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Concentrations at interfaces in binary alloys
CONCENTRATIONS
AT INTERFACES
IN BINARY
ALLOYS*
J. L. MEIJERINGt Concentration profiles across interfaces and at a free surface of 8 binary f.c.c. alloy are computed, using the regular neerest-neighbour approximation. The y-Fe-Cu system serves as example, and most celculations 8re made for 8 temper8ture where the mutual solubilities are 4 et. o/o The coherent interface free energy is slightly anisotropic (but rather strongly at low temperatures) and nearly indifferent to 8 shift of the concentration-profile curve across the lattice planes. Computed concentration pro&s et the free surface of Fe-Cu solid solutions show substantial enrichment of Cu. Incoherent interfaces 8re approximated by planes containing 25% vacent. lattice sites. One then finds also 8ppreci8ble enrichment of Cu at the grain boundaries 8nd the incoherent interphase interface. The chemical and the crystallographic phase boundary are about 8 A apart. It is suggested that an electronic double layer is responsible for the low interface free energy found experimentally for the y-Fe-& interphase. CONCENTRATION
AUX
INTERFACES
DANS
DES ALLIAGES
BINAIRES
L’auteur a c&u16 les profils de concentration au travers d’interfaces et B une surface libre d’un allisge bin&e c.f.c., en utilisant l’approximstion r&+&e des plus proches voisins. Le syst6me y-Fe-Cu est utilise comme exemple, et la plupert des calculs sent feits pour une temperature & laquelle les eolubilites mutuelles sont de 4 8t. ‘A. L’6nergie libre d’interface cohbrente est leg6 16gbrement anisotopie (msis assez fortement aux basses temp6ratures) et presque indii%rente B.une veriation de 1s oourbe-profil de concentration au trevers des plans r&iculeires. Les profils de concentration calcul& $ la surface libre de solutions solides Fe-Cu montrent un enrichissement substantiel en Cu. Les interfaces incoh&entes sont consid&es comme &ant des plans contenant 26% de sites r&iculaires vacants. De cette fapon, on trouve Qgelement un enrichisaement appr6cieble en Cu aux frontibres de grains et B l’interf8ce entre phases inooh&ente. Les frontibres de phase chimique et cristallogrephique sont d&antes d’environ 8 A. L’auteur suggere qu’une double couche 6lectronique est responsable de 18 Bible Bnergied’interface trouv6e exp&imentalement pour l’entre-phase y-Fe-(%. KONZENTRATIONEN
AN INNEREN
OBERFLIiCHEN
IN BINPiREN
LEGIERUNGEN
Die Konzentrationsproflle an inneren und freien OberfWhen von k.f.z. Legierungen werden mit Hilfe der normalen Niiherung n&chster Nachbern bereohnet. Das System y-Fe-Cu dient 8113Beispiel. Die meisten Rechnungen werden fiir eine Temperstur gemacht, bei der die Liislichkeit etw8 4 at.- % betriigt. Die freie Energie kohiirenter innerer Oberfliichenist schwaoh anisotrop (bei tiefen Temperaturen starker) und fast indifferent gegen eine Versohiebung der Konzentrationsprofllkurve iiber die Gitterebenen. Die berechneten Konzentrationsprofile an der freien Oberfliiche von Fe-Cu-Legierungen zeigen eine starke Anreicherung von Kupfer. InkohItrente innere Oberf%iohen werden sngenithert durch Ebenen mit 25% leeren Gitterpl&tzen. Msn fmdet denn such eine merkliche Anreicherung von Kupfer an Korngrenzen und an der inkohiirenten ZwisohenphesenWohe. Die ohemische und die kristallogmphische Phesengrenze sind etwa 8 A voneinander entfernt. Es wird vermutat, d8B eine elektronische Doppelschicht fiir die experimentell gefundene niedrige freie Energie der Zwischenphaaenflitchein y-Fe-& verantwortlich ist. 1. INTRODUCTION
In a single-phase alloy in equilibrium the concentrations at the grain boundaries-and at the free surface-will in general deviate from the concentration inside the crystals. This effect may have practical consequences of some importance. The object of this paper is to try to calculate the concentration profile across such interfaces and also across an interphase interface in a two-phase alloy. The model used for these incoherent interfaces is closely related to that of a coherent interphase interface as used by One(l) and Hillert . t2) Therefore we will treat coherent interfaces first, and calculate some aspects not treated by these authors : different crystallographic directions, “even” and “odd” interfaces. Received July ‘7, 1965. emical L8boretory of Technical University, D&t%o%. t l
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2. COHERENT IN REGULAR
INTERPHASE INTERFACES BINARY F.C.C. ALLOYS
Consider a regular binary solid phase, with free enthalpy per mole : G = bx(l - x) + RT{x In x + (1 -
x) In (1 - x)}
(1) where x is atomic fraction, R the gas constant and T absolute temperature. If the interaction parameter b is positive, a miscibility gap results. The equation giving its boundaries can be written as r
T
c
l-x In --+4x=2 2
where T, is the critical temperature. For T < T, two values of x-say, x1 and 1 - x1obey equation (2) and stand for the concentrations of the two phases in equilibrium. In successive lattice planes parallel and close to a plane coherent
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interfaces the system is quite appropriate in view of the small difference in lattice parameter between the com~nen~. The mutual ~lub~ti~ of y-Fe and C?u are not symmetx%&, but may be rationalized at 4 at.% near the peritectic temperature. A slightly larger value was taken: 2 = 0.040025. This m&es TIT0 = 0.57906 = 413 In 10 simplifying numerical computation; e.g. equation (3) becomes h3lOW
Bas
2s
2 +----
1.5 distance
I.0 in iwlf Mice
0.5 pornmeters
0
FIQ. 1. Logarithmic plot of BXCBBB concentration (z - 2,) as a function of distance at one side of a coherent interphase interft~e, for 3 orientations in the f.c.c. lattice, computed with the n~~t-neigh~~ model of 0x10.~~’ azm- 0.040025,pertaining to T/T, = 0.67906. The cird at the to (z = 0.6 at the interfaoe) is oommon to the “odd” 80Putiona of all orient&ions. The drawn curve ia calculated with the method of Cahn and Hilliard,(**” where the difference equation is approximated by a differential equstion.
interface between the two phases z will vary monotonically between x1 and 1 - x1. Ifit is assumed that the energy term bzfl - 2) in (1) is only due to nearestneighbour interaction equation (2)-with a slight modification-is found to hold also(‘) for each of the successive x-values. The modification is that-while 2 in the loga~t~c term pertains to the lattice plane in question-in the term 4% it has to be replaced by the average of x for all nearest-neighbour positions of a site in that plane. Thus for the f.c.c. structure with interphase interface f/(100) one obtains the difference equation
3 % for all whole values of n, the number of the lattice plane. In the b.c.c. case treated by HillertP the second term is 2(~,+~ + x,.&. In this paper we examine not only (lOO), but also (111) and (110) interfaces, in the f.c.c. lattice. For (111) the seoond term in (3) is replaced by (z+i +
22, + xn+& and for (110) by &(x,s + 42,r 4 22, + 4x,1+ %I+3 ). We note in passing that the b.c.c. (110) case would be identical to the f.c.c. (111) we. For the &coherent interface model (cf. Section 5) an additional parameter will be necessary, depending on the specific alloy system. We had the y-Fe-& system in mind and chose the temperature acoordingly, using it for the coherent interfaces also. For such
-
%J/%J
+
G-1+
%
+
qa+r=
I.5
(4)
The approach to the asymptotio values 2, = 0.046025 and z_~ = 0.959975 is exponential. It is easily found that, for not too small values of n, x - x, is divided by a factor 10.20 for each successive (100) plane farther away from the interface, This factor is 12.99 for (Ill) and 4.807 for (110). It seems obvious that the set of z values, as a faction of n, will have a centir of symmetry at x = 0.6. Either x1 = 0.5 (the ‘“odd” solution) or zrr+ zs = 1 (the “even”’ solution). From one of these two starting points one calculates easily the course of 2 across the interface for the (100) and (111) orientations. For the even (100) solution, for instance, one begins with z, = 0.5 - log ((1 -
L&J/~&
(5)
With xa = 0.27 one 6nds, with the next equation, x4 < x, and with xs = 0.28, x4 > xa. Thus xa must lie between 0.27 and 0.28. With x2 = 0.27155 one finds 4 < x,, and with 0.2717, xs > x,. The computation of the successive x values is speeded up ~~ide~bly by the ~owl~e that (x%-r - x,)/ x,) must converge toward 10.20. In the (110) (% ease the computations are more tedious. The results are plotted in Figs. 1 and 2. It can be seen in Fig. 1 that the “even” and “odd” points of the same orientation lie quite well on one curve for (100) and (IlO), but less so for (111). For the different orientations together one common curve can be drawn only rather roughly. TABLE 1. Spctic ooherent interface free energies calculated for TIT, = 0.67906, of. Fig. 1 orien~tion
“even”
“odd”
I:::! (110)
0.701 0.714 0.709
0.708 0.714 0.709
unrelaxed 0.977 1.128 1.197
Unit: kT&*, where k ia Boltunarm’s constant, T* the critical mixing temperature and (i the f.c.c. lattice parameter assumed to be ~de~~dent of ~n~nt~tion. In Table 1 the calculated interface f&e energies y are given, with kTJaB as unit. They are compared to the results of the Becker treatment(s) where the conoentration is assumed to jump directly from 2, to 1 - 5,
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AT INTERFACES
263
achieve negative d~~~~ is quite critical. In the even (110) case the concentmtions in the “central” 2 x 4 planes had to be varied, in the same sense, the ooe5oients P,, beii about p~~~io~l to the slope of the curve which can be drawn through the (110) points in Fig. 2. Without being too inaccumte one can say that if the successive l~tti~pl~e ~n~ntr~tio~ lie somehow on the curve in question, the lowest free energy of the interface is virtually attained. A shift CJ of the onrve across the lattice planes yields only minor A i variations in G, which passes a maximum and a a2 e minimum at the two symmetric positions. For the t (100) interface it was checked (of. footnote Section 5) that no asymmetric solution of equations (4) exists. If the odd interface were me&stable, suoh a solution ~ at would be expected, ss a saddle position in between. Especially for interfaces like (100) and (110) I attainment of the stable equilibrium will be difficult because of the virtual indifference of G once the 1 o” 2 3 v distance in t&f lai tke pmamtws concentration-profile curve has been reached somehow. Fro. 2. II& of ~ncent~tion pro& it&f (center of This is &ggmv&ed by the necessity of tra~~rt~, symmetry at z = 0.6) rtaininSto Fig. I. Here only say, Cu atoms to the interface, as the ooncentrstions the &ableinterface pror les are given, “even” for (111) and (loo), “odd” for (110). The complete (100) profile near it change in the same direction when the stable is drawn in Fig. 4. For sip see Fig. 1. ~n~nt~tion profile forms from e.g. the unstable one. without any relaxation. It is seen that the anisotropy These atoms must come from rather distant planes. of y is much smaller than in Becker’s approximation, Taking into aocount the value of (l”G/d$ near the although not nil, as in the &pp~~~tion of C&n snd asymptotes, the semi-~n~tive conclusion was reached Hilliard.@) that the number of lattice planes involved in the Further, the even (111) interface is seen to be more change-over must be of the order of ten thousand. stable than the odd one. For (100) this is also the Evidently, if the crystal containing a plane coherent case, but for (110) the odd interface is the stable one. interphase interface is only, say, a micron thick, the For these two orientations the y values differ only in most stable set of concentrations across the interface the fourth decimal, not given in the Table because its may differ markedly from that derived above and rtccuracy is uncertain, Still, the correctness of the correspond to an asymmetric position of the profile assertions made is borne out by examining whether the curve with respect to the lattice planes. The exact less stable interface is nnstable or metastable. solution of the difference equation (3) is based on the We call the free energy of a large crystal containing availability of infinite numbers of lattice planes at both the interface simply G (pro~~io~l to y). The sides of the interface. If, however, the orystsl is rather equilibrium condition (9), for all n, means that thin, the exact thermodynamic “bulk” concentrations SG/&s,, = o* for all x,. Thus also dG/dy = 0, where y are not attained and the average cmwetiration of the stands for any line&r comb~&tion of the concentratotal crystal must be taken G(areof, Then a change in tions : this of the order of.) over the number of lattice planes (6) can alter the ooncentrcttion profile from one near the Y =&4&a A even solution to one near the odd solution. with arbitrary finite coefficients p,. If, however, y can be so chosen that daG/dya is negative, the free 3. LOWER TEMPERATURES energy is a ~~~~ in that “direction,’ given by the We made also computations at T/T, = 0.43429, but ratios of the different p,. The equilibrium is then only for the coherent f.c.c. (111) interfitoe. This T is unstable. 75% of that in the preceding section, and 2, is only This was indeed found to be the case for the odd 0.0109. Naturally the concentration proille is steeper (100) and the even (110) interfaces. The choice of y to now, but wemention only the valuesof y, the unit being * The largehomogeneousparts of the cry&al far away from ~~=~~* as in Ts,ble 1: ~(111, even) = 0.954; ~(111, the interfaceeatingsa sourceand sinkfor the atoms neceesery odd) = 0.999, while the Becker treatment yields 1.106. to change2*.
,&yy,
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Comparison with Table 1 shows that y increases by lowering !l’, as is to be expected. The relative difference between “odd” and “even” has risen to more than the four-fold value. The main object of the computations at this temperature was to have a look at cross-relaxation effects by segregation in the “central” plane of the odd interface (Z = a). In a separate (111) layer each atom has 6 neighbours against 12 in the complete lattice. In the regular nearest-neighbour approximation, therefore, the energy of mixing is half of that in bulk, while the Gibbs mixing entropy is the same. Consequently a homogeneous (Ill) layer, with equal concentrations of the two sorts of atoms, is stable in itself as long as T > +T,, but if T < &T, it should split up in two two-dimensional “phases” with unequal concentrations. For the (100) and (110) layers this is only the case for T smaller than T,/3 and TJS respectively. The “central” (111) layer at T = 0.43429 T, splits up in equal patches with x = 0.203 and 0.797. The average value of x remains 0.5, and thus the total interaction energy with the adjacent layers is not affected in the regular approximation. Only the interaction energy in the layer itself is lowered, and this is partly cancelled by the smaller entropy. One finds that y( 111, odd) is decreased from 0.999 to 0.985 units, so that the even interface remains the stable one. At T = 0 calculations are easy; the even solution (stable for all directions) is then identical with the Becker solution. For the interfaces (ill), (loo), (110) and (210) the y values are in the ratio 1:2/t: 48: di. The ( 111) and (210) interfaces have respectively the smallest and largest y of all orientations. When the reciprocal values of y are plotted in a polar diagram a tetrakishexaeder (210) results. In the case of a free surface, convexity or partial concavity of the polar I/y surface plot indicates whether the free surface is stable or not with respect to faceting.@*‘) The same holds true for interfaces. As, at T = 0, our l/y surface consists of plane triangles, each overall interface orientation (apart from (111) and (100)) is indifferent versus formation of “zigzags,” at least when edge free energies are neglected. For instance, a coherent plane (110) interface at T = 0 might just as well form undulations, the orientation varying in the zone between (111) and (lli). These indifferences at T = 0 are characteristic for pair-wise interaction energy models.@) If at a low but non-zero temperature the (even) concentration profile is calculated, one finds in first approximation log x1 = (m/12) log 5,. (7) Here m stands for the number of nearest neighbours of
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an atom that lie in its own lattice plane; z1 and 1 - x1 are the concentrations in the planes adjacent to the symmetry center, x, and 1 - x, those at the asymptotes. For example, if x, = 10d, x1 is about 0.1 for (1 lo), 0.01 for (100) and 0.001 for (111) interfaces. Now 7n/12 is proportional to the critical temperature pertaining to segregation in the lattice planes (see above). It follows that, as T approaches zero, x1 in (7) becomes equal to the miscibility limit for this crosssegregation. This is clearly connected with what has been said about indifference against “zigzagging” of plane interfaces at T = 0. It was checked, however, for the (110) interface that as T rises above zero, x1 becomes smaller than the cross-segregation limit. As we have seen, cross-segregation for the aEd (110) interface is possible as long as T < T,/6, but we presume that in this temperature region the even interface is still the stable one. A special situation arises when the lattice plane parallel to the interface does not contain bonds between nearest neighbours, as e.g. the (210) orientation, and also the b.c.c. (100) orientation. Then no cross-segregation is possible in the regular model used. Also, equation (7) cannot be valid for m = 0. Examination shows that in this case the even and odd interfaces are equally stable at T = 0. Aa T rises above zero, the odd interface becomes stable, thanks to the entropy. A tendency for GodhGevento decrease at increasing T appears both in thef.c.0. (110) case (where the difference is positive at T = 0 and negative at 0.57906 T,) and in the (111) case (where it is less positive at the latter temperature than at 0.43429 T,). Thus one might expect that the odd interface is stable up to TC when m = 0, as in the b.c.c. (100) interface examined by Hillertc2). His Fig. 3 (lower part) suggests that the even interface is the stable one. However, as he remarks in his Section 10, the difference in G will probably be quite small anyhow. Hillert was mainly interested in the periodic concentration profiles of intermediate states in coherently precipitating or ordering systems. It may be remarked that “long-period superstructures” like CuAu II will show periodic concentration profiles drr equilibrium. In the normal tetragonal superstructure CuAu I the Cu ooncentration in one (001) plane is constant, but in the orthorhombic CuAu II it goes up and down regularly in the [OlO] direction. According to Sato and Toth’s’ the adverse energy of the out-of-step boundaries is compensated by a Brillouin-zone effect. However, there can be no full energy compensation, as CuAu II only becomes stable at higher temperatures. Evidently it has a higher entropy than CuAu I, and this is probably due to the out-of-step boundaries being diffuse, just like coherent phase boundaries. The virtual indifference of the free energy with respect to a shift of the lattice planes relative to the concentration profile (see Section 2) makes it easier to understand that the “long” period of CuAu II is an irrational multiple of the distance between successive (010) planes.
MEIJERING: 4. FREE
CONCENTRATIONS
SURFACES
Looking at an f.c.c. Cu-Fe alloy, we call the interaction energies of Cu-Cu, Fe-Fe and Cu-Fe pairs ui, u, and ua respectively. In the thermodynamics of regular solutions and their coherent interfaces only 2u, - u1 - uz comes into play. Expressed in b (cf. equation (1)) this is equal to b/6N. The situation becomes less simple when there are atoms with missing neighbours (dangling bonds). For instance, a free surface can be expected to be enriched in atoms with the “cheapest” bonds, so that u1 - uz now plays a role too. For our calculations we took -6Nu, and -6Nu, equal to the sublimation energies of Cu and Fe: 82 and 97 kcal/mole. With b = 9 kcal (that is : T, about 2250°K) one finds -6Nua = 85 kcal. Two parameters define the problem completely: b/RT and the ratio between the difference in sublimation energy (here 15 kcal) and RT. The concentration profile-again at T = 0.57906 T,-below a (100) surface of the saturated Fe-rich phase was computed. The equations are the same as (4), apart from the first one (n = 1) which now is log ((1 -
~r)/~r} + xi+
52 = l/6
(3)
where x, is the atomic fraction of Cu in the outermost layer. Figure 3 shows a very substantial enrichment of Cu at the surface, x1 being 0.9835. For the saturated Curich phase (xc0 = 0.959975) the surface concentration is slightly higher : x1 = 0.9836. Naturally such computations can also be made for unsaturated solid solutions. For example, if = 0.01, the right-hand-side of equation (4), which :1og {( 1 - x,)/x,) + 3x oo, becomes 2.02564 and for n = 1 the r.h.s. of equation (8) is also increased by 0.52564. One finds x1 = 0.296. And for x, = 0.001 this becomes 0.0221. For dilute solutions the enrichment coefficient XJX~ goes’ to 10413= 21.54 at T/T, = 0.57906. Without specifying the values of T, ul, u2 and ua one can derive for (100) surfaces of dilute f.c.c. solutions kT In (x&c,)
= 4(ua -
u,),
(9)
k = R/N being Boltzmann’s constant. For non-dilute homogeneous solutions x1/x, cannot be expressed in a closed form. Then something general can only be said on the question which of the two elements will be enriched at the surface. It is found that the sign of (us -
u2w
-
AT
xm)
+
@I
is the same as that of log (x1/x,).
-
u2k.m
Here-if
(10)
z = 1 for
INTERFACES
266
‘rT----T
00 0
S”
AX
f
-
2
3 4 5 6 7 disfonce inhalf httice pammeters
B
FIQ. 3. Concentration profile at a free (100) surface of an f.o.c. saturated Fe-rich Fe-& solid solution, computed for same T/T, as Figs. 1 and 2; zw = 0.040026.
pure B-ur, u2 and ua are the energies of B-B, A-A and A-B bonds respectively. Only when us lies between u1 and u2 is the same element (the most volatile) enriched in the surface for all concentrations. Otherwise expression (10) changes sign somewhere. This possibility is clearly favoured by a relatively large (positive or negative) heat of mixing as compared to the difference in heat of vaporization of the elements. As the surface layer is thermodynamically intermediate between the bulk crystal and the vapour phase, it is not surprising that we have the same situation as in binary vaporization equilibria. There too, the vapour may be enriched in the least volatile component. The concentration where (10) goes through zero is comparable to an azeotrope. Expression (10) is essentially the same as equation (3, 14) of Pinea. This author has already derived the equations governing the concentrations near a free surface, leading to expressions for y in the cases of weak and strong adsorption. He does not compute concentration profiles. Similar effects as near a free surface are to be expected for the proportion of different sorts of atoms adjacent to lattice vacancies. (On the other hand, the equilibrium concentration of vacancies would depend on that proportion, more than on the bulk concentration of the alloy.) In diffusion experiments, the Kirkendall shift is usually in the direction of the constituent with the lower sublimation energy. This is in accord with the view that those atoms are preferential neighbours of vacancies, and thus more mobile than the other atoms. If the energy of formation of a vacancy and of a free surface are calculated for a pure metal according to this rough method of constant bond energies, the results are too high by, say a factor two, as is well known. In
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metals, if bonds are broken, some of the bonding strength is transferred to the remaining bonds.(ll) Although the crude picture used for a free surface crrnnot yield reliable values for y, it is believed that pronounced concentration effects should indeed be expected theoretically at the free surface of many alloys (and non-metallic solutions). Experimentally, Inman et uZ.(la)found at the surface of a Cu-Sb solid solution an excess quantity of Sb equivalent to 0.37 monatomic layers. And Sundquistos) concluded from equilibrium-shape measurements on small solid particles that excess quantities of the order of one monatomic layer of the solute are present on the surface of Ag with 4% Pb, Cu with 4% Ag and probably a-Fe with )% Cu. In all these cases the solute is more volatile than the solvent, and also the mixing enthalpy is positive, so that enrichment of the solute is to be expected theoretically. The same applies to the order of magnitude of the effect. The adjustment of the equilibrium concentration at a “fresh” surface-or after a temperature changemay give rise to transient surface effects, for instance in catalytic activity. Contamination by ambient gases may alter the surface-concentration profile of a solid solution drastically. 5. INCOHERENT
INTERFACES
We now turn to incoherent (high-angle) interfaces, not only between different phases, but also grain boundaries. As in Section 4, the system Cu-Fe is the specific example. To make computations along the lines of Sections 2-4 possible we use the following crude model of a high-angle interface: All successive planes, at both sides of the interface, are taken parallel to (100) ; in one of these planes, however (the interface), 1 of the atoms are missing. Thus the incoherent interface is treated as something in between a coherent interface and a free surface. The factor $ was chosen by comparing gram-boundary energy with the energy of formation of a vacancy in copper. It is taken to be independent of the Cu/Fe concentration. The defect plane is assumed to be filled by a random mixture of &, Cu, $( 1 - z,,) Fe and 4 vacancies, where a+,is the, as yet unknown; concentration in the interface. In the three “central” planes an atom has in the average one dangling bond. No (100) plane is crossed by a bond, and thus the defect plane effectively separates the two parts of the lattice. Therefore, although these parts are parallel in the model (and definitely not in reality) this should not be disturbing once one has accepted more or less the crude picture of the defect plane as an incoherent boundary. In view of Fig. 2 the
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results would not be expected to be very sensitive to the orientation chosen. For the numerical computations we take the same temperature (T 2 0.57906To)aa in Sections 2 and 4. The planes are numbered from - co to + co, with n = 0 at the defect plane. Equation (4) holds for wehave n > 1 and n < -1. Forn=l log (1 - z&i}
+ &s + 21 + zs = 716
(11)
and for n = - 1 the same one, with 1 and 2 substituted by - 1 and -2. Finally we have log ((1 - %j)/z,> + g%l + z-1 + Zl = 7/6.
(12)
The three sorts of incoherent interfaces occurring in an equilibrium two-phase C&Fe alloy were computed. For the grain boundaries this is facilitated by symmetry: x, = x_,,. Starting with equation (12) we have only one unknown “too many”, and by the procedure described in Section 2 the appropriate asymptotic value (xa, = 0.040025 in the Fe-rich phase, x, = 0.959975 in the Cu-rich phase) is taken care of. Fig. 4 shows a substantial increase in Cu-concentration in and near the grain boundary in the Fe-rich phase: x,, = 0.7221. In the Cu-rich phase the enrichment at the grain boundary is necessarily small: x,, = 0.96917. The computation of the concentration profile of the interphe interface is more diilicult, as it is not symmetric and has no center of symmetry either. One now has bwo unknowns “too many”. The data in Fig. 4 were found as follows.* We choose some value for x,, to start with, and compute in the normal way the set of xi, x2, etc. so that one asymptote is correctly approached. This is repeated for other values of x0, and xi is plotted as a function of x,,. A second curve is obtained similarly, by taking care only of the asymptote at the other side. The intersection of the curves-rendered more exact by successive approximation-yields the desired concentration profile. At the Cu-rich side this curve virtually coincides with that pertaining to the grain boundary in the C&rich phase. At the defect plane the two values of x0 differ only 0.00001. For the next four planes towards Fe these differences are about 0.0002, 0.003, 0.03 and 0.2. Because of the vacancies in the defect plane, the Cu atoms (whose dangling bonds are “cheaper”) are enriched there. Then, it takes some distance for x to fall towards 0.04. This part of the curve is seen to be * This method was 8180 used to show that the coherent interface hes no esymmetric equilibrium, but only the “odd” and “even” ones, see Section 2.
MEIJERINU:
I
-“8
-7
-6
I
-5
CONCENTRATIONS
I
I
-4
I
I
-3
-2 -
I -1 distunce
AT
267
INTERFACES
I I I I 0 1 2 3 in half htttce pmn&-s4
I 5
Fro. 4. Concentr8tion prof%a WXOBBthree incoherent interfaces in a
two-phase Fe-Cu alloy: the grain bounderies in the Fe-rich and the Cu-rich phase, and the interphase interface. Computed for sane T/T, es Figs. 1,2 end 3, the incoherent boundary being 8pprOXkn8ted by 8 (100) lattice plane with 26% vacant lattice sites. The coherent interface profile (cf. Fig. 2) is drawn for comparison.
very similar to the coherent interface profile, but shifted over some 8 A. We can say that the chemical phase boundary and the crystallographic phase boundary (simulated by the defect plane) do not coincide. The incoherent phase boundary is Bplit up in a cdtment phme boundary and a grain boundary in that phase where it is cheapest. 6. DISCUSSION
The model used in Section 5 for an incoherent boundary is of course very crude, and we will not calculate y values. However, the splitting up of the incoherent interphase boundary into a coherent one pl& a grain boundary in one of the phases is rather striking and could give a more or less true picture, also for boundaries between phases with appreciably different lattice parameters, or even different crystal structures altogether. The general expectation that at a grain boundary in a solid solution the concentration may deviate markedly from that in bulk is put on a semi-quantitative basis by the calculations. One might well expect, however, a tendency towards enrichment of the lower 4neltin.g component rather than the most volatile component.. But in most cases that amounts qualitatively to the same. When the mixing enthalpy is relatively important, the enrichment at the interface may change sign with
concentration, as at a free surface (cf. Section 4). In that case the solute (in dilute solution) is enriched if the mixing enthalpy is positive, but impoverished if it is negative. One of the reasons of the author’s choice of the y-Fe-Cu system as example was the circumstance that the (incoherent) interface between the f.c.c. Fe-rich and C&rich phases has been found to have a smaller free energy than the grain boundaries in both phases. At 1000°C the y values for Vu-Fe”, “Cu-Cu” and “F-Fe” ‘were found(l4) to be in the ratio 0.61: 0.70: 1, the last one being about 850 erg/cm%. It has been found for several other metallic interfaces also that the interphase-interface free energy is smaller than the grain-boundary free energy in either phase,(l*) but the y-Fe-Cu case is particularly striking, because of the identical crystal structure of the two phases and the small difference in lattice parameter. Possibly an electronic double-layer effect causes a substantial reduction of the interphase free energy. This would be in accord with the state of affairs in alkali halides, where according to Spitzer’“) the interphase free energy is almost invariably between the two grain-boundary free energies. If the double-layer effect is important, it should also decrease the coherent interface free energy considerably. Indeed this appears to be the case. According to
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Table 1 Ycoh of Y-F&L at 1000°C would be about 170 erg/cm2 and this does not include elastic energy. Now for the Cu-Co system (which is quite comparable to Cu-Fe) the free energy of an incoherent lowangle interphase interface was estimated(ls) to be 50 erg/cm2. The coherent interface (with elastic interactions neglected) would thus appear to be of this order of magnitude only. According to the first *Dar&graph of this Section one would expect yCu_Fe to be somewhat larger than ycu_cu (both large-angle). It is difficult to see whether elastic effects due to the small difference in size of Cu and Fe atoms could bring Y,&_& below Ycu-CU. It should be remarked that Levin and Ivantsov(17) found 0.72 for YCu_Fe/YPe-Fe at lOOO”C,so that it is possible that ycU_cUis somewhat smaller than v_
YCu-Fe.
Grain-boundary segregation is important for several properties, for instance grain-boundary migration, (l*--~
diffusion
along
grain
boundrtries,(21)
em-
brittlement.(le-20) In many cases, however, the width of the concentration profiles across the grain boundaries is reported to be several hundreds of AngstrBms, or even several microns. Such zones cannot represent thermal equilibrium in a single-phase crystal aggregate. Only if the temperature is quite close to a critical mixing temperature could the effective width of the zone become so largec3) but even then the integrated excess concentration associated with the
VOL.
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boundary will remain small. Hilliard et aZ.(22)deduced from measurements on C&Au alloys that the grainboundary energies could be explained by excess quantities equivalent to about a monatomic layer of Cu or Au. REFERENCES
1. S. ONO, Mem. Fat. Eng. Kyuahu Univ. 10,195 (1947).
2. M. HILLERT, Acta Met. 9, 525 (1961). 3. J. W. CAHN and J. E. HILLIARD, J. Chew Phys. 28,258 (1958). 4. k. K&WEIT, 2. Phys. Chem. a, 163 (1962). 5. R. BECKER, 2. Met&k. 29, 245 (1937). 6. F. C. FRANK, Metal Surfacea, p. 1. Am. Soo. Metals (1963). 7. J. L. MEXJERINCI, Acta Met. 11,847 (1963). 8. C. HERRINQ, Phy8. Rev. 82, 87 (1951). 9. H. SATOand R. S. TOTE, Phys. Rev. 124,1833 (1961). 10. B. Y. PINES, Zh. tekh.$z. 22,190s (1952); V. K. SEMENCHENKO,Surface Phenomena in Metals and Alloys, transl. from Russian. Pergamon Press (1961). 11. C.HERRINO, MetaEInterjaces, p. 1. Am. Sot. Metals (1952). 12. M. C. INMAN, D. MCLEAN and H. R. TIPLER, Proc. Roy. rSoc. A 275, 538 (1963). 13. B. E. SUNDQUIST, Acta Met. 12, 685 (1964). 14. L. H. VAN VLACK, Trans. Am. Inst. Min. Met&. Engre 191,251 (1951); C. S. SMITE, Imperfections in Nearly Perfeet Crystals, p. 377. Wiley (New York), Chapman and Hall, London (1952). 15. D. p. SPITZER, J. p&s. Chem. 66,31 (1962). 16. V. A. PHILLIPS, Trans. Am. Inst. Min. Metall. Engrs 230, 967 (1964). 17. Y. N. LEVM cmd G. I. IVANTSOV, Phys. Met. Metallogr. 16, (nr. 4), 37 (1963). 18. H. M. MIEEK-OJA, J. Inst. Met. Land. 80, 569 (1952). 19. M. C. INMAN and H. R. TIPLER, Metallurg. Revs. 8, 105 (1963). 20. J. H. WESTBROOK,Metallurg. Revs. 9, 415 (1964). 21. V. I. ARKHAR~V et a~.,severalpaperscited h Raf. 19 and 20. 22. J. E. HILLIARD, M. COHENand B. L. AVERBACH,Acta Met. 8, 26 (1960).
AT INTERFACES
IN BINARY
ALLOYS*
J. L. MEIJERINGt Concentration profiles across interfaces and at a free surface of 8 binary f.c.c. alloy are computed, using the regular neerest-neighbour approximation. The y-Fe-Cu system serves as example, and most celculations 8re made for 8 temper8ture where the mutual solubilities are 4 et. o/o The coherent interface free energy is slightly anisotropic (but rather strongly at low temperatures) and nearly indifferent to 8 shift of the concentration-profile curve across the lattice planes. Computed concentration pro&s et the free surface of Fe-Cu solid solutions show substantial enrichment of Cu. Incoherent interfaces 8re approximated by planes containing 25% vacent. lattice sites. One then finds also 8ppreci8ble enrichment of Cu at the grain boundaries 8nd the incoherent interphase interface. The chemical and the crystallographic phase boundary are about 8 A apart. It is suggested that an electronic double layer is responsible for the low interface free energy found experimentally for the y-Fe-& interphase. CONCENTRATION
AUX
INTERFACES
DANS
DES ALLIAGES
BINAIRES
L’auteur a c&u16 les profils de concentration au travers d’interfaces et B une surface libre d’un allisge bin&e c.f.c., en utilisant l’approximstion r&+&e des plus proches voisins. Le syst6me y-Fe-Cu est utilise comme exemple, et la plupert des calculs sent feits pour une temperature & laquelle les eolubilites mutuelles sont de 4 8t. ‘A. L’6nergie libre d’interface cohbrente est leg6 16gbrement anisotopie (msis assez fortement aux basses temp6ratures) et presque indii%rente B.une veriation de 1s oourbe-profil de concentration au trevers des plans r&iculeires. Les profils de concentration calcul& $ la surface libre de solutions solides Fe-Cu montrent un enrichissement substantiel en Cu. Les interfaces incoh&entes sont consid&es comme &ant des plans contenant 26% de sites r&iculaires vacants. De cette fapon, on trouve Qgelement un enrichisaement appr6cieble en Cu aux frontibres de grains et B l’interf8ce entre phases inooh&ente. Les frontibres de phase chimique et cristallogrephique sont d&antes d’environ 8 A. L’auteur suggere qu’une double couche 6lectronique est responsable de 18 Bible Bnergied’interface trouv6e exp&imentalement pour l’entre-phase y-Fe-(%. KONZENTRATIONEN
AN INNEREN
OBERFLIiCHEN
IN BINPiREN
LEGIERUNGEN
Die Konzentrationsproflle an inneren und freien OberfWhen von k.f.z. Legierungen werden mit Hilfe der normalen Niiherung n&chster Nachbern bereohnet. Das System y-Fe-Cu dient 8113Beispiel. Die meisten Rechnungen werden fiir eine Temperstur gemacht, bei der die Liislichkeit etw8 4 at.- % betriigt. Die freie Energie kohiirenter innerer Oberfliichenist schwaoh anisotrop (bei tiefen Temperaturen starker) und fast indifferent gegen eine Versohiebung der Konzentrationsprofllkurve iiber die Gitterebenen. Die berechneten Konzentrationsprofile an der freien Oberfliiche von Fe-Cu-Legierungen zeigen eine starke Anreicherung von Kupfer. InkohItrente innere Oberf%iohen werden sngenithert durch Ebenen mit 25% leeren Gitterpl&tzen. Msn fmdet denn such eine merkliche Anreicherung von Kupfer an Korngrenzen und an der inkohiirenten ZwisohenphesenWohe. Die ohemische und die kristallogmphische Phesengrenze sind etwa 8 A voneinander entfernt. Es wird vermutat, d8B eine elektronische Doppelschicht fiir die experimentell gefundene niedrige freie Energie der Zwischenphaaenflitchein y-Fe-& verantwortlich ist. 1. INTRODUCTION
In a single-phase alloy in equilibrium the concentrations at the grain boundaries-and at the free surface-will in general deviate from the concentration inside the crystals. This effect may have practical consequences of some importance. The object of this paper is to try to calculate the concentration profile across such interfaces and also across an interphase interface in a two-phase alloy. The model used for these incoherent interfaces is closely related to that of a coherent interphase interface as used by One(l) and Hillert . t2) Therefore we will treat coherent interfaces first, and calculate some aspects not treated by these authors : different crystallographic directions, “even” and “odd” interfaces. Received July ‘7, 1965. emical L8boretory of Technical University, D&t%o%. t l
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2. COHERENT IN REGULAR
INTERPHASE INTERFACES BINARY F.C.C. ALLOYS
Consider a regular binary solid phase, with free enthalpy per mole : G = bx(l - x) + RT{x In x + (1 -
x) In (1 - x)}
(1) where x is atomic fraction, R the gas constant and T absolute temperature. If the interaction parameter b is positive, a miscibility gap results. The equation giving its boundaries can be written as r
T
c
l-x In --+4x=2 2
where T, is the critical temperature. For T < T, two values of x-say, x1 and 1 - x1obey equation (2) and stand for the concentrations of the two phases in equilibrium. In successive lattice planes parallel and close to a plane coherent
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interfaces the system is quite appropriate in view of the small difference in lattice parameter between the com~nen~. The mutual ~lub~ti~ of y-Fe and C?u are not symmetx%&, but may be rationalized at 4 at.% near the peritectic temperature. A slightly larger value was taken: 2 = 0.040025. This m&es TIT0 = 0.57906 = 413 In 10 simplifying numerical computation; e.g. equation (3) becomes h3lOW
Bas
2s
2 +----
1.5 distance
I.0 in iwlf Mice
0.5 pornmeters
0
FIQ. 1. Logarithmic plot of BXCBBB concentration (z - 2,) as a function of distance at one side of a coherent interphase interft~e, for 3 orientations in the f.c.c. lattice, computed with the n~~t-neigh~~ model of 0x10.~~’ azm- 0.040025,pertaining to T/T, = 0.67906. The cird at the to (z = 0.6 at the interfaoe) is oommon to the “odd” 80Putiona of all orient&ions. The drawn curve ia calculated with the method of Cahn and Hilliard,(**” where the difference equation is approximated by a differential equstion.
interface between the two phases z will vary monotonically between x1 and 1 - x1. Ifit is assumed that the energy term bzfl - 2) in (1) is only due to nearestneighbour interaction equation (2)-with a slight modification-is found to hold also(‘) for each of the successive x-values. The modification is that-while 2 in the loga~t~c term pertains to the lattice plane in question-in the term 4% it has to be replaced by the average of x for all nearest-neighbour positions of a site in that plane. Thus for the f.c.c. structure with interphase interface f/(100) one obtains the difference equation
3 % for all whole values of n, the number of the lattice plane. In the b.c.c. case treated by HillertP the second term is 2(~,+~ + x,.&. In this paper we examine not only (lOO), but also (111) and (110) interfaces, in the f.c.c. lattice. For (111) the seoond term in (3) is replaced by (z+i +
22, + xn+& and for (110) by &(x,s + 42,r 4 22, + 4x,1+ %I+3 ). We note in passing that the b.c.c. (110) case would be identical to the f.c.c. (111) we. For the &coherent interface model (cf. Section 5) an additional parameter will be necessary, depending on the specific alloy system. We had the y-Fe-& system in mind and chose the temperature acoordingly, using it for the coherent interfaces also. For such
-
%J/%J
+
G-1+
%
+
qa+r=
I.5
(4)
The approach to the asymptotio values 2, = 0.046025 and z_~ = 0.959975 is exponential. It is easily found that, for not too small values of n, x - x, is divided by a factor 10.20 for each successive (100) plane farther away from the interface, This factor is 12.99 for (Ill) and 4.807 for (110). It seems obvious that the set of z values, as a faction of n, will have a centir of symmetry at x = 0.6. Either x1 = 0.5 (the ‘“odd” solution) or zrr+ zs = 1 (the “even”’ solution). From one of these two starting points one calculates easily the course of 2 across the interface for the (100) and (111) orientations. For the even (100) solution, for instance, one begins with z, = 0.5 - log ((1 -
L&J/~&
(5)
With xa = 0.27 one 6nds, with the next equation, x4 < x, and with xs = 0.28, x4 > xa. Thus xa must lie between 0.27 and 0.28. With x2 = 0.27155 one finds 4 < x,, and with 0.2717, xs > x,. The computation of the successive x values is speeded up ~~ide~bly by the ~owl~e that (x%-r - x,)/ x,) must converge toward 10.20. In the (110) (% ease the computations are more tedious. The results are plotted in Figs. 1 and 2. It can be seen in Fig. 1 that the “even” and “odd” points of the same orientation lie quite well on one curve for (100) and (IlO), but less so for (111). For the different orientations together one common curve can be drawn only rather roughly. TABLE 1. Spctic ooherent interface free energies calculated for TIT, = 0.67906, of. Fig. 1 orien~tion
“even”
“odd”
I:::! (110)
0.701 0.714 0.709
0.708 0.714 0.709
unrelaxed 0.977 1.128 1.197
Unit: kT&*, where k ia Boltunarm’s constant, T* the critical mixing temperature and (i the f.c.c. lattice parameter assumed to be ~de~~dent of ~n~nt~tion. In Table 1 the calculated interface f&e energies y are given, with kTJaB as unit. They are compared to the results of the Becker treatment(s) where the conoentration is assumed to jump directly from 2, to 1 - 5,
MEIJERINC:
CONCENTRATIONS
AT INTERFACES
263
achieve negative d~~~~ is quite critical. In the even (110) case the concentmtions in the “central” 2 x 4 planes had to be varied, in the same sense, the ooe5oients P,, beii about p~~~io~l to the slope of the curve which can be drawn through the (110) points in Fig. 2. Without being too inaccumte one can say that if the successive l~tti~pl~e ~n~ntr~tio~ lie somehow on the curve in question, the lowest free energy of the interface is virtually attained. A shift CJ of the onrve across the lattice planes yields only minor A i variations in G, which passes a maximum and a a2 e minimum at the two symmetric positions. For the t (100) interface it was checked (of. footnote Section 5) that no asymmetric solution of equations (4) exists. If the odd interface were me&stable, suoh a solution ~ at would be expected, ss a saddle position in between. Especially for interfaces like (100) and (110) I attainment of the stable equilibrium will be difficult because of the virtual indifference of G once the 1 o” 2 3 v distance in t&f lai tke pmamtws concentration-profile curve has been reached somehow. Fro. 2. II& of ~ncent~tion pro& it&f (center of This is &ggmv&ed by the necessity of tra~~rt~, symmetry at z = 0.6) rtaininSto Fig. I. Here only say, Cu atoms to the interface, as the ooncentrstions the &ableinterface pror les are given, “even” for (111) and (loo), “odd” for (110). The complete (100) profile near it change in the same direction when the stable is drawn in Fig. 4. For sip see Fig. 1. ~n~nt~tion profile forms from e.g. the unstable one. without any relaxation. It is seen that the anisotropy These atoms must come from rather distant planes. of y is much smaller than in Becker’s approximation, Taking into aocount the value of (l”G/d$ near the although not nil, as in the &pp~~~tion of C&n snd asymptotes, the semi-~n~tive conclusion was reached Hilliard.@) that the number of lattice planes involved in the Further, the even (111) interface is seen to be more change-over must be of the order of ten thousand. stable than the odd one. For (100) this is also the Evidently, if the crystal containing a plane coherent case, but for (110) the odd interface is the stable one. interphase interface is only, say, a micron thick, the For these two orientations the y values differ only in most stable set of concentrations across the interface the fourth decimal, not given in the Table because its may differ markedly from that derived above and rtccuracy is uncertain, Still, the correctness of the correspond to an asymmetric position of the profile assertions made is borne out by examining whether the curve with respect to the lattice planes. The exact less stable interface is nnstable or metastable. solution of the difference equation (3) is based on the We call the free energy of a large crystal containing availability of infinite numbers of lattice planes at both the interface simply G (pro~~io~l to y). The sides of the interface. If, however, the orystsl is rather equilibrium condition (9), for all n, means that thin, the exact thermodynamic “bulk” concentrations SG/&s,, = o* for all x,. Thus also dG/dy = 0, where y are not attained and the average cmwetiration of the stands for any line&r comb~&tion of the concentratotal crystal must be taken G(areof, Then a change in tions : this of the order of.) over the number of lattice planes (6) can alter the ooncentrcttion profile from one near the Y =&4&a A even solution to one near the odd solution. with arbitrary finite coefficients p,. If, however, y can be so chosen that daG/dya is negative, the free 3. LOWER TEMPERATURES energy is a ~~~~ in that “direction,’ given by the We made also computations at T/T, = 0.43429, but ratios of the different p,. The equilibrium is then only for the coherent f.c.c. (111) interfitoe. This T is unstable. 75% of that in the preceding section, and 2, is only This was indeed found to be the case for the odd 0.0109. Naturally the concentration proille is steeper (100) and the even (110) interfaces. The choice of y to now, but wemention only the valuesof y, the unit being * The largehomogeneousparts of the cry&al far away from ~~=~~* as in Ts,ble 1: ~(111, even) = 0.954; ~(111, the interfaceeatingsa sourceand sinkfor the atoms neceesery odd) = 0.999, while the Becker treatment yields 1.106. to change2*.
,&yy,
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METALLURGICA,
Comparison with Table 1 shows that y increases by lowering !l’, as is to be expected. The relative difference between “odd” and “even” has risen to more than the four-fold value. The main object of the computations at this temperature was to have a look at cross-relaxation effects by segregation in the “central” plane of the odd interface (Z = a). In a separate (111) layer each atom has 6 neighbours against 12 in the complete lattice. In the regular nearest-neighbour approximation, therefore, the energy of mixing is half of that in bulk, while the Gibbs mixing entropy is the same. Consequently a homogeneous (Ill) layer, with equal concentrations of the two sorts of atoms, is stable in itself as long as T > +T,, but if T < &T, it should split up in two two-dimensional “phases” with unequal concentrations. For the (100) and (110) layers this is only the case for T smaller than T,/3 and TJS respectively. The “central” (111) layer at T = 0.43429 T, splits up in equal patches with x = 0.203 and 0.797. The average value of x remains 0.5, and thus the total interaction energy with the adjacent layers is not affected in the regular approximation. Only the interaction energy in the layer itself is lowered, and this is partly cancelled by the smaller entropy. One finds that y( 111, odd) is decreased from 0.999 to 0.985 units, so that the even interface remains the stable one. At T = 0 calculations are easy; the even solution (stable for all directions) is then identical with the Becker solution. For the interfaces (ill), (loo), (110) and (210) the y values are in the ratio 1:2/t: 48: di. The ( 111) and (210) interfaces have respectively the smallest and largest y of all orientations. When the reciprocal values of y are plotted in a polar diagram a tetrakishexaeder (210) results. In the case of a free surface, convexity or partial concavity of the polar I/y surface plot indicates whether the free surface is stable or not with respect to faceting.@*‘) The same holds true for interfaces. As, at T = 0, our l/y surface consists of plane triangles, each overall interface orientation (apart from (111) and (100)) is indifferent versus formation of “zigzags,” at least when edge free energies are neglected. For instance, a coherent plane (110) interface at T = 0 might just as well form undulations, the orientation varying in the zone between (111) and (lli). These indifferences at T = 0 are characteristic for pair-wise interaction energy models.@) If at a low but non-zero temperature the (even) concentration profile is calculated, one finds in first approximation log x1 = (m/12) log 5,. (7) Here m stands for the number of nearest neighbours of
VOL.
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1966
an atom that lie in its own lattice plane; z1 and 1 - x1 are the concentrations in the planes adjacent to the symmetry center, x, and 1 - x, those at the asymptotes. For example, if x, = 10d, x1 is about 0.1 for (1 lo), 0.01 for (100) and 0.001 for (111) interfaces. Now 7n/12 is proportional to the critical temperature pertaining to segregation in the lattice planes (see above). It follows that, as T approaches zero, x1 in (7) becomes equal to the miscibility limit for this crosssegregation. This is clearly connected with what has been said about indifference against “zigzagging” of plane interfaces at T = 0. It was checked, however, for the (110) interface that as T rises above zero, x1 becomes smaller than the cross-segregation limit. As we have seen, cross-segregation for the aEd (110) interface is possible as long as T < T,/6, but we presume that in this temperature region the even interface is still the stable one. A special situation arises when the lattice plane parallel to the interface does not contain bonds between nearest neighbours, as e.g. the (210) orientation, and also the b.c.c. (100) orientation. Then no cross-segregation is possible in the regular model used. Also, equation (7) cannot be valid for m = 0. Examination shows that in this case the even and odd interfaces are equally stable at T = 0. Aa T rises above zero, the odd interface becomes stable, thanks to the entropy. A tendency for GodhGevento decrease at increasing T appears both in thef.c.0. (110) case (where the difference is positive at T = 0 and negative at 0.57906 T,) and in the (111) case (where it is less positive at the latter temperature than at 0.43429 T,). Thus one might expect that the odd interface is stable up to TC when m = 0, as in the b.c.c. (100) interface examined by Hillertc2). His Fig. 3 (lower part) suggests that the even interface is the stable one. However, as he remarks in his Section 10, the difference in G will probably be quite small anyhow. Hillert was mainly interested in the periodic concentration profiles of intermediate states in coherently precipitating or ordering systems. It may be remarked that “long-period superstructures” like CuAu II will show periodic concentration profiles drr equilibrium. In the normal tetragonal superstructure CuAu I the Cu ooncentration in one (001) plane is constant, but in the orthorhombic CuAu II it goes up and down regularly in the [OlO] direction. According to Sato and Toth’s’ the adverse energy of the out-of-step boundaries is compensated by a Brillouin-zone effect. However, there can be no full energy compensation, as CuAu II only becomes stable at higher temperatures. Evidently it has a higher entropy than CuAu I, and this is probably due to the out-of-step boundaries being diffuse, just like coherent phase boundaries. The virtual indifference of the free energy with respect to a shift of the lattice planes relative to the concentration profile (see Section 2) makes it easier to understand that the “long” period of CuAu II is an irrational multiple of the distance between successive (010) planes.
MEIJERING: 4. FREE
CONCENTRATIONS
SURFACES
Looking at an f.c.c. Cu-Fe alloy, we call the interaction energies of Cu-Cu, Fe-Fe and Cu-Fe pairs ui, u, and ua respectively. In the thermodynamics of regular solutions and their coherent interfaces only 2u, - u1 - uz comes into play. Expressed in b (cf. equation (1)) this is equal to b/6N. The situation becomes less simple when there are atoms with missing neighbours (dangling bonds). For instance, a free surface can be expected to be enriched in atoms with the “cheapest” bonds, so that u1 - uz now plays a role too. For our calculations we took -6Nu, and -6Nu, equal to the sublimation energies of Cu and Fe: 82 and 97 kcal/mole. With b = 9 kcal (that is : T, about 2250°K) one finds -6Nua = 85 kcal. Two parameters define the problem completely: b/RT and the ratio between the difference in sublimation energy (here 15 kcal) and RT. The concentration profile-again at T = 0.57906 T,-below a (100) surface of the saturated Fe-rich phase was computed. The equations are the same as (4), apart from the first one (n = 1) which now is log ((1 -
~r)/~r} + xi+
52 = l/6
(3)
where x, is the atomic fraction of Cu in the outermost layer. Figure 3 shows a very substantial enrichment of Cu at the surface, x1 being 0.9835. For the saturated Curich phase (xc0 = 0.959975) the surface concentration is slightly higher : x1 = 0.9836. Naturally such computations can also be made for unsaturated solid solutions. For example, if = 0.01, the right-hand-side of equation (4), which :1og {( 1 - x,)/x,) + 3x oo, becomes 2.02564 and for n = 1 the r.h.s. of equation (8) is also increased by 0.52564. One finds x1 = 0.296. And for x, = 0.001 this becomes 0.0221. For dilute solutions the enrichment coefficient XJX~ goes’ to 10413= 21.54 at T/T, = 0.57906. Without specifying the values of T, ul, u2 and ua one can derive for (100) surfaces of dilute f.c.c. solutions kT In (x&c,)
= 4(ua -
u,),
(9)
k = R/N being Boltzmann’s constant. For non-dilute homogeneous solutions x1/x, cannot be expressed in a closed form. Then something general can only be said on the question which of the two elements will be enriched at the surface. It is found that the sign of (us -
u2w
-
AT
xm)
+
@I
is the same as that of log (x1/x,).
-
u2k.m
Here-if
(10)
z = 1 for
INTERFACES
266
‘rT----T
00 0
S”
AX
f
-
2
3 4 5 6 7 disfonce inhalf httice pammeters
B
FIQ. 3. Concentration profile at a free (100) surface of an f.o.c. saturated Fe-rich Fe-& solid solution, computed for same T/T, as Figs. 1 and 2; zw = 0.040026.
pure B-ur, u2 and ua are the energies of B-B, A-A and A-B bonds respectively. Only when us lies between u1 and u2 is the same element (the most volatile) enriched in the surface for all concentrations. Otherwise expression (10) changes sign somewhere. This possibility is clearly favoured by a relatively large (positive or negative) heat of mixing as compared to the difference in heat of vaporization of the elements. As the surface layer is thermodynamically intermediate between the bulk crystal and the vapour phase, it is not surprising that we have the same situation as in binary vaporization equilibria. There too, the vapour may be enriched in the least volatile component. The concentration where (10) goes through zero is comparable to an azeotrope. Expression (10) is essentially the same as equation (3, 14) of Pinea. This author has already derived the equations governing the concentrations near a free surface, leading to expressions for y in the cases of weak and strong adsorption. He does not compute concentration profiles. Similar effects as near a free surface are to be expected for the proportion of different sorts of atoms adjacent to lattice vacancies. (On the other hand, the equilibrium concentration of vacancies would depend on that proportion, more than on the bulk concentration of the alloy.) In diffusion experiments, the Kirkendall shift is usually in the direction of the constituent with the lower sublimation energy. This is in accord with the view that those atoms are preferential neighbours of vacancies, and thus more mobile than the other atoms. If the energy of formation of a vacancy and of a free surface are calculated for a pure metal according to this rough method of constant bond energies, the results are too high by, say a factor two, as is well known. In
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metals, if bonds are broken, some of the bonding strength is transferred to the remaining bonds.(ll) Although the crude picture used for a free surface crrnnot yield reliable values for y, it is believed that pronounced concentration effects should indeed be expected theoretically at the free surface of many alloys (and non-metallic solutions). Experimentally, Inman et uZ.(la)found at the surface of a Cu-Sb solid solution an excess quantity of Sb equivalent to 0.37 monatomic layers. And Sundquistos) concluded from equilibrium-shape measurements on small solid particles that excess quantities of the order of one monatomic layer of the solute are present on the surface of Ag with 4% Pb, Cu with 4% Ag and probably a-Fe with )% Cu. In all these cases the solute is more volatile than the solvent, and also the mixing enthalpy is positive, so that enrichment of the solute is to be expected theoretically. The same applies to the order of magnitude of the effect. The adjustment of the equilibrium concentration at a “fresh” surface-or after a temperature changemay give rise to transient surface effects, for instance in catalytic activity. Contamination by ambient gases may alter the surface-concentration profile of a solid solution drastically. 5. INCOHERENT
INTERFACES
We now turn to incoherent (high-angle) interfaces, not only between different phases, but also grain boundaries. As in Section 4, the system Cu-Fe is the specific example. To make computations along the lines of Sections 2-4 possible we use the following crude model of a high-angle interface: All successive planes, at both sides of the interface, are taken parallel to (100) ; in one of these planes, however (the interface), 1 of the atoms are missing. Thus the incoherent interface is treated as something in between a coherent interface and a free surface. The factor $ was chosen by comparing gram-boundary energy with the energy of formation of a vacancy in copper. It is taken to be independent of the Cu/Fe concentration. The defect plane is assumed to be filled by a random mixture of &, Cu, $( 1 - z,,) Fe and 4 vacancies, where a+,is the, as yet unknown; concentration in the interface. In the three “central” planes an atom has in the average one dangling bond. No (100) plane is crossed by a bond, and thus the defect plane effectively separates the two parts of the lattice. Therefore, although these parts are parallel in the model (and definitely not in reality) this should not be disturbing once one has accepted more or less the crude picture of the defect plane as an incoherent boundary. In view of Fig. 2 the
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1966
results would not be expected to be very sensitive to the orientation chosen. For the numerical computations we take the same temperature (T 2 0.57906To)aa in Sections 2 and 4. The planes are numbered from - co to + co, with n = 0 at the defect plane. Equation (4) holds for wehave n > 1 and n < -1. Forn=l log (1 - z&i}
+ &s + 21 + zs = 716
(11)
and for n = - 1 the same one, with 1 and 2 substituted by - 1 and -2. Finally we have log ((1 - %j)/z,> + g%l + z-1 + Zl = 7/6.
(12)
The three sorts of incoherent interfaces occurring in an equilibrium two-phase C&Fe alloy were computed. For the grain boundaries this is facilitated by symmetry: x, = x_,,. Starting with equation (12) we have only one unknown “too many”, and by the procedure described in Section 2 the appropriate asymptotic value (xa, = 0.040025 in the Fe-rich phase, x, = 0.959975 in the Cu-rich phase) is taken care of. Fig. 4 shows a substantial increase in Cu-concentration in and near the grain boundary in the Fe-rich phase: x,, = 0.7221. In the Cu-rich phase the enrichment at the grain boundary is necessarily small: x,, = 0.96917. The computation of the concentration profile of the interphe interface is more diilicult, as it is not symmetric and has no center of symmetry either. One now has bwo unknowns “too many”. The data in Fig. 4 were found as follows.* We choose some value for x,, to start with, and compute in the normal way the set of xi, x2, etc. so that one asymptote is correctly approached. This is repeated for other values of x0, and xi is plotted as a function of x,,. A second curve is obtained similarly, by taking care only of the asymptote at the other side. The intersection of the curves-rendered more exact by successive approximation-yields the desired concentration profile. At the Cu-rich side this curve virtually coincides with that pertaining to the grain boundary in the C&rich phase. At the defect plane the two values of x0 differ only 0.00001. For the next four planes towards Fe these differences are about 0.0002, 0.003, 0.03 and 0.2. Because of the vacancies in the defect plane, the Cu atoms (whose dangling bonds are “cheaper”) are enriched there. Then, it takes some distance for x to fall towards 0.04. This part of the curve is seen to be * This method was 8180 used to show that the coherent interface hes no esymmetric equilibrium, but only the “odd” and “even” ones, see Section 2.
MEIJERINU:
I
-“8
-7
-6
I
-5
CONCENTRATIONS
I
I
-4
I
I
-3
-2 -
I -1 distunce
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267
INTERFACES
I I I I 0 1 2 3 in half htttce pmn&-s4
I 5
Fro. 4. Concentr8tion prof%a WXOBBthree incoherent interfaces in a
two-phase Fe-Cu alloy: the grain bounderies in the Fe-rich and the Cu-rich phase, and the interphase interface. Computed for sane T/T, es Figs. 1,2 end 3, the incoherent boundary being 8pprOXkn8ted by 8 (100) lattice plane with 26% vacant lattice sites. The coherent interface profile (cf. Fig. 2) is drawn for comparison.
very similar to the coherent interface profile, but shifted over some 8 A. We can say that the chemical phase boundary and the crystallographic phase boundary (simulated by the defect plane) do not coincide. The incoherent phase boundary is Bplit up in a cdtment phme boundary and a grain boundary in that phase where it is cheapest. 6. DISCUSSION
The model used in Section 5 for an incoherent boundary is of course very crude, and we will not calculate y values. However, the splitting up of the incoherent interphase boundary into a coherent one pl& a grain boundary in one of the phases is rather striking and could give a more or less true picture, also for boundaries between phases with appreciably different lattice parameters, or even different crystal structures altogether. The general expectation that at a grain boundary in a solid solution the concentration may deviate markedly from that in bulk is put on a semi-quantitative basis by the calculations. One might well expect, however, a tendency towards enrichment of the lower 4neltin.g component rather than the most volatile component.. But in most cases that amounts qualitatively to the same. When the mixing enthalpy is relatively important, the enrichment at the interface may change sign with
concentration, as at a free surface (cf. Section 4). In that case the solute (in dilute solution) is enriched if the mixing enthalpy is positive, but impoverished if it is negative. One of the reasons of the author’s choice of the y-Fe-Cu system as example was the circumstance that the (incoherent) interface between the f.c.c. Fe-rich and C&rich phases has been found to have a smaller free energy than the grain boundaries in both phases. At 1000°C the y values for Vu-Fe”, “Cu-Cu” and “F-Fe” ‘were found(l4) to be in the ratio 0.61: 0.70: 1, the last one being about 850 erg/cm%. It has been found for several other metallic interfaces also that the interphase-interface free energy is smaller than the grain-boundary free energy in either phase,(l*) but the y-Fe-Cu case is particularly striking, because of the identical crystal structure of the two phases and the small difference in lattice parameter. Possibly an electronic double-layer effect causes a substantial reduction of the interphase free energy. This would be in accord with the state of affairs in alkali halides, where according to Spitzer’“) the interphase free energy is almost invariably between the two grain-boundary free energies. If the double-layer effect is important, it should also decrease the coherent interface free energy considerably. Indeed this appears to be the case. According to
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Table 1 Ycoh of Y-F&L at 1000°C would be about 170 erg/cm2 and this does not include elastic energy. Now for the Cu-Co system (which is quite comparable to Cu-Fe) the free energy of an incoherent lowangle interphase interface was estimated(ls) to be 50 erg/cm2. The coherent interface (with elastic interactions neglected) would thus appear to be of this order of magnitude only. According to the first *Dar&graph of this Section one would expect yCu_Fe to be somewhat larger than ycu_cu (both large-angle). It is difficult to see whether elastic effects due to the small difference in size of Cu and Fe atoms could bring Y,&_& below Ycu-CU. It should be remarked that Levin and Ivantsov(17) found 0.72 for YCu_Fe/YPe-Fe at lOOO”C,so that it is possible that ycU_cUis somewhat smaller than v_
YCu-Fe.
Grain-boundary segregation is important for several properties, for instance grain-boundary migration, (l*--~
diffusion
along
grain
boundrtries,(21)
em-
brittlement.(le-20) In many cases, however, the width of the concentration profiles across the grain boundaries is reported to be several hundreds of AngstrBms, or even several microns. Such zones cannot represent thermal equilibrium in a single-phase crystal aggregate. Only if the temperature is quite close to a critical mixing temperature could the effective width of the zone become so largec3) but even then the integrated excess concentration associated with the
VOL.
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boundary will remain small. Hilliard et aZ.(22)deduced from measurements on C&Au alloys that the grainboundary energies could be explained by excess quantities equivalent to about a monatomic layer of Cu or Au. REFERENCES
1. S. ONO, Mem. Fat. Eng. Kyuahu Univ. 10,195 (1947).
2. M. HILLERT, Acta Met. 9, 525 (1961). 3. J. W. CAHN and J. E. HILLIARD, J. Chew Phys. 28,258 (1958). 4. k. K&WEIT, 2. Phys. Chem. a, 163 (1962). 5. R. BECKER, 2. Met&k. 29, 245 (1937). 6. F. C. FRANK, Metal Surfacea, p. 1. Am. Soo. Metals (1963). 7. J. L. MEXJERINCI, Acta Met. 11,847 (1963). 8. C. HERRINQ, Phy8. Rev. 82, 87 (1951). 9. H. SATOand R. S. TOTE, Phys. Rev. 124,1833 (1961). 10. B. Y. PINES, Zh. tekh.$z. 22,190s (1952); V. K. SEMENCHENKO,Surface Phenomena in Metals and Alloys, transl. from Russian. Pergamon Press (1961). 11. C.HERRINO, MetaEInterjaces, p. 1. Am. Sot. Metals (1952). 12. M. C. INMAN, D. MCLEAN and H. R. TIPLER, Proc. Roy. rSoc. A 275, 538 (1963). 13. B. E. SUNDQUIST, Acta Met. 12, 685 (1964). 14. L. H. VAN VLACK, Trans. Am. Inst. Min. Met&. Engre 191,251 (1951); C. S. SMITE, Imperfections in Nearly Perfeet Crystals, p. 377. Wiley (New York), Chapman and Hall, London (1952). 15. D. p. SPITZER, J. p&s. Chem. 66,31 (1962). 16. V. A. PHILLIPS, Trans. Am. Inst. Min. Metall. Engrs 230, 967 (1964). 17. Y. N. LEVM cmd G. I. IVANTSOV, Phys. Met. Metallogr. 16, (nr. 4), 37 (1963). 18. H. M. MIEEK-OJA, J. Inst. Met. Land. 80, 569 (1952). 19. M. C. INMAN and H. R. TIPLER, Metallurg. Revs. 8, 105 (1963). 20. J. H. WESTBROOK,Metallurg. Revs. 9, 415 (1964). 21. V. I. ARKHAR~V et a~.,severalpaperscited h Raf. 19 and 20. 22. J. E. HILLIARD, M. COHENand B. L. AVERBACH,Acta Met. 8, 26 (1960).