- Email: [email protected]
On the determination of diffusion coefficients in chemical diffusion
ON THE
DETERMINATION
OF DIFFUSION
COEFFICIENTS
IN CHEMICAL
DIFFUSION*
R. W. BALLUFFIT The equations describing chemical diffusion in the generalcase where the partial molal volumes of the diffusing components may differ and may vary with composition are reviewed and discussed. Exact and easily applied relations for obtaining chemical and intrinsic diffusivities ape derived. The results should replace the standard Darken equations and Boltzmann-Matano analysis whenever there is any possibility that differences and variations of the partial molal volumes are of significance. SUR
LA DETERMINATION
DES
COEFFICIENTS CHIMIQUE
DE
DIFFUSION
DAN9
LA DIFFUSION
L’auteur examine et discute les equations relatives au coefficient de diffusion chimique da,ns le cas general ou les volumes partiels molaires des constituants peuvent differer ou varier suivant lacomposition. 11 deduit des relations exactes et facilement applicables qui donnent le coefficient de diffusion chimique et le coefficient de diffusion intrinseque. Ces relations pourraient les equations de Darken et de BoltzmannMatano chaque fois qu’il y a possibilite de differences et de variations du volume partiel molaire. UBER
DIE
vos
BEsTIMMUNG
DIEFUSIONSKOEEFIZIENTEN DIFFUSION
BEI
CHEMISCHER
Die Gleichungen, die die chemische Diffusion im allgemeinen Fall beschreiben, wo die partiellen molaren Volurnina der diffundierenden Komponenten verschieden und van der Zusamrnensetzung abhiingig sein k&men, werden zusammengestellt und diskutiert. Zur Bestimmung chemischer und innerer Diffusionskonstant,en werden exakte und leicht anwendbare Beziehungen hergeleitet. Wenn die Mogliehkeit besteht, da13 Unterschiede und mangelnde Konstanz der partiellen molaren Volumina von Bedeutung sein konnen, sollten die Ergebnisse an die Stelle der Gleichungen van Darken und der Durchrechung nach Boltzmann-Matano treten. 1. INTRODUCTION
Most analyses
of chemical
due to differences
diffusion
have ignored
possible effects due to differences in the partial molal
2. GENERAL
volumes of the diffusing components and variations of these partial molal volumes with composition. For example,
the standard
Darken(l)
that the partial molal volumes and equal.
this
not
condition
does
then
becomes
hold
stant cross section.
com-
and
the
analysis
complicated.
diffusion
of
with the development
However,
of more accurate techniques
involved
equations
in the
data.
It turns
application
the
x < 0
t=o
Cl = cl(+co)
x > 0
t=o
with respect
\
I.
/I
\
\A)
to the non-
COfor all t. Also,
2
=
-_D.
.%
lax
+
c.v
z
2)
where i = 1 or 2. The Di are the intrinsic diffusivities of Darken(l) and Hartley and Crankc6) and measure the fluxes relative to the crystal which is mechanically flowing with a velocity, u, relative to x = 0. As Bardeen and Herring(‘) point out, equation (2) holds for a vacancy diffusion model as long as vacancy supersaturation effects are negligible. This condition
* This research was supported by the United States Air Force through the Air Force Office of Scientific Research. Received February 15, 1960. t Department of Mining and Metallurgical Engineering, University of Illinois, Urbana, Illinois. 1960
when
is
Cl = C1(-co)
J.
general
is not much greater than in the case where
VOL. 8, DECEMBER
constant
the origin are
constant and equal partial molal volumes are assumed. It is strongly recommended that the general relations be used whenever there is any possibility that effects
ACTA METALLURGICA,
essentially
W,/ax = 0 at 2 = *co. The fluxes across any section fixed with respect to
out that the
of
on actual diffusion
diffused end of the couple at x = -
of the present paper is to present a set of exact and easily applied relations for obtaining chemical diffusivities from experimental
remains
The x = 0 plane is fixed
for
measuring diffusion coefficients, these effects should be of significance in a number of systems. The purpose
labor
couple
The initial condition
by workers in the field since they are
usually small (at least in metallic systems).
Measurements
the couple is sufficiently massive in the x directions.(3-5)
An
excellent discussion of this problem has been given by Crank@). These complicating effects generally have been ignored
EQUATIONS
couples have shown that the cross section of such a
In the general case,
more
of the partial molal
We shall limit our discussion to unidimensional diffusion in a binary sandwich-type couple of con-
assume
of the diffusing
ponents are constant diffusion
equations
and variations
volumes are significant.
should be met in most diffusion 871
couples at diffusion
872
METALLURGICA,
ACTA
times of interest.(s)
The Ci are measured in moles per
+ CJ,
= 0
We first note that
of 1 of the form
(3) Since the Bi are functions
F1 dC, + v, dC, = 0 1 Pi are partial
C, = C,(A) is assumed.
= 1
C,dEp, + C&P2
where the
solution
8, 1960
the Bi are functions
unit volume and obey the standard equations CJ,
VOL.
of the composition
molal volumes.
Prager@) and Crank@) the accumulation
Following
equations
are
coefficient
However,
the 0,
are not functions of composition in the general sense since the relation 1 = 3L(C1) depends upon the particular
boundary
under
investigation.
equation where D is the chemical diffusion
of 1, they are also functions
since I = A(C,).
conditions of the diffusion couple The Matano integration of
(9) yields
which is
x dCi.
(12)
given by
D = D&7, The velocity v=
(5)
+ D,V,C,.
of flow is
Using equations (12), (10) and (3) we solve for D in the convenient form
D)%+ 2 ax
V(D,1
(6) We note that the macroscopic terms
of
However, not
a
single
diffusion is described in
chemical
diffusion
coefficient.
the usual simple forms of Pick’s
pertain.
The
is a small velocity
integral
term
in
laws do
equation
term arising from variations
partial molal volumes tions will generally
with composition.
cause the couple,
(6)
of the
Such variaas a whole, to
swell or shrink. 3. THE
DETERMINATION AND INTRINSIC
expedited
scaling down the x coordinate by the factor (VI
of the penetration
curve
It is stressed that the origin
- V,)/V,.
is located at the x = 0 plane under all circumstances. C,( +m)
OF THE CHEMICAL DIFFUSIVITIES
the integral in equation
two approximate the following
small compared to the first and is readily evaluated by
The plane delined by
The determination of the chemical diffusivity, D, is somewhat complicated by the fact that it appears behind
Equation (13) may be solved graphically by the usual means. The second integral will generally be quite
methods
(4).
we solve for D exactly.
by introduction
Crank(s) reviews
for its determination.
In
The solution is
of the functions
s C,( - m) will not generally Inspection
xdC,=O
coincide
with the x =
0 plane.
of equations
(7) and (12) shows that coinci-
dence is only obtained
in the special case when the
couple as a whole does not expand or shrink. This result may also be demonstrated
by integrating
under the
diffusion curve to find the amount of either component which has crossed the x = 0 plane during the diffusion. When intrinsic
Then
j. z = __8.
a at
-z
a
data are available, be obtained from
the the
relations
!!!3
8
marker movement diffusivities may
ax ’
( 1 eiz,
(9)
and
D = c,v,e,
+ c,v,e,.
(10)
For the present boundary conditions, the Bi may be determined by the usual Boltzmann-Matano method
The slopes and integrals
where the variable
are the same as in equation
il E xt-*
is introduced
and the
in equations
(14) and (15)
(13) ; and, therefore, the
BALLUFFI:
Di are readily determined
DIFFUSION
COEFFICIENTS
if the marker velocity,
vn, is
IN
CHEMICAL
approximation.
by
showing
that
an inert
marker
initially placed at x = 0 will move parabolically
with
and
V, are
quite simple. vanishes,
displacement:
V2 is 2tv, = xm.
The instantaneous equation
(6).
velocity
Making
VP, - 4)
v=
(16)
of any marker is given by
the A substitution
we have
unknown,
it is suggested
forms
of
behavior
of
constant
method
system
of coping
with volume
is to employ where
equal
is satisfied
by
a particular
marker
traveling at constant composition at a constant value of 1, say il,, whose motion is given by xm = ;l,t*. In this case v, = dx,&?t &(D,
+/ym&)
-
increments
of diffusion
ordinate
in the direction
numbers
of atoms.(296J10) In this type of space, the
contain
(2)
when
previous
may
coefficients
is not markedly Darken
be readily
applied
as functions of composition.
In such a case, the labor standard
= constant.
REMARKS
equations
VI and Vz are known
d1]
equation0
in determining greater
diffusion
than when the
are employed
must
be known ;
gration
must be carried
lations are required.
However,
D by this method,
a Boltzmann-Matano out;
VI
and
type
and further
The advantages
equal
V,
inte-
manipu-
of this method,
therefore, appear open to question. 1. L. S. DARKEN, Trans.
D,)%‘(U
(2)
coor-
of the co-
REFERENCES
;l,,,/2tg where
=
4. CONCLUDING
The
(18)
changes
a non-laboratory
simple form of Fick’s laws always pertain. equation
and
that the investigator
VI and V2.
Another
in order to determine This
Fick’s
VI
measure the density of the two terminal alloys employed in the couple and use the approximation of
dinate
an
and the usual
When the detailed
during diffusion
acl
VI
The equations then become
The integral term in equation (4) and (6)
19~= Di,
laws pertain.
i.e.
is obtained when it is assumed that
each constant.
time. For this marker then, the term 2tv, in equations (14) and (15) is simply represented by the marker
873
In many cases, an excellent first order
approximation
known. We conclude
DIFFUSION
as an
Awwr. In& Min. (Metall.) Engrs. 175, 184 (1948). 2. J. CRANK, The Mathematics qf Diffusion p. 219. Oxford University Press (1956). _ __ _ 3. L. C. C. DA SILVA and R. F. MERL, Trans. Awaer. Inst. Min. (Metall.) Engrs. 191, 155 (1951). 4. R. W. BALLUFFI and L. L. SEIQLE, J. Appl. Phys. 25,607 (1954).
5. R. RESNICK and R. W. BALLUFFI, Trans. Awser. Inst. Min. (M&all.) Engrs. 203, 1004 (1955). 6. G. S. HARTLEY and J. CRANK, Trans. Faraday Sot. B42, 12 (1946). 7. J. &D&EN and C. HERRING, Atom. Movements p. 87. American Society for Metals, Cleveland (1951). 8. H. FARA and R. W. BALLUFFI, J. AppZ. Phys.
30, 325 (1959). 9. S. PRAGER, J. Chem. Phys. 21, 1344 (1953). 10. M. COHEN, C. WAQNER and J. E. REYNOLDS, Trans. Amer. Inst. Min. (Metall.) Engrs. 197,1534 (1953); 200,702 (1954).
DETERMINATION
OF DIFFUSION
COEFFICIENTS
IN CHEMICAL
DIFFUSION*
R. W. BALLUFFIT The equations describing chemical diffusion in the generalcase where the partial molal volumes of the diffusing components may differ and may vary with composition are reviewed and discussed. Exact and easily applied relations for obtaining chemical and intrinsic diffusivities ape derived. The results should replace the standard Darken equations and Boltzmann-Matano analysis whenever there is any possibility that differences and variations of the partial molal volumes are of significance. SUR
LA DETERMINATION
DES
COEFFICIENTS CHIMIQUE
DE
DIFFUSION
DAN9
LA DIFFUSION
L’auteur examine et discute les equations relatives au coefficient de diffusion chimique da,ns le cas general ou les volumes partiels molaires des constituants peuvent differer ou varier suivant lacomposition. 11 deduit des relations exactes et facilement applicables qui donnent le coefficient de diffusion chimique et le coefficient de diffusion intrinseque. Ces relations pourraient les equations de Darken et de BoltzmannMatano chaque fois qu’il y a possibilite de differences et de variations du volume partiel molaire. UBER
DIE
vos
BEsTIMMUNG
DIEFUSIONSKOEEFIZIENTEN DIFFUSION
BEI
CHEMISCHER
Die Gleichungen, die die chemische Diffusion im allgemeinen Fall beschreiben, wo die partiellen molaren Volurnina der diffundierenden Komponenten verschieden und van der Zusamrnensetzung abhiingig sein k&men, werden zusammengestellt und diskutiert. Zur Bestimmung chemischer und innerer Diffusionskonstant,en werden exakte und leicht anwendbare Beziehungen hergeleitet. Wenn die Mogliehkeit besteht, da13 Unterschiede und mangelnde Konstanz der partiellen molaren Volumina von Bedeutung sein konnen, sollten die Ergebnisse an die Stelle der Gleichungen van Darken und der Durchrechung nach Boltzmann-Matano treten. 1. INTRODUCTION
Most analyses
of chemical
due to differences
diffusion
have ignored
possible effects due to differences in the partial molal
2. GENERAL
volumes of the diffusing components and variations of these partial molal volumes with composition. For example,
the standard
Darken(l)
that the partial molal volumes and equal.
this
not
condition
does
then
becomes
hold
stant cross section.
com-
and
the
analysis
complicated.
diffusion
of
with the development
However,
of more accurate techniques
involved
equations
in the
data.
It turns
application
the
x < 0
t=o
Cl = cl(+co)
x > 0
t=o
with respect
\
I.
/I
\
\A)
to the non-
COfor all t. Also,
2
=
-_D.
.%
lax
+
c.v
z
2)
where i = 1 or 2. The Di are the intrinsic diffusivities of Darken(l) and Hartley and Crankc6) and measure the fluxes relative to the crystal which is mechanically flowing with a velocity, u, relative to x = 0. As Bardeen and Herring(‘) point out, equation (2) holds for a vacancy diffusion model as long as vacancy supersaturation effects are negligible. This condition
* This research was supported by the United States Air Force through the Air Force Office of Scientific Research. Received February 15, 1960. t Department of Mining and Metallurgical Engineering, University of Illinois, Urbana, Illinois. 1960
when
is
Cl = C1(-co)
J.
general
is not much greater than in the case where
VOL. 8, DECEMBER
constant
the origin are
constant and equal partial molal volumes are assumed. It is strongly recommended that the general relations be used whenever there is any possibility that effects
ACTA METALLURGICA,
essentially
W,/ax = 0 at 2 = *co. The fluxes across any section fixed with respect to
out that the
of
on actual diffusion
diffused end of the couple at x = -
of the present paper is to present a set of exact and easily applied relations for obtaining chemical diffusivities from experimental
remains
The x = 0 plane is fixed
for
measuring diffusion coefficients, these effects should be of significance in a number of systems. The purpose
labor
couple
The initial condition
by workers in the field since they are
usually small (at least in metallic systems).
Measurements
the couple is sufficiently massive in the x directions.(3-5)
An
excellent discussion of this problem has been given by Crank@). These complicating effects generally have been ignored
EQUATIONS
couples have shown that the cross section of such a
In the general case,
more
of the partial molal
We shall limit our discussion to unidimensional diffusion in a binary sandwich-type couple of con-
assume
of the diffusing
ponents are constant diffusion
equations
and variations
volumes are significant.
should be met in most diffusion 871
couples at diffusion
872
METALLURGICA,
ACTA
times of interest.(s)
The Ci are measured in moles per
+ CJ,
= 0
We first note that
of 1 of the form
(3) Since the Bi are functions
F1 dC, + v, dC, = 0 1 Pi are partial
C, = C,(A) is assumed.
= 1
C,dEp, + C&P2
where the
solution
8, 1960
the Bi are functions
unit volume and obey the standard equations CJ,
VOL.
of the composition
molal volumes.
Prager@) and Crank@) the accumulation
Following
equations
are
coefficient
However,
the 0,
are not functions of composition in the general sense since the relation 1 = 3L(C1) depends upon the particular
boundary
under
investigation.
equation where D is the chemical diffusion
of 1, they are also functions
since I = A(C,).
conditions of the diffusion couple The Matano integration of
(9) yields
which is
x dCi.
(12)
given by
D = D&7, The velocity v=
(5)
+ D,V,C,.
of flow is
Using equations (12), (10) and (3) we solve for D in the convenient form
D)%+ 2 ax
V(D,1
(6) We note that the macroscopic terms
of
However, not
a
single
diffusion is described in
chemical
diffusion
coefficient.
the usual simple forms of Pick’s
pertain.
The
is a small velocity
integral
term
in
laws do
equation
term arising from variations
partial molal volumes tions will generally
with composition.
cause the couple,
(6)
of the
Such variaas a whole, to
swell or shrink. 3. THE
DETERMINATION AND INTRINSIC
expedited
scaling down the x coordinate by the factor (VI
of the penetration
curve
It is stressed that the origin
- V,)/V,.
is located at the x = 0 plane under all circumstances. C,( +m)
OF THE CHEMICAL DIFFUSIVITIES
the integral in equation
two approximate the following
small compared to the first and is readily evaluated by
The plane delined by
The determination of the chemical diffusivity, D, is somewhat complicated by the fact that it appears behind
Equation (13) may be solved graphically by the usual means. The second integral will generally be quite
methods
(4).
we solve for D exactly.
by introduction
Crank(s) reviews
for its determination.
In
The solution is
of the functions
s C,( - m) will not generally Inspection
xdC,=O
coincide
with the x =
0 plane.
of equations
(7) and (12) shows that coinci-
dence is only obtained
in the special case when the
couple as a whole does not expand or shrink. This result may also be demonstrated
by integrating
under the
diffusion curve to find the amount of either component which has crossed the x = 0 plane during the diffusion. When intrinsic
Then
j. z = __8.
a at
-z
a
data are available, be obtained from
the the
relations
!!!3
8
marker movement diffusivities may
ax ’
( 1 eiz,
(9)
and
D = c,v,e,
+ c,v,e,.
(10)
For the present boundary conditions, the Bi may be determined by the usual Boltzmann-Matano method
The slopes and integrals
where the variable
are the same as in equation
il E xt-*
is introduced
and the
in equations
(14) and (15)
(13) ; and, therefore, the
BALLUFFI:
Di are readily determined
DIFFUSION
COEFFICIENTS
if the marker velocity,
vn, is
IN
CHEMICAL
approximation.
by
showing
that
an inert
marker
initially placed at x = 0 will move parabolically
with
and
V, are
quite simple. vanishes,
displacement:
V2 is 2tv, = xm.
The instantaneous equation
(6).
velocity
Making
VP, - 4)
v=
(16)
of any marker is given by
the A substitution
we have
unknown,
it is suggested
forms
of
behavior
of
constant
method
system
of coping
with volume
is to employ where
equal
is satisfied
by
a particular
marker
traveling at constant composition at a constant value of 1, say il,, whose motion is given by xm = ;l,t*. In this case v, = dx,&?t &(D,
+/ym&)
-
increments
of diffusion
ordinate
in the direction
numbers
of atoms.(296J10) In this type of space, the
contain
(2)
when
previous
may
coefficients
is not markedly Darken
be readily
applied
as functions of composition.
In such a case, the labor standard
= constant.
REMARKS
equations
VI and Vz are known
d1]
equation0
in determining greater
diffusion
than when the
are employed
must
be known ;
gration
must be carried
lations are required.
However,
D by this method,
a Boltzmann-Matano out;
VI
and
type
and further
The advantages
equal
V,
inte-
manipu-
of this method,
therefore, appear open to question. 1. L. S. DARKEN, Trans.
D,)%‘(U
(2)
coor-
of the co-
REFERENCES
;l,,,/2tg where
=
4. CONCLUDING
The
(18)
changes
a non-laboratory
simple form of Fick’s laws always pertain. equation
and
that the investigator
VI and V2.
Another
in order to determine This
Fick’s
VI
measure the density of the two terminal alloys employed in the couple and use the approximation of
dinate
an
and the usual
When the detailed
during diffusion
acl
VI
The equations then become
The integral term in equation (4) and (6)
19~= Di,
laws pertain.
i.e.
is obtained when it is assumed that
each constant.
time. For this marker then, the term 2tv, in equations (14) and (15) is simply represented by the marker
873
In many cases, an excellent first order
approximation
known. We conclude
DIFFUSION
as an
Awwr. In& Min. (Metall.) Engrs. 175, 184 (1948). 2. J. CRANK, The Mathematics qf Diffusion p. 219. Oxford University Press (1956). _ __ _ 3. L. C. C. DA SILVA and R. F. MERL, Trans. Awaer. Inst. Min. (Metall.) Engrs. 191, 155 (1951). 4. R. W. BALLUFFI and L. L. SEIQLE, J. Appl. Phys. 25,607 (1954).
5. R. RESNICK and R. W. BALLUFFI, Trans. Awser. Inst. Min. (M&all.) Engrs. 203, 1004 (1955). 6. G. S. HARTLEY and J. CRANK, Trans. Faraday Sot. B42, 12 (1946). 7. J. &D&EN and C. HERRING, Atom. Movements p. 87. American Society for Metals, Cleveland (1951). 8. H. FARA and R. W. BALLUFFI, J. AppZ. Phys.
30, 325 (1959). 9. S. PRAGER, J. Chem. Phys. 21, 1344 (1953). 10. M. COHEN, C. WAQNER and J. E. REYNOLDS, Trans. Amer. Inst. Min. (Metall.) Engrs. 197,1534 (1953); 200,702 (1954).