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 txD 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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Paris, 1822.

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