energy is given by gẵe; e ẳ g0 ỵ 2mS ỵ lS ị2e ỵ eịe ẵ6 Here, g0 is the specific surface energy on a surface with no strains in the underlying bulk material Fundamental Properties of Defects in Metals Table Crystal and vacancy properties Metal Crystal Melt temp (K) KO (eV) Debye temp (K) EfV (eV) Em v (eV) g0 (J mÀ2) Ag Al Au Be Co Cr Cs Cu Fe Hf Ir K Li Mg Mn Mo Na Nb Nd Ni Os Pb Pd Pt Re Rh Ru Sb Sr Ta Ti Tl U V W Zn Zr fcc fcc fcc hcp hcp bcc bcc fcc bcc hcp fcc bcc bcc hcp bcc bcc bcc bcc hcp fcc hcp fcc fcc fcc hcp fcc hcp rbh* fcc bcc hcp hcp bco** bcc bcc hcp hcp 1235 933.5 1337 1560 1768 2180 301.6 1358 1811 2506 2719 336.7 453.7 923 1519 2896 371 2750 1289 1728 3306 600.6 1828 2041 3458 2237 2607 904 1050 3290 1941 577.2 1408 2183 3695 693 2128 10.9 7.89 18.1 6.57 13.1 12.1 41.2 10.1 12.3 15.3 31.3 1.55 1.63 5.13 9.15 25.4 1.70 19.3 6.8 12.5 36.7 8.46 17.7 26.7 34 32.6 18.6 7.9 229.2 430.6 162.7 1.11 0.67 0.93 0.8 588.4 2.1 1.09 1.02 1.33 1.30 2.22 2.01 349.6 483.3 1.28 1.90 0.66 0.61 0.71 0.87 0.72 0.95 0.084 0.70 0.55 25.3 11.8 7.7 13 13.5 30.8 6.49 13.8 92.7 369.5 0.48 0.80 473.4 157.1 254.6 3.10 0.34 2.70 481.4 1.79 0.038 0.50 1.30 1.35 0.03 0.55 0.81 1.04 106.6 277.9 0.58 1.85 1.35 3.10 2.50 0.43 1.03 1.43 2.20 1.50 264.7 3.10 0.70 399.4 384.3 2.10 3.60 0.54 0.50 1.70 0.42 0.58 1.57 2.12 1.92 2.65 0.129 0.472 0.688 2.51 0.234 2.31 2.38 2.95 0.54 1.74 2.20 3.13 2.33 2.65 0.461 0.358 2.49 1.75 0.55 1.78 2.30 2.77 0.896 1.69 * rbh: rhombohedral bco: body-centered orthorhombic ** However, such a surface possesses an intrinsic, residual surface strain e*, because the interatomic bonding between surface atoms differs from that in the bulk, and for metals, the surface bond length would be shorter if the underlying bulk material would allow the surface to relax Partial relaxation is possible for small voids as well as for nanosized objects In addition to the different bond length at the surface, the elastic constants, mS and lS, are also different from the corresponding bulk elastic constants However, they can be related by a surface layer thickness, h, to bulk elastic constants such that mS ỵ lS ẳ m ỵ lịh ẳ mh=1 2nị ẵ7 where l is the Lames constant and n is Poisson’s ratio for the bulk solid Computer simulations on freestanding thin films have shown10 that the surface layer is effectively a monolayer, and h can be approximated by the Burgers vector b For planar crystal surfaces, the residual surface strain parameter e* is found to be between and 5%, depending on the surface orientation relative to the crystal lattice On surfaces with high curvature, however, e* is expected to be larger The relaxation of the void surface can now be obtained as follows We seek the minimum of the void energy as defined by eqn [5] by solving @FC =@e ẳ The result is eRị ẳ mS ỵ lS ịe h e ẳ mR þ ðmS þ lS Þ ð1 À 2nÞR þ h ½8Š and this relaxation strain changes the initially unrelaxed void volume Fundamental Properties of Defects in Metals 4 bcc Hf/v, eV fcc Hf/v, eV hcp Hf/v, eV 2.5 1.5 2.5 1.5 0.5 0.5 0 Surface energy of a vacancy (eV) Figure Correlation between the surface energy of a vacancy void and the vacancy formation energy 3.5 Vacancy formation energy and its bulk contribution (eV ) fcc Hf/v, eV bcc Hf/v, eV hcp Hf/v, eV 3.5 2.5 1.5 0.5 0 500 1000 1500 2000 2500 3000 3500 4000 Melting temperature (K) Figure Vacancy formation energies as a function of melting temperature Vacancy formation enthalpy (eV) fcc Hf/v, eV bcc Hf/v, eV hcp Hf/v, eV 3.5 Vacancy formation enthalpy (eV) Vacancy formation energy (eV) 3.5 10 15 20 25 30 35 40 Exp value Computed values Bulk strain energy 2.5 Ni 1.5 0.5 –0.35 –0.3 Bulk modulus * atomic volume (eV) 4p ½9Š R consisting of n aggregated vacancies, by the amount nO ẳ ẵ10 Applying these equations to a vacancy, for which n ¼ and R % b, we obtain VVrel O ẳ 3e 21 nị and for the vacancy formation energy –0.2 –0.15 –0.1 –0.05 Vacancy relaxation volume Figure Vacancy formation energy versus the product of bulk modulus and atomic volume V rel ðRÞ ẳ 3nOeRị 0.25 ẵ11 Figure Vacancy formation energy and its dependence on the relaxation volume ( EVf ¼ 4pR2  2 ) 2ð3 À 4nÞ VVrel g0 À mb 91 2nị O  rel 2 V ỵ mO V O ½12Š This equation is evaluated for Ni and the results are shown in Figure as a function of the vacancy relaxa tion volume VVrel =O It is seen that relaxation volumes of À0.2 to À0.3 predict a vacancy formation energy comparable to the experimental value of 1.8 eV Fundamental Properties of Defects in Metals Few experimentally determined values are available for the vacancy relaxation volume, and their accuracy is often in doubt In contrast, vacancy formation energies are better known Therefore, we use eqn [12] to determine the vacancy relaxation volumes from experimentally determined vacancy formation energies The values so obtained are listed in Table 3, and for the few cases7 where this is possible, they are compared with the values reported from experiments Computed values for the vacancy relaxation volumes are between À0.2O and À0.3O for both fcc and bcc metals The low experimental values for Al, Fe, and Mo then appear suspect The surface energy model employed here to derive eqn [12] is based on several approximations: isotropic, linear elasticity, a surface energy parameter, g0, that represents an average over different crystal orientations, and extrapolation of the energy of large voids to the energy of a vacancy Nevertheless, this approximate model provides satisfactory results and captures an important connection between the vacancy relaxation volume and the vacancy formation energy that has also been noted in atomistic calculations Finally, a few remarks about the vacancy formation entropy, SVf , are in order It originates from the change in the vibrational frequencies of atoms surrounding the vacancy Theoretical estimates based on empirical potentials provide values that range from 0.4k to about 3.0k, where k is the Boltzmann constant As a result, the effect of the vacancy formation entropy on the magnitude of the thermal equilibrium vacancy coneq centration, CV , is of the same magnitude as the statistical uncertainty in the vacancy formation enthalpy Table 1.01.3.2 Vacancy Migration The atomistic process of vacancy migration consists of one atom next to the vacant site jumping into this site and leaving behind another vacant site The jump is thermally activated, and transition state theory predicts a diffusion coefficient for vacancy migration in cubic crystals of the form DV ẳ nLV d02 expSVm ịexpEVm =kB T ị ẳ DV0 expEVm =kB T ị ẵ13 Here, nLV is an average frequency for lattice vibrations, d0 is the nearest neighbor distance between atoms, SVm is the vacancy migration entropy, and EVm is the energy for vacancy migration It is in fact the energy of an activation barrier that the jumping atom must overcome, and when it temporarily occupies a position at the height of this barrier, the atomic configuration is referred to as the saddle point of the vacancy It will be considered in greater detail momentarily Values obtained for EVm from experimental measurements are shown in Figure as a function of the melting point While we notice again a trend similar to that for the vacancy formation energy, we find that EVm for fcc and bcc metals apparently follow different correlations However, the correlation for bcc metals is rather poor, and it indicates that EVm may be related to fundamental properties of the metals other than the melting point The saddle point configuration of the vacancy involves not just the displacement of the jumping atom but also the coordinated motion of other atoms that are nearest neighbors of the vacancy and of the jumping atom These nearest neighbor atoms Vacancy relaxation volumes for metals Metal g (J mÀ2) m (GPa) n HfV (eV) V rel V =V (model) Ag Al Au Cu Ni Pb Pd Pt Cr a-Fe Mo Nb Ta V W 1.19 1.1 1.45 1.71 2.28 0.57 1.91 2.40 2.23 2.31 2.77 2.54 2.76 2.51 3.09 33.38 26.18 31.18 54.7 94.6 10.38 53.02 65.1 117.0 90.4 125.8 39.6 89.9 47.9 160.2 0.354 0.347 0.412 0.324 0.276 0.387 0.374 0.393 0.209 0.278 0.293 0.397 0.324 0.361 0.280 1.11 Ỉ 0.05 0.67 Ỉ 0.03 0.93 Ỉ 0.04 1.28 Ỉ 0.05 1.79 Ỉ 0.05 0.58 Æ 0.04 1.7, 1.85 1.35 Æ 0.05 2.0 Æ 0.3 1.4, 1.89 3.2 Ỉ 0.09 2.6, 3.07 2.2, 3.1 2.2 Æ 0.4 3.1, 4.1 À0.247 Æ 0.005 À0.311 Æ 0.003 À0.262 Ỉ 0.003 À0.259 Ỉ 0.005 À0.236 Ỉ 0.004 À0.282 Æ 0.005 À0.239, À0.225 À0.260 Æ 0.003 À0.218 Æ 0.02 À0.278, À0.245 À0.191 Ỉ 0.004 À0.284, À0.258 À0.264, À0.228 À0.298 Æ 0.028 À0.201, À0.161 V rel V =V (experiment) À0.05, À0.38 À0.15 to À0.5 À0.25 À0.2 À0.24, À0.42 À0.05 À0.1 Fundamental Properties of Defects in Metals lie at the corners of a rectangular plane as shown in Figure As the jumping atom crosses this plane, they are displaced such as to open the channel This coordinated motion can be viewed as a particular strain fluctuation and described in terms of phonon excitations In this manner, Flynn11 has derived the following formula for the energy of vacancy migration in cubic crystals EVm ẳ 15C11 C44 C11 C12 ịa3 w 2ẵC11 C11 C12 ị ỵ C44 5C11 3C12 ފ ½14Š Here, a is the lattice parameter, C11, C12, and C44 are elastic moduli, and w is an empirical parameter that characterizes the shape of the activation barrier and can be determined by comparing experimental vacancy migration energies with values predicted by eqn [14] Ehrhart et al.7,12 recommend that w ¼ 0.022 for fcc metals and w ¼ 0.020 for bcc metals Vacancy migration energy (eV) 2.5 In the derivation of Flynn,11 only the four nearest neighbor atoms are supposed to move, while all other atoms are assumed to remain in their normal lattice positions On the other hand, Kornblit et al.13 treat the expansion of the diffusion channel as a quasistatic elastic deformation of the entire surrounding material The extent of the expansion is such that the opened channel is equal to the cross-section of the jumping atom, and a linear anisotropic elasticity calculation is carried out by a variational method to determine the energy involved in the channel expansion A vacancy migration energy is obtained for fcc metals of EVm ¼ 0:01727a3 C11 p02 p0 p1 À p22 ỵ 29 p0 p2 ỵ 19 p22 ẵ15 and the parameters pi will be defined momentarily For bcc metals,14 the activation barrier consist of two peaks of equal height EVmax with a shallow valley in between with an elevation of EVmin , where fcc Hm/v, eV hcp Hm/v, eV bcc Hm/v, eV q0 q1 À q22 q02 112 q0 p2 ỵ 883 q22 ẵ16 s0 s1 s22 s0 0:29232s0 s2 ỵ 0:0413s22 ẵ17 EVmax ¼ 0:003905a C11 and 1.5 EVmin ¼ 0:002403a C11 The parameters pi, qi, and si are linear functions of the elastic moduli with coefficients listed in Table For example, 0.5 q1 ¼ 3:45C11 À 0:75C12 þ 4:35C44 0 500 1000 1500 2000 2500 3000 3500 4000 Melt temperature (K) Figure Vacancy migration energy as a function of melting temperature If the depth of the valley is greater than the thermal energy of the jumping atom, that is, greater than 32 kT , then it will be trapped and requires an additional activation to overcome the remaining barrier of Table expressions Figure Second nearest neighbor atom (blue) jumping through the ring of four next-nearest atoms (green) into adjacent vacancy in a fcc structure Coefficients for the Kornblit energy Function C11 C12 C44 p0 p1 p2 q0 q1 q2 s0 s1 s2 5.29833 0.86667 1.41903 6.36429 3.45 3.32143 3.62621 1.57190 1.46564 À4.76499 À0.3333 À0.88570 À3.66429 À0.75 À0.62143 À2.88241 À0.82810 À0.72184 9.35238 1.9111 1.64444 12.92142 4.35 3.70714 11.30366 4.21984 3.68855 10 Fundamental Properties of Defects in Metals À max Á EV À EVmin As a result, Kornblit14 assumes that the vacancy migration energy for bcc metals is given by & EVm ¼ EVmax ; max 2EV À EVmin ; if EVmax À EVmin 32 kT if EVmax À EVmin > 32 kT ½18Š Using the formulae of Flynn and Kornblit, we compute the vacancy migration energies and compare them with experimental values in Figure 10 With a few exceptions, both the Flynn and the Kornblit values are in good agreement with the experimental results The self-diffusion coefficient determines the transport of atoms through the crystal under conditions near the thermodynamic equilibrium, and it is defined as ÀÀ Á Á m eq DSD ¼ DV CV ẳ nLV a exp SVf ỵ SVm =k expQSD =kT ị ẵ19 expQSD =kT ị ẳ DSD where the activation energy for self-diffusion is Q SD ẳ EVf ỵ EVm ẵ20 The most accurate measurements of diffusion coefficients are done with a radioactive tracer isotope of the metal under investigation, and in this case one obtains values for the tracer self-diffusion coeffiT ¼ fD cient DSD SD that involves the correlation factor f For pure elemental metals of cubic structure, f is a constant and can be determined exactly by computation.15 For fcc crystals, f ¼ 0.78145, and for bcc crystals, f ¼ 0.72149 Theoretical vacancy migration energy (eV) Evm, Flynn, fcc Evm, Kornblit, fcc Evm, Flynn, bcc Evm, Kornblit, bcc 1.5 To determine the preexponential factor for selfdiffusion ÀÀ Á Á ¼ nLV a exp SVf ỵ SVm =k ẵ21 DSD requires the values for the entropy SVf ỵ SVm and for the attempt frequency nLV Based on theoretical estimates, Seeger and Mehrer5 recommend a value of 2.5 k for the former The atomic vibration of nearest neighbor atoms to the vacancy is treated within a sinusoidal potential energy profile that has a maximum height of EVm For small-amplitude vibrations, the attempt frequency is then given by rffiffiffiffiffiffi EVm for fcc and by nLV ¼ a M rffiffiffiffiffiffiffiffi 2EVm ẵ22 nLV ẳ for bcc a 3M crystals where M is the atomic mass In contrast, Flynn11 assumes that the atomic vibrations can be derived from the Debye model for which the average vibration frequency is rffiffiffi kYD ½23Š nLV ¼ h where YD is the Debye temperature and h is Planck’s constant The calculated preexponential factors for some fcc metals according to the models by Seeger and Mehrer5 and by Flynn are listed in Table together with experimental values They are also shown in Figure 11 While the computed values are of the right order of magnitude, there exists no clear correlation between the experimental and theoretical values In fact, the computed values change little from one metal to another, and the Flynn model predicts values about twice as large as the model by Seeger and Mehrer Either model can therefore be used to provide a reasonable estimate of the preexponential factor where no experimental value is available 1.01.3.3 Activation Volume for Self-Diffusion 0.5 When the crystal lattice is under pressure p, the selfdiffusion coefficient changes and is then given by 0 0.5 1.5 Experimental vacancy migration energy (eV) Figure 10 Comparison of computed vacancy migration energies according to models by Flynn and Kornblit with measured values DSD ðT ; pÞ ẳ DSD expQSD =kT ịexppVSD =kT ị ẵ24 The activation volume VSD can be obtained experimentally by measuring the self-diffusion coefficient as a function of an externally applied pressure Such measurements have been carried out only for a few Fundamental Properties of Defects in Metals This boundary condition replaces the incorrect one stated in eqn [110] The cavity must always satisfy this mechanical equilibrium condition expressed by eqn [122], regardless of whether the thermodynamic equilibrium condition is satisfied or not In other words, thermodynamic equilibrium and mechanical equilibrium obey two different and separate conditions From eqns [121] and [122] it follows that the surface strains eyy and e’’ are both equal to ẳ A pR 4mS ỵ lS ịe ẳ R3 4mM R ỵ 4mS ỵ lS ị pR 2g 4mM R ỵ 4mS ỵ lS ị ½123Š Using this result we can determine from eqn [118] the surface energy of the cavity as a function of its radius R: geRịị ẳ g0 ỵ 2mS ỵ lS ịẵ2e ỵ eRịeRị ẵ124 Associated with the surface strain e(R) is a stress and a strain field in the surrounding material given by eqn [120] It gives rise to the strain energy ððð sij eij d r ¼ 8pR3 mM e2 Rị ẵ125 U Rị ẳ The reference cavity radius R defined by eqn [106] undergoes a small change as the surface strains adjust to their mechanical equilibrium values given by eqn [120] As a result, the change in cavity volume, its relaxation volume, is DV R Rị ẳ 4pR2 uRị ẳ 4pR3 eRị ẵ126 When gas is present in the cavity at a pressure p, it performs the work ÀpDVR when the surface relaxes Therefore, the total free energy associated with the creation of a cavity is FC Rị ẳ 4pR2 geRịị ỵ 8pR3 mM e2 Rị 4pR3 peRị biaxial surface stretch modulus 2(mS ỵ lS) As mentioned above, the latter has the dimension of N mÀ1, and we may then relate it to the corresponding bulk modulus 2mM/(1–2nM) by multiplying the latter with a surface layer thickness parameter h The surface energy g0 has been determined both experimentally and from ab initio calculations and can be considered as known The surface layer has been determined by Hamilton and Wolfer10 from atomistic simulations on Cu thin films to be one monolayer thick; hence d ¼ b A value for the residual surface strain parameter e* has been chosen in Section 1.01.3.1 such that it reproduces the relaxation volume of a vacancy according to eqn [11] What if one selects the same value for voids containing n vacancies? The relative relaxation volume, R =nOị ẳ 3eRnịị, can now be that is, the ratio VnV computed with eqn [123] and the results are shown in Figure 23 by the solid curve As it must, for n ¼ it reproduces the vacancy relaxation volume of À0.25O In addition, it also agrees with the overall trend of the atomistic results of Shimomura.48 Of course, the atomistic results for small vacancy cluster vary in a discontinuous manner with the cluster size The surface stress model gives not only a reasonable approximation to these atomistic results, but also a valid extrapolation to relaxation volumes of large voids The chemical potential of vacancies for voids can now be computed with eqn [127] as FC(R(n ỵ 1)) FC(R(n)) Figure 24 shows the results for Ni as the solid curve The vacancy chemical potentials for ½127Š FC(R(N)) replaces now the surface free energy FS(N) used in eqn [105] to arrive at the cavity surface tension 2g0/R The latter is now given by FC(N ỵ 1) FC(N) It will be evaluated in the next section and compared with 2g0/R 1.01.7.4.4 Chemical potential of vacancies at cavities The free energy of a void or bubble, according to eqn [127], depends now on three surface parameters instead of just one as in eqn [105]: it depends on the surface energy g0 of a planar surface, on the residual surface strain e* for such a planar surface, and on the EAM, Shimomura Surface stress model 0.05 Relaxation volume/void volume eRị ẳ 31 Cu Planar surface energy: 1.71 J m–2 –0.1 –0.15 –0.2 –0.25 –0.3 –0.35 10 100 1000 Number of vacancies in void 104 Figure 23 The relaxation volume of voids in Cu according to atomistic simulations by Shimamura48 and according to the surface stress model 32 Fundamental Properties of Defects in Metals The task is then to divide the solid into cells, each containing one individual sink, and to solve in each diffusion equations of the following type 2.5 Ni Vacancy chemical potential (eV) r Á j ¼ P À recombination Capillary approx Surface stress model EAM simulations 1.5 Planar surface energy: 2.38 J m–2 0.5 10 Void radius, R/b Figure 24 Vacancy chemical potentials for voids in Ni voids are significantly lower than the capillary approximation predicts with a fixed surface energy (dashed curve) The chemical potentials from atomistic simulations of voids in Ni have been obtained by Adams and Wolfer49 using the Ni-EAM potential of Foiles et al.50 These results converge to those predicted with the surface stress model 1.01.8 Sink Strengths and Biases 1.01.8.1 Effective Medium Approach The fate of the radiation-produced atomic defects, namely self-interstitials and vacancies, is mainly determined at elevated temperatures by their diffusion from the places where they were created to the sinks where they are absorbed or annihilated, as in the encounter of an interstitial and a vacancy As there are many sinks within each grain of an irradiated material, the spatial distribution of the atomic defects requires the solution of a very complex diffusion problem Clearly, some approximations must be sought to arrive at an acceptable solution First, we assume that the rate of defect generation, that is, the rate of displacements, is constant, and the defect concentrations between the sinks no longer changes with time or adjust rapidly when the number of sinks and their arrangement changes Then within the regions between sinks, the diffusion fluxes are stationary, that is, @ ji ẳ Dij rịCrị ỵ Fi rịCrị @xj is independent of time ẵ128 ẵ129 for each mobile defect Here, P is the rate of defect production per unit volume generated by the radiation, and the other terms represent the rates of defect disappearance by recombination with other migrating defects On the outer boundary of this cell, the defect concentration C must then match the concentration in adjacent cells occupied by other sinks, and its gradient must vanish This cellular approach has been pioneered by Bullough and collaborators,51 but the drift term, the second term in eqn [128], is omitted when solving the diffusion equation Its effect is subsequently taken into account by changing the actual sink boundary into another, effective boundary at which the interaction energy between the sink and the approaching defect becomes of the order of kT, k being the Boltzmann constant An alternate approach52,53 is to view a particular sink as embedded in an effective medium that maintains an average concentration of mobile defects at a distance far from this particular sink, and to neglect production and losses of mobile defects nearby In this approximation, the r.h.s of eqn [129] is set to zero, and the outer boundary condition far from the sink is that C approaches a constant value C that remains to be determined later from average rate equations The diffusion equation is now solved with and without the drift term and the resulting defect current to the sink is evaluated The ratio of the currents with and without the drift defines the sink bias factor It is thereby possible to define unambiguously bias factors for each type of sink, and with these determine the net bias for a given microstructure It is this embedding approach that we follow here to evaluate the bias of a sink The first attempt to determine the dislocation bias by solving the diffusion equation with drift appears to have been made by Foreman.54 He employed a cellular approach retaining only the defect production term Furthermore, anticipating small bias values, the drift term was treated by perturbation theory and numerous approximations were introduced in the derivation The intent was to obtain rough estimates; nevertheless, Foreman concluded that the bias was larger than the empirical estimate Shortly thereafter, Heald55 employed the embedding approach and used the solution of Ham56 in the form presented by Margvalashvili and Saralidze.57 We shall return to this solution below, as it is the only analytical one Fundamental Properties of Defects in Metals 33 known However, this solution applies only when the mobile defect is modeled as a center of dilatation (CD) For more general interactions, Wolfer and Askin58 developed a rigorous perturbation theory that includes also the interaction induced by externally applied stresses The latter was shown to result in radiation-induced creep They also compared the bias obtained from the perturbation theory carried out to second order with the bias from Ham’s solution, and demonstrated that both results agree only for weak centers of dilatation Vacancies can be considered as such weak centers, but interstitials cannot As a result, perturbation theory to second order is in general insufficient to evaluate the dislocation bias expressed in polar coordinates (r, ’) The solution of the diffusion equation with drift determined by the size interaction energy [133] can be obtained in terms of products of modified Bessel functions with cosine functions, Kn cosðn’Þ and In cosðn’Þ,56 and the edge dislocation bias factor is then obtained 57 in the form 1.01.8.2 Zedge ẳ nR=r0 ị Dislocation Sink Strength and Bias In order to obtain the sink strength, one first solves the steady state diffusion equation r j ẳ r2 DCị ẳ per unit length of the dislocation Here, R is an outer cut-off radius taken as half the average distance between dislocations, and rd is the dislocation core radius It is assumed that at this radius the defect concentration becomes equal to the local, thermal equilibrium concentration The total defect current to all dislocations is then proportional to the ½131Š where r is the dislocation density When the drift term is now included, the defect current changes to J ¼ ZJ0 One is for the diffusion to an edge dislocation when the interaction energy is given by eqn [45] and the stress field is that in an isotropic material In this case W1 r ; ị ẳ mb ỵ n rel sin’ V 3p À n r ½133Š ! Kn ðrC =RÞ Kn ðrC =rd Þ À1 À In ðrC =Rị In rC =rd ị nẳ0 ! K0 rC =Rị K0 rC =rd ị % nR=r0 ị ẵ134 I0 ðrC =RÞ I0 ðrC =rd Þ X ð2 À dn0 Þ where the capture radius is defined as without a drift term For the case of a straight dislocation, the solution depends only on the radial direction in a cylindrical coordinate system with the dislocation as the axis The total current of defects is then given by à 2p DC DC eq ẵ130 J0 ẳ nR=rd Þ 2pr Dislocation sink strength ¼ ‘nðR=rd Þ 1.01.8.2.1 The solution of Ham   ỵ nịb mV rel  rC ẳ 6p1 nị kT The series converges very rapidly for R ) rC, and it is sufficient, as the numerical evaluation shows, to retain only the zeroth order term As before, 2R is the average distance between dislocations If this distance becomes small as in a dense dislocation cell wall with narrow dislocation multipoles, the longrange stress fields of individual edge dislocations cancel each other, and the net interaction energy of eqn [133] is no longer valid For example, consider an edge dislocation dipole as shown in Figure 25 The interaction energy with the migrating defects is now given by E1 x; yị ẳ mb ỵ n rel V 3p n " yỵh x þ hÞ2 þ ðy þ hÞ2 À yÀh ðx À hị2 ỵ y hị2 ẵ132 Z is called the bias factor, and there are as many such factors as there are diffusing defects and different types of sinks The complexity of the interaction of a migrating defect with the strain field of the sink makes it difficult to find an analytical solution to the diffusion equation with drift However, there exist a few important solutions ½135Š r h a Figure 25 Edge dislocation dipole # ½136Š 34 Fundamental Properties of Defects in Metals pffiffiffiffiffiffiffiffiffiffiffiffiffiffi At distances r ¼ x ỵ y >> h, this interaction energy becomes mb ỵ n rel 3=2 sin2a W1 r ; aị % V h ẵ137 3p À n r when expressed in the polar coordinates indicated in Figure 25 Comparing this with the interaction energy for a singular edge dislocation, eqn [133], we see that interaction with an edge dislocation dipole falls off as r–2, and it has an angular periodicity of twice that for the single edge dislocation The solution of the diffusion equation can again be constructed in terms of products of cosine functions, cos(2na), and the modified Bessel functions, but now of a different argument, namely 2rCrD/r2, where rD ¼ 21/2h is the radius of the dipole If we take this radius to be the sink radius, then the bias factor for the dipole is Zdipole ẳ nR=rD ị K0 2rC rD =R2 Þ K0 ð2rC =rD Þ À I0 ð2rC rD =R2 Þ I0 ð2rC =rD Þ !À1 ½138Š Again, just as for the bias of a single edge dislocation, eqn [134], we find when evaluating eqn [138] numerically that the first term, as shown, in this series already provides accurate results The important material parameter that determines the bias factor of edge dislocation is the capture radius defined in eqn [135] At this radius, and at polar angles perpendicular to the direction of the Burgers vector, the interaction energy is of the magnitude   p  ½139Š W1 rC ; ặ  ẳ kT 2 We may then attach the following physical meaning to this capture radius: when a defect reaches the dislocation at this distance from its core, and the interaction is attractive, meaning negative, it is inevitably pulled into the core, and when it is repulsive, that is, positive, the defect is definitively repelled Table 14 lists values for both the interstitial and vacancy capture radii evaluated at half the melting points for various metals, and the bias factors for rd ¼ 2b and R/b ¼ 2250, corresponding to a dislocation density of 1012 mÀ2 The vacancy capture radii are small for all metals, and about equal to what one would expect for the sum of the dislocation core radius plus the point defect radius, namely rd ’ 1–2b In contrast, the interstitial capture radii are significantly larger, in particular for fcc metals when compared with bcc metals The evaluation of eqn [134] gives the solid curve displayed in Figure 26 As already mentioned above, terms in the sum with n ! contribute less than 0.00025 to the dislocation bias factors In addition, for large values of R/rC, the term that depends on rC/r0 can be neglected, and the series expansions can be used for the modified Bessel functions K0 and I0 As a result, one then obtains the asymptotic approximation59 Zedge % ‘nðR=r0 Þ n2R=rC ị g ẵ140 where g ẳ 0.577216 is the Euler constant This approximation is also shown in Figure 26, and it is seen that it coincides with the exact results for rC/b ! However, for rC/b 2, eqn [140] gives incorrect bias factors less than one For small values of rC/b, Wolfer and Ashkin58 have obtained from perturbation theory the following expression: Zedge % ỵ ẵrC =2r0 ị2 OẵrC =2r0 ị4 ị nR=r0 ị ẵ141 As indicated, extension of this perturbation theory to higher orders shows that an alternating series is obtained with poor convergence This then suggests to seek a Pade approximation that may extend the usefulness of eqn [141] For example,58 Table 14 Capture radii and bias factors for interstitials and vacancies evaluated at half the melting points and with the size interaction only Element Vrel I =V Vrel V =V rC =b SIA rC =b vacancies ZdI ZdV Net bias ¼ ZdI =ZdV À Al Cr Cu Fe Mo Nb Ni Ta V W 1.9 1.21 1.55 1.1 1.1 0.76 1.8 1.05 À0.25 1.03 0.92 À0.31 À0.22 À0.25 À0.27 À0.19 À0.27 À0.20 6.43 À0.30 À0.18 14.32 8.31 13.46 8.69 10.62 5.08 14.77 1.56 5.06 14.91 2.36 1.11 1.80 1.94 1.84 1.24 1.64 1.177 1.47 1.70 1.358 1.229 1.342 1.239 1.284 1.135 1.366 1.020 1.134 1.369 1.043 1.011 1.026 1.030 1.027 1.013 1.022 0.154 1.020 1.024 0.302 0.216 0.308 0.208 0.250 0.120 0.337 0.112 0.337 Fundamental Properties of Defects in Metals where Ei is the exponential integral function Evaluation of eqn [144] gives the curve labeled as ‘Average’ in Figure 26 The angular average approximation [144] compares well with the exact result for large values of rC/b, that is, for interstitial capture radii, but slightly over-predicts vacancy bias factors Edge dislocation bias factor 1.5 Ham’s solution Asymptote Pade approx 3.0 Angular average 1.4 35 1.3 1.01.8.2.2 Dislocation bias with size and modulus interactions 1.2 1.1 10 15 Capture radius/Burgers vector 20 Figure 26 Edge dislocation bias factors based on Ham’s solution and various approximations to it The modulus interaction has been discussed in Section 1.01.5.3 Treating both the material as well as the defect inclusion as elastically isotropic, the modulus interaction depends on two diaelastic polarizabilities, aK and aG, for which values are provided in Table 11 For this isotropic case, the modulus interaction for edge dislocations is58 ½rC =2r0 ị nR=r0 ị ỵ mẵrC =2r0 ị2 rd nR=rd Þ exp½bW1 ðr ފd ‘nðr Þ ‘nðR=r0 Þ EiðÀrC =rd Þ À EiðÀrC =RÞ ½146Š ½142Š ½4pð1 À nފ2 and ẵ143 A2 ẳ aK 23am ị1 2nị2 ỵ 4am n1 nị ẵ144 ẵ4p1 nị2 ẵ147 The perturbation theory of Wolfer and Ashkin58 with the sum of the size interaction [130] and the modulus interaction [145] gives the result ! 2A0 edge ẵ148 %1ỵ rC þ Z kT ð2rd Þ2 ‘nðR=r0 Þ This suggests that an effective capture radius can be defined as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2A0 rC;eff ẳ rC2 ỵ ẵ149 kT and replacing rC with it in eqn [144] yields bias factors that include both the effects of size and modulus interactions Using the values given in Table 11 for the diaelastic polarizabilities, effective capture radii and new bias factors are obtained and presented in Table 15 Comparing these results with the corresponding ones in Table 14 shows that the modulus interaction contributes to the net bias about 25% for fcc metals, and about 10–15% for bcc metals 1.01.8.3 we obtain58 Zd % aK ð1 2nị2 ỵ 43am n ỵ n2 ị A0 ¼ with m ¼ fits the exact results for rC/b as seen in Figure 26 Very accurate results for the bias factors of edge dislocations can therefore be produced with eqn [142] for small capture radii and with eqn [141] for large capture radii The two approximations transition extremely well at rC/b ¼ These approximations suggest a way to proceed when the interaction energy assumes a more complicated form than in eqn [133] It is therefore tempting to see if an angular average of the size interaction energy, eqn [134], could be used to evaluate at least approximately the bias factors Obviously, the angular average of sinj is zero The diffusion flux will wind around the dislocation as it approaches the core in order to avoid regions where the interaction energy W1 becomes strongly repulsive Therefore, an average should only be taken over the angular range where W1 is attractive, that is, negative So, if we replace W1(r, ’) in eqn [134] with W1(r, p/2)/2 in the case of interstitials and with W1(r, 3p/2)/2 in the case of vacancies and evaluate the equation Zd % é R ẵ145 with Zedge ẳ ỵ W2 r ; ị ẳ A0 ỵ A2 cos2ịb=r ị2 Bias of Voids and Bubbles In elastically isotropic materials, the interaction of vacancies as well as interstitials with spherical cavities depends only on the radial distance r to the 36 Fundamental Properties of Defects in Metals Table 15 Capture radii and bias factors of edge dislocations with size and modulus interactions Element rC, eff /b SIA rC, eff /b vacancy ZdI ZdV Net bias ¼ ZdI =ZdV À Al Cu Ni Cr Fe Mo 20.38 18.30 20.09 10.47 10.75 12.73 3.55 2.89 2.98 2.69 3.12 3.27 1.70 1.66 1.69 1.46 1.47 1.53 1.21 1.18 1.18 1.17 1.19 1.20 0.405 0.408 0.434 0.256 0.240 0.276 cavity center Denoting by WS(r) this interaction energy at the saddle point configurations, the radial defect flux that includes the drift can be written as jr ẳ expẵbW S r ị ẫ dẩ DCr ịexpẵbW S r ị ẵ150 dr The defect current J through each concentric sphere around the cavity is a constant, that is, 4pr2jr ¼ J Integration of [150] then leads to DCðr ịexpẵbW S r ị ẳDCRịexpẵbW S Rị r J dr expẵbW S rị r2 Next, the modulus interaction with the strain field of the cavity is60 !  3aG 2gðRÞ R W2 R=r ị ẳ p À ½153Š 8m R r The strain field around the cavity is caused by both a pressure p when gas resides in the cavity, and by the surface stress g(R) Note that this surface stress is not equal to the surface energy or a constant as usually assumed As shown in Section 1.01.7.4.3, it changes with the cavity radius and the gas pressure p For the case of voids, when p ¼ 0, the surface tension is found to be R Matching this solution to the boundary conditions at r ! 1, where Cðr Þ ! C, and at r ¼ R, where CðRÞ ¼ C is the defect concentration in local equilibrium with the cavity, gives the defect current 4pRDC DC ị J ẳ é R=Rỵhị expẵbW S R=r ịd R=r ị ẳ 4pRZ0 DC DC ị ỵ nị2 mV rel ị2 U im r ; Rị ẳ 36p1 nịR3 e ẳ nịVVrel =O ẵ155 With eqns [154] and [155], the modulus interaction then takes the final form  rel 2  2 V 1n W2 R=r ị ẳ aG V O ỵ 2nịR=bị The evaluation of the void bias factor " Z Rị ẳ R=Rỵhị expẵbW R=r ފd ðR=r Þ S ½156Š #À1 ½157Š   X nn 1ị2n 1ị2n ỵ 1ị R 2n ỵ n2 ỵ 2nịn ỵ À n ½154Š where e* is the residual surface strain of a planar surface It is shown in Section 1.01.3.1, that this residual surface strain parameter can be related to the relaxation volume of the vacancy, and according to eqn [11] ½151Š The integration extends up to the distance h(R) from the cavity surface where the strain energy of the defect becomes zero, as discussed in Section 1.01.5.3 and shown in Figure 17 The interaction energy W S(R/r) consists of two parts, the image interaction and a modulus interaction The former has been discussed in Section 1.01.5.3, and it has been given by Moon and Pao32 in the form n¼2 2gRị 2me ẳ ỵ 2nịR=bị R r ½152Š The series converges slowly as r approaches the cavity radius R However, when R/r 0.99, no more than about 1000 terms are required to obtain accurate results requires numerical integration As an example, the bias factors for self-interstitials and vacancies are computed for Ni, at a temperature of 773 K, and using the defect parameters in their stable configurations as given in Tables 2, 7, and 11 The results are presented in Figure 27 The solid curves are obtained when both the image and the modulus interactions are included, while the dashed curves neglect the contribution of Fundamental Properties of Defects in Metals 2.2 SIA total bias SIA image bias Vac total bias Vac image bias Void bias factors 1.8 Ni 1.6 1.4 1.2 1 10 Void radius, R/b 100 Figure 27 Bias factors for voids in nickel as a function of the void radius in units of the Burgers vector the modulus interaction to the void bias factors It is seen that the modulus interaction contributes little to the void bias for vacancies, while it enhances the void bias for interstitials significantly 1.01.9 Conclusions and Outlook Several past decades of intense research have resulted in a good understanding of the fundamental properties of vacancies and self-interstitials in pure metals We have reviewed this understanding from the following point of view: how these fundamental properties affect radiation damage at elevated temperatures that exist in nuclear reactors The key parameters that emerge from this perspective are the displacement energy of Frenkel pairs, the formation and migration energies of vacancies and selfinterstitials, and their relaxation volumes and elastic polarizabilities While the physical basis for these key parameters is understood, obtaining precise values for them by experimental and theoretical means remains a formidable challenge In particular, this is true for the type of alloys that are used for components in the core of nuclear reactors In general, these are complex alloys For example, the austenitic stainless steels are composed of major alloy constituents, namely iron, nickel, and chromium, and many minor alloy elements such as molybdenum, titanium, manganese, carbon, and silicon Therefore, many different types of vacancies and self-interstitials can exist in 37 these alloys with potentially different properties It is unlikely that all these different properties can actually be measured Rather, only effective average properties, such as the self-diffusion coefficient, can be determined experimentally, and theoretical models must be employed to relate effective properties to the individual properties of the different vacancies At the present time, electronic structure method still require further development before effective properties of defects in complex alloys can be calculated In fact, only recently has it become possible to calculate, for example, accurate values for the formation energies of mono- and divacancies in pure metals Two advances have been responsible for the progress First, density functional theory requires different implementations when applied to bulk and to surface properties of metals The uniform electron gas that serves as a starting point for the electron density functional in the bulk interior of solids is not suitable to formulate the corresponding functional for the ‘electron edge gas.’ As shown by Kohn and Mattsson,61,62 functionals must be developed that join the edge and the interior bulk regions, and Armiento and Mattsson63,64 have proposed and tested functionals for these two regions and how to join them When applied to vacancies in metals,65,66 formation energies are predicted that are in much better agreement with experimental results But there are also differences For example, Carling et al.65 obtain a repulsive (positive) binding energy for divacancies in Al However, this result may be due to the limited number of atoms employed in the calculations and the periodic boundary conditions This brings us to the second advance that has recently been made, namely the implementation of an orbital-free electron density functional theory based on finite-element methods.67 With this approach, much larger systems containing effectively millions of atoms can be treated; these systems are truly finite, and realistic boundary conditions can be applied to them Figures 28 and 29 reproduced from Gavini et al.67 reveal a surprisingly large effect of the system size, that is, the effective number of atoms in a finite crystal into which one or two vacancies have been introduced As demonstrated by these results, in order to obtain defect properties that are independent of the size of a finite system requires thousands of atoms as well as their full relaxation In other words, electronic structure calculations need to be combined with continuum elasticity descriptions to predict radiation effects and to develop better alloys for nuclear power generation 38 Fundamental Properties of Defects in Metals clusters of several hundreds of atomic defects, and the volume fraction they occupy ranges from a part per million to perhaps a tenth of a percent So if r is the typical radius of a defect under consideration and 2R the average distance between them, then the defect volume fraction 0.9 Vacancy formation energy (eV) Unrelaxed Relaxed 0.85 S ẳ =Rị3 0.75 0.7 10 100 1000 104 Number of atoms 105 106 Figure 28 Vacancy formation energy for Al as a function of system size, and with and without relaxing the atomic positions when removing a central atom to create a vacancy Reproduced from Gavini, V.; Bhattacharya, K.; Ortiz, M Mech J Phys Solids 2007, 55, 697 0.1 Di-vacancy binding energy in Al (eV) ½A1Š 0.8 0.05 –0.05 –0.1 –0.15 –0.2 –0.25 10 100 1000 104 Number of atoms 105 106 Figure 29 Binding energy of a divacancy in Al as a function of system size and orientation Reproduced from Gavini, V.; Bhattacharya, K.; Ortiz, M Mech J Phys Solids 2007, 55, 697 Appendix A Elasticity Models: Defects at the Center of a Spherical Body A1 An Effective Medium Approximation The crystal defects that we consider here are mainly of microscopic size ranging from single atoms to is a small number As a result, each defect is surrounded by a cell that contains no other defect, and the solid can be viewed as composed of these cells If the solid as a whole is free of external stresses on its macroscopic surface, then the normal stress component averaged over the surface of each cell is also zero, since on the cell boundary, the stress fields of two neighboring defects overlap and cancel on average In the final step in this cellular approach, we approximate the typical cell by a spherical region with radius R with a defect of radius r at its center It is this effective medium approximation that elevates the analysis of a sphere with a defect at its center to more than just an academic exercise, and its results are more generally valid for defects in solids of finite extent Atomistic computer simulations using either semi-empirical potentials or first-principle methods are usually carried out in a periodically repeated cell that contains a finite number of atoms However, in this case the medium is in fact infinite, and the results are different from those obtained for a finite crystal on whose external surface the stress field created by the defect satisfies the boundary condition of zero tractions, that is, the vanishing of the stress component normal to the external surface Satisfying this boundary condition is different from having a defect stress field that approaches zero at infinity This difference is often attributed to the so-called image stresses that exist in a finite crystal, but not in an infinite one The model of a defect at the center of a finite sphere of radius R captures this difference, and as we shall see, this difference becomes independent of the radius R as it approaches infinity but such that S, the defect concentration, remains small but constant A2 The Isotropic, Elastic Sphere with a Defect at Its Center Consider then a small defect with a radius r situated at the center of an elastically isotropic sphere of radius R The defect is modeled in four different ways: Fundamental Properties of Defects in Metals As a center of dilatation (CD): the defect is created by displacing the inner radius r of the surrounding matrix by an amount urị ẳ cr ẵA2 39 partial differentiation Hooke’s law then connects the stresses with the elastic part of the strains, which for the isotropic case reads as sij ẳ mui;j ỵ uj ;i ị þ ldij uk;k À 2meij À ldij ekk ½A5Š Here, the parameter c is referred to as the strength of the defect As a cavity (CA): the inner surface is loaded by a pressure p, giving rise to a radial stress component in the surrounding medium, which has the boundary value Here, repeated indices imply a summation and a comma before an index, say j, means a partial derivative with regard to xj The mechanical equilibrium equations are sij ; j ¼ 0, or after substitutions of Hooke’s law sr rị ẳ p m ỵ lịuj ; ji ỵ mui; jj ẳ ẵA3 The pressure p can also represent radial forces that the defect exerts on its nearest neighbor atoms of the surrounding matrix, and they are then referred to as Kanzaki forces As an inclusion (INC): the material within the defect region is subject to a transformation strain eij as if it had endured a phase transformation We shall treat here only the simple case of an isotropic transformation strain eij ¼ dij where dij is the Kronecker matrix As an inhomogeneity (IHG): In addition to a transformation strain, the defect region has also acquired different elastic constants from the surrounding matrix The boundary conditions at the interface between the defect and its surrounding matrix are that the radial displacement and the radial stress component must be continuous These different models of defects correspond in essence to the three possible boundary conditions on the defect–matrix interface, namely the Dirichlet boundary condition for the CD, the von Neuman condition for the CA, and the mixed one for the INC and IHG One could also consider these three different boundary conditions for the external surface However, prescribing a value for the displacement of the external surface or a mixed boundary condition implies the presence of yet another surrounding medium, in which the defect-containing sphere would be embedded Here, these cases will not be considered We then restrict the following treatment to the boundary condition of a free external surface for which sr Rị ẳ ẵA4 The solution of the elasticity problems associated with all defect models is rather elementary and can be found in many textbooks However, as a way to introduce our notation, we sketch the procedure The deformation is described by a displacement r Þ, from which the strain tensor is obtained by field ui ð~ ½A6Š We assumed here that the transformation strains eij are constant within the defect region For a spherically symmetric problem, the displacement field possesses only one radial component u, which satisfies the differential equation " # d d uị ẳ0 ẵA7 r l ỵ 2mị dr r dr The solution to this equation is ur ị ẳ Ar ỵ B=r ẵA8 with two unknown constants A and B for each region, for the matrix that surrounds the defect and for the defect region itself However, for the defect region B ¼ The remaining constants are to be determined by the boundary conditions For later use we give the radial stress component sr ẳ 2m ỵ 3lịA ị 4mB=r ẳ 3K A ị 4mB=r ẵA9 for the case of an isotropically transformed inclusion, that is, when eij ¼ dij Furthermore, instead of the Lame’s constants m and l, we introduced the bulk modulus K and the shear modulus m The following ratio involving the two elastic constants will appear often, and we reserve the symbol o¼ 4m 2ð1 À 2nị ẳ ẳ gE 3K 1ỵn ẵA10 for it, where n is Poison’s ratio and gE is the Eshelby factor The hydrostatic stress is related to the elastic dilatation or the lattice parameter change by the equation Da ẵA11 sH ẳ sii ẳ 3K A ị ¼ 3K a We see that for a defect in the center of an isotropic sphere the lattice parameter changes are uniform throughout the matrix 40 Fundamental Properties of Defects in Metals The density of strain energy, on the other hand, is strongly peaked near the defect, as can be seen from the formula fðr Þ ẳ sij ui; j ẳ K A ị2 ỵ 6mB =r 2 ½A12Š Integrating this function over the entire sphere gives the strain energy associated with the defect, ððð ðR sij ui; j dV ¼ 4p fr ịr dr U0 ẳ ẵA13 Finally, let us state the formulae for volume changes The external volume change is computed from the equation DV ¼ 4pR2 uRị ẳ 4pR3 A ỵ B=R3 ị or DV ẳ 3A ỵ B=R3 ị V ẵA14 average by < > brackets However, if the defect region consists of a very different crystal structure or atoms with much lower form factors, then the defect region does not contribute to the diffraction pattern, and the lattice parameter change is due to the matrix only When the defect region is excluded, the lattice parameter change is indicated by ( ) parentheses We obtain the coefficients A and B by solving the equations for the boundary conditions For example, for the case of an inhomogeneity, the radial stress components vanish on the external surface, sr Rị ẳ 3KA 4mB=R3 ẳ ẵA16 On the interface between the inclusion region and the matrix, that is, at r ¼ r, the displacements and the radial stress components must be continuous, or AI r ¼ Ar ỵ B=r2 ẵA17 In a similar fashion, one defines the change of the defect volume due to the constraints imposed by the surrounding matrix, as and Du ẵA15 ẳ 3A ỵ B=r3 ị u A last point must be made regarding the measurement of the lattice parameter change If the atoms in the defect region contribute to the diffraction pattern just as the matrix atoms, we must calculate the appropriate average lattice parameter change over the entire volume V We denote this Here, the subscript ‘I’ designates a parameter for the inclusion region, while parameters for the matrix are without a subscript Solution of the three eqns [A17]-[A19] determines the three parameters A, B, and AI listed in the first three rows and the last column of Table A1 We note that the stress within the inhomogeneous inclusion is purely hydrostatic (because BI ¼ 0), and the results not depend on 3KI AI ị ẳ 3KA 4mB=r3 ½A18Š Table A1 The solutions for the integration constants A and B, and volume changes, lattice parameter changes, and defect strain energies that derive from them Defect type CD CA INC IHG A for matrix coS ỵ oS pS 3K1 Sị oS 1ỵo oS ỵ oS þ koð1 À SÞ B for matrix cr3 þ oS pr3 4m1 Sị r3 1ỵo r3 ỵ oS ỵ ko1 Sị 1 ỵ oSị 1ỵo 31 þ oSÞ 1þo ð1 þ oSÞ þ oS þ ko1 Sị 31 ỵ oSị ỵ oS ỵ ko1 Sị 31 ỵ oSị ỵ oS ỵ ko1 Sị 1 ỵ oịS ỵ oS ỵ koð1 À SÞ AI for defect   Du of defect u   DV of solid V Da for solid a   Da for matrix a Defect energy U0 NA NA 3c 3p1 ỵ oSị 4m1 Sị 1ỵo 3c S ỵ oS 1ỵo c S þ oS 3pð1 þ oÞ S 4mð1 À SÞ pS 3K 3S Sịo S ỵ oS oỵS 9c2 uk 1S pS 3K o1 SịS 1ỵo 1S 62 um 1ỵo c 3p2 u1 ỵ oSị 8m1 Sị S o1 SịS ỵ oS ỵ ko1 Sị 62 um1 Sị ỵ oS ỵ koð1 À SÞ Fundamental Properties of Defects in Metals Table A2 41 Simplified expressions obtained from the general ones in Table for small defect concentrations, S