^{1}

^{*}

^{2}

^{3}

In accordance with the definition of diffusivity, the origin of coordinate system of the original diffusion equation is set at a point in the solvent material. Kirkendall revealed that Cu atoms, Zn atoms and vacancies move simultaneously in the interdiffusion region. This indicates that the original diffusion equation is a moving coordinate system for the experimentation system outside the diffusion region. The diffusion region space which means vacancies and interstices among atoms plays an important role in the diffusion phenomena. The theoretical equation of the Kirkendall effect is reasonably obtained as a shift between coordinate systems of the diffusion equation. The situation is similar to the well-known Doppler effect in the wave equation. Boltzmann transformed the original diffusion equation of a binary system into the nonlinear ordinary differential equation in accordance with the parabolic law. In the previous works, the solutions of the diffusion equation transformed by Boltzmann were analytically obtained and we found that the well-known Darken equation is mathematically wrong. In the present study, we found that the so-called intrinsic diffusivity corresponds in appearance to the physical solution obtained previously. However, the intrinsic diffusivity itself conceived in the diffusion research history is essentially nonexistent.

The heat conduction equation proposed by Fourier in 1822 has been applied to investigating the temperature distribution in materials [

When the Fick’s laws were proposed, the Gauss’s divergence theorem had been already reported in 1840 [

It seems that the new fundamental findings different from the existing diffusion theories may exert a great influence on the actual diffusion problems, just because of the fundamental findings themselves. For example, one of them reveals that the well-known intrinsic diffusion concept is unnecessary for understanding the interdiffusion phenomena [

The solutions of a typical interdiffusion problem were already obtained as analytical expressions in accordance with results of the well-known Boltzmann Matano method [

In accordance with the definition of diffusivity, the origin of coordinate system of the original diffusion equation is set at a point in the solvent material. In 1947, Kirkendall revealed that Cu atoms, Zn atoms and vacancies move simultaneously in the interdiffusion region [

The water in a pond is at rest and a rectangle raft of mass

start walking with an arbitrary velocity for

In the following, we discuss the motion of person A in the isolated system composed of the raft and persons using each of the coordinate systems,

In an arbitrary time between

in the isolated system of the raft and persons, where

The velocities of persons A and B,

The relative velocity

and it does not depend on the coordinate systems

The migration length of raft between

When the persons on the raft stop walking at the same time

the raft returns to the initial position and it has been at rest ever since. The mass center of the isolated system is shifted from the initial state

In the present model, it is thus indispensable for understanding the motion of person A against the point R to investigate the behavior of the raft and water in the pond. In other words, it is apparent that we cannot understand the motion of person A against the point R only from the relative motion between the persons A and B on the raft.

As is known from the unified theory of diffusion problems, the basic concept of diffusion phenomena is in the interdiffusion problems [

Metal plates A and B with a uniform cross section

As shown in

Since the diffusivity depends on an interaction between a diffusion particle and the diffusion field near the diffusion particle itself, the basic diffusion equation with the diffusivity is expressed by the time and space coordinate

We denote the concentrations and diffusivities of I and II by

differential equations of diffusion.

The basic diffusion equations of I and II are expressed as

Since the shape variation of diffusion couple is negligible in the usual experimentation, the total particle numbers of I and II on an arbitrary cross section in the diffusion region are considered to be constant with a good approximation. In the typical interdiffusion problems, the relation of

is thus generally accepted regardless of the coordinate systems, where the normalized concentrations are used.

Substituting (4) into (3) yields

and it is rewritten as

In accordance with the theory of partial differential equation, (5) means that the relation of

must be valid using an arbitrary function

Their diffusivities are thus commonly expressed as

only in the partial differential equation (3). However, note that it is not generally valid in the actual diffusion phenomena when we substitute the initial and/or boundary values,

where the suffix

The Gauss’s divergence theory shows that the diffusion flux of basic diffusion equation (7) is obtained as

by integrating it with respect to

where

Substituting (4) into the Fickian first law

is valid only in the partial differential equation (3). Further, it is clarified that the relation of

is valid in the self-diffusion theory in [

which is valid only in the partial differential equation (3).

When the origin

in the fixed coordinate system outside the diffusion system. Here, the relations of differential operators yielding

are used for (7), where the relations of

are valid between the coordinate systems

It is generally considered that the diffusivity

In 1894, Boltzmann transformed (7) into the nonlinear ordinary differential equation of

using the relation

where the relation of

is used [

The general solutions of (13) were obtained as analytical expressions yielding

where (14) and (15) were simultaneously calculated using the integral constants

When we substitute the boundary values,

between

Substituting the initial and/or boundary values into the general solutions of the diffusion equation (13) i.e., (16) and (17), the physical solutions of the diffusion equation (3) were reasonably obtained as (18) and (19). As can be seen from figure 3 and figure 4 in [

The diffusivities

is valid between

is generally valid between the diffusion fluxes obtained by using (18) and (19).

Here, note that the difference between (6) and (20) or (9) and (21) has been ignored in the diffusion history. We believe that the Kirkendall effect (K effect) is caused by the relation of (21) [

In a word, the usual diffusivities

The so-called Darken equation relevant to the interdiffusion coefficient and the intrinsic diffusion coefficients has been used for analyzing interdiffusion problems for a long time [

The concept of diffusion flux is useful for understanding of diffusion phenomena. The diffusion flux of the diffusion region space plays an important role to understand interdiffusion phenomena like the motion of persons in the raft model depends on the motion of raft.

The Gauss’s divergence theory shows that the diffusion flux is obtained as

by integrating the diffusion equation (12) with respect to

Using the velocity

and the diffusion flux is

Here, note that the diffusion fluxes of (8), (22) and (24) are valid only in the partial differential equations (7), (12) and (23), respectively. When (8), (22) and (24) are applied to the elements I and II, the relation among diffusion fluxes of

is physically valid as a relation between the inside flux of the diffusion region and the outside one, where

Equation (25) yields the relation of

since

is thus valid, since the diffusion region space moves in the opposite direction against the diffusion particles.

Equations (26) and (27) yield a physically reasonable relation of

to be valid among the velocities of the coordinate systems.

In general, the diffusion experiments are performed at a high temperature between

The interface

It is considered that the diffusion region is an isolated system and the mass center is immobile between

The coordinate origin set at a point P on the M interface in the initial state is also immovable, since the M interface is immovable. In other words, the relation of

because of

Here, using the initial and/or boundary values for (24), the relation yielding

is reasonably obtained (See Appendix).

An inert marker set at the K interface moves in accordance with the flow of diffusion region space between

Based on the above mentioned, the migration length of an inert marker is expressed as

at

For an arbitrary time

is well known, where

Equation (33) shows that the K effect depends not only on the initial diffusivity values but also on the initial concentration values. Here, (33) is derived by using

so as to agree with the empirical equation (32).

Hereinbefore, the Kirkendall effect was reasonably explained by the present interdiffusion theory, regardless of the intrinsic diffusion concept. It was thus revealed that the diffusion region space plays an important role in the interdiffusion problems. At the same time, the discussions of setting coordinate system of diffusion equation are essentially dispensable for understanding the diffusion phenomena.

The diffusion study is an important subject relevant to basic problems in the fabrication process of materials such as alloys, semiconductors, functional materials, and so on. Their diffusion problems in detail have been thus widely investigated in accordance with the industrial requirement for a long time. However, some unsolved problems relevant to the fundamental nonlinear diffusion equation, which are extremely dominant in mathematical physics, have been still leaved in the diffusion history. In the cause of them, the physical interpretation of the K effect has been misunderstood for a long time.

As is well known, the physical solutions of a partial differential equation are determined from substituting the given initial and boundary values into the general solutions in mathematics. The difference between the physical solutions and the general solutions in mathematics has been confused in the diffusion history. For example, the expression of the diffusion flux

Based on the present mathematical theory, (35) is apparently wrong, since the left-hand side is a general solution whereas the right-hand is expressed by using physical solutions. Further, it is also revealed that the Darken equation is mathematically wrong in the derivation process [

It had been considered for a long time that the analytical solutions of (3) are impossible. However, we could reasonably obtain the general solutions of (13) as (16) and (17). In other words, the physical solutions of (3) were analytically obtained as (18) and (19). In the present work, the fundamental concept of interdiffusion phenomena was reasonably clarified in accordance with the mathematical theory. Further, the analytical method discussed in the present work is reasonably applicable to an interdiffusion problem of many elements system [

Here, the present results yield the following conclusions.

1) In general, the discussion about setting the coordinate systems of diffusion equation in the diffusion problems is essentially necessary for understanding of the diffusion phenomena.

2) An element in the interdiffusion region has only one diffusivity value. The so-called interdiffusion coefficient means the unsolved one in the partial differential equation. On the other hand, the intrinsic diffusion coefficient corresponds to the solved one using the given initial and boundary values for the general solutions. Therefore, such an especial intrinsic diffusion coefficient conceived in the diffusion history is essentially nonexistent in accordance with the mathematical theory.

In view of the influence of misunderstanding problems pointed out here on the younger, we hope that the conclusions are universally known in the concerned research field as soon as possible, just because of the fundamental matters themselves.

Okino, T., Cho, H. and Yamada, K.-M. (2017) Fundamental Concept of Interdiffusion Problems. Journal of Modern Physics, 8, 903-918. https://doi.org/10.4236/jmp.2017.86056

Even if the diffusion couple satisfies

where

Using (A-1) and the concentration difference of boundary values

Equation (A-2) yields

The diffusion flux of (A-3) caused by the coordinate transformation corresponds to the flux of diffusion region space given by

since the flux of diffusion region space moves in the opposite direction to the diffusion flux of (A-3). Substituting (A-4) into (29) yields

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