The correlation between the Schr ödinger equation and the diffusion equation revealed that the relation of material wave is not a hypothesis but an actual one valid in a material regardless of the photon energy. Using the relations of material wave and uncertain principle, the quantum effect on elementary process of diffusion is discussed. As a result, the diffusivity is obtained as a universal expression applicable to any problem of diffusion phenomena. The Gauss theorem in theory and the Kirkendall effect in experimentation reveal the necessity of the coordinate transformation for a diffusion equation. The mathematical method for solving an interdiffusion problem of many elements system is established. The phase shift of the obtained analytical solution indicates the correlation between the solutions of each diffusion equation expressed by a fixed coordinate system and by a moving coordinate system. Based on the coordinate transformation theory, some unsolved problems of diffusion theory are reasonably solved and also some new important findings are discussed in relation to matters in the existing diffusion theory.
The diffusion equation is one of the most fundamental and important equations in physics. In history, Fick [
The behavior of a micro particle should be essentially investigated by using the Schrödinger [
In 1923, de Broglie [
Using the relations of material wave and the uncertain principle, we could reasonably understand the elementary process of diffusion from a new viewpoint. We incorporated a kinetic potential into the Boltzmann factor [
For the well-known diffusion equation with a diffusivity D given by
∂ C ∂ t = ∂ ∂ x { D ∂ C ∂ x } (1)
for a concentration C ( t , x ) in the time and space ( t , x ) , the Gauss theorem indicates that the diffusion flux of
J ( t , x ) = − D ∂ C ∂ x + J ( t ) + J e q (2)
is mathematically valid because of ∂ / ∂ x { J ( t ) + J e q } = 0 [
The definition of diffusivity relevant to a relative motion between micro particles indicates that the coordinate origin of diffusion equation should be usually set at a point in the diffusion region space. The existence of J ( t ) indicates that the diffusion equation should be generally expressed by a moving coordinate system. On the other hand, the concentration profiles obtained in a laboratory are always observed by a fixed coordinate system in the diffusion region outside. The experimental result known as Kirkendall [
Solving the interdiffusion problems of many elements system is extremely important for the development of new useful materials. In general, the diffusivity depends on the concentration in the interdiffusion problems and the diffusion Equation (1) becomes then generally nonlinear differential equation. It had been thus considered that the mathematical solutions of Equation (1) are then impossible. However, recently the general solutions of Equation (1) were obtained as analytical expressions [
The coordinate transformation theory of a diffusion equation established that the diffusion equation of a moving coordinate system in the diffusion region inside is expressed as
∂ C ∂ t = D ∂ 2 C ∂ x 2 , (3)
even if the diffusivity depends on the concentration. Further, it was concretely revealed that the solutions of Equation (3) agree with those of Equation (1) if we take account of the phase shift between their analytical solutions, and vice versa.
Some new findings resulting from the coordinate transformation theory of diffusion equation and also resulting from the general solutions obtained as analytical expressions are concretely and systematically discussed. Thus, the new findings obtained here will be not only widely useful but also indispensable for analyzing the actual diffusion problems in future, just because of extremely fundamental ones.
Hereafter, an abbreviate differential notation for an arbitrary independent variable x and the well-known bracket notation of Dirac [
∂ ξ = ∂ ∂ ξ , 〈 ∇ ˜ | = ( ∂ x , ∂ y , ∂ z ) , 〈 r | = ( x , y , z )
If an operator Q is Hermite one, 〈 Q | = { | Q 〉 } † is valid in the Hermite conjugate † . Here, the notation 〈 ∇ ˜ | is thus defined as 〈 ∇ ˜ | = − { | ∇ 〉 } † because of ∂ ξ = − { ∂ ξ } † .
The function C ( t j , | r j 〉 ) is defined as a normalized concentration where a diffusion particle in the initial time and space ( t 0 , | r 0 〉 ) exists in the time and space ( t j , | r j 〉 ) after j times jumps. Since it is considered that the jump-probability from ( t j − 1 , | r j − 1 〉 ) to ( t j , | r j − 2 〉 ) is equivalent to that from ( t j − 1 , | r j − 1 〉 ) to ( t j , | r j 〉 ) in the isotropic space, the relation of
C ( t + Δ t , | r 〉 ) = { C ( t , | r 〉 − | Δ r 〉 ) + C ( t , | r 〉 + | Δ r 〉 ) } / 2 (4)
is then valid in the Markov process [
∂ t C = ( Δ r ) 2 2 Δ t 〈 ∇ ˜ | ∇ C 〉 , (5)
where Δ r = | | r j 〉 − | r j − 1 〉 | = | | Δ r 〉 | [
From the averaged Δ t and Δ r for a micro particle with mass m in a material, it is considered that ( Δ r ) 2 / 2 Δ t becomes a finite value of
D = ( Δ r ) 2 2 Δ t = Δ r p 2 m , (6)
where p is a momentum of the micro particle, and D called the diffusivity may depend on the configuration of micro particles near the time and space ( t j , | r j 〉 ) . The basic equation of diffusion is then expressed as
∂ t C = D 〈 ∇ ˜ | ∇ C 〉 , (7)
where the coordinate origin of Equation (7) expressing the relative motion between diffusion particles should be generally set at a point of the diffusion region space.
The behavior of a micro particle should be generally controlled by the quantum theory. The correlation between the Schrödinger equation and the diffusion equation was thus investigated in the previous work as shown in
D 0 = ℏ / 2 m (8)
for a micro particle with mass m is then obtained using the Planck constant ℏ ( = h / 2 π ) of the characteristic constant of a micro particle.
The collective motion of micro particles in a material results essentially from the statistical behavior of collision between a micro particle and the nearest particles surrounding the micro particle itself, where the statistical behavior is directly relevant to a jumping problem of a micro particle in a material. Since the diffusivity value of Equation (6) obtained from a jumping problem of a micro particle is equivalent to that of Equation (8) obtained from corresponding the diffusion equation to the Schrödinger equation in a collision problem, the relation of
Δ r p = ℏ (9)
is valid. When a micro particle exists as a wave packet of wave length l in the smallest local space Δ r , the relation of
λ = 2 π Δ r (10)
must be valid, while it exists as a particle if 0 < Δ r < λ / 2 π . Here, the relation of
material wave yielding
p = h / λ . (11)
was reasonably obtained by using Equations (9) and (10) [
The relation of uncertain principle proposed by Heisenberg [
Δ r Δ p ≥ ℏ / 2 (12)
is well known. The uncertain principle shows that a micro particle in a local space Δ r moves randomly with a momentum Δ p ( ≥ ℏ / 2 Δ r ) near the center of local space if 0 < Δ r < λ / 2 π , because of the relation of material wave. It is thus considered that a Brown particle moves with a finite momentum Δ p in a local space Δ r and instantly jumps at a time from the local space to the nearest local space. In that situation, the above momentum p of Equation (6) seems to be statistically equal to the momentum Δ p . Thus, Equations (6) and (12) yield
D ≥ D E / 2 n ( = ℏ / 4 m ) for D E = 5 × 10 2 N A ℏ = 3.18 × 10 − 8 [ m 2 ⋅ s − 1 ] , (13)
where N A and n ( = 10 3 m N A ) are the Avogadro constant and a molecular weight. Here, the value of D E is an elementary quantity of diffusivity which is valid regardless of a kind of material and the thermodynamic state. Equation (13) shows that a micro particle in a material has an essential possibility for jumping from a local space to the nearest-neighbor one in accordance with the thermodynamic state.
In the existing theory of diffusion, it has been considered that the random movement of micro particles in a material is caused by a thermal fluctuation. In other words, it has been considered that the diffusion occurs when a vacancy is formed at the nearest-neighbor of a diffusion particle in a material through the thermal fluctuation. Here, when the radius Δ r of local space satisfies Δ r = λ / 2 π for a micro particle through the uncertain principle of Equation (12), the relation of material wave shows that the micro particle passes smoothly through an interstice between the nearest micro particles as not a particle image but as a wave image, even if the nearest vacancy does not exist [
For the actual diffusivity, the dependences of a temperature T and the coordinate system ( x , y , z ) should be incorporated into D 0 = ℏ / 2 m . The existence probability of a micro particle with a thermodynamic activation energy Q in a material is well known as the Boltzmann factor exp [ − Q / k B T ] , where k B is the Boltzmann constant [
D = D N exp [ ( U − Q ) / k B T ] for D N = D E / n = 5 × 10 2 N A ℏ / n . (14)
In the physical system for a material composed of N ( j = 1 , 2 , ⋯ , N ) elements in the region V ( x , y , z ) within the closed surface S ( x , y , z ) , the Gauss theorem yields
∬ S 〈 n | J j ( t , x , y , z ) 〉 d S = ∭ V 〈 ∇ ˜ | J j ( t , x , y , z ) 〉 d V (15)
for an arbitrary differentiable vector
| J j ( t , x , y , z ) 〉 = − D j | ∇ C j ( t , x , y , z ) 〉 . (16)
Here, 〈 n | is a unit vector perpendicular to the surface element dS, and C j ( t , x , y , z ) and D j are the concentration and diffusivity for a material element j.
The Gauss theorem shows no more than a relation between the surface integral and volume integral in mathematics. In physics, however, the left-hand side of Equation (15) means the outflow of a material element j per unit time from the closed surface S ( x , y , z ) , and thus it should be relevant to the increasing rate of ∂ t C j ( t , x , y , z ) as far as the material element j is conserved within the closed surface S ( x , y , z ) . As a physical relation, therefore, using Equations (15) and (16) and the relation of
∬ S 〈 n | J j ( t , x , y , z ) 〉 d S + ∭ V ∂ t C j d x d y d z = 0 ,
the relation of material conservation law is obtained as
∬ S { ∫ ∂ t C j d x } d y d z = ∬ S { ∫ 〈 ∇ ˜ | | D j ∇ C j 〉 d x } d y d z (17)
or
∂ t C = 〈 ∇ ˜ | D ∇ C 〉 (18)
under the condition of no sink and source of the element j within the closed surface S ( x , y , z ) .
The integral calculation of { } in Equation (17) shows that the variables y and z are accepted as constant values because of the characteristic of multiple integral calculations. Thus, Equation (17) yields the diffusion equation of
∂ t C j ( t , x ) = ∂ x { D j ∂ x C j ( t , x ) } . (19)
The bracket { } in Equation (17) means an inflow flux from the surface element dydz. By defining the outflow flux as a plus value, the x component J x j of 〈 J ˜ j | = ( J x j , J y j , J z j ) is then expressed as
J x j = J j ( t , x ) = − ∫ ∂ t C j ( t , x ) d x = − ∫ ∂ x { D j ∂ x C j ( t , x ) } d x . (20)
In mathematics, Equation (20) yields the diffusion flux of
J j ( t , x ) = J F j ( t , x ) + J j ( t ) + J eq j for J F j ( t , x ) = − D j ∂ x C j ( t , x ) , (21)
where J F j ( t , x ) means the well-known Fick first law and J j ( t ) + J eq j is then a mathematical integral constant against x. In physics, J j ( t ) means a movement of a diffusion region space caused by the movement of diffusion particles. In a moving coordinate system setting the coordinate origin at a point in the diffusion region space, the flux should be physically accepted as J j ( t ) = 0 . However, the case of J j ( t ) ≠ 0 must be considered for the fixed coordinate system of diffusion region outside. The flux J eq j independent of t and x is an intrinsic one relevant to the Brown motion in the thermal equilibrium state. It plays an important role for understanding a self-diffusion mechanism [
Hereafter, we discuss such an interdiffusion problem in the time and space ( t , x ) for a diffusion couple, since the generality of mathematical physics for the diffusion theory is still kept even if the simplest coordinate system of the time and space ( t , x ) is used. We conceive a diffusion couple composed of a material A and a material B with a uniform shape and the same cross section as shown in
The diffusion depth proportional to D t indicates that the concentration profile of diffusion at a low temperature agrees with that of diffusion at a high temperature, if the diffusion time is suitably long. This means that we can essentially analyze a diffusion problem under the condition of temperature where the variation of total particles on a cross section in the diffusion region is negligible during a thermal treatment. In the research field concerned, for an arbitrary k between 1 ≤ k ≤ N , it is widely accepted that the relations of
∑ j = 1 N C j ( t , x ) = 1 and ∑ j = 1 N ∂ t C j ( t , x ) = 0 , ∂ x C k ( t , x ) = − ∑ j = 1 , j ≠ k N ∂ x C j ( t , x ) (22)
is valid between the normalized concentrations of N elements on a cross section in the diffusion region of x A ≤ x ≤ x B . Substituting Equation (22) into the diffusion Equation (19), the relation of
∂ x C k ( t , x ) = − ∑ j = 1 , j ≠ k N D j D k ∂ x C j ( t , x ) (23)
is valid in the differential Equation (19).
Since Equations (22) and (23) are simultaneously valid as identical equations, the relation of
D ˜ = D 1 = D 2 = ⋅ ⋅ ⋅ = D N (24)
must be valid in the differential Equation (19) then [
∂ t C j ( t , x ) = ∂ x { D ˜ ∂ x C j ( t , x ) } = { D ˜ ∂ x + ( ∂ x D ˜ ) } ∂ x C j ( t , x ) for j = I , I I , ⋯ , N . (25)
Here, the interdiffusion coefficient D ˜ has been accepted as an actual one in the existing theory of diffusion. However, D ˜ of Equation (24) is not a real quantity but a mathematical operator valid only in the differential Equation (25) and acts in common on each element in the diffusion field under the condition of Equation (22).
For the development of new useful materials, solving the interdiffusion problems of many elements system is extremely important. For the interdiffusion problem of many elements system, the present analytical method resulted in solving Equation (25) in accordance with the initial and/or boundary conditions of each element under the bound condition of Equation (22) [
Using the initial and/or boundary values D A j , C A j at x = x A and D B j , C B j at x = x B in the diffusion region x A ≤ x ≤ x B for D ˜ and C j ( t , x ) in the diffusion Equation (25), the general solutions were for the first time mathematically obtained as (see Ref. [
D j ( t , x ) = D A j + D B j 2 − D A j − D B j 2 erf ( x 2 D int j t + α j ) , (26)
C j ( t , x ) = C A j + C B j 2 − C A j − C B j 2 erf ( x 2 D int j t + β j ) , (27)
where D int j = D int + j = ( D A j + D B j ) / 2 for x ≥ 0 , D int j = D int − j = D A j D B j for x < 0 and
α j = erf − 1 ( D A j + D B j D A j − D B j − 2 ln D A j − ln D B j ) , β j = α j − ( D A j − D B j ) / ( D A j + D B j ) .
We confirmed that the solutions of Equations (26) and (27) agree well with results of the empirical Boltzmann Matano method [
In case of a binary system interdiffusion, for example, the relations between diffusivities and between diffusion fluxes yielding
D ˜ = D I = D I I , J ˜ I + J ˜ I I = − D ˜ ∂ x { C I + C I I } = 0 (28)
had been misunderstood as actual ones in the existing diffusion theory. However, they should be valid only in the differential Equation (25), since the initial values have not yet been taken into account. On the other hand, the relations of
D I ≠ D I I , J I + J I I = − { D I ∂ x C I + D I I ∂ x C I I } ≠ 0 (29)
taking account of initial values are naturally valid in case of using the solutions of Equations (26) and (27). It is obvious that the Kirkendall effect is caused by Equation (29) [
As discussed in Section 2, the diffusivity correlates with a jumping frequency of one micro particle in a material. There is only one diffusion particle at a point of time and space ( t , x , y , z ) and the diffusion particle has then statistically a value of jumping frequency. Thus, one element has only one diffusivity value in the diffusion field concerned. In the present diffusion field, therefore, the diffusivities of elements I and II are just D I and D I I and the other diffusivities relevant to such diffusion mechanisms as an interdiffusion and an intrinsic diffusion are actually nonexistent from the beginning.
The existence of J j ( t ) in Equation (21) indicates that we must take account of the difference between the diffusion equation of coordinate system set a point in the diffusion region space expressing the relative motion between micro particles and that of coordinate system set a point in the laboratory system outside the diffusion region. Based on the mathematical and/or physical consideration, the problems of coordinate system of diffusion equation are discussed in the following.
In relation to the interdiffusion problem, we conceived a raft model [
As can be easily seen, we can let the raft model correspond to interdiffusion phenomena in the diffusion couple shown in
When the interaction f in the diffusion field of an isolated system influences on diffusion particles under the condition of an external force F = 0 , a velocity v i ( t ) of the coordinate origin x ˜ = 0 against the coordinate origin x = 0 is defined as
x ˜ = x + x s f t for t ˜ = t and x s f t = − ∫ 0 t ˜ v i d t .
Here, the coordinate origin ( t , x ) set at a point of mass center of diffusion particles on the initial interface is then immovable because of F = 0 . As a matter of convenience, the suffix j expressing an element is removed because of no meaning in the essential theory in the following. Equation (19) is then transformed into
∂ t ˜ C ( t ˜ , x ˜ ) = ∂ x ˜ { D ∂ x ˜ C ( t ˜ , x ˜ ) − v i C ( t ˜ , x ˜ ) } . (30)
Here, the term v i C ( t ˜ , x ˜ ) means that the concentration-distance curve moves in parallel to the x axis. The Gauss theorem shows that the diffusion flux of
J ( t ˜ , x ˜ ) = − D ∂ x ˜ C ( t ˜ , x ˜ ) + v i C ( t ˜ , x ˜ ) + J e q (31)
is valid then. Further, the Gauss theorem shows that the diffusion flux of the coordinate ( t , x ) is expressed as
J ( t , x ) = − D ∂ x C ( t , x ) + J ( t , x ) + J e q , (32)
since the coordinate origin x = 0 is immovable.
Substituting Equation (14) into Equation (30) yields
∂ t ˜ C ( t ˜ , x ˜ ) = D ∂ x ˜ 2 C ( t ˜ , x ˜ ) + { ( ∂ x ˜ D ) − v i } ∂ x ˜ C ( t ˜ , x ˜ ) = D ∂ x ˜ 2 C ( t ˜ , x ˜ ) − ( D f / k B T + v i ) ∂ x ˜ C ( t ˜ , x ˜ ) (33)
because of f = − ∂ x ˜ U and F = 0 . Equation (6) shows that the diffusivity gradient ∂ x ˜ D is a velocity itself of a diffusion particle. This means that the relation of
( D f / k B T + v i ) ∂ x ˜ C ( t ˜ , x ˜ ) = ∂ x ˜ ( D f / k B T + v i ) C ( t ˜ , x ˜ )
is valid. A diffusion particle and a local space in the diffusion field change places with each other in the one-to-one correspondence. Thus, the diffusion region space moves with the quite reverse velocity of what each diffusion particle moves with a velocity at each point ( t , x ) on the concentration-distance curve. As a result, the relation of
D f / k B T + v i = 0 (34)
should be valid in the diffusion field. The diffusion particles then randomly move in accordance with the relation of
∂ t ˜ C ( t ˜ , x ˜ ) = D ∂ x ˜ 2 C ( t ˜ , x ˜ ) , (35)
satisfying the parabolic law. The relations between solutions of Equations (19) and (35) given by
C ( t ˜ , x ˜ ) = C ( t , x + x s f t ) , D ( t ˜ , x ˜ ) = D ( t , x + x s f t ) and J ( t ˜ , x ˜ ) = J ( t , x + x s f t ) (36)
are valid then.
Further, as an especial case, if we accept the same relation of f = − k v i that Langevin [
D = k B T / k (37)
called the Einstein equation. Equation (37) gives evidence for the validity of the present diffusion theory, as far as equation (37) is valid in a material.
When an external force F influences on the diffusion region from the diffusion region outside under the condition of f = 0 , a velocity v e ( τ ) of the coordinate origin x = 0 against the coordinate origin ξ = 0 is defined as
x = ξ + ξ s f t for t = τ and ξ s f t = − ∫ 0 t v e d τ , because of the movement of initial
mass center through the external force. The diffusion equation of a fixed coordinate system in the diffusion region outside yielding
∂ τ C ( τ , ξ ) = ∂ ξ { D ∂ ξ C ( τ , ξ ) } (38)
is transformed into the moving coordinate system ( t , x ) given by
∂ t C ( t , x ) = ∂ x { D ∂ x − v e } C ( t , x ) = D ∂ x 2 ( t , x ) − ( D F / k B T + v e ) ∂ x C ( t , x ) , (39)
because of F = − ∂ x ˜ U and f = 0 . In accordance with the same mechanism as the above position exchange between a diffusion particle and a local space, Equation (39) is also rewritten as
∂ t C ( t , x ) = D ∂ x 2 C ( t , x ) . (40)
Equation (39) shows that the diffusion flux of
J ( t , x ) = − D ∂ x C ( t , x ) + v e C ( t , x ) + J e q (41)
is valid then. Equation (41) corresponds to the diffusion flux in the laboratory system yielding
J ( τ , ξ ) = − D ∂ ξ C ( τ , ξ ) + J ( τ ) + J e q . (42)
When a large external force F satisfying the relation of
F ≫ k B T | ∂ x 2 ( t , x ) / ∂ x C ( t , x ) | / D
exists in the diffusion field, substituting Equation (14) into Equation (38) shows that the physical system concerned is not a diffusion problem from the start.
Based on the above analysis, it is found that there are 4 cases for relations between solutions of Equations (19), (35) and (38) as follows:
{ ( a ) C ( t ˜ , x ˜ ) = C ( t , x ) , ∂ x ˜ D ( t ˜ , x ˜ ) = ∂ x D ( t , x ) = 0 for f = F = 0 ( b ) C ( t ˜ , x ˜ ) = C ( t , x + x s f t ) , D ( t ˜ , x ˜ ) = D ( t , x + x s f t ) for f ≠ 0 , F = 0 ( c ) C ( t , x ) = C ( τ , ξ + ξ s f t ) , D ( t , x ) = D ( τ , ξ + ξ s f t ) for f = 0 , F ≠ 0 ( d ) C ( t ˜ , x ˜ ) = C ( τ , ξ + x s f t + ξ s f t ) , D ( t ˜ , x ˜ ) = D ( τ , ξ + x s f t + ξ s f t ) for f ≠ 0 , F ≠ 0 (43)
The basis of diffusion problems results from the relative motion between diffusion particles in the diffusion region space. Equation (43) reveals that the basic equation of diffusion should be expressed by a moving coordinate system. Here, the summarization in this section is shown in the following.
Hereinbefore, the coordinate transformation theory of diffusion equation was discussed using the coordinate systems ( t ˜ , x ˜ ) , ( t , x ) and ( τ , ξ ) . In accordance with the usual notation, however, if we redefine the coordinate system ( t , x , y , z ) as a moving one in the diffusion region inside, Equation (43) shows that the basic equation of diffusion should be expressed as
∂ t C = D 〈 ∇ ˜ | ∇ C 〉 , (7)
regardless of whether D depends on a concentration or not, and of whether an internal or an external force influences on the diffusion field or not. As discussed in relation to the derivation of Equations (35) or (40), Equation (7) shows the so-called Brown motion of the isotropic jumping iteration then.
On the other hand, if we also redefine the coordinate system ( t , x , y , z ) as a fixed one, the diffusion equation is expressed as
∂ t C = 〈 ∇ ˜ | D ∇ C 〉 . (18)
In case of the one dimensional space of Equations (7) and (18), the relations between solutions of Equation (7) C m ( t , x ) , D m ( t , x ) and those of Equation (18) C f ( t , x ) , D f ( t , x ) , and further between their diffusion fluxes, J m ( t , x ) and J f ( t , x ) are then obtained as
C m ( t , x ) = C f ( t , x + σ s f t ) , D m ( t , x ) = D f ( t , x + σ s f t ) and
J m ( t , x ) = J f ( t , x + σ s f t ) , (44)
where σ s f t has a value of σ s f t = 0 , σ s f t = x s f t , σ s f t = ξ s f t , σ s f t = x s f t + ξ s f t in accordance with Equation (43).
Under the condition of F ≪ k B T | ∂ x 2 ( t , x ) / ∂ x C ( t , x ) | / D , the behavior of diffusion particles shows the isotropic random movement. In that case, the shift effect x s f t of Equation (43) should depend only on the initial values of diffusivity and concentration in the isolated diffusion region. In the previous work [
x s f t = 2 μ ∑ j = I N D ω j ( C A j − C B j ) t (45)
satisfying the parabolic law was obtained, where ω → A if C A j > C B j and ω → B if C A j < C B j and the relation of diffusion length μ D t is used here. The velocity v i between the coordinate origins of a moving coordinate system and that of a fixed one becomes
v i = 1 μ ∑ j = I N D ω j ( C A j − C B j ) t − 0.5 . (46)
In general, the diffusion experiment is performed at a high temperature of T = T H during a time interval of 0 ≤ t ≤ t F . The solutions of Equations (26) and (27) are ones at the temperature of T = T H and the time t = t F then. On the other hand, the diffusion region space in experimental results is generally in the thermal equilibrium state at a room temperature of T = T R after the diffusion treatment. When we compare the solutions of diffusion equation with the experimental results, therefore, we must take account of a behavior of the diffusion region space caused by the temperature fall from T = T H to T = T R , since the concentration profile is then shifted by its behavior.
The coordinate ( t , x ) shown in
As shown in
The validity of theory mentioned here was concretely confirmed in comparison with the experimental results of diffusion couple with an inert marker [
Although the theoretical equation of Kirkendall effect Δ x e f f is given by
Δ x eff = 2 μ ∑ j = I N D ω j ( C A j − C B j ) t eff ,
we cannot understand a reasonable method at present for determining t e f f from the given diffusion problems.
Equation (38) is applicable to the coordinate system ( τ , ξ ) in diffusion region outside. In that case, Boltzmann revealed that the equation of
− ζ 2 d C d ζ = d d ζ ( D d C d ζ ) (47)
is valid in the parabolic space under the condition of parabolic law ζ = ξ / τ [
J ( ζ ) = − D ( ζ ) d C ( ζ ) d ζ , (48)
where
J ( ζ ) = − J 0 exp [ − ∫ 0 ζ η 2 D ( η ) d η ] for J 0 = D ( ζ ) d C ( ζ ) d ζ | ζ = 0 .
On the other hand, the dependence of diffusivity on the concentration means
d C d ζ = ∂ C ∂ ζ + ∂ C ∂ D ∂ D ∂ ζ (49)
in mathematics. By solving simultaneously Equations (48) and (49), as a result, the solutions of Equations (26) and (27) were mathematically obtained [
Here, it was revealed in Section 4 that Equation (38) corresponds to Equation (40) expressed by a moving coordinate system ( t , x ) in the diffusion region inside, even if the diffusivity depends on the concentration. In that case, Equation (40) is transformed into
− λ 2 d C d λ = D d 2 C d λ 2 for λ = x / t . (50)
Integrating directly Equation (50) with respect to l and using a integral constant A 2 , the relation of
d C d λ = A 2 exp [ − ∫ λ 2 D d λ ] (51)
is obtained. Further, integrating Equation (51) yields
C = A 1 + A 2 ∫ exp [ − ∫ λ 2 D d λ ] d λ , (52)
where A 1 is an integral constant. Using a diffusivity D int independent of l and a correction parameter ε 1 for Equation (52), the approximate relation of
∫ λ 2 D d λ = ( λ + ε 1 ) 2 4 D int
is derived (see Ref. [
C ( λ ) = C A + C B 2 − C A − C B 2 erf { λ 2 D int + ε } for ε 1 = 2 D int ε (53)
is obtained.
The solution of diffusivity of
D ( λ ) = D A + D B 2 − D A − D B 2 erf ( λ 2 D int + ε + ( D A − D B ) / ( D A + D B ) ) (54)
is also obtained by substituting Equation (53) into the equation expressing the dependence of diffusivity on concentration [
D = D A + D B 2 − D A − D B 2 erf ( f ( C ) ) ,
where
f ( C ) = erf − 1 { C A + C B C A − C B − 2 C ( λ ) C A − C B } + ( D A − D B ) / ( D A + D B ) .
When we solved Equation (48) [
ε s f t = σ s f t / 2 D int t for σ s f t = ∫ 0 t v d τ
in the parabolic space. In accordance with the previous investigation, therefore, the e value should be accepted as
ε = erf − 1 ( D A + D B D A − D B − 2 ln D A − ln D B ) − ε sft − ( D A − D B ) / ( D A + D B ) .
In other words, the following relations
C ( λ − ε s f t ) = C ( ζ ) and D ( λ − ε s f t ) = D ( ζ ) .
are valid in the present case.
By analyzing the diffusion Equations (48) and (50) in the parabolic space corresponding to Equations (38) and (40), it was revealed that Equation (44) is reasonably valid. Thus, it was found that analyzing the diffusion problem is satisfied by solving either Equation (7) or (18). In other words, the validity of coordinate transformation theory discussed here was revealed.
In the diffusion field, the diffusivities relevant to such diffusion mechanisms as the self-diffusion, the impurity diffusion, the one-way diffusion, the interdiffusion, the intrinsic diffusion, and so on had been investigated in accordance with the given diffusion problems. In relation to the diffusivities mentioned here, some misconceptions in the existing diffusion theory are revealed in the following.
Based on the mathematical theory of differential equation, it was revealed in relation to Equation (24) that the diffusivity called the interdiffusion coefficient is not an actual one but a mathematical operator valid only in the differential equation, which acts in common on each of diffusion elements in the diffusion system. At the same time, it was revealed that there is no reasonable necessity for introducing the concept of intrinsic diffusion into the diffusion theory. In Section 3, it was revealed that we cannot essentially accept such diffusion concept also in view of the definition of diffusivity in a material. Nevertheless, the relation of
D ˜ = C I D I N T I I + C I I D I N T I (55)
proposed by Darken [
The Gauss theorem shows that Equation (20) gives the definition of diffusion flux and that the general formula of diffusion flux is expressed as Equation (21) in a fixed coordinate system. The Fick first law is thus incomplete one without an initial value. It is, therefore, obvious that the Fick first law is not worthy as a universal law now. Using not the Fick first law but the redefined diffusion flux of Equation (21), we can first reasonably understand the self-diffusion mechanism [
The problem of coordinate system for diffusion equation had never been discussed in the existing diffusion theory. The Gauss theorem reveals that the diffusion region space moves as shown in the relation between Equations (20) and (21). In other words, the discussion of coordinate transformation about the diffusion equations is indispensable for investigating the diffusion phenomena meaning the relative motion between micro particles. In relation to the matter mentioned here, the misconception of diffusion flux is discussed in the following.
It had been considered for a long time that the mathematical solutions of diffusion equation are impossible when the diffusivity depends on the concentration. Using the diffusion flux, therefore, the diffusion phenomena had been discussed until recently as there was no other choice except a numerical analysis. A diffusion flux similar to Equation (31) or (41) is proposed as
J ( t , x ) = − D ∂ x C ( t , x ) + v F C ( t , x ) , (56)
without a discussion about the coordinate system of diffusion equation [
The matters relevant to the misconceptions pointed out here are extremely fundamental and they have been mistaken for such a long time in the history of diffusion. In fact, a lot of research papers based on the misconceptions are still published in the famous journals. Further, the misconceptions are also still plausibly described in many usual textbooks [
The definition of diffusivity expressed by Equation (6) indicates that the quantum effect on the elementary process of diffusion should be essentially incorporated into the diffusivity as a characteristic of micro particle. The correlation between the Schrödinger equation and the diffusion equation revealed that the relation of material wave is reasonably valid regardless of the photon energy. Using the relation of material wave and the uncertain principle, we could reasonably understand the elementary process of diffusion through the different mechanism from the existing theory. At the same time, the universal diffusivity expression of Equation (14) was reasonably obtained using the elementary quantity ( 5 × 10 2 N A ℏ = 3.18 × 10 − 8 [ m 2 ⋅ s − 1 ] ) of diffusivity including the characteristic constant of a micro particle called the Planck constant ℏ and that of a unit group of micro particles called the Avogadro constant N A .
The coordinate origin of basic diffusion equation expressing a relative motion between micro particles should be essentially set at a point of diffusion region space (or a virtual inert marker) on the initial interface shown in
∑ j = I N D ω j ( C A j − C B j ) ≠ 0 . (57)
In that case, the basic diffusion equation is expressed by a moving coordinate system as discussed in Section 4. Here, we concluded again that the basic diffusion equation is
∂ t C = D 〈 ∇ ˜ | ∇ C 〉 , (7)
even if 〈 ∇ ˜ | D ≠ 0 is valid.
On the other hand, the diffusion equation of a fixed coordinate system is expressed as
∂ t C = 〈 ∇ ˜ | D ∇ C 〉 . (18)
Here, it was concretely confirmed in Section 6 that solutions of Equations (7) and (18) are then transformable to each other by using Equation (44). In general, therefore, the basic equation of diffusion phenomena should be certainly defined as Equation (7).
Hereafter, Equation (7) will be widely applicable to the interdiffusion problems for the development of useful materials in future. In that case, however, we must then examine a phase shift resulting from the coordinate transformation of diffusion equation and further a phase shift caused by difference between a temperature during diffusion treatment and a room temperature after diffusion treatment.
In addition, as an especial case, if the relation of
∑ j = I N D ω j ( C A j − C B j ) = 0 (58)
is valid, f = 0 is also valid because of Equation (34). In that case, the Fick first law is then valid under the condition of J e q = 0 . At the same time, the Fick second law of Equation (18) is equivalent to Equation (7). In other words, the Fick laws are valid only under the extremely limited condition for diffusion problems.
In history, Fick applied the heat conduction equation to the diffusion phenomena as it had been. From a mathematical viewpoint on the Gauss theorem, however, we should have taken account of the difference between the state quantity of heat distribution and the real quantity of concentration profile of diffusion particles in those days. The mathematical misconceptions resulting from its thoughtlessness caused subsequently physical misconceptions on the fundamental theory in the diffusion problems as discussed in Section 7. They had been left untouched for such a long history of diffusion theory until recently.
In the present work, it was confirmed that the coordinate transformation of diffusion equation is indispensable for understanding the diffusion theory. The coordinate transformation theory shows that the general solutions (26) and (27) will be extremely useful for investigating the interdiffusion problems of many elements system in future.
The historical misconceptions such as an intrinsic diffusion and a drift velocity, which are actually nonexistent in the recent diffusion theory, were reasonably solved as concretely discussed in the present work. The new findings discussed here in accordance with the mathematical physics will be useful for understanding various diffusion problems in future, just because of the matters relevant to the fundamental theory.
Okino, T. (2018) Quantum Effect on Elementary Process of Diffusion and Collective Motion of Brown Particles. Journal of Modern Physics, 9, 1007-1028. https://doi.org/10.4236/jmp.2018.95063