Chapter 19 Voids and bubbles in metals
19.1 Introduction..............................................................................................................................1
19.2 The equilibrium bubble............................................................................................................2
19.3 Equations of state of helium and xenon .................................................................................4
19.3.1 Van der Waals EOS ................................................................................................4
19.3.2 Interatomic potentials ..............................................................................................4
19.3.3 Hard-sphere .............................................................................................................6
19.4 Nucleation and growth of cavities - the dislocation-bias model............................................8
19.4.1 Cavity sink strength.................................................................................................9
19.4.2 Coupling of diffusion and surface reaction during unsteady-state absorption of
point defects by cavities.......................................................................................16
19.4.3 Void nucleation.....................................................................................................21
19.4.4 Simplified point-defect balances..........................................................................25
19.4.6 Void growth..........................................................................................................27
19.4.7 Transition from nucleation to growth (example)..................................................28
References.....................................................................................................................................32
1
19.1 Introduction
Cavities play an exceedingly important role in the performance of nuclear fuel and its ancillary
structural components. In the fuel, cavities include the pores that remain following fabrication
and bubbles resulting from agglomeration of fission-gas atoms created during irradiation.
Although cavities do not develop in LWR cladding (Zircaloy), they appear during irradiation of
other irradiated metals and alloys used in reactor systems. In fast reactors and fusion devices,
high-energy neutrons displace atoms from their lattice sites and the remaining vacancies can
assemble into cavities with essentially no gas in them. Helium ions driven into the first wall of
fusion reactors agglomerate into tiny bubbles with concomitant degradation of mechanical
properties.
Cavities are characterized by the number of gas atoms they contain relative to the number of
vacancies needed to produce the cavity volume. When the cavity contains only vacancies but no
gas atoms, it is termed a void. At the other extreme are cavities containing rare-gas atoms at
densities characteristic of the solid form of the element. Figure 19.1 shows the continuum of
conditions between these two bounding cases.

Fig. 19.1 The variety of cavities in nuclear materials.
At the extreme left in this figure are voids in metals, which attain sizes easily measured by
transmission-electron microscopy (TEM). In this figure, they are a few tenths of a micron in
diameter, and are faceted, meaning that the surfaces are crystallographic planes of the lowest
surface energy. A true void has no gas in it. Moving to the right, the pores that remain in UO2
fuel following conventional fabrication methods (see Chap 16) can be quite large and contain
helium at roughly 1 atm (0.1 MPa). Helium is used as a cover gas during sintering. Because of
the low gas pressure, the pores are irregularly-shaped.
2
The third picture shows depressions in a grain boundary on a fracture surface of irradiated UO2.
These were formerly fission-gas-containing bubbles lying in the intact grain boundary of
irradiated fuel. They are termed intergranular bubbles and can attain diameters as large as a few
microns.
The fourth image in Fig. 19.1 shows tiny helium bubbles in the grain boundary of steel. The
source of the gas can be accelerator implantation or by (n,) on components of the alloy (usually
Ni). The gas contained in the bubbles on the grain boundaries of this photomicrograph and the
preceding one originated inside the grains as single atoms produced by the nuclear process. If
the temperature is sufficiently high, the dissolved gas diffuses towards the grain boundaries,
which are perfect sinks for the gas atoms.
The TEM image on the right shows nanometer-size fission-gas bubbles that have nucleated
within the grains of irradiated UO2. These are denoted as intragranular bubbles.
The pressure in the cavities shown in Fig. 19.1 increases from left to right. The arrows pointing
to each image originate at locations along the pressure line that approximate the pressure of the
gas in the cavity. The true void has zero pressure and the tiny intragranular bubbles can be
occupied by fission gas at pressures of hundreds of MPa.
19.2 The equilibrium bubble
The two diagrams in Fig. 19.2 describe the mechanical stress balance at the surface of a cavity
containing gas at pressure p. The diagram on the left demonstrates that the inward stress exerted
by the surface is 2/R, where R is the cavity radius and  is the surface tension of the solid
(N/m), or more commonly, the surface energy (J/m2). For UO2,  = 0.6 - 1 N/m; for iron,  ~ 2
N/m.
Fig. 19.2 Stresses at the surface of a cavity
The balance of radial stresses at the surface of a bubble is depicted in the right-hand sketch. Here
r(R) is the radial stress component in the solid at the surface of the bubble (positive in tension).
3
The lower scale in Fig. 19.1 shows the variation of cavity radius with gas pressure for a
particular situation called the equilibrium bubble. In this condition, the radial stress at the bubble
surface is equal to the hydrostatic stress in the bulk of the solid far from the bubble, or r(R) =
bulk, and the stress balance becomes:
p + bulk = 2/R
(19.1)
For cavities that are not in mechanical equilibrium with the bulk solid, Eq (19.1) is replaced by
the general case given by the equation below the right-hand diagram of Fig. 19.2. For the
nonequilibrium case, the stress increases or decreases with the inverse cube of the distance from
the bubble surface. The stress distributions for the three cases are depicted in Fig. 19.3.
Fig. 19.3 Stresses in the solid near a cavity
When p < 2/R - bulk, the cavity is pressure-deficient. p >2/R - bulk characterizes a pressureexcess cavity.
In Fig. 19.1, the voids and the pores are pressure-deficient, the intergranular fission-gas bubbles
in UO2 are probably close to equilibrium, and the small bubbles of He in steel and intragranular
fission-gas bubbles in UO2 are likely to be pressure-excess.
In general, a cavity is characterized by the number of gas atoms (n) and by the number of
vacancy volumes (m) it contains. The latter is related to the cavity radius by:
 3 
Rm  

 4 
1/ 3
m1 / 3
(19.2)
where  is the volume of an atom (or equivalently, the volume of a vacancy) 1 and Rm is the
radius of a cavity containing m vacancies.
In ceramics such as UO2,  refers to the volume of one U4+ and two O2- ions, which is a UO2 molecule. This
combination is required for electrical neutrality. With the density of UO2 equal to 10.98 g/cm3, the molecular
volume is  = [10.98(10-7)3(61023)/270]-1 = 0.041 nm3/molecule UO2.
1
4
The other factor controlling the properties of the cavity is n, the number of gas atoms it contains.
For a cavity of a specified radius, m is determined by Eq (19.2). The pressure inside the cavity
containing n gas atoms is fixed by the equation of state of the gas (actually, the fluid) along with
the temperature.
19.3 Equations of state of helium and xenon
An equation of state (EOS) provides the link between the number of gas atoms in a cavity (n),
the number vacancies (m), the temperature T and the pressure p. An EOS is generally expressed
in the functional form p(v,T), where v is the specific volume. According to Eq (19.2), the cavity
volume is m, so the specific volume is:
m
v
N Av
(19.3)
n
where NAv = 61023 is Avogadro's number (atoms per mole). The appropriate form of the EOS
depends on the pressure, or, equivalently, on the specific volume. At low pressure, the ideal gas
law applies:
RT
p
v
(19.4)
Where R = 8.314 J/mole-K is the gas constant. The pressure is in units of Pascals (N/m2), the
unit of temperature is Kelvins the specific volume is in m3/mole2.
19.3.1 Van der Waals EOS
At higher pressure, or smaller molar volume, a reduced form of Van der Waals equation is
generally employed:
RT
p
vb
(19.5)
The constant b is a property of the gas that accounts for the repulsive component of the
interatomic potential. The other constant in the Van der Waals equation reflects the attractive
portion of the potential. It is neglected for the present purpose because attraction between two
rare gas atoms is very small and because the repulsive portion of the potential dominates as the
gas becomes dense.
for He, b = 0.039 nm3/atom
for Xe, b = 0.085 nm3/atom (19.6)
Rather than deal with the p(v,T) form of the EOS, deviations from ideality are more readily
expressed in terms of the compressibility:
pv
Z
RT
(19.7)
ideal gas:
2
Z=1
Van der Waals gas: Z = (1 - b/v)-1
a more convenient unit for the specific volume is nm3/atom, which differs from m3/mole by a factor of
(10-9)3(6x1023) = 610-4
(19.8)
5
19.3.2 Interatomic potentials
The Van der Waals EOS suffices for modest deviations from ideality, but as seen from Eq (19.8),
the compressibility approaches infinity as v  b. However, very small pressure-excess bubbles
can reach atomic volumes significantly smaller than the values given in Eq (19.6). In these highdensity states, the collection of atoms is closer to a liquid than a gas, and the electron clouds of
neighboring atoms overlap. In this state, adjacent atoms strongly repel each other. This situation
is shown in Fig. 19.4.
Fig. 19.4 The interatomic potential and the hard-sphere approximation
Typical representations of the interatomic potential function include:

(r )  A  / r    / r 
Lennard-Jones potential
n
6

(19.9a)
 and A are properties of the gas, whereby the second term in the brackets accounts for dipoledipole interactions.3 The first term empirically represents the repulsive portion of the potential
due to overlapping electron clouds. The exponents n controls the steepness of the repulsive
potential. For the rare gases, a typical value of n is 12.
Morse potential
(r) = E{exp[-2(r-re)] - 2exp[-(r-re)]}
(19.9b)
the parameters are , re and E.
Buckingham Potential
(r ) 
 
 min 
r

6 exp 1 
6
  rmin

6


r  
   min  
 r  


(19.9c)
min and rmin are the value and position of the potential minimum (point A in Fig. 19.4), and  is
a measure of the steepness of the repulsive portion.
3
separation of positive (nucleus) and negative (electrons) charges in an atom creates a dipole. Such separation
occurs continuously, setting up a fluctuating dipole. Interaction of the dipole moments of adjacent atoms creates an
attractive force between the two.
6
The repulsive components of the above interatomic potentials rise rapidly with decreasing
separation of the two atoms. The extreme expression of this feature is the hard-sphere
potential, which is zero until a separation  at which point the potential becomes infinite. This
simplification is shown by the heavy line in Fig. 19.4. Physically,  is the hard-sphere diameter
and as shown in the inset of the figure, corresponds to the minimum separation of the nuclei of
the two atoms and where the interatomic potential changes sign. This interatomic potential can
be written as:
Hard-sphere potential
 =  for r < 
 = 0 for r > 
(19.9d)
This potential has only a single parameter, which is related to the minimum specific volume, vo,
that the rare gas can achieve. This minimum occurs when the gas has been compressed into a
liquid or a solid.
Suppose that the solid adopts an fcc structure (Fig. 3.1). When the atoms are compressed so that
they "touch", the structure is shown in Fig. 19.5
Fig. 19.5 Atoms at maximum compression in an fcc lattice structure
The hard-sphere diameter is drawn along a face diagonal between the centers of neighboring
atoms. In terms of the lattice constant ao,  = ao/ 2 and the effective volume per atom is
v o  a 3o / 4 . (see footnote4 ). Eliminating ao between these two equations gives:
v o  3 / 2
(19.10)
If the minimum specific volume vo can be measured (e.g., by X-ray diffraction), the hard-sphere
diameter follows from Eq (19.10).
19.3.3 Hard-sphere EOS
It would seem to be a straightforward matter to determine the EOS once the hard-sphere
diameter is given (usually as a fitting parameter). However, such is not the case; there is a vast
literature seeking to do just this. The hard-sphere model is directly connected to the virial EOS,
which, in terms of the compressibility as a function of atomic volume v, is written as:
Z = 1 + B2/v + B3/v2 + .............
4
(19.11)
The factor of 4 is the number of atoms in the unit cell: 8 corner atoms shared among 8 unit cells gives 1 atom to
each; 6 face-centered atoms each shared with another unit cell gives 3 atoms. The total is 4 atoms per unit cell.
7
Detailed calculations that permit the coefficients Bn to be expressed in terms of the hard-sphere
diameter yield:
B2  2 3 3
2
2
B3/B = 0.625
3
2
B4/B = 0.287
4
2
B5/B = 0.110
(19.12)
5
2
9
2
B6/B = 0.0389 .... B10/B = 0.000404
The coefficients up to B6 were calculated in 1964 and it was not until 2006 that terms up to B10
were calculated. The significant effort expended to determine these coefficients stems from the
limitation of the series; it works only for the gas phase up to the v-3 term. To accurately
reproduce the EOS (in the form of Z vs v) for the liquid state requires many more terms.
When Eq (19.11) is rewritten in terms of the dimensionless variable:
y = B2/4v
the result is:
Z = 1 + 4 y + 10y2 + 18.36y3 + 28.22y4 + 319.81y5 + ........ 105.8y9 + ......
(19.13)
In a very fortunate mathematical discovery [1], the coefficients of yn, when rounded off to 4, 10,
18, 28, 40, were found to be reproduced by n2 + 3n. The above equation then can be extended to
an infinite series:

Z  1   (n 2  3n ) y n
n 1
which has the following closed form:
Z
1  y  y 2  y3
(1  y) 3
(19.14)
combining Eqs (19.12) and (19.13), the y variable is expressed as:
 3
y
6 v
(19.15)
So, if the hard-sphere diameter is known, the equation of state from the ideal-gas region to the
dense liquid region can be computed. Usually,  is picked to provide the best fit of Eqs (19.14)
and (19.15) to available EOS data. The best estimates are:
Helium
 ~ 0.20 nm
Xenon
 ~ 0.36 nm
(19.16)
The approximately-equal-to sign in these values reflect the need to estimate  by fitting to the
equation-of-state.
Figures 19.6a and 19.6b show the EOS of helium and xenon. For the Van der Waa1s EOS, v in
Eq (19.8) is expressed in terms of y by use of Eq (19.15). The Van der Waals equation fails well
before the specific volume attains the dense gas or liquid regime. In Fig. 19.6b, the analysis by
Ronchi [2] is included. The temperature dependence of this EOS results from use of a perturbed
hard-sphere model and the Lennard-Jones interatomic potential function for Xe; agreement with
the Carnahan-Starling (C&S) EOS is very good.
8
Example: Using Eq (19.1) with bulk = 0, the pressure in 2-nm-radius fission-gas bubble in equilibrium
with stress-free UO2 is (21)/210-9 = 109 Pa = 1000 MPa. At 1000oC, the specific volume of xenon (v) is
determined from the Carnahan-Starling EOS as follows. The unknown v is made dimensionless by:
x = v/3
With  taken from Eq (19.16), the compressibility is:
Z
p 3
109 (0.36 10 9 ) 3 (6 10 23 )
x
x  2.65x
RT
8.314 1273
and from Eq (19.15), y = 0.52/x. Solving Eq (19.14) by trial-and-error yields x = 1.65 and Z = 4.36. The
specific volume v = 1.65(0.36)3= 0.077 nm3/atom, which is slightly smaller than the non-ideality
constant in the Van der Waals EOS (Eq (19.6)). However, the VdW EOS is clearly not applicable to this
condition.
From Eq (19.2) and  = 0.041 nm3/vacancy, the 2-nm-radius bubble is equivalent to m = 820 vacancies.
From Eq (19.3) (omitting NAv), the bubble contains 435 xenon atoms.
80
(a)
140
Compressibility, Z
60
Compressibility, Z
(b)
Carnahan-Starling
Ronchi - 800 K
Ronchi - 1400 K
Van der Waals
120
40
VdW
C&S
20
100
80
60
40
20
0.0
0.1
0.2
0.3
0.4
0.5
0.6
y = 3 /6v
0.7
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
y = 3 /6v
Fig. 19.6 Equations of state for helium (a)
and xenon (b)
19.4 Nucleation and growth of cavities - the dislocation-bias model
Nucleation can be either homogeneous or heterogeneous. The latter refers to the formation of the
condensed phase on disturbances in the medium; the trails of bubbles along the track of a
nuclear particle in a bubble chamber is a well-known example of heterogeneous nucleation. The
most common example of homogeneous nucleation is the formation of droplets of water from
clouds in the atmosphere. An essential feature of these examples is that a single species is
responsible for nucleation of the second phase. In irradiated solids, both mechanisms create the
0.8
9
cavities shown in Fig. 19.1. However, contrary to second-phase nucleation in unirradiated solids,
two species are responsible for the addition of volume to the cavities in materials subject
to irradiation by neutrons or fission fragments: the vacancy increases the volume and the
interstitial decreases it. Both of these point defects are created simultaneously by irradiation and
the relative rates of their arrival at the
embryo cavities determines the rates of nucleation and growth. The arrival rates depend upon
two characteristics of the system: the rates of creation of the two types of point defects, and the
efficiency, or strength of the sinks whereby they are eliminated from the microstructure.
The earliest treatments of cavity nucleation in irradiated metals were directed at the explanation
of the development of the population of voids. Since equal numbers of vacancies and interstitials
are produced by irradiation, it was natural to assume that equal numbers were available for
elimination at sinks in the microstructure. However, if the rates of absorption of vacancies and
interstitials are equal, nucleation and growth of cavities could not occur.
In the simplest picture, the microstructure is assumed to consist of cavities and network
dislocations. Cavities exhibit no preference for either point defect, and are termed neutral sinks.
Because of the stress fields in the vicinity of the interstitial and the dislocations, the two are
slightly more strongly attracted to each other than are the vacancies to dislocations. This
preference is expressed quantitatively as a bias factor. The resulting theory, termed the
dislocation-bias model, was initially developed in the 1970s [3,4,5]. In order to quantitatively
express nucleation and growth of cavities, a property called the cavity sink strength is required.
This is a measure of the rate at which the cavity absorbs point defects from the surrounding solid.
In irradiated solids, the mobile species are vacancies (Vs) and interstitials, or more specifically,
self-interstitial atoms (SIAs). In addition, high-energy neutrons in metals generate helium by
(n,) reactions and thermal-neutron irradiation of uranium dioxide produces xenon and krypton
by fission of 235U. These rare gases are totally insoluble in the solid in which they were created,
and so are readily absorbed by the cavities generated by the point defects. In metals, the cavities
are termed voids even though they invariably contain some gas whose presence significantly
accelerates nucleation. This three-species (V, SIA and He) nucleation process is very difficult to
treat theoretically, and in the following analysis, the effect of the gas is neglected.
19.4.1 Cavity sink strength
A cavity can consist of anywhere from a few vacancies to a sufficiently-large number to be
observable by electron microscopy. The behavior of cavities in irradiated solids depends upon
the rates at which various extended defects (e.g., cavities, dislocations, precipitates, grain
boundaries) absorb the point defects created by the high-energy displacement process. This topic
has been treated in Sect. 13.4, but in this section, attention is focused on how cavity absorption
of point defects affects the processes of nucleation and growth.
The rates (per cavity) at which point defects are absorbed by size-m cavities are:
- vacancies: rate =  D V (C V  C surf
(19.17a)
V )
- interstitials: rate =  D I C I
(19.17b)
DV, DI = diffusivities of V and SIA, m2/s
CV, CI = volumetric concentrations of V and SIA in the bulk of the solid, m-3
10
 = sink strength of the cavity , m
C suef
= vacancy concentration in solid adjacent to cavity, m-3
V
5
The rate can be controlled by diffusion from the bulk, by "reaction" at the surface of the cavity
or by a combination of the two. The individual sink strengths for these two mechanisms are
given below and the overall sink strength for the two processes acting in series is described in
Sect. 9.4.2. The driving force for this process is the difference between the point-defect
concentration in the bulk solid and that adjacent to the surface of the cavity.
Diffusion Control
For a rate completely controlled by diffusion, m represents control of the point-defect sink
strength by diffusion in the medium surrounding the cavity. In the diffusion-controlled case, the
reaction rate at the cavity surface is assumed to be very fast, so that the V concentration at the
 C eq
cavity surface is the equilibrium value, C surf
V
V The latter is the equilibrium value consistent
with the stress state of the solid here (for interstitials, C eq
I  0 ). Comparing Eq (19.17a) to Eq
(13.27) shows that the sink strength for steady-state diffusion-controlled vacancy absorption by
cavities is:
diff  4R m
(19.18)
where Rm is the radius of a cavity equivalent to m vacancies (from Eq (19.2).
Because the cavity is a neutral sink, Eq (19.18) applies equally well to interstitial absorption by
cavities.
Reaction Control
In this limit, diffusion from the bulk is very fast and the point-defect concentration at the cavity
 C V . Reaction-rate control refers to the absorption mechanism
surface is the bulk value, or C surf
V
by which point defects jump from supply sites into capture sites surrounding the cavity. Once a
point defect hops into one of the latter sites, absorption by the cavity is assured. Diffusion of the
point-defect around the cavity is not involved; only the concentration of point defects within
jumping distance of the capture sites affects the rate.
Other quantities required for the computation of the sink strength include:
1. The number of vacancies in the cavity.
2. The probability that a supply site contains a point defect. If the volumetric concentration of Vs
in the solid is CV. The vacancy site fraction is CV, where  is the atomic volume, or,
equivalently, the volume of a vacancy ( a 3o / 4 in the fcc lattice).
 is related to the conventional designation of sink strength by:
concentration of size-m cavities, m-3
5
k 2m   N m , where Nm = volumetric
11
The rate at which a vacancies from the supply sites become attached to (or "react" with) the
capture sites and thereby become incorporated into the cavity is:
rate 
V
supply sites jumps reactions
reactions



 ( C V )  Z  w  1
supply site
cavity
V s
jump
cavity  s
w is the probability per unit time that a V jumps a particular nearest-neighbor site. It is the "oneway" jump frequency described in Sect. 4.4 and defined by Eq (4.26). The relation of w to the
vacancy diffusivity DV is given by the Einstein equation, Eq (4.28). In the fcc lattice, the V jump
distance is = ao/2, and each V site is surrounded by 12 equivalent atom sites into which it can
jump (total jump frequency =12w). The connection between w and DV is:
DV 
1
6
2  
1
6
a
2
o

/ 2 12w   a o2 w
Eliminating w from the above two equations yields:
rate  Z( / a o2 )D V C V
Comparing this equation to Eq (19.17a), the sink strength of a cavity is:
 react  Z( / a o2 ) 
1
4
Za o (fcc )
(19.20)
The parameter Z (called the combinatorial number) is the number of single-jump links between
the supply sites and all of the capture sites surrounding the cavity. This parameter is best
determined by representing the lattice as a series of spherical shells around a site chosen as the
origin.
Shell representation of the lattice
As with diffusion-controlled absorption of point defects, the cavity is regarded as an empty
sphere. However, since the mechanism explicitly models hopping of point defects in the vicinity
of the cavity, the capture sites and the supply sites surrounding the cavity must be identified. For
this, the lattice sites surrounding the cavity are apportioned into spherical shells; nearestneighbor, next-nearest-neighbor, etc. The shells are designated by the index "n", with n = 0 being
the center of the cavity. The shell radius is denoted by Rn and the number of sites in the shell is
jn. The dependence of Rn and jn on shell number depends on the lattice type.
The first six shells of the fcc structure are depicted in Fig. 19.7. In each of these diagrams, the
gray cubes represent the unit cell, with the shell's origin shown at the lower-left corner. Complete
representation of the shell requires inclusion of all eight unit cells that share the origin. Instead,
all of the sites in the complete shells are displayed (minus the unit cells) to the right of the unit
cells. The origin is located at the center of each shell.
For the fcc lattice, the shell radius as a function of shell number is given by:
Rn / ao  n / 2
(19.21)
(19.19)
12
The number of sites in each shell (jn) requires counting all atoms in the eight unit cells that share
the small cube at the origin. The unit cell for shell no.1 (upper left in Fig. 19.7) contains three
atoms at the nearest-neighbor distance from the origin. However, each of these atoms is shared
between two adjacent unit cells, so the first shell contains j1 = 3 x 1/2 x 8 = 12 sites.
In the second shell (n = 2), each of the 3 sites in the unit cell shown in the figure is shared with 3
other unit cells (not shown) so that the number of sites is: j2 = 3 x 1/4 x 8 = 6. The remaining
sites per shell are determined by analogous counting.
The two characteristics of a shell, Rn and jn, are plotted in Fig. 19.8. The discrete shell radii
shown as points in the upper graph increase with shell number according to Eq (19.20). There are
no sites with radii between these points. However, as shown in the lower plot, the number of
atoms per shell varies widely with shell number; for example, shell 13 contains 72 atoms but
shell 14 is empty6
Two structural parameters need to be specified in order to determine the reaction-rate sink
strength of a cavity.
1. The number of shells surrounding the cavity that contain capture sites. For example, if the
cavity surface corresponds to shell no. 8, shell no. 9 probably also provides capture sites; that is,
if a V jumps into a site in shell no. 9, a radial force pulls it first into shell 8 and then into the
cavity. This long-range attraction is due to the stress fields surrounding the cavity and the nearby
point defect. The latter, whether a V or an SIA, is energetically more stable inside the cavity than
isolated in the bulk solid. Just how far from the cavity surface this attraction persists can only be
determined by detailed atomic-scale computer simulation (Chap. 14). However, the probability
that a shell contains capture sites decreases rapidly with its distance from the cavity surface.
2. The sites outside the ring of capture sites from which a point defect can reach a capture site in
one jump. Such sites are termed supply sites. The number depends on the jump distance of the
point defect (e. g. ao/2 for a vacancy in the fcc lattice) and mk,n, the number of capture sites in
shell n accessible from supply site k. Suppose that the capture sites around a cavity are
surrounding by ktot supply sites. The total number of routes by which Vs in supply shells can
jump into the capture shells is:
Z   m k ,n jk
n tot k tot
(19.22).
Applications of Eq (19.22) to a single vacancy, a divacancy, and a 13-vacancy cavity are shown
below.
A single vacancy treated as a cavity
Although all vacancies are equivalent, for the present analysis one of them, termed the sink
vacancy, is considered to be fixed and to act as a cavity. Furthermore, we assume that only the
sites in the nearest-neighbor shell (shell no. 1) around the sink vacancy contains capture sites.
6
The shell radii are defined by Eq (19.21), but this does not guarantee that every shell contains atom sites
13
Referring to the top left-hand unit cell in Fig. 19.7, the capture sites are the face-centered
positions nearest to the origin (shell No. 1). As shown in Fig. 19.9, shells nos. 2, 3 and 4 are
supply shells. The arrows emanating from one of the sites in a supply shell represent the
allowable jumps into the capture shell.
- each of the sites in supply shell no. 2 (j2 = 6) accesses 4 shell-1 capture sites (m2,1 = 4)
- each of the sites in supply shell no. 3 (j3 = 24) links to 2 shell-1 capture sites (m3,1 = 2)
- supply shell no. 4 (j4 = 12) can reach only 1 shell-1 capture site (m4,1 = 1)
Point defects in supply shells further out than shell 4 cannot reach shell no.1 in a single jump of
length ao/2. Using Eq (19.22), the combinatorial number for the sink vacancy is:
Z = 46 + 224 + 112 = 84
The combinatorial number 84 depends upon the assumption concerning the number of shells
acting as capture sites around each vacancy. The actual number probably includes sites in shell
no. 2 as well as those in shell no. 1, so the combinatorial number could be considerably greater
than 84. The same equation applies to the V - SIA reaction, although the combinatorial number
would be different.
R4 / ao  2
R1 / a o  1/ 2
R5 / ao  5/ 2
R3 / ao  3/ 2
R6 / ao  3
Fig. 19.7 Nearest-neighbor shells in the fcc lattice.unit cells (in gray) and atom structures
of shells (right). The side length of the cubes is the lattice parameter ao.
34
14
3.5
2.5
2.0
1.5
1.0
0.5
0
2
4
6
8
10
12
14
12
14
16
18
shell number, n
80
70
60
number in shell, jn
shell radius, Rn/ao
3.0
50
40
30
20
10
0
2
4
6
8
10
16
shell number, n
Fig. 19.8 Properties of atom shells in the fcc lattice
15
35
Fig. 19.9 Supply-shell routes to capture sites in shell no. 1 in the fcc structure
Divacancy sink
Figure 19.10 shows a vacancy pair in the (100) plane pointing in the <110> direction. The (100) plane in
the figure consists of four of the faces of the unit cell. The divacancy in the fcc structure occupies corner
and face-centered positions. In this plane, the divacancy could be oriented in any of the four <110>
directions in this plane. Counting the three {100} planes shown in Fig. 19.10, there are 12 possible
orientations for the divacancy. In the 12-site diagram in the
Fig. 19.10 A divacancy in the fcc structure
upper-left-hand corner of Fig. 19.7, one vacancy is at the center (as shown) and the second vacancy
occupies any one of the 12 positions in shell no. 1. In fact, the second vacancy moves readily from one
site to another in this shell. In this sense, the divacancy appears as a spherical cavity of radius R1.
Calculation of the combinatorial number Z of the divacancy is not done here; it is significantly larger
than the value of 84 for a single vacancy.
16
The 13-vacancy cavity
The cavity includes the central vacancy and the 12 vacant sites of shell no. 1. For this cavity, the
capture shells are assumed to consist of the cavity surface (shell no. 2) and the adjacent shell
36
(no. 3). The paths for vacancies to reach the two capture shells from the surrounding the supply shells
(nos. 4-7 ) are determined in a manner similar to that used for the single-vacancy sink. Diagrams of the
shells are shown in Fig. 19.11. Table 19.1 summarizes the resulting values of qn. The combinatorial
number is given by:
6
7
k 4
k 4
Z   m k , 2 jk   m k ,3 jk
(19.23)
The first summation accounts for the links from supply shells nos. 4 - 6 to capture shell no. 2. The
second sum covers the paths from supply shells 4 - 7 to capture shell no. 3. Supply shell
no. 7
has a single link to capture shell no. 3, but no path to capture cell no. 2. Also, supply shell no. 4 has no
pathway to capture cell no. 2. Adding the numbers in the last column, the combinatorial number is Z =
192.
Table 19.1 Supply and capture cells for a 13-vacancy cavity in the fcc structure
Supply shell (jk)
4 (12)
5 (24)
6 (8)
7 (48)
4 (12)
5 (24)
6 (8)
7 (48)
Capture shell (n )
2
2
2
2
3
3
3
3
Links (mk,n)
0
1
0
0
4
2
3
1
mk,njk
0
24
0
0
48
48
24
48
19.4.2 Comparison of reaction-controlled and diffusion-controlled sink strengths
At steady state, diffusion-controlled absorption of vacancies by a cavity is given by a combination of
Eqs (19.2) and (19.18):
 diff / a o  (12 2 )1 / 3 m1 / 3
while for reaction control:
react/ao = 84 for m = 1
and
(19.24)
react/ao = 192 for m = 13
Figure 19.12 compares the sink strengths for diffusion control from Eq (19.24) with the two points
calculated above for reaction control. This plot shows that the absorption process is diffusion-controlled
for all cavity sizes, including m = 1. If this were true, calculation of the reaction-controlled sink
strengths would be a useless exercise. The fallacy behind this interpretation lies in the underlying
assumption of steady-state. In order to properly couple the reaction- and diffusion-control steps, the
system must be treated in the unsteady-state condition.
17
37
Fig. 19.11 Supply and capture sites for a 13-vacancy cavity in the fcc structure
The conventional diffusional sink strength of a caavity is obtained by solving the steady- diffusion
equation, Eq (13.22) with the boundary conditions of Eq (13.23). The absorption rate by all cavities in a
unit volume of solid is given by Eq (13.27), from which the sink strength of a single cavity, Eq (19.18),
is obtained.
18
38
Fig. 19.12 Steady-state cavity sink strengths for surface-reaction and diffusion rate control
To show where reaction-control is important, consider the transient that follows the binding of two
vacancies (treated as a cavity) to form an immobile divacancy. This is equivalent to the appearance at t =
0 of a small cavity in a solid with a uniform vacancy concentration. Thereafter, growth occurs as
vacancies are absorbed by the cavity at a rate influenced by both the surface reaction and diffusion. The
problem is simplified by assuming that the cavity radius remains constant in time despite the accretion
of vacancies. This is not as serious an assumption as it first appears because the simultaneous absorption
of SIAs (which are not included in the present analysis) reduces the net volume addition to the cavity by
about 98%.
Vacancy diffusion from the bulk of the solid to the cavity surface is governed by:
 CV
1   2  CV 
r

 DV 2
t
 r 
r  r 
with the initial condition:
and the boundary condition:
CV = C bulk
at t = 0, all r
V
(19.25a)
(19.26a)
CV = C bulk
at r = , all t
V
(19.27a)
The boundary condition at the cavity surface equates the rate at which Vs arrive at the cavity surface to
the rate at which they are incorporated into the cavity by the surface reaction:
  CV
 react D V Csurf
  4R 2 D V 
V
 r
where R is the radius of the divacancy cavity and C
(i.e., in the supply shells).
surf
V


R
(19.28)
19
is the vacancy concentration at the cavity surface
39
The mathematical analysis is simplified if the system is made dimensionless with:
  C V / C bulk
V
 r/R
  DV t / R 2
(19.29)
which converts Eq (19.25a) to:
 1   2  



  2      
=1
and the initial condition to:
=1
the boundary conditions become:
h 
and


h
where
(19.25b)
at  = 0, all 
at  = , all 
at  = 1
(19.26b)
(19.27b)
(19.28b)
 react  react

4R  diff
(19.30)
The sink strengths in Eq (19.30) are the steady-state values given Eqs (19.18) and (19.20).
The solution is given in Ref. 6, from which the dimensionless vacancy concentration at the cavity
bulk
surface (C surf
V / C V ) is obtained as:


2
1
1  h e ( h 1)  erfc (h  1) 
h 1
The rate at which a cavity absorbs vacancies is:
(1, ) 

bulk
rate   react D V C surf
  react D V C bulk
V
V (1, )  D V C V
Using Eq (19.31), the time-dependent sink strength in Eq (19.32) is found to be:
   react (1, )  SS 1  h  F()
where

F()  e ( h 1)  erfc (h  1) 
2

(19.31)
(19.32)
(19.33)
(19.34)
gives the time dependence of the sink strength and SS is the cavity sink strength for mixed-rate control
at steady-state:
 SS
 1

1
 react  

h  1   diff  react



1
(19.35)
20
Initially, F(0) = 1 and  = react; or, when the cavity first appears in the uniform sea of vacancies, V
absorption is totally reaction-rate controlled.
For large times, F() = 0 and  = SS, meaning that steady-state has been achieved and the cavity's sink
strength reflects the series resistances of diffusion and surface reaction. If the reaction-rate sink strength
react is much larger than the diffusion-controlled sink strength,
Eq (19.35) shows that SS =
40
diff. This cavity sink strength, expressed by Eq (19.18), is used in most models involving point defects
interacting with cavities.
Example: What is the time-dependence of the sink strength?
The time variation of the cavity sink strength is calculated from Eqs (19.33) and (19.34).
First, dimensionless time  is converted to real time t by Eq (19.29). Taking react and diff from Fig. 19.12 for a
divacancy cavity (m = 2), the time-dependence of the sink strength  can be calculated using the above equations.
To determine the rate of vacancy absorption by the cavity using Eq (19.32), the diffusivity is expressed in the
usual form:
 E 
(19.36)
D V  D oV exp   m 
 RT 
R = 8.314 J/mole-K is the gas constant and T is the temperature in Kelvins. D oV is the pre-exponential
factor and Em is the migration energy barrier. These are listed in Table 19.2 and the last column gives
the resulting DV at 900 K. Unfortunately the values of these parameters are far from self-consistent.
Restricting attention to 316 stainless steel and nickel, at 900 K, two sources give DV = 6104 nm2/s and
another pair are centered on 5106 nm2/s.
These two values, and R = 0.2 nm by Eq (19.2) for m = 2, are used in Eq (19.29) to convert
dimensionless time to real time The dimensionless sink strength in the ordinate of Fig. (19.12) is
converted to diff and react with ao = 0.36 nm for steel. The result is depicted in Fig. 19.13 for the above
two diffusivities. The diffusion-controlled steady-state sink strength is attained in a very short time
following formation of the divacancy void embryo.
Table 19.2 Vacancy diffusion coefficients
Ref.
Metal
D oV , nm2/s
Em, kJ/mole
3
9, 12
13
16
19
22
Ni
316 SS
316 SS
?
Cu
316 SS
31012
81013
11013
21012
11013
11012
133
134
105
67
77
126
DV(900),
nm2/s
6104
1.3106
8106
3108
3108
6104
21
41
time, s
1e-14 1e-13 1e-12 1e-11 1e-10 1e-9 1e-8 1e-7 1e-6 1e-5 1e-4
10
react
sink strength , nm
8
DV = 5x106 nm2/s
6
4
2
DV = 6x104 nm2/s
diff
0
1e-11
1e-10
1e-9
1e-8
1e-7
1e-6
1e-5
1e-4
1e-3
time, s
Fig. 19.13 Sink strength of a divacancy cavity at 900 K for two values of DV
19.4.3 Void nucleation
Nucleation consists of the net absorption of Vs by very small cavities, or by the formation of divacancies
as void embyos. This process is described by a flux in size space and is illustrated in Fig. 19.7. This flux
(J) is not to be confused with the standard meaning of flux as the rate at which particles pass through a
unit area. Rather, it is akin to the slowing-down density in reactor physics, whereby neutrons move
downward in energy space.
The figure shows the three processes that drive the flux J, which is the rate per unit volume at which
clusters pass from size m to size m+1. The V and I absorption rates are given by
Eqs
(19.17a) and (19.17b), respectively.
The vacancy emission process shown in Fig. 19.14 is driven by the equilibrium vacancy concentration at
the cavity surface. Expressing Eq (3.4) as a volumetric concentration of Vs,
 2

 2  
eq



C eq
 p    (C eq
V  (C V ) o exp 
V ) o exp 
R
kT
R
kT
 
 m

 m
(19.37)
The surface tension stress in Eq (19.1) has been added to the -p term in Eq (3.4). The internal pressure p
is due to gas trapped in the cavity. This feature is neglected in what follows (i.e.¸ p = 0), which is
therefore valid only for a true void. (C eq
V ) o is the volumetric vacancy concentration in the solid in the
absence of stress.
22
The three processes in size space represented by the arrows are:
42
- vacancy absorption converts size-m cavities to size m+1
- interstitial absorption changes size m+1 cavities to size m
- emission of vacancies by cavities of size m+1 produces cavities containing m vacancies
The flux in size space is given by:
J =  m D V C V N m - [  m1D I CI N m1 +  m 1D V C eq
V N m 1 ]
(19.38)
For very small clusters, the growth and shrinkage terms in this equation are individually much larger
than their difference, which permits of the approximation J = 0. Equation (19.38) then becomes a
recursion formula:
mDVCV
N om 1 
N om
eq
 m 1 (D V C V  D I C I )
(19.39)
Fig. 19.14 Flux of clusters (cavities) in size space
The superscript indicates that this distribution applies to the case J = 0. For this reason, the above
formula is called the constrained distribution. Assuming the sink strengths of the void clusters are
diffusion-controlled, from Eqs (19.18) and (19.2):
o
m
R
 m 
 m 

 m1 R m1  m  1 
1/ 3
(19.40)
In terms of m, Eq (19.39) is:
1/ 3
N om1
 m 


m 1


N om
exp(  m 1 / 3 ) / SS  arr
(19.39a)
23
where
S S  C V /( C eq
V )o
is the vacancy supersaturation of the solid and:
(19.41)
43
arr 
DICI
DVCV
(19.42)
is the arrival-rate ratio, so named because it closely approximates the ratio of the fluxes of SIAs and Vs
to cavities or clusters.
The parameter  contains the effect of the C eq
V term in Eq (19.39). The latter is given by
Eq (19.37) and Rm is expressed in terms of m by use of Eq (19.2). The result is:
 4   2   2 / 3
  

 3   kT 
1/ 3
(19.43)
For most solids, the surface tension  is between 1 and 2 J/m2, so kT/2 is approximately 0.005 and 0.01
nm2. Using the atomic volume of typical metals,   0.01 nm3,   8 - 16.
For m = 1 and N1o  C V , Eq (19.39a) is:
N o2
2 1 / 3
 
C V e / SS  arr
The general formula is:
N om
m 1 / 3

1 / 3
1 / 3
C V (e  / SS  arr )(e 2  / SS  arr )......( e ( m 1)  / SS  arr )
(19.44)
Figure 19.15 is a plot of Eq (19.44) for various values of SS and arr. The curves decreases with
increasing m until a minimum is reached at a critical size mmin. For larger cavities, Eq (19.44) no longer
applies; nucleation has finished and growth has begun.
In order to calculate the nucleation rate with the aid of Eq (19.44), Eq (19.38) is rewritten as:


D C eq  D I C I

 m D V C V  N m  m1 V V
N m1 
m
DVCV


J=
The coefficient of Nm+1 can be expressed by Eq (19.39), giving:
N
N 
J   m D V C V N om  om  om1 
 N m N m1 
(19.45)
Rearranging and summing:
 (
m 1
m
N
N 
N om ) 1  D V C V   om  om1 
N m1 
m 1  N m
24
Expanding the infinite sum on the right-hand side:
J

 Nm
  N
m 1

o
m

N m1   N1 N 2   N 2 N 3 
N
N
   o  o    o  o   ......  1o  o  1  0  1
o
N m1   N1 N 2   N 2 N 3 
N1 N 
44
The last equality arises from the assignments N1 = N1o = CV and N = 0 (because there are no very large
voids). The nucleation rate is thus:
J
DVCV
 

m 1
m
N

o 1
m

D V C 2V
  N
m 1
m
(19.46)
1

o
m
/ CV

where N om / C V is given by Eq (19.44) and, from Eqs (19.2) and (19.18):
In order to calculate the nucleation rate, a number of parameters need to be known. These include:
- The V and I concentrations (CV and CI) ,
- the equilibrium vacancy concentration (C eq
V )o
- the diffusion coefficients DI and DV.
From these properties and KNRT, the vacancy supersaturation SS can be computed from
(19.41) and the arrival-rate ratio arr from Eq (19.42).
 m  (48 2 )1 / 3 m1 / 3
Eq
(19.47)
19.4.4 Simplified point-defect balances
Determination of CV and CI is discussed in detail in Sect. 13.5, but here we use a simplified form of the
point-defect balances. These balances equate the rates of production of the point defects (KNRT, dpa/s)7
to the rates of their removal at sinks in the microstructure. For the present analysis, V-I recombination is
neglected and the microstructure contains only network
dislocations, a neutral sink (unspecified) and the voids created by the nucleation process. Thus, Eqs
(13.36) and (13.37) at quasi-steady-state are:
for vacancies:
KNRT/= ( + k 2P + 4  R m N m )DVCV
(19.48) for
m
interstitials:
KNRT/= (zI + k 2P + 4  R m N m )DICI
(19.49)
m
25
dpa is the acronym for displacements per atom. Displacements per unit volume per unit time is KNRT/, where  is the
atomic volume. NRT are the initials of the authors whose theory is used. See Sect. 13.4.4 for details
7
45
SS = 300
Fig. 19.15 Constrained void-size distribution a function of vacancy and arrival-rate ratio -  = 10
26
 is the network dislocation density in units of m . Note that for the SIA balance,  is
multiplied by a factor zI called the bias factor. It is the all-important parameter in permitting void
nucleation. If zI were unity, comparison of Eqs (19.48) and (19.49) shows that DVCV = DICI, or arr =
1.0. As will shortly be seen, nucleation in this case is impossible.
-2
46
k 2P in Eqs (19.48) and (19.49) is the strength of the other neutral sink (in addition to voids). In terms of
the notation used to represent sink strength in Eqs (19.17a) and (19.17b) (see footnote No. 5), it is given
by:
k 2P   P N P
(19.50)
Use of k 2p to describe sink strength avoids the need to specify the structural nature of the sink.
The summation in Eqs (19.48) and (19.49) accounts for the sink strength of the existing void distribution
(not the constrained version given by Eq (19.44)). The size distribution Nm needs to be followed in time
after the nucleation rate J has delivered void embryos past the critical size (the minima of the curves in
Fig. 19.15). These voids then enter the growth phase.
In principle, the CV term in Eq (19.48) should be replaced with C V  C eq
V in order to account for
emission of vacancies from all the sinks. This term has been omitted from Eq (19.48) because in most
cases the vacancy supersaturation is very large, so C V  C eq
Eq
V . The analogous term in
(19.49) is not included in the void sink strength for interstitials because the equilibrium interstitial
concentration C eq
I is very small.
KNRT needs to be divided by the atomic volume  in order to convert the point-defect production rate
from a per-atom basis (dpa) to a per-unit-volume basis.
Equating the above point-defect balances yields the arrival-rate ratio:
  k 2P  4 R m N m
DI CI
arr 

D V C V z I   k 2P  4 R m N m
(19.51)
The arrival-rate ratio is less than unity to the extent that the dislocation-bias factor zI is greater than
unity. Estimates of zI range from 1.02 to 1.1.
The vacancy supersaturation obtained from Eq (19.48) is:
SS 
CV
K NRT / 

eq
2
CV
  k P  4 R m N m D V C eq
V


(19.52)
SS and arr are used in Eq (19.42) to determine the constrained distribution. In addition to the
microstructural parameters , k 2P , the atomic properties DV and C eq
V and the point-defect production rate
KNRT are required. Armed with these quantities, the nucleation rate
27
J is computed from Eq (19.46) with N / C V obtained from Eq (19.44). Figure 19.16 graphs the result
of such a calculation for nickel as a function of the supersaturation and the arrival-rate ratio. Notable in
this figure is the extreme sensitivity of the nucleation rate to both parameters, as will be demonstrated in
the example given below. The curves march to the right (higher supersaturations for the same nucleation
rate) as arr  1. In effect, arr = 1.0 is unreachable in terms of physically-accessible supersaturations.
o
m
47
1e+18
1e+17
arr = 0.95
J = nucleation rate, m-3s-1
1e+16
0.97
1e+15
0.99
1e+14
1e+13
1e+12
1e+11
1e+10
1e+9
1e+8
1e+7
1e+6
1e+5
50
100
150
200
250
300
350
400
450
500
supersaturation, SS
Fig. 19.16 Nucleation rate as a function of vacancy supersaturation and arrival-rate ratio
19.4.6 Void growth
Upon exceeding the critical size, the voids continue to grow at a rate given by the following volume
balance on a single void:
d 4
3
2 

(19.53a)
3 R m   4R m R m  4R m D V C V  D I C I  
dt
The last equality is obtained by inserting Eq (19.18) into Eqs (19.17a) and (19.17b) and neglecting the
vacancy-emission term.
Expressing the void-growth law in more convenient terms yields:
  D C  (1  arr )  D C eq  (SS) (1  arr )
Rm R
m
V V
V V
or,:
R 2m  2D V C eq
V (SS) (1  arr ) t
(19.53b)
A void nucleus of size Rmi grows to size Rmf in the time interval t:

R m f  R 2mi  R 2m

1/ 2
(19.54)
19.4.7 Transition from nucleation to growth
Nucleation theory as presented in Sect. 19.4.3 determines: i) the rate at which nuclei enter the solid at a
size sufficient to prevent shrinkage (Fig. 19.16) and ii) the size of the stabilized nuclei (minima in the
curves of Fig. 19.15). The essential result of nucleation is continual increase in
28
the void number density at a roughly constant void size. At the end of the nucleation period, growth of
the void continues but few voids are produced.
This description implies a sharp transition from nucleation to growth, but as usual, nature is not so
accommodating. The transition is gradual, with nucleation decreasing with time while growth takes
over. In this section, a simple model of this transition is presented and application to specific conditions
given as an example.
The scheme for treating simultaneous nucleation to growth is shown in Fig. 19.17. The step-function
method shown in the figure is intended to capture the continuous decrease of void nucleation with time
48
and the increasing dominance of void growth. Time is (arbitrarily) divided into fixed intervals denoted
by t. The end of each interval is indicated by the integer j, or
t = jt. The groups, labeled by
the integer m, represent voids that are nucleated during their first t and continue to grow thereafter.
Each group (m) is activated following the end of the nucleation period of the preceding group (m-1). At
the beginning of each interval, the current values of the vacancy supersaturation SS and the arrival-rate
ratio arr determine the nucleation rate J from Eq (19.46). The radius of the nuclei produced during this
period correspond to the minimum of the curve in Fig. 19.15 for the same values of SS and arr.
Following the termination of the nucleation period, growth continues at a constant number density Nm
for the group. Both nucleation and growth diminish with time because the increasing number density
and sizes provides a void sink for vacancies that depresses the vacancy supersaturation SS upon which
both depend.
Fig. 19.17 Method of analyzing the evolution of void nucleation to growth. The thickness of the
arrows reflects the magnitudes of the nucleation and growth rates of each group.
Example: neutron irradiation of iron at 900 K
The parameters chosen for this example are:
 = 1.510-6 nm-2
k 2p = 1.010-6 nm-2
6
C eq
nm 3
V  2  10
 = 0.01 nm3
KNRT = 1.010-7 s-1
DV = 1104 nm2/s
t = 104 s
zI = 1.1
with these values, Eqs (19.52) and (19.51) become:
29
SS 
4
k
2
void
5  10
 2.5  10 6
where:
arr 
k
k
2
void
2
void
6
 2.5  10
 2.7  10 6
k 2void  4 R m, j N m
m 1
is the sink strength of all voids present at time t. Initially, k 2void = 0.
nucleation (m = j): Eq (19.46) gives Jm and mmin is the critical void size given by the minimum of Eq
(19.44). The void density and size produced by nucleation over period j-1 to j are:
Nm = Jjt
/3
R m, j  0.134  m1min
49
growth (m < j): from Eqs (19.53b) and (19.54):


1/ 2
0 R 2m  4  10 4  SS(1  arr )  t
R m, j  R 2m, j1  R 2m
______________________________________________________________________________
The calculation proceeds as follows:
j = 1 SS = 200, arr = 0.943
nucleation of group 1: mmin = 79  R1,1 = 0.6 nm; J1= 1.210-13 nm-3s-1 N1 = 1.210-9 nm-3
The sink strength at the end of the first time period is k 2void = 8.810-9 nm-2. This is over two orders of
magnitude smaller than the combined sink strengths of the network dislocations and the neutral sink (the
numbers following k 2void in the above equations for SS and arr), so for the next time interval:
j = 2 SS = 199, arr = 0.944 (the slight reduction of SS is due to nonzero k 2void )
growth of group-1 voids nucleated in the previous time period:
m=1
N1 = 1.210-9 nm-3
R1,2 = 6.7 nm
This group of voids retains its number density but in ~3 hr, its radius has grown from less than
nm to 7 nm.
1
nucleation of group-2 voids
mmin = 79  R2,2 = 0.6 nm, J2 = 1.210-13 nm-3s-1  N2 = 1.110-9 nm-3
void sink strength, including groups 1 and 2 voids, is k 2void = 1.110-7 nm-2.
.
.
.
.
j = 6 SS = 172 arr = 0.951
growth
m = 1 N1 = 1.210-9 nm-3 R1,6 = 14 nm
m = 2 N2 = 1.110-9 nm-3 R2,6 = 12 nm
m = 3 N3 = 4.910-10 nm-3 R3,6 = 10 nm
m = 4 N4 = 1.710-10 nm-3 R4,6 = 8 nm
30
-11
-3
m = 5 N5 = 7.410 nm
R5,6 = 6 nm
nucleation
mmin = 115  R6,6 = 0.7 nm J6 = 3.910-14 nm-3s-1  N6 = 3.910-11 nm-3; k 2void = 4.810-7 nm-2
.
.
_____________________________________________________________
Following to the method outlined above, the time variation of several important characteristics of the
void nucleation/growth process are calculated and shown in Figs. 19.18 and 19.19.
Figure 19.18 depicts the variation of the void sink strength and the vacancy supersaturation over a
period of about two and one-half days. The vacancy supersaturation (SS) decreases from its initial value
of 200 to about 170 over this period. k 2void increases by about two orders of magnitude and attains a
value that is ~ 40% of that of the pre-existing vacancy sinks  + k 2P (gray line in the graph). These two
curves are closely coupled: the high SS causes voids to nucleate and grow, which in turn drives down
the vacancy supersaturation. The void sink strength appears to be approaching a plateau where SS is so
low that voids neither nucleate or grow at appreciable rates. The difference between the dashed and solid
50
curves for k 2P is a measure of the effect of time-step size on the accuracy in attempting to represent what
is actually a continuous curve. The smaller of the two t values is the more accurate of the two.
Fig.
19.18 Time variations of the void sink strength and vacancy supersaturation
Figure 19.19 shows the evolution of the void size distribution (number density per unit radius).
31
Fig. 19.19 Change of the void number-density distribution with time
The distribution moves to the right with increasing time, reflecting the growth of voids nucleated early
on. The highest-density voids have the largest radius because early nucleation occurs at the highest
vacancy supersaturation. The maximum number density (tops of vertical lines) increases slowly with
time but is never much different from 10-9 nm-3 /nm.
51
Void Swelling
The practical objective of void nucleation and growth modeling is to permit prediction of the increase of
volume of a piece of metal due to the presence of the voids. The consequences of void swelling are
numerous and all deleterious. Distension of a component such as a fuel rod or a control rod could
interfere with its removal; nonuniform swelling can result in bowing of a long component such as a fuel
rod; swelling of one component but not another with which it is in contact can increase stresses in both.
Void swelling is simply the volume of all the voids in a unit volume of the original metal:
V
  4 3 R 3m N m
V
m
(19.55)
The fractional swelling at the time corresponding to j = 6 in the previous example is 310-5, or 0.003%,
attained in ~ 17 hours.
52
9.5 Growth of cavities - the production-bias model (Sects. 9.5 - 9.6 - 11-2-09.doc)
The void nucleation and growth theory described in the preceding section relies solely on the preference
of the dislocations for interstitials. This bias leaves an excess of vacancies available to nucleate and
grow voids. If zI in Eq (19.51) were 1.0, Eqs (19.48) and (19.49) show that DVCV = DICI and hence the
arrival-rate ratio would also be unity. Figure 19.15 would require an infinite critical void size and Fig.
19.16 would require an infinite V supersaturation to nucleate voids.
In addition, this model assumes that single Vs and SIAs are produced in equal amounts by irradiation.
This production rate is related to the energy and flux of the irradiating particles (electrons, ions,
neutrons). For neutrons, KNRT (given by Eq (12.64)) is on the order of 10-7 dpa/s.
However, KNRT is the actual point-defect creation rate only for electron irradiation. As seen in Fig.
19.20, the damage structure created by neutrons colliding with atoms in a metal (and fission fragments
slowing down in ceramic nuclear fuels) is dramatically different from the single- V+SIA - per-collision
process that electrons produce. The collision of the nuclear particle with a lattice atom creates an
energized atom called a primary knockon atom (PKA) of tens to hundreds of keV. Subsequent
displacements (called cascades) caused by the PKA have been calculated by an atomic-level simulation
method called molecular dynamics (MD). The examples shown on the left in Fig. 19.20 show only
displacements, without distinguishing between vacancies and interstitials. Note the difference in size
between the cascades produced by the 10 keV and 50 keV PKAs. The diagram show the condition of
the damaged region before intracascade recombination of vacancies and interstitials (cooling) has
occurred.
Fig. 19.20 Molecular Dynamics simulation of
cascades produced by PKAs of various energies.
Ref. 31
35
The point-defect production rate before intracascade recombination is denoted by KNRT with units of
displacements-per-atom-per-second, or dpa/s, and is given by Eq (12.58).
A fraction  of the V-SIA pairs survive immediate recombination, and only these are involved in
altering the microstructure of the solid, mainly by creating and growing voids. For electron irradiation, 
= 1, but ion or fast-neutron bombardment produces dense cascades depicted in the left of Fig. 19.20. For
these cascades,   0.1. The effective point-defect production rate for the present situation is:
G = KNRT/
(19.56)
The atomic volume  in this equation bestows on the production rate units of point defects per second
per unit volume.  = 0.01 nm3 is used throughout this section.
Example: How many point-defect pairs are created in a 20 keV cascade?8
The number of point-defect pairs created by a cascade is given by Eq (12.58):
 NRT  0.8
E PKA
2E d
(12.58)
where Ed is the displacement energy, for which 30 eV is assumed (see Sect. 12.4.1). For  = 0.1, the point-defect production
from this cascade is 0.1(0.820103/230) = 27 V-SIA pairs.
After cooling, the damaged region is characterized by the spatially-restricted and clustered Vs and SIAs
shown in the right-hand portion of Fig. 19.20. The behavior of the clusters of Vs and SIAs radically
changes the consequences of irradiation damage from that predicted by the dislocation-bias model
described in Sect. 9.4. After a few attempts to explain this phenomenon and its effect on void formation
in the 1970s [7 - 9], a new theory, called the production-bias model, was introduced in the 1990s [10 19] and has undergone considerable refinement in the 2000s [20 - 29].
9.5.1 Point-defect cluster formation
Figure 19.21 shows the stages of cascade evolution viewed along its axis. The top sketch represents the
V-SIA recombination that occurs almost immediately after the cascade has been produced and is
responsible for the survival fraction . In the middle of the diagram are the clusters of point defects that
remain after the cascade has vanished. Finally, at the bottom of the drawing are the few single point
defects that are free to migrate in the solid.
Of the surviving point defects, a fraction form clusters close to the site of the original cascade:
- fraction of unrecombined Vs appearing as vacancy clusters = V
- fraction of unrecombined SIAs appearing as interstitial clusters = I
The spread of clustering fractions shown on the right-hand side of Fig. 19.21 represents roughly the
range of literature values obtained by various computational methods. Of the surviving point defects,
G(1- V) of Vs and G(1- I) of SIAs escape to become single species migrating freely in the bulk solid
(see Fig. 12.12).
8
that is, produced by a 20 keV PKA. For comparison, a head-on elastic collision of a 1-MeV neutron with an iron atom
creates a 70 keV PKA
36
A significant feature of the clusters is the separation of the two types. As suggested in Fig. 19.21, the V
clusters congregate near the center of the cascade track while the SIA clusters condense further out (see
Fig. 12.11a). The latter feature is due to the ability of interstitials created in the cascade to move outward
as "crowdions", which is a line structure containing an extra atom in a close-packed row (Sect. 17.8 of
Ref. 30).
As shown in Fig. 19.22, idealized clusters consist of a single layer of point defects that have condensed
between close-packed planes ((111) in the fcc lattice and (110) in the bcc structure). The loops are
termed faulted or prismatic because the stacking sequence of the close-packed planes (ABCABC... in
the fcc lattice) is interrupted by insertion of the disk of SIAs or by removal of part of a plane ( insertion
of vacancies). The periphery of each type of loop is an edge dislocation with the Burgers vector

perpendicular to the plane; in the fcc structure, b  a o [111] , where ao is the lattice constant and ao/3 is
3
the spacing between the (111) planes. There are ( 4 a o2 ) 1 atoms per unit area in the (111) plane in the
fcc lattice, so the number of interstitials in a loop of radius RIL is9:
3
n IL
4  R IL 


 (R )( 4 a ) 
3  a o 
2
IL
3
2
2 1
o
(19.57)
The corresponding equation for V loops is obtained from Eq (19.57) by replacing IL by VL.

The a o [111] orientation of b does not permit glide of the loop because the direction is not one of the
3
three <110> directions of the fcc slip system (see Fig. 6.8). For this reason, in this state are called
sessile, meaning immobile. The a o [111] dislocation that constitutes the loop's periphery is called a
3
Frank partial dislocation. Literature values of loop sizes fall in the ranges 30 < nVL <50 for vacancy
loops and 6 < nIL < 25 for interstitial loops. The number of loops produced by a single cascade depends
on the energy of the primary knockon atom (PKA) that generates the cascade.
Rearranging Eq (19.57) for the loops in the fcc structure :
1/ 2
R IL
 3

 

4



a o n 1IL/ 2
(19.58)
Taking ao = 0.36 nm and n IL  15 gives R IL  0.52 nm . The analogous calculation for vacancy clusters,
using n oVL  40 gives R oVL  0.85 nm. The superscript o denotes an as-formed value.
Example: For an NRT displacement rate of 5x10-6 dpa/s, estimate the radii and production rates of V and SIA
clusters. The atomic volume of the metal is  = 0.01 nm3.
Taking the cascade-cooldown survival fraction of the point defects () to be 0.1, the production rate of V- SIA
pairs per unit volume (before clustering) is:
G ~ 0.1(510-6)/0.01 = 510-5 V-I pairs/nm3-s
Taking V ~ 0.5 from the middle of the range in Fig. 19.21, the production rate of vacancy clusters is:
9
At this point, the SIA cluster is considered as an interstitial loop, so the subscript designation IL. Subsequently, however,
the picture of the SIA cluster changes, and with it, its designation.
37
Fig. 19.21 Debris from a cascade and a void
Fig. 19.22 Faulted loops in the fcc structure
V cluster production rate 
 V G 0.5  5  10 5
V clusters

 0.6  10 6
o
40
n VL
nm 3  s
and, with I ~ 0.4,
 I G 0.4  5  10 5
SIA clusters
SIA cluster production rate 

 1.3  10 6
15
n IL
nm 3  s
Determination of the cluster densities, NVL and NIL, requires application of the balance equations for the clusters,
which is treated in Sect. 9.5.4.
9.5.2 Point-defect clusters
38
The clusters of vacancies are usually identified as circular loops (Fig. 19.21). There is no evidence that
they can move. The original version of the production-bias model was based on immobilization of a
portion of the vacancies and SIAs in clusters (10, 12, 15). Because point defects are created in equal
quantities in the cascade, the Vs corresponding to the SIAs locked in clusters provides an excess of free
Vs over free SIAs, thereby enhancing void growth. The SIA cluster is mobile; the V cluster remains
where formed. The mobility of the SIA clusters could cause them to attack voids in the same way that
free SIAs do, namely by 3-dimensional (3D) diffusion. However, the principal mode of transport of the
SIA clusters is in a line, or one-dimensional (1D). This greatly reduces their tendency to impinge on
voids; rather, they tend to be absorbed by network dislocations, vacancy loops, or even grain boundaries.
This aspect of the production-bias model is described in Refs. 21, 22, 23 and 29.
SIA Loops
Interstitial clusters can be viewed in two ways: as loops or collections of crowdions.
As shown in Fig. 19.23, SIA loops start out as roughly circular disks of atoms that have been deposited
between close-packed planes in the crystal lattice. Being faulted, these loops are thermodynamically
unstable. They are converted to unfaulted loops by a reaction whereby the peripheral Frank partial
dislocation decomposes into two mobile dislocations:
ao
3
[111] 
ao
2
[110] 
ao
6
[112]
The second dislocation on the right is a Shockley partial. The reaction is depicted on the right in
Fig. 19.23; the ao/2[110] dislocation replaces the Frank partial on the loop periphery while the Shockley
partial sweeps over the loop, removing the fault in the process.
The Burgers vector of the a o [110] dislocation of the unfaulted loop points in a direction along which
2
slip, or glide, is possible (see Fig. 6.7). However, the new dislocation is a closed circle, and the <110>
direction in which it glides makes an angle of 54.7o with respect to the (111) plane. The mobile
dislocation is termed glissile and the loop moves as a unit in one dimension. Figure 7.8 shows three of
the twelve <110> directions in the fcc structure on which the new dislocation loop can glide.
Crowdion clusters
In both the fcc and bcc structures, individual interstitials assume two forms. The most stable is a
dumbbell configuration (see Fig. 3.2), which, however, is immobile (sessile). The other form is the
crowdion, which is formed by an interstitial is squeezed into a close-packed atom row. Because the
interstitial clusters are formed in the highly unthermodynamic manner of a cascade, they appear as a
closely-packed group of crowdions, as shown at the bottom of Fig. 19.23. When the extra atoms fall in
the same plane, they form the stacking fault shown in the upper left of the figure. A cluster of > ~ 10
crowdions is sufficiently resistant to reversion to dumbbells that its lifetime is controlled by removal at a
sink. As suggested by the shaded ovals in Fig 9.23, the glissile loop and the crowdion cluster are just
different ways of looking at a single structure. The interstitial loop is a circular extra layer of atoms; the
crowdion cluster is also an extra layer of atoms in a roughly circular shape. When considered as a loop,
the periphery is an edge dislocation with the same properties as network dislocations. When considered
as a cluster of crowdions, the movement of the unit can be determined. For simplicity, the interstitial
loop/crowdion cluster unit will henceforth be termed a croop (crowdion/loop).
A more realistic picture of the croop in bcc Fe than the sketch in Fig. 19.23 is available from molecular
dynamics simulations (MD - Chap. 14), such as the study by Wirth et al [32]. As shown in Fig. 19.24,
the interior of the croop is a combination of crowdions and split dumbbells. The structure is much more
39
irregular than suggested by Figs. 19.22 and 19.23. As seen in the cross-section views of the croop in
Fig. 19.24, the interstitials do not even occupy a single (110) plane.
Fig 9.23 in the fcc crystal structure: Top: unfaulting of a Frank partial dislocation to form a
glissile (mobile) loop. Bottom: interstials as a group of neighboring crowdions; gray circles are
regular lattice atoms; the crowdions are shown as open circles
1D Croop movement
The crucial property of the croop is the ease with which its constituent crowdions can move (hop) either
backward or forward in the same direction along adjacent rows. This type of motion can be simulated
by the molecular dynamics computational technique (see Chap. 14), in which a cubical crystallite
containing as many as 105 atoms is seeded with a cluster like the one shown in Fig. 19.24. For a bcc
metal, all SIAs are placed in a disk between (110) planes. Temperatures ranging from 200 - 1000 K
impart random thermal motion of all atoms in the block, including the SIA cluster. The atoms move
more-or-less en bloc in a [111] direction. The positions of the atoms are followed as a function of time,
giving typical trajectories shown in Fig. 19.25.
The initial disk of interstitials has become quite ragged, but it is hanging together as it moves along the
[111] direction (for bcc). The reason that it does not disintegrate is thermodynamic: a clump of
interstitials is more stable than the same number dispersed in the crystal as single SIAs.
Croop movement is thermally-activated. The croop must overcome a small potential barrier in hopping
along its trajectory. The barrier is due to a small stress called the Peierls stress that all dislocations must
overcome in order to glide. In most situations, dislocations glide because of a shear stress in the
direction of the Burgers vector (Sect. 6.8). However, in the present case, no stress is present and
movement requires thermal agitation for the loop to move (much like the random 3D jumping of a point
defect). Consequently, its 1D movement can be represented by a diffusion coefficient D1D.
For any species undergoing 1D diffusion, displacement is related to time by:
x 2  2D1D t
(19.59)
40
Fig. 19.24 Cluster containing 19 SIAs in bcc Fe. Solid circles represent <111> crowdions; open
circles are split dumbbell interstitials (Ref. 32)
Rewriting the above as:
D1D
( x / a o ) 2 a o2

t
2
and plotting the numbers from Fig. 19.25 as ( x / a o ) 2 vs t, drawing the best straight line through the 3
points (the points don't line up well) and using ao = 0.28 nm for bcc Fe, yields D1D ~ 10-8 m2/s.
The diffusivities computed in this way are shown in Fig. 19.26 for bcc Fe from Refs. 28 and 32.
The data from Ref. 28 exhibit no discernable temperature dependence, and the line from Ref. 32
represents an activation energy of ~ 4 kJ/mole. Such weak temperature dependence is very different
from that of most single-particle diffusivities. To understand this peculiar behavior, as well as the
dependence of D1D on its size, the Einstein equation for 1D motion is invoked:
D1D  1 2  croop2croop
where croop is the vibration frequency of the croop and croop is its jump distance. If there are ncr (same
as nIL) crowdions in the cluster, each vibrating independently, the croop vibrates ncr times the frequency
cr with which each crowdion vibrates: croop = ncrcr. Also, when a crowdion moves by its hop distance
cr, the center of gravity of the croop only moves by croop = cr/ncr. The croop diffusivity is thus given
by:
D
1
D1D  1 2 (n c r  c r )( c r / n c r ) 2 
 c r 2c r  I
(19.60)
2n c r
ncr
where DI is the diffusivity of a single SIA. The croop diffusivity is approximately inversely proportional
to the number of crowdions it contains.
The activation energy for movement of a croop is approximately equal to that of a single crowdion.
Because this form of SIA moves very easily, E cmr is very small, about 2 kJ/mole. This is midway between
the zero activation energy suggested by the data of Ref. 28 and the 4 kJ/mole from the plot in Fig. 19.26
for Ref. 32.
Croop movement
T he moving croop occasionally switches from one close-packed direction to another. As depicted in
Fig. 19.27 the croop undergoes a zigzag motion. The directions of the lines refer to one of the 12
possible<110> directions in the fcc structure.
41
Fig. 19.25 MD simulation of a 19-SIA cluster in bcc iron at 260 K. Gray dots are lattice sites and
black dots are SIAs (Ref. 28)
Following creation from the cascade, the crowdions in the cluster move more-or-less in unison in a 1D
back-and-forth random walk indicated by the arrows. After a certain number of jumps, a switch to
another <110> direction occurs (in the literature, this is termed "change of Burgers vector"). These
events, indicated by the crosses, characterize the 3D aspect of the cluster's migration (in the figure, the
<110> direction changes are shown in two dimensions). Finally, its life ends by entering a void or a
contacting a dislocation.
Croop motion is characterized by two characteristic lengths:
1. Lch = average distance between direction changes (between crosses in figure)
2. Lcroop = average total distance before absorption by a cavity or dislocation (star  dot in figure)
If Lch >> Lcroop , motion is 1D; if Lch << Lcroop, motion is purely 3D.
9.5.3 Reaction rates of croops
Pure 1D motion
As analyzed in Ref.13, the absorption mean-free path for 1D croop movement is analogous to the
collision of a moving particle (the croop) with a stationary one (the sink) - each presents a "cross
section". Here, only three sinks are considered: voids, vacancy loops and network dislocations.
42
1e-6
2
D1D, m /s
1e-7
1e-8
Col 132
vs Col 2
Ref.
Ref. 28
32
Ref.
1e-9
0
2
4
6
8
10
12
3
10 /T
Fig. 19.26 Croop diffusivities vs temperature 19-SIA croops in Fe.
Fig. 19.27 Movement of a croop
By analogy with the particle - particle reactions, the mean free path of the mobile croop is given by:
1
Lcroop
  void N void  d  (   VL )
(19.61 )
VL is the length of the peripheral edge dislocations that comprise the vacancy loops in a unit volume of
solid. This term will be discussed in detail later.
d is the effective diameter of the interaction between the croop and a dislocation. It is a complicated
function of the stress fields surrounding the croop and the dislocation, and depends on the size of the
former as well (13,16):
d = 7(/4)ncr(Tm/T)b
(19.62)
b is the Burgers vector of the dislocation representing the croop periphery (e.g, (3/2)ao for bcc), Tm is
the melting point of the metal and T is the temperature (both in Kelvins)
43
The cross section for croop-void interaction is analogous to that for the hard-sphere cross section
between colliding atoms,  void   R 2void , so Eq (19.61) becomes:
1
Lcroop
 R 2void N void  d  (   VL )
(19.63)
Example: For a croop containing 15 crowdions interacting with dislocations in iron (ao = 0.287 nm) at 900 K, this
equation gives an effective interaction diameter of 40 nm. This is more than an order-of-magnitude greater than
the commonly-assumed core diameter of an edge dislocation. The mean free path of a croop in iron with a
dislocation density of 1014 m-2 is [(4010-9)(1014)]-1 = 2.510-7 m = 250 nm. The croop moves a good fraction of
the grain diameter before it encounters a dislocation line.
Example: Compare the relative effectiveness of the dislocations in the previous example as targets for 1-nm
diameter croops to that of a void population characterized by: Nvoid = 31018 m-3 and Rvoid = 9 nm.
void = (9)2 = 254 nm2. The mean free path in due to the voids alone is [(25410-18)(31018)]-1 = 1.310-3 m
=1300 m. The dislocations provide a target for croops that is 1300/0.250 = 5200 times larger than for voids.
This result implies that nearly all croops moving in 1D are absorbed by dislocations, which frees up Vs for
growing voids.
Barashev et al [18] present a method for translating the 1D-mean free-path to a croop-sink reaction rate.
The rate of reaction (per unit volume) is written in a form analogous to radioactive decay:
Jcroop = Ncroop/tcroop
where Ncroop is the volumetric concentration of croops and tcroop is the lifetime of the croop. For an object
undergoing a random walk in 1D, the mean-square displacement at time t is 2D1D t, where D1D is the 1D
diffusion coefficient. As in Eq (19.59), this can be expressed as:
L2croop  2D1D t croop
(19.64)
Eliminating tcroop between these two equations yields:
J croop 
2D1D N croop
L2croop
(19.65)
expressing Lcroop by Eq (19.63), Eq (19.64) yields the total reaction rate:


J croop  2D1D R 2void N void  d  (   VL ) 2 N croop
(19.66)
Equation (19.65) is a 3rd-order rate equation (second order in Nvoid and , first order in Ncroop).
The reaction rate of croops with particular sinks (voids, dislocations or vacancy loops) is given by [22]:
1
1
J croopsin k  2D1D Lcroop
Lsin
k N croop
1
where Lsin
k is one of the terms in the brackets of Eq (19.66).
9.5.4 Defect balance equations
(19.67)
44
The heart of the production-bias method of rationalizing void growth in metals are the conservation
equations for the four species involved: two point defects (vacancies and SIAs) and two extended
defects (vacancy loops and croops). The balance equations (also called conservation statements) for the
four defects are quasi-steady-state. Changes in their concentrations are slow enough compared to the
individual rates of production and destruction that such a simplification is warranted. Consequently, the
rate of production of a defect is equated to its rates of removal at all sinks.
Voids, however, are extended defects whose size increases with time. They must be treated in the
unsteady state, as in Eq (19.53a)
Approximations in the analysis include neglect of:
- V-SIA recombination
- formation of sessile SIA clusters from the cascade (all clusters are glissile - croops)
- each sink (voids, vacancy loops, croops) is a single size10
- change in network dislocation density
- 3D motion of croops (they move in 1D only, but undergo occasional direction changes)
The notation used in these balances is summarized below:
G = rate of point-defect pair creation after intracascade recombination, dpa/s
I = fraction of remaining interstitials that form clusters (croops)
V = fraction of remaining vacancies that form clusters (vacancy loops)
 = atomic volume, (taken to be 0.01 nm3 for all metals)
zI = bias factor for dislocations
 = network dislocation density, m-2
DI = diffusion coefficient of SIAs, nm2/s
DV = diffusion coefficient of vacancies, nm2/s
CI = concentration of SIAs in bulk solid, nm-3
CV = concentration of vacancies in bulk solid, nm-3
-3
C eq
V = equilibrium vacancy concentration, nm
-3
C VL
V = vacancy concentration in solid adjacent to vacancy loop, nm
JX = rate of removal of defect X by all extended defects, nm-3s-1
JX-Y = rate of removal of defect X by sink Y, nm-3s-1
Rvoid = void radius, nm
Nvoid = void number density, nm-3
RVL = radius of a particular vacancy loop, nm
R VL = mean radius of vacancy loops, nm
NVL = vacancy-loop number density, nm-3
 VL  dislocation density contributed by vacancy loops, nm-2
Ncroop = number density of croops nm-3
ncr = number of SIAs in a croop (the same as nIL in Eqs (19.56) and (19.57))
Of the three extended defect radii, R VL , Rcroop and Rvoid, with time, the first and the second remain
constant and the last increases. Individual vacancy loops are born with a radius of R oVL and in a short
10
Size distributions of these extended defects are treated in Ref. 26.
45
time disappear. However, the average V loop radius in the solid, R VL , changes with time at a rate
comparable to that of the voids. The remaining extended defect, the network dislocations, is considered
to remain at a constant density. This is justified because absorption of point defects by edge dislocations
causes them to climb, not necessarily to increase or decrease in length.
The interactions between the four defects with each other and with the voids and dislocations are shown
in Fig. 19.29.
Fig 19.29 Diagram of defect flows in an irradiated metal
The arrows under the cascade icon are the production rates of the two types of point defects and their
clusters. The defect production rate per unit volume is given by Eq (19.56).
The black arrows in Fig. 19.29 represent the movement of vacancies to and from the three extended
defects: voids, dislocations and V loops. Solid arrows mean absorption of Vs; dashed arrows indicate
emission (evaporation) of Vs. The gray arrows depict the absorption of SIAs by the same three extended
defects. Because of the extremely low equilibrium interstitial concentration, evaporation of SIAs from
the extended defects is negligible. The thick gray arrows represent absorption of croops by the extended
defects. The flows are identified by letters or numbers.
The arrows crossing the shaded ovals around the four radiation-produced defects indicate the point
defect flows that need to be included in the balance equations.
SIAs: This balance applies to the single interstitial atoms that escape both recombination in the cascade
and clustering into croops.
(1   I )G      
(19.68)


(1   I )G  4R void N void  z I [   VL ] D I C I
(19.68a)
46
Vacancies: The V balance includes the same three extended defects as the SIA balance:
(1 - V)G = (1 - 1') + (2 - 2') + (3 - 3')
VL
(1   V )G  (4R void N void  ) D V (C V  C eq
V )   VL D V (C V  C V )
(19.69)
(19.69a)
The V concentration that controls evaporation of Vs from the V loops (C VL
V ) is different from the
equilibrium concentration that applies to the voids and network dislocations (C eq
V ).
Constructing balance equations for the croops and the vacancy loops poses issues that are not
encountered with the point-defect balance equations. In the latter, the only characteristic of the
population is its concentration , CI or CV; the croops and the vacancy loops, on the other hand, are
defined by their concentrations (NVL and Ncroop) and by the number of point defects they contain
(nVL and ncr). The kinetics of these two entities are not described by the usual rate equations (i.e., the
product of a sink strength, a diffusivity, and a concentration driving force). Rather, the balance is of the
form applied to radioactive decay: production rate = concentrationlifetime.
Croops:
The croops, born with ncr SIAs, undoubtedly shrink or grow as they move through the cloud of Vs and
SIAs en route to destruction at a void or a dislocation. However, this possibility is neglected, and ncr is
assumed to remain at its initial value from birth to death by absorption.
Equating the rate of croop removal given by Eq (19.65) to the rate of production results in the croop
balance:
 I G 2D1D N croop
(19.70)

n cr
L2croop
Division of the production rate of clustered SIAs by ncr converts the left-hand side to a production rate
of croops. According to Eq (19.60), ncr D1D = DI, the SIA diffusion coefficient. The croop mean-free
path, Lcroop, is related to the loss rates of the processes A, B and C in Fig. 19.29 by Eq (19.63), so the
above equation becomes:
2
(19.70a)
 I G  2D I Lcroop
N croop
Vacancy loops: The sink strength of the vacancy loops is based on the length of the loop's peripheral
edge dislocation, which provides a dislocation density of:
(19.71)
 VL  2R VL N VL
where R VL is the mean radius of the vacancy loops, a slowly varying function of time. At steady state,
new vacancy loops consisting initially of n oVL vacancies (or radius R oVL ) are created at a rate  V G / n oVL
loops/nm3-s. Once created, they shrink by the net effect of processes 3, 3', C and  shown in Fig. 19.29
and eventually disappear. These processes determine the loop's lifetime, tVL. The balance on the V loop
concentration is:
 V G N VL

t VL
n oVL
(19.72)
The relation between the number of vacancies in a loop and its radius is given by a simplified version of
Eq (19.56). The number of vacancies per unit area of the loop is approximated by -2/3 instead of
47
(
3
4
a o2 ) 1 , which is the exact value for the fcc lattice. Since   a 3o / 4 for this structure, the error
incurred by the approximation is less than 10%. Approximating the area of an atom by 2/3, Eq (19.56)
becomes :
n VL   R 2VL /  2 / 3
(19.73)
Replacement of NVL in Eq (19.72) by VL using Eq (19.71) requires a relationship between RVL at any
time and the initial value R oVL . Because of net V loss from the vacancy loops, RVL ranges from the
initial value to zero, the average value of all VLs in the solid is denoted by R VL  qR oVL . The value of q
will be determined later. With this relation, Eq (19.72) and Eq (19.73), Eq (19.71) becomes:
 VL  2q V G  2 / 3 t VL / R oVL
(19.74)
Lifetime of a vacancy loop
The V-loop lifetime is determined by interactions 3, 3',  and C in Fig. 19.29. The loss rate of vacancies
from a single vacancy loop is given by:
dnVL/dt = - [ + (3' - 3) +ncrC]
(19.75)
The fluxes of point defects to a unit length of dislocation are given by Eqs (13.18) and (13.20). Since
the length of line in a loop of radius RVL is 2RVL, the portion of the vacancy loss due to the first three
terms in dnVL/dt is:
 d n VL 


 2R VL [z I D I C I  D V (C V  C VL
(19.75a)
V )]
 d t   ,3',3
1
The rate at which croops attack a vacancy loop is given by Eq (19.67) with Lcroop
VL  d  2R VL and
1
given by Eq (19.63)
Lcroop
 d n VL

 dt

1
  2Lcroop
(d  2R VL )D I N croop
C
(19.76b)
where according to Eq (19.60), ncrD1D = DI
Substituting Eqs (19.70) and (19.81) into the above equation yields the rate of shrinkage of a vacancy
loop due to croop impingement:
 d n VL 
 dG

  (2R VL ) I 1
(19.76c)
L croop
 d t C
Replacing nVL by RVL by Eq (19.73) and adding the contributions given by Eqs (19.75a) and (19.75c)
yields
Id  G
1 d R VL
 (z I D I C I  D V C V )  D V C VL
V 
2/3
1
dt

Labs
(19.77)
The concentration of vacancies in the solid at the loop periphery, C VL
V , is different from the normal
equilibrium vacancy concentration C eq
V that appears in the corresponding terms for voids and network
dislocations (e.g., Eq (19.17a)). C VL
V is responsible for the "evaporation" or "emission" of vacancies
from vacancy loops. As shown in Sect. 19.5.8 of Ref. 30, the reason for the difference between C VL
V and
48
C eq
V is the change in loop energy as a vacancy is added or removed. Assuming that the loop is unfaulted,
the edge dislocation that forms its periphery generates strain energy in the adjacent solid. The energy per
unit length of the dislocation is Gb2, where G is the shear modulus of the metal and b is the magnitude of
the Burgers vector of the dislocation. The energy of a V loop of radius RLV is:
EVL = 2RVLGb2 = 2Gb21/3nVL
(19.78)
where the second form is obtained using Eq (19.73). The change in loop energy per V added or removed
is dEVL/dnVL. As shown in Sect 19.5.8 of Ref. 30, the loop-energy effect results in:
  Gb 2 1 / 3 
 
 dE VL / dn VL 
eq
eq
eq



C VL

C
exp

C
exp

C
exp


V
V
V
V


 n
kT
n VL kT 



 VL




(19.79)
For stainless steel, the shear modulus 50 GPa, and the magnitude of the (ao/2)[111] Burgers vector is
0.287 nm. The Boltzmann constant k = 1.3810-23 J/atom-K. Using these values and T = 900 K in Eq
(19.79) gives  = 165. A smaller estimate of  = 33 has been obtained by Ref. 35 using an analysis
similar to the current one. The consequences of these very large values of  are explored below.
VL
eq
From Eq (19.73), nVL = 7/8RVL and C V  C V exp(  / R VL )
,
where  = /7.8
eq
From Eq (19.81) with  = 0.01 nm3, nVL = 8.2RVL, and C VL
V = C V exp(4/RVL). The factor  has units of
eq
nm, as does RVL. Substituting this expression for C VL
V into Eq (19.84) and dividing by DV C V yields:
1
DVC 
where A is the dimensionless quantity:
eq
V
A
2/3
 
dR VL
 A  exp 
dt
 R VL



1
z I D I C I  D V C V   I d  G / Lcroop
D V C eq
V
(19.80)
(19.81)
1
DICI, DVCV and Lcroop
are all functions of VL which, according to Eq (19.71), is proportional to R VL .
However, there is a distinction between RVL in Eq (19.80) and R VL in the quantities in Eq (19.81). The
former refers to a single vacancy loop, whereas the latter is an average characterizing the entire
population of vacancy loops in the solid. On the time scale of the disappearance of a vacancy loop after
its formation from the cascade, R VL varies not at all. Thus, A is a constant and determination of the VL
lifetime requires integration of Eq (19.80) from RVL = R oVL at t = 0 to RVL = 0 at t = tVL. This is best
accomplished by rendering Eq (19.80) dimensionless with the definitions:
2/3
x = /RVL and  = t/t*, where t*   / D V C eq
V
with these new variables, Eq (19.80) becomes:
1 dx
 (A  e x )
2
d

x
49
A typical value of A is < 1, so that this parameter can be neglected compared to ex. For loops that
contain 40 Vs at birth, or a radius of 0.83 nm, the above equation integrates to:

 
t
dx '
e x'
 
  2

dx '  Ei (4.9)  e 4.9 / 4.9  Ei (x )  e  x / x
t * 4.8 x ' (A  e x ' ) 4.8 x ' 2
x
x

(19.82)
and the integral reduces to the result shown. Here Ei(x) is the exponential integral:

e x'
dx '
x'
x
Ei ( x )   
Converting x back to RVL, Fig. 19.30 shows the shrinkage of a vacancy loop of initial radius of 0.83 nm.
Fig. 19.30 Vacancy-loop radius as a function of the dimensionless time since creation. T = 900 K,
 = 33
The loop shrinks slowly at first but for RVL < 0.5 nm, the loop vanishes practically instantaneously.
2/3
Such is the nature of the function in the last bracketed term in Eq (19.82). For the values of D V C eq
V
given in Table 9.3 and  = 33 ( = 33/7.8 = 4.2) , t* = 4600 s and the lifetime of the vacancy loop is
tVL = t*VL = 4600(2.310-4) ~ 1 s. The dashed line is an eyeball estimate of the average VL size in the
solid. It is R VL ~ 0.7, or in Eq (19.74), q ~ 0.7/0.83 = 0.85.
50
Table 9.3 Properties of irradiated iron at 900 K
property
symbol
Initial void radius
R ovoid
Void density
Nvoid
Dose rate
Vacancy clustering fraction
SIA clustering fraction
Network dislocation density
Effective diam. of dislocation
For absorbing SIA loop
Bias factor for dislocation
V diffusivity
SIA diffusivity
Equilibrium vacancy conc.
As-produced V-loop radius+
@
As-produced croop radius
Lattice parameter
Atomic volume
o
+
based on n VL = 40 Vs and Eq (19.58)
@
Value
10 nm
210-10 nm-3
G
V
I

d
2.5x10-7 dpa/s-nm3
0.5
0.4
1.510-6 nm-2
40 nm
zI
DV
DI
C eq
V
1.1
105 nm2/s
11012 nm2/s
210-7 nm-3
R oVL
Rcr
ao

0.83 nm
0.52 nm
0.287 nm
0.01 nm3
based on n I L = ncr = 15 SIA and Eq (19.58)
Example: What are the point-defect concentrations in irradiated iron for  = 33 and the parameters of
Table 9.3?
Using the VL lifetime estimated in the text (tVL ~ 1 s),
Eq (19.74):
VL = 2(0.85)(0.5)(2.510-7)(0.012/3)(1)/0.83 = 1.210-8 nm-2
Eq (19.63):
1
Lcroop
 (10) 2 (2 10 10 )  40  (1.5 10 6  1.2 10 8 )  6.0 10 5
nm 1
DINcroop = (0.4)(2.510-7)/2(6.010-5)2 = 14 nm-1 s-1
Eq (19.70a):
Eq (19.72a):
k 2V '  4(10)(10 6 )  1.1(2  10 4 )  3.5  10 4
Eq (19.68a):
DICI = (0.6)( 2.510-7)/[4(10)(210-10)+1.1(1.510-6 +1.210-8)] = 8.810-2 nm-1s-1
Rearranging Eq (19.69a):
from Eq (19.79):

nm 2
eq
(1   V )G   VL D V (C VL
V  CV )
D V (C V  C ) 
4R void N void     VL
eq
V



(19.69b)

eq
eq
eq
o
eq
eq
C VL
V  C V  C V exp(  / n VL )  1  C V exp(  / qn VL )  1  exp( 33 / 0.85  40 )  1 C V  287C V
Note that the vacancy concentration in the solid adjacent to the V loop is nearly 300 times larger than
the equilibrium concentration!
51
0.5(2.5  10 7 )  (1.2  10 8 )(10 5 )( 287)( 2  10 7 ) 1.94  10 7
D V (C V  C ) 

 0.126 nm-1 s-1
4(10)( 2  10 10) )  1.5  10 6  1.2  10 8
1.54  10 6
eq
V
eq
5
7
DVCV = D V (C V  C eq
V )  D V C V  0.13  (10 )( 2  10 )  0.15
C V D V C V 0.15


 7.5
0.02
C eq
D V C eq
V
V
DICI
0.088
arrival-rate ratio = arr =

 0.59
DVCV
0.15
supersaturation: SS =
The difference D V (C V  C eq
V )  D I C I drives void growth, so it is worthwhile presenting. Equation
(19.69b) gives the first term and Eq (19.68a) gives the second, and the difference is:
D V (C V  C )  D I C I 
eq
V


 VL D V C eq
V exp  / n VL  G V   I   W 
4R void N void     VL
(19.83)
where
W
(z I  1)(1   I )(   VL )
4R void N void     VL
(19.84)
The important feature of Eq (19.83) is that the driving force for void growth is determined principally by
the evaporation of vacancies from vacancy loops (first term in the numerator) and the bias of the
dislocations for interstitials (the W term).
19.5.5 Void swelling
The rate of metal swelling due to growth of internal voids is determined by the rates at which point
defects enter the voids in a unit volume of metal:
d  V 

  (J V void  n cr J croop void )
dt  V 
(19.85)
The first term on the right accounts for the effect of the single point defects on void volume:
J V  void  4R void N void[D V (C V  C eq
V )  DICI ]
(19.86)
and the second term represents the rate at which croops interact with the voids. The rate is given by
Eq (19.67):
1
1
(19.87)
J croopvoid  2Lcroop
Lcroop
 voidD1D N croop
1
2
1
where Lcroop
is given by Eq (19.63).  enters Eq (19.85) because each
Lcroop
 void  R void N void and
point defect that is absorbed by a void changes the its size by one atomic volume.
Example (con't) What is the void swelling rate?
52
The quantities in the above equations are taken from the previous example. Also, DI = ncrD1D.
Equation (19.86) becomes:
JV-void = 4(10)(210-10)(0.126 - 0.088) = 1.110-9 nm-3s-1
Equation (19.87) becomes:
Jcroop-void = 2(610-5)((10)2)(210-10)(14) = 0.110-9 nm-3s-1
and the swelling rate is:
d  V 
9
9
11
s 1

  (1.1 10  0.1 10 )(0.01)  1.0  10
dt  V 
In one year of irradiation, the void swelling would be 0.03%.
19.6 Void lattices
At this point, it is worthwhile to summarize the sequence of events that voids pass through.
The first stage is nucleation, during which small voids are produced by agglomeration of vacancies in
the bulk.
The second stage is growth, which follows after the nucleation period has essentially terminated. In this
stage, the existing voids enlarge by net absorption of vacancies while maintaining a roughly constant
number density.
The third stage is the beginning of a formation of a void lattice, described in detail below. This stage
begins at about 1 dpa.
The fourth stage is limitation of the growth of the voids in the void lattice.
In this section, the third and forth stages are analyzed in detail. The starting condition is a random array
of voids of radius R ovoid and number density Nvoid.
According to Evans, who first observed them [33], void lattices are self-organizing nanostructures. They
exhibit two quite unique properties:
1. The void lattice mimics the underlying crystal lattice, albeit with a much larger unit cell
2. All voids are essentially the same size
These properties are clearly to be seen in the TEM pictures of Fig. 19.31. The three views show, from
left to right: the (100) face with the characteristic atom placement clearly revealed; the (110) cut in
which the empty rectangle with the ratio of side lengths of 2; the hexagonal (111) arrangement11.
Table 19.5 summarizes the pertinent data concerning void lattices as of 1993. The distance between
neighboring voids, Lvoid, is 50 - 500 times the crystal lattice parameter (ao) and 2 - 7 times Rvoid, the void
radius (the ratio is about 4 in Fig. 19.31). The beginnings of a regular structure appears at ~ 1 dpa and
the void lattice is fully organized at doses between 10 and 100 dpa. It is stable at temperatures between
1/4 and 1/2 of the melting point (in K).
19.6.1 Origin of void lattices
11
These are called projections, they reveal the voids in two planes of the structure when viewed in the <ijk> directions
perpendicular to the (ijk) planes indicated here.
53
Why do void lattices form? The reason has been called Darwinian, in the sense that an assembly of
voids with the lattice structure of the underlying crystal is better able than a random collection to
withstand destruction from the onslaught of SIAs in the croops. Figure 19.32 illustrates this feature. For
the same number density of spheres (voids), a projectile incident as shown by the arrows has a far better
chance of hitting a sphere in the random distribution on the left than in the rows of spheres aligned one
behind the other in the direction of a projectile. The question is: by what mechanism does the mobile
SIA debris from the cascade change the disorganized collection of voids to a pattern resembling a
military parade?
100 nm
Fig. 19.31 Transmission electron micrographs of the void lattice of Nb-O
from B. A. Loomis et al, J. Nucl. Mater. 68 (1977) 19
Table 19.5 Characteristics of void lattices in various pure metals produced by different projectiles
after W. Jäger & H. Trinkhaus, J. Nucl. Mater. 205 (1993) 395
Metal
fcc Al
fcc Al
fcc Ni
bcc Nb
bcc Nb
bcc Mo
bcc Mo
Source Energy
of PKAs MeV
Al ions
0.4
neutrons
> 0.1
Ni ions
5
Ta ions
7.5
Ni ions
3.2
neutrons
> 0.1
neutrons
> 0.1
Metal
temp, K
323
328
800
1073
1283
858
1193
Dose Void radius, Void spacing,
dpa
Rvoid, nm
Lvoid nm
40
5
60
6
32(?)
250(?)
360
13
66
140
13
34
30
38
145
36
3
27
45
3
124
(?) questionable data
Lvoid/Rvoid
6
4
3
3
2
4
7
In Fig. 19.32, the 1-D migrating croops (arrows) are all moving in a <110> direction (in an fcc lattice).
As they impinge on the random array on the left in the figure, they first remove all spheres with the
smallest linear density in the direction of the arrow. What survives are the densest rows of spheres in the
same direction as the SIA cluster. This results in a random array of rows of spheres aligned in the same
direction as the arrows (right-hand diagram in Fig. 19.32). Considering that the same process is
occurring in all of the <110> directions, the end result is a 3D spatial array with the same lattice
structure as the underlying crystal. Note that this void realignment model in no way involves an
interaction between the voids and the underlying lattice atoms. This basic model of void-lattice
54
formation was suggested by Foreman in 1972 [32] and has since been enlarged and quantified by many
papers.
9.6.2 How voids react with croops
In subsequent discussion of void lattices, two simplifying assumption are made:
1. voids are represented as cubes with the same volume as a spherical void of radius Rvoid.
2. the void lattice has a simple-cubic structure with a void number density Nvoid
These are by no means realistic restrictions. However, they greatly simplify the explanation of the
processes in which void lattices take part.
Fig. 19.32 Unidirectional croops impinging on a group of voids.
Such a void lattice is shown in Fig. 19.33. The cube dimension is s = (/6)1/3Rvoid and the void spacing is
1/ 3
Lvoid ~ N void
.
The spaces between voids are of two types. The regions joining two voids are called supply volumes,
because the largest input of croops to the voids originates here. The other volumes called corridors are
zones where the croops can travel long distances. The only events are removal by dislocations or, with a
change in direction, absorption by a void.
The rate per unit volume at which SIAs (in the form of croops) are generated in these two regions is IG,
where G is given by Eq (19.56) and I is the fraction of the SIA leaving the cascade as clusters
9.6.3 Voids as sinks for croops
In a void lattice SIAs bunched in croops enter voids from the supply volumes and the adjacent corridors.
Croops from supply volumes (Fig. 19.34)
The rate at which croops are generated in the volume element (2s)2dx in Fig. 19.34 is (2s)2Gdx. Of
these, one out of six are headed towards void A. The probability that croop 1 in the figure arrives at
void A without a direction change or removal by a dislocation is exp(-x/Ltot), where Ltot is the mean
distance traveled before one of these interruptions occurs. It has the form of Eq (19.63) with the void
absorption term replaced by the direction-change term and the V loops ignored:
Ltot1  Lch1  d  
The rate at which SAIs in the form of croops enter void A is:
(19.87)
55
j1 = rate of SIA absorption in void A from croops = 6  (2s) (G / 6)
2
( L void 2s )
 exp( x / L
tot
)dx
0
The factor of 6 is the number of supply volumes associated with each void, one for each face.
Integrating the above yields:

j1  4 s 2 L tot  I G 1  e  ( L void2s ) / L tot

(19.88)
corridor
corridor
croop supply volumes
corridor
A
2s
Lvoid
Fig. 19.33 Simplified representation of a void lattice
Fig. 19.34 Croops entering a void from one of its supply volumes
As (Lvoid -2s)/Ltot  0, J1  4s2G(Lvoid -2s). That is, none of the croops generated in the supply volumes
change directions before entering the void. As (Lvoid -2s)/Ltot  , J1  4s2GLtot. In this limit, the
length of the supply volumes appears infinite as far as croop absorption is concerned.
Croops from corridors
Figure 19.27 suggests that croops can change direction many times before being absorbed at a sink, one
of which may be a void. To render this process quantitative, the following analysis considers only
croops from the corridor nearest the void.
56
Figure 19.35 shows the trajectory of a croop produced in the volume element d2V = (2s)dxdy in one of
the half-corridors next to void A.
For a croop generated in the corridor to enter a particular void, it must:
1. be aimed towards dV, probability p1 = 1/6
2. interact in dV, probability p 2  e  y / L tot  e ( y 2s ) / Ltot
3. change direction (not be absorbed by a dislocation) probability p3 = Lch1 / Ltot1  L tot / L ch
4. head towards void A, probability p4 = 1/6
5. reach void A, probability p5 = e  x / L to t
Fig. 19.35 Croops entering a void from a half-corridor
The probability that a croop produced in d2V enters void A is the product of the five probabilities. The
rate at which SIAs produced as croops in d2V reach void A is p1p2p3p4p5(IG)d2V. Over a distances y in
Fig. 19.35, this is:

d j2   I G (2s)dx )  p1p 2 p 3 p 4 p 5 dy 
0
1
36

 I G (2s)( L tot / L ch ) 1  e

 2 s / L tot

 e
 2 y / L tot
dy  e  x / L tot dx
0

 136  I G  s  (L2tot / L ch ) 1  e 2s / L tot e  x / L tot dx
The rate at which void A absorbs SIAs generated (as croops) in all 24 adjacent corridors is:
j2 
24
36

 I G  s  (L / L ch ) 1  e
2
tot

 2 3 G  s  (L / L ch ) 1  e
3
tot
 2 s / L tot
L void 2 s
 x / L tot
 e
dx
0
 2 s / L toot
1  e
( L void 2 s ) / L tot

(19.89)
The factor of 24 in Eq (19.94) accounts for all of the half-corridors that can supply croops to void A.
There are six directions emanating from a void (x, y, z), each of which contains 4 half-corridors. The
multiplying factor for the croop production rate from a single half-corridor that gives the appropriate
total production rate is 6 (void faces)  4 (half-corridors/void face) = 24 half-corridors per void.
57
Example: Compare the absorption rates of croops from the supply volumes and the corridors that feed
void A. Use the following conditions: Rvoid = 5 nm (s = 4 nm), Lvoid = 60 nm, d = 310-3 nm-1 (see
example following Eq (19.63)).
Figure 19.36 shows plots of J1/G (Eq (19.93)) and J2/G (Eq (19.94)) against the mean direction-change
length Lch. As expected, the absorption rate from supply volumes increase as Lch increases and
approaches that of pure 1D migration. For small Lch, the corridors immediately surrounding the void
contribute about 20% to the total absorption rate, but this percentage decreases towards 10% for large
Lch and the probability for direction change in the supply volume (p3) decreases. It is likely that if
corridors further removed from the void than those treated above were included, their contribution
would not be significant.
3000
supply volumes
J/G, nm3
2500
2000
1500
1000
corridors
500
0
0
100
200
300
400
500
Lch, nm
Fig. 19.36 Croop sink strength of a void in a lattice for s = 4 nm Lvoid = 60 nm and  = 1014 m-2
9.6.4 Void size in a void lattice
Three sources provide the vacancies and interstitials that regulate the size of voids in a void lattice. The
first two are the single Vs and SIAs, which migrate to voids from the bulk solid by ordinary 3D
diffusion. The third source is the SIA clusters, or croops, described above.
Absorption of single point defects by the void provides a net vacancy inflow to balance the SIAs
introduced by the croops. To maintain a stable void size, the point-defect balance on a void in the void
lattice is:
j1  j2  4R void (D V [C V  C eq
(19.90 )
V ]  DICI )
j1 and j2 are given by Eqs (19.88) and (19.89), respectively. The right hand side is taken from
Eq (19.83).
58
Example: find the void radii in a void lattice with the following parameter range:
3
3
  3 R void
(all length units in nanometers). Lvoid/Rvoid = , N void  Lvoid
The other parameters are those in the problem in the previous section.
From Eqs (19.89) and (19.88), the left hand side of Eq (19.90) is:
 8s 3 4  L3tot 

(1  e  Y ) (1  e  Z )
LHS  j1  j2   I G 
 s 

 Y 3  L ch 

where
Y = 2s/Ltot
Z = [ - 2(/6)1/3(Rvoid/Ltot)]
(19.91)
(19.92)
Ltot is given by Eq(19.87) with d = 40 nm.


Use exp  / n VL  287 and VL = 1.210-8.
The right-hand side of Eq (19.90) is obtained from Eq (19.83).
287(1.2  10 8 )(10 5 )( 2  10 7 )  G[(0.6  0.4)  W]
RHS  4R void (D V [C V  C ]  D I C I ) 
3
 3 R void
 (  1.2  10 8 ) / 4R void
(19.93)
8
0.1  (1  0.4)(  1.2  10 )
(19.94)
W
2
43 R void
   1.2  10 8
eq
V
The solution is obtained by trial-and-error. An Rvoid trial is substituted into Eq (19.94), then into
Eq (19.93) to give RHS. The same value of Rvoid (as s = (/6)1/3Rvoid) is inserted into Eqs (19.92),
then into Eq (19.91) to determine LHS. When RHS = LHS, the solution for Rvoid has been obtained.
The four parameters of the problem network dislocation density, croop direction-change mean free path,
void spacing-to-void radius ratio and point-defect production rate. Realistic ranges of these variables
are:
10-6   10-4
50  Lch  1000
3   30
10-7  G  10-6
It is not feasible to graph the solutions for the above ranges of all four parameters. Figure (19.37) gives
typical results for two , G pairs with  and Lch as the two parameters that are independently varied. No
solution could be found for the combinations of  and Lch which do not have an associated Rvoid on the
plots.
On the (a) plot for  = 3.0, solutions were obtained only for the starting value Lch = 50 nm and for
Lch = 100 nm. As  was increased to 3.1, 3.2, and 3.3, solutions were found up to Lch = 200 nm. For
 = 3.4 and 3.5, the upper limit was Lch = 500 nm. At  = 3.6, Rvoid was determined for the entire range
of Lch. Solution was not possible for  > 3.6, irrespective of the Lch parameter. These restrictions on the
allowable parameter values may be due to the inapplicability of the steady-state assumption on which
the calculation is based. That is, the voids in the lattice either undergo unlimited growth or shrink to
nothing. The void sizes for Lch = 50 nm started at Rvoid = 11 nm when  = 3.0 and decreased regularly to
Rvoid = 1 nm for  = 3.6. At each value of , the void size increased roughly linearly with Lch until the
latter's maximum value was reached. At this location, the void radius varied in an irregular fashion from
11 nm to 35 nm.
59
For the ,G pair in Fig. 19.37b, solutions were found for nearly all , Lch combinations. Except for a
spike at 50 nm, the Rvoid deter mined were roughly independent of Lch. Solutions were not found for
 < 10, but no upper limit of this variable was found up to a chosen maximum of  = 25.
According to the data in Table 19.5, the radii of the voids in a lattice range from 3 to 40 nm, which is
within the extent of the calculated values in Fig. 19.37. However, the observed range of  in Table 19.5
(last column) is 2 - 7, into which the values in Fig. 19.37a fall. However, for the large dislocation
density used for Fig. 19.37b, the region of  where the calculation gave solutions for Rvoid was distinctly
outside the observed range of this parameter.
Fig. 19.37 Void sizes in a void lattice
35
References
1. N. Carnahan and K. Starling, "Equation of State for Non-attracting Rigid Spheres"
J. Chem. Phys. 51 (1969) 635
2. C. Ronchi, "Extrapolated EOS for rare gases at high temps & densities"
J. Nucl. Mater. 96 (1981) 314
3. J. Katz & H. Wiedersich, "Nucleation of voids in materials supersaturated with Vs & Is",
J. Chem. Phys. 55 (1971) 1414
4. H. Wiedersich & J. Katz "Nucleation of voids & other irradiation-produced defects",
Adv. in Colloid & Interface Sci., 10 (1979) 33
5. K. Russell, "Nucleation of voids in irradiated metals", Acta Metall. 19 (1971) 753
6. H. Carlslaw & J. Jaeger, "Conduction of heat in solids", 2nd Ed., p. 248, Oxford (1959)
7. R. Bullough, B. Eyre and K Krishan, "Cascade Damage Effects on the Swelling of Irradiated
Materials", Proc. Roy. Soc A346 (1975) 81
8 P. Heald and M. V. Speight, "The influence of cascade damage on irradiation creep and
swelling", J. Nucl. Mater. 64 (1977) 139
9. A. Foreman and M. Makin, "The effect of vacancy loops on the swelling of irradiated
materials"
J. Nucl. Mater. 79 (1979) 43
10. C. Woo & B. Singh, “Concept of Production Bias and its role in defect accumulation in
cascades”, Phys. Stat. Sol. B159 (1990) 609
11. C. Woo & B. Singh, "Production bias due to clustering of point defects in cascades" ,
Phil. Mag., A65 (1992) 889
12. B. Singh & A. Foreman, "Production bias & void swelling in transient cascade-damage
conditions" Phil. Mag., A66 (1992) 975
13. H. Trinkhaus et al, “Glide of SIA loops produced by cascade damage – effect on void
formation” J. Nucl. Mater. 199 (1992) 1
14. S. Zinkle & B. Singh, “Displacement damage & defect production in cascade damage”
J. Nucl. Mater. 199 (1993) 173
15. C. Woo et al, “Microstructural evolution driven by production bias”,
J. Nucl. Mater. 206 (1993) 170
16. H. Trinkhaus et al, “Glissile I loop production in cascades – defects accumulation in
transient,
J. Nucl. Mater. 206 (1993) 200
36
17. H. Trinkhaus et al, “Defect accumulation under cascade-damage conditions”
J. Nucl. Mater. 212-215 (1994) 18
18. B. Singh et al, “Production bias & cluster annihilation: why necessary?”,
J. Nucl. Mater. 212-215 (1994) 200
19. B. Singh et al, "Microstructure evolution under cascade-damage conditions"
J. Nucl. Mater. 251 (1997) 107
20. S. Golubov et al, "Defect accumulation in metals & alloys in cascade-damage conditions",
J. Nucl. Mater. 276 (2000) 78
21. S. Golubov et al, Recoil-energy-dependent defect accumulation in pure copper"
Phil. Mag. A 81 (2001) 2533
22. A. Barashev, S. Gollubov & H. Trinkhaus, "Reaction kinetics of glissile interstitial clusters",
Phil. Mag. A 81 (2001) 2515
23. H. Trinkhaus et al, "Modeling microstructural evolution in metals under cascade-damage
conditions" J. Nucl. Mater. 283-287 (2000) 89
24. A. Semenov & C. Woo, "Void nucleation under cascade-damage conditions"
Phys. Rev. B66 (2002) 024118
25. B. Singh et al, "Recoil-energy-dependent void swelling in Cu: I Experimental results"
Phil. Mag. A 80 (2000) 2629, 2533
26. S. Golubov et al, "Grouping method of kinetic equation describing the evolution of point
defect clusters" Phil. Mag. A 81 (2001) 643
27. B. Singh et al, "Radiation-damage theory", Encycl. of Mater. Sci. & Technol. (2001) 7957
28. Y. Osetsky et al, "1D transport by SIA clusters in Fe & Cu", Phil. Mag. 83 (2003) 61
29. A. Barashev & S. Golubov, "Steady-state size distribution of voids under cascade
irradiation"
J. Nucl. Mater. 389 (2009) 407
30. D. Olander "Fundamental aspects of nuclear reactor fuel elements",
National Technical Information Center (NTIS) (1976)
31. R. Stoller, " The role of cascade energy in primary-defect formation in Fe",
J. Nucl. Mater. 276 (2000) 22
32. B. Wirth et al, "Dislocation-loop structure; energy & mobility of SIA clusters in bcc Fe"
J. Nucl. Mater. 276 (2000) 33
31. J. H. Evans, "Observations of a Regular Void Array in High Purity Molybdenum irradiated
with
2 MeV Nitrogen Ions", Nature 229 (1971) 403
37
32. A. J. E. Foreman, Harwell Report AERE-R 7135 (1972)
33. C. Woo & W. Frank, "Theory of void-lattice formation", J. Nucl. Mater. 137 (1985) 7
34. C. Woo & W. Frank," Limited growth, displacive stability and size uniformity of voids in a
void lattice" J. Nucl. Mater. 140 (1986) 214