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-
MICROCOPY RESOLUTION TEST CHAFT
NAP?VAL BUREAU OF STAYDAROS - 1963
LEGAL NOTICE
This report was prepared as an account of
Government sponsored work. Neither the
United States, nor the Commission, nor any
person acting on behalf of the Commission:
ORNI - AEC - OFFICIAL
A. Makes any warranty or representa-
tion, expressed or implied, with respect to
the accuracy, completeness, or usefulness
of the information contained in this report,
or that the use of any information, appa-
ratus, method, or process disclosed in this
report may not infringe privately owned
rights; or
B. Assumes any liabilities with respect
to the use of, or for damages resulting from
the use of any information, apparatus,
method, or process disclosed in this report.
As used in the above, “person acting on
behalf of the Commission” includes any em-
ployee or contractor of the Commission, or
employee of such contractor, to the extent
that such employee or contractor of the
Commission, or employee of such contractor
prepares, disseminates, or provides access
to, any information pursuant to his employ-
ment or contract with the Commission, or
his employment with such contractor.
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OLNE -
उमा
سا رے نے
زین - لیاری کے
l
in.
.
io. isinin
ORNI - AC - OFFICIA
Paper for presentation at the IAEA Symposium "Thermodynamics with
Emphasis on Nuclear Materials and Atomic Transport in Solids," Vienna,
July 22-27, 1965.
SM-66/59
-LEGAL NOTICE ---
The report was prepared as an account of (overomeot spunsored work, Nellher who Uulled
Suales, cor ide Commission, 108 any perwa aching on he ball of us Coo'nission:
A. Makes any warraoly or reprcarolllon, enpressed or implied, with reopere di Wie acero
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use of say Information, apparatus, molhod, or pr xros disclosed in the report.
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Such employee or contractor ul We Commission, or employee of such cuotractor prepares.
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with ebe Comass100, or to employment with ouod contraser.

ATOMIC TRANSPORT PROBLEMS OF INTEREST IN NUCLEAR SYSTEMS*
Ted S. Lundy and Fitz R. Winslow
Oak Ridge National Laboratory, USA
.
FITLITE""37975 The
PHE FIRE :" ..::.. ';.
- ORNI – AEC - OFFICIAL
*Research sponsored by the U. S. Atomic Energy Commission under
contract with the Union Carbide Corporation.
ORNI - AEC - OFFICIAL
ORNL-ACC-OMHICIAL
ATOMIC TRANSPORT PROBLEMS OF INTEREST IN NUCLEAR SYSTEMS
Ted S. Lundy and Fitz R. Winslow
Metals and Ceramics Division
Oak Ridge National Laboratory
Oak Ridge, Tennessee, USA
ABSTRACT
Several problems encountered in the application of basic diffusion
data to systems of practical interest such as those encountered in
nuclear engineering will be discussed. These problems will include
.isothermal diffusion experiments, thermal gradient diffusion, and radia-
tion effects on diffusion processes.
Most of the reliable information on diffusion in solids has been
obtained from cases where there is essentially no concentration gradient
of the diffusing species. In systems of interest to nuclear engineers,
however, concentration gradients are almost invariably present. Thus,
a working knowledge of the relation between self- and chemical-diffusion
coefficients is necessary in order to estimate the proper coefficients
to use in a given situation. Knowledge of thermodynamic activity coef-
ficients as functions of composition is necessary additional information.
Recently it has been clearly demonstrated that indiscriminate use of
an Arrhenius-type expression to describe the temperature dependence of
diffusion data may lead to large errors in the prediction of diffusion,
Vigten
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coefficients. The ramifications of this finding will be discussed with
emphasis on refractory, body-centered cubic systems.
We have studied, both theoretically and experimentally, the effect
of a thermal gradient on the redistribution of substitutional impurities
in a metal. This work has led to an increased appreciation of the impor-
tance of this effect in systems of interest in the nuclear field.
The enhancement of diffusion by radiation fields remains a problem
of justified interest. We have made the first successful direct measure-
ments of the effect of fast-particle bombardment on diffusion in a metal.
Our results correlate well with predictions based on a model of the
annihilation of excess point defects (vacancies and interstitials) by
two mechanisms – migration to homogeneously-distributed fixed sinks
such as dislocations, and recombination of defects.
INTRODUCTION
In this paper we will discuss several problems that might be
encountered in the application of basic diffusion data to systems of
prac';ical interest. Specifically, we will consider (1) the relations
between self- and chemical-diffusion coefficients and the information
necessary for use of isotope diffusion data for predictions of the
behavior of real systems, (2) the pitfalls which might be encountered in
using the well-known Arrhenius-type expression to predict diffusion
coefficients outside the range of measurements, (3) effects of thermal
gradients on the distribution of elemental components in alloys, and
(4) effects of fast-particle bombardment on the diffusion process.
CHEMICAL DIFFUSION FROM TRACER MEASUREMENTS
Most of the problems of matter transport in nuclear systems occur
in components having a chemical concentration gradient. These gradients
could be built-in as in the mating of dissimilar alloys or could occur
because of interactions irith the operating environment. Unfortunately,
the study of interdiffusion, under the influence of a concentration
CIAL
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gradient, is neither as well understood nor as widely studied as is
tracer diffusion in the absence of a concentration gradient. Thie lack
of knowledge can be remedied by the use of self-diffusion coefficients,
some of which are well known, only if certain other information is
available. The theory by which a connection can be made between the
chemical-diffusion coefficient, D, needed to describe or predict a
chemical concentration gradient and the self-diffusion coefficients of
the alloy (this discussion will be limited to binary alloys) was derived
by Darken. (1) The assumptions which he made are, in many cases, not
completely valid, but experimental findings have tended to confirm the
form of the equations. These assumptions are: (1) the molar volume of
the alloy 18 independent ve concentration, (2) diffusion is by a vacancy
mechanism, and (3) vacancies are present at equilibrium concentration at
all times and positions. In luany systems deviations from constancy of
the molar volume are small so that errors introduced by (1) are often
small. For large deviations the theory has been extended hy Sauer
and Freise [2] to include changes in mclar volume. The assumption of
a vacancy mechanism of diffusion is generally considered valid for sub-
stitutional diffusion in metallic phases. However, there is convincing
evidence for other mechanisms of diffusion in some intermetallic phases
and in some nonmetal.s. The third assumption, that the vacancies are in
local equilibrium, can be seriously in error because of the diffusion
process itself. The Darken equations then seem to be only fair approx-
imations to reality. For a binary system the Darken equation for the
chemical-diffusion coefficient is
D = D2N2 + D2N1 ,
:
(1)
where 1 and 2 refer to the two components, N is the mole fraction, and
D is the intrinsic diffusivity. The D's can be obtained from a chemical
diffusion experiment only by using a second relation which involves the
velocity of motion, v, of a constant concentration point in the sample,
usually taken as the interface:
ORNL - AEC - OFFICIAL
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.
:,:
=. (D) – D2) ON
.
.
131 130 - 21h - Ho
10131510 - DIV-INIO
There is another relation originally derived by Bardeen and
Herring (3) connecting the intrinsic coefficients with the commonly
measured tracer coefficients, D*. They assumed that the chemical
potential of vacancies, Hve is almost zero everywhere and that cross terms
in an irreversible thermodynamic treatment do not contribute to matter
transport. The first assumption 18 essentially equivalent to assumption
(3) of Darken; the error introduced by the second is probably small.
Bardeen and Herring showed that for the ith component of the system,
De = D (1 + à in *),
where yt is the activity coefficient.
Thus, if all the assumptions are met and if thermodynamic data
is available for the syster of interest, chemical diffusion effects can
be calculated from tracer data. Unfortunately, there are many cases
where this is not possible.
THE ARRHENIUS CONCEPT
The usual way of reporting the temperature dependence of dif-
fusion data is to give values of Do and Q contained in the Arrhenius-
type equation,
... D = Do exp(-Q/RT),
where Do is called the frequency factor and Q the activation energy.
The original Arrhenius equation, proposed in 1889 [4) as an empirical
law for the inversion of sucrose, is given by
ü ln K
ΛΗ
(5)
RTZ
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Equation (5) may be readily integrated to give an equation having the
form of Eq. (4) for constant Sh. Early investigators in the diffusion
area realized this fact and coupled it with the success of absolute
reaction-rate theory in identifying the empirical constants, Do and Q,
in terms of the thermodynamic properties om the moving species. These
developments. led to the almost universal acceptance of Eq. (4) as a law
for diffusion processes. The purpose of th? 3 section will be to diecubb
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some of the deviations from Arrhenius-type behavior recently reported
for diffusion in body-centered cubic metals. This subject is of direct
interest to nuclear designers because it is often necessary for them
to extrapolate diffusion data taken at high temperatures to lower
reactor-operating temperatures. It will be shown, at least for some
materials, that such extrapolations lead to unrealistic conclusions.
It is fairly well known that the Arrhenius equation cannot, in
general, hold for chemical diffusion. It is also well known that grain-
boundary effects, magnetic effects, and surface effects can drastically
increase material transport rates over that for volume diffusion. This
section will discuss only material transport which is tentatively iden-
tified as volume diffusion.
Figure I shows Arrhenius-type plots for the diffusion of 95 Zr
and 95 Nb in zirconium over the entire range of stability of the beta
phase. The curvature has come to be called "anomalous" with respect to
the "normal" materials that have straight-line plots such as that illus-
trated in Fig. 2 for the diffusion of 95 Nb in niobium. Both figures were
from studies conducted at ORNL. [5,6] For niobium, D was measured by a
direct sectioning and counting technique over a temperature range of
878 to 2400°C and a range in D of almost eleven orders of magnitude.
Yet, there is no trace of curvature on the plot. The few high data
points can be explained on the basis of a limited annealing time which,
because of lºss of resolution of the sectioning technique, gives rise to
a systematic error in D.
"Normal" self-diffusion behavior can be defined in terms of
Eq. (4) with constant values for Do and Q. The activation energy, Q, is
empirically given by
. Q = 34 Tm ,
where To is the melting temperature in degrees Kelvin. This approximation,
recently improved by Sherby and Simnad (7) and by LeClaire(8), is usually
correct to within 10 to 20%. The pre-exponential factor, Do, has been
extensively discussed by Zener. [9] It 18 sufficient here to say that in
• "normal" metals Do lies between 0.05 and 5 cin2/sec.
(6)
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The volume-diffusion data for all face-centered cubic and hexagonal
metals seem to follow the "normal" empirical rules; but some body-centered
cubic metals including B-Zr, B-Ti, y-u, and B-Hf behave differently. Ihe
body-centered cubic phases of these metals havı Arrhenius plcts similar
to those shown in Fig. 1. They are characterized by an apparent activa-
tion energy which is not only a function of temperature but is also con-
siderably lower than would be predicted by Eq. (6) and an apparent Do
several orders of magnitude lower than 0.05 to 5 cm2 , sec.
Anomalous metals are apparently anomalous for both self- and
impurity-diffusion. Figure 3 illustrates the diffusion behavior for a
number of impurities in B-titanium. [10] Note that there is curvature for
all isotopes studied except possibly C-14, but that the point of maximum
curvature tends systematically toward higher temperatures as the diffu.
sivity increases. This curvature may, in fact, be absent in the data
with the highest diffusivity (and lowest activation energy), C-14.[11]
This behavior certainly suggests that some intrinsic property of B-Ti,
and by inference of B-Zr and y-V, is involved in the anomaly.
The appearance and acceptance of the anomalous results have naturally
led to several speculations. We first wondered if it is possible that
most metals are anomalous but that the anomalies have not been seen because
of the narrow temperature ranges covered by the data. This possibility :
can be assessed by doing more refined diffusion experiments in order to
extend the range of directly measureà volume diffusion coefficients.
This has been done for self-diffusion in niobium and tantalum with the
results noted previously. [6,12)
Speculations on the mechanism of the anomalous behavior in B-Zr,
B-Ti, y-U, arid B-Hf may be divided into four categories:
1. There is no fundamental reason why the activation energy must
· be temperature independent. In fact, so little is known about the nature
and properties of the so-called activated complex, even for a process so
basically simple as sell-diffusion, that temperature dependence must be
considered as a real possibility, even in the face of a multitude of
· experimental data to the contrary.
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2.. Grain-boundary diffusion may obscure the volume-diffusion
mechanism at lower temperatures. This would lead to nonlinear pene-
tration plots which have not been observed in these systems. Further,
although ail samples were polycrystalline, the grain size was large
enough so that this possibility does not seem reasonable.
3. Possibly, diffusion along dislocations is contributing to extra :
cransport at lower temperatures. In contrast with grain boundaries,
dis).ocation diffusion would not necessarily lead to changes in the pene-
tration plots and so might be missed. Calculations which have been made,
however, show that about 1010 to 1011 dislocations/cm2 are necessary to
account for the effect. (8,23] This number secms unreasonably high for a
metal annealed at temperatures such as those used in the diffusion runs.
One possible source of a large number of dislocations in the anomalous
metals is the phase transformation that occurs in each. Peterson and
Rothman (14) have discounted this possibility by showing that pre-annealing
the specimens for long times at temperatures higher than the diffusion
temperature before depositing the isotope and diffusion annealing the
specimens causes no marked changes in D. In spite of this, dislocation
diffusion must be regarded as a real possibility until more is known
about the behavior of dislocations in the various body-centered cubic
metals.
4. The final suggestion, originally made by Kidson (15), is that
some very mobile impurity is attracted to vacancies in these metals. The
reaction to l'orm a vacancy-impurity complex could leave large concentra-
tions of mobile complexes in the metal at low temperatures. Excess dir-
fusion would be a natural consequence of these "extrinsic" vacancies.
Kidson's calculations indicated that the effects in zirconium could be
explained by impurity contents of the order of the actual oxygen impurity
· content in the samples used. Recent work by Askill,[16] however, indi-
cites no difference in diffusion coefficient (for 18 ?Ta in B-Ti) for:
specimens annealed in high-purity argon, and at vacuums of 1 x 10-5 and
8 x 10-9 torr. . This, of course, does not rule out the possibility that
impurities already in the specimen might cause the effect.
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In summary, an upward curvature of the Arrhenius-type plot has
occurred for certain body-centered cubic metals, B-Zr, B-Ti, and y-U.
This evidence has in turn cast doubt on the validity of the Arrhen'us
equation as an empirical relation for design purposes. It should be
noted that the effect, an enhancement of transport at low temperatures,
is in the wrong direction for design purposes. That is, more material
than predicted would be transported through a fuel-element cladding.
The materials in which the effect has been observed are also ones of
interest in nuclear systems. These facts make consideration of non-
Arrhenius behavior important for nuclear design.
REDISTRIBUTION OF ELEMENTS IN A TEMPERATURE GRADIENT
It is well known that certain nuclear reactor components are sub-
jected to extremely large thermal gradients. Such gradients are especially
important in the nuclear fuel and in the cladding materials. At present
little is knor. about the redistribution of elements contained within
such components, but it is surmised that such redistribution may signili-
cantly alter some of the physical and mechanical properties and cause
corresponding changes in reactor operating performance. This section will
review the various aspects of diffusion in a thermal gradient and will
suggest possible magnitudes of elemental redistributions. Particular
reference will be made to a recent paper by Biermann, Heitkamp, and
Lundy (17) on the redistribution of tracer elements in pure silver.
The theoretical problem of diffusion in solids under the influence
of a temperature gradient has been treated by approaches making use of the
thermodynamics of irreversible processes and by kinetic calculations.
However, neither thermodynamic nor atomistic theories have successfully
predicted thermal diffusion effects. The approach in this paper will be
to consider the thermodynamic approach first and then to examine some
of the kinetic models. •
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í 10
The phenonenological equations for fluxes J, in a two-component
system under a temperature gradient are: (18)
J1 = L11X1 + 112X2 + L13X3 ,
J2 = L21X1 + L22X2 + L23X3 , and
(7)
J = L31X1 + L32X2 + 133X3 ,
where the subscripts 1, 2, and 3 refer to solvent atoms, solute atoms,
and energy, respectively, and vacancies are neglected. The Lake are
phenomenological coefficients and the X, are thermodynamic driving
forces.
By choveing a reduced energy flow Jas (19)
JS = J3 - (hiji + h2J2) ,
where hy and ha are the partial molal enthalpies and j3 18 the total
energy flow, the forces are
X1 = - D (M2)T,
Xa = - V (Hz)T;
X3 = - T.
(8)
(9)
For a coordinate system referred to the center of gravity
J] + J2 = 0
(10)
1.11 = – L21,
L12 =:- L22 , and
= -- L23 ,
:
(11)
Y-INIO
IVIDUA10-
weden
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11
The Onsager relations are
L12 = L 21
(12)
L1 3 = 131 , and
L 23 = L32
Using the definition of the heat of transport as
(13)
gives
(14)
12 = - L22 10 (H 2 – M2), + v r) .
It can be shown that
Q* = ( ) T = 0,
(15)
80 Q* 16 the thermal energy transported by the reduced energy flow per
unit solute flow in isothermal diffusion relative to the center of
mass. (20) Since
My = M1° + RT In al , and
(16)
My = us RT In az ,
where ai and 22 are activities of components 1 and 2, respectively, and ·
aj = fNi , and
(17)
82 = f 2N2 ,
with Ni and N2 mole fractions of components 1 and 2, and Ni + N2 = 1, then
Ja = -1 22 [ T v N2 + q* v T ]
(18)
for very dilute solutions.
At steady state, Ji = Ja = 0, 80
. N2 + V T = 0.
(19)
If the temperature gradient is zero except along the x-axis, it follows
that
(20)
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------
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12
This equation for the center of mass system is applicable for a two-
component system where the solute 18 present in very dilute quantities.
For example, it might be applied to interstitial solute diffusion in an
interstitial regular solid-solution alloy. We note here that the sign of
Q* gives the direction of thermal diffusion and the magnitude of Q* 18
proportional to the magnitude of the effect. If Q* 18 negative, the
solute tends to migrate to the hot portion of the specimen, while a
positive Q* implies migration to colder regions.
It can be shown that the heat of transport q* (defined in the center
of mass system) 18 related to heats of transport of the solvent and solute,
Q* and Q2, respectively. The latter quantities are defined as
at - , and
(21)
where the primed phenomenological coefficients are defined in a' coordinate
system relative to the lattice planes. The relationship is
(22)
where Di and D2 are diffusion coefficients of the solvent and solute,
respectively. This expression 18 useful in relating predictions of
kinetic models to the theory derived by the thermodynamic approach.
So far we have applied no particular mechanism of diffusion in
deriving the equations. If we now consider that the diffusion process
occurs by a vacancy mechanism, we can show that
Qi = Q8 – Eo ,
(23)
where QX 18 the heat of transport of the pure metal and E, is the forma-
tion energy for vacancies in the pure metal. For consistency we will
define a Q** such that
aš = Q* - Ez
(24)
where E2 18 the formation energy of a vacancy next to a solute atom.
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ir1.01330-DIV-IXIO
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13
Then, we may rewrite the previous expression for Q* to give
Q* = (Q* - Ez) - P Q7 Eo).
(25)
Two kinetic models will be considered. The simple kinetic model
assumes that the jump frequency of an atom depends only on the tempera-
ture of the initial site. Thus,
QQ = Ho ,
(26)
where Ho is the activation energy for migration of vacacies in the pure
metal, and
. Q2 = U2 – E2
(27)
where U2 18 the migration energy of the solute. So
Q* = (U2 – E2) - BA (Ho – Eo) .
(28)
the activation energy of migration, Ho, can be split into three parts or
Ho = H4 + Hp + Hf,
(29)
where Hy is the exces's kinetic energy which must be supplied to the atom
at the initial site to initiate jumping, Hy the energy which must be
supplied to the barrier atoms so they will allow the atom to pass through,
and Hf the energy required to open up the relaxed vacancy at the final
position of the jumping atom. Wirtz' model limits aš to the interval
Ho s Q S Ho.
(30)
Thus, the two kinetic models may be summarized by
Qp = BOHO
(31)
with Bo = 1 for the simple model, and -1 = Bo s +1 for the Wirtz model. .
If, similar relations for thermal diffusion in a dilute alloy are assumed,
Q* = (B2 (Q2 - Ez) - E
(Q0 – En) - Eo]
(32)
from which estimates of Q* may be made.
..
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In the experimental part of the work by Biermann et al., (17) pure
silver and the isotopes 124 Sb and 103Ru were chosen because all of the
information for estimating thermal effects by the various models was
available. Antimony diffuses much faster in silver than does silver so
in these experiments the silver lattice acted as a marker for thermal
effects; while ruthenium diffuses much blower than silver and thus may
be considered as a marker. By this choice of materials, it was hoped
to determine how silver atoms and the two isotopes behave in a thermal
gradient.
The experimental apparatus 18 shown in Fig. 4. The specimens con-
sisted of silver uniformly doped with either 124 Sb or 103Ru to about
300 or 2 in 10', respectively. The most novel feature of this apparatus
was the use of internal tungsten-silver thermocouples to measure tempera-
tures without disturbing the heat flow.
Figure 5 shows one set of results for 124 Sb. Note that for 528 hr
at temperatures of 822 to 886°C with a specimen 0.37 cm long, the concen-
tration of the isotope at the hot end is more than 30% greater than that
at the cold end. Thus, Q* 18. negative and from plots of in (c/co) vs T-1,
shown in Fig. 6, it was found to be (-29 + 3)kcal/mole, a value too large
to be explained by the simple kinetic model which predicted a value of
-7 kca.2/mol.e. The Wirtz model, however, allows a Q* range of
O to -37 kcal/mole and may therefore be use? in explaining the data.
Experiments using 103 Ru were unsuccessful and allowed no definite
conclusions. They did, however, suggest that Qð has a negative value
(1.e., pure silver in a temperature gradient tends to migrate to the
hotter regions.)
. The few experiments which have been carried out on the effect of a
thermal gradient on material transport have indicated that a simple
kinetic model of the jump process is not adequate to explain the results.
There is at present no theory which can even qualitatively predict the
effects. Therefore in nuclear systems where thermal gradients are present,
exact design information can only be obtained under the expected service
conditions.
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15
EFFECT OF FAST PARTICLE BOMBARDMENT ON DIFFUSION
·
Much has been reported about the effects of radiation on various
mechanical and physical properties of materials. These effects have
generally been explained in terms of additional defects produced by the
radiation and annealed out by atomic migration. The theories of produc-
tion of lattice defects by different kinds of radiation and annihilation
of such defects by thermal effects seem to be in reasonably good shape.
There have been, however, .o previous quantitative measurements of the
effect of fast-particle bombardment on the atomic migration process
itself. Biermann and Heitkamp (22,23) have described a method of measure-
ment of the enhancement of self-diffusion in lead by alpha bombardment.
Their work, though yielding only qualitative results, has led to addi-
tional experiments (on diffusion of 212Pb in silver) (24) which are the
first direct quantitative measurements of the effect of fast-particle
bombardment on diffusion in a solid. The remainder of this paper will
first examine various theories and then report the experimental results
of Heitkamp, Biermann, and Lundy. (24)
Theoretical aspects of radiation-enhanced diffusion have been dis-
cussed by Lomer (25,26), Dienes and Damask (27), Damask (28,29), Larsen
and Damask (30), and Wechsler. (31) They considered mechanisms of pro-
duction and annihilation of point defects and developed equations appli-
cable to various simple cases of defect annealing. Cases which were
considered are: (1) annihilation of vacancies and interstitials by migra-
tion to homogeneously-distributed fixed sinks, such as dislocations;
(2) recombination of vacancies and interstitials by atomic migration; (3) a
combination of cases 1 and 2; and (4) reconbination of the interstitial
atom with the vacancy created at the lattice site from which the atom was
displaced. Case 4 is not likely to be important for enhancement of dif-
fusion due to highly-energized charged particles.
The theory is developed as follows: Consider a constant production
rate K of lattice vacancies and interstitials and defect annihilation
resulting from defect interactions at homogeneously-distributed fixed
sinks and from recombination. Then the rates of change with time, t, of
the concentrations of the two types of defects are:
-
- -
-
-
-
.
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16
(33)
= 0v, 12c, - vz(Cro + C) ,
ove = K – dv, 12cy – v (Cve + c),
(34)
where
C. and C, are concentrations of vacancies and interstitials in excess
of thermal-equilibrium values, respectively,
d 18 a factor proportional to the concentration of fixed sinks,
V, and v, are jump frequencies of vacancies and interstitiala,
respectively,
1 18 the jump distance, and
C 18 the concentration of vacancies at thermal equilibrium (with-
out radiation).
Using Eqs. (33) and (34) it is easy to show that at equilibrium,
, A. [
ve + ( ° • vo ) + ]
(35)
and (y =
(36)
If the excess self-diffusion coefficient Dp in proportional to the excess
vacancy concentration (implyir:g a vacancy mechanism of diffusion), then
212 + C,
. 2
K
L
(37)
DE = 812v
v/ -
v l
ORNL-AEC - OFFICIAL
where p is the correlation factor for self-diffusion. For impurity dif-
fusion by a vacancy mechanism the same expression with fin substituted
for f should hold.
In the experimental work by Heitkamp, Biermann, and Lundy (24) a
surface-decrease alpha-recoil method of measuring the diffusion coefficients
was used. This method allowed values as low as 10-18 cm2/sec to be measured.
Since diffusion coefficient measurements were made in regions of specimens
OIN -.1EC-OFF
17
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very close to the surfaces, it was thought necessary to have additional
terms for annihilation of defects at the surface in the differential
equations describing the process. The equations were therefore
ot" = K - dv, 12C, - v(Cho + CV)C, + V,1872C, , and
(38)
88
o = K - dv, 12C4 - V (Cro + C) + v; 1972C%. (39)
Solutions to Eqs. (38) and (39) for the steady state, presented in the
paper by Heitkamp et al. (24) are rather complicated and will not be
repeated here. They predict that radiation-enhanced diffusion in silver
in the near-surface region X ~ 10-8 cm) is significantly smaller (about
2 orders of magnitude) than one would expect it to be in the bulk material.
We will briefly review the experimental details of the measurements.
First the alpha-recoil method of measuring diffusion coefficients was
used. In this method the surface concentration of 21 2Pb was found by
measuring the beta activity of the 208T1 recoil atoms. The decrease in
212Pb activity with time at temperature was related to diffusion coef-
ficients through use of appropriate solutions to Fick's second law. Such
measurements were made both with and without a field of bombarding alpha
particles from an 8.5-curie 210Po source. Thus, normal and enhanced
diffusion coefficients were measured and excess coefficients were obtained
by subtraction. Results of the study are summarized in Fig. 7. Here we
see that the isothermal diffusion may be described by an Arrhenius-type
plot with
D = 150 exp (- 60,000) cm2/sec.
(40)
The excess D values, obtained by subtraction of isothermal from enhanced
values shown in the shaded area of the figure, were all in the range of
(5 to 8) x 10-18 cm/sec.
If the measured values of De are compared with theoretical predic-
tions, there is reasonable agreement with predicted bulk values and
disagreement of about 102 with near-surface predictions. We do not at
present understand why the surface does not act as a sink for exce88
vacancies.
nts
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red.
.1.- iiuil -
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18
SUMMARY
In conclusi on, we have examined four of the important problems
that might be encountered in attempting to apply basic diffusion data
to nuclear bystem 8. We have not attempted to cover the whole problem
area, since several other probl«ems, such 28 near-surface effects,
short-circuiting paths of easy diffusion, etc., exist and merit
detailed consideration.
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19
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ๆ
ง า
REFERENCES
1. 'L. 8. Darken, Trans. AIME 174, 184 (1948).
2. F. Sauer and V. Freise, z. Elektrochem. 66, 353 (1962).
J. Bardeen and C. Herring, pp. 87-L11, Atom Movements, American
Society for Metals, Cleveland, Ohio, 1951.
S. Arrhenius, 2. Phys. Chemie 4, 226 (1889).
J. I. Federer and T. S. Lundy, Trans. Met. Soc. AIME 227, 592–97
(1963).
T. S. Lundy, F. R. Winslow, R. E. Bawel, and C. J. Mcllargue, to be
published in Transactions of the Metallurgical Society of AI.ME.
7. 0. D. Sherby and M. T. Simnad, Trams. ASM 54, 227-40 (1961).
A. D. Inclaire, "Diffusion in Body-Centered Cubic Materiels," paper
presented at the Symposium on Body-Centered Cubic Materials,
Gatlinburg, Tennessee, Sept. 16-18, 1964, to be published. Also:
Acta Met., 1, 438–47 (1953).
C. Zeier, J. Appl. Phys. 22, 372-75 (1951).
10. J. Askill and G. B. Gibbs, persona. I communication.
I. I. Kovenskii, Ukr. Fiz. Zn. 8, 797–98 (1963).
R. E. Pawel and T. S. Lundy, Acta Met. 13, 345–51 (1965).
13. J. Askill, personal communication.
N. I. Peterson and S. J. Rothman, Phys. Rev. 136, A842 (1964).
G. V. Kidson, Can. J. Phys. 41, 15.63–70 (1963).
16. J. Askill, personal communication.
W. Biermann, D. Heitkamp, and T. S. Lundy, Acta Met. 13, 71–78 (1965).
S. R. deGroot, Thermodynamics of Irreversible Processes, Interscience,
New York, 1951.
19. S. R. deGroot and P. Mazur, Non-Equilibrium Thermodynamics, North-
Holland, Amsterdam, 1962.
K. G. Denbigh, The Thermodynamics of the Steady St.te, Methune,
London, 1951.
21. K. Wirtz, Phys. 2. 44, 221 (1943), :
22. W. Biermann and D. Heitkamp, Z. Phys. Chemie Neue Folge 34, 265-68
(1962).
W. Biermann and D. Heitkamp, 2. Phys. Chemie Neue Folge 37, 137449 ..
(1963).
TY1310-
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3 V-INIO
23.
IVdou - stainau
OINI-AC- wifiCIAL
20
24.
25.
26.
27.
D. Heitkamp, W. Biermann, and T. S. Lundy, to be published.
W. M. Lamer, Diffusion coefficients in Copper Under Fast Neutron
Irradiation, AERE-TR-1540 (Dec. 1954).
W. M. Lomer, "Defects in Pure Metals," p. 255, Progress in Metal
Physics, Vol 8, Pergamon Press, New York, 1959.
G. J. Dienes and A. C. Damask, J. Appl. Phys. 29, 1713 (1958).
A. C. Damask, "Effect of Radiation on Rate Procecses in Alloys,"
p. 479, Reactivity in Solids, Elsevier, Ameterdam, 1960.
A. C. Damask, "Effect of Neutron Irradiation on Non-Fissionable
Alloys," p. 3, Radiation Damage in Solias, II, International Atomic
Energy Agency, Vienna, 1962.
R. E. Larsen and A. C. Damask, Acta Met. 12, 1131 (1964).
M. S. Wechsler, "Fundamental Aspects of Radiation Effects on
Diffusion," Symposium on Radiation Effects on Metals and Neutron
Dosimetry, Philadelphia, 1963.
29.
30.
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ORNL-ACC-vitician
OINC - AIC-OISICIAL
FIGURES
Fig. 1.
Fig. 2.
oo
Fig. 4.
Fig. 5.
(ORNL-LR-DWG 70182R) Diffusion of 95 Zr and 95 Nb in 8-Zr.
(ORNL-DWG 64-9858) Diffusion of 95 Nb and 18 21a in Niobium.
(ORNL-DWG 65-5949) Tracer Diffusion in B-Titanium.
(ORIVL-DWG 63-7309) Experimental Apparatus for Diffusion in a
Thermal Gradient.
(ORNL-DWG 63-7308) Concentration of 1245b as a Function of
Distance.
(ORNL-DWG 63-7307) Concentration of 124 Sb as a function of
the Reciprocal Absolute Temperature.
(ORNL-DWG 65-2363)' Measured Diffusion Coefficients of 212Pb
in Silver With and Without Alpha Bombardment.
· Fig. 6.
Fig. 7.
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.
22
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TEMPERATURE (°C)
1500
2000
1000
900
800

2 x 10-7 En
DIFFUSION COEFFICIENT (cm2/sec)
tohoto
No 950
10"
L
I
6
.
8
10
7
104/110K)
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Fig. 1. Diffusion of 95 Zr and 95 N
in B-Zr.
23
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ORNL-DWG 64-9858
TEMPERATURE (°C)
1400
1000
2400
4800
i-a FORMI - Alcornicial
A
10-7
LIV

-
-
-
-
-
---
-
10-8
--
..
.......
10-9
g5Nb=1.14
UND
1.10 exp (-96,050/RT)
10-10
10-14
-DNO TO =1.02 exp (-99,300/RT)
10-12
.
.....
D (cm2/sec)
....
..
.
...
10-44
.
-
.
-
-
-
.
-
-
. .
-
--
-
-
.
-
10-15
.
10-16
. ...
-
-
.
.
10-17
:
10-18
10-19
3.5
4.5
5.5
6.5
7.5
8.5
10,000/ (OK)
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Fig. 2. Diffusion of 95 Mo and 18 21a in Niobium.
ORNL-DWG 65-5949
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10-5
PHASE
CHANGE
10-6
C14
p32
DIFFUSION COEFFICIENT (cm2/sec)
Nj63
C060
| MELTING
10-8L POINT
Fe55
Mn 54
10-9
Sn 113
148
N695
To182
. 10-10
M099
TEMPERATURE (°C)
4200 4400
4000
900
1600
L
1
1400
l
0.6
L
0.5
0.7
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0.8
100%(OK)
Fig. 3.
Tracer Diffusion in B-Titanium.
..
.
.-..-..........
•
-
..
-
-
........................................... ..
. ORNI-AIC - OFFICIAL
ORNL-DWG 63-7309

ognani
SPRING
............
WIMMI
NINITIIIIIIIIIIIIIIIIIIIIIIIIIIIII
CERAMIC TUBE
.....
POWER LEAD
FAMILIUKWD
...
POWER LEAD
-
-
-
- -
- STAINLESS STEEL
ROD
-
.
.
. .
ALUMINA CEMENT
60/40 Pt-Rh WIRE
. . .
.
.
RADIATION
SHIELDS
-
00000000
bOO 000 do
-
-
-
-
....~
W CYLINDER
CU ROD
W WIRE
- Ag WIRE
W WIRE
-W FOILS
.
.
W FOILS
SPECIMEN
W FOIL
Cr-Al-TC-
Cu ROD-
...
..
com.....
-...
ti
STAINLESS STEEL
CYLINDER
.
CU PLATES
OT WATER COOLING
ORNI - AIC - OFFICIAL ...
Fig. 4. Experimental Apparatus for Diffusion in a Thermal Gradient
..
..
.
.
.
...
.-..
-.-..
....-
..-.-
.-.....
..
.
......
..
.
.
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1.2
SPECIMEN II
528 hr
% 1.0 ---
886 °C
822 °C
.8
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0.1
:
0.3
0.2
x (cm)
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.
Fig. 5. Concentration of 124Sb as a Function of Distance.
..--------
--
-----------..
27
27
ORNL-DWG 63-7307
0.10
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1528.hr
240 hr
10g %
-0.05
: -0.40
•
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0.92
0.90
0.88
0.86
4000/710K)
Fig. 6. Concentration of 124 Sb as a. Function of the Reciprocal
Absolute Temperature.
ORNL-DWG 65-2363
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T (°C)
400
-16.550 500
450
350
300
lo

L
-O WITHOUT IRRADIATION
• IRRADIATED BY
'o
S
1.02x10"
alphas
cm2. sec
ODL
D(cm?/sec)
'o
O
O
'o
RT
to= (150 + 100) exp(- 60,000 £5000) cm2/sect
LL
1.4 1 1.6
-
1.2
1.8
i
1000/T (OK)
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Fig. 7. Measured Diffusion coefficients of 21 2Pb in Silver With
. and Without Alpha Bombardment.
----
-
.-
-
.--.
-.
-
.
-
..
-
..
..
.
..
..
-
END
-
.
DATE FILMED
9/ 2 / 65