Melting of CuIn solid solutions at small superheating by droplet formation and liquid film migration

~ctn metaff. Vol. 37, No. 2, pp. 643-613, 1989 Printed in Great Britain. All rights reserved

Copyright

~00~-61~~89 $3.00 + 0.00 Q 1989 Pergamon Press pit

MELTING OF Cu-In SOLID SOLUTIONS AT SMALL SUPERHEATING BY DROPLET FORMATION AND LIQUID FILM MIGRATION T. MUSCHIK; W. A. KAYSSER’ and T. HEHENKAMP2 ’ Max-Planck-Institut fiir Metallforschung, Institut fur Werkstoffwissenschaften, Stuttgart, F.R.G. and 21nstitut fiir Metallphysik, Universitat GGttingen, GBttingen, F.R.G.

(Received

18 Apt-2 1988)

Abstract-The melting of homogenized, polycrystalline a-Cu-In solid solutions was studied by Differential Thermal Analysis and metallographic methods. At small superheating melt forms by two mechanisms. The formation of homogeneously distributed droplets in the interior of the mains and the subsequent diffusion of the excess indium to these nuclei dominates, if the solid solution is heated up rapidly ( x: 20 K s-i) up to 50 K above its solidus temperature. At small heating rates of = 8.3.10-’ K s-i, homogenization takes place by the formation of thin liquid films at the grain boundaries and their subsequent migration. The moving liquid films accumulate the excess indium from the dissolved supersaturated solid solution and leave a solid solution of equilibrium concentration in their wake. From the correspondence between the experimental migration velocities and calculated values it is concluded, that coherency strains in the solid adjacent to the liquid film provide the driving force for the second mechanism. R&un&-La fusion d’une solution homogene Cu-In a polycristalline a 6tt Ctudibe par analyse thermique differentielle et par metallographie. Pour de faibles surchauffes, le bain se forme selon deux mecanismes. La formation de gouttelettes riparties d’une fa9on homogbne a l’interieur des grains et la diffusion ulterieure de l’indium vers ces germes dominent si la solution solide est chauffee rapidement (u 20 K.s-‘) jusqu’a 50 K au dessus de la temperature du solidus. Aux faibles vitesses de chauffage (%8,3. 10-2K.s-‘) ~homog~n~isation se produit par la formation de films liquides fins aux joints de grains et par leur migration ulterieure. Les films liquides mobiles accumulent l’exccs d’indium de la solution solide sursaturee et laissent dans leur sillage une solution solide ayant la concentration d’equilibre. A partir de la correspondance entre les vitesses de migration experimentales et les valeurs calculees, on conclut que les deformations de coherence dans le solide adjacent au film liquide fournissent la force motrice pour le second mecanisme.

Z~~~nf~~Das Anf~~~tadium des Schmelzprozesses homogenisierter, polyk~stalliner a-Cu-In Mischkristalle wurde mit Differentialthermoanalyse, metallografischen und mikroanalytischen Untersuchungen studiert. Bei kleinen Uberhitzungen existieren zwei Schmelzmechanismen. Die Bildung homogen verteilter Schmelztriipfchen im Kominneren und die Diffusion tiberschtissigen Indiums zu diesen “Keimstellen” dominiert, wenn der Mischkristall schnell ( = 20 IS s-l) auf ca. 50 K tiber seine Solidustemperatur erhitzt wird. Bei einer Aulheizgeschwindigkeit von 8.3.10-* K s-’ erfolgt die Homogenisierung i&r die Bildung und Wanderung von Schmelztilmen an den Komgrenzen. Die wandernden fliissigen Filme nehmen dabei das iiherschiissige Indium der iibersattigten sich aufliisenden festen Liisung auf und lassen im du~hwanderten Bereich einen Mischk~stall mit Gleichgewichtskonzentration zuruck. Aus der Ubereinstimmung experimenteller und berechneter Werte fur die Wanderungsgeschwindigkeit der fltissigen Filme wird geschlossen, dag Kohlirenzspannungen in dem an den Schmelzfilm angrenzenden Mischkristall die treibende Kraft fur die Wanderung des fliissigen Filmes darstellen.

Investigations of the melting of solid, singlecomponent materials revealed the surface properties of the melting solid to influence strongly the early stages of the solid-liquid transition [l]. Melting at free surfaces was proposed to start at considerably lower temperatures than in the bulk [2]. Very small Pb and Au particles (mean diameter 2-3 nm), for example, melted up to 0.3 T,,, below the equilibrium

bulk melting point T,,,. The observation was explained by the fact that approximately every second atom may be considered as a surface atom with different energy states than the bulk atoms [2]. On the other hand it was reported that melting at phase boundaries can also be retarded. Small solid Arinclusions (mean diameter 2.6 nm) being embedded epitaxially in a surrounding Al-matrix could be superheated up to 480 K before melting [3]. The Ar atoms at the interface are thought to have properties 603

604

MUSCHIK et al.:

MELTING

similar to those of solid aluminum (smaller oscillation amplitudes, higher Debye-temperature and thus a higher melting temperature than bulk Ar) due to the epitaxy between the solid f.c.c. Ar-bubbles and the surrounding f.c.c. Al-matrix. Interestingly, melting in the interior of the bubbles was suppressed despite the considerable superheating, until the Ar inclusions obviously started to melt at the interfaces [3]. Both types of experiments emphasize the importance of surface and interface properties on the melting process of pure elements. Similar sophisticated experiments on the melting process of alloys have not been published yet. A small number of microstructural observations, however, indicates that melting in alloys can also be controlled by heterogeneous nucleation of the liquid phase. A Cu-3.81 wt%Cd alloy was homogenized at 923 K and aged at 828 and 803 K, respectively [4]. The retrograde solubility above the peritectic temperature (822 K) and the metastable retrograde extension of the solidus line below the peritectic temperature require the formation of melt during aging at 828 K and the precipitation of Cu,Cd below the peritectic. During aging at 828 K liquid phase formed at moving grain boundaries, which was preferentially collected along the triple junctions. During annealing at 803 K the formation of melt films at moving boundaries was also observed instead of the formation of the equilibrium phase Cu,Cd. The boundaries moved stepwise leaving rows of insulated melt inclusions behind. X-ray analysis yielded two different lattice parameters after the migration of the boundaries had started, indicating the presence of areas of solid solution with the initial composition and other areas with a distinct, but different Cd concentration. Sulonen [4] discussed his results with respect to the formation and the continued presence of the metastable liquid phase during aging. He concluded that the nucleation rate of the melt at grain boundaries is large compared with the nucleation rate of the intermetallic phase which should form. Baik and Yoon [5] observed liquid droplets forming at moving grain boundaries when cooling a saturated Mo-Ni solid solution from 1753 to 1663 K. The formation of the droplets was attributed to the retrograde solubility of Ni in solid MO, which causes a melting process during cooling down below the solidus. The boundary migration was suggested to be driven by coherency strains caused by concentration changes in the vicinity of the moving interfaces. Earlier observations of a similar phenomenon occurring during rapid superheating of Sb-rich Bi-Sb alloys about 3&70 K above their solidus temperatures were reported by Kucharenko [6]. Besides nucleation of plate-like shaped precipitates of the liquid phase in the interior of the grains the formation of liquid layers at grain boundaries and the subsequent migration of these layers were observed. The migrating layers left a structure of alternating liquid and solid lamellae in their wake similar to lamellae of two different solids left during discontinuous precip-

OF Cu-In SOLID SOLUTIONS

itation during solid-solid transformations behind migrating grain boundaries. A preferred melt nucleation at lattice defects was also found in supersolidus sintering experiments. During sintering of homogenized solid solutions in the liquid-solid two phase field a few degrees above their solidus temperature, melt formed initially at the particle’s free surfaces [7-91 and at high angle boundaries at sinter necks [7,8]. In the present work it was tried to obtain direct insight into melting at small degrees of superheating of a well homogenized a-Cu-In solid solution. Three kinds of experiments were performed. (1) Direct observations of melting specimens in a hot stage microscope (HSM experiments). (2) Determination of the heat consumption of melting specimens in a differential thermal analyzer (DTA experiments). (3) Investigation of partly molten alloys, that were quenched down from different states of melting, by metallographic methods and electron microprobe (EMP) analysis. EXPERIMENTS Alloys from copper (99.9998 wt%) and indium . (99.999 wt%) with compositions up to 9.45 at% In were molten in graphite crucibles using an induction furnace. The prealloys were homogenized in evacuated quartz glass tubes for 14 d at r, = 873 K. The homogeneity and final composition were compared with Cu-In alloy standards by electron microprobe analysis. The measurements are accurate to relative deviations in composition below 1.5%. 1. HSM experiments: in a hot stage microscope samples were flxed on a graphite holder which was heated directly by an electric current. The evaporation of In was partly suppressed by lo5 PaAr in the hot stage chamber. Observations were performed with a light microscope at a magnification of 30 x through a quartz glass window. The heating rate was 5 K/min. The temperature was controlled by Ni-NiCr thermocouples directly fixed to the specimen. 2. DTA experiments: samples were heated at 5 K/min in a differential thermal analyzer (10) to determine the starting temperature T,* of melt formation and the rate of melt evolution during further heating. The obtained temperatures T: were compared with equilibrium solidus temperatures T, determined in an independent investigation [lo], where the solubility limits of In in Cu on the Cu-rich side of the Cu-In phase diagram were obtained from the compositions of the solid phase after quenching alloys annealed for prolonged times in the solid-liquid two phase-field. 3. Quenching experiments: the initial stage of melting was investigated in detail with samples containing 3.3 at.% (set l), 7.9 and 8.3 at.% In (set 2). Turned alloy rods were cut into 2mm thick discs by spark erosion. Specimens of set 1 were attached to a W wire

MUSCHIK

er al.:

MELTING

and inserted in a vertical quenching furnace which had been preheated to a final annealing temperature approximately 50 K above T,. After isothermal annealing up to 4min the samples were quenched, falling freely into a brine kept at 293 K. Specimens of set 2 were rapidly quenched from different quenching temperatures T,, that means from different states of melting, in a newly developed quenching apparatus [l 11, which allowed heating up of the homogenized specimens at 5 K/min under flowing argon before quenching down to 293 K. A comparable quenching method described by Lengeler [ 121yielded quenching rates in the order of 4. lo4 K/s for slightly smaller specimens of pure material. In the present experiments the cooling rate is thought to be above 5, lo3 K/s. Metallographic sections of both sets of quenched samples were prepared by standard metallographic techniques. After grinding and polishing with diamond paste they were etched in a solution of 1 part H,PO,, 2 parts CH,COOH and 6 parts H,O for 10 s.

OF Cu-In

SOLID

I

SOLUTIONS

Solidus

L

I

fempemture

I

I

1220

1240

605

I

1260

1260

I

I

1300

1320

T(K)

(b) ----__.

T, = 1223

---

r,

-

T, = 1003

= to73

EXPERIMENTAL 0

The electron microprobe analysis of the homogenized samples (starting material) showed relative deviations of the sample compositions from the composition of the melted alloys to be less than 1.5%. The variation of the compositions of different slices cut from one bar were less than 0.03 at.% In. The grain size varied between 75 and 500 pm. During heating of the homogenized Cu-8.3 at.%. In samples in the hot stage microscope silvergray stripes developed along the grain boundaries at temperatures T = T, + (10 + 5) K. They broadened rapidly while the melt came out of the boundary. After l-2 s the entire surface was covered by the liquid. Only few spots of independent melt formation at the surface could be seen. For some alloys heated in the differential thermal analyzer the measured temperature of initial melting, T: , did agree with the solidus temperature T,, determined by the different method described above, within an experimental error of +4 K. Some alloys, however, superheated up to 17 K before melting was detectable. In the latter cases a strong endothermic reaction was indicated by a vault in the AT curve after the start of melting. AT is the temperature difference between sample and a reference sample of pure copper. This effect was especially pronounced in the composition ranges of l-3 at.% In and 7-9 at.% In. As an example Fig. l(a) shows the melting effect in Cu-1.62 at.% In. The tendency for superheating and the subsequent strong consumption of melting enthalpy increases with increasing holding temperature T, as shown in Fig. l(b). The homogenized specimens were held at Tp for several hours before heating up. The following section describes typical microstructures of specimens annealed in the quenching

1; 20

I 1240

I 1260

I 1280

I 1300

1320

r(K)

Fig. 1. Differential thermal analysis of Cu-1.62 at.% In. (a) DTA of the melting process during heating up at 5 Kmin-‘. T, is the equilibrium solidus temperature, T,* is the temperature, where melting starts. AT is the temperature difference between the sample and a reference sample of pure Cu. (b) AT at different holding temperatures Tr,. The calculated curve (see footnote? on p. 6) shows the amount of melt being formed in unit temperature intervals.

furnace. Figure 2 shows the microstructures of Cu-3.3 at.% In samples which were heated at 20 K/s from the solid state to the final temperature at 50 K above T, and annealed isothermally for 2 and 4 min, respectively (set 1). The grain boundaries, which were smooth after homogenization, faceted into areas containing liquid phase inclusions. These areas are connected by straight and inclusion-free areas [Fig. 2(a), arrow in Fig. 2(b)]. The interior of the grains shows a low density of larger inclusions (mean diameter 5 pm) and a higher density of smaller inclusions (mean diameter approximately 0.1 pm; density in 2D sections 2.10” and 4.10” mm2) after annealing for 2 and 4 min, respectively. In many cases a precipitation free zone with a width of 20 pm was observed at one side of the grain boundaries. The migration of some grain boundaries during slight superheating (maximum 10 K) of Cu7.9 at.% In (heated at a rate of 5 K/min, set 2) is documented by photographs of identical regions of the specimens taken before and after the melting experiments [Fig. 3(a, b), two of the identical pores are marked by the numbers 4 and 51. Most of the boundaries migrated against their center of curva-

606

MWSCHIK ef al.:

MELTING OF C&In SOLID SOLUTIONS

1 pm) indicate a depletion of In in areas of the matrix swept by the moving boundary [Fig. 4(b)]. Regions I and IV show the initial composition of the solid solution of 8.3 at.% In. They appear to be homogeneous. In regions II and III the In concentration was determined to be 8.08 and 8.18 at.% In which means a depletion of In in the wake of the migrating

Fig. 2. Microstructure of Cu-3.3 at.% In annealed at T4 5: T, + 50 K (quenched, etched). (a) Annealed for 2 min. Homogeneously distributed nucleation sites for the melt in the interior of the grains. Precipitate-free zone on the left hand side of the faceted interface. (b) Annealed for 4min.

ture. The average velocity of the migrating boundaries was approximately 10 pm/mm At some areas of the boundaries thin regions of melt phase [typical width 5 ,um] can be seen [an example is marked by 2 in Fig. 3(b)]. In other areas of the boundaries liquid films could not be detected by optical metalIography (marked I in Fig. 3b). Some of the pores which were present at the grain boundaries in the initial state are filled with liquid phase after superheating [marked 3 in Fig. 3(b)].+ Figure 3(c) shows typical X-ray line scans crossing boundaries of the Cu-7.9 at.% In sample. For all types of boundaries the counting rate indicates an increase of the In con~ntration in the interface regi0n.S Figure 4(a) shows the microstructure of a Cu-8.3 at.% In sample after superheating by 8 K (set 2). The originally smooth boundary migrated developing sharp bulges [arrows in Fig. 4(a)] indicate the direction of interface motion]. The micropro~ measurements (100 nA primary beam current, step width TThe pores are thought to be caused by a local Kirkendall effect during homogeni~tion due to the dBerent intrinsic diffusion coefhcients of In and Cu, respectively [21]. jThe counting rate represents the real volume concentration of indium in a thin film only for films thicker than 7 pm at 20 kV. At smaller precipitates the microprobe always measures also parts of the surrounding matrix with lower In content simultaneously. This limits the resolution of EMP and it is, therefore, not possible to measure the width of thin liquid films or narrow concentration profiles near the solid-liquid interface q~ntitati~ly.

IOOpm

(Cl TYPE 1

0

IO

Oistonce

20

(pm)

Fig. 3. Microstructure and In concentration protile at interfaces of Cu-7.9 at.% In. (a) After homogenj~tion (etched). Same region as shown in (b). Two identical pores in both pictures are marked 4 and 5. (b) Quenched from Tp = T, + 10 K, etched. Same region as shown in (a). Typical areas 1, 2, 3 are schematically depicted in (c). (c) Indium-La-X-ray line-scans crossing interfaces of sample described in (b) (20 kV accelerating voltage, 2001~4 beam current). An In enrichment is detectable in all three types of moving interfaces [types are indicated by areas 1,2,3 in (b)]. The resolution of the EMP for the concentration steps is approximately 3.5 pm.

MUSCHIK et al.: MELTING

OF Cu-In SOLID SOLUTIONS

601

Fig. 5. Cu-8.3 at.% In, quenched from Tq = T, + 10 K (etched). A part of a network of precipitates besides a nearly precipitate-free matrix. The mechanism of decomposition seems to be closely related to that of the discontinuous precipitation reaction. E 25000 :

.,,,,.. .,....;... .:..;__... I-

J 0

.‘,..

c . .........,.,..... . . ,_

‘.“...’

_,.,,__,,

I

t

100

50 Distance

. . .. ..,.; . . . .

.’

(pm

1

Fig. 4. Microstructure and In concentration profile at interfaces of Cu-8.3 at.% In. (a) After quenching from T, = T, + 8 K (etched). The same region is schematically depicted in (b). (b) EMP profile crossing the interface shown in (a). The initial composition X0 = 8.30 at.% In is found in the regions I and IV, 8.08 at.% In in II and 8.18 at.% In in III. The arrows indicate the direction of the interface motion concluded from this measurement. (c) After quenching from r, = T, + IO K (etched). Development of pocket-like precipitates with low or no mobility.

boundary. From the frequency of regions of liquid phase at the boundaries and the depletion of In at one side of the boundaries it can be concluded that boundary migration is connected with the formation of solid solution with In concentration close to tin addition the driving force for the formation of the melt phase increases with increasing superheating accelerating the growth of the melt areas after nucleation.

equilibrium under these experimental conditions. The rapid formation of liquid films along moving grain boundaries is consistent with the results of the hot stage examinations. The development of the microstructure during prolonged annealing is characterized by the formation of pocket-like precipitates? in the moving interfaces [see Fig. 4(c)]. They appear to move slower than other areas of the boundaries which contain thinner or no liquid films. According to this an increasing thickness of liquid films reduces their velocity. Our results describe the kinetics of melting in the first 2min after melting has started. After this time the specimens consist of networks of liquid films enclosing regions of depleted matrix material besides nearly precipitation-free regions containing only few “seas of melt”. There is no clear indication of homogeneous nucleation of the liquid phase at 5 K/min. Figure 5 shows a part of a network with essential features of the specimen morphology in this state. The reaction kinetics seem to be closely related to those observed in the discontinuous precipitation reaction which is also occurring in Cu-In [13]. DISCUSSION

1. Nucleation of melt phase Grain boundaries and their triple junctions were observed to be the preferential sites of the first formation of melt in a-Cu-In solid solution at small superheating of up to 10 K [Fig. 3(b)], whereas in the interior of the grains formation of melt was not observed. After annealing at high superheating, homogeneously distributed fine liquid phase inclusions were present in the interior of the grains in addition to broadened melt phase layers at the boundaries between the grains (Fig. 2). It is concluded that grain boundaries and triple junctions are the most potent heterogeneous nucleation sites, but a high density of nucleation sites of lower activity exists in the interior of the grains.

MUSCHIK et al.:

Indium

MELTING

concentration

Fig. 6. Schematic phase diagram of Cu-In type. The kinetics of melt formation depends on the density and the activity of the different nucleation sites as suggested by the differential thermal analysis of the Cu-1.62 at.% In alloy (Fig. 1). Melt formation, indicated by AT, started at smaller superheating, AT,, = T_ - T, (compare Fig. 6), when the homogenization temperature Tp had been lower. T,, is the actual temperature of the specimen. The maximum rate of melt formation, indicated by d AT/dT, increased with increasing Tp [Fig. l(b)]. Both effects are explained by the reduction of the density of highly potent heterogeneous nucleation sites during annealing at higher homogenization temperatures Tp. If the density of highly potent nucleation sites is reduced, nucleation sites of lower potency, which appear to be present in the interior of the grains in large numbers [Fig. 2(a)], may become active at increased su~rheating and lead to fast melt formation at many sites simultaneously (large d AT/df).t 2. Growth of melt phase The experiments in a-Cu-In solid solution with the starting concentration X, show, that during the initial stage of superheating by AT,, (Fig. 6), a-Cu-In solid solution with concentration gXS with X* & gXS < X0 is formed in distinct areas whereas other areas tempo-

OF Cu-In SOLID SOLUTIONS

rarily at the concentration of the homogenized solid solution, X,. Preferentially areas where solid solutions with reduced In content are formed are found to be zones in the wake of a number of initially straight boundaries, which bulged out into a wavy shape. Many of these parts of the boundaries contained visible liquid phase layers or at least an increased amount of In, detectable with electron microprobe analysis. The enrichment of In in the boundaries and the reduced In content of the solid solution in the wake of migrating boundaries indicate the boundary migration to be treatable as a solution precipitation process, where the boundary itself acts as a reservoir for In. When the boundary migrates with the velocity v into a grain of supersaturated solid solution of the initial composition X0, solid solution of the composition gXs is left behind the migrating boundary and the boundary takes up In proportional to the product of the concentration difference X’ - gXS and the migration velocity v. The concentration gX” was found to be close to the ~~lib~urn solid solubility XS in accordance with previous investigations of concentration changes during dissolution and reprecipitation of solids into and from a melt [16]. The migration of the boundaries observed in the present experiments is thus one mechanism of homogenization of solid solutions at small su~rheating which results in areas where the solid solution has compositions close to the solid solubility. A second mechanism for the depletion of In from the supersaturated a-Cu-In solid solution crystals into the liquid phase is the solid state diffusion of indium to sites where the melt phase has already formed. Those static melt sites are non-migrating thin liquid films, larger melt phase areas of triangular shape and the homogeneously distributed fine liquid phase inclusions which were present in the interior of the grains after increased superheating.$ Main emphasis will be given to the discussion of the first mechanism where formation of a close to

tThe measured free enthalpy consumption during heating up to temperature is compared with the calculated free enthalpy consumption of the Cu-In solid solution. The theoretical (c~c~at~) AT - T curve is quali~tively described by the amounts of liquid being formed in unit temperature intervals [14, 151. Melting is treated as a series of equilibrium states obeying the lever rule. The melting enthalpy is supposed to be independent of composition. The difference between the calculated and the theoretical AT curves indicates the delay in the start of the melt formation and the kinetic effect of the rapid increase in the amount of melt after initial melt formation (Fig. 1). $l.,arge melt areas of triangular shape may also form due to the flow of liquid phase from boundaries between two adjacent grains into the triple junctions. This movement of the liquid phase would be connected to the dissolution of Cu-In solid solution at the triple junctions and its repr~ipitation at planar interface areas where two adjacent grains are separated by a thin liquid layer.

Fig. 7. Model to estimate of the homogenization time of a su~rsatumt~ solid solution sphere surrounds by an infinite liquid of equilibrium concentration.

MUSCHIK et al.:

MELTING

equilibrium solid solution is connected to the migration of a boundary. The dissolution and reprecipitation of a-Cu-In solid solution at the interfaces of thin melt films between two crystals and the resulting migration of the melt films will be considered on the basis of the coherency strain theory which was recently developed by Yoon et al. [17-191. The first mechanism which is based on the solid state diffusion in the vicinity of fixed areas of the liquid phase will be discussed first, however, in order to estimate the limits in which the second mechanism is important. 3. Depletion of the supersaturated a-Cu-In solid tion by solid state dzflusion of Indium

SO/U-

The depletion rate of the superheated solid solution is determined by the supersaturation, the diffusion coefficients and the distance between the melt phase and the supersaturated solid solution volume elements. All three parameters depend on the superheating AT,,, and on T, . The supersaturation and the diffusion coefficients increase with temperature whereas the distances between the melt areas decrease with temperature due to the greater amount of liquid which is thought to form and due to the increasing density of critical nuclei. The distances between melt areas which nucleate at the most potent heterogeneities, the grain boundaries, are large compared to the distances between melt areas which form by homogeneous nucleation or by the nucleation at heterogeneities of low nucleation potency in the interior of the grains. At small superheating of 10 degrees the microstructures showed melt areas to have an average distance of 100 pm, which is approximately the average grain size [Fig. 3(a, b)]. At superheating of 50 deg homogeneous nucleation or at least nucleation at very homogeneously distributed nuclei of low potency had occurred forming melt inclusions with an average distance dh of approximately 1.5 pm [Fig. 2(b)]. The depletion rate of indium from the a-Cu-In solid solution will be determined for both cases, to obtain an estimate for the degree of supersaturation which is maintained during prolonged annealing at temperature. As a model of the case of the formation of melt restricted to the grain boundaries the depletion of a solid sphere with initial concentration X0 was calculated with the assumption of a surrounding liquid of a time independent concentration X’. The concentration at the periphery of the sphere was maintained at X” during annealing (see Fig. 7). Figure 8 shows the concentration profiles after various annealing times after Crank [20]. The interdiffusion coefficient of a Cu-3.3 at.% In alloy at 1203 K (times in brackets in Fig. 8) and of a Cu-7.9 at.% In alloy at 1021 K were extrapolated from data of [21]. The calculations indicate that at lower temperatures, where the Cu-7.9 at.% In alloy was superheated by 10 K (which corresponds to a final temperature of 1021 K), con-

609

OF Cu-In SOLID SOLUTIONS

siderable concentration differences between the peripheral and central regions of the solid sphere are maintained during annealing for up to 2min. Calculations assuming the presence of homogeneously distributed melt inclusions with small average distance (Cu-3.3 at.% In alloy after annealing at 1203 K) result in an almost complete homogenization during annealing for less than 1 s. Similarly short homogenization times were obtained for annealing of the Cu-7.9 at.% In at 1021 K. It should be noted once again, however, that formation of homogeneously distributed melt inclusions was not observed in the second case. As a summary it is concluded that fast homogenization by solid state diffusion occurs at higher superheating when the distances between the melt inclusions are small. Homogenization, where formation of a close to equilibrium solid solution is connected to the migration of a boundary, dominates at small superheating of coarse grained material when melt phase is formed mainly at the grain boundaries. Homogenization by liquid film migration is favored by higher In concentrations, resulting in lower solidus temperatures

and thus smaller

diffusion

4. Depletion of the supersaturated solution by liquid jilm migration

coefficients.

a-Cu-In

solid

4.1. The concentration profile in the vicinity of the liquidfilm. Figure 4(b) shows a schematic description of the microstructure shown in Fig. 4(a). The initially straight boundary bulged out thus increasing the interface area and the energy connected to the interfaces. The indium concentration in crystal II behind the migrating liquid film is clearly lower than the concentration in crystal I at a certain distance from the boundary. Up to that point the measurements are similar to the concentration profiles measured earlier during liquid phase sintering of W-Ni, MeNi and 0

Xc

I 1.0

0.8 Solid / Liqutd interface

0.6

0.4

t 1x5 0 0.2 center of the sphere

r/a Fig. 8. Concentration isochrones of a solid sphere (radius = 50 pm) at constant homogenization temperatures after different homogenization times (in s, calculated after Crank [20]). Cu-3.3 at.% In at 1203 K (times in brackets); Cu-7.9 at.% In at 1021 K.

MUSCHIK et

610

al.:

MELTING

Fe--& [22-241. In all these cases the liquid films which separated crystals of a solid solution of approx. equilib~um Ni or Cu con~nt~tion from W, MO or Fe crystals of a different solute concentration, were migrating into the direction of the latter crystals. Considering the moving boundary problem of the melt layer in the case of the slightly superheated or-C&In solid solution it is obvious, that the depletion of the su~rsaturated a-G-In solid solution [concentration profile in crystal I near to the liquid film in Fig. 9(a)] into the melt will occur. Under the steady state condition of a boundary moving with a constant velocity v in the x direction, the concentration profile of indium in crystal I is given by

X”(x) = X,+(x”(O)

- X~)exp(-~~~~~~)

(1)

where D$ is the diffusion coefficient of indium in the a-Cu-In solid solution (Df, is assumed to be independent of the small variation of the indium concentration in the a-Cu-In solid solution due to the depletion). The mi~mum concentration of the solid solution at the front interface liquidlsolid of the

x0

Fig. 9. Concentration profile and corresponding free enthalpy diagram of a migrating liquid film. (a) Indium concentration in the vicinity of the liquid film with width d and a migration velocity v. Coherency strains arise in front of the liquid film, where the concentration in the solid ranges between A’, and cxB. (b) Common tangent construction for an ~tmined (us) and a strained (st) planar solid solution in ~uilib~um with its respective liquid.

OF Cu-In SOLID SOLUTIONS migrating difference

melt for small superheating X0 - Xs is taken as an upper

con~ntration

is X”. The limit of the

difference AX”. X8 is appro~ma~d

by

XS = X, + AST(dX/dT),,

(2)

with AT = 10 K, (dX/dT), = -4.O*1O-4/K and X0 = 0.083 to 0.079. Typical migration velocities were 4.0*10-‘ms-‘. The diffusion coefficient is Z& = 1.25*10-i3m2 s-l at 1021 K. At a distance of 0.312 pm from the solidiliquid interface the concentration difference AX, is reduced to l/e of the value at the interface. It is obvious from this estimate that the microprobe with a resolution 2 1 pm cannot resolve the concentration change in front of the moving boundary. The steep gradient in the indium concentration in front of the migrating liquid layer makes the coherency strain theory applicable to this moving boundary problem. The theory was initially proposed by Hillert [17] and later refined by Yoon and Cahn for liquid film migration [18,19]. &?. ~5~5gen~~ati5~ by pique jilm ~igrati5~. If the depletion from indium of the a-Cu-In solid solution in front of the migrating liquid film would not change the stress state of the solid solution, the concentration of the a-Cu-In solid solution at the interface would be equal to X” [Fig. 9(b j-the free energy change is set equal to the free enthalpy change for the present considerations]. As a consequence of the local equilibria assumed to exist between the solid and the liquid phase at the interfaces, the indium concentration in the melt would be X’ at both interfaces. In the absence of the concentration gradient of indium in the melt no liquid film migration by dissolution and reprecipitation would occur. The coherency strain theory is based on the assumption that strains arise due to the concentration difference in the a-Cu-In solid solution between material at the interface and material in the interior of the E-Cu-In solid solution crystal. Strains are thought to result when the lattice parameter changes with the concentration and when the coherency between the areas of different concentration in crystal I is maintained. A change of the lattice parameter of the cx-Cu-In solid solution with the indium concentration of da/dX results in a coherency strain energy of the a-Cu-In solid solution at the solid/liquid interface of E, if Vegard’s law is valid [see Fig. 9(b)]. The dashed line in Fig. 9(b) shows the increase in the free enthalpy of the cc-Cu-In solid solution right at the solid/liquid interface due to the coherency strain energy increment which changes with the square of the concentration difference from the initial concentration of the a-Cu-In solid solution X0. As a consequence, the In concentration of the melt in contact with the strained solid solution, cX’, differs from that in contact with the unstrained material at the opposite side of the liquid film, Xi. The value of eXi is determined following the treatment of Yoon and Cahn [18] (see Appendix).

MUSCHIK et nl.: MELTING OF Cu-In SOLID SOLUTIONS Despite the fact that Vega&s law is not valid in Cu-In, the data of [25] indicate a constant du/dX resulting in the direct applicability of Yoon and Cahn’s approximations. In order to determine the extent of equilibration of the supersaturated solid solution by liquid film migration, the velocity of the liquid film is calculated as a function of time. The velocity of the film is given by Fick’s first law. v = D &*(dc jdx) which is approximated AcaR 2: AX’ to

by dc/dx = Ac*R/d

ds/dt z L)+AX’jd

(3) and (3a)

where D is the diffusion coefficient of In in the melt, AX’ is the difference of the In concentration in the melt at the opposite liquid-solid interfaces, and d is the thickness of the film [see Fig. 9(a)]. According to the experimental situation, equation (3a) is written as (Appendix} ds(r)jdt =

rr d, +

D*K,+t2 lu,*t+(ds/dt)dt

J0 where s(t) is the position of the liquid film at time t, d, is the thickness of the liquid film at t = 0, and 1(1 and & are constants. Numerical integration of equation (4) yields s(t) (Fig. 10). Assuming physically meaningful values for do between 1. lo-“rn and l.lO-‘m, we find as a main feature that the liquid film migrates with a constant velocity of about 3.3. lo-’ ms-’ after a time of acceleration. The acceleration time interval increases with increasing values of do from about 2 s for do = 1. lo-” m to about 80 s for do = 1. IO-’ m. Since the lower values of d, are more likely the constant velocity is thought to be reached within 5 s once T has exceeded the solidus temperature T,. The results compare well with the experimentally observed typical migration velocities of the liquid films of (1.7-4). low7 ms-‘. The proposed

modification therefore, be homogenized superheating

611

of the coherency strain theory may, well applied to the melting process in ~lycrystalline Cu-In alloys at small and constant heating rate. CONCLUSIONS

At small superheating of homogenized polycrystalline Cu-In solid solutions melt is formed by two mechanisms. The formation of small homogeneously distributed melt droplets in the interior of the grains with a subsequent solid-state diffusion of In to the melt sites dominates when the solid solution is heated up rapidly f x 20 KS-‘) up to 50 K above its solidus tem~rature and then held between 2 and 4min at this temperature. Calculations show that after formation of the melt nuclei homogenization is completed within one second. At small constant heating rates (8.3. 10m2KS-‘) droplet formation does not occur up to 10 K above T,. ~omogeni~tion takes place in this case by the formation of thin melt films in the prior grain boundaries and their subsequent migration (liquid film migration). The moving liquid films accumulate excess solute atoms from the swept supersaturated solid solution thus leaving a homogenized region in their wake. The migration velocities of the liquid films calculated with a modification of the coherency strain theory of 3.3. IO-‘ms-’ agree well with typical experimental values of (1.74). 10e7 ms-‘. It is concluded that coherency strains at solid-liquid interfaces provide the driving force for liquid film migration during the melting process. Since moving liquid films were also observed in rapidly heated specimens, the second mechanism seems to be active in any case as long as the chemical driving force for decomposition is small. Acknowledgement-We

thank Professor D. N. Yoon for his fruitful comments on the final manuscript. This work was in part subject of a diploma thesis of T. Muschik at the University of Gijttingen (1985).

REFERENCES 40 1

do = lO%l

2

d,=10-9m

3

+,=10-*m

4

do =W7m

Slope : 0.33pms“

2 2

20

rg

0

t(s)

Fig. 10. Results of the numerical integration of equation (4). liquid film migrates with a constant velocity of 0.33 grns-’ after a time of initial acceleration. The initial transient time increases with increasing initial thickness d, of the liquid film. The

1. R. W. Cahn, Nature News Views 323, 668 (1986). 2. PH. Buffat and J.-P. Borel, Phys. Rev. A 13, 2287 (1976). 3. C. J. Rossouw and S. E. Donelly, Phys. Rev. Lert. 55, 2960 (1985). 4. M. Sulonen, Z. Metallk. 55, 543 (1964). 5. Y.-J. Baik and D. N. Yoon, Acta metall. 33, 1911 (1985). 6. Y. S. Kucharenko, Phys. MetaN. Metallogr. 39, 121 (1975). 7. J. A. Lund and S. R. Bala, Modern ~~elo~rnent~ in PM Vol. 6, (edited by H. H. Hausner, H. W. Antes and G. D. Smith), Vol. 6. MPIF, APMI Princeton, N.J. (1974). 8. S. R. Bala and J. A. Lund, Z. MeraNk. 70, 185 (1979). 9. L. Cambal and J. A. Lund, Int. J. Powder Metall. 8, 131 (1972). 10. T. Muschik and T. Hehenkamp, Z. Metallk. 78, 358 (1987). 11. T. Muschik. Diploma thesis, Univ. of Giittingen (1985). 12. B. Lengeler, Phi!. Mag. 34, 259 (1976).

612

MUSCHIK ef al.:

MELTING OF Cu-In SOLID SOLUTIONS

13. R. A. Fournelle and J. B. Clark, MelaN. Trans. 3C, 2757 (1972). 14. T. Hehenkamp and R. Kossak, 2. Me&k. 74, 195 (1983). 15. T. Heumann and 0. Alpaut, .I. less-common. Met. 6, 108 (1964). 16. S. J. L. Kang, Y. D. Song, W. A. Kaysser and H. Hofmann, Z.Metallk. 75, 86 (1984). _ 17. M. Hillert. Scrinta me&/f. 17. 237 119831. IS. D. N. Yokn, i. W. Cahn, i. W. handwerker, J. E. Blendell and Y. J. Baik, in Znterface ~igraf~on and Conrroi of Microstructures. Am. Sot. Metals. Metals Park, Ohio (1985). 19. C. A. Handwerker, J. W. Cahn, D. N. Yoon and J. E. BlendelI, in D@sian in Solids: Recent Developments, T.M.S./A.I.M.E. Warrendale. Pa (1985). Clarendon 20. J. Crank, The Mathematics. of &ii&z. Press, Oxford (1976). 21. K. Hoshino, Y. Iijima and K.-I. Hirano, Acta metall. 30, 265 (1982). 22. D. N. Yoon and W. J. Huppmann, Acta metall. 27,973 (1979). 23. M. Hofmann-Amtenbrink, W. A. Kaysser and G. Petzow, J. Physique C4, 543 (1985). 24. W. A. Kaysser and G. Petzow, Powder Metall. 28, 145 (1985). 25. E. A. Owen and E. A. O’Donnell Roberts, J. Inst. Merals 81, 479 (1952-53). APPENDIX ~er~ation of equation (4)from equation &,I in~iud~g the present experimental condition Following the treatment of Yoon and Cahn [1X]the free energy increase of a strained solid solution is E,, = ( Y/a2)*(da/dX)2*(cXs - X,)z.

(Al)

The additional energy changes the melt concentration local equilibrium with the solid to

in

cX’(r) = X’ + (( Y/a2)(d~~dX)’

Xs(t) - X0= (dX”ldT)*(dT/dt)*t.

AX’ = l’{ Y*(du/dX)*(l~~) +(dX’/dT)*[(dT/dt)]2*t}2 *{(n/RT)*[X’*(l - X’)]/(X’ - X’)}

(A21

The present experiments require a modification of equation (A2). During heating up in the temperature interval between T, and T,, at a constant heating rate dT/dt the concentration difference X”(t) - X,, changes continuously from

(A4)

Since, except t2, all terms of equation (A4) are constant the equation transforms to AX’ = t2*K,.

@4a)

For the present experimental condition we obtain K, = 1.58*10-* s-~. During migration the thickness of the liquid film increases. The thickening rate is determined by the actual temperature and the velocity. Due to the concentration difference of In between the dissolving and the reprecipitating material, X”(T) - X,, additional In is accumulaied in the liquid phase as the liquid film migrates with the velocity d.s/dt. The accumulated In provides additional liquid phase with the concentration X’. Tbe volume ratio of dissolved and reprecipitated solid solution to the additionally formed liquid phase is approximated by [X*(f) - X,]/Xi, with the assumption of a constant molar volume for all phases involved. The thickness of the liquid film then increases with dd/dt = {[X’(f) - X,]/X;,}*(ds/dt) = [(dXs/dT)*(dT/dr)/(X;,)]*t*(ds/dr).

(A51

The term in the brackets is constant which reduces equation (A5) to

with K2= 1.71+10-4 s-l. Integration results in the thickness d

+ 2uK}*[(Q/RT)

(A31

For the present considerations the effects of curvature are neglected, that means K is set to zero in equation (A2). Since X’ is much larger than Xl(t) the term (X’ (1 - A”)]/@” - A’“) from equation (A2) is treated as a constant. With AX’ = cX’ - XLequations (A2) and (A3) combine to

ddjdt = K2*t*ds/dt

x [(dXJ/dT)*(dT/dt)*t]’

x (X’(1 - X’)]/(X’ - XS).

zero to a maximum value at T, . A’,(1)is approximated from the slope dX,/dT of the solidus at X0 and the time i resulting in

(Asa) of equation (ASa)

K,*I *(ds/dr) dt (A6) s0 where d, is the initial thickness of the liquid film at t = 0. Combining equations (3a), (A4a) and (A6) results in equation (4). d(I) = dO+

Nomenclature on facing page

MUSCHIK

et al.:

MELTING

OF C&In

SOLID

SOLUTIONS

NOMENCLATURE Svmbol

Dimension

EC, Y

daYdx

Nm-’ Nm-* m m

dX’jdT

K-’

dT/dr

KS-’ s

X0 X' X” CX’ CX”

AX’ R R T K 0

A.M. 37,*-s

m3 mol-’ JK-i mol-’ K m-l Jme2

Meanine Strain energy Elastic modulus of Cu [19] Lattice parameter of Cu Change of lattice parameter of Q-In between 0 and 10 at.% In. da/dx is assumed to be constant [25] Slope of the solidus line at X,. It is assumed to be constant in the temperature range of the experiments Heating rate (constant) Time Initial In concentration in the solid solution Liquidus concentration in equilibrium with the unstrained solid solution Solidus concentration in the unstrained solid solution Liquidus concentration in equilibrium with the strained solid solution Solidus concentration in the strained solid solution Difference between X’ and cX’ Molar volume of Cu Gas constant Absolute temperature Curvature Interfacial energy

Value 2.10” 3.6. lo-”

9.1 lo-”

-4.10-d 8.3.10-2

0.083 0.195

1.10-6 8.31

613