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EQUILIBRIUM COMPOSITION OF INTERPHASE BOUNDARIES
P. Wynblatt, S. Dregia
To cite this version:
P. Wynblatt, S. Dregia. EQUILIBRIUM COMPOSITION OF INTERPHASE BOUNDARIES. Jour-
nal de Physique Colloques, 1990, 51 (C1), pp.C1-757-C1-766. �10.1051/jphyscol:19901119�. �jpa-
00230027�
COLLOQUE DE PHYSIQUE
Colloque Cl, supplement au n o l , Tome 51, janvier 1990
EQUILIBRIUM COMPOSITION OF INTERPHASE BOUNDARIES
P. WYNBLATT and S.A. DREGIA"
Carnegie Mellon University, Department of Metallurgical Engineering and Vaterials Science, Pittsburgh, PA 15213, U.S.A.
Ohio State University, Department of Materials Science and Engineering, Columbus, OH 43210, U.S.A.
Resume - Alors que de nombreuses recherches ont 6t6 entreprises sur la composition de certains types --
d'interface, tels que les surfaces libres et les joints de grains, la composition des "joints de phases" a suscite beaucoup moins dlinte'r$t. Cet article r6sume, premibrement quelques r6sultats de mod6lisation Monte Carlo concernant la composition d'interfaces semi-cohergntes dans les systemes binaires. Les calculs ont montrk que la skgre'gation
&
ce type d'interface est li6e aux int6ractions entre, d'une part les deux composants du systhe, et d'autre part les dislocations interfaciales. Deuxiemement, une m'ethode, b a d e sur le modble des liaisons entre voisins proches et I'approximation des solutions rggulieres, a 6t6 employee pour 6valuer la composition d'interfaces cohgrentes dans le syst6me ternaire Ag-Cu-Au. En conclusion, sont discute's et cornpar& avec les prgdictions thgoriques, les rgsultats de mesures experimentales de compositions d'interfaces obtenues par croissance Qpitaxique dans le systbme Ag-CU- Au.Abstract - Whereas there has been substantial research devoted to the composition of certain types of interface, such as free surfaces and grain boundaries, much less attention has been paid to the composition of interphase boundaries. This paper summarizes some recent Monte Carlo modeling of compositional effects at semicoherent interphase boundaries in two-component systems, where it has been shown that segregation effects can arise from differences in the interaction of the two components with misfit dislocations at the interface. It also discusses a treatment of compositional effects at coherent interphase boundaries in three-component systems, which makes use of the nearest neighbor bond model in conjuction with the regular solution approximation. This treatment is employed to assess the effects of several variables on interphase boundary composition in the Ag-Cu-Au system. Finally, the results of experimental measurements of boundary composition in epitaxially grown interfaces in Ag-Cu-Au are discussed and compared with theoretical predictions.
l INTRODUCTION
The equilibrium composition of an interface generally differs from that of the adjacent bulk phases. This phenomenon was treated by Gibbs 111 over a century ago, and is commonly referred to as interfacial segregation.
Most of the work on interfacial segregation has been performed either on free surfaces 12, 31 (i.e. solid-vapor interfaces) or on grain boundaries 14, 5/ (i.e. solid-solid interfaces separating bulk phases of identical structure and composition which differ only in relative crystallographic orientation). In contrast, relatively few studies have been performed on the composition of interphase boundaries. In this paper we will consider only interphase boundaries which separate two solid phases. For simplicity, we will deal with cases where the two solid phases have the same crystal structure, although some of the approaches outlined below are applicable, in principle, to more complex situations. We will also limit dicussion to equilibrium boundary composition. Thus, we will ignore kinetic boundary segregation effects, which can arise in migrating boundaries when the rate of accumulation of certain species swept up by a boundary as it moves through a material exceeds the rate at which those species diffuse away.
Some theoretical treatments of the equilibrium composition of interphase boundaries are available. These include Gibbs' early work Ill, where details on the variation of properties in the vicinity of an interface, including compositional characteristics, were deliberately ignored. In the late fifties, Cahn and Hilliard I61 developed a continuum theory to describe the thermodynamics of inhomogeneous systems. They applied this approach, among others, to the calculation of the composition profile across an interphase boundary separating equilibrium phases in binary (i.e. two-component) systems. This treatment is appropriate for coherent interfaces (i.e. for cases where the lattice planes of the two phases are continuous across the interface). In addition, a number of discrete-lattice models were used to obtain composition profiles across a coherent interphase boundary 17-1 01. These treatments all made use of a nearest-neighbor regular solution model, and their equivalence to the continuum Cahn-Hilliard theory was demonstrated by Lee and Aaronson 1101.
Both the continuum and discrete lattice formalisms are most suitable for the treatment of coherent interfaces. While many interfaces are coherent, there are numerous interesting cases where interphase boundaries are semicoherent (i.e. where the interface contains periodic arrays of interfacial or misfit dislocations). The role of misfit
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19901119
Cl-758 COLLOQUE DE PHYSIQUE
dislocations is to accomodate the mismatch between the lattices of the two phases which meet at the interface. In Section II of this paper we discuss the results of recent discrete interatomic potential calculations of the composition variations across a semicoherent interphase boundary in a binary solution I1 11.
Thus far, the continuum and discrete lattice approaches have been used for estimating the composition profile in the vicinity of interphase boundaries in binary systems. In section Ill of this paper, we discuss recent extensions I121 of the nearest-neighbor regular solution model to interphase boundaries in ternary (i.e. three-component) systems where more interesting segregation effects are likely to occur.
Most of the experimental evidence for segregation to interphase boundaries has been obtained by indirect means.
For example, the presence of segregation to interphase boundaries has been inferred from measurements of precipitate coarsening in alloys doped with interface-active impurities, where the changes in coarsening rate were attributed to changes in interfacial energy resulting from impurity segregation /13,14/. A few direct measurements of interphase boundary composition, obtained by the use of microanalytical techniques, have also been reported 115,161. Those studies probably represent examples of kinetic rather than equilibrium measurements. In Section IV, we summarize some recent measurements of equilibrium interphase boundary composition 112,171. Some general conclusions on equilibrium segregation at interphase boundaries are drawn in Section V of the paper.
II MODELING OF SEMICOHERENT INTERPHASE BOUNDARIES IN BINARY SYSTEMS
In this section we describe the results of computer simulations of (001) semicoherent interphase boundaries separating copper-rich and silver-rich phases in Ag-CU alloys /11/. The phase diagram for the Ag-CU system displays a eutectic at high temperatures, and the coexistence of two dilute fcc terminal solid solutions at lower temperatures in the solid state. It thus provides a simple framework for modeling interphase boundaries.
The simulations were performed by Monte Carlo modeling techniques, with the Embedded Atom Method (EAM) 118,191 used to model the interatomic interactions. The EAM constitutes a powerful new tool for rnodeling interatomic interactions in metallic systems containing different atomic species. It has been applied to a range of important problems in metals and alloys, including surface segregation phenomena I201 as well as several aspects of structure and composition of interphase boundaries 11 7,21,22/.
In the EAM, each atom is treated as an "impurity" embedded in the electron density resulting from all the other atoms in the solid. The energy of the nth embedded atom, E,, is the sum of two contributions:
The first term in Ea. (1) is a summation of oaiwise electrostatic repulsions between the ionic core of the embedded atom and all other ionic cores in the host solid. The second term is attractive and is determined by the electron density of the host at the embedding site, ph,,. The EAM functions used in the Monte Carlo simulations described below were obtained by fitting the functions to propelties of pure Cu, Ag, and Au, as well as the enthalpies of mixing of the binary solutions formed by pairs of those elements l 1 li.
The Monte Carlo modeling approach employed in the simulations was based on a scheme developed by Foiles 1231. In this approach, two types of configurational changes were allowed: (a) small displacements in the positions of atoms (which simulate atomic vibrations, and atomic relaxations in the vicinity of solute atoms and of the interphase boundary), and (b), spatial expansion or compression of the entire system (which minimizes stresses, and allows for thermal expansion effects). In addition, arbitrary changes in the chemical type (i.e. atomic number) of the atoms were considered. Thus, although the total number of atoms in the system remained fixed, the relative numbers of Ag and Cu atoms could change, subject to a specified difference in chemical potential between the two species.
The computational cell used in these calculations consisted of 19 (002) atomic layers of the Ag-rich phase, in contact with an equal number of layers of the Cu-rich phase. The lattice parameters of pure Ag and Cu at 300K (4.09
A
and 3.61 5A,
respectively) are approximately in the ratio 9 to 8. In view of the low mutual solubilities of Ag and Cu, coupled with thermal expansion coefficients which tend to compensate for compositionally induced lattice parameter changes, close registry between the two phases across the interface was maintained by matching 8 lattice units of the Ag-rich phase against 9 lattice units of the Cu-rich phase in all computations. Thus each layer of the Ag-rich phase contained 64 (8x8) atoms, while the layers of the Cu-rich phase consisted of 81 (9x9) atoms, for a total of 2755 atoms.Because of the stacking sequence of (002) planes in the face-centered cubic structure (ABABA---) there are two possible limiting coincidence structures for a (001) interface: one where coincidence can be characterized by a copper atom centered on a square of silver atoms, and another where coincidence occurs when a silver atom is centered on copper. Previous energy minimizing calculations (corresponding to equilibration at T=O K) have shown
that the energies of these two interfaces are very similar 11 21. Furthermore, preliminary Monte Carlo simulations showed only minor differences between these two interfaces. Thus only one of these interfaces was studied in detail, namely the interface where a copper atom is centered on a square of silver atoms.
Cyclic (i.e., periodic) boundary conditions were used in the three directions perpendicular to the surfaces of the cell.
Consequently, in the plane of the interface, the simulation represented an infinitely repeating sequence of periodic units of interface structure. In the direction perpendicular to the interface, the simulation represented an infinite stack of interfaces separated by the thicknesses of the Cu-rich and Ag- rich crystals.
Before beginning the Monte Carlo simuiations, the system was "equilibrated" at T=O K by allowing all the atoms in the cell to relax to positions which minimized the total energy. During the equilibration process, the periodic vectors (i.e., edge lengths of the cell) in each direction were allowed to vary independently. This procedure was effected in order to obtain a well defined initial configuration for the system. After equilibrating the system at T=O K, the Monte Carlo calculations were carried out at the desired temperatures.
The atom fraction of Ag in each layer of the periodic cell, averaged over several million attempted Monte Carlo steps, for temperatures of 600, 700, 800 and 900 K, are summarized in Fig. 1. The "physical" interfaces (where discontinuities occur in the numbers of atoms per plane) lie between layers 19 and 20, and between layer 0 (which is equivalent to layer 38) and layer 1, as indicated by the dashed lines. Judging from the composition profiles displayed in Fig. 1, it is evident that each 19-layer phase is sufficiently thick that there exists a- bulk- like region in the interior. Thus, the two interfaces are far enough apart that their interaction is negligible. The compositions of the bulk-like regions, calculated by averaging over the five central atom layers in each phase, were used to obtain estimates of the phase boundary compositions in the (bulk) Cu-Ag system. These computed compositions were found to be in good agreement with the experimental phase boundaries.
Figure 1 shows that the width of the composition profile associated with the interface increases with increasing temperature. This trend is in accordance with previous predictions of both continuum 161 and discrete lattice models I101 of coherent interfaces. The interface composition profiles obtained when those earlier models are combined with the regular solution approximation 1101 are perfectly symmetric, a result which is dictated by that approximation. In contrast, the present model shows that the compositional profile can be asymmetric in the case of semicoherent boundaries. In particular, it should be noted that silver appears to penetrate into the copper-rich side of the interface.
The above result is consistent with ideas put forward recently by Kamat, Hirth, and Carnahan 1241. They postulated that misfit dislocations are repelled into the elastically softer of the two phases by image forces arising from the stiffer phase, but attracted back towards the interface by coherency forces. The balance of these forces leads to an equilibrium "stand-off' of the dislocations from the interface into the softer phase. In the simulations described here, the computational approach does not provide a means for displacement of the misfit dislocations. However, the observation that the lower-stiffness silver pentrates the copper phase so as to envelop the misfit dislocations, is completely analogous to the stand-off phenomenon.
One of the attractive features of Monte Carlo simulation is its ability to yield configurational information not readily accessible by other modeling approaches. In the present instance, it is possible to extract information about the spatial distribution of the two components within the lattice planes in the vicinity of the interface. This was done for the layers adjacent to the central interface. Each of these layers was divided into a large number (72 X 72) of small rectangular sub-cells. The number of atoms (either 0 or 1) of each type occupying each sub-cell was counted and added to a running total of the number of atoms of each type observed in the sub-cells. In this way the probability of finding either type of atom at various locations in the vicinity of the central interface was determined. These probabilities are represented in Fig. 2 by the degree of shading of the sub-cells, with the darkest shading indicating the highest probability of occupancy.
Displayed in Figs. 2a and 2b are the probabilities of occupancy of Cu and Ag in layer 20 at 600K. (Layer 20 represents the first layer of the Cu-rich phase.) These figures show that the distribution of atomic species is not uniform. In this layer there is a higher probability of finding Ag atoms near the corners (Fig. 2b), whereas Gu atoms are distributed with considerably higher concentration in a cross-like pattern centered on the center of the plane (Fig. 2a). The cross-like pattern corresponds to the locus of the misfit dislocations across the plane whereas the corners of the plane correspond to points of coincidence between the copper- and silver-rich phases. These results are readily explained on the basis of earlier, energy minimizing calculations (corresponding to a temperature of absolute zero) for the CU-Ag interface 1221. Those calculations showed that the atomic relaxation field in the Cu layer adjoining the interface is such as to lead to a state of tension near points of coincidence, and a state of compression along the path of interfacial dislocations, as shown in Fig. 3. The present results are consistent with those atomic displacements in that the smaller Cu atoms tend to cluster where the structure is compressed, whereas the larger Ag atoms tend to occupy sites in the expanded regions along the interfacial dislocations.
COLLOQUE DE PHYSIQUE
Layer
Fig. 1. Ag concentration (atom fraction) as a function of layer number across a Ag-Cu semicoherent interface.
The dashed lines represent the "physical" interfaces, and are located between atom layers containing 64 and 81 atoms, respectively 11 I/.
(c)
Fig. 2. Position probabilities for atoms in layer 20 (i.e., the first layer on the copper-rich side of the interphase boundary): (a) Gu at 600K, (b) Ag at 600K, (c) Cu at 800K, (d) Cu at 900K /l l/.
The effects of temperature on both interface structure and composition can be assessed by comparing Figs. 2a, 2c and 2d representing computations at 600, 800 and 900K, respectively. These figures schematically illustrate the probability of finding Cu atoms in layer 20. Considering purely structural aspects (i.e., the locations at which the atoms tend to be found), it is clear that even at a temperature as low as 600K, the periodic arrangement of atoms has been disrupted, with the largest displacements occuring in the vicinity of the interfacial dislocations. It is also apparent that the disordering increases with temperature, possibly reflecting interfacial phenomena akin to surface roughening andlor surface premelting. It is convenient to use the highest concentrations of Cu atoms as a means of locating the interfacial dislocations. Thus it can be seen that, whereas the dislocations remain reasonably well localized at 600K (Fig. 2a) in a cross-like pattern running through the center of the computational cell, thermal disorder leads to a wandering of the interfacial dislocations as the temperature is increased (Fig. 2c). At the highest temperature (Fig. 2d) the identity of the interfacial dislocations is lost, indicating the possible existence of a transition from a semicoherent to an incoherent interface between 800 and 900K.
In summary, the results discussed in this section indicate that a number of interesting compositional effects can arise at semicoherent interphase boundaries in binary systems, as a consequence of the different interactions of the two species with defects present at the interface. However, the compositional effects are rather subtle, and may be difficult to verify experimentally. In contrast, compositional effects at interphase boundaries can be more spectacular when a third component is present in the system, as dicussed in the next section.
Ill MODELING OF COHERENT INTERPHASE BOUNDARIES IN TERNARY SYSTEMS
In this section we summarize a model for equilibrium segregation at a planar interface separating two fcc ternary phases 1121. The model will be applied primarily to the Ag-Cu-Au system (dilute in Au) where a silver-rich and a copper-rich phase coexist at equilibrium, and for which, experimental information is available, as discussed in Section IV. The segregation problem is treated under the assumption that the two phase system is entirely coherent, with a single lattice parameter and no atomic size misfit. This approach is an extension of the discrete analog of the Cahn-Hilliard model developed by Lee and Aaronson I1 01.
Consider a three-component system which consists of two phases in equilibrium at constant temperature and pressure. If the existence of the interface is neglected and only the equilibrium between the two bulk phases is considered, then the conditions of equilibrium demand equality of the chemical potentials of each component in the two phases. Denoting the two phases by a and
P,
and the chemical potential of the ith component by pi, the conditions of equilibrium may be stated as:Since four composition variables are required to specify the system (two atom fractions in each phase) and only three relations are available from Eq. 2, one composition variable may therefore be selected arbitrarily, i.e. there is one compositional degree of freedom in the system.
For ternary regular solutions, the chemical potential of component 1 may be expressed as 1251:
p1 = RT1nC1 + Z["12C2 + ml3C3 -(ml2C1C2 + "13C1C3 + %3C2C3)1 (3) where Ci(i = 1,2,3) are the bulk atom fractions of the three components in the solution, a,, are the regular solution parameters for the three binary solutions underlying the ternary system, Z (=l 2 for fcc solutions) is the coordination number of an atom in the solution, and k and T are the Boltzmann constant and the absolute temperature, respectively. Furthermore, the regular solution parameters may be expressed in terms of nearest-neighbor bond energies as:
where @ji
(4;;
or $jj) are the energies of the bonds connecting nearest neighbor pairs of atoms of types i and,j (i and f, o r i and j). By substituting relatidns such as Eq. 3, for each of the components, into Eqs. 2, and selecting a value for any C; in one of the phases, the bulk compositions of the coexisting phases can be determined.Our primary interest here is to obtain the composition profile perpendicular to a planar interface between two semi- infinite phases. This profile can be calculated by minimizing the total free energy of the solid, subject to constraints of mass conservation, in a crystal considered to consist of a stack of atomic layers each of which is of distinct composition. Two somewhat different but equivalent approaches can be used. One has been described in detail for the case of interphase boundaries by Dregia /12/, while the other is a simple extension of the approach developed by Wynblatt and Hoffmann for the case of segregation to ternaw alloy surfaces 1261. Onlv the results of
these treatments will be presented here.
Cl-762 COLLOQUE DE PHYSIQUE
The results of those approaches are valid for a one-dimensional compositional variation along a <loo> or < l 1 l >
direction in substitutional solid solutions based on an fcc lattice. In those directions, the problem is simplified somewhat by the fact that each atom in a given layer interacts only with neighbors which are distributed either in the layer itself or with atoms in the immediately adjoining layers. The coordination number of an atom within its own layer is denoted as Z,, while the number of neighbors in each of the adjacent layers is denoted by Z,. Thus, the total coordination number of an atom, Z, is given by:
z
=z, +
22, ( 5 )The modifications required to apply this approach to an interface of arbitrary orientation have been discussed by Lee and Aaronson I1 01.
The composition of the pth atomic layer in the system is given by:
where
rand
cpare the atom fractions of component i in the pth layer and in the bulk of the a phase, respectively;and the A H ~ are enthalpies of segregation, and may be expressed as:
where
For convenience, Eqs. 6 and 7 have been written using the bulk composition of the a phase as reference.
However, any plane in the two phase system could have been used as reference, including of course a plane in the bulk $ phase. The terms containing m' in Eqs. 7 are explicitly ternary contributions to the enthalpies of segregation, and vanish in the limit of a binary system. Because the pairwise nearest neighbor bond model was used, the ternary term, m, is made up of three binary regular solution parameters. Thus, in order to apply this model one need only extract values of these parameters from,ryreasured values of the heats of mixing of the three underlying bulk binary systems.
Despite its limitations, the above regular solution model is useful because it provides a simple framework for predicting interfacial compositions from thermodynamic properties of bulk phases, when these properties are available. This scheme is used next to provide order of magnitude estimates of interfacial segregation in the Ag- Cu-Au system.
We use the model described above to calculate the equilibrium composition profiles normal to planar (001) and (1 11) interfaces. In the model, these interfaces differ only in the values of the vertical and laterial coordination numbers (Z,,(001) = 4, Z..(111) = 3). The required allov parameters were extracted from measured enthalpies of mixing of the Ag-Cu, Cu-Au and Ag-Au binary alloy systems 1271. The regular solution parameter for the Ag-CU interaction was obtained by averaging values for copper alloys dilute in silver, and silver alloys dilute in copper. For Cu-Au and Ag-Au, the alloy parameters were determined from the enthalpies of mixing of alloys dilute in gold.
Because the experimental values of mCuAu and oAgAu are nearly equal, exact equality was forced here for the sake of simplicity. The consequences of this choice are discussed below. Values of the parameters used were: ZwAgCu
= 0.32 eV, and ZocuAu = ZwAgAu = -0.1 75 eV.
The first step in the determination of the segregation profile is the solution of the bulk equilibrium equations. First, bulk equilibrium is determined in the binary Ag-Cu system at a given temperature. Secondly, the atom fraction of Au in one of the phases is fixed and the remaining atom fractions are determined by solution of the equations of bulk equilibrium in the ternary system. The available degree of freedom is consumed by fixing the atom fraction of Au in one of the phases; hence the second step consists of finding three atom fractions from three equations expressing uniformity of the chemical potentials of the three components (Eqs. 2). This step is simplffied by the equality of oCuAu and O,qgAu, which dictates that the atom fractions of Au in the two phases be equal. Thus, the ternary problem is reduced to a modified binary problem; i.e., a "binary" problem where the atom fractions of Ag
Fig. 3. Periodic relaxation field on the copper side of the interphase boundary at 0 K. The atoms are represented by circles, and the in-plane component of the displacements, magnified eight times, is represented by arrows/22/.
Layer Number
Fig. 4. Calculated equilibrium profile showing segregation of Au at a coherent (001) interphase boundary in Ag- Cu-Au at 580K. The bulk compositions of the two phases are (CAg = 0.003, CAu = 0.1, Cc, = 0.897) and (C Ag
- -
0.897, CA, = 0.1 CC, = 0.003) 11 21.Fig. 5. Variation of interfacial excess of gold at (001) and (1 11) interphase boundaries in Ag-Cu-Au as a function of bulk gold atom fraction at 800K I1 U.
Cl-764 COLLOQUE DE PHYSIQUE
and Cu add up to (1
-
CA,) instead of 1.The equilibrium bulk phases are used as a starting point in the solution for the interfacial composition profiles. One begins with a set comprising an even number of contiguous lattice planes. One half of the set is assigned the bulk composition of the Cu-rich phase, the other is assigned the composition of the Ag-rich phase. The individual layer compositions are adjusted iteratively by continued substitution in the equations of equilibrium, Eqs. 6, with free energies of segregation calculated by Eqs. 7. This iterative process is continued until the composition profiles converge to a stationary value.
The equilibrium composition profiles calculated for the Ag-Cu-Au system at 580 K are illustrated in Fig. 4. The following features of the compositions profiles should be noted. The Au profile is symmetric, while the Cu and Ag profiles are antisymmetric. This is a direct consequence of the equality of the alloy interaction parameters ~c,A,, and oA The profiles show negligible solubility of Cu in Ag and vice versa, which is due to the large positive value oythe enthalpy of mixing in the Ag-Cu system. The profiles also show composition oscillations extending in both phases, which reflect a tendency for ordering that results from the negative enthalpies of mixing in the Cu-Au and Ag-Au systems. As can be seen, gold is segregated to the interface, although only four atomic layers are substantially different in composition from the bulk phases at this temperature. This indicates that the total number of layers (32) used in the numerical calculation is sufficiently large to represent an infinite system.
The Gibbsian interfacial excess of gold was calculated from the composition profiles by the following relation:
l- =
z (G" - C:,,) + C (G,' -
(9)
f's
where the summation extends over all layers in each phase. Note that the interfacial excess is measured in units of monolayers and that it is independent of the location of the dividing surface because of the symmetry of bulk compositions.
The dependence of the gold excess on temperature, bulk composition, and interface orientation was investigated.
The variation of the interfacial excess of gold at (001) and (1 11) interfaces at 800K is illustrated in Fig. 5 as a function of the bulk gold atom fraction. The segregation of gold at the interface is higher for the (001) interface.
Also, it can be seen that the interfacial excess of gold initially increases, reaches a maximum, and then vanishes as the bulk atom fraction increases from zero to about 0.58. The disappearance of gold interfacial excess at a bulk concentration of 0.58 corresponds to the limit of stability of the two phase region for Ag-Cu-Au in the regular solution approximation. In other words, at 800K, only a single phase exists in the system for compositions exceeding CA,, = 0.58. Thus, the interface vanishes, and so does the interfacial excess of gold.
While not illustrated in a figure, the interfacial excess of gold increases with decreasing temperature, as expected from Eqs. 6 and 7 when the enthalpies of segregation are negative. However, plots of 111
r
versus reciprocal temperature are non-linear because of the dependence of the enthalpies of segregation on composition.It is possible to draw some general conclusions from the results presented in this section. This model of segregation to coherent interphase boundaries in ternary systems displays a number of interesting features. It shows, as illustrated by Fig. 4, that when an impurity (such as Au) is introduced into a simple two-phase system (such as Ag-Cu), the impurity can segregate strongly to interphase boundaries. Some features of the segregation behavior are similar to segregation at simpler interfaces, such as free surfaces and grain boundaries, but others are not. As in the case of simpler interfaces, the intensity of segregation is a function of temperature and increases with decreasing temperature. In contrast to simple interfaces, the intensity of segregation with increasing bulk concentration of impurity does not increase monotonically (Fig. 5), but peaks out as a result of changing phase equilibria with composition.
The driving force for segregation depends on a complex interplay between the various binary solution parameters (as illustrated in Eqs. 7) and is not intuitively obvious. However, the overall segregation of gold at the interface can be rationalized as arising from the replacement of unfavorable Ag-CU bonds (positive wAgcu) by more favorable Cu-Au and Ag-AU bonds (negative ,,,,,,,&-C! and wAgAu). Clearly, the effects shown here for an assumed coherent interface would be significantly modified in a semicoherent interphase boundary by the interaction of all three components with misfit dislocations. Work is currently undelway to address this issue by means of Monte Carlo simulations of the type described in Section II.
The intensity of the segregation is also expected to depend on interfacial orientation (Fig. 5). Thus, relative changes in interfacial energy resulting from segregation will affect the equilibrium shapes of second phase particles in two-phase materials. Furthermore, decreases in interfacial energy due to impurity segregation might also affect kinetic behavior, such as critical nucleus size and nucleation rate, during precipitation reactions or other phase transformations. Thus, the model presented here might also be useful in estimating the importance of related
kinetic effects.
IV MEASUREMENT OF SEGREGATION AT AN INTERPHASE BOUNDARY IN Ag-CU-AU
In this section, we report on an experimental study of segregation at an interphase boundary separating silver-rich and copper-rich phases in a Ag-Cu-Au alloy 1171. For the purposes of this study, a model experimental system was constructed which conformed closely to the geometry of the models described in Sections II and Ill, in order to facilitate later comparisons between experimental results and calculations.
The types of samples used consisted of epitaxial thin films produced by vapor deposition onto monocrystalline sodium chloride substrates cleaved along (001) planes. First, alternating layers of silver and gold were deposited onto the substrate, with silver as the last layer. The total thickness of this multilayered film was 80 nm. Next, alternating layers of copper and gold were deposited, starting with copper. The total thickness of the second multilayered film was 52 nm. Each multilayer contained about 15% of gold by volume. With this configuration, no gold was present at the Ag-Cu interface in the as-deposited sample. Subsequent to deposition, the films were annealed on their substrates in pure hydrogen at 580K for 46 hours. The purpose of this thermal treatment was to enable gold to achieve a near-equilibrium distribution in the system.
The structure of the annealed sample was investigated by transmission electron microscopy and electron diffraction. This showed that the films were monocrystalline and oriented so that (OO1)Ag// (OO1)Cu and [I 1OJASN [l 10],,. A lattice parameter misfit of about 10% was measured, indicating that the interphase boundary was not coherent. In addition, the distribution of gold in the direction normal to the interface was determined by scanning Auger electron spe,lroscopy, using line scans along the edge of a sputtered crater.
The crater edge profiling technique employed in that study is a useful method for investigating interfacial composition profiles 1161. With a film having a total thickness of the order of 1
o2
nm, and a sputtered crater about 106 nm across, one obtains an interface which is magnified by a factor of about 104 along the edge of the crater.The results of an Auger line scan along the surface of the sputtered crater is shown in Fig. 6. It can be seen that the experimental results are qualitatively similar to the calculated profile displayed in Fig. 4, i.e. gold is clearly segregated to the interface, and is partitioned about equally between the copper-rich and silver-rich phases. The experimental profile, however, is much broader than the predicted profile, a result which most likely stems from experimental artifacts associated with sputter-induced mixing and deviations from planarity of the interface.
The results described above represent a first attempt at direct experimental measurement of equilibrium interphase boundary composition. They illustrate some of the experimental difficulties which face highly quantitative determinations of equilibrium composition profiles associated with interphase boundaries. While some progress has been made, it is clear that more precise experimental approaches still need to be developed, in order to establish a truly quantitative picture of the compositional changes in the vicinity of interphase boundaries.
Fig. 6. Auger intensity profiles obtained after equilibration of silverlcopper thin films containing gold 11 71.
Cl-766 COLLOQUE DE PHYSIQUE
V CONCLUSIONS
Some progress has been made on the theoretical aspects of compositional effects at interphase boundaries.
Simple nearest-neighbor bond concepts, used in conjunction with the regular solution approximation, can be useful for assessing qualitative compositional trends at coherent boundaries, when the degree of misfit across the boundary is negligible. Although not discussed in this paper, compositional effects arising from elastic constraints due to misfit at coherent boundaries can be treated by the Cahn-Hilliard theory. Semicoherent boundaries cannot readily be addressed by the above approaches, but are amenable to treatment by computer intensive simulation techniques. These latter have excellent potential for quantitative predictions if reliable descriptions of interatomic interactions are available.
In contrast to the progress in theory, experimental measurements of compositional profiles across interphase boundaries have thus far provided only qualitative evidence of segregation effects. Thus, novel experimental approaches are sorely needed to provide both a better picture of the magnitude of compositional effects at interphase boundaries, and a data base against which the predictive ability of theoretical models may be tested.
Acknowledgement
Support from the Department of Energy under Grant D€-FG02-88ER45358 is gratefully acknowledged.
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