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Diffusion of carbon in alpha iron
DIFFUSION
OF
CARBON C.
G.
IN
ALPHA
IRON*
HOMAN?
The diffusion of carbon in alpha iron using a tracer technique was performed in the temperature range of 616°C to 844°C. 9 plot of the In D versus l/T in this temperature range deviates from the usual straight line behaviour. An empirical model which includes the possibility of a carbon-vacancy interaction is proposed to explain this deviation.
DIFFUSION DU CARBONE DANS LE FER ALPHA L’auteur Btudie par une technique de traceur, la diffusion du carbone dans le fer alpha pour une gamme de temp6ratures s’6tendant de 816°C it 844°C. la relation ln D = f(l/T) ne poss&de pas les caract&istiques Dans cette gamme de tempkratures, d’une droite. Pour expliquer ce r&,ultat, l’auteur propose un modAle empirique qui fait intervenir la possibilite d’une interaction entre les atomes de carbone et les lacunes. DIFFUSION
VON
KOHLENSTOFF
IN
ALPHA-EISEN
Die Diffusion van Kohlenstoff in alpha-Eisen wurde im Temperaturbereich 616°C bis 844°C mittels Trilgt man In D gegen l/T auf, so erhtilt man Abweichungen vom einer Isotopentechnik verfolgt. Zur Erkliirung der Abweichungen wird ein empirisches Model1 vorgeschlagen, iiblichen linearen Verlauf. das die Miiglichkeit einer Wechselwirkung zwischen Kohlenstoff und Leerstellen einschlieot.
agreement
INTRODUCTION
By implicitly
assuming that every diffusional
made by carbon performed
atoms in an alpha iron matrix
with identical
local surroundings,
perature
jump was
Wert’l’
of our results with the previous high temresults of Smithc4) indicated
extrapolation questionable. The implicit
from
low
that a linear
temperature
assumption
made
data
by Wert
was
may
be
was able to derive an elegant theory based on a simple random walk model. This model leads to an
correct for low temperature
studies in pure iron, but
expression
it is perhaps not completely
accurate in the tempera-
for the diffusivity characterized by a single parameter 7. Wert was able to qualitatively
relaxation fit experimental
data representing
approximately However, which
fourteen
the low
has been
decades
not be extrapolated
of
An empirical
diffusivity.
Furthermore,
data, could
atoms could have two different surrounding
data(3$4) at
anelastic
techniques(5)
result of an extremely
of carbon
a t’racer
technique
simple single relaxation
to examine
model and to provide
of t’his investigation
showed
with
a
st)raight
line.
Furthermore,
neighbor
iron sites is
The agreement of this model with the low temperasmall, is assured. However,
and room pressure
was impossible to fit the high temperature
configuration
exist on all the nearest neighbor
ture results,(2) where the population
was to redetermine
in order
configu-
vacant.
the anomalous
a zero that
of vacancies
A calculation
it
diffusivities quantitative
indicates
is
at the higher temperature,
the proposed model includes the possibility vacancy interactions.
the
point for a high pressure diffusion experiment. The results
The first one is the normal
is when one of these nearest
iron by
small pressure effect.
the diffusirit)y at high temperature
in
in which the moving carbon
sites of the carbon atoms and the second configuration
of the pressure
in alpha
have indicated
The purpose of this experiment using
rations.
where iron atoms
recent measurements
iron
of carbon
experimental
to the experimental
on t’he diffusion
model for the diffusion
determined,c2)
the high temperature. effect
neighbor
sites to a diffusing carbon atom may be vacant. alpha iron was developed
t,emperature
quant.itatively
measurements of the
ture range where one of the nearest
based on this model
a high temperature
of carbon
(Appendix
A)
pressure effect on the
diffusivity
of carbon in alpha iron proportional
activation
volume of formation
to the
of vacancies.
THEORY
* Received, Janua,ry 20, 1964; revised February 20, 1964. Submitted to the Physics Department, Rensselaer Polytechnic Institute, Troy, N.Y. in partial fulfillment of the requirements
In developing an empirical model for the diffusion of C in a-Fe which would include the possibility of
of the lMast)er of Science degree. ‘j’ Work performrad at Watervliet York.
vacancy-carbon interaction, the following conditions.
ACTA
METALLURGICA,
VOL.
Arsenal,
Watervliet,
12, SEPTEMBER
1964
New
1071
it is necessary
to satisfy
SCT_4
1072
1. The
diffusivity
independent
carbon
in
alpha
iron
is
derived
with
from
changes
of
VOL.
3. A good empirical
the model vacancy
should
be
fit be obtained
range.
x exp -
equation
diffusion couple
-
I = erf “-z.
i
‘22/m
tracers are used, equation
(2) may
2 22/11)t
+%&+?x])~~ ‘c
in the free state; (All energies carbon
atoms
the
(6) v2
vacancy
concentration
in atom
percent *B -
?,=6:exp
E,,
RT
(
(7
i
where E,, is the energy of forn~ation of a vacancy.
The
radioactivities
to CO, C,, and C$. Therefore,
effe&ive
formulated
diffusion
coefficient
De
may
be
to be:
from a
plot of
De = (1 -
qP,
+ rlD,
(8)
where erf-l
~oncentratioll
(9)
’
it is possible to determine
if D is independent
and if this is true,
its value
of the at the
annealing temperature.
is the diffusivity
II,=
The :proposed reaction is c+v+c...v The forward
reaction
atom
assumed
requires the presence of a free
bimolecular
(Experimental
(3)
(C) and a “free” and
evidence
carbon-vacancy
hence,
also suggests
The reverse reaction
differential
vacancy
governing
(V), and is
second
with unpaired motion
Ds,exp-
unimolecular.
The
this model is :
From
anelastic
the concentration
vacancy
concentrations,
respectively.
and
If one
g
relaxation
(19)
measurements,
of vacancies
D, = 0.008 exp -
where
is negligible’s’
19,800 cal/mole
(11)
)
RT
i
And for the case when q < I; (q < 1O-2 at SO%.) ,__ _^_ _, . I 19,WO cal/mole 6V1 ~... .0.008 exp -+
i
where C,, C, and C, are the pair, total carbon
of
where E,, is the motional energy of the carbon atom when paired with a vacancy.
order.
this assump-
requires only a paired
and is assumed
equation
associated
a carbon atom and for the other eon~guration,
B ~on~e~tr~,t~o~ sf ~r~or~-~~~~~e~ pa+s
total
of E,,,
C. Effective difSusion coeficient
where oO, ab and a, are the measured
tion.ti))
frequencies
con~gurations;
energy for a carbon
or expressing on the
t,o read 1 = erf-
carbon
in t,he two
At equilibrium, the ratio of paired to total carbon concentration, rj, is
tration at distance x from the interface.
corresponding
(5)
“)
and B is the binding energy of tOhepair.
high and low side of the couple and C, is the concenIf radioactive
atom
(2)
where (7, and CO are the initial concentrations
be modified
(Em;,
(I) can be shown to be S6)
C, c,
C?,v,
are expressed in kcaljmole.) of the concen-
In the case of a semi-infinite
C 2 _-5 ( c;, -
the carbon
(I)
D is independent
the solution of equation
-
!
where y1 and V, are the vibrational the motional
!z.=Dazc at ax2
tration.
(
E *g
for measured
Pick’s second law of diffusion may be written as:
when t,he diffusivity
(4) ma,y be written:
dC -2 = Gv,C,C, exp clt
concentration
over the entire ~m~)erature
A. Solution qf d$usion
1964
assumes that the carbon atoms are the more mobile
at different temperatures. diffusi~ities
12,
defect, then equation
of carbon concentration.
2. Equations consistent
of
METALLURGICA,
B ex!?
!
E”,, RT
RT
E,,
1
,
y2
112)
HOMAN:
A plot
D,)
of In (11, -
literature
value
vs.
l/T
DIFFUSION
together
OF
with the
of Bc8) and Efvcg) should
yield
the
value of [SCv,/v,) D,,] and E,,.
C IN
ALPHA
evacuated
2 ,u, radiomethane
to about
mitted
to a pressure
vessel
was
sealed.
A. Counting An
procedure
end
GM.
PROCEDURE
window
tube
(window
was mounted
thickness
as shown
1 mg/cm2)
in Fig.
1.
This
fixture insured constant geometry for surface counting sample
ends.
The
counting
end shield was concentric diameter for
of 0.120 in.
counting
counting
geometry was
chi-square
tests.
in the aluminum
with the samples and had a
The V block held the samples
and grinding.
equipment
port
The reproducibility
and the reliability
checked
of
of the counting
periodically
by
means
of
Cylinders
of 0.187
in. dia. and from
99.98%
18 in. long
were
zone refined
iron
machined
stock.oO)
The cylinders were then etched in a solution
of 10 g of oxalic acid, 20 cc of 30 ‘A hydrogen peroxide 300 ml
of
H,O
at 45°C
0.020 in. were removed
until
approximately
from the diameter
and the
surface was very bright and shiny. The cylinders
Carbon
was deposited
surface
heating
with
by
approximately
burized
vessel
the gases remaining and cylinders
furnace
which
to be carburized
Mettler
Grammatic
microbalance
quartz
carburizing
vessel.
were weighed on a
After
and placed the
in a
vessel
was
butt
in an
The car-
and shiny
to determine
to insure a uniform
The uniform
welding,
cylinder
to
in the vessel,
were placed
were bright
were reweighed
gain and counted tribution.
furnace
at 750°C for 72 hr.(l”)
cylinders,
appearance,
induction
The temperat’ure was t)hen raised to 850°C for about
90 min. After evacuating annealing
an
in the
on the cylinder
500°C for 30 min.
of the cylinder
the entire
of the methane
concentrations
cylinders.)
tion treatment.
carefully
and
pressure
different
was ad-
cm of Hg and the
in
the weight carbon
dis-
count rate of a cylinder
was
between 50 and 200 cpm depending
Sample preparation
B.
of 0.5-4.0 (The
was varied to obtain EXPERIMENTAL
1073
Fe
The diffusion in a helium
to a carburized
on the carburiza-
couple was formed
atmosphere, cylinder
by
a pure iron
(using
a special
fixture and after suitably preparing the weld surfaces). The weld was made amperes load.
through
The time
Metallographic
by passing
3-5
pulses of 900
the sample under a small pressure of a pulse never examination
exceeded
of similar
5 sec.
cylinders
of
Armco iron welded with the same procedure indicat,ed good
welds with no discernible
entrapments
at the
interface. Since this diffusion experiment high pressures,
a new technique
sample was developed.03)
will be extended
Because
the
(1) the solubility
of Carbon in Cu is low,(14) (2) the diffusivity a-Fe is low with respect to the diffusivity Feo5) and (3) Cu is an extremely
to
to encapsulate
of Cu in of C in a-
good “getter”
of O,,
it was decided
to plate the diffusion couples with 0.003 in. of Cu. After a diffusion anneal, it was possible to remove part of the copper oxide layer to obtain a bright Cu color. The solubility of C in Cu was GM END
qualitatively
investigated
C. Diflusion
anneal
by a counting
technique.
‘WIN”C)ul .. ..__.. COUNTING
A temperature 05°C GM TUBE SUPPORT
control
over the sample length of
at 725°C during a diffusion anneal was possible
by means of a diffusion furnace and controller. temperature
fluctuations
during
the
measured by means of a chromel-alumel ENDWINDOW SHIELD AND COUNTING PORT V-BLOCK
of counting
assembly.
The were
thermocouple
imbedded in the Cu heat bath of the diffusion furnace. The thermocouple potential was measured with a L&N K-3 potentiometer against a cold junction of glycerine in a dewar. The chromel-alumel couple was calibrated
FIG. 1. Schematic
anneal
before and after a diffusion run with
a Leeds and Northrup standard Pt-Pt 10% Rh thermocouple using an ice bath cold junction. The
1074
ACTA
METALLURGICA,
temperature of the cold junction was measured with a ~alibrate~l thermometer. The sample was inserted into the furnace with a chrome]-alumel thermocouple welded to the low activity side of the couple. Measurements of the heating time of this thermocouple as compared to the furnace thermocouple indicated that a maximum time of 1 minute was needed to heat from 0.8 T, to TD for diffusion runs lasting from 60 to 90 min. After the anneal, the sample was quenched within 10 set to R.T. The data has not been corrected for the heat-up time.
The sample was mounted in the V block and the Cu plating was removed from the end with a precision surfacegrinder. The grinder has an accuracy of 0.0001 in. and produced a #lS finish on the sample end. Sectioning and counting from the high carbon (activity) end showed that the original cpm measured before the anneal could be obtained within 0.005 in. of the end. This result qualitatively confirmed the fact of the extremely low solubility of C in Cu. At least three sections were made to within 0.2 in. of the weld to be sure t,hat the concentration in the carburized portion of the sample was uniform after the anneal. All samples reported in this paper were within 1 “/;; of the original value of a0 determined before the anneal in this region. The penetration was measured from this point. For samples #I & #lo, every section was counted for at least 10,000 counts; samples fA, 2A. 3A, 5 and 7 have at least three
VOL.
12,
1964
sections counted for at least 10,000 counts; and for sample #6, every section was counted for at least 3,000 counts. Background measurements were made periodically during the counting procedure and varied between 23.4 and 24.3 over several months of counting. A summary of experimental data has been tabulated in Table #I. TABLE 1. Diffusion annealing data -. ._ -
z.-. Couple no. 1 1A 2A 2B 3A 5 6 7 10
&vt.y$J
a0 iwm)
(%!
0.022 0.022 0.021 0.021 0.009 0.006 0.006 0.006 0.008
191 191 184 184 80 55 53 52 209
721.0 f 721.0 f 732.0 * 732.0 * 730.9 & 814.9 + 823.5 & 616.0 * 844.2 i -
tn x 10s sec. 0.3 0.3 0.3 0.3 0.4 1.0 1.0 0.3 1.0
4.56 4.56 4.20 4.20 3.60 3.54 4.08 5.40 3.60
RESULTS
A. Measurement of Dexp. Figure 2 is a typical penetration curve which was obtained from the measurements of the activity of Sample #l after a diffusion anneal of 76 min at 721.0%. If one plots (erf)-r
% ~ a0 -
‘b
%
vs. distance
from the interface on probability paper, the data may be fit with a straight line. Results of this experiment, some of which are plotted in Figs. 3-8 indicates that in the ranges of carbon concentration and temperature st,udied the diffusivity is independent of carbon
HOMAN:
DIFFUSION
OF
C IN
THE
ALPHA
1075
Fe
99.99
ox-(lb (lo-oh
98
9e
95
95
yo
90
ox-Ob (lo-(lb
e.
so
70
70
60
60
50
50
40
40
30
30
20
20
10
10
5
5
2
2
98
9.9
95
95
90
so
80
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
5
5
2
2
1
-30.0
-15.0
DISTANCE
FIG. 3.
0.0
FROM
I5.0
INTERFACE
( lu3
Probability plot of (a, - ab)/(q, from weld: sample #lA.
-60
30.0
-
a,) vs. distance
9999
3A
FIG. 5.
t20
0.0 FROM
WELD
(1631N.)
Probability plot of (a, - ab)/(rzO from weld: sample #5.
ub) vs. distance
99.99
99 99
DC*
6
To=730.9+.4”C
To =823.5
to=3.6Xl03sec
1D =4.OBXlO3sec
Cl4
IN
Ci41N
w-Fe
D = Ii 6 X10-’
i
I.O°C
d-Fe
D = 4 7.1 X 10-7cmz/sec
cm2/sec
98
96
95
95
90
90
80
80
‘0
70
70
70
60
60
60
60 50 40
a*-Ob oa-Ob
-20 DISTANCE
99.99
DC#
-40
IN.)
50
98
98
95
95 90
-90 armgo
80
50
M
40
40
‘la
30
30
30
30
20
20
20
20
10
IO
10
5
5
5
2
2
IO
5t-+
2
-80
-60
DISTANCE
FIG. 4.
-40 FROM
-20 WELD
(10-31N.)
Probability plot of (a, - aJ/(aO from weld: sample #3A.
0.0
+10
’ q,) vs. distance
-80
-60
DISTANCE
-20
-40 FROM
WELD
(lO-3
0.0
+10
IN.1
FIG. 6. Probability plot of (a, - ~+,)/(a,, from weld: sample #6.
aa) vs. distance
1056
ACTA
99 99
DC*
METALLURGICA,
1
7
VOL.
99 99
12.
1964 99.99
99.99
To =616.0-+0.5°C to=5.4x103 set C’41Noc-Fe D= 2.36X IO-‘cm hec
=644.2f
D =62.0 98
98
95
90
90
8o
EO O_
70
‘0
ao-(lb
60 50 40
60 50 40 30
30
20
20
10
IO
5
5
2
2
I
WELD
( lCj3
IN.)
Fro. 7. Probability plot of (a, - aa)/(ao from weld: sample #7.
concentration.
Table
experimental
#2
values
I*
is a summary
of all the
of the diffusivities of the diffusivity
the temperature
to study
pre-exponential
determined
activation
of carbon
indicates
(D, Hence
studied
a subsidiary
0.0
WELD
(lO-3
+20
IN.)
to
(B -
be
2.2 cm2/sec
Efv -
E,,)
to
and
the
be 29,300
DISCUSSION
is not
A. Comparisovb with the results of other measurements
of the
Comparison
two
Rewriting
plot was made of
diffusivities
good
competing equation
Figure
mechanisms
10
RT
3.
Summary
to x 103 SW
Couple no.
carbon indicate
effect,
if any, is not detectable.
Most
for
C-14
of the low temperature a quenching
approximately
1
721.0 * 0.3 721.0
f
0.3
4.56 4.56
+0.044 +0.032
;?q 3A 5 6 7 10
732.0 730.9 814.9 823.5 616.0 844.2
f & * + + +
0.3 0.4 1.0 1.0 0.3 1.0
4.20 3.60 3.54 4.08 5.40 3.60
-0.035 10.063 +0.060 +0.060 + 0.080 +0.045 +0.100
c, -
cbt
c0 -
cb
0.850 0.800 0.820 0.200 0.950 0.860 0.850 0.988 0.88*5
erf l#J
diffusion
technique
0.700 0.600 0.640 0.600 0.900 0.720 0.700 0.976 0.770
4 0.733 0.595 0.647 0.595 0.617 0.831 0.733 1.596 0.849
7 x,
carbon. measure-
in the sample
of diffusion calculations
X* (in.)
It
that an isotope effect would yield
of diffusivities
ments involve
1A
* z is some arbitrary distance from weld. i_ Values determined from straight line fit at r. $ See appendix B.
616”C-
Smith’s14) work using nonradioactive
smaller than for the nonradioactive
(13)
range
of this data with
values
from Fig. 10, one may find the value of the TABLE
D values(314) indicate
temperature agreement
would be expected
(12)
reported
in the
The quantitative
that an isotope
are in
B - E,, - En%,
D,) == 6 ii D,, exp
agreement
844°C.
D, from low
measurements.@)
of the D, values obtained in this experi-
ment with previously
shown in Fig. 9 minus the
extrapolated anelastic
that
operation.
range
diffusivities
corresponding temperature
terms
energy
in
form as shown in Fig. 9. In order
this effect
the measured
-20 FROM
5
Cal/mole.
a plot
simple exponential
I
FIG. 8. Probability plot, of (a, - ab)/(aO - (lb) vs. distance from weld: sample # 10.
a*) vs. distance
alpha iron measured in this and other(3s4) experiments over
-40
DISTANCE
in this experiment. However,
I
I
-60
FROM
X 10-7cm%ec
98
95
DISTANCE
I.O’C
lo-’ 12.0 10.3 11.5 12.9 11.8 32.6 47.1 2.4 62.0
11 cm2/sec f i_ * k $+ + f i
0.6 0.5 0.5 0.6 0.6 1.6 2.4 0.1 3.1
HOMAN:
DIFFUSION
OF
The
C IN
proposed
formation abrupt
model
indicates
transition. magnetic
magnetically
the vacancy
changes
occurring
the
Curie
of a possibly
factor [6(v1/v2) Do,]
magnitude
of the
pre-exponential
several orders of magnitude normally
at
the mobility
by the that the
inert C-12 or C-14 atom.
B. Pre-exponential
DIcmt&,
that
One is led to speculate
may not influence
The
1075
Fe
energy of alpha iron is unaffected
magnetic point
ALPHA
measured.(2,3)
factor
is
greater than the value
However,
this value
is of
the same order for the Do of self-diffusion measurements in alpha iron. %lg) This would appear to mean
4 83 IO-
that
2
the
relaxation
mechanism
associated
of self-diffusion
with
the
partially
diffusion
influences
the
carbon mobility. If one accepts to estimate entropy
changes.
vJv2 would IO
proposed,
of the factor
A reasonable
probably
corresponding
IO-
the model
the value
be
it is possible containing
value
between
the
of the ratio
1 and
10, the
range of Do, being 0.37 too.037
cm2/sec.
This compares to a value for Do, of 0.008 cm2/sec. C. Activation 1.50
1.40
1.30
1.20
1.10
1.00
.90
.50
9.5 kcal/mol
FIG. 9. Plot of diffusivity vs. 103/ToK for carbon in alpha iron. A Chemical technique (Stanley); 0 tracer technique E] Phase boundary migration (Smith).
preparation
fraction
Depending
stage.
quenching
should
the dispersion the
-
B of a carbon
B) vacancy
may be assigned from radiation
pair of damage
individual
present at the quench tem-
be ‘(frozen”
effect of these excess vacancies in
the
energy (E,, + E,, energy
statisttics ; e.g., the quench rate, a certain
of the vacancies
perature
on
A binding
noticed
anelastic
annealing
into
by several
relaxation
technique
the lattice.
was
The
may be the cause of investigators(2J6)
measurements.
developed
by
lO-63
An
KW6)
2
to
eliminate this dispersion. It has been shown series
of
damaged carbon pair.
on
atoms
indicating
Furthermore, annealing
atom may encounter precipitation
iron
radiation
that vacancies
a binding
energy
G,)
trap
7 6
it was suggested in these radiation kinetic
studies
that
many vacancies
the
I 9 8
for the
5
carbon
4
on its way to a
site.
The smoothness the
et &.(7~8J7) in a
alpha
at low temperatures,
damage
over
by Damask
measurements
3
of the curves
temperature
range
in Figs. 9 and 10
6OO”C-850°C
implies
lO-7 2
that the diflusivity of carbon in alpha iron is unaffected by the magnetic change at 77O”C, the Curie temperature. This result is in striking contrast to the abrupt changes in the self-diffusion coefficientW in alpha iron, the diffusivity of nickel in iron,(l5) and the diffusivities of hydrogen in nickel which occur at the Curie temperature.
and iron@O)
I IO’
FIG. 10. Plot of De,,, alpha, iron.
-
D, vs.
/ To tl
103/ToK
for carbon
I
.
in
ACTA
1078
calorimetric
studies at Brookhaven;@)
value may be appreciably solution
does
not
either t,o previous of vacancy The
participate
in the
precipitation
microscopy
however, reaction
studies
by various
of the reaction.
at Brookhaven(17)
activation 10 kcal/mol. activation or vacancy
A
has been
investigators.(s~1a~~9) From the
of this experiment,
an upper
energy for the motion
of the
of the pair is about
This energy is sufficiently energies of motion
limit
of either
less than the
13, 1964
results.c5) equation
value for AV,,
is pressure
on a plot of In D, versus l/T
at high temperatures. be
The value
diffusivities
may
fit
expression.
An empirical
of the measured
with
a
model
which includes
of a carbon-vacancy
two
interaction
mechanism the
is proposed
to explain this deviation.
A possible counting bution rate.
error in the radioactive
low energy /3 particles of subsurface
The
(14)
l3, has been shown to be pressure independentc5) pressure independent,
to the surface
counting
thickness
of
count
b of
each
6
f0
n, exp - (FK) (LX
(171
where a, is the countrate measured at the surface due to the distribution
a, in the section.
If ,u is the bulk
then b will be in the order of the range
absorptivity,
of cc particles in the material. All the count rates measured in this experiment were looking
into a decreasing
the exception
gradient
of the a0 determination.
of activity, with In the extreme
gradient, is very
steep, our
the a0 determina.tion
would appear to be a
This error may be cxpressed
by
(8) may be rewritten
B, = D, + @, D,,,
technique
section can be expressed by the relation
However,
Y&,
in
is due to the contri-
activity
effective
volume effect.
Assuming
the decrease
B. Nonstatistical experimental errors
case where the activity
For the case 7 < 1, equation
then
a
Furthermore,
count rate would be due primarily to surface activity.
APPENDICES
4000 atm.
sensitive,
from
assuming
D, will be larger.
in alpha iron.
a straight line behavior
of 0.5 atomic volumes.
a, =
The diffusion of carbon in alpha iron deviates from
in D, is predicted
A 15 ‘A decrease
(16) at 82B’C and 4000 atm.
the carbon
CONCLUSIONS
possibility
T’OL.
if E,,
or to an insu&ieney
range of EfWfrom 29 kca,l/mol to 40 kcal/mol results
due
that the latter effect may be occurring.
suggested
this
higher if all the carbon in
traps at the temperature
electron
indicate
METALLURCICA,
B and E,,
to
to be
it is clear that
(18) for the conditions Using
of this experiment.
equation
(18)
we
= I -
2
may
write,
neglecting
background, X
erf --.2dtDe,,
2a exl,
a0 X
=
erf I___ 21/tD,,t
(19)
D exp I D,,t
(20)
or (15) under the conditions The room pressure values are indicated script 0. Integrating
by the sub-
one obtains
D,(P. 117) = D, + yoD,, exp -
of this experiment.
This effect has been investigated the diffusivity activity
side.
by redetermining
of sample #2B
looking
In this counting
situation
from the low (21)
D exp 2 Dact
The results of this determination indicate a diffusivity approxima~ly 10 % higher than the results of 3A. Another source of error is in neglecting to correct the diffusion time for the heat-up time. Such a correction In the low temperature region where D, 4 D, the model agrees qualitatively with the high pressure
would
any temperature. Comparison of
tend to increase the diffusivity all
the
errors
involved
in
at the
HOMAN:
measurement to
the
indicated
subsurface
that the constant
activity
contribution
OF
DIFFUSION
error due described
above is the largest, hence the reported limit of error
10.
reflects this uncertainty. ACKNOWLEDGMENTS
I would
like
to
acknowledge
many Prof.
helpful
Huntington
and to thank him for the encouragement
he provided to
colleagues
throughout
acknowledge at Watervliet
thesis
the
with
like
my
adviser,
discussions
this research. the
help
B.
I would also
received
Arsenal, especially
Cox for his aid and ideas,
H.
from
my
Mr. J. F.
REFERENCES
6. 7. 8. 9.
12. 13.
and Dr. L. Meisel and
Mr. J. Frankel for their discussions.
1. 2. 3. 4. 5.
11.
C. A. WERT, Phys. Rev. 79, 601 (1950). C. A. WERT and C. ZENER, Phys. Rev. 76, 1169 (1949). J.K. STANLEY, Trans. AIME 185,(1949). R. P. SMITH, Trans. AZME 224, 105 (1962). J. BASS and D. LAZARUS,J. Phys. Chem. Solids 23, 1820 (1962). Also A. ROSMAN et al.,Physica 23, 1001 (1957); Physica 26, 533 (1960). P. SHEL~MON,D@&on in Solids, Chap. 1. McGraw-Hill (1963). F. FUJITA and A. DAMASK, Acta Met. 12, 331 (1964). R. ARNI)T and A. DAMASK, Acta Met. 12, 341 (1964). H. BROOKS, Impurities and Zmperfections. ASM Monograph (1957). R. MEHI. et rtl., A& Met. 9, 256 (1961).
14. 15. 16. 17.
18. 19.
20.
C IN
ALPHA
Fe
1059
R. DOREMUSand E. KOCH, Trans. AZME 218,591(1960). E. SMITH, Indirect Observations of Imperfections in Crystals, p. 207. Interscience (1962). The Batelle Iron used in this experiment was provided through the kind offices of J. W. Halley, Chairman of the Pure Iron Subcommittee of the American Iron and Steel Institute Research Committee. The analysis provided is Al-15 ppm; 0-5 ppm; Co-5 ppm; Cu-7 ppm; Ni-20 ppm; P-9 ppm; Si-10 ppm; C-10 ppm; O-10 ppm; N-2 ppm; S-18 ppm. Other metallic impurities were not detected. The radioactive methane used to carburize the specimen was purchased from two soumes, Tracerlab and Nuclear Chicago. The activity of the three ampules used was 4.9, 2.0 and 2.0 mc/mM. See L. 8. DARKEN, in Atom Movements. SSM Monograph (1951). The author is indebted to Prof. F. Lenel, Rensselaer Polytechnic Institute, for suggesting this method of reducing decarburization effects. M. HANSEN, Constitution of Binary Alloys, 2nd edition, p. 353. McGraw-Hill (1958). C. J. SMITHELLS Metals Reference Book, 3rd edition, Vol. 9. Butterworths (1962). T. S. Kfi Metals Tech. T.P. No. 2370 (June 1948); Phys. Rev. 74, 9 (1948). H. WAGENBLASTand A. DAMASK, J. Phys. Ch,em. Solids 23, 221 (1962). H. WAGENBLAST,F. FUJITA and A. DAMASK, Actu Met. 12, 347 (1964). D. LAZARUS, in Solid State Physics, Vol. 10. Academic Press (1960). R. BORG and C. BIRCHENALL, Trans. AZME 218. 980 (1960). P. BUFFINGTON,K. HIRANO and &M.COHEN, Actn Met. 9, 434 (1961). See F. N. RHINES, in Atom Novem7nt.v. 8SM Monograph (1951).
OF
CARBON C.
G.
IN
ALPHA
IRON*
HOMAN?
The diffusion of carbon in alpha iron using a tracer technique was performed in the temperature range of 616°C to 844°C. 9 plot of the In D versus l/T in this temperature range deviates from the usual straight line behaviour. An empirical model which includes the possibility of a carbon-vacancy interaction is proposed to explain this deviation.
DIFFUSION DU CARBONE DANS LE FER ALPHA L’auteur Btudie par une technique de traceur, la diffusion du carbone dans le fer alpha pour une gamme de temp6ratures s’6tendant de 816°C it 844°C. la relation ln D = f(l/T) ne poss&de pas les caract&istiques Dans cette gamme de tempkratures, d’une droite. Pour expliquer ce r&,ultat, l’auteur propose un modAle empirique qui fait intervenir la possibilite d’une interaction entre les atomes de carbone et les lacunes. DIFFUSION
VON
KOHLENSTOFF
IN
ALPHA-EISEN
Die Diffusion van Kohlenstoff in alpha-Eisen wurde im Temperaturbereich 616°C bis 844°C mittels Trilgt man In D gegen l/T auf, so erhtilt man Abweichungen vom einer Isotopentechnik verfolgt. Zur Erkliirung der Abweichungen wird ein empirisches Model1 vorgeschlagen, iiblichen linearen Verlauf. das die Miiglichkeit einer Wechselwirkung zwischen Kohlenstoff und Leerstellen einschlieot.
agreement
INTRODUCTION
By implicitly
assuming that every diffusional
made by carbon performed
atoms in an alpha iron matrix
with identical
local surroundings,
perature
jump was
Wert’l’
of our results with the previous high temresults of Smithc4) indicated
extrapolation questionable. The implicit
from
low
that a linear
temperature
assumption
made
data
by Wert
was
may
be
was able to derive an elegant theory based on a simple random walk model. This model leads to an
correct for low temperature
studies in pure iron, but
expression
it is perhaps not completely
accurate in the tempera-
for the diffusivity characterized by a single parameter 7. Wert was able to qualitatively
relaxation fit experimental
data representing
approximately However, which
fourteen
the low
has been
decades
not be extrapolated
of
An empirical
diffusivity.
Furthermore,
data, could
atoms could have two different surrounding
data(3$4) at
anelastic
techniques(5)
result of an extremely
of carbon
a t’racer
technique
simple single relaxation
to examine
model and to provide
of t’his investigation
showed
with
a
st)raight
line.
Furthermore,
neighbor
iron sites is
The agreement of this model with the low temperasmall, is assured. However,
and room pressure
was impossible to fit the high temperature
configuration
exist on all the nearest neighbor
ture results,(2) where the population
was to redetermine
in order
configu-
vacant.
the anomalous
a zero that
of vacancies
A calculation
it
diffusivities quantitative
indicates
is
at the higher temperature,
the proposed model includes the possibility vacancy interactions.
the
point for a high pressure diffusion experiment. The results
The first one is the normal
is when one of these nearest
iron by
small pressure effect.
the diffusirit)y at high temperature
in
in which the moving carbon
sites of the carbon atoms and the second configuration
of the pressure
in alpha
have indicated
The purpose of this experiment using
rations.
where iron atoms
recent measurements
iron
of carbon
experimental
to the experimental
on t’he diffusion
model for the diffusion
determined,c2)
the high temperature. effect
neighbor
sites to a diffusing carbon atom may be vacant. alpha iron was developed
t,emperature
quant.itatively
measurements of the
ture range where one of the nearest
based on this model
a high temperature
of carbon
(Appendix
A)
pressure effect on the
diffusivity
of carbon in alpha iron proportional
activation
volume of formation
to the
of vacancies.
THEORY
* Received, Janua,ry 20, 1964; revised February 20, 1964. Submitted to the Physics Department, Rensselaer Polytechnic Institute, Troy, N.Y. in partial fulfillment of the requirements
In developing an empirical model for the diffusion of C in a-Fe which would include the possibility of
of the lMast)er of Science degree. ‘j’ Work performrad at Watervliet York.
vacancy-carbon interaction, the following conditions.
ACTA
METALLURGICA,
VOL.
Arsenal,
Watervliet,
12, SEPTEMBER
1964
New
1071
it is necessary
to satisfy
SCT_4
1072
1. The
diffusivity
independent
carbon
in
alpha
iron
is
derived
with
from
changes
of
VOL.
3. A good empirical
the model vacancy
should
be
fit be obtained
range.
x exp -
equation
diffusion couple
-
I = erf “-z.
i
‘22/m
tracers are used, equation
(2) may
2 22/11)t
+%&+?x])~~ ‘c
in the free state; (All energies carbon
atoms
the
(6) v2
vacancy
concentration
in atom
percent *B -
?,=6:exp
E,,
RT
(
(7
i
where E,, is the energy of forn~ation of a vacancy.
The
radioactivities
to CO, C,, and C$. Therefore,
effe&ive
formulated
diffusion
coefficient
De
may
be
to be:
from a
plot of
De = (1 -
qP,
+ rlD,
(8)
where erf-l
~oncentratioll
(9)
’
it is possible to determine
if D is independent
and if this is true,
its value
of the at the
annealing temperature.
is the diffusivity
II,=
The :proposed reaction is c+v+c...v The forward
reaction
atom
assumed
requires the presence of a free
bimolecular
(Experimental
(3)
(C) and a “free” and
evidence
carbon-vacancy
hence,
also suggests
The reverse reaction
differential
vacancy
governing
(V), and is
second
with unpaired motion
Ds,exp-
unimolecular.
The
this model is :
From
anelastic
the concentration
vacancy
concentrations,
respectively.
and
If one
g
relaxation
(19)
measurements,
of vacancies
D, = 0.008 exp -
where
is negligible’s’
19,800 cal/mole
(11)
)
RT
i
And for the case when q < I; (q < 1O-2 at SO%.) ,__ _^_ _, . I 19,WO cal/mole 6V1 ~... .0.008 exp -+
i
where C,, C, and C, are the pair, total carbon
of
where E,, is the motional energy of the carbon atom when paired with a vacancy.
order.
this assump-
requires only a paired
and is assumed
equation
associated
a carbon atom and for the other eon~guration,
B ~on~e~tr~,t~o~ sf ~r~or~-~~~~~e~ pa+s
total
of E,,,
C. Effective difSusion coeficient
where oO, ab and a, are the measured
tion.ti))
frequencies
con~gurations;
energy for a carbon
or expressing on the
t,o read 1 = erf-
carbon
in t,he two
At equilibrium, the ratio of paired to total carbon concentration, rj, is
tration at distance x from the interface.
corresponding
(5)
“)
and B is the binding energy of tOhepair.
high and low side of the couple and C, is the concenIf radioactive
atom
(2)
where (7, and CO are the initial concentrations
be modified
(Em;,
(I) can be shown to be S6)
C, c,
C?,v,
are expressed in kcaljmole.) of the concen-
In the case of a semi-infinite
C 2 _-5 ( c;, -
the carbon
(I)
D is independent
the solution of equation
-
!
where y1 and V, are the vibrational the motional
!z.=Dazc at ax2
tration.
(
E *g
for measured
Pick’s second law of diffusion may be written as:
when t,he diffusivity
(4) ma,y be written:
dC -2 = Gv,C,C, exp clt
concentration
over the entire ~m~)erature
A. Solution qf d$usion
1964
assumes that the carbon atoms are the more mobile
at different temperatures. diffusi~ities
12,
defect, then equation
of carbon concentration.
2. Equations consistent
of
METALLURGICA,
B ex!?
!
E”,, RT
RT
E,,
1
,
y2
112)
HOMAN:
A plot
D,)
of In (11, -
literature
value
vs.
l/T
DIFFUSION
together
OF
with the
of Bc8) and Efvcg) should
yield
the
value of [SCv,/v,) D,,] and E,,.
C IN
ALPHA
evacuated
2 ,u, radiomethane
to about
mitted
to a pressure
vessel
was
sealed.
A. Counting An
procedure
end
GM.
PROCEDURE
window
tube
(window
was mounted
thickness
as shown
1 mg/cm2)
in Fig.
1.
This
fixture insured constant geometry for surface counting sample
ends.
The
counting
end shield was concentric diameter for
of 0.120 in.
counting
counting
geometry was
chi-square
tests.
in the aluminum
with the samples and had a
The V block held the samples
and grinding.
equipment
port
The reproducibility
and the reliability
checked
of
of the counting
periodically
by
means
of
Cylinders
of 0.187
in. dia. and from
99.98%
18 in. long
were
zone refined
iron
machined
stock.oO)
The cylinders were then etched in a solution
of 10 g of oxalic acid, 20 cc of 30 ‘A hydrogen peroxide 300 ml
of
H,O
at 45°C
0.020 in. were removed
until
approximately
from the diameter
and the
surface was very bright and shiny. The cylinders
Carbon
was deposited
surface
heating
with
by
approximately
burized
vessel
the gases remaining and cylinders
furnace
which
to be carburized
Mettler
Grammatic
microbalance
quartz
carburizing
vessel.
were weighed on a
After
and placed the
in a
vessel
was
butt
in an
The car-
and shiny
to determine
to insure a uniform
The uniform
welding,
cylinder
to
in the vessel,
were placed
were bright
were reweighed
gain and counted tribution.
furnace
at 750°C for 72 hr.(l”)
cylinders,
appearance,
induction
The temperat’ure was t)hen raised to 850°C for about
90 min. After evacuating annealing
an
in the
on the cylinder
500°C for 30 min.
of the cylinder
the entire
of the methane
concentrations
cylinders.)
tion treatment.
carefully
and
pressure
different
was ad-
cm of Hg and the
in
the weight carbon
dis-
count rate of a cylinder
was
between 50 and 200 cpm depending
Sample preparation
B.
of 0.5-4.0 (The
was varied to obtain EXPERIMENTAL
1073
Fe
The diffusion in a helium
to a carburized
on the carburiza-
couple was formed
atmosphere, cylinder
by
a pure iron
(using
a special
fixture and after suitably preparing the weld surfaces). The weld was made amperes load.
through
The time
Metallographic
by passing
3-5
pulses of 900
the sample under a small pressure of a pulse never examination
exceeded
of similar
5 sec.
cylinders
of
Armco iron welded with the same procedure indicat,ed good
welds with no discernible
entrapments
at the
interface. Since this diffusion experiment high pressures,
a new technique
sample was developed.03)
will be extended
Because
the
(1) the solubility
of Carbon in Cu is low,(14) (2) the diffusivity a-Fe is low with respect to the diffusivity Feo5) and (3) Cu is an extremely
to
to encapsulate
of Cu in of C in a-
good “getter”
of O,,
it was decided
to plate the diffusion couples with 0.003 in. of Cu. After a diffusion anneal, it was possible to remove part of the copper oxide layer to obtain a bright Cu color. The solubility of C in Cu was GM END
qualitatively
investigated
C. Diflusion
anneal
by a counting
technique.
‘WIN”C)ul .. ..__.. COUNTING
A temperature 05°C GM TUBE SUPPORT
control
over the sample length of
at 725°C during a diffusion anneal was possible
by means of a diffusion furnace and controller. temperature
fluctuations
during
the
measured by means of a chromel-alumel ENDWINDOW SHIELD AND COUNTING PORT V-BLOCK
of counting
assembly.
The were
thermocouple
imbedded in the Cu heat bath of the diffusion furnace. The thermocouple potential was measured with a L&N K-3 potentiometer against a cold junction of glycerine in a dewar. The chromel-alumel couple was calibrated
FIG. 1. Schematic
anneal
before and after a diffusion run with
a Leeds and Northrup standard Pt-Pt 10% Rh thermocouple using an ice bath cold junction. The
1074
ACTA
METALLURGICA,
temperature of the cold junction was measured with a ~alibrate~l thermometer. The sample was inserted into the furnace with a chrome]-alumel thermocouple welded to the low activity side of the couple. Measurements of the heating time of this thermocouple as compared to the furnace thermocouple indicated that a maximum time of 1 minute was needed to heat from 0.8 T, to TD for diffusion runs lasting from 60 to 90 min. After the anneal, the sample was quenched within 10 set to R.T. The data has not been corrected for the heat-up time.
The sample was mounted in the V block and the Cu plating was removed from the end with a precision surfacegrinder. The grinder has an accuracy of 0.0001 in. and produced a #lS finish on the sample end. Sectioning and counting from the high carbon (activity) end showed that the original cpm measured before the anneal could be obtained within 0.005 in. of the end. This result qualitatively confirmed the fact of the extremely low solubility of C in Cu. At least three sections were made to within 0.2 in. of the weld to be sure t,hat the concentration in the carburized portion of the sample was uniform after the anneal. All samples reported in this paper were within 1 “/;; of the original value of a0 determined before the anneal in this region. The penetration was measured from this point. For samples #I & #lo, every section was counted for at least 10,000 counts; samples fA, 2A. 3A, 5 and 7 have at least three
VOL.
12,
1964
sections counted for at least 10,000 counts; and for sample #6, every section was counted for at least 3,000 counts. Background measurements were made periodically during the counting procedure and varied between 23.4 and 24.3 over several months of counting. A summary of experimental data has been tabulated in Table #I. TABLE 1. Diffusion annealing data -. ._ -
z.-. Couple no. 1 1A 2A 2B 3A 5 6 7 10
&vt.y$J
a0 iwm)
(%!
0.022 0.022 0.021 0.021 0.009 0.006 0.006 0.006 0.008
191 191 184 184 80 55 53 52 209
721.0 f 721.0 f 732.0 * 732.0 * 730.9 & 814.9 + 823.5 & 616.0 * 844.2 i -
tn x 10s sec. 0.3 0.3 0.3 0.3 0.4 1.0 1.0 0.3 1.0
4.56 4.56 4.20 4.20 3.60 3.54 4.08 5.40 3.60
RESULTS
A. Measurement of Dexp. Figure 2 is a typical penetration curve which was obtained from the measurements of the activity of Sample #l after a diffusion anneal of 76 min at 721.0%. If one plots (erf)-r
% ~ a0 -
‘b
%
vs. distance
from the interface on probability paper, the data may be fit with a straight line. Results of this experiment, some of which are plotted in Figs. 3-8 indicates that in the ranges of carbon concentration and temperature st,udied the diffusivity is independent of carbon
HOMAN:
DIFFUSION
OF
C IN
THE
ALPHA
1075
Fe
99.99
ox-(lb (lo-oh
98
9e
95
95
yo
90
ox-Ob (lo-(lb
e.
so
70
70
60
60
50
50
40
40
30
30
20
20
10
10
5
5
2
2
98
9.9
95
95
90
so
80
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
5
5
2
2
1
-30.0
-15.0
DISTANCE
FIG. 3.
0.0
FROM
I5.0
INTERFACE
( lu3
Probability plot of (a, - ab)/(q, from weld: sample #lA.
-60
30.0
-
a,) vs. distance
9999
3A
FIG. 5.
t20
0.0 FROM
WELD
(1631N.)
Probability plot of (a, - ab)/(rzO from weld: sample #5.
ub) vs. distance
99.99
99 99
DC*
6
To=730.9+.4”C
To =823.5
to=3.6Xl03sec
1D =4.OBXlO3sec
Cl4
IN
Ci41N
w-Fe
D = Ii 6 X10-’
i
I.O°C
d-Fe
D = 4 7.1 X 10-7cmz/sec
cm2/sec
98
96
95
95
90
90
80
80
‘0
70
70
70
60
60
60
60 50 40
a*-Ob oa-Ob
-20 DISTANCE
99.99
DC#
-40
IN.)
50
98
98
95
95 90
-90 armgo
80
50
M
40
40
‘la
30
30
30
30
20
20
20
20
10
IO
10
5
5
5
2
2
IO
5t-+
2
-80
-60
DISTANCE
FIG. 4.
-40 FROM
-20 WELD
(10-31N.)
Probability plot of (a, - aJ/(aO from weld: sample #3A.
0.0
+10
’ q,) vs. distance
-80
-60
DISTANCE
-20
-40 FROM
WELD
(lO-3
0.0
+10
IN.1
FIG. 6. Probability plot of (a, - ~+,)/(a,, from weld: sample #6.
aa) vs. distance
1056
ACTA
99 99
DC*
METALLURGICA,
1
7
VOL.
99 99
12.
1964 99.99
99.99
To =616.0-+0.5°C to=5.4x103 set C’41Noc-Fe D= 2.36X IO-‘cm hec
=644.2f
D =62.0 98
98
95
90
90
8o
EO O_
70
‘0
ao-(lb
60 50 40
60 50 40 30
30
20
20
10
IO
5
5
2
2
I
WELD
( lCj3
IN.)
Fro. 7. Probability plot of (a, - aa)/(ao from weld: sample #7.
concentration.
Table
experimental
#2
values
I*
is a summary
of all the
of the diffusivities of the diffusivity
the temperature
to study
pre-exponential
determined
activation
of carbon
indicates
(D, Hence
studied
a subsidiary
0.0
WELD
(lO-3
+20
IN.)
to
(B -
be
2.2 cm2/sec
Efv -
E,,)
to
and
the
be 29,300
DISCUSSION
is not
A. Comparisovb with the results of other measurements
of the
Comparison
two
Rewriting
plot was made of
diffusivities
good
competing equation
Figure
mechanisms
10
RT
3.
Summary
to x 103 SW
Couple no.
carbon indicate
effect,
if any, is not detectable.
Most
for
C-14
of the low temperature a quenching
approximately
1
721.0 * 0.3 721.0
f
0.3
4.56 4.56
+0.044 +0.032
;?q 3A 5 6 7 10
732.0 730.9 814.9 823.5 616.0 844.2
f & * + + +
0.3 0.4 1.0 1.0 0.3 1.0
4.20 3.60 3.54 4.08 5.40 3.60
-0.035 10.063 +0.060 +0.060 + 0.080 +0.045 +0.100
c, -
cbt
c0 -
cb
0.850 0.800 0.820 0.200 0.950 0.860 0.850 0.988 0.88*5
erf l#J
diffusion
technique
0.700 0.600 0.640 0.600 0.900 0.720 0.700 0.976 0.770
4 0.733 0.595 0.647 0.595 0.617 0.831 0.733 1.596 0.849
7 x,
carbon. measure-
in the sample
of diffusion calculations
X* (in.)
It
that an isotope effect would yield
of diffusivities
ments involve
1A
* z is some arbitrary distance from weld. i_ Values determined from straight line fit at r. $ See appendix B.
616”C-
Smith’s14) work using nonradioactive
smaller than for the nonradioactive
(13)
range
of this data with
values
from Fig. 10, one may find the value of the TABLE
D values(314) indicate
temperature agreement
would be expected
(12)
reported
in the
The quantitative
that an isotope
are in
B - E,, - En%,
D,) == 6 ii D,, exp
agreement
844°C.
D, from low
measurements.@)
of the D, values obtained in this experi-
ment with previously
shown in Fig. 9 minus the
extrapolated anelastic
that
operation.
range
diffusivities
corresponding temperature
terms
energy
in
form as shown in Fig. 9. In order
this effect
the measured
-20 FROM
5
Cal/mole.
a plot
simple exponential
I
FIG. 8. Probability plot, of (a, - ab)/(aO - (lb) vs. distance from weld: sample # 10.
a*) vs. distance
alpha iron measured in this and other(3s4) experiments over
-40
DISTANCE
in this experiment. However,
I
I
-60
FROM
X 10-7cm%ec
98
95
DISTANCE
I.O’C
lo-’ 12.0 10.3 11.5 12.9 11.8 32.6 47.1 2.4 62.0
11 cm2/sec f i_ * k $+ + f i
0.6 0.5 0.5 0.6 0.6 1.6 2.4 0.1 3.1
HOMAN:
DIFFUSION
OF
The
C IN
proposed
formation abrupt
model
indicates
transition. magnetic
magnetically
the vacancy
changes
occurring
the
Curie
of a possibly
factor [6(v1/v2) Do,]
magnitude
of the
pre-exponential
several orders of magnitude normally
at
the mobility
by the that the
inert C-12 or C-14 atom.
B. Pre-exponential
DIcmt&,
that
One is led to speculate
may not influence
The
1075
Fe
energy of alpha iron is unaffected
magnetic point
ALPHA
measured.(2,3)
factor
is
greater than the value
However,
this value
is of
the same order for the Do of self-diffusion measurements in alpha iron. %lg) This would appear to mean
4 83 IO-
that
2
the
relaxation
mechanism
associated
of self-diffusion
with
the
partially
diffusion
influences
the
carbon mobility. If one accepts to estimate entropy
changes.
vJv2 would IO
proposed,
of the factor
A reasonable
probably
corresponding
IO-
the model
the value
be
it is possible containing
value
between
the
of the ratio
1 and
10, the
range of Do, being 0.37 too.037
cm2/sec.
This compares to a value for Do, of 0.008 cm2/sec. C. Activation 1.50
1.40
1.30
1.20
1.10
1.00
.90
.50
9.5 kcal/mol
FIG. 9. Plot of diffusivity vs. 103/ToK for carbon in alpha iron. A Chemical technique (Stanley); 0 tracer technique E] Phase boundary migration (Smith).
preparation
fraction
Depending
stage.
quenching
should
the dispersion the
-
B of a carbon
B) vacancy
may be assigned from radiation
pair of damage
individual
present at the quench tem-
be ‘(frozen”
effect of these excess vacancies in
the
energy (E,, + E,, energy
statisttics ; e.g., the quench rate, a certain
of the vacancies
perature
on
A binding
noticed
anelastic
annealing
into
by several
relaxation
technique
the lattice.
was
The
may be the cause of investigators(2J6)
measurements.
developed
by
lO-63
An
KW6)
2
to
eliminate this dispersion. It has been shown series
of
damaged carbon pair.
on
atoms
indicating
Furthermore, annealing
atom may encounter precipitation
iron
radiation
that vacancies
a binding
energy
G,)
trap
7 6
it was suggested in these radiation kinetic
studies
that
many vacancies
the
I 9 8
for the
5
carbon
4
on its way to a
site.
The smoothness the
et &.(7~8J7) in a
alpha
at low temperatures,
damage
over
by Damask
measurements
3
of the curves
temperature
range
in Figs. 9 and 10
6OO”C-850°C
implies
lO-7 2
that the diflusivity of carbon in alpha iron is unaffected by the magnetic change at 77O”C, the Curie temperature. This result is in striking contrast to the abrupt changes in the self-diffusion coefficientW in alpha iron, the diffusivity of nickel in iron,(l5) and the diffusivities of hydrogen in nickel which occur at the Curie temperature.
and iron@O)
I IO’
FIG. 10. Plot of De,,, alpha, iron.
-
D, vs.
/ To tl
103/ToK
for carbon
I
.
in
ACTA
1078
calorimetric
studies at Brookhaven;@)
value may be appreciably solution
does
not
either t,o previous of vacancy The
participate
in the
precipitation
microscopy
however, reaction
studies
by various
of the reaction.
at Brookhaven(17)
activation 10 kcal/mol. activation or vacancy
A
has been
investigators.(s~1a~~9) From the
of this experiment,
an upper
energy for the motion
of the
of the pair is about
This energy is sufficiently energies of motion
limit
of either
less than the
13, 1964
results.c5) equation
value for AV,,
is pressure
on a plot of In D, versus l/T
at high temperatures. be
The value
diffusivities
may
fit
expression.
An empirical
of the measured
with
a
model
which includes
of a carbon-vacancy
two
interaction
mechanism the
is proposed
to explain this deviation.
A possible counting bution rate.
error in the radioactive
low energy /3 particles of subsurface
The
(14)
l3, has been shown to be pressure independentc5) pressure independent,
to the surface
counting
thickness
of
count
b of
each
6
f0
n, exp - (FK) (LX
(171
where a, is the countrate measured at the surface due to the distribution
a, in the section.
If ,u is the bulk
then b will be in the order of the range
absorptivity,
of cc particles in the material. All the count rates measured in this experiment were looking
into a decreasing
the exception
gradient
of the a0 determination.
of activity, with In the extreme
gradient, is very
steep, our
the a0 determina.tion
would appear to be a
This error may be cxpressed
by
(8) may be rewritten
B, = D, + @, D,,,
technique
section can be expressed by the relation
However,
Y&,
in
is due to the contri-
activity
effective
volume effect.
Assuming
the decrease
B. Nonstatistical experimental errors
case where the activity
For the case 7 < 1, equation
then
a
Furthermore,
count rate would be due primarily to surface activity.
APPENDICES
4000 atm.
sensitive,
from
assuming
D, will be larger.
in alpha iron.
a straight line behavior
of 0.5 atomic volumes.
a, =
The diffusion of carbon in alpha iron deviates from
in D, is predicted
A 15 ‘A decrease
(16) at 82B’C and 4000 atm.
the carbon
CONCLUSIONS
possibility
T’OL.
if E,,
or to an insu&ieney
range of EfWfrom 29 kca,l/mol to 40 kcal/mol results
due
that the latter effect may be occurring.
suggested
this
higher if all the carbon in
traps at the temperature
electron
indicate
METALLURCICA,
B and E,,
to
to be
it is clear that
(18) for the conditions Using
of this experiment.
equation
(18)
we
= I -
2
may
write,
neglecting
background, X
erf --.2dtDe,,
2a exl,
a0 X
=
erf I___ 21/tD,,t
(19)
D exp I D,,t
(20)
or (15) under the conditions The room pressure values are indicated script 0. Integrating
by the sub-
one obtains
D,(P. 117) = D, + yoD,, exp -
of this experiment.
This effect has been investigated the diffusivity activity
side.
by redetermining
of sample #2B
looking
In this counting
situation
from the low (21)
D exp 2 Dact
The results of this determination indicate a diffusivity approxima~ly 10 % higher than the results of 3A. Another source of error is in neglecting to correct the diffusion time for the heat-up time. Such a correction In the low temperature region where D, 4 D, the model agrees qualitatively with the high pressure
would
any temperature. Comparison of
tend to increase the diffusivity all
the
errors
involved
in
at the
HOMAN:
measurement to
the
indicated
subsurface
that the constant
activity
contribution
OF
DIFFUSION
error due described
above is the largest, hence the reported limit of error
10.
reflects this uncertainty. ACKNOWLEDGMENTS
I would
like
to
acknowledge
many Prof.
helpful
Huntington
and to thank him for the encouragement
he provided to
colleagues
throughout
acknowledge at Watervliet
thesis
the
with
like
my
adviser,
discussions
this research. the
help
B.
I would also
received
Arsenal, especially
Cox for his aid and ideas,
H.
from
my
Mr. J. F.
REFERENCES
6. 7. 8. 9.
12. 13.
and Dr. L. Meisel and
Mr. J. Frankel for their discussions.
1. 2. 3. 4. 5.
11.
C. A. WERT, Phys. Rev. 79, 601 (1950). C. A. WERT and C. ZENER, Phys. Rev. 76, 1169 (1949). J.K. STANLEY, Trans. AIME 185,(1949). R. P. SMITH, Trans. AZME 224, 105 (1962). J. BASS and D. LAZARUS,J. Phys. Chem. Solids 23, 1820 (1962). Also A. ROSMAN et al.,Physica 23, 1001 (1957); Physica 26, 533 (1960). P. SHEL~MON,D@&on in Solids, Chap. 1. McGraw-Hill (1963). F. FUJITA and A. DAMASK, Acta Met. 12, 331 (1964). R. ARNI)T and A. DAMASK, Acta Met. 12, 341 (1964). H. BROOKS, Impurities and Zmperfections. ASM Monograph (1957). R. MEHI. et rtl., A& Met. 9, 256 (1961).
14. 15. 16. 17.
18. 19.
20.
C IN
ALPHA
Fe
1059
R. DOREMUSand E. KOCH, Trans. AZME 218,591(1960). E. SMITH, Indirect Observations of Imperfections in Crystals, p. 207. Interscience (1962). The Batelle Iron used in this experiment was provided through the kind offices of J. W. Halley, Chairman of the Pure Iron Subcommittee of the American Iron and Steel Institute Research Committee. The analysis provided is Al-15 ppm; 0-5 ppm; Co-5 ppm; Cu-7 ppm; Ni-20 ppm; P-9 ppm; Si-10 ppm; C-10 ppm; O-10 ppm; N-2 ppm; S-18 ppm. Other metallic impurities were not detected. The radioactive methane used to carburize the specimen was purchased from two soumes, Tracerlab and Nuclear Chicago. The activity of the three ampules used was 4.9, 2.0 and 2.0 mc/mM. See L. 8. DARKEN, in Atom Movements. SSM Monograph (1951). The author is indebted to Prof. F. Lenel, Rensselaer Polytechnic Institute, for suggesting this method of reducing decarburization effects. M. HANSEN, Constitution of Binary Alloys, 2nd edition, p. 353. McGraw-Hill (1958). C. J. SMITHELLS Metals Reference Book, 3rd edition, Vol. 9. Butterworths (1962). T. S. Kfi Metals Tech. T.P. No. 2370 (June 1948); Phys. Rev. 74, 9 (1948). H. WAGENBLASTand A. DAMASK, J. Phys. Ch,em. Solids 23, 221 (1962). H. WAGENBLAST,F. FUJITA and A. DAMASK, Actu Met. 12, 347 (1964). D. LAZARUS, in Solid State Physics, Vol. 10. Academic Press (1960). R. BORG and C. BIRCHENALL, Trans. AZME 218. 980 (1960). P. BUFFINGTON,K. HIRANO and &M.COHEN, Actn Met. 9, 434 (1961). See F. N. RHINES, in Atom Novem7nt.v. 8SM Monograph (1951).