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 4 — tsV

kBT.

given in atomic fractions. Here, 4 is the vacancy formation enthalpy, and SV 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 4 have been reviewed and tabulated by Ehrhart and Schultz;7 they are listed in Table 2. When these values for the metallic elements are plotted versus the melt temperature in Figure 4, an approximate correlation is obtained, namely

4 ~ Tm/1067 [3]

4000 3500

2- 3000

о о

Ь 2500

0 >

2000

га

1 1500 “ 1000

500 0

0 500 1000 1500 2000 2500 3000 3500 4000

Melt temperature (K)

Figure 3 Leibfried’s empirical rule between melting temperature and the product of bulk modulus and atomic volume.

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 4 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 4nR2 times the specific surface energy g0. Taking the atomic volume as the vacancy volume, that is, O = 4pR 3/3, we show in Figure 6 the measured vacancy formation enthalpies as a function of 4nR2y0, using for g0 the values9 at half the melting temperatures. It is seen that 4 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 4 to something less than O. The difference

VVel = VV — O [4]

is referred to as the vacancy relaxation volume. The experimental value7 for the vacancy relaxation of Cu is —0.25O, 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 [e(R), £*] = 4-pR2g[e(R), £*] + 8nR3me2(R) [5]

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

g[e, e*]=g0 + 2(mS + 1s)(2£* + e)e [6]

Here, g0 is the specific surface energy on a surface with no strains in the underlying bulk material.

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, m and Is, 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

+ Is = (m т i)h = mh/ (1 — 2v) [7]

where l is the Lame’s constant and v 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 3 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 dFc/де = 0. The result is

(ms + ^s)e* h e*

m’R + (ms + Is) (1 — 2v)R + h and this relaxation strain changes the initially unre­laxed void volume

0

0 500 1000 1500 2000 2500 3000 3500 4000

Melting temperature (K)

Figure 4 Vacancy formation energies as a function of melting temperature.

0

0 1 2 3 4 5

Surface energy of a vacancy (eV)

Figure 6 Correlation between the surface energy of a vacancy void and the vacancy formation energy.

4

3.5 3

2.5 2

1.5 1

0.5 0

0 5 10 15 20 25 30 35 40

Bulk modulus * atomic volume (eV)

Figure 5 Vacancy formation energy versus the product of bulk modulus and atomic volume.

n O = R3 [9] consisting of n aggregated vacancies, by the amount Vrel (R) = 3nOe(R) [10]
4pR2
2 + 3mO
[12]
This equation is evaluated for Ni and the results are shown in Figure 7 as a function of the vacancy relaxa­tion volume I VVel/O I. It is seen that relaxation volumes of -0.2 to -0.3 predict a vacancy formation energy comparable to the experimental value of 1.8 eV.
and for the vacancy formation energy

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 —0.2O and —0.3O 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, SV, 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 con­centration, CVq, is of the same magnitude as the statisti­cal uncertainty in the vacancy formation enthalpy.

1.02.2.1 Point Defects Compared to Defects of Greater Spatial Extent

In crystallography, we learn that the atoms and ions of inorganic materials are, with the exception of glasses, arranged in well-defined planes and rows.3 This is, however, an idealized representation. In real­ity, crystals incorporate many types of imperfections or defects. These can be categorized into three types depending on their dimensional extent in the crystal:

1. Point defects, which include missing atoms (i. e., vacancies), incorrectly positioned atoms (e. g., inter­stitials), and chemically inappropriate atoms (dopants). Point defects may exist as single species or as small clusters consisting of a number of species.

2. Line defects or dislocations, which extend through the crystal in a line or chain. The dislocation line has a central core of atoms, which are located well away from the usual crystallographic sites (in cera­mics, this extends, in cylindrical terms, to a nano­meter or so). Most dislocations are of edge, screw, or mixed type.4

3. Planar defects, which extend in two dimensions and are atomic in only one direction. Many differ­ent types exist, the most common of which is the grain boundary. Other common types include stacking faults, inversion domains, and twins.1,2

The defect types described above are the chemical or simple structural models for the extent of defects. It is critical to bear in mind that all defect types, in all materials, may exert an influence via an elastic strain field that extends well beyond the chemical extent of the defect (i. e., beyond the atoms replaced or removed). This is because the lattice atoms sur­rounding the defect have had their bonds disrupted. Consequently, these atoms will accommodate the existence of the defect by moving slightly from their perfect lattice positions. These movements in the positions of the neighboring atoms are referred to as lattice relaxation.

As a result of the elastic strain and electrostatic potential (if the defect is not charge-neutral), defects can affect the mechanical properties of the lattice. In addition, defects have a chemical effect, changing the
oxidation/reduction properties. Defects also provide mechanisms that support or impede the movement of ions through the lattice. Finally, defects alter the way in which electrons interact with the lattice, as they can alter the potential energy profile ofthe lattice (whether or not the defect is charged). For example, this may lead to the trapping of electrons. Also, because dopant ions will have a different electronic configuration from that ofthe host atom, defects may donate an electron to a conduction band, resulting in n-type conduction, or a defect may introduce a hole into the electronic struc­ture, resulting in p-type conductivity.

At intermediate temperatures where SIAs and vacan­cies are mobile, significant solute segregation to point defect sinks can occur. This can lead to precipitation of new phases due to the local enrichment or deple­tion of solute. These radiation-induced or — enhanced precipitation reactions typically become predominant

phenomena in irradiated ferritic and austenitic steels at elevated temperatures for doses above about 10 dpa, and in irradiated reactor

pressure vessel steels at low dose rates for damage levels above 0.001-0.01 dpa.247,248 Some general aspects of radiation-induced and — enhanced solute segregation and precipitation were described previ­ously in Section 1.03.3.9. The solute segregation and precipitation associated with irradiation can lead to several deleterious effects including property degradation due to grain boundary or matrix embrit-

tlement224,247,249-252 and enhanced susceptibility for localized corrosion or stress corrosion cracking.253-256 Solute segregation and precipitation can lead to either enhanced or suppressed void swelling

behavior.149,257,258 For austenitic stainless steel,

undesirable precipitate phases that generally are associated with high void swelling include the radiation-induced phases M6Ni16Si7 (G), Ni3Si (g0), MP, M2P, and M3P, and the radiation-modified phases M6C, Laves, and M2P200 The undesirable radiation-induced and — modified phases generally are associated with undersized misfits with the lattice, which tends to preferentially attract SIAs and thereby enhance the interstitial bias effect. Figure 26 shows an example of enlarged cavity formation in association with G phase precipitates in neutron- irradiated austenitic stainless steel.106 Potentially desirable radiation-enhanced and — modified phases (when present in the form of finely dispersed precipi­tates) include M6C, Laves, M23C6, MC, s, and w200

To unfault a 1/4 [110] (110) dislocation loop in spinel, we must propagate a 1 /4[112] partial shear dislocation across the loop plane.12 This is described by the following dislocation reaction:

4[110] + i[H2] ! 1[101] [11]

faulted loop partial shear unfaulted loop

This reaction is shown graphically in Figure 5. When we pass a 1 /4[112] shear through a 1/4 [110] (110) dislocation loop, the atomic planes beneath the loop assume new registries, such that in eqn [6], ajpj and a2p2 commute as follows: a1p1 ! a2p2 ! a1p1. The anion layers beneath the loop are left unchanged (B! B, C! C). Also the Al p’ layers are left unchanged (p’ ! p’). Taking the faulted (110) stack­ing sequence in eqn [6] and assuming that the planes to the right are above the ones on the left, we perform the 1/4[112] partial shear operation as follows:

(P’S) (a, P,C) (P’S) KP2C) (P’S (a, b,C) |(P’S)| (a, P,C) (P’S) ^C) (faulted)

P’S a2P2C P’S a, P,C

(P’S) (a, P,q (P’S) (a2P2C) (P’S (a, P,C) (P’S) (a2P2C) (P’S) (a, P,C) (unfaulted)

[12]

After propagating the partial shear through the loop, we are left with an unfaulted layer stacking sequence. The Burgers vector of the resultant dislocation loop, 1/2[101], is a perfect lattice vector; therefore, the newly formed dislocation is a per­fect dislocation (equal to the Mg-Mg first nearest — neighbor spacing). The resultant 1/2[101](110) dislocation is a mixed dislocation, in the sense that the Burgers vector is canted (not normal) relative to the plane of the loop.

Complex structural materials, such as FMS and NFA, are composed of a wide range of types, morphologies, size scales, and associations of various microstuctural features. A general master model approach to treating He transport, fate, and consequences in such alloys is shown in Figure 3. He transport and clustering reac­tions within the matrix and in various microstructural regions are treated. In this framework, RT models, or KMC simulations, are used to transport and partition He to the various microstructural regions, such as dislocations, grain and subgrain boundaries, and het­erophase interfaces. He is further transported within a region to and from internal subregions. For exam­ple, dislocations are regions, and precipitates and jogs on dislocations constitute subregions. He can recycle within or between regions. The accumulation of He in the various sites results in the formation of He bubbles (and in some cases voids).

Preliminary development of a master model code to implement this framework is underway.178 In its current RT formulation, He is generated by transmu­tations as both Hei and Hes at relative fractions that can be based on a physically motivated parameter that needs to be established. These interstitial and substitutional forms can switch by interaction with vacancies and SIA, respectively. The Hei rapidly dif­fuses to the various regions where it is trapped or captured by a matrix vacancy to form Hes. Hes dif­fuses more slowly to the various regions or is dis­placed by a SIA to form Hei. The differences in the diffusion rate are reflected in the steady-state matrix concentrations of Hei (very low) and Hes (higher).

A CD model (see Section 1.06.3) is used to track the fate of He and bubble evolution by reactions He + Hem! Hem +1 in the various regions and subregions. Although the framework of the CD mod­els allows the disassociation ofclusters by He emission, this is not implemented in the results described below. That is, a He2 cluster is taken as a stable nucleus for the formation and evolution of larger bubbles. A further assumption is that the HemVn clusters grow with He addition as equilibrium bubbles (m/n < 1), along the lowest free-energy path, with a real gas equation of state p = 2g/rb, as described in Section 1.06.3. This approximation is valid at low damage rates and vacancy-rich environments associated with neutron irradiations. The current implementation focuses on bubble evolution, and since the dislocation bias is set to B = 0, the model does not directly treat void formation and swelling. However, this capability will be implemented in future versions of the code.

Helium atoms trapped at precipitates, disloca­tions, and GBs may either be emitted back to the matrix, at a rate determined by their binding ener­gies, or diffuse to be captured by deeper subregion traps. The He + He! He2 reactions form a bubble nucleus in all regions and subregions, either hetero­geneously with other trapped He or homogeneously with reactions between two freely diffusing He atoms. Nucleation of bubbles on dislocations is a very impor­tant process. Dislocations are modeled as a size distri­bution of segments bounded by deeper traps than the dislocation itself, such as junctions, jogs, and attached precipitates. The initial distribution of dislocation seg­ments is resegmented (split) as bubbles homogeneously nucleate on them.

The master model contains many parameters. Where possible, microstructural observations were used to provide microstructural parameters for grain sizes, dislocation densities, and precipitates, as summar­ized in Table 6. In general, the binding and activation energies were obtained from the models described in Section 1.06.5. Details are presented elsewhere.178

Figure 41 shows an example of the master model predictions of bubble radii (a, c) and number densi­ties (b, d) compared with the ISHI data described previously in this section for 40 appm He/dpa at 500 °C up to 10 dpa for the FMS F82H (a, b) and NFA MA957 (c, d) microstructure variables shown in Table 6. The data are shown for F82H in both as tempered (AT) and 20% CW conditions. The over­all agreement is quite good. The model predicts that almost all of the bubbles form on dislocations in F82H and on dislocations and dislocation associated NF in MA957, broadly consistent with observations. The model predicts a smaller number of larger

	(b) dPa
(c) dpa	(d) dpa

Figure 41 Model predictions corresponding in situ He-implanter data for the (a) average bubble radius; (b) number density in F82H; (c) average bubble radius; and (d) number density in MA957also show the observations in experiments.

bubbles than observed in the F82H in the AT condi­tion. The agreement is better for MA957, and the model predictions are consistent with the observation that a higher number density of smaller bubbles form in this case. The MA957 model predicts that there is a lower number of smaller bubbles in the matrix and especially on GBs. Note that the models do not yet contain lath boundaries that are observed to contain a high concentration of bubbles in F82H. The predicted size distribution of bubbles is shown in Figure 42. The agreement with the experimental results is again quite good and reflects the significant differences that are observed in the two alloys.

During the migration of a point defect through the crystal lattice, it traverses an energy landscape that is schematically shown in Figure 18. The energy minima are the stable configurations where the defect energy is equal to Ef(r), the formation energy, but modified by the interactions with inter­nal and external strain fields, which in general vary with the defect location r. In order to move to the adjacent energy minimum, the defect has to be thermally activated over the saddle point that has an energy

ES(r)=Ef (r)+Em(r) [69]

where Em(r) is the migration energy. As the prop­erties of the point defect, such as its dipole tensor and its diaelastic polarizability, are not necessarily the same in the saddle point configurations as in the stable configuration, the interactions with the strain fields are different, and the envelope of the saddle point energies follows a different curve than the envelope of the stable configuration energies, as indicated in Figure 18. For a self-interstitial, we

must also consider the different orientations that it may have in its stable configuration. Accordingly, let Cm(r, t) be the concentration of point defects at the location r and at time t with an orientation m. For instance, the point defect could be the self­interstitial in an fcc crystal, in which case, there are three possible orientations for the dumbbell axis and m may assume the three values 1, 2, or 3 if the axis is aligned in the xb x2, or x3 direction, respec­tively. The elementary process of diffusion consists now of a single jump to one adjacent site at r + R, where R is one of the possible jump vectors.

The rate of change with time of the concentration Cm (r, t) is now given by

—C

= Cn(r — R, t)Lnm(r — R | R)

r, v

‘У ‘ Cm(r, t)Lmn(r 1 R) [70]

r, v

Here, the first term sums up all jumps from neigh­boring sites to site r thereby leading to an increase of Cm (r, t), while the second term adds up all the jumps (really the probabilities of jumps) out of the site r. The frequency (or better the probability) of a partic­ular jump from r to r + R while changing the orien­tation from m to V is denoted by Lmn(r | R). The eqn [70] applies now to each of the possible orientations, and it appears that this leads to as many diffusion equations as there are possible orientations, and these equations may be coupled if the defect can change its orientation between jumps.

To circumvent this complication, one considers an ensemble of identical systems, all having identical microstructures, and identical internal and external stress fields. The ensemble average of the defect concentration at each site, denoted simply as C(r, t) without a subscript, is now assumed to be the ther­modynamic average such that exp(-bEm)

£exp(-bEvf)

= C(r, 0exp(-b4)/N(r)

where the normalization factor N only depends on the location r as do the energies for the stable defect configurations.

Substituting this into eqn [70] on both sides constitutes another assumption. To see this, suppose that the defect concentrations CV(r — R, t) on all neighbor sites happen, at the particular instance t, to be aligned in one direction. Since their new

alignments after the jump to site r is correlated with the jump vector R and the previous orienta­tion, the added defect population does not possess the equilibrium distribution of eqn [71]. However, Kronmuller et a/.33 argue that after several sub­sequent jumps of defects from the neighbors to this site r, the earlier deviation from the equilibrium distribution will have died out. Thus, introducing the thermodynamic averages on both sides of the eqn [70] is a plausible approximation. To proceed further requires a more specific form of the jump probability. For a jump from the site r to r + R, it is assumed that	and a drift force as
bElv(r+1r)+bE (r)
[77]
Fi (r)
The components of the jump vector R are denoted by capital letters Xi, while the components of the loca­tion vector r are given by the lower case letters x. The normalization factor N(r) is replaced in the eqns [76] and [77] with an exponential function of the average defect formation energy according to
—bf (r) [78]
exp
It is important to emphasize, as Dederichs and Schroder34 first did, that the above Taylor expansion does not remove the dependence of the saddle point energy on the jump direction R. To what degree it still depends on the jump direction is a function of the crystal lattice and magnitudes ofthe elastic strains.
Read More 22.08.2015   Comprehensive nuclear materials Diffusion Coefficient The temperature dependence of the diffusion coeffi­cient has an Arrhenius form: D = D"exp( “її where Ha is the activation enthalpy of diffusion, and D0 is the diffusion prefactor that contains all entropy terms and is related to the attempt frequency for migration. When diffusion involves only an interstitial migrating from one interstitial site to an adjacent interstitial site, the activation enthalpy of diffusion is composed mainly of the migration enthalpy. In com­parison, for vacancy-mediated diffusion, dopants are trapped in substitutional positions and form a cluster with one or more vacancies. In such a situation, diffu­sion requires the formation ofthe cluster that assists in diffusion, migration of the cluster, and finally, the dissociation of the cluster. It is common for experi­mental studies referring to vacancy-mediated diffu­sion to refer to the activation enthalpy of diffusion. The activation enthalpy is the sum of the formation enthalpy and the migration enthalpy. The formation energy represents the energetic cost to construct a defect in the lattice (which may well require a com­plete Frenkel or Schottky process to occur). The for­mation energy of a defect, Ef (defect), is defined by Ef (defect) = E (defect) + qme — E j where Ef (defect) is the total energy of the supercell containing the defect; q, the charge state of the defect; me, the electron chemical potential with respect to the top of the valence band of the pure material; the number of atoms of type j; and m the chemical poten­tial of atoms of type j. It should be noted that in this definition, contributions of entropy and phonons have been neglected. The migration energy is the energy barrier between an initial state and a final state of the diffusion process. For a system with a complex poten­tial energy landscape, there are a number of different paths that need to be considered. 1.02.4 Summary Point defects are ubiquitous: as intrinsic species, they are a consequence of equilibrium, but usually they are far more numerous incorporated as extrinsic spe­cies formed as a consequence of fabrication condi­tions. Slow kinetics mean that impurities are trapped in ceramic materials, typically once temperatures drop below 800 K, although this value is quite material-dependent. The intentional incorporation of dopants into a crystal lattice can be used to funda­mentally alter a whole range of processes: this includes the transport of ions, electrons, and holes. As a result, diffusion rates and electrical conductivity can be manipulated to increase or decrease by many orders of magnitude.1,2 Other mechanical or radia­tion tolerance-related properties can also be changed radically. This chapter has provided the framework for understanding the properties of point defects. In par­ticular, the understanding of the concentration of equilibrium-intrinsic species, dopant ions and their interdependence, defect association to form clusters and nonstoichiometry. In each case, these defects alter the lattice surrounding them, with atoms being shifted from their perfect lattice positions in response to the specific defect type. Electronic defects have been described: not only electrons and holes formed by doping, but also states formed by excitation. Struc­tural defects and electronic defects are considered together through Brouwer diagrams. Finally, we have also considered the transport of ions through the lattice via different processes, all of which require the formation of point defects. Read More 22.08.2015   Comprehensive nuclear materials Mechanisms of Irradiation Hardening Irradiation introduces obstacles to dislocation motion, which results in plastic deformation, in the form of defects resulting from atomic displacement and from transmutation products. Small Frank loops and defect clusters, known as black dots, large Frank loops (about an order of magnitude larger), precipitates, and cavities (either voids or bubbles) contribute to hardening in an irradiated alloy. Frank loops unfault and eventually contribute to the network dislocation density. Precipi­tates are certainly present in the unirradiated alloy, but additional precipitation results from the segrega­tion of elements during irradiation and from the irradiation-induced changes that shift the thermody­namic stability of phases. Transmutation production of new elements in the alloy can also result in the forma­tion of new precipitates. The production of insoluble species, most importantly helium, also results in pre­cipitation, especially in the form of bubbles. Defects are divided into two classes: long range and short range. Short-range obstacles are defined as those that influence moving dislocations only on the same slip plane as opposed to long-range obstacles, which impede dislocation motion on slip planes not containing the obstacle.1 Coherent precipitates and large loops are long-range obstacles, but for this analysis, only network dislocations will be considered as long-range obstacles, a reasonable simplification from observations. As recommended by Bement,2 the contributions from short-range obstacles are added directly, AFTs = DFlr + DFsr [1] where the quantities in eqn [1] are total stress, long — range contribution to stress, and short-range contribu­tion to stress. The contributions from the short-range obstacles are added in quadrature as follows3: (AFsr)2 = (AFSMloop)2 + (AFlgloop )2 + (Afprecip)2 + (Afcavity)2 [2] where the term on the left represents the contribution from all short-range obstacles, and the terms on the right represent the stress contributions from small loops, large loops, precipitates, and cavities, either voids or bubbles. The contribution to hardening by network dislo­cations may be expressed by Tnet a Gb / Pd [3] where tnet is the increment in shear stress, G is the shear modulus, b is the Burgers vector, and pd is the dislocation density. The constant a is dependent upon the geometry of the dislocation configuration and is usually determined experimentally. However, Taylor has calculated a to be between 0.15 and 0.3,4 and Seeger has determined the value to be 0.2, incor­porating the assumption of a random distribution of dislocation directions.5 Short-range defects such as small and large Frank loops and precipitates are treated as hard impenetrable obstacles where disloca­tions bow around them by the Orowan mechanism. The stress increment is expressed by At = GbJ~NdJJi [4] where N is the defect density and d is the diameter. The constant b ranges between 2 and 4 as suggested by Bement2 or 6 as suggested by Olander.6 Voids and bubbles are also treated as hard obstacles using the same expression. Precipitates and bubbles have been observed in austenitic stainless steels to nucleate and grow together.7 In this case, the bubbles and precipi­tates are considered as one obstacle where the hard­ening increment is calculated assuming rod geometry using a treatment by Kelly expressed by8: 0.16GbvNd 6d 1 — pVNd n 3b — where the parameters are the same as for eqn [4]. From the previous discussion, it can be inferred that because the nature of the irradiation-induced defects determines the degree of hardening, and because the nature, size, and density ofdefects is a strong function of temperature, radiation strengthening will be a strong function of irradiation temperature. Figure 1 illustrates 1   Temperature (°C) Figure 1 Relative contribution to strengthening from irradiation-induced defects in the austenitic stainless steel, PCA, irradiated to 7dpa in the Oak Ridge Research Reactor. Reproduced from Grossbeck, M. L.; Maziasz, P. J.; Rowcliffe, A. F. J. Nucl. Mater. 1992, 191-194, 808. Neutron fluence (ncm 2)(E>0.1 MeV) Figure 2 Yield strength of 20% cold-worked type 316 stainless steel irradiated in the EBR-II. Reproduced from Fish, R. L.; Cannon, N. S.; Wire, G. L. In Effects of Radiation on Structural Materials; Sprague, J. A., Dramer, K., Eds.; ASTM: Philadelphia, PA, 1979; ASTM STP 683, p 450. Reprinted, with permission, from Effects of Radiation on Structural Materials, copyright ASTM International, West Conshohocken, PA.   strengthening from individual types of defects as a function of irradiation temperature for the austenitic stainless steel PCA.7 As can be seen from Figure 1, the black dot damage characteristic of low temperatures vanishes at temperatures over 300 °C as Frank loops emerge. Bubbles and precipitates also become major contri­butors to hardening above 200 °C. Read More 22.08.2015   Comprehensive nuclear materials Spallation Proton-Neutron Irradiations, SPNI High fluxes of neutrons can be generated by high- energy and current (power) proton beams via spall­ation reactions that fragment the atomic nuclei heavily in a heavy metal target (like W, Pb, and Hg). At 500 MeV, these reactions produce «10 neu­trons per proton. Applications of spallation sources include neutron scattering, nuclear waste transmu­tation, and driving subcritical fission reactors. A key challenge to developing advanced high-power, long- lived spallation source targets is the ability of structural alloys to withstand severe radiation dam­age, corrosive fluids, and mechanical loading. Most notably, radiation damage in spallation source irradia­tions, produced by the neutrons and protons, results in both high dpa and concentrations of transmutation products, including He and H (see Table 1). As a consequence, there have been and continue to be international programs on radiation effects in SPNI environments, beginning with a large program in Los Alamos Neutron Science Center (LANSCE) in 1996 and 1997,55 followed by a continuing SINQ (Swiss Spallation Neutron Source) Target Irradiation Program — STIP, started in 1998, that con­tinues to this day, at the Paul Sherrer Institute, in Switzerland, involving an international collaboration of ten institutions in China, Europe, Japan, and the United States.56,57 Because of the accelerator production of tritium tar­get application, the irradiation temperature LANSCE experiment was up to «164 °C. The highest damage levels, mostly produced by protons, were «12 dpa and «180 appm He/dpa.4 About 20 materials were irradiated in a variety of specimen configurations in this study. The maximum damage levels in the STIP-I to — IV irradiations56,57 were « 25 dpa and 2000 appm He. The corresponding temperatures ranged from 80 to 800 °C, but most specimens were nominally irra­diated between 100 and 500 °C. The temperatures directly depend on the high nuclear heating rates in the target, and both varied by ±15% during the 2-year irradiation; and, in the case of STIP-I and-IV, some capsules experienced a significant overtem­perature transient. The high heating rates also result in fairly large uncertainties in the temperatures of individual specimens. Note that the temperature control in the most recent STIP-V experiment was significantly better than that in previous studies. Over 60 elemental metals and alloys, ceramics, and composites have been irradiated in the STIP-I to — V, in the form ofminiaturized specimens for both micro­structural studies and mechanical testing, including tensile, fatigue, fracture toughness, and Charpy V-notch (CVN) measures of the DBTT. Some speci­mens were irradiated in contact with stagnant liquid Hg, PbBi eutectic, and Pb. The STIP database is discussed in Section 1.06.4 Read More 22.08.2015   Comprehensive nuclear materials Dislocation Sink Strength and Bias In order to obtain the sink strength, one first solves the steady state diffusion equation V-j = V2(DC) =0 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 2p J0 = над [°C — ОСЧ1 I130′ 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 Dislocation sink strength = -—P [131] £n(R/rA) where p is the dislocation density. When the drift term is now included, the defect current changes to J = ZJo [132] 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. 1.01.8.2.1 The solution of Ham 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 M 1 + V Vrel sin’ 3% 1 — V r 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 cos(n’) and In cos(n’),56 and the edge dislocation bias factor is then obtained 57 in the form Kn(rC/rd ) fn(rc/rd ) 1 [134] where the capture radius is defined as _ (1 + n)b mvrel| c 6я(1 — v) 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 long — range 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 4b 1 + v V rel 3p 1 — v y + h y — h (x + h)2 + (y + h)2 (x — h)2 + (y — h)2 Figure 25 Edge dislocation dipole. At distances r energy becomes Wl(r; «)«   : Jx2 + y2 >> h, this interaction ub 1 + n = sin2a —A Vrel23/2h —- [1371 3я 1 — n r2   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 = 2b and R/b = 2250, corresponding to a disloca­tion density of 1012m—2 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 > 1 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 _ ‘n(R/ro) £n(2R/rc) — g where g = 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 rCJb> 6. 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 * 1 + mR/^2 — °([rc/(2ro)]4) [141] 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   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 = 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   1 [138]   Ко(2гс/гр) Io(2rc/?d)   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   W1( rC; ±P   [139]   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 Capture radii and bias factors for interstitials and vacancies evaluated at half the melting points and with the size interaction only Element Vf/П VV0l/V rc/b SIA rc/b vacancies zd Zd ZV Net bias = Zd/Zd — 1 Al 1.9 —0.31 14.32 2.36 1.358 1.043 0.302 Cr 1.21 —0.22 8.31 1.11 1.229 1.011 0.216 Cu 1.55 —0.25 13.46 1.80 1.342 1.026 0.308 Fe 1.1 —0.27 8.69 1.94 1.239 1.030 0.208 Mo 1.1 —0.19 10.62 1.84 1.284 1.027 0.250 Nb 0.76 —0.27 5.08 1.24 1.135 1.013 0.120 Ni 1.8 —0.20 14.77 1.64 1.366 1.022 0.337 Ta 1.05 —0.25 6.43 1.56 1.177 1.020 0.154 V 1.03 —0.30 5.06 1.47 1.134 1.020 0.112 W 0.92 —0.18 14.91 1.70 1.369 1.024 0.337   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.   Figure 26 Edge dislocation bias factors based on Ham’s solution and various approximations to it.   Read More Календарь Ноябрь 2024 Пн Вт Ср Чт Пт Сб Вс   1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30   Рубрики 1 BIOFUELS A Look Back at the U. S. Department of Energy’s Aquatic Species Program: Biodiesel from Algae A. Worrall ABOUT THE AUTHOR ACCELERATOR DRIVEN SUBCRITICAL REACTORS ADVANCED BIOFUELS Advanced Biofuels and Bioproducts Advanced separation techniques for nuclear fuel reprocessing and radioactive waste treatment ADVANCES IN Advances in Biochemical Engineering/Biotechnology Advances in Biofuels Advances in Biorefineries Alcoholic Fuels Algae Energy All alternative energy Alternative transportation An Indispensable Truth An Introduction to Nuclear Materials AN INTRODUCTION. TO THE ENGINEERING. OF FAST NUCLEAR REACTORS ANNEX III. AHWR. Bhabha Atomic Research Centre, India ATOMIC ACCIDENTS BACKGROUND Bioenergy Bioenergy — Realizing the Potential Bioenergy from Wood BIOENERGY. RESEARCH:. ADVANCES AND. APPLICATIONS BIOETHANOL BIOFUEL'S ENGINEERING PROCESS TECHNOLOGY BIOFUELS BIOFUELS 1 Biofuels and Bioenergy Biofuels for Road Transport Biofuels from Agricultural Wastes and Byproducts BIOFUELS FROM ALGAE Biofuels Refining and Performance BIOGAS BIOGAS 1 BIOMASS — DETECTION, PRODUCTION AND USAGE Biomass and Biofuels from Microalgae Biomass Conversion Biomass Gasification and Pyrolysis BIOMASS NOW — CULTIVATION AND UTILIZATION BIOMASS NOW — SUSTAINABLE GROWTH AND USE Biomass Recalcitrance Biomasses identities Biotechnological Applications of Hemicellulosic Derived Sugars: State-of-the-Art Biotechnological Applications of Microalgae Biotechnology. for Fuels and Chemicals Cellulosic Energy Cropping Systems Comprehensive nuclear materials DESIGN FEATURES TO ACHIEVE. DEFENCE IN DEPTH IN SMALL AND. MEDIUM SIZED REACTORS Design of Reactor Containment Systems for Nuclear Power Plants ENERGY Energy Efficiency ENERGY SERVICES IN THE HINDU KUSH HIMALAYAN REGION Energy Storage Estimating Loss-of-Coolant Accident (LOCA) Frequencies Through the Elicitation Process EUROSUN 2008 EuroSun2004 EuroSun2008-10 EuroSun2008-11 EuroSun2008-13 EuroSun2008-2 EuroSun2008-3 EuroSun2008-4 EuroSun2008-5 EuroSun2008-7 EuroSun2008-9 EXAMPLES OF REACTIVITY-CONTROL SYSTEMS Exploitation des creurs REP Fast Breeder Reactor Programs: History and Status Fast Reactor Safety. (Nuclear science. and technology) Fertilization Frequency Response Testing. in Nuclear Reactors Fuels and Chemicals. from Biomass Gebaudetypologie Bayern Geothermal Energy GREEN BIORENEWABLE. BIOCOMPOSITES Handbook Nuclear Terms Handbook of biofuels production Handbook of Small Modular Nuclear Reactors Hydrogen — Fuel Cells Hydropower IAEA RADIATION TECHNOLOGY SERIES No Infrastructure and methodologies for the. justification of nuclear power programmes Innovative Biofibers from Renewable Resources Integral design concepts of advanced water cooled reactors Introduction to Nuclear Power Liquid Biofuels: Emergence, Development and LIQUID, GASEOUS AND SOLID BIOFUELS — CONVERSION TECHNIQUES Management von Biogas-Projekten Materials' ageing and degradation in. light water reactors Microbes and biochemistry of gas fermentation Modern Power Station Practice Natural circulation data and methods for advanced water cooled nuclear power plant designs Natural gas Neutron Scattering Applications and Techniques Nuclear and Radiochemistry Nuclear Back-end and Transmutation Technology for Waste Disposal NUCLEAR CHEMICAL ENGINEERING NUCLEAR ELECTRIC POWER Nuclear energy Nuclear Power and the Environment Nuclear power plant life management processes: Guidelines and practices for heavy water reactors NUCLEAR POWER PLANTS NUCLEAR POWER the PUBLIC Nuclear Reactor Design NUCLEAR REACTOR ENGINEERING Nuclear Reactors 1 NUCLEAR REACTORS 2 Others Particle Image Velocimetry (PIV) Passive Safety Systems and Natural Circulation in Water Cooled Nuclear Power Plants PHYSICS OF. HIGH-TEMPERATURE. REACTORS POWER Pretreatment Techniques for Biofuels and Biorefineries Principles of Fusion Energy PROCESS SYNTHESIS. FOR FUEL ETHANOL. PRODUCTION Production of Biofuels and Chemicals with Ionic Liquids Progress, Challenges, and Opportunities for. Converting U. S. and Russian Research Reactors Pumping Pyrolysis Radioactive waste management and contaminated site clean-up Recycling of Biomass Ashes Renewable Energy Products Rudiger Meiswinkel, Julian Meyer, Jurgen Schnell Second Generation Biofuels and Biomass Small modular reactors (SMRs) the case of Russia solar energy Solar Thermal and. Biomass Energy Sonar-Collecttors SonSolar Springer Series in Materials Science Study on Neutron Spectrum of Pulsed Neutron Reactor Sustainable Biotechnology Switchgrass Technologies for Converting Biomass to Useful Energy The Asian Biomass Handbook The Experimental Analyze Of The Solar Energy Collector The Future of Nuclear Power Thermal biomass. conversion and utilization —. Biomass information. system Thoria-based Nuclear Fuels Type of Membrane Contactor Suitable for CO2-O2 Exchanges in Culture Broth Vereinfachtes Verfahren WAS INTENTIONALLY LEFT BLANK Why We Need Nuclear Power wind power WORKSHOP ON NUCLEAR REACTION DATA AND. NUCLEAR REACTORS:. PHYSICS, DESIGN AND SAFETY Автономное энергообеспечение Альтерантивная энергия Альтернативные источники энергии и энергосбережение Без рубрики Биотопливо Ветровая энергия Ветрогенераторы, солнечные батареи и другие полезные конструк­ции Возобновляемые источники энергии ВОПРОСЫ ТЕОРИИ. И ИННОВАЦИОННЫХ РЕШЕНИЙ. ПРИ ИСПОЛЬЗОВАНИИ. ГЕЛИОЭНЕРГЕТИЧЕСКИХ СИСТЕМ ГЕЛИОЭНЕРГЕТИКА Другая энергия Зарядные станции Источники питания Источники энергии ИСУ Концентраторы солнечного излучения КРЕМНИЕВЫЕ СОЛНЕЧНЫЕ БАТАРЕИ Получение кремниевых пластин для солнечной энергетики Применение солнечной энергии Разное Солнечная электроэнергия Солнечная энергия Солнечная энергия — использование Солнечные батареи Солнечные коллекторы Солнечный ДОМ ТОНКОПЛЕНОЧНЫЕ. СОЛНЕЧНЫЕ ЭЛЕМЕНТЫ. НА ОСНОВЕ КРЕМНИЯ ФОТОЭЛЕКТРИЧЕСКОЕ. ПРЕОБРАЗОВАНИЕ. 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