On the diffusional growth of compounds with narrow homogeneity range in multiphase binary systems

Acta Materialia 50 (2002) 525–535 www.elsevier.com/locate/actamat

On the diffusional growth of compounds with narrow homogeneity range in multiphase binary systems V. Buscaglia a

a,*

, U. Anselmi-Tamburini

b

Institute of Physical Chemistry of Materials, National Research Council, via De Marini 6, I-16149 Genoa, Italy b Dept. of Physical Chemistry and C.S.T.E./CNR, University of Pavia, v.le Taramelli 16, I-27100 Pavia, Italy Received 2 August 2001; received in revised form 28 September 2001; accepted 30 September 2001

Abstract A general analysis of the problem of diffusional growth of n compounds with narrow homogeneity range in a binary system is presented. Both constituents are assumed to be mobile. The end members of the diffusion couple can be any combination of pure elements (when terminal solubility is negligible), saturated terminal solid solutions and binary compounds. The kinetic equations follow from the coupling between chemical reactions at phase boundaries and partitioning of the diffusion flux between two adjacent layers. Different kinds of parabolic rate constant are used to describe the growth of compound layers in various conditions and the relationships between these quantities are established. In particular, the rate constant of the second kind for the exclusive growth of a given compound is a function of the n rate constants of the first kind measured on the complete couple where all the intermediate phases grow simultaneously. The rate constant of the second kind is related to the diffusion properties and the thermodynamic stability of the phase. The equivalence between the present approach and the purely diffusional model of Wagner is shown. Most of the models proposed in the literature can be obtained as special cases of the present analysis.  2002 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Theory & modeling (kinetics, transport, diffusion); Diffusion (bulk); Solid-state reactions; Growth kinetics

1. Introduction When two different materials are put in intimate contact at high temperature, interdiffusion occurs under the effect of the gradient of the chemical potential. In many systems, interdiffusion is accompanied by the formation of intermediate phases and the overall process is described as

* Corresponding author. Tel. +39-010-6475708; fax: +39010-6475700. E-mail address: [email protected] (V. Buscaglia).

reactive diffusion [1–4]. There are many technologically important processes where multiphase diffusional growth is involved. Well known examples include high-temperature corrosion of metals and alloys [5], realization of intermetallic protective coatings by diffusional annealing [5], modification of the surface properties of titanium alloys by nitriding or carbonitriding [6], welding and joining of dissimilar materials [7]. In general, reactive phase formation has a strong influence on fabrication, properties and eventually in the failure of materials and devices. Diffusional growth of a single binary phase from

1359-6454/02/$22.00  2002 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 6 4 5 4 ( 0 1 ) 0 0 3 6 7 - 6

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its constituting elements is described by Wagner scaling theory [5, 8] later extended to the formation of ternary oxides by Schmalzried [9]. The theory relates the growth rate of the intermediate compound to the diffusive and thermodynamic properties of the new phase under the hypothesis that local equilibrium is established at phase boundaries. Wagner’s theory has been verified for many systems and in particular for the oxidation of metals. Multilayer diffusional growth in binary and pseudobinary systems has been studied by many authors [9–30]. In any case, the main goal was to relate the observed growth rate to one or more fundamental properties (parabolic constant for the exclusive growth of the given phase, chemical diffusion coefficient, self-diffusion coefficient, etc.) of the phases which constitute the system. In addition to the hypothesis of local equilibrium, the different compound layers are usually assumed to grow simultaneously with parabolic kinetics. In fact, when formation of several intermediate phases is possible, some complications may arise [2, 31–34]. The number of equilibrium phases as shown by the phase diagram is not always observed on diffusion couples and this can be an indication of nucleation difficulty in the case of missing phases [34]. The different compounds can appear in sequential order rather than simultaneously. This is often observed in the case of thin films and can be related to the small thickness of the compound layer [31–34]. Usually the growth of each compound in a multiphase couple follows the parabolic law, as in the case of a single layer, meaning that interface processes are all faster than diffusion through the various layers. However, the growth of some phases can be non-parabolic [2, 33]. Hence, when analyzing data from multiple phase growth by means of a suitable model, a preliminary investigation of the given system is required to assess that the boundary conditions are well satisfied. Deviations from the “ideal” behaviour are hardly described by a general theory and usually require an “ad-hoc” treatment. The first general treatment of multilayer growth was given by Kidson [10]. The displacement velocity of the phase boundary was defined by the difference of the fluxes of the diffusing species arriving at and leaving the interface and the fluxes are

expressed as a function of the chemical diffusion coefficient (supposed independent of composition) and of the concentration range of the phases (Fick’s law). The approach of Kidson was later improved and successfully applied to different systems, as show for example in Refs. [19, 24, 28]. More recently, an extension of this method to diffusion couples with finite geometry has been adopted for the calculation of carbon and nitrogen chemical diffusion coefficients in carbides and nitrides of transition metals [29]. However, Kidson’s analysis can be applied only to intermediate phases with a relatively broad homogeneity range. When the deviation from stoichiomety is ⬍2–3 at.%, the concentration range through the layer can not be longer measured by conventional microanalytical techniques. The approach presented by Wagner [12] is based on the generalization to multiphase systems of the Boltzmann–Matano method for the calculation of the interdiffusion coefficient [1, 35]. The theory is given for a generic number n of intermediate compounds and can be applied to any system, irrespective of the homogeneity range of the phases, although the final equations take a simpler form when all intermediate compounds have a narrow homogeneity range and the solubility in the terminal phases is limited. The importance and completeness of Wagner’s analysis was sometimes underestimated by other investigators. The reason for this circumstance lies probably in the fact that Wagner made no use of the concept of chemical reaction at phase boundary. The introduction of chemical reactions in multiphase growth is a natural choice when line compounds are involved, because reactions immediately account for the movement of the interfaces and the resulting picture of the whole process is significantly simplified. Chemical reactions are not required in Wagner’s treatment because the theory stands on the general formulation of the continuity equation. Many papers were devoted to the study of multilayer growth using as basic concept the coupling of chemical reaction and flux partitioning at the phase boundaries [11, 13–18, 20–23, 30], in particular for what concerns the formation of multilayered scales during the oxidation of metals [13–15, 17, 18, 20, 23]. The rate constant for the exclusive growth of each

V. Buscaglia, U. Anselmi-Tamburini / Acta Materialia 50 (2002) 525–535

phase was related to the rate constants measured on the complete couple where all intermediate phases grow simultaneously or to the overall growth constant. These models have largely contributed to a better understanding of the multiphase growth process in binary systems. However, they lack of generality because they were developed to treat only the cases n = 2 and n = 3. More recently a general and elegant approach to the problem of multilayer growth was presented by Wang et al. [25]. Although the phenomenological picture of the process is different from the flux partitioning view, the conclusions are equivalent. The relatively large number of models published in the literature and the different formalisms adopted can generate some confusion, in particular if the reader is not strictly familiar with the problems of reactive diffusion. The possibility to have at one’s disposal some general relationships would greatly improve the situation. An additional complication is that different kinds of rate constants were adopted by different Authors to describe the exclusive growth of a compound and the relationships between the various constants are not always given. Furthermore, comparisons among different approaches were seldom reported. One significant exception is represented by the work of Williams et al. [21] where the equations proposed by Kidson [10], Wagner [12] and Shatynski et al. [16] were applied to describe multiphase diffusion in the AgZn system and the results critically compared. However the analysis is limited to the case n = 2. As far as the authors know, a general and critical comparison between Wagner’s analysis and models incorporating chemical reactions at the phase boundaries was not yet given. The objective of the investigation reported herein was to develop a general model incorporating chemical reactions and flux partitioning at the interfaces for the diffusional growth of n intermediate phases with narrow homogeneity range. The resulting equations are compared with previous models and the differences discussed. The comparison is carried out also by defining the relationships between different types of rate constants.

527

2. Theory 2.1. Statement of the problem Let us consider a binary diffusion couple consisting initially of pure A (phase 0) and pure B (phase n + 1) where n intermediate phases with a narrow homogeneity range grow simultaneously. For sake of simplicity, the stoichiometry of each compound is normalized to 1 mole of component B; therefore the ith phase is denoted as AniB. At a generic time t, the sequence of phases can be indicated as A/An1B/…/AniB/…/AnnB/B, with n1⬎n2⬎ …⬎nn. For pure component B is nn + 1 = 0 and for pure component A is n0 = ⬁. The system is assumed to satisfy to some general assumptions. The reaction takes place isothermally and the products grow as parallel, compact layers. Local thermodynamic equilibrium is established within the reaction layers as well as at the phase boundaries, where ideal contact is assumed. If growth is controlled by solid-state diffusion, the thickness, ⌬xi, of each intermediate phase will obey the parabolic growth rate ⌬x2i ⫽ 2kIit

(1)

I i

where k is the parabolic rate constant of the first kind [12] for the formation of phase i in a diffusion couple comprising all phases from 0 to n + 1. If exclusive formation of the intermediate phase i (i = 1,..,n) in a diffusion couple consisting initially of phases (i⫺1) and (i + 1) is considered, Eq. (1) will rewrite as ⌬x2i ⫽ 2kIIit II i

(2)

where k is the parabolic rate constant of the second kind [12] for formation of phase i. The rate constant kIIi can be related to the self-diffusion coefficients of the constituents in phase i and to the variation of the Gibbs’ free energy for the formation of phase i from phases i⫺1 and i + 1. This can be done by means of the Nernst-Einstein equation and the Wagner scaling theory [5, 8], as shown later. As noted by Wagner [12], the rate constant kiII is a fundamental property of the material and is independent of the number of phases in the diffusion couple. On the contrary, the rate constants kIi are determined by the properties

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of all phases involved, as also pointed out by Kidson [10]. Thus, the problem of multilayer growth can be reduced to the calculation of the constants kIIi from the constants kIi. Both kind of constants can be easily measured by means of suitable diffusion couples. 2.2. Treatment of multilayer growth Let us consider the system A/An1B/…/AniB/…/AnnB/B. For the sake of generality, both species A and B are assumed to be mobile without coupling between the diffusion fluxes. However, the introduction of coupling conditions does not affect the general result of the proposed analysis. In addition it is assumed that the diffusion fluxes in the terminal components of the couple are negligible. Diffusion in the terminal phases can be neglected when the terminal solubility of A in B and of B in A is limited or when the diffusion in the element is slower than in the reaction products, situations often encountered, for example, in the formation of transition metal silicides and other intermetallics. However, the aforementioned restrictive assumption about the nature of the end components will be later dropped out and more general boundary conditions allowed. The movement of each phase boundary implies formation and consumption of the adjacent phases and this can be related to interface reactions. For the generic i/i + 1 (i = 1,..,n⫺1) interface the following reactions can be considered (vi⫺vi ⫹ 1)A ⫹ Avi ⫹ 1B → AviB (Diffusion of component A) vi (vi⫺vi ⫹ 1) B ⫹ AviB → A B vi ⫹ 1 vi ⫹ 1 vi ⫹ 1 (Diffusion of component B)

(3a)

(3b)

Reactions at the first (A/An1B) and at the last (AnnB/B) phase boundary directly involve pure components A and B, respectively v1A ⫹ B → Av1B

(4)

vnA ⫹ B → AvnB

(5)

The phase boundary has a very important role in diffusional growth because it must be able to act

as either a source or a sink of vacancies or interstitials, as the point defects flux is generally discontinuous at the interfaces. A discussion on this subject is presented in Ref. [36]. In general, if JA,i and JA,i+1 are the fluxes (mol cm⫺2 s⫺1) of A through layers i and i + 1 respectively, it will be JA,i⬎JA,i + 1 and the difference contributes to the growth of phase i at the expenses of phase i + 1 by means of Reaction (3a). Similarly, JB,i + 1⬎JB,i for B diffusion, and the difference contributes to the growth of phase i + 1 at the expenses of phase i by means of Reaction (3b). The above inequalities holds if the growth of all the intermediate phases start simultaneously at a certain time, the system is left undisturbed and the end components are not yet completely consumed. The growth rate (cm s⫺1) of the ith layer can be obtained from the mass balance at phase boundaries i⫺1/i and i/i + 1 dni Vi d⌬xi (|J | ⫽ Vi ⫽ ⫺ dt dt vi⫺1⫺vi A,i⫺1 ⫹ vi⫺1|JB,i⫺1|) Vi(vi⫺1⫺vi ⫹ 1) (|J | ⫹ vi|JB,i|) ⫹ (vi⫺1⫺vi)(vi⫺vi ⫹ 1) A,i Vi ⫺ (|J | ⫹ vi ⫹ 1|JB,i ⫹ 1|) vi⫺vi ⫹ 1 A,i ⫹ 1

(6)

where Vi is the molar volume (supposed constant) of AniB and the module, |J|, of the flux terms has been introduced to account for the diffusion of A and B in opposite directions. For the first intermediate layer (An1B) only the second and third term of the right hand side of Eq. (6) contribute to the growth. Likewise, for the last intermediate layer (AnnB) only the first and second term of the right hand side of Eq. (6) contribute to the growth. This is because diffusion in the terminal phases (A and B) has been assumed to be negligible. To find a relationship between the rate constants of the first kind and the rate constants of the second kind the invariance of the diffusion flux is assumed. The flux through any layer will be same, irrespective of the system used to define the rate constant, if variables such as layer thickness, temperature and activity of the components at the interfaces are the same. For the growth of phase i in a diffusion couple initially consisting of phases i⫺1 and i + 1, the

V. Buscaglia, U. Anselmi-Tamburini / Acta Materialia 50 (2002) 525–535

decomposition of the parent phases must be considered, as the diffusing species are not directly available. For what concerns A diffusion, decomposition of compound Ani⫺1B at interface Ani⫺1B/AniB provides A atoms Avi⫺1B → (vi⫺1⫺vi)A ⫹ AviB

(7)

and the formation of new AniB phase occurs at interface AniB/Ani + 1B by means of Reaction (3a). Likewise, in the case of B diffusion, decomposition of Ani + 1B at interface AniB/Ani + 1B provides B atoms vi

Avi ⫹ 1B →

vi ⫹ 1

vi⫺vi ⫹ 1 B ⫹ AviB vi ⫹ 1

vi⫺1 vi⫺1⫺vi B ⫹ Avi⫺1B → A B vi vi vi

(9)

Reaction (9) is equivalent to Reaction (3b). The growth rate of AniB can be written as d⌬xi ⫽ V˜ i(|JA,i| ⫹ vi|JB,i|) dt

(10)

where V˜ i is the reaction volume, i.e. the volume of phase AniB formed for each mole of A transported in a Ani⫺1B/AniB/Ani + 1B couple. The factor ni which multiplies the term JB,i takes into account that the number of moles of AniB formed for each mole of B is ni times greater than the number of moles formed for each mole of A. Differentiation of Eq. (2) and insertion into Eq. (10), gives (11)

Substitution of the differential form of Eq. (1) and of Eq. (11) into Eq. (6) for each intermediate phase (i = 1,..,n) leads to the system of linear equations V1 V1 kII1 kII2 ⫺ n1⫺n2 V˜ 1C1 n1⫺n2 V˜ 2C2

Ci ⫽ ⫺

i⫽1

Vi Vi(ni⫺ni+1) Vi kIIi⫺1 kIIi+1 kIIi ⫹ ⫺ i ⫽ 2,…,n⫺1 vi⫺1⫺ni V˜ i⫺1Ci⫺1 (vi⫺1⫺vi)(vi⫺vi+1) V˜ iCi ni⫺ni+1 V˜ i+1Ci+1

Vn Vnnn⫺1 kIIn⫺1 kIIn Cn ⫽ ⫺ ⫹ nn⫺1⫺nn V˜ n⫺1Cn⫺1 nn(nn⫺1⫺nn) V˜ nCn

i⫽n

(12) where

(13) II i

The rate constants k can be easily obtained by inversion of the matrix of the coefficients of the system Eqs. (12), and results in

冘 冋冘 冘 册 冘 n

k ⫽ V˜ 1C21 II 1

nj Cj V C1 j⫽1 j

(14a)

i⫺1

kIIi ⫽ V˜ iC2i ni

1 Cj ni ⫹ V Ci Vi j⫽1 j

(14b)

n

⫹

nj Cj i ⫽ 2,…,n⫺1 V Ci j⫽i⫹1 j n

kIIn ⫽ nnV˜ nC2n

1 Cj V Cn j⫽1 j

(14c)

The expression of the reaction volume V˜ i can be obtained by considering the reactions which occur at both interfaces of the Ani⫺1B/AniB/Ani + 1B couple (see Eqs. (3a), (7)–(9)) V˜ i ⫽

ni⫺1⫺ni ⫹ 1 V (ni⫺1⫺ni)(ni⫺ni ⫹ 1) i

(15)

This relationship holds also for the cases i = 1 and i = n, taking into account that n0 = ⬁ (pure A) and nn + 1 = 0 (pure B). Upon substitution of Eq. (15) into Eqs. (14) one obtains kIIi ⫽

(16)

冋冘

ni Cj (ni⫺1⫺ni ⫹ 1)Vi ⫹ C2i ni (ni⫺1⫺ni)(ni⫺ni ⫹ 1) VC Vi j⫽1 j i i⫺1

冘 册 n

kIIi |JA,i| ⫹ ni|JB,i| ⫽ ˜ Vi⌬xi

C1 ⫽

Ci ⫽ √kIi

(8)

for formation of AniB at the Ani⫺1B/AniB interface

529

⫹

njCj i ⫽ 1,…,n VC j⫽i⫹1 j i

Thus, the rate constants kIIi can be calculated from the rate constants kIi if the stoichiometry of the growing compounds and their molar volume are known. Eq. (16) has been obtained assuming that the reaction occurs between the two elements A and B, with negligible terminal solubility. However, if the diffusion couple initially consists of the saturated terminal solid solutions or of two binary compounds, herein both denoted as An0B and Ann + 1B, interfacial reactions different from Eqs. (4) and (5)

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V. Buscaglia, U. Anselmi-Tamburini / Acta Materialia 50 (2002) 525–535

have to be considered. At phase boundary An0B / An1B, the reactions are Av0B → (n0⫺n1)A ⫹ Av1B

(17a)

n0 (n0⫺n1) B ⫹ Av0B → An1B n1 n1

(17b)

while, at phase boundary AnnB/Ann + 1B, one has (nn⫺nn ⫹ 1)A ⫹ Ann ⫹ 1B → AvnB nn

Ann ⫹ 1B →

nn ⫹ 1 ⫹ AvnB

(nn⫺nn ⫹ 1) B nn ⫹ 1

(18a) (18b)

aII B

The different nature of the end components of the couple reflects in the relationship between kIIi and kIi. Following the former approach, one obtains kIIi ⫽

冋

(ni⫺1⫺ni ⫹ 1)Vi C2i × (ni⫺1⫺ni)(ni⫺ni ⫹ 1) (n0⫺nn ⫹ 1)

冘

i⫺1

(ni⫺nn ⫹ 1)

i ⫽ 1,→,n

j⫽1

rate of phase i from phase i⫺1 and element B at an activity corresponding to the equilibrium between phases i and i + 1. For historical reasons, the use of a different kind of rate constant, called Tammann parabolic rate constant [8] or practical rate constant [9], is often preferred. The practical rate constant, ki, can be derived from the general expression of the diffusion flux in a chemical potential gradient (i.e the Nernst–Einstein equation) and directly reflects the diffusion properties of phase AniB. Following the approach of Refs. [5, 9] one obtains

冘

Cj (n0⫺ni)(ni⫺nn ⫹ 1) Cj (n0⫺nj) ⫹ ⫹ (n0⫺ni) (nj⫺nn ⫹ 1) VjCi Vi VjCi j⫽i⫹1 n

册

(19)

Eq. (19) is completely general and describes all possible situations where pure elements, saturated terminal solid solutions and binary compounds can be differently combined as end components of the diffusion couple. Eq. (16) can be obtained from Eq. (19) in the limiting case n0 = ⬁ and nn ⫹ 1 ⫽ 0. 2.3. Relationships between the rate constant of the second kind, other parabolic rate constants and diffusion properties It is useful to establish the relationship between the rate constant of the second kind and other kinds of rate constants for the growth of a compound for two reasons: (i) to compare the results of calculations performed by Eq. (16) or Eq. (19) with data available in the literature; (ii) to compare the present model with existing models. In particular, the comparison among different models can be no straightforward because different definitions of the rate constant for the exclusive growth of a given intermediate phase were used by different Authors. For example, in the case of solid–gas reactions, Wagner [12] suggested the use of the parabolic rate constant of the third kind, kIII i , giving the growth

冕冉

ki ⫽

冊

DA,i ⫹ DB,i dlnaB ni

I

(20)

aB

where DA,i and DB,i are the self-diffusion coefficients of the components, while aIB and aIIB are the activity of component B at the Ani⫺1B/AniB and AniB/Ani + 1B interfaces, respectively. Eq. (20) is obtained by implicitly assuming that formation of the compound occurs from the constituting elements (see, for example, reactions (4) and (5)). On the contrary, reactions (3a) and (7)–(9) involve the participation of the adjacent binary compounds and, consequently the rate constant of the second kind is different from the practical rate constant. The difference between the practical rate constant and other rate constants for the exclusive growth of a binary compounds was sometimes misunderstood in the literature [14, 17]. If dni∗ is the number of moles of formula units of phase AniB formed in the time dt at the interfaces by reactions analogous to Eqs. (4) or (5), the growth rate described by ki will be defined as Vi

冉

冊

ki dni∗ |JA,i| ⫹ |JB,i| ⫽ ⫽ Vi dt ni ⌬xi

(21)

Making use again of the concept of flux invariance, comparison between Eq. (21) and Eq. (11) and insertion of Eq. (15) yields ni(ni⫺1⫺ni ⫹ 1) k (ni⫺1⫺ni)(ni⫺ni ⫹ 1) i i ⫽ 1,→,n

kIIi ⫽

(22)

This equation defines the general relationships

V. Buscaglia, U. Anselmi-Tamburini / Acta Materialia 50 (2002) 525–535

between the rate constant of the second kind and the corresponding practical rate constant. Following the above procedure, one can obtain the relationship between the rate constant of the third kind, kIII i , and the practical rate constant kIII i ⫽

ni⫺1 k ni⫺1⫺ni i

(23)

Comparing Eq. (22) and Eq. (23) immediately gives the relationship between the rate constant of the third kind and the rate constant of the second kind kIII i ⫽

ni⫺1(ni⫺ni ⫹ 1) II k ni(ni⫺1⫺ni ⫹ 1) i

(24)

It is worth noting that, when i = 1, it is kiIII = ki, II and, when i = n, it is kIII i = ki . Eq. (24) is equivalent to the definition of the rate constant of the third kind given by Wagner (Eq. (30) in Ref. [12]) in terms of mol fractions. In spite of the generality of Eq. (20), for practical use it is convenient to perform the integration. Even if the self-diffusion coefficient can be related to point defect concentration and defect mobility, these quantities are known only for a limited number of systems. In general, it is more useful to define Eq. (20) in terms of the average value of ¯ B,i, ¯ A,i and D the self-diffusion coefficients, D through each layer. Introducing the definition of chemical potential, mB = m0B + RTlnaB and using the mean-value theorem for integrals, Eq. (20) can be rewritten as mII B

ki ⫽

1 RT

冕冉 冊

冊

冉

¯ A,i DA,i D ⫹ DB,i dmB ⫽ ni ni

I

¯ B,i ⫹D

mIIB⫺mIB RT

and, after some rearrangements, the expression of the practical rate constant is

冊

final

冉

¯ A,i ni(ni⫺1⫺ni ⫹ 1) D (ni⫺1⫺ni)(ni⫺ni ⫹ 1) ni |⌬G0i | ¯ B,i ⫹D RT

ki ⫽

ni(ni⫺1⫺ni ⫹ 1) |⌬G0i | ⫽ Deff,i (ni⫺1⫺ni)(ni⫺ni ⫹ 1) RT i ⫽ 1,%,n where Deff,i is an effective diffusion coefficient and ⌬G0i is the Gibbs free energy variation for the formation of one mole of phase AniB from the adjacent phases. Namely, the reactions to take into account are A / Av1B / Av2B couple (n1⫺n2)A ⫹ An2B → An1B i ⫽ 1 Ani⫺1B / AniB / Ani+1B couple ni⫺1⫺ni ni⫺ni A B⫹ A ni⫺1⫺ni+1 vi⫺1 ni⫺1⫺ni+1 vi+1 B → AniB i ⫽ 2,…,n⫺1 Ann⫺1B / AnnB / B couple nn⫺1⫺nn nn Avn⫺1B ⫹ B → AnnB nn⫺1 nn⫺1

(25)

i⫽n

Finally, upon substitution of Eq. (25) into Eq. (22), one can directly relate the rate constant of the second kind to the diffusion properties of the compound ¯ A,i D ¯ B,i ⫹D ni n2i (ni⫺1⫺ni ⫹ 1)2 |⌬G0i | ⫽ kIIi 2 2 (ni⫺1⫺ni) (ni⫺ni ⫹ 1) RT

Deff,i ⫽

冋

册

(26)

⫺1

It is worth noting that the effective diffusion coefficient is closely related to the Darken inter˜ i. diffusion coefficient D ˜i ⫽ D

mB

531

1 ni ni ¯ A,i ⫹ ¯ B,i ⫽ D D D 1 ⫹ ni 1 ⫹ ni 1 ⫹ ni eff,i

When the mobility of the two species is very different, Eq. (26) yields the average value of the selfdiffusion coefficient of the fastest diffusing atom. For example, the expressions reported by van Loo et al. [37] for the growth of nickel silicides can be easily reproduced. Expressions similar to Eq. (25) were also given by Barge et al. [38] and applied to the Co-Si system to obtain the effective diffusion coefficient in cobalt silicides. The “effective diffusivity”, a, defined in Ref. [38] should be equivalent to the practical rate constant.

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3. Discussion Eqs. (16) and (19) represent general relationships for the interpretation of the results of reactive diffusion experiments. The adopted formalism is quite simple and only the stoichiometry of the compounds, the molar volumes and the rate constants of the first kind are required for calculation. The rate constants of the first kind can be readily measured on diffusion couples where pure elements (in the case of negligible terminal solubility), saturated terminal solid solutions and binary compounds can be differently combined as end components. In this way, the rate constant of the second kind for n intermediate phases can be determined by means of a single diffusion experiment. However, one has to be reasonably sure that the boundary conditions of the model are satisfied by the given system. In particular, equilibrium conditions should be attained or closely approached at the phase boundaries. This does not always occur when the reaction is carried out at relatively low temperatures (i.e. in the case of thin films), while equilibrium conditions should be more easily established at high temperatures and long reaction times. Also the presence of impurities can unpredictably affect the layer sequence and phase composition. Other useful quantities (Eqs. (22)–(26)) can be obtained from the rate constant of the second kind. In particular, the average value of the self-diffusion coefficient of the rate-determining species can be calculated if the Gibbs free energy of formation of the compound from the adjacent phases is known (Eqs. (25) and (26)). Relationships (16) and (19) can be quite safely applied even when the homogeneity range of the intermediate phases is relatively broad provided that average values are adopted for variables such as stoichiometric coefficients and molar volumes. In particular, the error should be small if the concentration profile through each phase is close to linearity. In addition, the experimental uncertainty can be often larger than the errors arising from the approximations of the model. On the contrary, the assumption of limited solubility in the terminal phases A and B should be more strictly verified or, alternatively, the terminal phases should be preventively saturated. This is because the above analysis does

not account for diffusion in the end components. In general, if diffusion in the terminal phases is ignored, the rate constants of the second kind resulting from Eqs. (16) and (19) will be underestimated. Diffusion in the terminal phases can be treated with the same formalism previously developed, but the final relationship is not particularly useful because the knowledge of the chemical diffusion coefficient in the terminal phases is required (for a discussion on this subject see, for example, Refs. [19] and [22]). The situation can be even more complex if the diffusion process is influenced by the finite geometry of the samples because the parabolic growth constants of the first kind depend on the saturation level of the terminal phases [19, 29]. In such a case, only numerical solutions are possible [29]. When n = 1, from Eq. (16) it is kII1 = C21 = k1I , as expected. As a special case of Eq. (16), it is worth considering the situation CiCj, corresponding to the growth of one intermediate layer much thicker than the others. In such a case, the limiting expression of the rate constant kIIi is limkIIi(CiCj) ni(ni⫺1⫺ni ⫹ 1) ⫽ C2i (ni⫺1⫺vi)(ni⫺ni ⫹ 1)

(27)

The resulting value is greater than C2i and this is the difference in comparison to the case n = 1. The ratio of the flux in layer j to the flux in layer i can be obtained by combining Eqs. (11) and (14) and takes a simple form when the formation of compound i is predominant nj |JA,j| ⫹ nj|JB,j| (C Cj) ⫽ for i⬍j and lim |JA,i| ⫹ ni|JB,i| i ni ⫽ 1 for i⬎j Even if the growth rate of all but one layers approaches zero (CjCi), the flux through each of them remains comparable to that in the thicker layer. This is a consequence of the coupling between diffusion fluxes and reactions at phase boundaries; the net growth rate of a layer is related to the difference between formation and consumption rate at the interfaces. As a general consequence of Eqs. (16) and (22)–(24), it can easily be inferred that I kIIiⱖkIII i ⱖkiⱖki

V. Buscaglia, U. Anselmi-Tamburini / Acta Materialia 50 (2002) 525–535

The equalities hold when the formation of a single compound layer (n = 1) occurs from the pure elements. Since the above rate constants define the growth rate in different types of diffusion couples, it is immediately clear that, for a given temperature and reaction time, the thickness of a compound layer will be minimum when all the intermediate compounds grow simultaneously from the elements (kIi) and maximum when the phase grows in a diffusion couple initially consisting of its adjacent phases (kIIi). This observation indicates that a compound missed or hardly detectable in a complete couple can be more easily observed with an appropriate experimental set-up. The assumption of flux invariance adopted in the present analysis of phase growth is of general validity, irrespective of the predominant diffusion mechanism, either lattice or grain boundary diffusion. However, in the intermediate situations, when both diffusion mechanisms contribute to growth, the assumption of flux invariance also implies comparable microstructures for the same compound in different conditions and attention must be paid in the interpretation of the results. For example, formation of a given compound in various types of diffusion couples can result in the development of different microstructures because of different nucleation conditions. As a first approximation, the effective diffusion coefficient in a polycrystalline layer can be expressed as the sum of the contributes of lattice diffusion (subscript “l”) and grain boundary diffusion (subscript “gb”) [31] Deff ⫽ Dl ⫹

冉

冉

冊

2d DA D ⫽ ⫹ DB a gb n

冊

2d DA ⫹ ⫹ DB a n

l

gb

where a is the grain size and d is the grain boundary width. Consequently, the contribution of grain boundary diffusion to the overall transport process can be different according to different grain sizes. The knowledge of lattice and grain boundary diffusion coefficients is thus required for a full understanding of reactive diffusion experiments, which provide only information on Deff. Reliable information on the relative mobility of the components can be obtained in Ani⫺1B/AniB/Ani + 1B couples

533

where suitable markers are used to define the location of the Kirkendall plane. For some systems, different microstructures develop at either side of the Kirkendall plane and its location can be easily defined without using markers [3, 39]. In the case of non isotropic crystals, the diffusion coefficient depends on the crystallographic direction. Therefore, the effective diffusion coefficient given by Eq. (26) for a polycrystalline layer represents an average value. Often, a strong texture is observed on layers obtained by reactive diffusion because the single grains have the tendency to growth with a well-defined orientation. A detailed knowledge of the microstructure is thus required for a more complete interpretation of the experimental results. The treatment of multilayer growth given in the present analysis in terms of phase boundary reactions and including the partitioning of diffusing species in two adjacent layers is strictly equivalent to the classic diffusional approach. The proof of this equivalence is given in the Appendix A. For what concerns many other models available in the literature, most of them can be obtained as special cases of Eqs. (16) and (19). The model of Wang et al. [25–27], deserves a particular mention, for his generality. They adopted an approach different from that of flux partitioning and the exclusive growth of each intermediate phase, in the case of diffusion-controlled reactions, was defined in terms of the practical rate constant. Nevertheless, the final result is equivalent to the present analysis; if Eq. (20) in Ref. [25] is multiplied by the ratio kIIi/ki given by Eq. (22), an expression of kIIi identical to Eq. (16) is immediately obtained. It is worth noting that Wang et al. [25] have denoted as “apparent rate constant” the rate constant of the first kind and as “intrinsic rate constant” the practical rate constant. The more general case corresponding to Eq. (19) was not treated in Ref. [25]. The results of Yurek et al. [15] and of Hsu [23] for the oxidation of metals forming two and three oxide layers, respectively, can be reproduced in the same way. The equations reported by Viani and Gesmundo [20] for n = 2 and n = 3 can be obtained by means of the definition of the rate constant of the third kind (Eq. (23)) and taking into account that, in Ref. [20], the rate constants were defined in terms of oxygen mass contained in the

534

V. Buscaglia, U. Anselmi-Tamburini / Acta Materialia 50 (2002) 525–535

oxide layers. Last but not least, Eqs. (63) and (64) and (65)–(67) in Ref. [22] can be obtained from Eq. (19) putting n = 2 and n = 3, respectively. 4. Summary and conclusions The diffusional growth of n intermediate compounds with narrow homogeneity range in a multiphase A/B binary system as presented herein is described in terms of chemical reactions and partitioning of the diffusion flux at phase boundaries. Both atomic species, A and B, are assumed to be mobile. For each compound, the growth constant of the second kind is related to the growth constants of the first kind of all the intermediate phases which are being formed. The end components of the diffusion couple can be any combination of pure elements (when terminal solubility is negligible), saturated solid solutions and binary compounds. Only the thickness, the stoichiometry (expressed as AniB) and the molar volume (referred to 1 mole of component B in the formula unit) of the different phases are required in the calculation, provided that all the compounds grow simultaneously with parabolic law and local equilibrium is established at phase boundaries. The rate constant of the second kind was related to the diffusion and thermodynamic properties of the given phase as well as to other kinds of rate constants. The average value of the self-diffusion coefficient of the most mobile constituent can be obtained if the Gibbs free energy of formation of the compound from the adjacent phases is known. Even if the components have comparable mobility or the relative mobility is unknown, the treatment can be applied for the calculation of an effective diffusion coefficient. In any case, fundamental transport properties of the n intermediate phases can be obtained from a single diffusion experiment. The relationships among the various rate constants also point out that the growth rate of a given compound can be maximized in a diffusion couple initially consisting of the adjacent phases. The above analysis is based on the concept of flux invariance: the diffusion flux through a given intermediate phase is the same in two different (in the sense that they were assembled from different end phases) dif-

fusion couples provided that variables such as layer thickness, temperature and activity of the components at the phase boundaries are the same. This assumption holds even when grain boundary diffusion gives a significant contribution to the growth process, provided that the microstructure developed in different conditions is comparable. However, reactive diffusion experiments do not provide any information on the relative importance of lattice and grain boundary diffusion. The present approach was compared to the purely diffusional treatment of multilayered growth given by Wagner. It is shown that the two models are strictly equivalent in the case of compounds with narrow homogeneity range. Most of the models available in the literature can be obtained as special cases of the present analysis. Appendix A The reaction volume (Eq. (15)) can be written as N2i (Ni ⫹ 1⫺Ni⫺1) V (Ni⫺Ni⫺1)(Ni ⫹ 1⫺Ni) i i ⫽ 1,…n

V˜ i ⫽

(A1)

where Ni is the molar fraction of component B Ni ⫽

1 1 ⫹ ni

(A2)

If the terminal phases are the pure elements, it will be N0 = 0 and Nn + 1 = 1. In addition, it is essential to recognise that the molar volumes, ni, introduced by Wagner [12] refer to one mole of atoms, while the molar volumes Vi refer to 1 mole of component B in the formula unit. Consequently it is ni = NiVi

(A3)

By substitution of Eqs. (A1–A3), Eq. (16) rewrites as

冋

Ni+1⫺Ni⫺1 (1 (Ni⫺Ni⫺1)(Ni+1⫺Ni) i⫺1 ni Cj ⫺Ni) Nj ⫹ Ni(1⫺Ni) n Ci j⫽1 j kIIi ⫽ C2i

冘 冘 n

⫹ Ni

Cj ni (1⫺Nj) n C i j⫽i⫹1 j

册

(A4)

V. Buscaglia, U. Anselmi-Tamburini / Acta Materialia 50 (2002) 525–535

From Eq. (13) and the definition (Eq. (1)) of kIi, it is Cj ⫽

⌬xj √2t

and substitution into Eq. (A4) gives an expression identical to Eq. (24) of Wagner’s original paper [12].

冋

Ni+1⫺Ni⫺1 × (1 (Ni⫺Ni⫺1)(Ni+1⫺Ni) i⫺1 ⌬x2i ni ⌬xi⌬xj ⫺Ni) Nj ⫹ Ni(1⫺Ni) n 2t 2t j⫽1 j

kIIi ⫽

冘 冘

(A5)

册

⌬xi⌬xj ni (1⫺Nj) n 2t j⫽i⫹1 j n

⫹ Ni

The only difference is that the intermediate phases are enumerated from 2 to n⫺1 in Ref. [12] rather than from 1 to n. It is worth noting that Eqs. (20) and (24) of Ref. [12] contains some mistakes of probable typographic nature; however these mistakes can be easily identified by following Wagner’s analysis. The equations were later correctly reported by other authors [9, 39]. Following the same procedure as above, the equivalence between Eq. (19) and the classic diffusional approach can be established.

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