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# Chapter 5

Diffusion in Solid‐State Systems

## 5.1 Introduction

Diffusion as the process of mitigation and mixing due to irregular movement of particles (atoms, ions, molecules) is one of the basic and ubiquitous phenomena in nature. As this process shows up in all states of matter over very large time and length scales, the subject is still very general involving a large variety of natural sciences such as physics, chemistry, biology, and geology and their interfacial disciplines. Besides its scientific interest, diffusion is of enormous practical relevance for industry and life, ranging from steelmaking, growth of oxide scales, sintering, and high temperature creep of metals to oxide/carbon dioxide exchange in the human lung. It therefore comes as no surprise that the early history of the subject is marked by scientists from diverse communities, e.g. the scottish botanist R. Brown (1773–1858), the scottish chemist T. Graham (1805–1869), the german physiologist A. Fick (1829–1901), the english metallurgist W.C. Roberts‐Austen (1843–1902), and the german physicist A. Einstein (1879–1955). Today, exactly 162 and 112 years after the seminal publications by Fick and Einstein, respectively, the field is flourishing more than ever with more than 10 000 scientific papers per year.

The present chapter is confined, of course, to diffusion in condensed matter, namely, in metals, binary alloys, and oxides. Emphasis is on very basic fundamental aspects, the contents being roughly characterized by the headings general theory of diffusion, diffusion coefficients, Matano–Boltzmann analysis, Kirkendall effect, Darken analysis, factors influencing diffusion, impurity diffusion in metals, grain boundary (GB) diffusion in metals, diffusion in solid oxides, morphology of reaction products, and measurement of diffusion parameters.

Many references are added at the end of the chapter for readers in the forefront of the subject.

## 5.2 General Theory of Diffusion

### 5.2.1 Basic Concepts, Laws, and Mechanisms

Diffusion is defined as “a process whereby particles intermingle as the result of their spontaneous movement caused by thermal agitation.”

In the solid state, this manifests itself as the migration of “solute” atoms through a host lattice via the defect structure of that lattice. Since the rate of formation of defects increases with the vibrational energy of the atoms, the diffusion process is enhanced at elevated temperatures. However, the inherently rigid structure of a solid leads to a slow diffusion process as compared with liquids and gases. The basic point defects within a crystal lattice are vacancies and interstitial atoms, and it is via these, and a combination of them, that solid‐state diffusion occurs, in general.

Various atomic mechanisms of diffusion in crystals have been identified and are catalogued as follows.

#### 5.2.1.1 Interstitial Mechanism

Solute atoms that are considerably smaller than the solvent (lattice) atoms (e.g. hydrogen, carbon, nitrogen, and oxygen) are usually incorporated in interstitial sites of a metal. In this way, an interstitial solid solution is formed. Interstitial solutes usually occupy octahedral or tetrahedral sites of the lattice. Octahedral and tetrahedral interstitial sites in the FCC and BCC lattices are illustrated in Figure 5.1. Interstitial solutes can diffuse by jumping from one interstitial site to the next as shown in Figure 5.2. This mechanism is sometimes also denoted as direct interstitial mechanism in order to distinguish it more clearly from the interstitialcy mechanism discussed below.

#### 5.2.1.2 Vacancy Mechanism

Self‐atoms or substitutional solute atoms migrate by jumping into a neighboring vacant site as illustrated in Figure 5.3. In thermal equilibrium, the atomic fraction of vacancies in a monoatomic crystal is given by

where *S*^{F} and *H*^{F} denote formation entropy and enthalpy of a vacancy (superscript F). Self‐diffusion in metals and alloys and in many ionic crystals (e.g. alkali halides) and ceramic materials occurs by the vacancy mechanism.

#### 5.2.1.3 Divacancy Mechanism

Diffusion of self‐atoms or substitutional solute atoms can also occur via bound pairs of vacancies (denoted as divacancies or as vacancy pairs) as illustrated in Figure 5.4. At thermal equilibrium, divacancies in an elemental crystal are formed from monovacancies according to the reaction

As a consequence of the law of mass action, we have for the mole fractions *C*_{V} and *C*_{2V} of mono‐ and divacancies

The quantity *K* contains the Gibbs free binding energy of the vacancy pair. Since the monovacancy population under equilibrium conditions increases with temperature, the concentration of divacancies becomes more significant at high temperatures. Divacancies in FCC metals have a higher mobility than monovacancies (Cahn and Haasen 1996). Therefore, self‐diffusion of FCC metals usually has some divacancy contribution in addition to the vacancy mechanism. The latter is, however, the dominating mechanism at temperatures below 2/3 of the melting temperature (Seeger et al. 1970).

#### 5.2.1.4 Interstitialcy Mechanism

In this case, self‐interstitials – extra atoms located between lattice sites – act as diffusion vehicles. As illustrated in Figure 5.5, a self‐interstitial replaces an atom on a substitutional site, which then replaces again a neighboring lattice atom. Self‐interstitials are responsible for diffusion in the silver sublattice of silver halides. In silicon, the base material of microelectronic devices, the interstitialcy mechanism dominates self‐diffusion and plays a prominent role in the diffusion of some solute atoms including important doping elements (Murch and Nowick 1984). This is not surprising since the diamond lattice (coordination number 4) provides sufficient space for interstitial species.

#### 5.2.1.5 Interstitial–Substitutional Exchange Mechanisms

Some solute atoms (B) can be dissolved on interstitial (B_{i}) and substitutional (B_{s}) sites of a solvent crystal (A) and diffuse via an interstitial–substitutional exchange mechanism (see Figure 5.6). For some of these so‐called hybrid solutes, the diffusivity of B_{i} is much higher than the diffusivity of B_{s}, whereas the opposite is true for the solubilities. Under such conditions, the incorporation of B atoms can occur by the fast diffusion of B_{i} and the subsequent changeover to B_{s}. Two types of interstitial–substitutional exchange mechanisms can be distinguished:

If the changeover involves vacancies (V) according to

the mechanism is denoted as *dissociative mechanism* (sometimes also *Frank–Turnbull mechanism* or *Longini mechanism*). The rapid diffusion of Cu in germanium and of some foreign metallic elements in polyvalent metals such as lead, tin, niobium, titanium, and zirconium has been attributed to this mechanism (see Section 5.10).

If the changeover involves self‐interstitials (A_{i}) according to

the mechanism is denoted as *kickout mechanism*. The fast diffusion of Au, Pt, and Zn in silicon has been attributed to this mechanism (Bracht et al. 1995).

The considerations above illustrate the main diffusion mechanisms. In the case of semiconducting materials, the simple picture may be complicated to some extent by the wide range of energy values available up to the Fermi level. This range leads to the possibility that the given lattice defect may occur with different states of ionization. However, a useful consequence of this phenomenon is the opportunity to investigate diffusion processes in more detail by using electrical methods.

The equation that governs the relationship between the flux *J* of the diffusing species and the concentration gradient of that species at any point is

where *D* is called the *diffusion coefficient* of that species. The negative sign indicates a flow from a region of high concentration to that one of lower concentration in an isotropic medium. *D* may be defined as “the quantity of substance that, in diffusing from one region to another, passes through each unit of cross‐section per unit of time, when the volume‐concentration gradient is unity.” Equation 5.6 is the *first Fick’s law*.

The actual diffusion mechanism, which is operating in a given situation, may be established by the position and the charge state of the diffused impurity, if this information can be obtained. If not, the variation of *D* in the particular situation may lead to a solution. In particular, the temperature dependence of *D* is often given by Chitraub et al. (2000)

where *Q* is the activation energy for the jump mechanism while each mechanism has a unique value of *Q*.

Using now the continuity equation for the flow of atoms through a given volume

we have with 5.6

which is known as *second Fick’s law*.

For the case of diffusion in one dimension only, this becomes

and if *D* is independent of *x*, then

In order to see which of the above two equations should be applied in any particular case and, further, which boundary conditions must be used in order to find a solution to that equation, it is necessary to consider the possible types of diffusion processes and the ways in which diffusion may be carried out in practical terms.

### 5.2.2 Diffusion at Chemical Equilibrium

In this case, diffusion occurs in a uniform chemical and defect environment. It is termed *self‐diffusion* when dealing with solute atoms of the same species as the host lattice and *isoconcentration diffusion* for the case of foreign atoms. In the latter situation, the crystal must be pre‐diffused to a homogeneous level of impurity concentration prior to the experimental diffusion (Huggins 2001). It is clear that in both cases the solute and solvent atoms must be mutually distinguishable, and this is best achieved by the use of radioisotopes (Section 5.12.3). Under these diffusion conditions, the concentration of the diffusing species is constant, and, consequently, if *D* is considered to be exclusively a function of *C* at a given temperature, then *D* will be independent of position. Therefore, in this case, Eq. 5.11 is always valid.

### 5.2.3 Diffusion in a Net Chemical Flux

The system is now in a nonequilibrium situation with chemical fluxes being formed by the presence of chemical potential gradients. Diffusion under these conditions is often termed *chemical diffusion*. Since, in this situation, there exists a concentration gradient ∂*C*/∂*x*, Eq. 5.11 is only valid if *D* is independent of *C*. Otherwise, one must use the form of Fick’s law shown in Eq. 5.10. However, if it is for the present state assumed that it is valid to use Eq. 5.11, then the boundary conditions, necessary for a solution, are specified by the experimental conditions under which the diffusion is carried out (Vuci and Gladi 1999).

### 5.2.4 Condition of Limited Source

In this situation, the total amount of impurity present in the system is small, and the appropriate boundary condition is that the total number of impurity atoms in the diffused region is constant. Experimentally, the impurity is often present as a thin layer, deposited on the surface of the host crystal. The corresponding solution to Fick’s law is

where *Q*^{i} is the number of impurity atoms, per unit area, initially contained in the layer (Labid et al. 1997).

### 5.2.5 Conditions of Infinite Source

Here we have the case in which the number of atoms already diffused into the lattice at any time is small compared with their number available in the source. The appropriate boundary condition is now that the number of impurity atoms per unit volume, *C*_{0}, at the surface of the crystal is constant for all diffusion times. The corresponding solution is

To be more precise, in practice *C*_{0} is the solubility of the impurity under the particular conditions of the experiment in question. erfc(*y*) is a complementary error function given by

The most common way of obtaining this condition is to diffuse from a vapor source. This is the method employed for most of the experimental diffusions described in the open literature, and hence the solution shown in Eq. 5.13 is of considerable interest. Provided that the infinite source condition is satisfied, it indicates the theoretical form of the impurity profile for all concentration experiments and those chemical diffusions, where *D* is independent of *C*. Any deviation from this profile shape is thus usually an indication that *D* is a function of *C*, and Eq. 5.10 must be solved to give the relationship between *D* and *C*. The discussion of this problem is presented in a later section.

## 5.3 Diffusion Coefficients

Diffusion in materials is characterized by several diffusion coefficients, which depend on the experimental situation. Our focus is on bulk diffusion in simple binary systems. In this section, we will distinguish the various diffusion coefficients by lower and upper indices. We will drop the indices in the following sections again, whenever it is clear which diffusion coefficient is meant.

### 5.3.1 Tracer Diffusion Coefficients

If the diffusion of A atoms in a solid element A is studied, one speaks of self‐diffusion. Studies of self‐diffusion use a tracer isotope A* of the same element. A typical initial configuration for a tracer self‐diffusion experiment is illustrated in Figure 5.7a. If the applied tracer layer is very thin as compared to the average diffusion length, the tracer *self‐diffusion coefficient* is obtained from such an experiment.

The connection between the macroscopically defined tracer self‐diffusion coefficient and the atomistic picture of diffusion is the famous *Einstein–Smoluchowski relation*. In simple cases, it reads

where *l* denotes the jump length and *τ* the mean residence time of an atom on a certain site of the crystal. The quantity *f* is the correlation factor. For self‐diffusion in cubic crystals, *f* is a numeric factor. Its value is characteristic of the lattice geometry and the diffusion mechanism. In some textbooks, the quantity *D*_{E} is denoted as the *Einstein diffusion coefficient*. Equation 5.15 considers only the simplest case: cubic structure, all sites are energetically equivalent, and only jumps to nearest neighbors are allowed.

In a homogeneous binary A_{x}B_{1 − x} alloy or compound, two tracer diffusion coefficients for both, A* and B* tracer atoms, can be measured. A typical experimental starting configuration is displayed in Figure 5.7b. We denote the tracer diffusion coefficients by and . Both tracer diffusion coefficients will in general be different.

This diffusion asymmetry depends on the crystal structure of the material and on the atomic mechanisms that mediate diffusion. Both diffusivities, of course, also depend on temperature and composition of the alloy or compound and for anisotropic media on the direction of diffusion.

When the diffusion of a trace solute *C** in a monoatomic solvent A or in a homogeneous binary solvent A_{x}B_{1 − x} (Figure 5.7) is measured, the tracer diffusion coefficients

are obtained. These diffusion coefficients are denoted as *impurity diffusion coefficients* or sometimes also as foreign atom diffusion coefficients.

Figure 5.8 shows collected data of self‐diffusion coefficients in sulfides in relation to analogous results obtained for several metal oxides (Mrowec and Przybylski 1984). It becomes clear from this comparison that the rate of self‐diffusion in metal sulfides is generally much higher than in the corresponding oxides. As chemical diffusion coefficients in metal oxides and sulfides are comparable, as shown in Figure 5.9, the significantly higher self‐diffusion rates of cations in the majority of transition metal sulfides result mainly from much higher defect concentration and not their mobilities (see Chapter ). The only known exception is niobium sulfide.

### 5.3.2 Chemical Diffusion (or Interdiffusion) Coefficient

So far, we have considered in this section cases where the concentration gradient is the only cause for the flow of matter. We have seen that such situations can be studied using tiny amounts of trace elements in an otherwise homogeneous material. However, from a general viewpoint, a diffusion flux is proportional to the gradient of the chemical potential.

The chemical potential of a species *i* in a binary alloy is given by

In Eq. 5.16, *G* denotes Gibbs free energy, *n*_{i} the number of moles of species *i*, *T* the temperature, and *p* the hydrostatic pressure. The chemical potential depends on the alloy composition. For ideal solutions, the chemical potentials are

where depend on *T* and *p* only. In this case, the gradient of the chemical potential is directly proportional to the logarithmic gradient of the concentration. In nonideal solutions, the gradient of the chemical potential gives rise to an “internal” driving force. As a consequence, the *interdiffusion coefficient* is concentration dependent, and Fick’s equation in the form of 5.10 must be used.

Examples of diffusion couples that entail an interdiffusion coefficient are (see Figure 5.7):

- Pure end‐member diffusion couples consisting of two samples of pure elements joined together (e.g. Ni/Pd, Cu/Ag, etc.).

- Incremental diffusion couples consisting of two samples of homogeneous alloys joined together (e.g. Ni
_{50}Pd_{50}/Ni_{70}Pd_{30}, Ni/Ni_{70}Pd_{30}, etc.).

- Diffusion couples that involve solutions of hydrogen in a metal (e.g. Pd‐–H/Pd, Ag
_{1−x}H_{x}/Ag_{1−Y}H_{Y}, etc.). Binary metal–hydrogen systems are often special in the sense that hydrogen is the only mobile component.

Interdiffusion results in a composition gradient in the diffusion zone. Interdiffusion profiles are analyzed by the Boltzmann–Matano method or related procedures, as described below. It allows to deduce the concentration dependence of the interdiffusion coefficient

from the experimental diffusion profile.

As already mentioned, the rate of chemical diffusion and thus the rate of defect mobility in the majority of transition metal sulfides are generally higher than in analogous oxides, but these differences do not exceed 1 order of magnitude (Figure 5.10). Some of the most important results concerning chemical diffusion in transition metal sulfides obtained by Grzesik and Mrowec (2006) are the data describing the mobility of predominant defects in niobium sulfide, being the main product of niobium sulfidation. As it is shown in Figure 5.8, the mobility of these defects is several orders of magnitude lower than that in all other metal sulfides and oxides. It should be noted that one of the most fascinating problems in the case of niobium sulfidation is its extremely high resistance to sulfur attack at high temperatures, in spite of very high concentration of point defects in the growing scale on this metal. The only hypothetical explanation of this unexpected phenomenon was based on the assumption that the mobility of defects in niobium sulfide scale is several orders of magnitude lower than in other transition metal sulfides (Gesmundo et al. 1992). Thus, in spite of a very high concentration of defects, the sulfide scale on niobium shows very good protective properties, comparable with those of the Cr_{2}O_{3} scale on chromium.

### 5.3.3 Intrinsic Diffusion Coefficients

The *intrinsic diffusion coefficients* (sometimes also component diffusion coefficients) *D*_{A} and *D*_{B} of a binary A–B alloy describe the diffusion of the components A and B in relation to the lattice planes. The diffusion rates of A and B atoms are usually not equal. Therefore, in an interdiffusion experiment, a net flux of atoms across any lattice plane exists. The shift of lattice planes with respect to a sample fixed axis is denoted as *Kirkendall effect* (see Section 5.7).

The Kirkendall shift can be observed by incorporating inert markers at the initial interface of a diffusion couple. This shift was observed for the first time for the Cu/Cu–Zn diffusion couples by Smigelskas and Kirkendall (1947). In the following decades, work on many different alloy systems and a variety of markers demonstrated that the Kirkendall effect is a widespread phenomenon of interdiffusion.

The intrinsic diffusion coefficients *D*_{A} and *D*_{B} of a substitutional binary A–B alloy are related to the interdiffusion coefficient and the marker velocity *v*_{K} (*Kirkendall velocity*). These relations were deduced for the first time by Darken (1948) and refined later on by Manning (1968). They will be discussed in Section 5.8. If the quantities and *v*_{K} are known from experiment, the intrinsic diffusion coefficients can be deduced.

We emphasize that the intrinsic diffusion coefficients and the tracer diffusion coefficients are different. *D*_{A} and *D*_{B} pertain to diffusion in a composition gradient, whereas and are determined in a homogeneous alloy. In a metal–hydrogen system usually only H atoms are mobile. Then the intrinsic diffusion coefficient and the chemical diffusion coefficient of hydrogen are identical.

## 5.4 Matano–Boltzmann Analysis

The Matano–Boltzmann analysis is used frequently by researchers to study diffusion in the solid state. By this method, one can measure interdiffusion coefficients, , at different compositions from the concentration profile measured by microprobe analysis. However, this method is restricted to systems where the partial molar volumes of the components are constant, i.e. the total volume does not change with reaction and mixing. Consider the case when two materials with initial compositions and are coupled and annealed for reasonably short time, *t*, such that after annealing, still some part of the end‐members is not affected by the diffusion process as shown in Figure 5.11a. Boundary conditions can be written as

where “−” and “+” represent the left‐ and right‐hand end of the reaction couple.

Boltzmann (1894) introduced the variable

which means that *C*_{B} is a function of *λ* only. The relation states that all compositions in a diffusion zone move parabolically in time with respect to one fixed frame of reference.

By using the definition of *λ*, and transforming *∂C*_{B}/*∂*_{t} and *∂C*_{B}/*∂*_{x} in Fick’s second law, using also Eq. 5.20, one can write

This treatment is known as the Boltzmann transformation, and this transformation was used for the first time by Matano (1933) to study interdiffusion in the solid state.

Initial conditions at time *t* = 0 can be written as, considering Eq. 5.20,

Equation 5.21 contains only total differentials and *∂λ* can be canceled from both sides. Integrating from initial composition to the concentration of interest to measure the interdiffusion coefficient,, at () leads to

The data is always measured at some fixed time so that *t* is constant. If we assume that after annealing the ends of the couple are not affected, then d*C*_{B}/d*x* = 0 at and . Using Eq. 5.20 one can write

and

Equation 5.25 defines the plane *x*_{M} = 0, and the initial contact plane between the end‐members is called *Matano plane*.

The Matano plane position, *x*_{M}, can be determined from the concentration–penetration curve of the system measured by X‐ray microanalysis by equalizing the areas *P* and *Q* as shown in Figure 5.11b. After integrating by parts, Eq. 5.24 can be modified, and, from Figure 5.11c, the interdiffusion coefficient can be expressed in terms of shaded areas as

The main disadvantage of this analysis is that one has to find the position of the Matano plane, *x*_{M}. When the total volume does not change with reaction/mixing, this is easy to determine. However, when the total volume changes, determining the initial contact plane (denoted by *x*_{0}) is rather confusing.

## 5.5 Kirkendall Effect

By the Matano–Boltzmann analysis, one can quantify the interdiffusion coefficient, , which is, in fact, a kind of average diffusivity of the elements, and it does not shed light on the diffusivities of the species, separately. In the early stage, it was common belief among researchers that diffusivities of the species are the same.

Hartley (1946) was the first to use purposely foreign inert particles, titanium dioxide, in an organic acetone/cellulose acetate system to study the inequality of the diffusing species. Shortly after that, Smigelskas and Kirkendall (1947) used the same technique to examine the inequality of diffusivities of the species in the Cu–Zn system by introducing molybdenum as an inert marker. Researchers dealing with metallic systems at that time were not familiar with Hartley’s work, and the effect of inequality of diffusivities on the inert marker was named the Kirkendall effect (Darken and Gurry 1953).

In the experiment by Smigelskas and Kirkendall, a rectangular bar (18 × 1.9 cm^{2}) of 70/30 wrought brass (70 wt%Cu/30 wt%Zn) was taken. This bar was ground and polished, and then 130 µm diameter molybdenum wires, which are inert to the system, were placed on opposite sides of the surfaces. Then a copper layer of 2500 µm was deposited on that, as shown in Figure 5.12. This couple was subjected to annealing at 785 °C. After annealing for a certain time, one small piece was cross‐sectioned to examine, and the rest of the part was further annealed. Following this method, it was possible to get specimens at different annealing times. With annealing, α‐brass grows in between, and after etching, the distance between the markers was measured. If the diffusivities of copper and zinc are the same and there is no change in volume during diffusion/reaction, marker should not move and stay at the original position.

However, after measuring, it was clear that with increasing annealing time, the distance between markers decreases parabolically with time. Considering the change in the lattice parameter, it was found that only 1/5 of the displacement occurred because of molar volume change. This shift was explained by Smigelskas and Kirkendall (1947), being possible to draw two conclusions of enormous impact at the time on solid‐state diffusion:

- The rate of diffusion of zinc is much greater than that of copper in α‐brass.

- When zinc diffuses more rapidly than copper in α‐brass, the interface shifts to compensate, at least partially, the diffusion rate.

Till then, direct exchange or ring mechanisms were accepted as diffusion mechanism in the solid state as shown in Figure 5.13a,b. If any of these mechanisms would be true, then diffusivities of the species should be the same. However, from Kirkendall’s experiment, it is evident that Zn diffuses faster than Cu, which results into the movement of the markers. When zinc diffuses away, all the sites are not occupied by the flow of Cu from opposite direction, and, because of that, vacant sites are left unoccupied. In other sense, there should be a flow of vacancies opposite to the faster diffusing species Zn to compensate for the difference between the Zn and Cu flux. Vacancies will flow toward the brass side, and excess Zn will diffuse toward the Cu side. Ultimately, this results into shrinking in the brass side and swelling in the copper side so that markers move to the brass side. In some diffusion reactions, pores can be found in the product phase. If there is not enough plastic relaxation during the process, vacancies will coalesce to form pores or voids in the reaction layer.

From this experiment, it was clear that diffusion occurs by a vacancy mechanism (Figure 5.13c), and after that the direct exchange and ring mechanisms were abandoned. At first, this work was highly criticized, but later this phenomenon was confirmed from experiments on many other systems (da Silva and Mehl 1951; Nakajima 1997; Sequeira and Amaral 2014). The impact of Kirkendall’s work, at that time, can be realized from Mehl’s comment on his work (Mehl 1947).

## 5.6 Darken Analysis

From Kirkendall’s experiment, it was clear that the diffusion process in solid solutions cannot be described by one diffusion coefficient; rather one has to determine the diffusivity of both species. This was treated mathematically by Darken (1948). Almost at the same time, Hartley and Crank (1949) studied the same subject, and they named the diffusivities of species as intrinsic diffusion coefficient. Seitz (1948) studied the solid‐state diffusion process more extensively.

Let us consider a binary diffusion couple of species A and B of the compositions and as shown in Figure 5.14. Before annealing fiducial (inert) markers are introduced at the initial bonding interface and annealed at elevated temperature so that interdiffusion takes place. When interdiffusion starts, the markers will be trapped at a certain fixed composition and cannot escape at a later stage so that they move along with that fixed composition. If the intrinsic diffusivity of B (*D*_{B}) is higher than the intrinsic diffusivity of A (*D*_{A}) at that marker plane (called Kirkendall plane), then the Kirkendall marker plane will move to the right‐hand side from the initial contact interface, *x*_{M/0}. The Matano plane, *x*_{M}, is the initial contact interface when there is no change in total volume and is fixed with respect to the ends of the diffusion couple. This initial contact plane is denoted by *x*_{0} when volume changes upon reaction/mixing.

The intrinsic molar flux at the Kirkendall plane can be expressed by Fick’s first law as