c0 ls0 ws1f">interdiffusion problems between metal plates as an ex-
ample of the diffusion problems, its validity was con-
firmed in the present study. The new findings obtained
here may make a fundamental change to the existin g dif-
fusion theory.
2. Summary of Interdiffusion Problems
Since the physical essence is kept even if we investigate
the diffusion problems of time and space , the FSL
of is investigated in this section. It will be clari-
fied later that we can systematically understand the one-
way diffusion, the impurity diffusion and the self-diffu-
sion as a special case of the interdiffusion. Therefore, we
briefly summarize the interdiffusion problems for a dif-
fusion couple between metal plates A and B, where the
plate A is the alloy composed of I atom and II atom and
the plate B is the pure metal of II atom. The coordinate is
then defined as at the interface between the plate
A and the plate B
0x
0
AB
x
t. Further, the inter-
diffusion region at a time is defined as
x
xx
0
n
and at .
x0t
In the interdiffusion problems between metal plates, it
is generally accepted that the deformation of specimen
between diffusion before-and-after is almost negligible.
In other words, the number of total atoms on an ar bi-
trary crystal cross section perpendicular to
x
axis is
considered as a constant value during the diffusion proc-
ess. The relation III
nn n
n
nI
C
II
C
III
1CC
is thus valid, where I or
II is the number of I atoms or II atoms on the same
cross section. Using the normalized concentration
for I atom and for II atom, the relation o f
(1)
is thus widely accepted in this field.
In the following, the abbreviated notations of
2
22
,

 
,,andtxy z for
I
CII
C
.
are used. The FSL for or is
III
txx
CDC 

IIII II
txx
CDC 
D D
, , (2)
where I and II are diffusivities of I atom and II
atom. Equations (1) and (2) yield
III II
0
xx
DD C
 
III
DD D
. (3)
Equation (3) sh ows
D
I
, (4)
where is the so-called interdiffusion coefficient. In
this case, the diffusion flux FFL of
J
for I atom or
II
J
for II atom is
III I
IIII IIII
,
.
xx
xx
JDCDC
J
DC DC
 
 

III III
0
x
JJ DCC
(5)
Equations (1) and (5) yield
 
III
0JJ
II IIIIIII
for 1DCD CDCC
. (6)
Here, (6) shows that the number of I atoms which dif-
fuse from the plate A into the plate B is equal to that of II
atoms which diffuse from the plate B into the plate A.
The K-effect shows that must be v alid in
the interdiffusion problems [9]. It is obvious that (6) is
inconsistent with the K-effect. In order to solve the in-
consistency, Darken proposed the interdiffusion coeffi-
cient of

III
DDandDD
III
0JJ
(7)
instead of (4), assuming inconsistent with (4)
so that (6) is not valid [10]. Here, III
were then
designated as “intrinsic diffusion coefficient.” Equation
(7) has been widely used for the analysis of interdiffusion
problems. However, the author thinks that the FFL
should be revised so RR
is valid for the new
diffusion flux R
J
under the condition of (4). In other
words, we can understand the K-effect without the in-
trinsic diffusion conception. In the next section, the NDF
to replace the FFL will be defined, and also it will be
Copyright © 2012 SciRes. JMP
T. OKINO
1390
clarified that the NDF is mathematically and physically
reasonable.
3. Divergence Theory and New
Diffusion Flux
For an arbitrary differentiable vector

,,,
J
txyz
V in a
space closed in a surface , the divergence theo-
rem between the volume integral and surface integral is
defined as
S
dd
SV
J
VJS

 , (8)
where Dirac’s vector representation is used and
a
normal unit vector perpendicular to a surface element
and
dS

,,
x
yz
 . Here, applying the relation
defined as

R
,,,,, 0
,
J
txyz JtxyzJ t (9)
to (8), the relation of
R
dd
VS 0
d
S
J
VJ


SJS

(10)
is valid, where 0
J
tS
is defined as a vector on . In
the present physical system, the first term of the right-
hand side of (10) means a physical quantity Q which
outflows through S per unit time. On the other hand, the
second term is relevant to an inflow rate of Q caused by a
material source on S. The decrease rate of Q is thus ex-
pressed as
0d
t
SS
d
J
SJ


SQ. (11)
Substituting (11) into the right-hand side of (10), the
relation of
Rdt
V
J
VQ
(12)
is generally valid.
In the conventional diffusion problems, the flux
J
correlates with the FFL of
,,,
J
DCtxyz
D
,,Ctx
 , (13)
using the diffusivity and the concentration
of the material quantity Q. Equation (13)
has been used only for the diffusion problems where the
concentration gradient is not zero in the initial state under
the condition of no material sink or source. The substitu-
tion of (13) into (9) yields the NDF of
,yz


R
,,,,, 0
,
J
txyzD Ctxy zJ t

,,, dtxyzV
, (14)
taking the source effect into account. Further, substitut-
ing (14) and the relation of
tt
V
QC
into (12), the FSL is thus obtained as

R,,, ,,,
t
J
DCtxyz Ctxyz, (15)
where 00Jt .
J
t
In mathematics, 0 is relevant to the integral
constant of
d
V
J
V
because of 00Jt
S
. In physics, it is relevant to the
material inflow caused by a material source on . The
present theory reveals that the FSL is applicable to the
diffusion problems as it is, even if a material source in-
dependent of
,,
x
yz is contained in the diffusion sys-
tem.
4. Application of NDF to Interdiffusion
Problems
The so-called Kirkendall interface is defined at K
x
x
where the number of I atoms which diffuse from the
plate A into the plate B is equal to that of II atoms which
diffuse from the plate B into the plate A. On the other
hand, the original interface between the plate A
0x
and the plate B
0x0xx at M is the so-called
Matano interface. The K-effect means
effKM 0xxx
. In the following, we define the
NDF under the condition of (4) and (6) and investigate
the diffusion problems in accordance with the NDF.
As is well known, the diffusion junction depth jun
x
at a diffusion time for a material of diffusivity D is
expressed as t
x
jun Dt . (16)
x
We use notations of jun Dt

and
ABA B
CCC

 A
CCIA
DDA
x
x for , at
and B
CC
, at
II B
DDB
x
x with the superscript/
subscript
of
IA or
. For the concen-
tration gradient of
II B
AB jun
Cx
, the diffusion flux
0t

is defined as
J

AB
0eqABeq
jun
AB
for ,
C
tDJ DCCtJ
x
xx
x


 


J
(17)
0J
eq
where
is a constant value relevant to the
Brownian motion in the thermal equilibrium state, and
III
eq 0JJ
eq must be then physically valid.
Under the condition of (1), (4), (6) and (17), (9) or (14)
yields


IIIIII
RR 00
II
ABAB
,,
,
Jtx JtxJtJt
DDCCt


 (18)
II IIII
AB AB
CC CC and where
Copyright © 2012 SciRes. JMP
T. OKINO 1391

Jtx

III
,,0J tx
t
because of (1) and (6). In consid-
eration of the atomic migration caused by the interdiffu-
sion, the integral calculation of (18) with respect to
correlates with the K-effect as follows:

 

III
eff0 0
0
II
ABA
1d
2
.
t
xJtJt
DDC t
 
 
 B
t
C
(19)
Here, note that (19) is consistent with the parabolic
law. On the other hand, the experimental analysis of in-
terdiffusion problems also shows the parabolic law
yielding
eff
x
mt
mI
, (20)
where the slope is experimentally determined [9].
In the present diffusion system, C is defined as
at B
I
B0C
x
xD
. In the interdiffusion problems be-
tween metal plates, it is widely accepted that B in the
present diffusion system can be approximately replaced
by the impurity diffusivity near B
I
imp
D
x
x
A
D
because
of (4). Therefore, the interdiffusion coefficient at
A
x
x is obtained as
II
Aimp A
DDmC
10 Zn 0.3C
, (21)
by using (19) and (20). For example, substituting
and A in Ref. [9] and
imp near
80.5
m s

14 21
10m s
4.9m
Zn 1.D26 B
x
x14 2
10ms
in Cu plate into
(21), the interdiffusion coefficient 1
A7.6D
 
1058
D
D

III
,,1C tx
D
D
is obtained at the absolute temperature T be-
cause of in the present case. Therefore, var-
ious A values are obtained through the interdiffusion
experimentation by using diffusion couples between the
plate A (various compositio n rates of I and II atoms) and
the plate B (the pure metal of II atom) for various com-
binations of I and II ato ms. After a large number of A
data were thus accumulated, using those A data at the
same for the initial and/or boundary values of the
general solutions of Refs. [12,13], the solutions of inter-
diffusion problems between alloy plates are possible. On
the contrary, using those A data for (19), we can also
predict the behavior of K-effect. As can be seen from the
above discussion, the intrinsic diffusion conception in-
consistent with the FSL is not unnecessary for under-
standing the interdiffusion problems.
Zn
Aimp
D
D
T
II
DD
In the present diffusion system, using the NDF of (14)
as an additional condition equation for the FSL, the
problems of interdiffusion, one-way diffusion, impurity
diffusio n an d self -d if fu sion are systematicall y und er stood
as follows:
1) Interdiffusion: For and
,

Ctx
I
 

III
RA
,,
xIII
ABeq
,DCtxDCCtJ 
 
IIII IIIIIIII
RBABeq
,,
x
J
txDCtxD CCtJ 
2) One-way diffusion:
J
tx
.

 

III
AABeq
II II
Req
,
,.
III
R
,,
x
J
txDC txD CCt J
JtxJ
 

The one-way diffusion corresponds to II I
DD a
DD

in the above 1). In this case, the plate B is
3) Impu
nd
BA
considered as a solvent material.
rity diffusion:
II II
,

II II
eqReq
,,
R
,
x
J
txDC txJJtxJ
 .
The impurity diffusion also corresponds to II I
DD
and DD

in the above 1), and I1C and B0C
I
BA A
ered as a
nt materi
in (17). In this case, the plate B is also consid
solveal.
4) Self-diffusion: For,
 
III
,,1CtxCtx and
III
DD
IIII II
A
CC
BA B
CC,
I
,,

 
III
Req
IIII IIII
Req
,
,,.
x
x
J
txDC txJ
J
txDC txJ
 
 
The self-diffusion is considered as a special interdiffu-
sion where the concentration gradient is zero in the initial

state.
In consideration of the NDF in time and space
,tx,
(15) becomes

,,d
Rt
J
txCtx x
. ) (22
Further, the self-diffusion behavio
follows. For convenience, a pure material is divi
tw
r is understood as
ded into
o regions 0and 0xx
at 0t in the present
diffusion system. The diffusion region is between
AB
x
xx
ahe bory condition of t a time t. Tunda
III0
,,Ctx Ctx C
is then used for A
x
x
or
x
B
x in the present diffusion system, where 00.5C
is valid because of

III
,,1txCtx. Fure
al between AB
Cor a p
materi
x
xx
, (22) yields

II II
,o ,
xx
DCtxD CtxJ
I IIII
eq eq
rJ
 (23)
,0
tCtxbecause of
. In this case, even if
III
RR
0JJ
is valid, the random movement of a
le occurs an3) in the
present diff
n atom
or a molecud it is governed by (2
usion system. Equation (23) thus shows the
correlation between D and eq
J
, where D is relevant to a
jump frequency of an atom or a molecule and eq
J
is
relevant to its thermal motio
The integral calculation of (23) gives
n.
I
andCx xCC

0
for ,
x
xxx
II
0
AB
xC
 

(24)
where II IIII
eq eq
J
DJD

for III
DD and I
eq
J
. Equation (24) shows that

I
Cx and
II
Cx
II
eq 0J
Copyright © 2012 SciRes. JMP
T. OKINO
1392
depend on
x
and that

x Cis valid

AB
0xxxx . If we pay attention to only
one side of I
C or II
C, thtion occurs via
the ran the other hand, if we pay at-
tention to boI
C and II
C,
 
III0
,,Ctx Ctx C is actually valid, because we
cannot know in a pure mrial whher a
AB
III
x be-
atomic migra
etn atom between
C
e
movement. On
th sides of
ate
tween
dom
x
xx
concentratio
in th
e
is one of
se
e early stage, we
I
C or II
C in the initial state.
That (24) is independent of t means the time-averaged
n profile caud bye Brownian motion.
Therefore, the diffusion occus as a result of the random
movement even in a pure material. Equation (24) thus
gives the evid ence of the Brownian motion in a pure ma-
terial.
From the historical point of view, if the FFL had been
revised
th
r
might have understood the
bhavior of Brownian motion before the Einstein theory.
As can be seen from the above 1) - 4), the diffusivity
depends on the concentration when

0
J
t depends on
t. The concentration dependence of diffusivity is thus
caused by the material source on the dion boundary
terface at A
iffus
in
x
x or B
x
x. In consideration of the
above 1) - 4), the FFL is obviously incomplete without

0
J
t for FSversal.
Hereinbefore, it was clarified that we can revise the
o it is applicable to various diffu
L and it is not u
sions
with the
conserva
ni
F
wFL s
lu
sion problem
divergence theorem sh
tion law. On the other
s. It
as also shown that the NDF plays an extremely impor-
tant role to understan d diffusion problems. Hereafter, the
diffusion problems should be analyzed by using the NDF
for a diffusion system as an additional condition equation
of (15).
5. Conc
The FSL consistent
be exactly valid as aould
hand,
the FFL should be replaced by the NDF which is not only
exactly valid in mathematics but also extremely useful
for physics. The obtained novel results in the present
study are as follows:
1) Even if a material source independent of the space

,,
x
yz is contained in the diffusion system, the FSL is
applicable to analyzing the diffusion problems as it is.
terial source

0
The ma
J
t plays an extremely impor-
tant role in the diffusion problems.
2) A law must be universal. The NDF is systematically
applicable to the problems of interdiffusion, one-way
n coefficients can be
ob
ion study. Equation (21) is dominant in the techno-
lo
[1] J. B. J. Fourier la Chaleur,” Chez
Firmin Didot,
, 1855, pp. 31-39.
hte, Vol. 66,
rte Bewegung von in Ruhenden Flus-
diffusion, impurity diffusion and self-diffusion. Further,
the NDF of (14) or (22) is applicable to analyzing diffu-
sion problems, for instance as seen from the derivations
of (21) and (24). However, the FFL has not ever been
used for analyzing the diffusion problems because of its
incompleteness. The NDF is thus universal to the diffu-
sion problems, but the FFL is not.
3) The NDF reveals that the K-effect is reasonably
obtained as (19). The interdiffusio
tained by applying (21) to the experimental results. On
the contrary, (19) can predict the behavior of K-effect
using various combinations of the obtained diffusivities.
As a result, the intrinsic diffusion conception inconsistent
with the FSL is thus not only unnecessary but also un-
real.
The NDF derived here is a fundamental equation in the
diffus
gical material science, since the atomic diffusivity val-
ues in alloy are obtained by using it for the interdiffusion
experimentation. Although the present study was dis-
cussed in relation to diffusion problems between metal
plates, the results obtained here are also applicable to
various material problems described by the FSL. Hereaf-
ter, the new findings obtained here may make a funda-
mental change to the existing diffusi on theory.
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