SIMULATION OF OXIDE DISPERSOID STABILITY
IN IRRADIATED ALLOYS
by
Mohammad Seyed Saiedfar
B.E.,
(Met. E.) Arya-Mehr University of Technology
Tehran-Iran (1973)
Submitted in partial fulfillment of
the requirements for the degree of
Master of Science in Nuclear Engineering
at the
Massachusetts Institute of Technology
May, 1978
0
Massachusetts Institute of Technology 1978
Signature of Author
Department of Nuclear ngineering
May 12, 1978
A I
Certified by
Thesis Supervisor
Accepted by
Chairman, Department Committee
Archives
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
SLr 8 1978
LIBRARIES
SIMULATION OF OXIDE DISPERSOID STABILITY
IN IRRADIATED ALLOYS
by
MOHAMMAD SEYED SAIEDFAR
Submitted to the Department of Nuclear Engineering
on May 12, 1978 in partial fulfillment of the requirements
for the Degree of Master of Science
in Nuclear Engineering
ABSTRACT
A theory which has been developed by Maydet and Russell for
stability of incoherent precipitates under irradiation is
evaluated for austenitic stainless steel with chromium oxide
precipitate and aluminum with aluminum oxide precipitate.
The arrival rate ratio of interstitials and vacancies 01/k
is mapped over the temperature range 0.25 & T/TM • 0.60 for
two different displacement rates (K = 10-6 dpa/sec and K = 10
dpa/sec) and two various dislocation density expressions. It
is seen that the ratio falls off and is essentially zero at a
higher temperature extreme.
The vacancy supersaturations versus homologous temperature are
plotted for those alloys with the above irradiation and
material parameter condition. It 'is shown that at a high
temperature T % 0.6 TM there is no vacancy supersaturation and
Sv = 1 (Cv = Cveq).
The critical oxide particle size is calculated and graphed
over a range of temperatures 0.25 • T/TM • 0.60 for positive
and negative atomic misfits for both systems. It is seen that
for positive misfit there always exist critical particles for
solute supersaturation Sx > 1 for a whole range of temperatures.
The particle trajectories of a precipitate under thermal,
irradiation and cyclic irradiation are calculated and plotted
for various conditions of irradiation and temperatures.
Calculations show that particles which are greater than
critical size grow and those which are under critical size or
with Aý > 0 decay.
Thesis Supervisor:
Title:
Kenneth C. Russell
Associate Professor
of Metallurgy
ACKNOWLEDGMENTS
I wish to express my appreciation to my thesis
advisor, Dr. Kenneth C. Russell for his helpful discussions
and guidance during this work.
I would like to thank Professor John E. Meyer for
taking the time to read my thesis, and Steve I. Maydet for
his helpful suggestions regarding some of the computational
work.
I wish most heartily to thank my wife Fahimeh, who
has encouraged me through my graduate studies, and my parents
who have patiently guided me through my life.
The financial support of the Atomic Energy Organization of Iran (AEOI) is gratefully acknowledged.
I wish to acknowledge Barbara Harris for her assistance
in the typing of this paper.
TABLE OF CONTENTS
ABSTRACT . .
. .
. . .
. .
ACKNOWLEDGMENTS
. . .
. . . ..
LIST OF FIGURES
... .
LIST OF TABLES . . . .
Chapter
1 INTRODUCTION
2
2.2
. . .
......
. .
. . ..
. ....
3.5
3.6
4.4
. . . . . .
2
.
4
....
...
. .
.
7
.
. .
....
.
10
. .
11
. ...
....
.
....
.. . . . . .
Short Time Experimental Techniques . . . . .
2.1.1 High Voltage Electron Microscope
Technique . . . . . . . .
.....
2.1.2 Ion Bombardment Technique ..
.....
Theoretical Analysis . ..........
.
2.2.1 Enhanced Diffusion . ...
.
....
2.2.2 Solute Drag . . . . . . . . . . . . .
2.2.3 Chemical Vacancy Effect . .......
2.2.4 Recoil Dissolution . . . . . . . . .
2.2.5 Disordering Dissolution . . . . . . .
.
14
18
.
.
18
18
19
19
19
20
20
21
. .
23
Equation of Motion.......
......
Nodal Line Formalism . . . . . ............
Nodal Line Equations . ............
Critical Point and Irradiation-Modified
. . .
Potential . .. . . . . . . . . . . . .
Defect Conservation Equations.. . . . . . ..
Point Defect Arrival Rates . . . . . . . . . .
23
29
33
. . .
39
CALCULATIONS AND RESULTS
4.1
4.2
4.3
. . .
... .......
THEORETICAL MODEL . . . .
3.1
3.2
3.3
3.4
4
........................
LITERATURE REVIEW . . . . . .
2.1
3
.
. .
. . .
. .
. . .
. . .
. . .
.
.
.
.....
Material Parameters ... . . . . . . . .
..
Dislocation Densities . . . . . . . . .... .
Irradiation Conditions . . . .
.........
4.3.1 High Displacement Rate . . . . . . . .
4.3.2 Low Displacement Rate . ........
Point Defect Arrival Rate and Concentration .
34
35
37
40
40
43
44
44
45
4.5
4.6
5
Critical Particle Size
. ...
......
4.5.1 Thermal Equilibrium Condition . . . ..
4.5.2 Substitutional Solute Atoms . . . .
.
4.5.3 Interstitial Solute Atoms .
......
Particle Trajectories
. ...
... . . . . . .
4.6.1 Thermal Decay . .......
..... .
4.6.2 Irradiation Induced Growth . .. .. .
4.6.3 Cyclic Irradiation
......... .
SUMMARY . . .
. .
. .
. . .....................
APPENDIX A: PROTOTYPE COMPUTER ALGORITHMS
A.1
A.2
A.3
A.4
A.5
Algorithm A
Algorithm B
Algorithm C
Algorithm D
Prototypes
........
.....
.. .
..........
.... ..
..
. . .
.
.
.. .
... . .
..
. . . . . ..................
.
. . . . . . ..................
. ...
..
...........
.
..
BIBLIOGRAPHY . . . . . . . . . . . . ......
53
54
54
62
69
69
70
75
79
82
82
83
83
83
84
100
LIST OF FIGURES
Page
1.
Ability of Dispersion-Strengthened Alloys to
Retain Strength at Higher Temperatures.
15
2.
Phase Space for Particle Trajectories, Showing
Processes Giving Rise to Motion. Bx = Rate of
Solute Capture, ax = Rate of Solute Emission,
y= Rate of Vacancy Capture, av = Rate of
Vacancy Emission, 8i = Rate of Self-Interstitial
Capture.
25
3.
Schematic Illustration of Nodal Lines, Critical
Point, and Particle Trajectories.
30
4.
Critical Points: (a): Stable Node, (b): Stable
Focus, (c): Saddle, (d): Center.
32
5.
Point Defect Bias versus T/TM in 316 SS with
Constant Dislocation Density.
46
6.
Point Defect Bias versus T/TM in 316 SS Using
Dislocation Density Expression from Table 4.
46
7.
Same as Figure 5 except for Aluminum.
47
8.
Same as Figure 6 except for Aluminum.
47
9.
Rate of Defect Creation or Losses versus T/TM in
316 SS with Constant Dislocation Density and
K = 10-6 dpa/s.
49
10. Rate of Defect Creation or Losses versus T/TM in
316 SS Using Dislocation Density Expression from
Table 4 and K=10 - 6 dpa/s.
49
11. Same as Figure 9 except with K = 10 - 3 dpa/s.
50
12. Same as Figure 10 except with K = 10-3 dpa/s.
50
13. Vacancy Supersaturation versus T/TM in 316 SS
with K = 10-6 dpa/s.
51
8
Page
14. Vacancy Supersaturation versus T/TM in 316 SS
with K = 10- 3 dpa/s.
51
15. Same as Figure 13 except for Aluminum.
52
16. Same as Figure 14 except for Aluminum.
52
17. Critical Particle Size versus T/TM in 316 SS
Under Thermal Condition.
55
18. Same as Figure 17 except for Aluminum.
55
19. Critical Particle Size versus T/TM in 316 SS
Under Irradiation K = 10-6 dPa/s with Negative
Misfit and Constant Dislocation Density.
57
20. Critical Excess Vacancy versus T/TM in 316 SS
Under Irradiation K = 10- 6 dpa/s, with Negative
Misfit and Constant Dislocation Density.
57
21. Critical Particle Size versus T/TM in 316 SS
under Irradiation K = 10-6 dpa/s with Negative
Misfit and Using Dislocation Density Expression
from Table 4.
57
22. Same as Figure 19 except for Aluminum.
58
23. Same as Figure 20 except for Aluminum.
58
24. Same as Figure 21 except for Aluminum.
58
25. Same as Figure 19 except K = 10- 3 dpa/s.
60
26. Same as Figure 20 except K = 10- 3 dpa/s.
60
27. Same as Figure 21 except K = 10- 3 dpa/s.
60
28. Same as Figure 19 except for Aluminum with
K = 10- 3 dpa/s.
61
29. Same as Figure 20 except for Aluminum with
K = 10- 3 dpa/s.
61
30. Same as Figure 21 except for Aluminum with
K = 10- 3 dpa/s.
61
31. Critical Particle Size versus T/TM in 316 SS
under Irradiation K = 10- 6 dpa/s with Positive
Misfit and Constant Dislocation Density.
65
9
Page
32. Critical Excess Vacancy versus T/TM in 316 SS
Under Irradiation K = 10-6 dpa/s with Positive
Misfit and Constant Dislocation Density.
65
33. Critical Particle Size ve sus T/TM in 316 SS
under Irradiation K = 10 - dpa/s with Positive
Misfit and Using Dislocation Density Expression
from Table 4.
65
34. Same as Figure 31 except for Aluminum.
35. Same as Figure 32 except for Aluminum.
36. Same as Figure 33 except for Aluminum.
37. Same as Figure 31 except K = 10 -
3
dpa/s.
38. Same as Figure 32 except K = 10 -
3
dpa/s.
39. Same as Figure 33 except K = 10 -
3
dpa/s.
40. Same as Figure 31 except for Aluminum with
K = 10- dpa/s.
41. Same as Figure 32 except for Aluminum with
K = 10- 3 dpa/s.
42. Same as Figure 33 except for Aluminum with
K = 10- 3 dpa/s.
43. Thermal Decay of Cr203 in 316 SS with Sx = 2
and Negative Misfit at T = 0.35 TM.
44. Irradiation Induced Growth of an Over Critical
Particle Size Under Irradiation K = 10- 3 dpa/s
at T = 0.35 TM with Positive Misfit for Various
Solute Supersaturation Conditions.
45. Same as Figure 44 except K = 10-6 dpa/s.
73
46. Same as Figure 44 except for Aluminum.
74
47. Particle Trajectory of a Cr 2 0 3 Precipitate in
316 SS Under Cyclic Irradiation.
78
LIST OF TABLES
Page
1.
Nominal Alloy Compositions (w/o).
40
2.
Material Parameters of 316 SS and Cr20 3 .
41
3.
Material Parameters of Aluminum and A1 2 03 .
42
4.
Dislocation Density Expressions.
43
5.
Typical Value of Damage Rates.
44
6.
Anticipated Structural Materials Requirement
for Fission Breeders and Fusion Reactors.
76
10
CHAPTER 1
INTRODUCTION
The thermally activated growth and coarsening of
precipitates in metals has been studied for many years and is
now well understood, particularly in thermal conditions under
which the stability of precipitates is controlled by the
balance between the diffusion of solute atoms to the precipitate and the thermally-activated release of atoms from the
precipitate surfaces.
When the rate of dissolution procesiSes
exceeds the rate of diffusion to precipitates, the
precipitates disappear to maintain the solute atoms in
solution.
However, at lower temperatures where a reduced
solute concentration is in equilibrium with precipitates,
the system attempts to lower its total free energy by the
growth of large precipitates at the expense of smaller ones.(1)
In an irradiation environment, thermal equilibrium is
disturbed by the production of thermal spikes and displacement cascades.
The high energetic atoms can easily produce
nonequilibrium phases or cause to dissolve other phases which
could exist in an equilibrium situation.
Since the dissolution
or redistribution of phases which serve as barriers to
dislocation, such as oxide precipitates, alter the mechanical
properties of structural components during operation of a
12
reactor, therefore, the study of alloys with oxide dispersoid
in their matrix becomes important.
Al-A1 2 0 3 alloys, consisting of an Almatrix with.A12 03 particles, considered interesting
reactor core materials since they-.have a sm&ll absorption
cross-section for thermal neutrons, and they preserve good
strength and resistance to corrosion up to high temperatures.
These alloys, for instance, are to be used as canning
material in an organic-cooled, heavy water-moderated reactor
operating at temperatures from 400 to 4600C.
On the other
hand, at these temperatures the ductility of the Al-A1 2 0 3
alloys is very much reduced with respect to the ductility
at room temperature.
During recent years extensive studies
were therefore made of the mechanical properties of Al-Al,2 03
alloys in order to better understand and possibly improve
their ductility. (2)
Another system which is considered here is 316SS with
dispersed Cr 2 0 3 particles.
The 316SS is selected material
for the liquid metal fast breeder reactors (LMFBR) and is
candidate material for controlled thermonuclear reactor (CTR)
first walls.
The addition of Cr 2O 3 particles in the 316SS is
to preserve the stability of the alloy's dislocation structure
at high temperatures and, as a result, the superior strength
above 0.55 to 0.65 of the melting point of the matrix. (3)
The purpose of the calculation is to predict the
particle critical size under various irradiation conditions
13
and to study the growth or dissolution of such an oxide
particle.
The calculations are based on the theory which
has been developed by Maydet and Russell for incoherent
particles.
CHAPTER 2
LITERATURE REVIEW
The strengthening of materials by a-dispersed second
phase has been the subject of considerable investigation, and
theoretical approaches show that the hardening of alloys
depends principally upon whether the discrete second phase
particles are sheared by dislocations moving in the matrix.
The degree of coherency of the matrix-precipitate interface
is one of the controlling conditions for the shearing, such as
is obtained, for example, in materials containing GuinierPreston zones.
The incoherent precipitates, on the other
hand, are usually not sheared by dislocation moving in the
matrix (except possibly at high deformation and for a small
size of precipitates).
Oxide dispersions are generally
incoherent, and so we can include them in the general group of
alloys with incoherent precipitates.
(5 )
The primary interest in dispersion-strengthened
material is based, of course, on their stability at very high
temperatures (due to the high free energy of the formation of
the oxides), as shown in Figure 1. (3 )
The graph shows (1)
nickel plus a dispersed oxide of thorium compared to an
outstanding wrought nickel alloy, and (2) aluminum plus a
60,000
50,000
-- '-'
UI(X2;
-·
--
-·
40,000
30,000
20,000
x
.1PERSION
THENED
-
10,000
I
4
Fig.
I
I
-~I
.9
.8
.7
METAL
FRACT ION OF MP OF BASE
5.
.6
1 Ability of Disnersion-Stren-thened
Alloys to Retain Strength at Hijher
Temnreratures.
(
Kef. 3)
Rfter
16
dispersed oxide of aluminum to a high temperature aluminum
alloy.
They are compared on the basis of their 1000-hour
rupture strength at various absolute temperatures expressed
as a fraction of the melting point of the base metal.
In
both cases, although the dispersion-strengthened pure metal
is weaker than the precipitate-strengthened solid-solution
matrix alloy at low temperatures, the dispersion-strengthened
materials are less sensitive to increases in temperature and
thus are superior in strength above 0.55 to 0.65 of the
melting point of the pure metal.
This strength and stability
to unusually high temperatures are the most important characteristics of inert particle-dispersion-strengthened metal
systems.
Another potential advantage is that dispersionstrengthened pure metals will have a higher thermal and
electrical conductivity than the same metal when solid
solution strengthened.
Thus, dispers ion-strengthened
materials may find application where both strength and
thermal conductivity are required, e.g., for components of
heat exchangers' parts, and reactor cladding; and the higher
electrical conductivity would be advantageous in electrical
conductors or electrical contacts.
Therefore, the study of microstructural stability of
oxides which are insoluble, hence thermodynamically stable,
becomes influential in the prevention of the deleterious
changes in the mechanical properties of steels and aluminum
alloys caused by the bombardment of neutrons within the
nuclear reactor.
There is some experimental evidence which shows that
some phases tend to dissolve while some others tend to grow
during neutron or heavy ion bombardment.
work of Jones
(6 )
For instance, the
shows that the shells of small (< 50A)
thoria particles have been congregated around larger (>> 50A)
thoria particles in 5 Mev Ni++ irradiated Ni/ThO 2 alloy
specimens.
In addition, the work of Vaidya and Bohm (7 ) shows
that incoherent precipitates of Mg 2 Si in an Al-Mg-Si alloy
were dissolved during 250 Kev Al+ bombardment at a damage rate
of about 10- 2 dpa/sec at room temperature.
In another
example, Al-2 %Ge foils have been bombarded by both 1 Mev
electrons and low energy Al + ions (100 and 200 Kev) over a
range of temperatures.
In both damaging regimes copious
precipitation of Ge has been observed at temperatures where
little or no such precipitation occurs during prolonged
thermal aging.(8)
The fact that reactor tests require a long-time
neutron irradiation of nearly ten years, however, has stimulated the development of techniques for simulating highfluence neutron irradiation effects in metals.
(9 )
As a
result, the long-term irradiative microstructural changes
can be predicted by two ways:
1) short time experimental techniques, and
2) theoretical analysis.
2.1 Short Time Experimental Techniques
The experimental simulation techniques typically
involve bombardment of the material with high energy particles
(electrons, ions, and self-ions) wherein high effective dose
levels can be achieved in a relatively short time.
2.1.1 High Voltage Electron Microscope Technique
Irradiation experiments involving the use of high
energy charged particles have been utilized to increase atom
displacement rates by an order of magnitude relative to inreactor displacement rates.
With such techniques it is
possible, in principle, to attain in a matter of hours the
state of damage which exists in a metal after residing for
several years in the core of a fast reactor.
High voltage
electron microscope (HVEM) electron irradiation methods, in
particular, have added a new dimension to the study of such
phenomena by providing the capability for continuous, direct
observation of the developing microstructure during
irradiation.
2.1.2 Ion Bombardment Technique
Bombardment of metals by energetic heavy ions has
proven to be a useful tool for compressing the time scale of
irradiation tests by many orders of magnitude.
Reasonable
currents of H + , C+ , and metal-ion beams of energies from 1
to 10 Mev can be obtained from a accelerators.
Because the
range of heavy ions in solids is quite small (typically
19
1
pm), all the initial energy of the ion can be dissipated
in a small volume of the specimen producing high damage rates.(10)
Since the number of displaced atoms in an irradiated experiment is a reasonable measure of the extent of radiation
damage, the time scale may be highly compressed.
2.2 Theoretical Analysis
There are several mechanisms which have been proposed
for alternation of phase stability.
2.2.1 Enhanced Diffusion
A simple method by which irradiation may alter a
microstructure is by enhancement of the diffusion coefficient. (11 )
The rate of diffusion of substitutional atoms is
proportional to the vacancy concentration, which in turn may
be increased many fold by irradiation.
It must be emphasized,
however, that enhanced diffusion may only speed up the rate of
reaction.
It is therefore not expected to give rise to phases
which would not appear under extended thermal aging.
2.2.2 Solute Drag
In irradiated metals there is a new flow of vacancies
and interstitials to dislocations, grain, and interphase
boundaries, voids, and other fixed sinks.
There are point
defect exchange mechanisms which may cause the composition of
matter arriving at the sinks to differ from the overall
composition of the alloy.
As a result, the regions near the
sinks tend to become enriched in one (or more) components and
depleted in one (or more) others.
The balance between the
solute drag and back diffusion dictates the degree of
segregation.
2.2.3 Chemical Vacancy Effect
Vacancies have been observed to act as a chemical
component in the nucleation of voids and dislocation loops
in quenched or irradiated material,
(1 2 - 1 3 )
and in assisting
in the precipitation of Si and Ge particles from aluminumbased solid solutions.(14)
The contribution to driving force
for the reaction comes from the annihilation of the highly
supersaturated vacancies at the defect aggregate.
This
concept has been extended on a quantitative basis to the
stability of incoherent or semi-coherent precipitates under
irradiation.(4)
The interfaces of such precipitates may
serve as sinks for vacancies and interstitials and allow them
to enter into reaction.
2.2.4 Recoil Dissolution
The dynamic collision events which occur as a result
of atomic displacement within collision cascades cause atoms
within the precipitate to recoil into the surrounding matrix.
The flux of such recoils is readily estimated from our
knowledge of the energy spectrum within collision cascades
and the number of atoms sputtered from solid surfaces during
irradiation.(1)
the damage rate.
For convenience, this flux can be related to
Calculations based on existing theories
21
suggest that for a damage rate of K displacement/atom/sec this
flux of atoms is
4 K/cm 2 -sec.
xs101-
Furthermore, as the
dissolution rate is directly proportional to damage rate, it
can be scaled to the particular irradiation environment.
Thus, the volume change (dv) of a sphere of radius r
is simply
dv/dt = -4rr2 /N
where N is the number of atoms per unit volume.
2.2.5 Disordering Dissolution
An alternative mechanism for dissolution, particularly
pertinent to ordered precipitates, is that the disordering
effect of the displacement cascades essentially destroys the
ordered precipitate lattice, so that localized regions of high
solute concentration are created.
In the absence of diffusion
such a state will persist, as during irradiation of PE16
alloy at room temperature, for example.
When diffusion occurs,
however, the small disordered regions created within the
precipitate will reorder, while those near the surface will
result in the loss of solute, by diffusion to the surrounding
matrix.
Suppose that only those displacements which occur in
a shell of thickness (k) at the precipitate surface can result
in the loss of solute atoms by diffusion to the matrix.
In
the case of heavy ion or fast neutron irradiation (Z) will be
of the order of the cascade size, i.e., 100 A.
Furthermore,
22
suppose that only a fraction, f, of such solute atoms
actually become dissolved.
Then the dissolution rate is
simply given by
dv/dt = -47r 2 Zfk
CHAPTER 3
THEORETICAL MODEL
3.1 Equations of Motion
The following section follows Maydet and Russell.(4)
They considered spherical precipitates which are incoherent
with the matrix, and are therefore good sinks for vacancies
and interstitials.
The matrix and the precipitate are binary
substitutional solutions dilute in solute and in solvent,
respectively.
The precipitate may then be characterized by
two variables: the number of solute atoms (x) and the number
of excess vacancies (n).
Thus
n = a-x
(1)
where (a) is the number of matrix atoms displaced by the
precipitate.
For example, in a dilute Al-Si alloy the Si
precipitate has a 20% volume difference and every 4 free Si
atoms require a vacancy in forming a Si precipitate.
A
precipitate with a greater atomic volume than the matrix would
thus have n>O to relieve the strain energy, and an undersized
precipitate would have n<O.
The behavior of a precipitate
particle may then be described by its movement in a phase
space of coordinates n and x.
The processes giving rise to motion are shown in
Figure 2.
A particle moves in the x direction by transfer
of solute atoms across the incoherent matrix: precipitate
interface.
The arrival of vacancies or self-interstitials
at the particle or the loss of vacancies gives movement in
the n direction.
The rate of mass transfer due to irradia-
tion sputtering is expected to be low for incoherent
precipitates, and is ignored.
The particle moves with a
velocity equal to the frequency of addition times the jump
distance- in this case, unity.
k = Bx (n,x) - ax (n,x)
v, (n,x) - av (n,x) - Bi (n,x)
A =
where Bx (n,x),
8,
(2)
(3)
(n,x) and Bi (n,x) are the arrival rates
of solute, vacancies, and interstitials, respectively.
ax (n,x) and av (n,x) are the rates of loss of solute and
vacancies, respectively.
The B's are determined by the
concentrations and mobilities of the respective point defects.
The rate of emission of vacancies and solute atoms
may be determined by a method developed in analyzing void
nucleation.(16)
First, it is assumed that these emission
rates are unaffected by the presence of the nonequilibrium
irradiation-induced interstitials.
The interstitial concen-
trations are very small, so the assumption should be valid
unless the arrival of an interstitial at the precipitate
somehow "triggers" the emission of a vacancy or solute atom.
This being unlikely, av and ax may be calculated from the
8V
W
(I)
z
U
C
ýGx
ax
VT W'e
x (ATOMS)
Fig. 2.
Phase space for Particle Trajectories, Showing
Processes Giving Rise to Motion.
= Rate
-B
of
Solute Capture, a, = Pate of Solute Emission,
Bv = Pate of Vacancy Capture, av = Rate of
Vacancy Emission, Bi = Rate of Self-Interstitial
Capture.
26
principle of detailed balancing applied to a system without
First, one envisions the prevention
excess interstitials. (17)
of particle growth beyond some size.
Then, particles below
this size will become equilibrated with one another, in which
case each process and its inverse occurs at the same rate,
i.e., detailed balancing obtains.
mer
Then, the number (n,x-l)-
gaining a solute atom to become (n,x)-mer is exactly
balanced by the reverse process, as are the (n-l,x)-mer
gaining a vacancy to become (n,x)-mer.
Denoting the equili-
brium number of (n,x)-mer as
-1
pO(n,x) = 9m lexp(-AG (n,x)/kT)
(4)
where Qm = matrix atomic volume and AGO(n,x) = free energy
of forming the particle from n matrix vacancies and x solute
atoms.
Then
8v (n-l,x)po(n-l,x) = av (nx)po(n,x)
(5)
8x (n,x-l)po(n,x-l) = ax (n,x)p°(n,x)
(6)
Replacing differences in AGO(n,x) by derivatives gives
av (n,x) % Rv(n,x) exp(l/kr(aAGO(n,x)/an)
(7)
ax (n,x) R 8x(n,x) exp(l/kT)(aAG*(n,x)/ax)
(8)
where the slowly varying Bi(n-l,x) and 8x(n,x-l) have for
simplicity's sake been replaced by their values at (n,x).
The dependence of
8v
,
Bx and AGo on (n) and (x) will
hence forth be understood.
Standard nomenclature in nucleation theory adopted
from polymer chemistry (cf. dimer, trimer, etc.), here (n,x-l)mer
hpere signifying a prec:ipit:ate -cQntaining--n -:e-xcessvacancies and
xr-1 'olute. atoms.
The free energy change on forming a precipitate
particle from a solid solution supersaturated with solute
and vacancies has been calculated from the capillary model.(18)
AGO = - xkT In Sx - nkT In Sv + (367rQ2)1/
yx3 + xAgs
(9)
where Sx = ratio of the actual and saturation concentrations
of solute, Sv = ratio of actual and saturation concentrations
of vacancies, Q= atomic volume of precipitates, Y = particlesmatrix surface energy, Ag. = strain energy per molecule of
precipitate, where this strain energy can be calculated (19)
from physical properties of matrix and precipitate as follows:
Ag s = 2pmC, (VP _ Vm) 2 /3VP
(10)
where
Um
C, =
Em
2(vm + 1)
3KP
3KP + 4pm
3P -2P)
3 (1-2vP)
(11)
(12)
(13)
where m and p superscripts denote the matrix and precipitate
quantities, respectively,
E = Young's modulus,
v = Poisson's ratio,
V = specific volume,
KP = effective bulk modulus, and
Pm= shear modulus.
Writing strain energy in terms of the number of molecule (x)
and number of excess vacancy the result becomes for an oxide
dispersoid M 2 0 3
AgS = 2AQmcC
6
(6-n/x) 2
(strain energy/molecule of PPt.)
(14)
where
6
-
m and 6 =
50s
,
m
250,
for substitutional and interstitial oxygen atoms, respectively.
In the case where elastic properties of matrix and precipitate
are equal, the strain energy per molecule of PPt. becomes
Ags =
gE(6-n/x) 2
9(1-v)
(15)
(15)
In Equation 9, the first two terms reflect the effects
of solute and vacancy supersaturations, the third the surface
energy contribution, and the fourth accounts for the strain
energy associated with the particle having either more or
fewer vacancies than are required to render it stress-free
(as occurs at n/x = 6).
We may now write
R = 8x( 1 - exp(l/k1(3aAGo/Dx)])
1
= Bv( 1 -Si/ýv-
exp[(l/kT)(DAGo/On)])
(16)
(17)
where
(l/kT)aAGO/ax = -lnS x + AX-1/ + B(6 2 -n 2 /x 2 )
(18)
(1/kT)aAGO/an = -InSv - 2B(6-n/x)
(19)
and
A =
2/ 3
(20)
(367q2)/ 3 (Y/kT)
(21)
B = 2/3 QpmC /kT
Equations 16 and 17 may now be used to calculate particle
trajectories in (n,x) space in terms of defect concentrations
and mobilities, surface energies, and other quantities which
may be either measured or calculated.
3.2 Nodal Line Formalism
Nodal lines are obtained by setting Equations 16 and
17 for k and A to zero, and plotting the results; this is done
schematically in Figure 3.
The arrows show the direction of
motion of particles at various points in the plane.
*
If the
*
nodal lines intersect (as at n, x, Figure 3),
the point of
intersection is known as a critical point.
Linearizing the velocities at this point gives (after
Somarji
(20 ) )
= a(x-x ) + b(n-n )
h = c(x-x ) + d(n-n )
(22)
(23)
If the roots are equal or if x or n are highly nonlinear near the critical point, other more complex configurations are possible.
n
n'
0Fig. 3. Schematic Illustration of Nodal Lines, Critical
Point, and Particle Trajectories. (After Ref. 4)
31
11, X2 of the characteristic equation
The roots,
X2 - (a+d)X + ad-bc = 0
(24)
X 1 X2 = 1/2 {(a+d) ±[(a-d)2 + 4bc] 2}
(25)
dictate which of four basic critical points shown in Figure 4
will occur.
(The critical points in Figure 4 are arranged
so that the nodal lines are along the coordinate axes.)
The
classification of critical points is:
if XX 2>0 : X1 and X
if X 1X 2<0 : 1
if
X,
and X
real : node,
real : Saddle Point,
= P+iv,
= p-iv
X
focus if
P00
center if p=O
The arrows in Figure 4 indicate particle velocities in the
various parts of the (n,x) plane.
Particles migrate fairly
directly into the stable node, spiral into the stable focus,
and circle about the center.
Particles approach the saddle
point in two quadrants and retreat in others.
(The critical
point in Figure 3 is thus a saddle.)
The critical points are characterized by the Poincare
index, j, which is defined as the number of clockwise rotations of the velocity vector during a clockwise circuit of
the critical point.
The Poincare index is given by:
j = (ad-bc)/jad-bcl
It is seen from Figure 3 that j = +1 for the node, focus,
(26)
X2
X2
6
______
(b)
X2
X2
/i
XI
(c)
·
I ·
i
(a)
----~
Fig. 4.
·
I
.
(d)
Critical Point"- (a) Stable Mode
(b) Stable Focus (c) Saddle Point
(C) Center (After Ref. 4) .
33
and center, and j = -1 for the saddle.
Reversing the signs
of coefficients in equations 22 and 23 (which reverses the
direction of the arrows) does not change the sign of j.
The
sign change reverses the stability of the node and focus, but
not the saddle (always unstable) or the center (always stable).
3.3 Nodal Line Equations
Nodal lines are obtained by setting A and h individually equal to zero.
Any particle on an k nodal line may
have a velocity only in the n direction, and vice versa.
The
particle has zero net velocity at a critical point.
For A = 0
n = x{6+(1/2B) In
For
[Sv(l-Si/v)]}
(27)
= 0
n = x6{l+A/B 62 x1
3
_(l/B6 2 ) In Sx
(28)
12
The equation for the nodal lines are seen to be
independent of the kinetic parameters, other than
actually energetic in nature.
i/89 which is
Furthermore, Sv and Bi/8 v
appear only in A = 0 and Sx appears only in k = 0.
Figure 3 refers to a hypothetical system in which
6>0.
In the region above the f nodal line the particle
contains a surfeit of vacancies and has a greater tendency
for emission than for capture, so that A<0.
The situation is
reversed for the k nodal line in that the vacancy-rich
particle would increase its strain energy by solute emission,
34
which is thus relatively unlikely, and k>0.
Similar reasoning
applies for 6<0.
3.4 Critical Point and IrradiationModified Potential
Simultaneous solution of Equations 27 and 28 shows that
the critical point is located at an x
= -32ry3 0 2 /3(A4)
x
n
*
*
*
= x [~-(1/2B) In
*
and n
given by
3
Sv ( 1 -i/ýv)]
(29)
(30)
or a radius of
r
= -2yQ/Aq
(31)
where Aý is an irradiation-modified potential given by
AQ = -kT In Sx
[Sv(1-Bi/ýv)]6-(kT/4B) [ln Sv(1-Si/ýv)] 2
(32)
If A4>0, the nodal lines do not intersect (for x>0) and all
particles will eventually decay.
In the absence of excess
defects Aý = -kTlnSx and familiar Gibbs-Thomson equation is
recovered.
The n,x phase is divided into four quadrants which
meet at the critical point.
As seen in Figure 3, in quadrant
I all particles will grow and in quadrant III all particles
will decay; in quadrants II and IV particle survival depends
on the magnitudes of the kinetic parameters, Bv, 8x and Bi .
35
From the definitions, the * nodal line must be crossed
vertically (so that * = 0) and the h nodal line must be
= 0).
0
crossed horizontally (so that
This means that parti-
cles which find themselves between the nodal lines (either
above or below x) cannot escape.
grow, and those with x < x
Particles with x > x
will
will decay; in each case the path
lies somewhere between the nodal lines.
Particles not between
the lines will approach them, curving in to cross in the
appropriate direction.
A particle which is located at (or very
near): the saddle poink will be iimmoibilized at that:.point. .(This
is in the absence of statistical fluctuations, which are not
included in the model.)
Such a particle resembles the
critical nucleus in nucleation theory that it is a zero
growth rate situation.
These various trajectories are illus-
trated by particle flow lines in Figure 3.
3.5 Defect Conservation Equations
Expressions for the steady-state defect conservation
have been given by Brailsford and Bullough.(21)
They express
these as
K + Kth - DvCvvv2
K - DiCiJi
2
-
CvCi = 0
- cCvCi = 0
Solving this system of equations for the point defect
concentrations they obtain
(33)
(34)
36
c v = (Diki 2 /2a)[-(1-p)
+ {(l+p) 2 + n}2]
c i =(Dvkv 2 /2a)[-(l1+l)
+ {(l+p)2
+ n})
2]
(35)
(36)
with
n = 4 K/DiDvki 2kv 2
(37)
2
" = aKth/DiDvki kv2
(38)
Kth = DVCve kv2
(39)
where (kv)-1 and (ki)-1 are the mean distance, a free vacancy
or interstitial moves in the medium before becoming trapped.
So kv 2 = the sink strength for vacancies, per unit area
ki2 = the sink strength for interstitials, per unit area
Dy, D i = the diffisivity of vacancies and self-interstitials,
respectively, Cv,Ci = the concentration of vacancies and
self-interstitials, respectively, a = 1017Di = the recombination
probability, K = the atomic displacement rate, dpa/sec, and
Kth = the rate of thermal vacancy production. (2 2 )
Therefore, the vacancy supersaturation which has a
chemical effect on the stability can easily be calculated
Sv = Cv/Cveq
(40)
where
Cveq = exp(-Efv/RT)
(41)
with
Efv = energy of formation of vacancy,
R = gas constant, and
T = temperature in OK.
3.6 Point Defect Arrival Rates
The defect arrival rates at a hypothetical, nonbiased
sink one atom in size are:
B6 = DvCv/a2
(42)
2
(43)
= DiCi/a
8
with a = jump distance.
So the relative arrival rates of interstitials and
vacancies, ý9/84 (the point defect bias) may be expressed
through Equations 35 and 36 as
iCi/OvC v = k v-(l+P)+{(l+P) 2+n2 ]
DiCi/DvCv = ki 22 [v(1P)+{ (1+P)
ki
[-(i-)+{(i+p)'+n}
(44)
]
It is obvious when thermal emission of vacancies is negligible
the above equation simplifies to a
go
=
kv
k i2
Zd
ZiPd
v
(45)
Zi
where Zv,Zi = the bias factors for vacancies and interstitials,
respectively.
Zv and Zi are both of the order of unity, with
38
Zi being a few percent larger owing to the slight preference
for interstitials at dislocations.
The effective addition step for solute addition is
diffusion through the matrix, rather than across the incoherent
particle-matrix interface.
(2 3 )
Assuming the precipitate to be
a nonbiased sink, the gross rates of point defect addition
under steady-state irradiation conditions are
v= 8O
1k~
(46)
Bi = 8v Zv/Zi
(47)
Bx = Bv Cx
(48)
where Cx = solute concentration in the matrix.
CHAPTER 4
CALCULATIONS AND RESULTS
The theory of precipitate stability under irradiation
outlined here was applied to a number of different materials.
The analysis was confined to two materials: austenitic type
316 stainless steel (316SS) and aluminum with chromium oxide
and aluminum oxide particles dispersed in those matrixes,
respectively.
These alloys, as discussed before, are used in
fast breeder reactor and thermal reactor structural materials.
The neutron economy in reactors, especially thermal reactors,
is important in order to maintain the chain reaction in the
reactor.
So the oxides are selected so that they do not dis-
turb neutron population inside the reactor.
In general, 1-3w%
oxides are added to alloys in order to improve their mechanical
properties at higher temperatures (0.55 TM-0.65TM).(2r5)
These particles play as unbiased (neutral) sinks in the matrix
which have no preference for capturing one type of defect over
the other, where their effect was not taken into account in
our calculation.
Prototype computer algorithms used in the analysis are
presented in Appendix A.1, together with a short explanation
as to their utility.
39
40
4.1 Material Parameters
Where possible, actual material values were used in
the analysis.
In the case of (316SS) alloy, the nominal
composition of major alloying additions were assumed and
weighted averages of the appropriate quantities used.
For
316SS, composition is given in Table 1.
TABLE 1: Nominal Composition (WZ) of 316 SS
Fe
Ni
Cr
Mo
Mn
Si
64.95
13.30
17.30
2.33
1.72
0.40
C
.08 Max
The material parameters used in the investigation are listed
for each of the materials in Tables 2 and 3.
4.2 Dislocation Densities
Dislocation density plays important role to determine
vacancy supersaturation.
At high displacement rates the
vacancy supersaturation depends only slightly on the dislocation density, whereas at low displacement rates the vacancy
concentration decreases for high dislocation density materials.
For a fixed dislocation density and displacement rate the
vacancy supersaturation depends on temperature.
This occurs
because the vacancy flux is determined by the steady state
balance of vacancy production, by displacement and annihilation
at sinks, and by recombination with interstitials.
So
analytical expressions for a given material are important
input parameters because they reflect the microstructural
41
TABLE 2:
(316SS) and (Cr203)
.....
Physical Properties
m
Material
Parameters
316 SS
Young's Modulus
(dyn/cm2 )
1.93x10 12
Poisson's Ratio
0.30
0.28
26,25
Melting Point (OK)
1675
2539±25
27,28
151.99
24
2.68x10 1
Atomic Weight
Density (gr/cm3 )
Atomic Volume (cm3 )
Vacancy Diffusion pre
Exponent (Cm2 /sec)
Ref
Cr 2 0 3
7.84
2
5.21
-23
24,25
cal,24
-23
1.185x10
4.832x10
Cal
0.58
29
Vacancy Formation
Energy (ev)
1.6
29
Vacancy Migration
Energy (ev)
1.4
29
Lattice Parameter (AO)
Interstitial Motion
Energy (ev)
3.51
5.38
0.2
30,24
29
Interstitial Diffusion
Pre Exponent (cm2 /sec)
.001
29
Jump Distance (A')
2.54
Cal.
Recombination Prob (S-1)
10' 7 Di
21
Equilibrium Solubility of
Oxygen (at. frac)
.003
31
Surface Energy* (erg/cm2 )
2000
assumed
42
TABLE 3: Aluminum and Aluminum Oxide Physical Properties
Material
Parameters
Aluminum
Aluminum
Oxide
Ref.
Young's Modulus
dyn/cm 2
6.89xl0 1 1
3.79xl01
32,22
Poisson's Ratio
0.33
0.30
32,28
934
2345
24,24
Atomic weight
26.98
101.96
24,24
Density (gr/cm3 )
2.692
3.965
32,24
1.658xl0 - 2 3
4.248x10 - 2
Cal
430
24,34
Melting Point (OK)
Atomic Volume (cm3 )
2
Vacancy Diffusion Pre
Exponent (cm2 /sec)
1.71
Vacancy Formation
Energy (ev)
0.75
35,24,-
Vacancy Migration
Energy (ev)
0.73
35,24
Lattice Parameter (AO)
4.049
Interstitial Motion
Energy ev
5.13
.08
32,24
36
Interstitial Diffusion
Pre Exponent (cm2 /sec)
.008
37
Jump Distance (Ao)
2.86
Cal
Recombination Probability
(S-1)
1017D i
assumed
.0001
31
Equilibrium Solubility of
oxygen (atom fraction)
Surface Eenrgy* erg/cm 2
2000
assumed
I
*NOTE: Assumed surface energy between matrixes and precipitates.
This chosen value is not too far from surface energy in
other systems.
response of the material to a particular irradiation dose.
Here the literature is somewhat lacking, and data is not
available for different temperatures and dose levels.
Maydet,(30) by using the experimental data which
are reported by Brager and Straaslund (3 8 ) for dislocation
densities in neutron irradiated 316SS, has determined a
simple temperature dependent expression for dislocation
density which is shown in Table 4.
Because of insufficient
data for aluminum the same dislocation density as 316SS is
scaled (equality of dislocation density at same homologous
temperature) for this metal under irradiation.
In eaah
system, the vacancy-supersaturatiQn.was calculated on the
basis of constant and temperature dependent.dislocation
density.
TABLE 4: Dislocation Density Expressions
Material
Dislocation Density (cm/cm3 )
3x10 1 2
316 SS
316 SS
6.72x101 3 exp(-0.0115T)
Al
lx1011
Al
6.72x1013exp(-0.0206T)
Ref.
assumed
(30)
assumed
Scaled from
316 SS
4.3 Irradiation Conditions
Calculations are based on two various conditions of
irradiation.
4.3.1 High Displacement Rate
High displacement rate (K = 10- 3dpa/sec), which is
associated with high energy electron bombardment or heavy ion
bombardment techniques, allows simulation of neutron irradiation in a matter of hours of an amount of irradiation damage
equivalent to several years exposure in a fast reactor core
environment.(39)
4.3.2 Low Displacement Rate
The displacement rate in fast breeder reactors are in
the %l-Odpa/sec range in the center of the core. (4 0 )
The
following Table (8) shows the typical values where the
accelerated damage rates in some of the regimes used to
simulate neutron bombardment are apparent:
TABLE 5: Typical Value of Damage Rates
Damage rates characteristic of:
(a) Van de Graaf
(b) Van de Graaf or Variable Energy Cyclotron
(c) High Voltage Electron Microscope
4.4 Point Defect Arrival Rate and Concentration
Any dislocation in the solid exhibits a preferential
attraction for interstitial compared with vacancies, due to
the nonrandom drift of interstitials down the stress gradient
near the dislocation core.(41)
The trend in 8i/4 v is
in Figures 5 through 8 for (316SS and Al for different kinds
of irradiation conditions and dislocation densities.
These
curves are plotted by using Equation (44), and are seen to
be constant and equal Zv/Zi up the the point at which
thermal emission becomes significant, where Bi/
v
decreases
sharply before leveling off at essentially zero at the higher
temperature extreme.
However, it is seen that with increasing
displacement rate the curve shifts to the right and begins to
drop at higher temperatures.
This temperature shift is due
to the annihilation of point defects in the dislocation sink.
(DvCvPdI in equation (33) becomes significant with respect
to other terms.)
On the other hand, this shift is vacancy
concentration dependent, where a higher displacement rate
produces higher vacancy supersaturation and may be
annihilated by other mechanisms, such as recombination
with the opposite type of defect.
However, from Figures
13 through 16 can be inferred that when
316 SS
K= 10- 3 dpa /S
Nd= 3 X O1'2 cm/cm3
I
--- K=lO-6dpo/S
I
F-
0.9
C
0.8
o~
0.7
oqO
0.6
S0.5
o
2
- 00.3
0_ 0.2
Fig. 5. Point Defect Bias
0.1
-
I
.
r
I
I
I
1
4,
.
.25 .30 .35 .40 45 .50 .55 .60
HOMOLOGOUS TEMPERATURE (T/TM)
316 SS
Nd=6
7
-K=1O-
Versus T/Tr, in
316SS with Constant
Dislocation Density.
3 dpa/S
2 X 103 exp(-0.0115T) --- K=10- 6 dpa/S
1~
I.0
-
0.9
08
--
-
-
S0.7
o0zL
( 0.6
F-
0.5
C
-0.3
o
0
n_
U.I
0.1
n\
I
.25 .30 .35 .40 45 .50 .55 .60
HOMOLOGOUS TEMPERATURE (T/TM)
Fig. 6 Point Defect Bias
Versus T/TM in
316SS Using Dislocation Density Expression from Table 4.
08
-3
Aluminum
K=10
Nd= I X 1011cm/cm 3
K10-6dpa/S
---
dpa/S
47
-
E
0
>
07
o
n 0.6
m
S0.5
U
LL0.4
S0.3
a-,U.Z
Fig. 7 Same as Figure 5
except for Aluminum.
0.1
I
I
I
.25 .30 .35 .40 .45 .50
HOMOLOGOUS TEMPERATURE
Aluminum
6 72
Nd= .
-K=lO-
.55 .60
(T/TM)
3
dpo/S
X IO 3exp(-0.0206T) --- K=10-6dpa/S
0.9
0.8
0.7
0.6
m
H-
0.5
LL
0.4
0.3
a0-
_
0.2
Fig. 8 Same as Figure 6
except for Aluminum.
0.1
I
____
___
.25 .30 .35 40 .45 .50 .55 .60
HOMOLOGOUS TEMPERATURE (T/TM)
48
vacancy supersaturation becomes
100 the Bi/B( begins to fall
off at that temperature, and dislocation sinks overcome other
sinks which exist in the system.
Figures 9 through 12 show the rate of creation or loss
of point defect in 316SS with different displacement rates
and dislocation densities, where the individual terms of
Equation (33) are plotted versus homologous temperatures.
These
curves are useful in order to understand the role of each sink
or source in the determination of defect concentration at
certain temperatures.
For instance, Figures 9 and 11 are
plotted for two different displacement rates where the
dislocation density is fixed.
The thermal vacancy emission
curve does not change because of its temperature and dislocation density dependence (Kth = CveqD pd), but the dislocation
sink and recombination terms on Figure 11 are shifted to the
right as a result of higher recombination probability in the
higher vacancy supersaturation system.
The same discussions
can be applied for Figures 10 and 12.
Figures 13 through 16 are vacancy supersaturations
versus homologous temperatures for 316SS and Al for two
different conditions of irradiation.
The supersaturation is
observed to decrease quite rapidly with increasing temperature
and is effectively unity (Cv = C th) at the higher temperature
extreme.
Decreasing of dislocation density with temperature
gives a higher vacancy supersaturation with respect to constant
316 SS
-
Recombination
Nd=3 X 1012 cm/cm3 ---- Dislocation Sink
K=10-6dpa/S
--- Therma mission
U)
r
U0
z
0
ii
-20
/
0
0J--
0
Fig. 9 Rate of Defect
Creation or Losses
Versus T/TM in 316SS
with Constant Dislocation Density and
K = 10-6dpa/s.
//
I
-25
I
I
I
I
I
I
.25 .30 .35 .40 45 .50 .55 .60
HOMOLOGOUS TEMPERATURE (T/TM)
316 SS
-I
t-
Recombination
-
Nd=6.72 X 10 3exp(-0.0115T) --.. Sink
K=lO-6dpa/S
-.-- Thermal Emission
rr
0
7
0
/
LU
I ---
0Y-20
0D
-J
-L3
/
/
/
/7
II
I
I
I
I
.25 .30 .35 .40 45 .50 .55 .60
HOMOLOGOUS TEMPERATURE (T/TM)
Fig. 10 Rate of Defect
Creation or Losses
Versus T/TA
M in 316SS
Using Dislocation
Density Expression
from Table 4 and
K = 10-6 dpa/s.
-I
N
-
SS
316
=
3
1012
X
50
Recombination
Sink
cm/cm3----Distocation
K=lO 3 dpo/S
--
Thermal Emission
00
L-J
cO -5
//
Or)
o0
z
-10
LU-I0
w
o
w
U_ -15
W
LU
LL
0
LU
'<-20
Fig. 11 Same as Figure 9 exceot
/
01
-Jr
I
with K = 10-dpa/s.
I
I
I
I
I
.25 .30 .35 .40 45 .50 .55 .60
HOMOLOGOUS TEMPERATURE (T/TM)
316 SS
-
Recombination
Nd=6.72 X10 exo(-O.O115T) _ocation
-I K=10- 3 dpo/S
--- Thermal Emission
cn
W
O -5
01
o
,/-
z
/
-10
/7
LU
O
0
< -20
r-
.9_
LJ
Fig. 12 Same as Figure 10 except
with K = l0-3dpa/s.
,,,,,,,,,,,,,,,,
-20 .5• .50 .4U 4- .U0 .55 .bU
HOMOLOGOUS TEMPERATURE (T/TM)
316 SS
K=li-6dnn/.
_
Nd=3 X 102 cm/cm3
79
y
pl3Rx
p'
(-0Oll5T)
Fig. 13 Vacancy Supersaturation
versus T/TM
1 in 316SS
with K = 10-6dpa/s.
.25 .30 .35 .40 .45 .50 55 60
HOMOLOGO US TEMPERATURE (T/TM)
316 SS
20
K=10
-3
dpa/S
3
- Nd=3X1012 cm/cm
--- Ndz672XIO13exp(-0.0115T)
F
15
U)10O
0-J
5
Fig. 14 Vacancy Supersaturation
Versus T/TM in 316SS
with K = 10-3dpa/s.
0
.25 .30 .35 .40 45 .50
HOMOLOGOUS TEMPERATURE
55 .60
(T/TM)
Aluminum
-
K=lO-6dpa/S
- -- Nd=6.72 X 10 3 exp(-0.0206T)
Nd= IX 1011cm/cm
3
Ir
I'
Fig. 15 Same as Figure
13 except for
Aluminum
.25
.30
.35
.40
.45 .50
.55
HOMOLOGOUS TEMPERATURE (T/TM)
.60
Aluminum
-
Nd= IX IOllcm/cm 3
K=lO-3dpa /S
---
Nd=6. 7 2 X 1013exp(-O.0206T)
Fig.16 Same as Figure
14 except for
Aluminum
0
HOMOLOGOUS
TEMPERATURE
(T/TM)
53
dislocation density at higher temperatures (because the dislocations are unsaturable sinks for point defects, and a
decreasing number of such a sink means lower sites for
capturing the defects, which leads to higher vacancy supersaturation).
On the other hand, at high displacement rates,
where the recombinations are dominant, the vacancy supersaturation is slightly dislocation density dependent,.
especially at lower temperatures, as can be seen in Figure
14 and 16.
4.5 Critical Particle Size
By using Equation 32 the irradiation modified free
energy
A4
=-kTlnSx[Sv(l-Si/av)]6-kT/4B[lnSv((1-i
i/v)]2
(32)
can be calculated, and as mentioned before, if Ad<O there
will be a stable particle whose critical size can be calculated
by Equation 29; and if
Aq>0 represents that the nodal lines
do not intersect, no stable nuclei can exist and all
particles will decay.
There are several factors which determine the sign
and amount of irradiation modified potential energy (dA).
One of them is 6 (precipitate misfit), where negative misfit
means that solute atoms are in substitutional sites in the
precipitate; and the positive misfit sign shows that solute
atoms are accommodated in interstitial sites.
4.5.1 Thermal Equilibrium Condition
Before studying the stability of particles under
irradiation it is useful to consider them under thermal
conditions and see how they behave.
Expanding the first term of Equation 32, which is the
determinative term in almost all cases,
A4
= -kT[lnSx+6lnSv+6ln(l-ai/Bv)]
shows that the bracket sign determines the sign of Aý.
(49)
In
thermal conditions where there is no excess vacancy due to
irradiation the Sv = 1 and Bi/8 v = 0, so the Potential function
becomes
Aý = -kTlnSx
(50)
It is obvious when Sx>l the A4<0, and as a result there will
be a critical size, which decreases with the increasing of
temperature.
The critical size versus homologous temperature
(T/TM) for Cr 2 03 PPt. and Al 2 03 PPt. in 316SS and Al matrixes
are depicted in Figures 17 and 18.
4.5.2 Substitutional Solute Atoms
When the two kinds of atoms are more nearly of the same
size, a substitutional solid solution is formed, in which the
atoms of solute replace those of the solvent, so that the two
occupy a common lattice.
In such cases there is a distorted
region around each solute atom, and the relative size of atoms
of solvent and solute determine whether the lattice expands
316 SS + Cr2 0 3 ppt.
Sv= I
8
r
SX=2
6
E
S x =10
Fig. 17 Critical Parti-
2
cle Size Versus
>
T/T, in 316SS
Under Thermal
Conditions
I
I
I
I
I
I
I
.25
.30
.35
.40 .45
.50
.55
HOMOLOGOUS TEMPERATURE (T/TM)
I
.60
Aluminum+A12 0 3 ppt.
8
,. .=,-
Sx =2
6
Sx=10
5
o4
-j
3
Fic..
1
Same as Figure
17 exce-t
hfor
Aluminum.
I
I
I
I
I
I
I
.25
.30
.35
.40
.45 .50 .55
HOMOLOGOUS TEMPERATURE (T/TM)
I
.60
or contracts.(42)
However, the case at hand, which is
negative misfit, means that there is a contraction around
the precipitate, and as seen from Equation 40 the high
vacancy supersaturation actually helps to shrink the precipitate by the chemical effect of vacancy and to aggravate the
stability of the phase.
The effective term then is solute
supersaturation, which must overcome other terms and factors
in order to stabilize the precipitate.
Figures 19 and 21 show the critical particle size
versus homologous temperature for 316SS with Cr 2 0,
PPt.
In
Figure 19, in spite of increasing critical particle size with
decreasing temperatures, it is seen that the critical size
diminishes, for Sx = 1 at high temperatures T>0.55 TM, which
is the effect of the second term in Equation (32):.where
Aý = -kTlnSx[Sv(1-Bi/
v ) ] 6 k T/ 4 B )
[lnSv (1-Bi/v)12 (32)
However, the critical particle size is very large: about 107
order of magnitude larger than the critical size for Sx = 2.
Figures 22 and 24 are the same as Figures 19 and 21,
except for the Al-Al 2 03 PPt. system.
Although the vacancy
supersaturation in aluminum is less than 316SS, the critical
size of Al 203 in the aluminum matrix is larger than Cr 2 03 in
316SS matrix.
This is merely because of the smaller percipi-
tate misfit in aluminum than chromium oxide in the 316SS
system.
316 SS + Cr2 0 3 ppt.
Nd= 3 X 1012cm/cm 3
II
K
8=-0.185
K=10
6
dDo/S
/-17
-16
15
-14
X- 2ý
ý
06
Fig. 19 Critical Particle Size
versus T/TM in 316SS
under Trradiation K=10 - 6
dpa/s with Negative Misfit
and Constant Dislocation
Density.
5
4
3
2
I
I .25I .30I .35i 40I .45I .50I .55I .60
HOMOLOGOUS TEMPERATURE (T/Tm)
13ý=-()
617
+
'203 VPp-
IQR
Nd 3 X IO'2 cm/cm3 K= 10-6dpa/s
10
16
.,=I /
9
-15
1
8
14
6
SL=2
5
5
4
3
2
i i 1 I i i 1 i
.25 .30 .35 .40 .45 .50 .55 .60
HOMOLOGOUS TEMPERATURE (T/TM)
I
7
316 SS + Cr2 0 3 ppt.
Nd=6.7 2 X 1013 exp(-O.0115T)
Fig. 20 Critical Excess Vacancy
Versus T/Tk
1 in 316SS
Under Irradiation K=10 - 6
dpa/s with Negative Misfit
and Constant Dislocation
Density.
8=-0.185
K=lO-6dpa/S
--
Sx=lO
x
0
2J
3
2
I
X=1
.25 .30 .35 40 .45 .50 .55 .60
HOMOLOGOUS TEMPERATURE (T/TM)
Fig. 21 Critical Particle Size
Versus T/TD, in 316SS
Under Irradiation K=10-6
dpa/s with Negative Mlisfit
and Using Dislocation Density
Exnression from Table 4.
..
b =-U .4
Aluminum + A12 0 3 ppt.
Nd=3
X IOl0cm/cm 3
f
K=lOGdpa/S
S =2
7
6
* 5
94
Fig. 22 Same as Figure 19
except for Aluminum
3
2
1
8
I
i
I
I
I
I
I
1
.25 30 .35 .40 .45 .50 .55 .60
HOMOLOGOUS TEMPERATURE (T/TM)
Aluminum-Al 2 0 3 ppt.
8=-0.4897
Nd=l X IOIcm/cm 3
K=10-6 dpa/S
7
Sx= 2
6
5
0
34
Fig. 23 Same as Figure 20
exceot for Aluminum
2
I
I
I
I
I
i
.25 .30 .35 .40 45 .50 .55 .60
HOMOLOGOUS TEMPERATURE (T/TM)
Aluminum +A12 0 3 ppt.
-
-i6d.
""pI
[2
/IIl•'
3 ..
./
f
•
S=-0.4897
•
AIu•exp-u.u•u20
'T
/
)
I
-6•
I_
I
/
=iU dapa/S
Sx2
7
6
*
5
34
3
2
.25 .30 .35 .40 .45 .50 .55 .60
HOMOLOGOUS TEMPERATURE (T/TM)
Fig. 24 Same as Figure 21
excePt for Aluminum
From Equation 30
*
n
*
= x [6+(1/2B) inS (1-ýi/ v) ]
(30)
the critical excess vacancy content can be calculated.
Figures 20 and 23 show the critical excess vacancy content
versus homologous temperature for 316SS-Cr 03 and Al-Al 03
2
2
systems, respectively.
Figures 19 and 21 and 22 and 24 are mapped to show
the critical particle size of chromium and aluminum oxide
versus homologous temperatures in their matrixes, respectively.
It is seen that particles exist at high temperature with
solute supersaturation (Sx>l) , the critical and partical
size increases with a decrease of temperature.
This is due
to the main effect of excess vacancies, which acts as a
chemical component or a driving force to remove the solute
atoms from the precipitates.
At lower temperatures T<0.45TM,
the vacancy supersaturation is high enough to change the sign
of irradiation modified free energy in order to unstable
precipitates.
Figures 25 through 30 are the same as Figures 19
through 24, except for the irradiation condition, which is
10-3dpa/sec.
It is obvious that this amount of displacement
rate produces higher vacancy supersaturation due to the more
severe irradiation condition.
As a result, for a certain
homologous temperature, the critical particle size increases
(when 6<0) compared to the previous condition of irradiation
-
0
8=-o.185
316 SS +Cr 20 3 ppt.
N==3
XIO 2 cm/cm 3
··
K=10-3dpa/S
r
7
Sx2
,:L=
6
;
5
94
Fig. 25 Same as Figure 19
exceot K=1063 dna/s.
3
2
I
I
I
I
I
I
I
I
25 .30 .35 40 .45 .50 .55 .60
HOMOLOGOUS TEMPERATURE (T/TM)
& -O.
185
Nd=.3X 1 12LCm m'
Nd:3X 1012cm/cm :5
ppt
7
+Cr
SS
13
6
K=lO-3dpa/S
6
5
Sx=IO
4
3
Fig. 26 Same as Figure 20
except K-10-3 dpa/s.
I
I
I
I
I
I
1
I
.25 .30 .35 .40 .45 .50 .55 .60
HOMOLOGOUS TEMPERATURE (T/TM)
316 SS tCr20 3 ppt.
iO .
f- 7
Z7
=
Nd
.
2
X
e
xp
s=-0.185
pa/S
(
8
7C
6
Fig. 27
I
1
I
I
I I
1
I
.25 .30 .35 .40 45 .50 .55 .60
HOMOLOGOUS TEMPERATURE (T/TM)
Same as Figure 21
except K=10-3dpa/s.
8=-0.4897
Aluminum tAI20 3 ppt.
0
Nd=l XIOll
61
K=lO- 3 dpa/S
/cm 3
7
Sx:IO
S
6
S4
Fig. 28 Same as Figure 19
except for Aluminum
with K=l0- 3 dpa/s.
3
2
I 1
I I
I
.30 .35 .40 .45 .50 .55
1
i25
I
60
HOMOLOGOUS TEMPERATURE (T/T M)
Aluminum-Al•:= nnt
s=-04897
Nd= I X101cm/cm 3
7
K=10 3dpa/S
S 2
6
5
04
3
Fig. 29 Same as Figure 20
exceot for Aluminum
with K=lO 3dpa/s.
2
I
(
1
(
I
I
1
1
.25 .30 .35 40 .45 .50 .55. .60
HOMOLOGOUS TEMPERATURE (T/TM)
Aluminum+AI 2 0 3 ppt.
s0.4897
hi-7) I131 AT)~\
6.72X0
exp(-0.0206T)
/I-3,,/
K=-R
pOda/S
7
Sx=2
6
SXZIO
*x
0 5
34
3
Fig. 30 Same as Figure 21
except for Aluminum
with K=10l3 doa/s.
2
1
I
I
r\r
I
I
~r
L
n~
I
nr
1
L
rr\
rr
I
r~
S.HOMOLOGOUS
.TEMPERAUTURE (T/T.b
HOMOLOGOUS TEMPERATURE (T/T
M)
(K = 10-6dpa/sec).
It also changes the sign of (A4) at
higher temperatures.
4.5.3 Interstitial Solute Atoms
When the solute atoms are very much smaller than those
of the solvent, the solute atoms enter the interstices or
"holes" in the solvent lattice.
This kind of interstitial
solid solution is nearly always accompanied, by an expansion,
of the lattice of the solvent, and there is a local distortion
of the lattice in the region of each solute atom.
However,
the atoms in their immediate neighborhood rearrange themselves into a configuration of minimum energy.
It is evident from Equation (49) that the interstitial
solute
A4 = -kT[lnSx + 61nSv + 6ln(l-Si/Sv)]
(49)
atom (6>0) has a positive effect to keep the irradiation
modified energy's :sign negative; with increment of vacancy
supersaturation the absolute amount of A4 increases and as
a result the critical size decreases.
On atomic scale,
the addition of interstitial atoms to precipitate in irradiated
metal does not change the number of atom sites which were in
a precipitate, and actually increases the volume of precipitate.
Thus, as discussed before, in irradiated metals there
are net flow of vacancies and interstitials to the sinks (such
as incoherent precipitates, therefore, there is a defect
mechanism exchange which leads to a stability of precipitates.
63
Figures 31 and 33, and 34 and 36, show the critical
particle size versus homologous temperature for 316SS-Cr 2 03
and Al-A1 2 03 systems, respectively.
The displacement rate as
indicated on top of each graph is 10-6dpa/sec.
As is seen,
there exist critical particle sizes for a whole range of
temperatures, except for SxI 1 at high temperatures TZ0.40TM
where the excess vacancy which decreases with temperature
cannot compete with solute undersaturation at higher
temperatures.
Figures 37 through 42 are the same as Figures 31
through 36, except for the displacement rate (K = 10-3dpa/sec),
which produces higher vacancy supersaturation in the alloy,
and as a result, a smaller critical particle size compared
to K = 10-6dpa/sec.
It is obvious that at high temperatures
the effect of solute supersaturation becomes significant
where (Sv and (1-Bi1/v)+ i) and the (Aý) becomes a function
of solute supersaturation and temperature Aý = -kTrlnSx.
It
is seen, however, that the critical particle size increases
with increasing temperature when (6>0).
For solute super-
saturated solids (Sx>l) the critical particle size increases
up to a certain temperature and after that decreases.
These
conditions are due to competition between Sv and (1-ai/Bv):
one term decreases (Sv) with increasing temperature, while
the other term increases (1-8i/8v).
The critical sizes for all figures are calculated
for particular homologous temperatures (i.e., 0.2 5 TM, 0.30 TM
64
0.60 TM),
and so the temperatures which the AA' sign changes
are not calculated.
where Ac
= 0 the x
It is obvious that at that temperature
= w, which is somehow unrealistic.
316 SS +Cr20 3 ppt.
Nd=3XIOI2cm/cm 3
8=1.0374
K=IO-6dpo/s
II-
I098-
S76-
Fig. 31 Critical Particle Size versus
T/TM in 316SS under Irradiation
K=10- 6 dpa/s with Positive Misfit
and Constant Dislocation Density.
54
3
2
j
I
I
I
I
I
I
I
I
.25 .30 .35 .40 .45 .50 .55 .60
HOMOLOGOUS TEMPERATURE (T/TM)
/
8= 1.0.574
/I
IO' dpa/s
K= 10"6dpa/s
-C r20 33 ppt
ppt
I0I -- 316SS
516SS -Cr2o
=
10-- Nrl
Nd =53 X I012
1012 cm/cm
8=1.0374
6
9
8
7
Fig. 32 Critical Excess Vacancy versus
T/T in 316SS under Irradiation
K=l - 6 dpa/s with Positive
Misfit and Constant Dislocation
Density.
5
4
3
2-
I
I
I
I I
I
I
I1
.25 .30 .35 .40 .45 .50 .55 .60
HOMOLOGOUS TEMPERATURE (T/TM)
316 SS +Cr20 3 ppt.
M,792
6i
Y 1•
•.
II
3
8=1.0374
,vn(-nl
6
lIT
Kv=n1 .I
nn/c
Jiii
V
p'
F
10
9
8
7
o6-
Fia. 33 Critical Particle Size Versus
5-
T/TU in 316SS under Irradiation
K=10 6 dpa/s with Positive
.isfit and Usina Dislocation
Density Exnression from Table 4.
4
3
2
I
I
I
I
I1 I11
.25 .30 .35 .40 .45 .50 .55 .60
HOMOLOGOUS TEMPERATURE (T/TM)
8=+0.2758
Aluminum +Al2 0~ ppt.
Nd=I XIO I 1cm/cm
66
K=lO- 6dpa /S
*x
O
0
-j
Fig. 34 Same as Figure 31 except for
Aluminum.
.25 30 .35 .40 45 .50 .55 .60
HOMOLOGOUS TEMPERATURE (T/TM)
, Aluminum+Alp,-pnt
8 =+0.2758
II
K=lO 6 dpa/S
3
Nd=IXXIO"cm/cm
10
9
8
7
1
t-
Sx=2
0
-J
5
-/
4
Fig.
3
35 Sal:e as Ficqure 32 excent for
Aluminum.
2
I
i
i
I
;
i
I
1
i
.25 .30 .35 40 45 .50 .55 .60
HOMOLOGOUS TEMPERATURE (T/TM)
Aluminum+AI 2 0 3PPt.
13
7
Nd=6. 2 X 10 exp(-0.0206T)
8=+0.2758
K=10dpa /S
- 22
20
j18
-16
10
9
S8
07
6
5
4
Fi-.
3
2
I
I
,25 .30 .35 .40 .45 .50 .55 .60
HOMOLOGOUS TEMPERATURE (T/TM)
I
36 Same as Figure 33 excert for
Aluminur.
8= O374
316 SS +Cr 20 3 ppt.
,,Nd=3 XIO 2cm/cm 3
I"0
K=10- 3 dpa/S
9
8
E
Sx=l
5
Sx =O.
Fig. 37 Same as Figure 31
except K=10 -dpa/s.
I
I I
I
I
.25 .30 .35 .40 45 .50 .55 .60
HOMOLOGOUS TEMPERATURE (T/TM)
8=+ .0374
316 SS-Gr20o
ppt.
2
Nd= 3XIO cm/cm3
K=10
v
7
-3
dpa/S
-
c:5
Sx=
34
3
Fig. 38 Same as Figure 32
except K=10-3dpa/s.
2 rI
S.25
I
I
I
I
I
I
I
.30 .35 .40 .45 .50 .55 .60
HOMOLOGOUS TEMPERATURE (T/TM)
8= 1.0374
K=lO-3dpa/S
ppt.
316 SS+ Cr
7INd= 6 .72 XIO'20 33 exp(-0.0115T)
Sx=l
5
E
Fig. 39 Same as Figure 33
excent K=10 3dpa/s.
I
I
I
I
I
I
I
25 .30 .35 .40 45 .50 .55 .60
HOMOLOGOUS TEMPERATURE (T/TM)
Aluminum+A1 20 3 ppt.
8=0.2758
Nd= I XIO"cm/cm3
K=lO-3dpa/s
C
I!
-I
10
9
Sx=O.!
8
7
o 6
0
_J
5
4
Fig. 40 Same as Figure 31 except for
Aluminum with K=10- 3doa/s.
1
I
j
1
1
1
1
.25 .30 .35 .40 .45 .50 .55 .60
HOMOLOGOUS TEMPERATU RE (T/TM)
AlAIminum-AlW.
nnt
O..m/203
= . X..
II
IXIO1cm/cm3
Na
u
10
8=+0.2758
K=lO-3dpa/s
9
8
7
r
6
0
5
4
Pici_
41
5Thmp a
Fi~rnIrP
32
cx~ni-
fnr
Aluminum with K=10-ldpa/s.
3
2
II
~
I
.25 .30 .35 .40 .45 .50 .55 .60
HOMOLOGOUS TEMPERATURE (T/TM)
AluminumtA120 3 ppt.
S=+0.2758
Nd=672 XIdexp(-O.O206T)
K=10-3dpo/s
II
10
9
8
7
-6
5
4-
Fig. 42 Same as Figure 33 exceot for
Aluminum with K=10 3dpa/s.
32
.25 .30 .35 .40 .45 .50 .55
.60
HOMOLOGOUS TEMPERATURE (T/TM)
69
4.6 Particle Trajectories
Qualitative evaluation of the velocity equation has
provided a general description of conditions for particle
growth or dissolution, but has not provided actual trajectories
(i.e., growth or decay paths).
These were calculated from
equations (16, 17) for a couple of displacement rates in each
system.
In each case the starting point was a stress free
particle (x) and n = 6x vacancies.
The procedure was to calculate x and n, multiply by the
time increment, calculate a new value of n and x, and iterate
up to a total elapsed time t = 10 s hr.
The time increment
for most of the irradiative cases was chosen so that xAt/x
and nAt/n were each less than or equal to 0.01, where for
thermal decay or growth, because of high amounts of x/x and
n/n, the amount of time increment was equal to 10 -0.
The
resulting plots, Figure (43-47), describe the evaluation of
the particle in n and x, but not in time, except for the end
points.
We note, however, from equations (16,17) that x and n
are exponential in the deviation of n and x from the respective
nodal lines.
As such, one would expect an initial transient
period of relatively rapid movement toward a nodal line,
followed by slower, relatively steady growth or decay between
the lines.
4.6.1 Thermal Decay
Applying the free energy model to determine the
critical particle size shows that at thermal equilibrium
conditions where sv = 1 and ai/8 v = 0 (i.e., because of a
fairly large formation energy of interstitial and hence the
thermal equilibrium concentration of interstitial is very
low).
As a consequence, the arrival rate 8i of interstitials
in equilibrium with the actual vacancy concentration is very
small and will be neglected.
(4 4 )
There would be critical
particle size for Sx>l as depicted in Figures (17,18) for
(316SS-CrO23) and (Al-A1203) systems.
To see how the under critical size particles dissolve,
the following condition has been chosen to plot the (n,x)
phase space of a Cr 2 03 in 316SS matrix.
x = 1x10 -
6
Molecules where x
= 3.5x10 6
6 = -0.185
Sx= 2
T = 0.35 TM
The calculation shows that after 3.8 x 1010 years the radius
of such a precipitate decreases from 226 A0
(x = 106 molecules)
to 209 AO, which is a very slow decay process.
Figure 43 shows
the behavior of this particle during thermal aging at T = 0.3 5TM.
In addition, the calculation for thermal decay of under critical
size of A1 2 03 PPt. shows that it tends to decay (x is negative),
but it does not shrink too much after 10s hr. of aging time.
4.6.2 Irradiation Induced Growth
Particle trajectories for a positive misfit for various
values of solute supersaturation are depicted in Figures
(44-46).
The growth rates are seen to increase with increasing
THERMAL DECAY
316 SS + Cr 2 0
3
Sx=2
8=-0.185
ppt
T=0.35 TM
-1.85
Cn
w
S
-
%
i
t=O
I.75
z0o
x
-1.65
c
-1.55
I
-1 IR
7.5
I
I
8.5
X(X10
5
I
4.5
) (MOLECULES)
Fig. 43 Thermal Decaly of Cr20 3 in 316SS with S =2
and Nerative Misfit at ?=0. 3 5 TyM.
K=IO - 3 dpa/S
Irradiation Time= 105 hr
316 SS + Cr2 0 3 ppt.
T=0.35 TM
8 =1.037
10
9
A
A
C,
0
z
8
o
7
0
c0
6
-1
5
4
3
2
--
3
4
5
6
7
8
9
10
LOGIoX ( MOLECULES)
Fig. 44 Irradiation Induced Growth of an over critical
Particle Size under Irradiation K=10- 3 dpa/s at
T=0.35Tk, with Positive Misfit for various
Solute Sunersaturation Conditions.
316 SS+ Cr 2 0 3 ppt.
T=0.35 TM
K=10-6 dpa/S
Irradiation Time= 105hr
8=1.037
I0
=10
9
S8
z
O
•-
0
6
x
c5
4
3
2
1
I
2
3
4
6
5
4
X(X10 ) (ATOMS)
Fig. 45 Same as Figure 44 except
7
8
9
-=1O0"dDa/s.
II
-3
dpa/S
Aluminum + Al 2 0 3 ppt.
K=10
T=0.35 TM
Irradiation Time= 105 hr
S= + ) 9 7r
Sx=10
10
9
(1)
Lii
0z
(0
0
8
7
6
x
c
5
Sx=2
4
f Sx=l
Sx =0.5
3
2
I
VIV
A\A
I'
I
\I
MAt"
( IV
crCI II
i.LL
L%.
EC \
I
Fig. 46 Same as Figure 44 excent for Aluminum
75
the solute supersaturation.
The arrows on the curves show
the (n,x) phase space of a precipitate after 105 hr. of
irradiation with a certain amount of solute supersaturation,
and as seen, particles with various amounts of solute
supersaturation grow on the same path in (n,x) phase space.
The irradiation condition and other factors are shown on
the top of each curve.
4.6.3 Cyclic Irradiation
The following table shows the anticipated structural
materials requirement for fission breeders and fusion
reactors.(40)
As is seen, the number of power cycles per
year varies from 103 - 109, depending on the kind of reactor.
It means that during such an irradiative period the beam of
irradiation is on and off alternately.
Therefore, it is
proper to adjudicate the particle trajectory under such an
irradiation condition.
To pursue this way two different
cases are considered:
Case 1:
Positive misfit 6>0
Solute under saturation Sx<0.5
Aý<0 during irradiation period (K = 10- 6 dpa/sec)
Aý>0 during thermal period (k = 0 dpa/sec)
The temperature which fits these conditions for 316SS
was extracted from Figures (17,33) and was found to be T =
0.5TM.
Then an over critical particle with x = 107 molecules
76
TABLE 6: Anticipated Structural Materials Requirement for
Fission Breeders and Fusion Reactors
Parameter
Fission
Breeder
(Steel)
Magnetically
confined
fusion
Inertially
confined
fusion
Temperature (oC)
300-600
300-500(steel)
500-1000
(refract.)
300-500(steel)
500-1000(refract.)
Maximum displacement
rate (instantaneous
dpa/s)
%10- 6
3-10xl0-7 (mirrors
and tokamaks)
1-10x10-5 (theta
pinch)
Average* (dpa/yr)
%50
10-30
10-30
Helium gas production (at.ppm/yr)
%10
200-600(steel)
25-150
(refract.)
200-500(steel)
25-150(refract.)
Number of power
cycles (yr-')
%10
l10 (mirror)
10 3 -105 (tokamak)
3x106 (0 pinch)
107-109
Stress level(MPa)
60-120
60-120
100-200
Desired lifetime
conditions (dpa)
He (at.ppm)
100-150
20-30
Acceptable
>20
>400(steel)
>50-100(refract.)
Reactor life
300-1000
6000-20,000
(800-5000 refract.)
AV/V0 % (lifetime)
<5
<5-10
<10
Creep %(lifetime)
<1
<1
<1
Ductility
(% elongation)
>1
>1
>1
1-10
* 70% PF
molecules was taken to irradiate for one year of cyclic
radiation with 2.5 x 10 s cycle/year.
The calculation shows
that there is no change in precipitate size after one year.
77
Case 2:
Negative misfit 6<0
Solute supersaturation Sx = 2
A4>0 during irradiation period (K=I0-6
dpa/sec)
A4<O during thermal period (K=O dpa/s)
The temperature in this case for 316SS - Cr 2 0 3 system was
found to be T=0.45TM (from Figures 17,21).
The overcritical
size particle with x=5xl0 7 molecules was selected.
Theoreti-
cally, it should decay during the irradiation period; this
happens, but first it tends to grow and the number of excess
vacancy decreases until after about 2x10 6 sec it starts to
decay.
Figure 47 shows the behavior of such a precipitate
under cyclic irradiation.
CYCLIC IRRADIATION
I0
316 SS + Cr20 3 ppt.
T=0.45 TM
S x =2
-
K=10 - 3 dpa /S
8= -0.185
9.9
9.8
9.7
9.6
9.5
0
9.4
9.3
C-)
x
c
I
9.2
9.1
9
8.9
8.8
8.7
8.6
8.5
8.4
8.3
8.2
8.1
8
4.92
4.9
Fig.
5
4.94 4.96 4.98
X(X107 ) (MOLECULES)
47 Particle Trajectory of a Cr20
in
3
Preciiitate
316Sr, under C,,clic Irradiation.
CHAPTER 5
SUMMARY
The subject of the calculations is the microstructural
stability of oxides which are insoluble, thermodynamically
stable, and in a matrix which has undergone intense displacement damage.
These alloys were of interest because the
excellent thermal stability which results from the low
diffusivity of oxides in matrix and low solubility of oxides
was expected to produce good radiation stability.
The Maydet and Russell
(4 )
Theory for incoherent
precipitates with a little change in strain energy term due
to different elastic properties of precipitate and matrix, was
used to evaluate the stability of chromium and aluminum
oxides in 316SS and aluminum matrix respectively.
The arrival rate ratio of interstitials and vacancies
(B4/B4) decreases sharply before levelling off at essentially
zero at the higher temperature extreme.
Because of higher
defect migration energy in 316 stainless steel the temperature
where the Bi/84 curve begins to drop is higher than in
aluminum with the same condition of displacement rate and
dislocation density.
80
The vacancy supersaturation is observed to decrease
quite rapidly with increasing temperature and is effectively
unity at the higher temperature extreme.
For the same reason
the vacancy supersaturation in 316 stainless steel becomes
unity at a higher temperature than aluminum.
The critical particle sizes are seen to depend
crucially on the sign of the precipitate misfit where the
vacancy supersaturation helps to stabilize the positive misfit
nuclei, and critical size of this kind increases with increasing
of temperature (because of decrement of vacancy supersaturation).
On the other hand, the vacancy supersaturation destabilizes
the negative misfit particles, and critical sizes increase with
decreasing temperature.
The solute supersaturation actually
helps to stabilize the particles, and as seen, higher solute
supersaturation leads to smaller critical size particles.
Computer calculations were used to calculate the
arrival rate ratio of interstitials and vacancies (ai/ v), the
vacancy supersaturation with different displacement rates
(K=10-6dpa/sec and K=l0 - 3 dpa/sec), and two different dislocation density expressions for a range of temperatures
0.25 < T/TM < 0.60 for 316 stainless steel and aluminum
system.
Then the critical particle size versus homologous
temperature was calculated for negative and positive precipitate:matrix misfits.
Finally, the particle trajectories of
a precipitate with and without irradiation are calculated at
T=0. 3 5TM where the growth or dissolution behavior was in
81
accord with predictions; particles were found to dissolve in
the presence of a solute supersaturation, or to grow in the
presence of an undersaturation, depending on the sign of the
particle:matrix misfit.
It is seen that at a particular
condition of irradiation with different amounts of solute
supersaturation they grow or decay on the same path, and as
a result a higher amount of solute supersaturation accelerates
the growth mechanism.
The behavior of a particle under cyclic irradiation
(2.5xl0 s cycle/year) at high temperature were calculated.
The temperature and particle size were chosen so that the
particle grow under irradiation and decay under thermal
condition and vice versa.
It was seen that the particle size
did not change too much after such a year of irradiation.
APPENDIX A
PROTOTYPE COMPUTER ALGORITHMS
The FORTRAN IV algorithms presented here were among
those run at the Information Processing Center of the
Massachusetts Institute of Technology to generate data for
the present analysis.
An alphabetical labeling scheme will be used to
identify the particular programs prototypes presented here
are for 316 stainless steel and Cr 2 03 PPt. system.
The same
prototypes have been used for Al-A1 2 03 system.
A.1 Algorithm A
This algorithm calculates the vacancy supersaturation
(sv = Sv) and arrival rate ratio of interstitials and vacancies
(/i
v
= BIAS) and irradiation modified free energy (4 = PHI)
and creation or losses defects rates such as (recombination=
REC),
(dislocation sink = SINK) (thermal emission = KTH) at
various homologous temperatures (i.e., 0.25 < T/TM • 0.60)
for two different displacement rates = R where the dislocation
density (pd = ND) is constant.
Data is output through a page
listing.
82
A.2 Algorithm B
This algorithm is same as algorithm A, except
dislocation density = ND which is temperature dependent.
A.3 Algorithm C
This algorithm calculates the (n=N) and (x=X) phase
space of a particle during TOP = 10 s hr. of irradiation.
Data is output through a page listing.
A.4 Algorithm D
This algorithm is same as algorithm C except for
cyclic irradiation and irradiation time TOP = 1 year.
A.5 Prototypes
Massachusetts Institute of Technology 1978
0
ALGORITHM
A
THIS PROGRAM CALCULATES THE VACANCY SUPERSATURATION AND
CRITICAL EXCESS VACANCY AND CRITICAL PARTICLE SIZE AT
VARIOUS TEMPERATURE,WHERE DISLOCATIO\N DENSITY IS CONSTANT.
316 S.S(MATRIX)+CR203
(PPT.).
************************+++++++**********************
REAL NKoNU0ONDNUMMUMKONSKTHMU
DIMENSION SX(4)•R(2),DELTA(2),SV(16),T(8)oDV( 8),CVE(8),KTH(8),
IMU(8) ETA(16),CV(16),A(8) 8(8),BIAS(16)
DATA E0,NUOKAOND,AVGAMMAEMNUM/2.687EI2,0.2891.38E-16,
12.54E-8,3.0E12,6.022E23,20O00..1.93E12,0.30/
READ(5,1)(SX(I) I=1,4)
1 FORMAT(4F10.2)
READ(5,2)(R(I)• I=1,2)
2 FORMAT(2F1o.6)
OMEGAM=1*18546E-?3
READ (5,5)WOgRO
5 FORMAT(2Fln.5)
PI=3.1415
THIRD=1.0/3.0
TWOTHD=2*THIRD
OMEGAO=WO/AV
DELTAS=((I.O/5.0)*OMEGAO-OMEGAM)/OMEGAM
DELTAI=((1.0/2. 0 )*OMEGAO-OMEGAM)/OMEGAM
DELTA(I)=DELTAS
DELTA(2)=DELTAI
MIM=EM/(2.f*(NUM+1.0))
KO=EO/(3.0*(1.0-.00*NUO))
C6=3,0*KO/(3.0*KO+4.0*MU4)
*
*
*
ý
0
READ(5,6)(T(I),I=198)
6 FORMAT(8F7.2)
DO 10 I=1,8
A(I)=2.0*(36.O*PI*OMEGAO**2)**THIPD*GAMMA/(3.0*K*T(I))
B(I)=2.0*MUM*C6*OMEGAO/(3.O*K*T(I))
10 CONTINUE
ENGF=1.6
ENGM=1.4
D0=0.58
EC=11604.854
ZI=1.02
ALPHA=1 .OE17
Q0=2000,0**3.
RR=OMEGAO**2.
WRITE(6,33)
33 FORMAT(2X,,SX',6Xt8IAS,,7x,DEFLTA',6X,'TEMP'912X,'SV',18X, XS',
ND,14X, DPA'//)
118X,'NS ° ,18X,
Dn 40 I=19•
DV(I)=DO*EXP(-ENGM*EC/T(I))
CVE(I)=EXP(-ENGF*EC/T(I))
KTH(I)=DV(I) *CVE(I)*ND
MU(I) =ALPHA*KTH(I) / (DV(I) *ZI*ND*ND)
ETA(I)=4.O*ALPHA*R(1)/(DV(I)*ZI*ND*ND)
40 CONTINUE
DO 50 J=9,16
I=J-8
ETA(J) =4.O*ALPHA*R (2)/ (DV (I)*ZI*ND*N))
50 CONTINUE
DO 60 I=1,8
Sc=((1.0 -MU(I))+ (ETA(I)*(1,0+MU(I))**2)**0.5)
0
C
C
C
C
TT=(-(1.0+MU(I))+(ETA(I),(1.0+MU(I))**2)**0.5)
CV (I)=ZI*ND*SS/(2.O*ALPHA)
BIAS(I)=TT/(ZI*SS)
SV(I)=CV(I)/CVE(T)
60 CONTINUE
DO 70 J=9,16
I=J-8
SS=(-(1.O-MU(I))+(ETA(J)*(1.0+MU(I))**2)**0.5)
TT=(-(il.0MU(I))+(ETA(J)*(1.MO*U(I))**2)**0.5)
CV(J) =ZI*ND*SS/(2.0*ALPHA)
BIAS(J)=TT/(ZI*SS)
SV(J)=CV(J)/CVE(I)
70 CONTINUE
FOLLOWING DO STATEMENT HAS BEEN USED TO CONSIDER THERMAL CONDITION
CALCULATION WHICH IS SV=1 AND BIAS=O.0 AND SIMPLY CAN CANCCEL FOR
THE REGULAR CALCULATIONS WHERR THE IRRADIATION DISTURB THFRMAL
CONDITION.
DO 83 I=1916
SV(I)=1.0
BIAS(I)=0.0
83 CONTINUE
DO 1001=1,4
DO 100 J=1,2
DO 100 II=1,8
PHI=-K*T(II)*ALOG(SX ( I)*(SV(II)*(.0BIAS(II) ) ) **DELTA(J))
1-K*T(II)*(ALOG(SV(II)*(1*0-BIAS(IT))))**2/(4 0*B(TII))
IF(PHI.EQO.00) GO TO ?9
XS=-32.O*PI*OO*RR/(3.0*PHI*PHI*PHI)
NS=XS*(DELTA(J)+(ALOG(SV(II)*1 (J-BIAS(II))))/(2,0*B(II)))
WRITE(6,22)SX(I),BIAS(II),DELTA(J),T(I I)SV(IT) ,XSNSNDP(1)
22 FORMAT(1X,F4.1,4XF7.4,4X,F7.4,4XF6.1,4XE16.8,4X•E16.8,4XE16.8
1 ,44,EX168,•1XPE10.2//)
GO TO 100
29 WRITE(6,32)
32 FORMAT(60X,'PHI=0')
100 CONTINUE
DO 200 I=1.4
DO 200 J=1,2
DO 200 II=9.16
JJ= I-8
PHI=-K*T(JJ)*ALOG(SX(T)*(SV(I)*(1.-BIAS(I)
) )**DELTA(J) )
1-K*T(JJ)*(ALOG(SV(II)*(1,0-BIAS(II))))**2/(4.0*B(JJ))
IF(PHI.EQ.0.0) GO TO 39
XS=-32.0*PI*QQ*RR/(3.O*PHfI*PHI*PHI)
NS=XS * (DELTA(J)+(ALOG(SV(II)*(1.0-BIAS(II))))/(2,0*B(JJ)))
WPITE(6,23)SX(I) BIAS( I)
DELTA(J),T (JJ) SV(II) ,XSNSND•(2)
23 FORMAT ( XF4
,4XF7.4,4XF7.4,4XF6.1 ,4XE16.894XE16.894XE16.8
1,4X,E16.89,4X, PE102//)
GO TO 200
39 WRITE(6,42)
42 FORMAT(60X,'PHI=0')
200 CONTINUE
STOP
END
0
0
ALGOITTH4
THIS
PROGRAM
CRITICAL
DENSITY
316
CALCULATES
EXCESS
IS
S.S(MATRIX)+CR203
THE
VACANCY
TEMPERATURE
8
AT
VACANCY
VARIOUS
SUPERSATURATION
TEMPERATURESWHERE
AND
BTAS
AND
*
DISLOCATION*
DEPENDENCE.
(PPT.).
******************************wo+++*******************************
I
I
I
REAL N,K,NUONDNUMMUM, ONSKTHMU
DIMENSION SX(4),R(2),DELTA(2),SV(16),T(8),DV(8) ,CVE(8) KTH(8),
IMU(8),ETA(16),CV(16),A(8)8, (8),ND(B),BIAS(16)
DATA EO,NUO, KAOAVGAMMA,EM,NUM/2.687E 12, O.28, 1.38E-169
12.54E-8,6.022E23,2000., 1.93E12,0.30/
READ(5,1)(SX(1),I=1I4)
1 FORMAT(4F10.2)
READ(5,2)(R(I),I=1,2)
2 FORMAT(2F10.6)
OMEGAM=1. 18546E-23
READ (5,5)WORO
5 FORMAT(2F1O.5)
PI=3.1415
THIRD=1.0/3.0
TWOTHD=2*THIRD
OMEGAO=WO/AV
DELTAS=((1.O/5.0)*OMEGAO-OMEGAM)/OMEGAM
DELTAI=((1.0/2.O)*OMEGAO-OMEGAM)/OMEGAM
DELTA (1)=DELTAS
DELTA(2)=DELTAI
MUM=EM/(2.0*(NUIM+1.0))
KO=EO/(3.0*(1.0-2.0*NUO))
C6 = 3 .0*KO/(3. 0*KO+4.oMU4)
1
"
0
0
0
0
READ(S,6) (T(I),I=1,8)
6 FORMAT(8F7.2)
DO 10 I=198
A(I)=2.0*(36.0*PI*OMEGAO**2)**THIPD*GAMMA/(3.*K*T (I))
B(I)=20O*MUM*C6*OMEGAO/(3.0*K*T(I))
ND(I)=6.72E13*EXP(-0O0115*T(I))
10 CONTINUE
ENGF=1.6
ENGM=1.4
DO=0.58
EC=11604.854
ZI=1.02
ALPHA=1.OE17
QQ=2000.0*o3.
RR=OMEGAO**2.
WRITE (6,33)
33 FORMAT(2X*,SX',6X,'BIAS', 7 X,*DELTA,96X,*TEMP*'12KXSV',18X,*XS'
118X 'NS* •l•X
, 9 NO, 14X DPA//)
DO 40 I=1,8
DV(I)=D0*EXP(-ENGM*EC/T(I))
CVE(I)=EXP(-ENGF*EC/T(I))
KTH(I)=DV ()*CVE(I)*NOD(I)
Pp=DV(I) *ZI*N (I)*ND (I)
MU(I) =ALPHA*KTH(I)/PP
ETA(I)=4.0*ALPHA*R( )/PP
40 CONTINUE
DO 50 J=9,16
I=J-8
ETA(J)=4.0*ALPHA*R (2)/(DV (I)*7*ND(I)*ND ())
50 CONTINUE
0
0
·
DO 60 I=1,I
SS=(- (1O-MU(I)) *(ETA (I)+(IO+MU(I))**2)**0.5)
TT=(( 1.0+MU(I))+ (ETA(I)+(1OMU(I)) **2)**0.5)
CV (I) =ZI*ND(I) *SS/(2.O*AL-PHA)
BIAS(I)=TT/(ZI*SS)
SV(I)=CV(I)/CVE(I)
60 CONTINUE
DO 70 J=9916
I=J-8
SS = (-(1.O-MU(I))+(ETA(J)+(10+MU(I))**2)**05)
TT= (-(1 .0 MU(I)) (ETA(J)*(1 0+MU( I))**2) **05)
CV(J)=ZI*ND(I)*SS/(2.O*ALPHA)
BIAS(J)=TT/(ZI*SS)
SV (J) =CV (J)/CVE (I)
70 CONTINUE
DO 1001=1,4
DO 100 J=1,2
DO 100 II=198
PHI=-K*T(II)*ALOG(SX(I)*(SV(II)*(1.0-81AS(II)))**DELTA(J))
1-K*T(II)*(ALOG(SV(II)*(1*0-BIAS(Il)))))*2/(4°0*B(II))
XS=-32.0*PI*QQ*RR/(3.0O*PI*PHI*PHI)
NS=XS*(DELTA(J)+(ALOG(SV(II
)*(1.0-81AS(II))))/(20*B(ITI)))
WRITE(6,22)SX(I),BIAS(II) ,DELTA(J),T(II),SV(II),XSNSND(II),R(1)
22 FORMAT(lX,F4,1t4XF7.4,4X.F74,4XF,
F6.1,4XE16.8,4XE16.889XE16.8
1,4X9E16.8#4X 1PE10O2//)
100 CONTINUE
DO 200 I=1.4
DO 200 J=1,2
DO 200 II=9916
JJ=I 1-8
0
PHI=-K*T(JJ)*ALOG(SX(T)* (SV(II)*(1.o-IAS(
) ))**DELTA(J))
1-K*T(JJ)*(ALOG(SV(I)*
1*(IO-BIAS(II))))**2/(4.0*B(JJ))
XS=-32.O*PTIQQ*RR/(3.0*PHI*PHI*PHI)
NS=XS*(DELTA(J)+(ALOG(SV(II)*(1.O0-1AS(II))))/(2.0*8(JJ)))
WRITE(6,23)SX(I),BIAS(II),DELTA(J)T(JJ)SV(If),XSNSoND(JJ),R(2)
23 FORMAT ( XF4. 1,4XF7.4,4X F7.4,4X
4XF6
F6. 1 4,EI6.8,4X
8
~E16.8,4X E16.8
1 4Xv,E16.84X IPE10.2//)
200 CONTINUE
STOP
END
0
0
·
·
ALGORITH'
C
C
C
C
*
*
*
*
·
·
·
C
THIS PROGRAM CALCULATES THE TRAJECTORIES IN NX-SPACE OF
PRECIPITATES IN A SUBSTITUTIONAL BINARY ALLOY SYSTEM (ALL
PARAMETERS ARE GIVEN IN C.G.S. UNITS)*
316 S.S(MATRIX)*CR203 (PPT.),
C
REAL NKNUO0INCRNDNUMMUPMKONS
DATA EO0NUOKAONDAVGAMMAEMNUM/2.687EI2,O.28l1.38E-16e
12.54E-8,3.0E12,6.022E?232000.0,1.93E12,•.30/tTOPTM/3.0E7,1675.0/
DIMENSION SX(4),R(2),DELTA(2),SV(2)9BOV(2)
READ (5,1) (SX(I),I=1,4)
1 FORMAT(4F10.2)
READ(5S2)(R(I),I=1,2)
2 FORMAT (2F10.6)
READ(5,3)(SV(I)I=1.?2)
3 FORMAT(2F12.1)
READ (5,5)WOtRO
5 FORMAT(2F1,.5)
OMEGAM=1.18546E-23
BIAS=098
T=0.35*TM
EC=11604.854
LTM=S5000
PT=3.1415
TS=1.OE-2
THIRD=1.O0/3.0
TWOTHD=2*THIRD
OMEGAO=WO/AV
DELTAS=((1.0/5.0)*OMEGAO-OMEGAM)/OMEGAM
*
*
*
*
O
0
0
0
0
0
0
0
DELTAI=((1.0/2.0)*OMEGAO-OMEGAM)/OMEGAM
DELTA(1)=DELTAS
DELTA(2)=DELTAI
MUM=EM/(2.0*(NUM+1.0))
KO=EO/(30*(1. 0-2.0*NUO))
C6=3.O*KO/(3.O*K0+4.0*MUM .)
A=2.0*(36.0*PI*OMEGAO**2)**THIRD*GAMMA/(3.*0K*T)
B=2.0*MUM*C6*OMEGAO/(3.0*K*T)
D=0.58*EXP(-3.0*EC/T)
WRITE(6,11)A989C69KOtMUM
11 FORMAT(SX9,A=*,E16.8,//5X•'3=tE16.89//5XtC6=,E1I6.8,//SX,
1'KO=',E16.89//5X,'MUM=' ,E16.8,//)
DO 76 I=192
BOV(I)=D*SV(I)/(AO*AO)
76 CONTINUE
Do 30 11=1,2
00 30 1=1,2
DO 30 J=1,4
PHI=-K*T*ALOG(SX(J)*(SV(1)*(1.0-PIAS))**DELTA(II))-K*T*(ALOG(
1SV(I)*(I.O-PIAS)))**2/(4*8)
XS=-32.0*PI*GAMMA**3*OME3AO**2/(3*PHI**3)
NS=XS*(DELTA(I1)+ALOG(SV(I)*(1.0-PIAS))/(2?*))
RS=-2.O*GAMMA*OMEGAO/PHI
WRITE(697)OELTA(II),R(I),SX(J),SV(1),0MEGAOOMEGAMPHI,XSNSRS
7 FORMAT(5X9 'PPT.MISMATCH=',F7.3,//5X*,DAMAGE RATE=,*1PE10.2,//5X*
1'SX=',lPE10.2,//SX,'SV=*19PE10.29//SX*'ATOMIC VOLUME OF OXIDE = t*
21PE10.3,//SX 'ATOMIC VOLJME OF MATRIX=',1PE10.3,//SX ,PHI=',
31PE10.2,//5X9,XS=O,1PFIO.3,//5X *NS='91PE10.3,//5X,*RS= ,1PE1O.3)
WRITE (6,27)
27 FORMAT(1HO,7X9,DXDT',14X9,DNDT*,14X9,INCR*, 14X,'SUM * ,
0
0
•0
0
w
A
a
116X.'Xl ,14X,'N'//)
X=1.OE4
N=DELTA(II)*X
IFLAG=O
INCR=O.0
SUM=0.0
DO 30 M=I1LIM
BV=BOV(I)*X**THIRD
BX=CX*BV*SX (J)
55 DNDT=BV*(1.0-PIAS-EXP(-2*.0*8*(DELTA(I)-N/X))/SV(I))
DXDT=BX*(1.O-EXP(A/X**THIRD*B*(DELTA(11)**2-(N/X)**2))/SX(J))
SUM=SUM+ INCR
IF(IFLAG.EQ50)IFLAG=0O
IF(IFLAG.EO.0) WRITE(6,8)DXDTDNDT,INCRSUMXN
8 FORMAT(IXE16.892X,E16.,82XE16.8,2X,E16.8,4X,1PE10.2,5X,
11PE10.2 /)
IFLAG=IFLAG+l
TI=TS*X/DXDT
IF(N.EQ.O.O)GO TO 40
T2=TS*N/DNDT
GO TO 50
40 T2=1.OE4
50 INCR=AMINI(ABS(TlI)ABS(T2))
X2=X
AN=N
X=X+DXDT*INCR
N=N+DNDT*INCR
DX=ABS(X2-X)
DN=AS (AN-N)
Q01=ABS(DXOT*INCR)
0
0
0
0
Q2=ABS (DNDT*INCR)
03=.O1*DX/X
Q4=.OI*DN/N
IF (Q1.LT.Q3) M=LIM
IF (Q2.LT.Q4) M=LIM
IF X.LT.1.O) M=LIM
IF(ABS(N)*LT.5.0)M=LIM
IF (SUM.GT.TOP)M=LIM
IF(M*EQ.LIM) WRITE(6,9) SUM
9 FORMAT(30X,'TOTAL TIME = 1,1PE9.2,'SEC')
30 CONTINUE
STOP
END
*
e
9
0
0
•
ALGORITHM
C
*
C
C
C
C
*
*
*
*
C
*
0
D
CYCLIC IRRADIATION
THIS PROGRAM CALCULATES THE TRAJECTORIES IN N,X-SPACE OF
PRECIPITATES IN A SUBSTITUTIONAL BINARY ALLOY SYSTEM (ALL
PARAMETERS ARE GIVEN IN C.G*S. UNITS)*
316 S.S(MATRIX)*CR203 (PPT.),
*****************************90*********
C
REAL N,KNUO,INCRND,NUM9MUM,KONS
DATA EO,NUOKgAONDAVGAMMAEMNUM/2.687E2.02891o.3RF-16,
12, 5 4E-89 3 .0E12,6.022E23,2000.0,1.93E1i,0.30/,TOPTM/3.0E7,1675.0/
DIMENSION SX(4),R(2),DELTA(2),SV(2),V
TS(2),BIAS(2)
READ (5,1) (SX(I),I=1,4)
1 FORMAT (F10.2)
READ(5,2)(R(I),I=1,2)
2 FORMAT (2F10.6)
READ(53)(SV(I)I=1,2)
3 FORMAT(2Fl2.1)
READ (5,5)WORO
5 FORMAT(2F10.5)
OMEGAM=1
18546E-23
BIAS(1)=O.O
BIAS(2)=0.98
CX=0.003
T=0.45*TM
EC=11604.854
LIM=5000
Mu=50
PI=3.1415
TS(1)=1.OE-9
*
*
00
0
00
TS(2)=1.OE-6
THIRD=1.0/3.0
TwOTHD=2*THIRD
OMEGAO=WO/AV
DELTAS=((1.0/5.0) *OMECAO-OMEGAM)/OMEGAM
DELTAI=((1.0/2.0)*OMEGAO-OMEGAM)/OMEGAM
DELTA () DELTAS
DELTA(2)=DELTAI
MUM=EM/(2.n*(NUM+1.O))
KO=EO/(3.0*(1.0-2.0*NUO))
C6=340*KO/(3.0*KO*4.0*MUM)
A=2.0*(36.0*PI*OMEGAO**2)**THIRD*GAM4A/(3*0*K*T)
B=2.0*MUM*C6*OMEGAO/(3.0O*KT)
D=0.58*EXP(-3.0*EC/T)
WRITE(6911)A,8,C6,KOMUM
11 FORMAT(5X,*A=',E16.8*//5X9'B=',E16.8,//SX,*C6= ,E16.8°//5X,
l*KO=*#EI6.89//5X,*MUM=*,EI6.8,//)
DO 76 I=1.r
BOV(I)=D*SV(I)/(AO*AO)
76 CONTINUE
11=1
J=3
WRITE (6927)
27 FORMAT(IHO97X,*DXDT9,14X,'DNDT0*14XIINCR',
116Xt'X'v14XqN'//)
X=1.0E7
N=DELTA(I1)*X
IFLAG=O
INCR=0.0
SUM=0.0
14X,'SUM',
0
S
O
·
·
0
0
0
0
0
DO 30 M=1,LIM
00 44 1=1,?
RT=O.O
47 BV=BOV(I)*X**THIRD
BX=CK*BV*SX(J)
DNDT=BV*(I.O-BIAS(I)-~XP(-2.0*B*(DELTA(I)-N/X))/SV(I))
DXDT=BX*(I,0-EXP(A/X**THIRDOB*(DELTA(Il)**2-(N/X)**2))/SX(J))
SUM=SUM+INCR
IF(IFLAG.EQ.MM)IFLAG=O
IF(IFLAG.EQOO) WRITE(6B,)DXDTDNDT,INCRSUMXN
8 FORMAT(1XE16.892XE16.892XE16.8,2X9E16.8,4XtlPE10.2,5X9
llFE10,2 /)
IFLAG=IFLAG+I
T1=TS(I)*X/DXDT
T2=TS(I)*N/DNDT
INCR=AMIN1(ABS(TI),ABS(T2))
X=XDXDT*INCR
N=N*DNDT*INCR
RT=RT+INCR
IF(RT.GT*60.0)GO TO 44
GO TO 47
44 CONTINUE
IF(X*LT.I.O)M=LIM
IF(ABS(N)*LT*5.O)M=LIM
IF(SUM.GT.TOP)M=LIM
IF(M*EQ.LIM) WRITE(6,9) SUM
9 FORMAT(30X,'TOTAL TIME = ',1PE9.2,'SECI)
30 CONTINUE
STOP
END
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R.S. Nelson, J.A. Hudson, Journal of Nuclear Materials
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(2)
E. Ruedl in Radiation Damage in Reactor Materials
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(3)
G. Mervin Ault and H.M. Burte in Oxide Dispersion
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(10)
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(11)
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(12)
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100
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"Radiation Produced
101
(14)
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(19)
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The Oxide Handbook, 1973.
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American Institute of Physics Handbook, Third Edition
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