The equations describing the interdiffusion process (mixing) in the general case where the components diffusivities vary with composition and the reactions of diffusing components within the diffusion zone are allowed (i.e

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D:\Basic Research & finished\GDM + Darken\HOLLY-MD_ZNAGH.WPD Korekta dla WP 6.0b: April 20, 2006

Interdiffusion in Solids, Free Boundary Problem for r - Component One Dimensional Mixture Showing Constant Concentration

K. Holly* and M. Danielewski**

*Jagiellonian University, Institute of Mathematics ul. Reymonta 4, 30-059 Cracow, Poland

**University of Mining and Metallurgy, Faculty of Materials Science and Ceramics Al. Mickiewicza 30, 30-059 Cracow, Poland

Key words: interdiffusion, Stefan problem


The concept of separation of diffusional and drift (i.e. the mass flow due to translation) flows and the postulate that the total mass flow is a sum of diffusion flux and translation only, are applied for the general case of diffusional transport in r-component compound (process defined as interdiffusion in one dimensional mixture).

The equations of mass conservation (continuity equations), the appropriate expressions describing the fluxes (drift flux and diffusional flux) and momentum conservation equation (equation of motion) allow a complete quantitative description of diffusional transport process (in one dimensional mixture showing constant concentration) to be formulated. The equations describing the interdiffusion process (mixing) in the general case where the components diffusivities vary with composition and the reactions of diffusing components within the diffusion zone are allowed (i.e. the new mixture formation), are derived. When certain regularity assumptions (concerning initial data) and, a quantitative condition (concerning the diffusion coefficients - providing a parabolic type of the final equation) are fulfilled, then there exists the unique solution of the interdiffusion problem. Numerical solution can be obtained by the Faedo - Galerkin method. An effective algebraic criterion allows to determine the type of particular problem. In general "the regular diffusion" (parabolic type equation) and "explosive diffusion" (hyperbolic type equation) can result from the presented model. The results extend the standard Darken approach. The phenomenology allows the quantitative data on the dynamics of the processes to be obtained within an interdiffusion zone. The limited applicability of the second Fick's law in the quantitative analysis of the transport processes is emphasized.


Several analyses of the interdiffusion have neglected the effects due to variation of medium properties with the composition, neglected the effects of differences in the partial molar volumes of the diffusing species and all of them ignored the possible reactions within the diffusion zone. For example, the fundamental Darken and Wagner equations assume that the partial molar volumes of the diffusing components are constant [1], [2] and equal [2].

The conservation of momentum is not included in all the models of interdiffusion [3], [4]. Thus, under these simplified assumptions, all the models of inter- diffusion neglect the dynamics of the transport process, e.g. the


* To avoid collisions with other mathem atical symbols, the intrinsic diffusion coefficients will be denoted as .

generation of stresses in solids and the variable gradient of pressure in gases and liquids [1]-[4].

This paper is an attempt to unify the interdiffusion phenomenology, to bridge the gap between the physico- chemical statements of the processes (which are inherently coupled with the reactive, mutual and interdiffusion processes, mass transport in general) and their reduction to diffusion problems. The essence of this attempt comprises:

i) the postulate that the total mass flow is a sum of diffusion and translation fluxes only and,

ii) a rigorous use of the fundamental Darken concept, the concept of the drift velocity [2] (which is a common velocity of all the mixture components, e.g. an alloy components).

The medium expansion as a result of accumulation and/or medium formation (e.g. an alloy production) within the transport zone are inherent processes of the diffusional mass transport. Consequently, the production of the host compound (or other phases in general) must be included in the general interpretation of the continuity equation. The complete description of the transport may not neglect the local momentum conservation.

Consequently, the Navier-Stokes equations are included in the presented analysis of the interdiffusion.

A more detailed analysis of the concepts of drift velocity, the choice of the proper reference frame for diffusion, as well as the other consequences of the proposed formalism have already been published [5, 6], [7].

A general phenomenological treatment of the interdiffusion problem is given below. No assumptions whatsoever are made as far as the mechanism of the diffusion processes is concerned.


It is essential to state the common aspects of the already mentioned fundamental models of diffusional transport [1]-[4]:

i) that seemingly attractive choice of internal reference frame is useless in the direct description of transport processes in the external reference frame (it is termed often as an observer or laboratory frame of reference). However, because the laws of conservation does not depend on the choice of the reference frame, this frame of reference can be used as a source of information about the dynamics of the system (allows for mathematical disjoining of dynamics and diffusion),

ii) that the reactions within the diffusion zone - e.g. medium (compound) production, and/or local accumulation - affect the local drift velocity of the medium in any fixed external reference frame, and

iii) a common, for all the models, unified approach to the relation between the mobility, diffusivity and activity (the Darken's concept of the variable intrinsic diffusion coefficient*, [2]).

When any compound (media) acts upon a field (e.g. chemical potential gradient), the different elements respond in different ways. In case of a multicomponent medium, the force arising from any concentration gradient causes the atoms of the particular component to move with a velocity ( ) which in general may differ from velocity of the atoms of some or all the other components. As the medium is common for all the transported species, all the fluxes are coupled and their local changes can affect the common compound drift velocity ( ).

All the fluxes of all elements can affect the common medium drift velocity and are coupled by this process, namely production and/or expansion/compression of the medium (the effect which is a result of all the local






accumulation processes). The above phenomenon is called interdiffusion and the adequate proportionality coefficient in Fick'ian formula describing this transport process in the fixed external reference frame (ERF) is a chemical interdiffusion coefficient. The coefficients are not characteristic material constant only. It follows:

where ERF-index emphasizes the fact that both, the flux and interdiffusion coefficient, depend on the choice of the reference frame.

The choice of the reference frame does not affect the local concentration distributions and mobilities of all the transported species and, consequently, their diffusional fluxes. The i-specie conservation in the external reference frame (the only reference frame available in the experimental conditions) is expressed by the equation of mass conservation (local continuity equation of an i-th component):

where the reaction term (local sink/source of mass) is neglected (formation of the new phases is not allowed in the course of the analyzed interdiffusion process).

The flux vector is a sum of the diffusional and drift (translation) flow:

Thus, upon substituting eqs.(2) and (3), the equation describing the mass conservation of the i-component in the ERF takes the usual form of the continuity equation, where the physical sense of all the terms is given below the equation:

It should be pointed out that the drift generation term can be the result of the all local processes. Without external force fields, only the accumulation or production (the medium or other phases) may affect locally the drift velocity, may generate drift. The drift production term as well as a local drift velocity are always common for all the transported components.

In the IRF (frame of reference which has drift velocity equal to that of the medium), the drift term is vanishing




and consequently, the local mass balance (continuity equation for an i-th component) becomes:

In the general case of diffusional transport in the r-component mixture, r-continuity equations for all diffusing components must be fulfilled in any elementary volume within an open system. The continuity equation, eq.(4), in the unidimensional mixture becomes:

The important assumption of the local equilibrium in a mixture (non-explicit assumption) is included in present analysis. This assumption is also a foundation of the majority of phenomenological models of transport (interdiffusion) [1], [2]. Decades of successful applications of these models in describing the mass transport in solids justify the incorporation of this fundamental concept of non-equilibrium thermodynamics into the transport equations in all media, in particular solids. The postulate of local equilibrium is as follows: "For a system in which irreversible processes are taking place, all thermodynamic functions of state exist for each element of the system. These thermodynamic quantities for the nonequilibrium system are the same functions of local state variables as the corresponding equilibrium quantities." [8].

From this assumption it follows that (in the course of the analyzed processes of interdiffusion in solids at constant temperature), the total concentration of the mixture is constant (as in the equilibrium state). In the other words it is assumed, that the transport processes do not affect the local medium properties. Namely it is assumed the transport processes do not affect the constant concentration of the mixture (total concentration of the solid compound, e.g. an alloy).

2.1. Mathematical Formulation of the Problem of Interdiffusion in the r-component One Dimensional Solid Solution

When we are not interested in the temperature effects (they can be neglected), the interdiffusion (mutual diffusion in a solid solution, e.g in an alloy) may be treated as a mixing (the process has low free energy of the reaction). In this section the interdiffusion in a r-component solid solution will be formulated.


- molecular mass of the i-th component of the mixture, e.g. an alloy, (i = 1, 2, ... , r);

- right border of the segment occupied by the mixture at the begin-


ning of the process ( );

- initial density distribution of the i-th component in the mixture (i = 1, ..., r). The initial global concentration of the mixture:

is constant and positive;

- diffusion coefficient of i-th component (i = 1, ..., r), where denotes the Cartesian product;

- the viscosity coefficient of the mixture;

- initial momentum of the mixture mass center;

- the examination time (time at which measurements were carried out);

- time evolution of the density derivative of the i-th component at the left boundary (i = 1, ..., r - 1);

- time evolution of a force acting on the mixture boundary;

- time evolution of a body force (evolution, e.g of gravity force).

The unknown:

- where and are the mixture boundaries;

- density of the i-th component (i = 1, ..., r);

- drift velocity;

- pressure of the mixture.

Physical laws:








The local mass conservation law for the i-th component:

where is the i-th component velocity, i.e.

the postulate of constant concentration of the mixture;

Notice: the mixture density, , assumes values in the interval

the local law of momentum conservation (equation of motion):


is the velocity of the mixture (more precisely, it is a distribution of the velocities of the local mass centers);

the total i-th component mass conservation law :

where (i = 1, ..., r).




Initial conditions:

- initial position of the mixture boundaries;

- initial distribution of the mixture components;

- initial momentum of the mixture.

Boundary conditions:

- the gradient of the components at the left boundary of the mixture;

- the resultant (the net) thrust force acting on the boundary of the mixture;

- the velocities of the mixture boundaries.

It is obvious that:

In particular:

Examples of possible modifications:

Instead of the second boundary condition, , one can introduce the global force acting on the mixture as a whole:

when the first law of dynamics results.






The measurement of the normal derivative of i-th density for some

, can be replaced by the measurement of the left boundary position . For example for the r-component alloy bar (mixture segment ) which is placed vertically on a rigid immovable surface, one can assume: .

2.2. Reformulating of the Problem

1) Upon summing up the total mass conservation formulae of the all components, eqs.(12), one gets:

i.e. the total mass conservation law of the mixture.

2) Upon summing up the local mass conservation formulae (continuity equations, eqs.(7)) of all the components, one gets:

i.e. the local conservation law of the mixture mass (continuity equation for the mixture).

3) Upon multiplying by the i-th component total mass conservation law, eq.(12), and adding all the

obtained equations, one gets: Hence

4) If are solutions of the differential equation:

(i.d. x1, x2 represent time evolution of position of two mixture particles), then, providing that , one have:

The above inequality and the Liouville theorem result in:






Consequently, from eq.(19) and the third initial condition it follows that

Upon applying the Liouville theorem, the equation of motion, equation of continuity for the mixture and the second boundary condition, one can calculate the total force acting on the center of mass (the mixture mass center) at a moment t :

Equation (21) can be integrated over the time range :

5) Multiplying by the equations of continuity of i-th component (partial continuity equation, eq.(7)) and

summing the all obtained formulae, one gets:








Hence, from eqs.(8) and (11),

Upon integration of the eq.(25) over the x range , using the conclusion of preceding step (eqs.(19) - (22))

and, the mass conservation law from the first step, eq.(15), one calculates:


is the total mass of the mixture.

Mathematical disjoining of dynamics and diffusion.

For a moment the following natural parametrization of the interval ,










will be useful. We denote:

where mi is the mass of the i-th component:













The last two terms in eq.(39) can be rearranged to minimize their visual dependence on , namely:









Thus, upon assuming that are known, the boundary can be calculated as a solution of the following integro-differential equation:

where K(t) is a function of given by eq.(33), (45).

Knowing and one can calculate the boundary from eq.(17), the densities from eqs.(34) and (39) and, the pressure p from the following explicit formula:

where the density of the mixture is given by the eq.(33), (45), while velocity u of the mixture is given be eqs.(38) and (39).

The rescaled densities are a solution of the initial-boundary value problem for the following system of differentio-functional equations:










where eqs.(49) - (51) represent the boundary conditions and initial condition, respectively. Note that has been eliminated. The symbol denotes the scalar multiplication in . Moreover:

for a fixed , is a linear operator defined by:




the mapping is given by the formula:


1. Theorem. If certain regularity assumptions (concerning initial data ) and a quantitative condition (concerning the diffusion coefficients - providing a parabolic type of the final eq.(48)) are fulfilled, then there exists the unique solution of the problem (48) - (51). In addition

, which means that the densities of all components can be only non-

negative .

2. Numerical solution of the above problem can be obtained by the Faedo - Galerkin method.

3. An effective algebraic criterion (concerning the components diffusivities, i.e. ), is derived.

The criterion allows to determine the type of particular phenomenon (the type of final eq.(48)). In general "the regular diffusion" (parabolic type equation) and "explosive diffusion" (hyperbolic type equations) can result from the presented model.

4. In the case of an interdiffusion process in which the total mass of all components in the mixture is constant and, there is no mass exchange with an surrounding environment, the additional boundary condition can be postulated.

Boundary condition:

- velocity of the i-th component at the boundary of the mixture

equals the local center of mass velocity, i.e. velocities of all








components at the boundary are equal.

It allows to decrease the number of necessary data (data ), namely it allows to calculate the gradient of the

components at the boundaries of the mixture .

The boundary condition can be written in the form:

which taking use of eq.(8) is

The sum of the boundary conditions for all the mixture components, eqs.(62), results in

which taking use of eq.(24) becomes

from which it follows

From eqs.(24), (62) and (65) it is also









thus, e.g. for the left boundary it is


Upon dividing eq.(68) by molecular mass of the i-th component one gets:

Summing up of the above formulae one gets


Consequently from the postulate , data and the above presented theorem, it follows

From the eqs.(68) and (72) it follows that (in the case of an interdiffusion process in which the mass of each


1. Wagner, C., Diffusion and High Temperature Oxidation of Metals, in "Atom Movements" (American Society for Metals, Cleveland, Ohio 1951).

2. Darken, L.S., Trans. AIME, 174, 184 (1948).

3. Balluffi, R.W., Acta Metallurgica, 8, 871 (1960).

4. Prager, S., J. Chem. Phys., 21, 1344 (1953).

5. Danielewski, M., Solid State Ionics, 45, 245 (1991).

6. Danielewski, M., Netsu Sokutei, 20, 7 (1993).



mixture component is conserved) the density gradients of all components are given by

or in general

Consequently the process is entirely determined by the initial components distribution (data ), mixture

properties (data ) and external parameters (data ).

The demonstrated unified methods of analysis of transport phenomena should not lead to the negligence of more advanced transport analysis (e.g. using other formulas of equation of motion and energy conservation formula - when temperature in not a constant). The second Fick's law, as a formula that neglects any interactions between diffusing components and the compound (accumulation does not result in expansion, production is not allowed), is at present only a minor importance and its use should be carefully analyzed. In the general case of transport in a r-component medium, the mass conservation equations, momentum equation, boundary conditions and initial conditions form an initial-boundary problem that allows a complete analysis of interdiffusion process to be carried out.


Support by the Polish State Committee for Scientific Research in the course of this work is acknowledged with gratitude, project no. 3 3662 92 03 (AGH no. 18.160.62).



7. Danielewski, M., Defect and Diffusion Forum, 95-98, 125 (1993).

8. Fitts, D.D., Nonequilibrium thermodynamics (McGraw-Hill, New York, 1962) pp.21, 44, 88.


Dyfuzja Wzajemna w Cia»ach Sta»ych, Jednowymiarowe Zagadnienie Dyfuzji w Mieszaninie r-Sk»adnikowej o Sta»ym Ðrednim St“óeniu i Swobodnym Brzegu


Rozdzia» ogólnego strumienia masy na cz“у dyfuzyjn i translacyjn (dryft), zastosowano dla opisu dyfuzji wzajemnej w mieszaninie (roztworze sta»ym) r-sk»adniów (proces zdefiniowany jako dyfuzja wzajemna w jednowymiarowej mieszaninie). Podstawowe prawa zachowania dla oÑrodka cig»ego (równania zachowania masy i p“du) oraz postulowane równania na ogólny strumie½ masy w mieszaninie,

pozwoli»y na pe»ny iloÑciowy opis procesu. Wyprowadzono ogólne równania opisujce proces mieszania (dyfuzji wzajemnej) dla ogólnego przypadku w którym wspó»czynniki chemicznej dyfuzji w»asnej (intrinsic diffusion coefficients) s funkcj st“óenia wszystkich sk»adników roztworu sta»ego oraz kiedy zachodzi tworzenie nowej mieszaniny w polu dyfuzji (zachodzi efekt Kirkendall`a). Dla tak sformu»owanego problemu wykazano, óe jeóeli spe»nione s pewne ogólne warunki o regularnoÑci danych pocztkowych oraz iloÑciowe warunki dotyczce wspó»czynników dyfuzji (warunki zapewniajce parabolicznoу równania b“dcego rozwizaniem zagadnienia), to wówczas istnieje jednoznaczne rozwizanie tak sformu»owanego problemu dyfuzji wzajemnej. Rozwizanie numeryczne moóna uzyskaƒ metod Faedo-Galerkina. Istnieje algebraiczne kryterium pozwalajce na okreÑlenie typu badanego procesu. W ogólnoÑci prezentowany model przewiduje moóliwoу dyfuzji regularnej (równania typu parabolicznego) i "wybuchowej" (równania typu hiperbolicznego).

Uzyskane wyniki rozszerzaj model zaproponowany po raz pierwszy przez Darkena. Prezentowany fenomenologiczny opis dyfuzji wzajemnej pozwala na iloÑciowy opis dynamiki procesów w polu dyfuzji.

PodkreÑlono ograniczenia stosowalnoÑci II prawa Fick`a dla opisu procesów transportu masy.




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