Ion exchange at a metal–ceramic interface

Acta Materialia 50 (2002) 1165–1176 www.actamat-journals.com

Ion exchange at a metal–ceramic interface R. Raj a,*, A. Saha a, Linan An b, D.P.H. Hasselman c, P. Ernst d b

a Department of Mechanical Engineering, University of Colorado, Boulder, CO 80309-0427, USA Advanced Materials Processing and Analysis Center, Department of Mechanical, Materials, Aerospace Engineering, University of Central Florida, Orlando, FL 32726, USA c Department of Materials Science and Engineering, Virginia Polytechnic Institute, Blacksburg, VA 24061, USA d Department of Materials Science and Engineering, Case Western Reserve University, 10900 Euclid Ave, Cleveland, OH 44106-7204, USA

Received 11 May 2001; received in revised form 13 November 2001; accepted 13 November 2001

Abstract We present the results for an interfacial reaction between magnesium aluminate spinel and aluminum metal at 640 °C which, for the first time, show the occurence of ion exchange at a metal–ceramic interface. The exchange reaction occurs by the diffusion of magnesium and aluminum ions across the interface in opposite directions. The enrichment of the spinel in aluminum cations, and the metal in magnesium is confirmed by the appearance of phases of aluminum oxide and magnesium–aluminum intermetallics in the X-ray diffraction spectra. Electron spectroscopic images of spatial elemental distribution provide further evidence for the reaction. A drop in the thermal and electronic conductivities are explained by enhanced scattering of the electrons by solid solution Mg in the aluminum metal. The experiments were carried out on composites consisting of an aluminum matrix with a fine dispersion of spinel particles, whose size was varied while the volume fraction was held constant. Measurable ion exchange reaction was seen only in the composites containing the smaller (one micrometer size) particles. The influence of surface to volume ratio of the particles on the reaction rate confirms that the reaction was interface controlled. The possibility of a nucleation and growth process for the conversion of spinel into alumina and solid solution magnesium was ruled out by quantitative thermodynamic analysis.  2002 Published by Elsevier Science Ltd on behalf of Acta Materialia Inc. Keywords: Metal–ceramic interfaces; Ion exchange; Aluminum; Spinel

1. Introduction Ion exchange is used to induce a compressive stress in the near surface regions of silicate glasses. In the early studies sodium ions in soda lime glass

* Corresponding author. Tel.: +1-303-492-1029; fax: +1303-492-3498. E-mail address: [email protected] (R. Raj).

were exchanged with larger ions of potassium by immersing the glass in a salt of potassium at elevated temperatures [1,2]. The compressive stress created in the surface layer increases the effective fracture strength of the glass which has made the process technologically important. Recently, ion exchange has also been employed to produce surface layers of graded refractive index for optical waveguides [3]. The principal driving force for the ion-exchange process is provided by entropy of mixing.

1359-6454/02/$22.00  2002 Published by Elsevier Science Ltd on behalf of Acta Materialia Inc. PII: S 1 3 5 9 - 6 4 5 4 ( 0 1 ) 0 0 4 1 8 - 9

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In this paper we show that an ion-exchange reaction can also take place at a metal–ceramic interface, with the potential for technological significance in metal–ceramic bonding. The ceramic was magnesium aluminum spinel and the metal nominally pure aluminum. This exchange reaction may be represented as follows (1) MgO.1.25Al2O3 ⫹ xAl→Mg1⫺xO.1.25Al2+x/1.25O3 ⫹ xMgSS The spinel contains two cations, Mg2+ and Al3+. In the reaction x moles of aluminum are exchanged with the same number of Mg ions in the ceramic. As a result the ceramic becomes enriched in Al3+ cations, while the Mg atoms form a solid solution with the aluminum metal. The reaction occurs at 625–640 °C, just below the melting point of aluminum. This temperature is apparently high enough for the mobility of the cations in the spinel [4]. The structure and the energetics of the spinels [5] permit them to tolerate a wide range of cation compositions. The ideal spinel structure consists of cubic closed packed arrays of oxygen ions with divalent and trivalent cations distributed in the tetrahedral (A) and octahedral (B) sites, respectively. However, the cations may exchange their positions by the following reaction [6]: MgA ⫹ AlB ⫽ Mg⬘B ⫹ Al•A

(2)

The terms on the left hand side denote the cation locations in the ideal crystal. The inverse cation locations are given on the right: the Mg divalent ion present in the octahedral site embodies an effective negative charge of one unit and the Al trivalent ion in the tetrahedral site carries a positive charge of equal magnitude. The enthalpy for the reaction in Eq. (2) becomes comparable to the entropy of mixing at approximately 625 °C [7,8]. In summary, these studies [4–7] suggest that energetics as well as the mobility of cation exchange reaction are feasible at moderate temperatures. We describe the experiments and the results from this study in chronological order. The initial objective of the work was to study the influence of spinel particle size on the thermal diffusivity of the aluminum composites; the expectation being that the thermal diffusivity would decrease with smaller particles because of the contribution from

the thermal resistance of the interfaces. The results however showed an anomalously large drop in the thermal conductivity which could not be explained by the continuum models. Since the system was metallic we sought to confirm if electron scattering in the metal was responsible for the drop in thermal conductivity, by measuring the electrical conductivity. The agreement between electrical and thermal conductivities suggested the scattering of electrons by magnesium atoms leached into the aluminum metal, a result which was confirmed by electron spectroscopic imaging of the elements. The ion-exchange reaction between spinel and aluminum was finally confirmed by X-ray diffraction studies. The rate of the ion-exchange reaction was found to depend greatly on the size of the spinel particles. The small particles, with their higher surface to volume ratio showed significant progress in the reaction while large particles did not. These results are consistent with the interfacial nature of this reaction.

2. Experimental methods and results This section is separated into two parts. Section 2.1 gives a description of the procedure used to prepare the samples containing a dispersion of spinel particles of different size. Section 2.2 contains descriptions of experiments used to characterize the microstructure of the composites as well as their electrical and thermal conductivities: it is divided into three sub-sections: Section 2.2.1 describes the conductivity experiments, Section 2.2.2 the electron-spectroscopic-imaging (ESI) experiments, and Section 2.2.3, the X-ray diffraction experiments. Each sub-section contains a description of the procedure followed by presentation of the results. 2.1. Material preparation The objective of the experiments was to vary the surface to volume ratio of the dispersed ceramic particles. Therefore composites with different particle size of spinel were prepared. The volume frac-

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tion of the particles was held constant at 10 vol%. The particle size ranges are listed in Table 1. The dispersions were prepared by the procedure described in Gustafson et al. [9]. The source materials were 99.5% pure aluminum powder, and spinel powder of composition (MgO).(1.25Al2O3). The average particle size of aluminum was 20 µm. The spinel powder had a very wide distribution with an average size of 45 µm; it was classified by sedimentation into the five size groups, as given in Table 1. The first step in the milling process was to mix the powders in acetone with a spin bar and ultrasonic dismembrator. Next, samples C and D were dried and then vibration-milled for 2 h with zirconia balls in a polyethylene bag. The powder was then dried in a rotating drier under vacuum at 40 °C until acetone was completely evaporated. The powder was stored under vacuum in a desiccator. Samples A and A2 were attrition milled for 6 h and dried. The mixed powders were cold pressed in a graphite die, 12.5 mm in diameter, using a graphite foil as a liner. These powder packs were hot pressed at 625 °C at 20 MPa in a tungsten furnace under slight argon overpressure. Densification was measured by the displacement of the pressing piston, and was completed in approximately 30 min. The ideal density from the rule of mixtures for 10 vol% spinel aluminum is 2.82 g/cm3, the density of spinel being 3.58 g/cm3 and that of aluminum 2.71 g/cm3 at room temperature. The relative density of the hot-pressed composites ranged from 95 to 99.6% of the theoretical. Disk shaped specimens

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were cut from the hot pressed billet for property measurements. The specimens were electropolished and optical photographs were used to measure the particle size. Examples of pictures from fine particle and coarse particle specimens are given in Fig. 1. For reference, samples of ‘pure aluminum’ were prepared by the powder route, using vibratory milling and hot pressing. They were used as a benchmark for understanding the electrical and thermal conductivities of the composite specimens. 2.2. Structure and properties 2.2.1. Thermal and electrical conductivity Specimens for thermal diffusivity and conductivity measurements were prepared by cutting disks, 1.00±0.05 mm thick, from the hot-pressed billet. They were polished on one side to a finish of 1 mm. The thermal diffusivity of the sample was measured from ambient temperature up to ⬇600 °C by the flash diffusivity technique [10,11]. A glass-Nd laser and a InSb detector were used. The detector was cooled with liquid nitrogen to enable measurement of the transient temperature response.

Table 1 Description of samples Sample number

Volume fraction Spinel particle of spinel size (nominal range as obtained after classifying by sedimentation) (µm)

Milling procedure (U: ultrasonic V: vibratory A: attrition)

A A2 C D

10% 10% 10% 10%

U+A U+A U+V U+V

1–2 2–3 6–8 10–15

Fig. 1. Optical micrographs from samples A and D showing the distribution of the spinel particles in the aluminum matrix.

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Measurements at elevated temperature were carried out by placing the specimen in a carbon resistance furnace with nitrogen protective atmosphere and sapphire windows. The thermal conductivities were deduced from the equation ␬ ⫽ K / (rCp) where ␬ is the thermal diffusivity, K is the thermal conductivity and r and Cp are density and specific heat at constant pressure. The latter two quantities were obtained by assuming rule of mixtures. The specific heat for the composite (10 vol%=13 wt% spinel) was calculated to be 0.887 J/gK using 0.9 J/gK for aluminum and 0.8 J/gK for magnesium aluminate spinel. The measurements of thermal conductivity were repeated on the same specimens three months after the initial measurements. The values were reproducible to within ±5%. The electrical resistance was measured on the same samples that were used for thermal diffusivity. A four point, constant current method, using a fixed geometry of the electrodes and their placement on the samples, was employed to obtain a relative measurement of the resistance of the samples. The measurements of thermal and electrical conductivities are plotted in Fig. 2. Both conductivities decrease as the particle size of the spinel becomes smaller. The first step in understanding such behavior is to consider a composite consisting of a metallic matrix of a given conductivity, containing

Fig. 2. Experimental measurements of thermal and electrical conductivity as a function of initial spinel particle size. The results have been normalized with respect to the values measured for pure aluminum, also prepared by the powder route.

particles of a lower conductivity and interfaces with a finite thermal boundary resistance. As the particles are made smaller the interfaces make a larger contribution to thermal resistance causing the thermal conductivity of the composite to decrease. This problem has been analyzed in the literature [12,13]. The range of conductivities for the composite predicted by the continuum models is shown by the shaded area in Fig. 2. The lower bound for the shaded area is given by the condition where the interfaces are ideally insulating. For this case, the composite conductivity, ␬, depends only on the volume fraction of the dispersed phase and is given by [11]: 1⫺f0 ␬ ⫽ ␬m 1 ⫹ 0.5f0

(3)

where f0 is the volume fraction of the dispersed phase and ␬m the thermal conductivity of the matrix phase. We note that in Fig. 2 all data lie below the lower bound, and the data for the small particles lie well below the lower bound. The unexpectedly large drop in the electrical and thermal conductivities can be explained by enhanced scattering of electrons by the magnesium atoms distributed as solid solution in the aluminum metal. The data in the Metals Handbook [14] provide a measure of the change in thermal and electrical conductivities in magnesium–aluminum alloys relative to nominally pure aluminum. The thermal conductivity of pure aluminum (at room temperature) is 247 W/mK, whereas alloy 514 containing 4 wt% Mg has a thermal conductivity of 146 W/mK, that of alloy 518 with 8 wt% Mg is 96.2 W/mK, and alloy 520 with 10 wt% Mg has a value of 87.9 W/mK. The electrical resistance follows a similar pattern increasing from 26.55 ν⍀m for pure aluminum to 49.3, 74.9 and 87.9 ν⍀m, respectively for these alloys. These changes are comparable to the results presented in Fig. 2. Finally, in order to rule out the possibility that the change in conductivities may have been caused by a residual stress (arising from the difference in the thermal expansion of spinel and aluminum) we measured the thermal conductivity as a function of temperature. Since the spatial extent of the residual stress would depend on the particle size, we would expect a change in the relative values of thermal

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conductivity between the fine particle and coarse particle composites, when the temperature is increased. The results are given in Table 2: they do not show such an effect, thus ruling out the possibility that the large drop in the conductivities may have been caused by residual stress. 2.2.2. Electron spectroscopic imaging (ESI) The spatial distribution of the elements, Al, O and Mg was imaged in an electron microscope fitted with an in-column imaging energy filter (Zeiss EM 912W). This microscope contains a slit aperture in the energy-dispersive plane of the imaging energy filter, which selects a narrow bandwidth of energy. Above 50 eV, the electron energy-loss spectrum of a material typically exhibits elementcharacteristic absorption edges, superimposed on a decaying background. To project the spatial distribution of an element a three-window technique is used (see, for example [15,16]). First, the image with the slit aperture centered on the maximum of the absorption edge is recorded. Next, two images with the filter placed just before the absorption edge are recorded; when extrapolated they provide the background for the energy loss from the first image. The net intensity distribution for the first image reveals the projected spatial distribution of the element with the characteristic absorption edge. ESI maps from specimens A (1–2 µm) and C (6–8 µm) are shown in Fig. 3. Most noteworthy feature of the results is the higher concentration of Mg in the aluminum matrix in Sample A. It is also possible to distinguish ceramic particles with different magnesium to aluminum cation Table 2 The influence of temperature on the relative thermal diffusivity of samples with fine particles and coarse particles T (°C)

21 204 300–302 405–408 511–514 593–599

Thermal diffusivity (cm2/s) Sample A2 Sample D Ratio (A2/D) 0.347 0.303 0.306 0.299 0.292 0.245

0.630 0.612 0.579 0.579 0.566 0.461

0.55 0.50 0.52 0.52 0.52 0.53

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ratios by comparing the relative contrasts in the same locations for Al, O and Mg. For example the highest contrast for aluminum is expected in metal aluminum regions, the next highest in alumina rich spinel particles, and the lowest contrast in magnesium rich particles. A similar ranking can be constructed for the O and the Mg maps. These trends are summarized in Table 3. They suggest that the medium-high contrast regions in the O map, accompanied by a medium contrast in the Al map indicate the presence of the alumina-rich ceramic phase. The conditions for alumina rich particles can be observed in sample A but much less so in sample C. The ESI maps and the measurement of conductivities suggest that spinel particles react with the aluminum matrix in such a way that the ceramic phase is enriched in aluminum while the matrix phase is enriched in magnesium. The X-ray diffraction experiments presented next provide further evidence for this conclusion.

2.2.3. 2.2.3 X-ray diffraction The X-ray experiments were used to compare the evolution of phases in specimens with large spinel particles (10 µm) and fine particles (1 µm) after both had been annealed at 640 °C for 4 h. The conductivity measurements would suggest that the fine particle composites would show considerable reaction while the large particles would not. The two types of specimens were prepared in the same way, except for the particle size of the spinel phase. The aluminum powders were mixed with the spinel particles and attrition milled. X-ray diffraction scans were obtained for each type of specimen. The specimens were then annealed at 640 °C for 4 h in ultrapure argon atmosphere. The diffraction spectra were repeated after the annealing treatment. The X-ray diffraction patterns were obtained with Cu–Ka radiation in Model PAD V, Scintag Inc., CA. The specimens were prepared by packing the milled powder in a plexiglass holder of dimensions 25×25×5 mm. The annealed specimen of the fine particle size exhibited broad peaks. The line broadening was used to estimate the crystallite size, d⬘, using the Scherrer equation:

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Fig. 3. ESI maps for Al, Mg and O from high resolution transmission electron microscopy. Typical images from sample A and C are shown below. The guidelines for interpretation of the images are described in Table 2. Table 3 Guidelines for interpretation of the ESI images Phase

Contrast in ESI images O Mg

Spinel High Alumina-rich spinel Medium– high Aluminum Low

d⬘ ⫽

0.9l Bcosq

Al

High Low

Low Medium

Low– medium

High

(4)

where l is the X-ray wavelength, q is the peak position and B is the corrected value of the line width given by: B ⫽ 冑B2s⫺B20

(5)

where Bs is the full width half maximum for the sample and B0 for a standard crystal (silicon). The results are given in Fig. 4. The specimens with 10 µm spinel particles did not show any change between the unannealed and annealed conditions; therefore only the diffraction scans for the annealed specimen are given. It shows a strong primary (311) spinel peak. The results from the 1 µm particle size specimen, before and after annealing, are shown in the upper two spectra. These show notable changes: most significant is the emergence of a prominent peak for γ-alumina, which is cubic, and β-alumina, which is a hexagonal phase, as well as a high intermetallic of aluminum and magnesium. The spinel peak which is quite broad in the unannealed specimen becomes narrower after annealing. Attrition milling of the powders is expected to have created fines of the spinel phase which contribute to peak

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Fig. 4.

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X-ray diffraction spectra showing the difference between specimens containing large spinel particles and fine spinel particles.

broadening in the unannealed specimen. It appears that these fines, because of their very small size are almost completely converted into aluminum oxide upon annealing. The primary peaks for γ-alumina, β-alumina and the intermetallic are quite broad. The peaks were deconvoluted and full width half maximum of the peaks were analyzed to obtain the crystallite size with the following results: γ-alumina, 22 nm, βalumina, 24 nm and Al12Mg17, 27 nm. The primary spinel peak in the 1 µm particle-size specimen shows a finite shift towards a smaller lattice parameter, which would be expected if the spinel were to become enriched in aluminum cations. It needs to be noted that attrition milling produces a wide range of spinel particles. The broad spinel peak in the unannealed specimen of fine particles is evidence that the milling created a population of superfine particles as well. It is likely that these superfine particles converted completely into aluminum oxide by the exchange reaction. It is also likely, however, that the ion-exchange reaction introduced significant elastic strains in the surface regions of the spinel particles (since the lattice parameter for alumina is smaller than for spinel) which could have caused them to spall, especially since the spinel particles were angular and the

sharp tips and edges of such particles would have been susceptible to fragmentation. 3. Discussion Taken as a whole, the three observations described in the preceding section, the conductivity measurements, the elemental maps and the X-ray experiments, provide credible evidence that there is a reaction between spinel particles and the aluminum matrix which occurs below the melting point of aluminum. The reaction not only produces enrichment of spinel in aluminum cations, illustrated by the shift of the spinel peak, but it also creates nanoscale particles of cubic and hexagonal phases of alumina. In the aluminum metal the reaction produces a solid solution of magnesium as well as precipitates of a high intermetallic. The extent of the reaction is strongly dependent on the surface to volume ratio of the spinel particles which shows that the reaction is interface controlled. There are two possible explanations for the above reaction: (i) the dissolution of spinel into the aluminum metal followed by reprecipitation as a new phase of aluminum oxide, or (ii) the ion exchange reaction.

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The thermodynamic driving force for a possible nucleation and growth process is discussed in the Appendix A. Such a reaction is possible only if the interfacial energy between the spinel particles and the aluminum matrix is added to the chemical driving force for the dissolution reaction. The analysis given in Appendix A concludes that such a scenario is possible only if the spinel particle size is less than 1 nm, which is far below the experimental values. The dissolution precipitation reaction would also require the diffusion of oxygen anions in the ceramic phase, at least to some extent. This is highly unlikely at 640 °C given the large size of the oxygen ions and their closed packed configuration in the ceramic. In contrast to the dissolution–precipitation reaction, the ion exchange reaction requires only that the cations in the ceramic should have significant mobility. We have been unable to find direct data for the diffusion of cations in magnesium aluminate spinel, but there are experiments where the order/disorder reaction among the cations has been studied as a function of temperature. References [4,17] describe experiments where the cations were first disordered by neutron irradiation; these specimens were then annealed at different temperatures, in steps of 100 °C to observe the onset of reordering of the cations. The transition was seen to take place between 600 and 700 °C. These experiments suggest that the cations possess enough mobility in this temperature range to enable the ion-exchange reaction in the present experiments. The diffusivity of magnesium in the aluminum metal at these temperatures is not expected to be the limiting kinetic condition; indeed, the ESI maps in Fig. 3 show a fairly uniform distribution of magnesium in the aluminum matrix. The wide range of solid solubility between spinel and aluminum oxide [18], and the eutectic nature of the binary Al–Mg phase diagram [19] suggest that there is a natural driving force for the ion-exchange reaction. In the spinel the exchange energy for inversion of cation positions, as described by Eq. (2) can be compensated by configurational entropy. In the metal, the entropy as well as the enthalpy favor the solubility of magnesium and the formation of intermetallics. Therefore, in broad terms, the ion exchange reaction is not

expected to be limited by thermodynamic driving force. However, the details of the interfacial reaction pathway are likely to be interesting and complex because they involve transfer of electrical charge as given by: MgA ⫹ Al→MgSS ⫹ Al•A ⫹ e

(6)

In order to conserve charge neutrality the following defect reactions must accompany Eq. (6): MgA ⫹ e ⫽ Mg⬘B

(7a)

or, Al•A ⫹ e ⫽ AlB

(7b)

The richness of the charge transfer reactions raises the following questions: (a) does the interface itself harbor a net charge in order to achieve electrochemical potential equilibrium of all the species on either side of the interface?, and (b) what is the spatial distribution of the charged defects just inside the ceramic near the interface? In view of the complexity of the reaction it is unlikely that the charge distribution near the interface is such that the electrical fields vanish. If these fields are indeed finite then they could provide a mechanism to explain the strong cohesion between aluminum and magnesium aluminate spinel [9] in the form of electrostatic attraction between the charges on either side of the interface [20]. Such an attractive force between charges in the ceramic and image charges in the metal have been proposed to explain a phenomenological observation that interfaces between metals and non-stoichiometric ceramics (e.g. spinel) exhibit strong bonding [21].

4. Summary We have presented experimental evidence for an ion-exchange reaction at interfaces between a nonstoichiometric ceramic, magnesium aluminate spinel, and aluminum at 625–640 °C. The reaction leads to enrichment of the metal in magnesium and the enrichment of the ceramic in aluminum cations. The possibility of a dissolution–precipitation reaction where the spinel phase dissolves completely into the metal and nucleates fresh crystallites of

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aluminum oxide is ruled out from thermodynamic considerations. The clearest evidence for the exchange reaction is provided by X-ray diffraction spectra which confirm the emergence of crystallites of the alumina phase, the slight decrease in the lattice parameter of the spinel phase consistent with enrichment of aluminum cations, and by the appearance of an Al– Mg intermetallic. A change in the electrical and thermal conductivities and elemental maps obtained by ESI further substantiate the reaction. The results presented here raise new questions about the distribution of charged defects at and near metal–ceramic interfaces, when the ceramic phase is non-stoichiometric. We propose that the reactions may lead to charge and electrical field gradients near the interface which may have an influence on the mechanical properties of such interfaces. The present experiments have been carried out on fine dispersions of spinel particles in an aluminum matrix. The next phase of this work should emphasize studies of planar interfaces in layered metal–ceramic architectures.

Acknowledgements This research was supported by a grant from the Division of Materials Research of the National Science Foundation, DMR-9610167. The ESI maps were obtained at the Electron Microscopy Laboratory (Director Dr Manfred Ru¨ hle) at the Max-Planck-Institute fu¨ r Metallforschung in Stuttgart Germany. F. Ernst thanks Kersten Hahn for the help in processing the ESI images. We are grateful to L. Nardone for the optical micrographs given in Fig. 2.

Appendix A

3{MgO.1.25Al2O3} ⫹ 2Al+4.75Al2O3 ⫹ 3Mg

view of chemical free energy, we include the effect of the interfacial energy of the spinel particles in the analysis. To analyze the problem we calculate the chemical potential of Mg when the spinel particles are in equilibrium with alumina and aluminum metal as given by Eq. (A1). If this chemical potential is greater than the chemical potential of Mg as a solid solution in aluminum then it is inferred that spinel would react with aluminum, otherwise not. In the following analysis we calculate these two chemical potentials of Mg and compare them to study the feasibility of the dissolution reaction. In the nomenclature, subscript j represents the species Mg. The chemical potential in the standard state is m0j . The chemical potential of Mg at the spinel interfaces is mdj , and that in a solid solution with the aluminum matrix is given by mm j . The interfacial energy of the spinel– aluminum interface is gi. The molar volume of Mg, on a per atom basis is ⍀j. And finally, the free energy of the reaction (A1) is ⌬G; this is a positive quantity since the reaction is unfavorable unless the interfacial energy of the spinel particles are assumed to increase the Gibbs Free energy of the left hand side of Eq. (A1). Both mdj and mm j will change if the spinel particles were to dissolve into aluminum. We denote the initial particle size as r0, and the size of a partially dissolved spinel particle as r. As the particle dissolves both chemical potentials would increase: mdj because the smaller radius increases the interfacial activity, and mm j because dissolution increases the concentration of Mg in aluminum. Our analytical approach is to calculate the relative changes in both chemical potentials as a function of r/r0. The reaction would continue only if mdj remains greater than mm j as r/r0 continues to become smaller. The chemical potential of Mg at the particle– matrix interface can be written as: mdj ⫽ m0j ⫺⌬G ⫹

We consider the following reaction: (A1)

Since the reaction is unfavorable from the point of

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2gi⍀j r

(A2)

Rearranging terms we may write the above equation as: ⌬mdj ⫽ ⫺B ⫹

C r / r0

(A3)

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Fig. 5. The change in the relative chemical potential of species j in the particle phase and the matrix phase, with parameters B and C. A lower value for B implies a smaller chemical barrier for dissolution, while a higher value for C implies a greater interfacial driving force for dissolution. In case (i) the particles are essentially non-reactive. Partial dissolution occurs in case (ii) until a stable condition is reached. In (iii) and (iv) the particles may react completely with the matrix.

where, mdj ⫺m0j ⌬G 2gi⍀j ⌬mdj ⫽ ,B⫽ , and C ⫽ kT kT r0kT

xj ⫽ (A4)

m j

The chemical potential, m of Mg in the solid solution alloy with aluminum can be written as: 0 mm j ⫽ mj ⫹ kTln(ajxj)

(A5)

where aj is the activity coefficient, and xj is the molar concentration of Mg. The molar concentration xj is related to r, the initial radius r0 and the volume fraction of the spinel phase, f0, by the equation:

1⫺(r / r0)3 rdMm · ·w (1 / f0)⫺(r / r0)3 rmMd j

(A6)

where rd and rm are the density of the spinel and the aluminum, and Md and Mm are the respective molecular weights. wj is the stoichiometric number for Mg in the composition MgO.1.25Al2O3, that is, wMg ⫽ 1. Substituting Eq. (A6) into Eq. (A5) and rearranging the terms we have: 3 ⌬mm j ⫽ A ⫹ ln{1⫺(r / r0) }⫺ln{(1 / f0) 3 ⫺(r / r0) }

where,

(A7)

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⌬mm j ⫽

冋

册

0 mm r dM m j ⫺mj ·w , and A ⫽ ln aj· kT rmMd j

(A8)

We now compare Eq. (A3) with Eq. (A7) as a function of r/r0. The results for different values of A, B and C are given in Fig. 5. The physical significance of these constants should be kept in mind. A represents the activity coefficient for solubility of j in the matrix, B the free energy cost of reacting the spinel with aluminum, and C the ratio of the interfacial energy and the initial particle radius. Four possible outcomes of the relative values of chemical potentials are presented in Fig. 5. In case (i) the particle remains essentially stable since the chemical potential in the matrix is greater than the chemical potential in the particle (there is an unstable situation at a particle size of about r / r0 ⫽ 0.15, but it is almost impossible that this state can be reached by thermal fluctuations). In case (ii) the particle chemical potential remains greater than the matrix until the particle size shown by the black dot is reached where the two chemical potentials become equal signalling an equilibrium state. A critical state shown in (iii) occurs when the two chemical potentials become equal to one another at the radius where the curves touch. Case (iv) signals instability since the particle potential is always greater than the matrix potential: therefore, the particle will continue to dissolve completely. A closed form relation for the critical condition shown in case (iii) would be useful in providing insights into the possible conditions under which particles can react with the matrix phase. An approximate form for this condition may be obtained by noting that when numerical plots of Eq. (A3)Eq. (A7) are made for different values of the parameters, the critical condition, shown in (iii) occurs in the range 0.75 ⬍ (r / r0) ⬍ 0.95. We can obtain an approximate expression for the critical particle size by substituting (r / r0) ⫽ 0.85 into Eq. (A3)Eq. (A7) and setting them equal to one another. This exercise leads to the following condition on the particle size for dissolution into the matrix phase: 1 2gi⍀j · rⱕ kT A ⫹ B ⫹ ln(f0)⫺0.95

(A9)

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where we have assumed that (1 / f0)(r / r0)3. The application of Eq. (A9) to the aluminum– spinel composite yields critical value for r⬇1 nm (where A ⫽ 0.1, f0 ⫽ 0.1, ⍀j ⫽ 0.01 nm3, gi ⫽ 1 Jm⫺2, and B ⫽ 10, that is ⌬G⬇1 eV per Mg atom), which is about two or three orders of magnitude smaller than the present experimental work would suggest.

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