R 87 Philips Re.s. -Rep. 3, 281-302; 1948

. I

RETROGRADE SOLUBILITY CURVESESPECIALLY IN ALLOY SOLID SOLUTIONS

~Y J. L. MEIJERING

SummaryUsing Gibbs's entropy of mixing m;'d Richards's rule for the entropyof fusion. of'metals, a graph is derived thermodynamically, whichserves to predict whether a solidus in a binary alloy system is retro-grade or not, knowing only e.g. the eutectic concentration of theliquid and the coexisting solid solubility at the eutectic temperature.No disagreement is found with the solidus curves determined experi-mentally up till now. When'the solid solubility is very low, a retro- .gràdc solidus is the normal thing to expect. •The graph can be tested in a more quantitative way by using thedata at thesolubility maxima found experimentally. Then systematic, deviations arise, at least when the solid solubility is rather low. Ananalysis of their possible causes leads to the conclusion that impor-tant deviations from the Gibbs entropy of mixing can occur in dilutesolid solutions when the solubility is small, owing to a loosening ofthe lattice by the substitution of solvent atoms by solute atoms ..Some remarks are made on solidus curves in non-metallic systems,transformation curves in iron alloys and retrograde solubilityin liquids.It is shown that Grube and Flàd's measurements in the Ni-Cr svstem,

- and the thermodynamic data calculated therefrom, arc ·ineor~ect.'

RésuméUtilisant l'entropie de mëlange de Gibbs et la règle de Richards pourI'entropie de fusion de mëtaux, l'auteur établit 'thermodynnmique-ment un graphique servant à prédire quand un solidus d'un systèmed'alliage binaire est rëtrograde ou non, en connaissant seulement p.e. laconcentration cutectique du liquide et la solubilité solide coexistanteà la température eutectique. On n'a pas encore trouvé d'écart aux "courbes solidus,' relevées expërimentalement. jusqu'ici. Lorsque la' ,solubilitë solide .est très faible, il faut s'attendre normalement à unsolidus rétrograde. .Le graphique peut être contrölë de manière plus quantitative enutilisant les donnëes à la solubilité maximum trouvé es expérimen-talement. Des ëcárts systömatiques ' apparaissent tout au moinslorsque la solubilité solide est plutöt faible. Une analyse de leurscauses possihles conduit à la conclusion que des écarts importants àl'entropie de mélange de Gibbs peuverit sé produire dans des solutionssolides diluées lorsque la solubilitë est petite, et dus à un relachementdu réseau Iors de la substitution d'atomes du solvant par des atomesdissous. Quelques remarques sont faites à propos de courbes solidus'dans des systèmes non-métalliques, de courbes de transformation dansdes alliages de fer et de la solubilité rétrograde dans des liquides.L'auteur ,montre que les mesures de Grube et Flad sur le systèmeNi-Cr et les données thermodynamiques calculêes à partir de cesmesures, sont incorrectes. ' •

1. Introduetion '..,-

In 1908 Van Laar 1) calculated several binary phase diagrams, startingfrom diff~rent melting points and enthalpies ("heats") of melting for the

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, .282 -, J. L. JI1EIJERING

components A. and B,and different mixing-enthalpy curves for the liquidand solid phases, while for the mixing entropies the Gihbs "paradoxen"expression was used. Among thc resulting synthetic diagrams were someof the type in fig. 1. Often a non-stable maximum solubilityin the soliduscurve occurred also below the eutectic temperature in diagrams of normal-type, like fig. 2. The possibility of fig. 1 seems to have been practicallyforgotten for a long time. This will he due partly to the fact thatVan Laar .did not stress' the 'point of the .retrograde solidus particularly

A x 8S.J640

.Fig. 1. Eutectic system with a retrograde solidus.

besides' the other 'results ,of his calculations *), but also to the absence." of an experimental example for fig. 1 until 1'926 (so far as the author is'

. aware], So when in that year Jeukins 3) found a'maximum in the Zn-richsolidus in the system Zn-Cd, which was confirmed by Stockdale 4), thiswas considered so abnor~al that a tentative interpretation was given,which made the maximum a diseontinuous one, requiring an allotropietransformation in Zn.

Hansen 5) disca~ded this interpretation, and gave a continuous solidusas infig. L, stating, however, that this course of the solidus was against

" ,

Fig. 2. "Normal" eutectic system. .

*) In his book 2) the maxima are shown in some diagrams, but not mentioned at aU inthe text. .'

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, .RETROGRADE SOLUBILiTY CURVES IN ÀLLOY SOLID SOLUTIONS ' 283

.'all '.previous' experience and that no adequate explanation for it was ;available. , , - ,

Only in 1942 .and afterwards were more examples found,. especially \by Raub and coworkers 6) 7) 8). The maxima in the silver-rich solidusc~ves of Ag-Pb, Ag-Bi and Ag-Tl were very pronounced indeed. Mean-while Scheil, apparently unaware of Ya~ Laar'!, results, stated in a shortpaper 9) that a solidus starting from the melting point 'bf A will mostlyend at x = 0 again at absolute zero temperature. The lower part ofthe curve is of course metastable, but when t~e eutectic te!llperature isrelatively low, there is a good chance that the maximum willlie in thestable part. Scheil expects that the system Al-Ga will be found to havea retrograde solidus. The same prediction is made for Fe-S, as it willexplain the fact that sulphur-bearing iron is more' succesfully forged at1200 °C than at lower temperatures (above the eutectic one), where itshows the phenomenon of red-shortness. .,We now propose to examine the matter in a more quantitative way, and

to derive an approximate relation which - in contrast with Van -Laar's: 'results - can be practically applied to experimental phase' diagrams,especially of alloys.

_'2. Enthalpy and entropy relations for the temperature dependence of '

solubilities

The sign of the temperature coefficient of a solubility is given by theprinciple o~,Le Chatelier. In fig. 3a ,and 3b two différent examples öfenthalpy ("heat content")~curves of two-phase systems at a certain tempe-rature are given. The sign of H is in 'the usual thermodynamical sense;thus in fig. 3a, on mixing, the system is absorbing heat from the surrpUnd~

, ings. When x stands for the atomic (or molecular] fraction of B, H is theenthalpy of one mole of the mixture. The linear centr~l portion gives H forthe heterogeneous mixtures.

853642

Fig. 3a and b. Two examples of enthalpy-concentration 'curves for a system with a two-phase region. . ,

.-

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If we take 'a concentratien just inside this two-phase region, the course'of the dashed extension of the one-phase ~-c~ve nearby teaches whether, the temperature must he raised or lowered to :make our mixture homo-ge~eous. According to Le Chatelier in fig. 3a both mutual solubilitiesincrease on heating; in ,fig. 3b the right-side solubility must decrease onheating. Ifwe call 'the difference in concentration of the coexistent phasesLlx and the difference in their respective If-values LlH, it is seen that dxJdTin the phase diagram has always the same sign as (dHJdx - LlHJLlx), "

, dHJdx pertaining to the same phase boundary as dxJd T of course. WritingS for the entropy of mixing for one mole, we can also substitute (dSJdx-LlSJLlx) for the above term in parentheses, since (dGJdx-LlGJLlx) is alwayszero .in equilibrium and G, the free enthalpy, is lI-TS (T is absolutetemperature) .

Also known since a long time are the quantitative versions of thesereTations: -

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,,'" 284 I • J, L, lI1EIJERING

./

dH LlH dS LIS·---- a_;-Llxdx dx Llx-----=

d,T d2G d2GT-dx2 dx2

(1)

.It is tohe noted that -d2Gfdx2 is always positive for stable and metastablephase~., '

• I .. \

-. (1) is ident}cal with ::van d~rWaals'~ coexistence equation for constant pressure. A sh~rt, ,derivation goes as follows:In equilibrium the tie-line (with slope LJG/LJx= (G'-G)/(X'-X»

f' must coincide with the double-tangent to the free-enthalpy curves of both.phases (fig. 4).When T is changed, oG/ax must remain equal to LJG/LJx, thus doG/ox = 'd IIG/Llx.

. " - a (aG) a (oG_\ ,,' ' a (LlG) , ,ec "Ox ox) liG Llx 'd ox ,=-ar- dT +~ dx, and d Llx =aT dT, because small changes in oX and

, x' (at constant T) do not affect 'the slope of the tie-line, owing to its tangency.

A oG s as dT ' o2G d' LJS dT hi hood ' I ° h (1).'.s aT = - , we get - ox + ox2 X - Llx ,w C IS 1 enttea WIt

These relations seem to be less universally known than they deserve. Grube and Flad 11)

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.Fig. 4. Free-enthalpy curves (at a certain temperature) of two phases; x and x' are their Iconcentrations in mutual equilibrium.

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, RETROG;RADE. SOLUBILITY CUR':ES IN ALLOY S~LII] SOLUTIONS .', . 285

-3·5 ...: ... ----------'-~M x ~f t ~3644

.,

hav~ measured the equilibrium 3 H2 + Cr203 ~ 3 H20 '+ 2 Cr (alloyed with Ni) at nóe-:and 1200 °C, and derived the (H, x)-curve for the Ni-Cr system therefrom. It is given infig. '5, turned upside down, as they take H positive when the surroundings absorb heatfrom the system. With the help of the above rules for the sign of dxjdT they would haveseen immediately that their results are in contradiction with the phase diagram ofNi-Cr(fig. 6), in which the solubilities increase with rising temperature. The trend of the solubi-Iities in it is surely right, and is also confirmed by Grube and Flad'.s measurements them-selves (in the heterogeneous region PH,O/PH, is constant for a given temperature).In a paper on the entropy of mixing in alloys Kubaschewski and Schneider 12) derive an

(S, x)-curve from Grube and F\ad~s data, which is negative over the whole range of concen-'

~--'

I •.,a

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Fig. 5. Enthalpy-concentration curve of Ni-Cr at about 1150 coC,according to Grube andFlad 11) *).

Ni x c,.<f;](US'

Fig. 6. Phase diagram of Ni-Cr (semi-schematical). '

*) The oalculation of the enthalpy by Grube and Flad was executed for round values of .;, "including the sequence 40, 50, 80 and '90 at. %. A straight line was drawn hetween :the points at 50 and 80 %. According to their own results, however; this straight partshould extend from 43 to 86 %, and the curved parts should be shorter and less .curved at the ends. Then the kinks, with their wrong directions, would have been stillmore striking. This also applies to fig. 7.

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286. J. L. MEIJERING

,trations (fig.'7)! This is termed to be unusual, but in qualitative agreement with a fallin the specific heat of Ni on alloying with Cr. Now the abnormal trend of this curve atx = 0 and 1 is practically impossible, as it would imply that e.g, the very first few Cr-atoms

. _substituted in the Ni lattice would take up ordered positions with respect to each other. Butsurprising is that these authorities on thermochemistry of alloys in spite of-the unusual.result fail to note that it is contradicted by the phase diagram,

o

-1~~i~------------x~------------~C~f 'f 536"6

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'Fig. 7. Entropy-coucentration curve of Ni-Cr at ± llSO oe, according to Kubaschewskiand Schneider 12) (see note on preceding page) ..

D,

-- -The discrepancy on the Cr-side of the system is not so surprising, as here the values, obtained for Hand S are extremely sensitive to small temperature deviations. E.g. a'difference of ouly 2°between the experiments at] 200°Cwith pure Cr and with alloys (atomicfraction of Cr = x) wouldentail an error in the entropy of not less than 1'8 x calfdegree,sufficient tö make it positive over the greater part of the system! In the alloys poor in Crperhaps equilibrium 'with Cr20a was not always attained.

3. Derivation' of a gx:aph for predicting whether an alloy solidus will heretrograde or not • ,

After this .digression, which serves to emphasize the usefulness of the _relations for the sigh of dx/dT in general, we apply them to the .case ofthesolidus. For a general survey the entropy relation is indicated, because •generally the Gibbs expression fo~ the entropy of mixing,

. . ~. ,

~R ~xInx + (l-x)ln (l-xH,

constitutes at 1east a reasonable approximation for both the liquid andthe solid phase, while' ·the enthalpies of mixing of course vary 'widely fordifferent systems. We will. restrict ourselves - firstly to (binary) alloys,which are the most important systems when we are concerned with solidsolutions; all cases of retrograde solidus curves now known experimentally

, are in f~ct alloy systems. A further schematization is now possible, asaccording to Richards's rule the entropy of melting of monatomic solidsis approximately ' 2'1 cal/degree. After a survey of these entropies formetals, calculated from Landolt-Börnstein's tables, wie chose the slightlygreater value of 1'1 R (2'2 cal/degree] as a mean stan_?ard value for "nor-mal" metals. The melting' entropies of Sb, Bi, Si,IS) and Ga are much

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"RETROGRADE SOLUBILITY CURVES IN ALLOY'SOLID SOLUTIONS' 287

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greater, about 2'5 fR. These elements have in the solid s~ate a structurewith low coordination number,. while in the liquid state the. structure is ;probably-more close-packed; at least Ga and Bi are knoWn to contract onmelting. The melting entropy of gray tin, by summation of that of whitetin and the transformation entropy, also turns out to he large (2'5 R-3 R). 'We assume therefore that this will also he the case with Ge, which hasdiamond structure like Si and gray Sn. We take the melting entropy permole, Sm, for Si, Ge, Sb, Bi and Ga to he (2'5 ± 0'2) R, for the other metals(1'1 ± 0'1) R *). We ignore the relatively small individual deviationsbe-

, cause of the scatter of the values of different authors for one metal, andthe temperature dependence of Srn; a consequen~e of the solid and liquidnot having the same specific heat. ', Now it is just an obvious step to assume that f~r metastablé (andevenunknown] modifications, too', the melting entropy lies hetween Rand 1'2 Ror, between 2'3 R ana 2'7 R, depend.ent on the kind of-structure and noton the individual metal. Thus Sm for e.g. body-centred cubic Ni and face-centred cubic Bi would fall hetween the lower limits, and that for diamond-structured Al hetween the higher ones., We now take the case that solid A and B (where B has the same crystalstructure as the stable modification of A, and thus in most cases will 'hemetastahle) have precisely the same melting entropy = CfR. Calling theatomic fractions of B in the solid phase and in the liquid phase x and yrespectively; we write for the entropy of the solid phase (which is thus the"phase with A·structure): . .

Sso·1= -1:;. ~xin x'+' (I-x) In (l"":'x)(

and for the entropy of the liquid phase:

Sliq = R ~Cf-y Iny-(l-y) In (I-y)(.

Thus'(dSjdx)sol,= R In ~(I-x)!x~ and'"

LIS=R Cf-y Iny-(l-y) In (I-y) + x Inx + (I-x) In (I-x) . 'Llx ' y-x

These two 'expressions must be equal when the soiidus (at the A~side) is ,,'just vertical (dTJdx = 00). After a simple transformation, and writing"Brigg's logarithms, one gets: '

" -

Cf l-y " I-y , ,log x = logy-O'434 - + -log (I-y) --, -,' log (I-x). (2)

Y Y y. ,

*) Se, Te and As are not considercd as basis metalshere.'I

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( In fig. 9 the four curves bordering the two shaded bands give this functionfor the values of th~ parameter a : 1'0, 1'2, 2'3 and 2'7 respectively (fromtop to bottom) . It is to be noted that the scale of x is logarithmic, that of y.linear, The calculation of log x for a given value of y is easy, as the last .-

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, 288 'J, L. MEIJERING

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atomic fraction

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Fig. 8. 'The curves give the entropies of the solid and the liquid phase, Gibbs's expressionfor the entropy of mixing being used, and 1'1 R taken as the entropy of melting per mole.The entropies of two-phase mixtures lie on a straight line joining the points repreaentingtbe coexisting solid, and liquid. In example I an alloy consisting of the saturated solidsolution plus a small amount of liquid has a smaller entropy than the solid solution of theparol' overall concentration. Therefore the solid solubility must, rise with tempernture,Jn c...ample II the opposite is the case.

RETROGRAI?E SOLUBILITY CURVES IN ALLOY' SOLID SOLUTIONS 289. I \ '. • • ...... • ."

term on the right-hand side is 0 bpth for x::;: 0 and for y ::;:1, an'd remainssmall there between'. Neglecting it, a provisional value for x, is firstlycalculated, ,;hich is substituted in the term in question. For the upper pairof curves this correction nowhere exceeds 0'02; a ~econd approximation isunnecessary. For the lower pair the correction can he, omitted, as it does,not exceed 6'003 h~re.

When (J forA and for B (with A-structure) vary e.g. between 1'0 and1'2, the curve for A-B lies inside the upper shaded band, also when thesetwo values of (J are not precisely the same.. Fig. 9 can be used for all binary alloy systems; .it is not applicable to .

. the solid solubility curves of any intermetallic phases, but only to thoseof the phases adjoining one of the pure metals. When this is -Sb, Bi, Si,Ge or Ga, we are concerned with the lower band, in other cases with,the upper band, which is thus by far the more important one. E.g., inthe system Al-Si we need the upper band for the Al-rich solidus, and.the lower one for the Si-rich solidus. Of course the values of y and x tobe plotted are atomic fractions ot Si in- the first case, and. atomic fractionsof Al in the second case.

Such a'b_and constitutes a boundary area, wherein the solidus is very_ steep, and - owing to.the uncertainties ofthe generalizations used - might

as well be sloping.to the right as to the left. In the area below the band we •must expect (with an increasing certainty when the point representing the

-,

'---I---t---I---I---t---I---j in sol

~ &w%o 04 0-5 0·6 0·7 0-8 oe "O-y

40% 60"10 80 % in liq 53648

Fig. 9. Graph used to predict wether a solidus is retrograde or not, Tlie four curves bordering,the two shaded bands represent equatiori (2) with (J = 1'0,1'2,2'3 and 2'7 respectively. Asfor the crosses, sec end of section 4 and table Ill.

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. ,\290 J. L. MElJERING

, .coexisting values of x and y is getting further away from the band) dTJdxto he positive, which is required for the "retrograde part" of a solidus. Onthe other hand in the area above and to the left of the band we must expectdTJdx to be negative, which is "normal". There is one restriction: when thecurve x = y is crosséd, dTJdx becomes zero and then positive again(LlSJLlx is negative for y < x). When the solid s~lution is more concentratedthan the liquid (as in Cu-Ni) the solidus must rise ~th concentration. Thisis trivial and has nothing to do with retrograde lines; so we have omittedthe curve x = y from fig. 9, and will only consider cases where x < y.

In general no precise knowledge of the solid solubility is needed, e.g.when at a certain temperature the concentration of the liquidus is 60 at.%B, the solidus can be predicted to have a positive dTJdx (at that tempe~rature), when we only know that (at that temperature) the solid solubility

'. is less than 4'6 at. % B, when A,is a normal metal.The three-phase temperature (eutectic, peritectic) where solidus and

" liquidus terminate on decreasing temperature, will be often the only onewhere x is known, e.g~ from an extrapolation of the solid solubilities atlower/temperatures, where the' A-rich phase coexists with another solidphase. Furthermore this temperature is crucial for classifying solidus'curv~s in the retrograde or the normal category. When dTJdx is negativehe~e, it will not become infinite at a higher temperature, because aminimum of x can be excluded. *)

When,on the other hand dTJdx is positive at the three-phase temperature,the solidus must' have a yertical tangent at some higher temperature, pro-vided it runs right through to the melting point; of pure A. There aretwo cases, however, where a solidus of a phase adjoining pure A terminates. '; ,

, '.

c

A X 8';3649

Fig. 10. System with miscibility gap in the liquid phase.

*), It would necessitate - with the assumptions made concerning the entropies - 'an,increase ofY.with T; such a retrograde liquidus has never been found in alloy systemsand is incompatible with the assumed Oibbs mixing entropy of the liquid. See lastsection. '

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RETROGRADE SOLUBILITY CURVES,IN ALLOY SOLID SQLUTIÓNS 29i

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\' \~In a three-phase equilibrium at risîng temperatures too, and will only beclassified as retrograde when dTfdx at this highest 'temperature is' already" 'negative .;gain. The first case is connected+with an allotropie transfor-mation in A; two examples predicted to be retrograde are the solidus curvesof yFe-èu ~nd yFe-S (table I). The second case is connected, with a'miscibility gap' in the liquid phase, and concerns the solidus CD in fig. 10.Practically always in such a case (example: Ni-Ag) x remains small alongCD, and y relatively large along EF, so that as a rule the solid solubilitywill increase steadily from C to D, and we will not call êo a retrogradesolidus: *) .

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4. Application to experimental phase diagrams

The phase diagrams in Hansen 5) and, sofar as possible, the subsequentliterature were examined on the' basis of fig. 9. Table I gives the soliduscurves, which are predicted to be retrograde; th~se in table IJ are labelled.as "possibly -retrograde", because here the points representing x and yat the three-phase temperature fall inside the, requisite .hand, the lower one,for th~ right-hand' column. , ..The eight solidus curves in the left-hand column of table I have- indeed

been found to be retrograde, 'while the other ones in this table (mostlygiven as dotted lines) are not known experimentally, !

As tbe two remaining solidus curves known to be retrograde, Cu"Cdand A~-Sb, are contained in table 11, no disagreement between theory andexperiments is found. The Cu-Tl solidus was found by Raub and Engel7)1;0 be just about retrograde when determined microscopically, but definitelynot so according to their X-ray measurements. Fig. 9 favours the secondpossibility. .

Systems where no solid solutions are reported' have not been includedin the tables; the majority of these.solidus curves with "zero" solubilitywill be retrograde. The number of experimentally known 'retrograde.solidus ,curves will increase ènormously with the application' of refined'techniques of determination 'of sm~ll solid solubilities. 'I'he smaller thesolubility, the higher teinperature (or rather: smaller concentration inthe coexisting liquid) at the maximum solubility. Very great Telativ~increases in solid solubility abov~ the eutectic temperature m~st beexpected when this temperaturp is low (in comparison with 'the meltingpoint of pure A) and the solubility in A is small. Voce and Hallowes 14)found a tenfold iricrease in the solid solubility 'of Bi in Cu even from' 600°(330° above the eutectic temperature) to 800°C; according to fig. 9' themaximum lies still. higher tip. .

*) The typical phenomenon of p~rtial liquefaction on cooling n'evertheless exists here of

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292. J. L. MEUERING

Very qualitatively we can say that it is only natural for a solubility to, increase with temperature, so long as the concentration of the coexistingphase (i.e. th~ liquid) is not coming "too near". '. '

The phenomenon of complete solidification 'at rising temp~ratures canhave practical consequences not only on hot working, but also on sinteringproperties., Of the last eight solidus curves in table II ("possibly retrograde"),the first four h~ve been determined experimentally; they are not foundretrograde, but Cu-Ag and Sb-Pb run practically vertical at thc eutectictemperature.P] Also very steep is the solidus of Ag-Hg 26), wher; log x,

'_lies only 0'09 above the upper band in fig. 9. It is to be noted that quan-titatively dTJdx is also controlled by the value of d2Gfdx2 (see equation (1)in section 2), which is normally great at small x. The Sn-In solidus was

( fo~nd by Rhines, Urquhart and Hoge 27) to be so steep at it's lower endthat they regard it with suspicion. Log x lies about 0'5 above the upper, band, but a value of 15 kcal for d2Gfdx2 was calculated to make that steepslope (± 50°,per at.%) possible. Subtracting RTfx(l-x) = 7'2 ]fcal forthe contribution of the entropy, there remains 8 kcal for d2Hfdx2• This is a

I 'normal value; when the enthalpy of mixing (in the Sn-structur~d phase]is assumed to be parabolic: H = -ax (l_:_x), d2Hfdx2 • 2a and the mini-mum of H (at x = t) = ':_l a, i.c. :_ 1 kcal/g.atom, ,

In' those nine cases where the solubility maximum has been determined,one can also plot this maximum.x against the value of Y at the tempera-ture where it occurs, and. expect these points to fall in the upper band offig. 9. The result can he seen from the crosses in fig. 9 and from table Ill. 'The fourth column giv,es the experimental y, derived from tmax and theliquidus curves' in Hansen; the fifth column gives the range of y theore-tically necessary to .make the solidus curve vertical, derived from Xmox

and the upp~r handjn.fig. 9. It is seen that -only with Ag-Tl falls Yexp

inside this range; in the other eight cases the deviations are all to the sameside. Owing to the uncertainty in log Xmax the range of Ytheor should be in." fact a little bit gre~ter for the lowest systems in ,table Ill, while a variationof several atomic per cent in Yexp ~spossible, owing 'to the uncertainty intmax' But in all eight cases the discrepancies are too great to be explainedin thisway, Thus our relation does not stand up to this more rigorous test:the increasè in solubility with temperature continues longer than expectedtheoretically. The success of the qualitative .test at the three-phase tempé-rature is due to the fact that this -temperature lies well below tmal< in allexperimental cases up till now, which by the way will be more or lessrequired for' a pronounced retrograde effect. 'I'hè chance that the 20 soliduscurves in the second and third columns of table I will indeed turn out tobe retrograde is more raised than lowered hy the deviati~n from theory.

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RETROGRADE SOLUBILITY CURVES IN ALLOY SOLID SOLUTIONS 293

TABLE I

Systems that theoretically should have a 'retrograde solidus

Zn-Cd Cd-Ph Co-AuAg-TI7)5) ca-ss Cr-AuAg-Pb 6)5) Zn-Sn 15)5) Au-~s 1~)5)Ag-Bi 7)5) Zn-In16Y Ni-BiAu-T17)5) Zn-Hg yFe-CuAu-Pb 7)5) Cu-Hg yFe-S *)Au-Bi 7)5) AI-Hg W-FeCu-Bi 14)5) Al-Ga *) W-Co 10)

Be-Al W-Ni 20)Be-Ag 17) Mo-Ni 21)

The solidus curves concerned ai:e those at the side of the first-named element in eachI

system. Where no reference is given, the experimental data are to be-found in 'Hans en 5),When a reference preceeds the reference s), the first one gives only solid solubility data,!I~d the Iiquidus from Hansen is used. -

':

TABLE H

Systems with a "possibly retrograde" solidus

•Cu-Cd 8)5)Au-Sb 18)5)**)Cu-AgPb-HgIn-Ga 22)

Sb-Pb 23.)Bi-Sn 24)Ge-Sh 25)Si-AgSi-Al

'TABLE HI

The subscript of table I also applies to this table. ~

yexp 8S

"Ag-Tl 0'075 5500 0'64 0'63-0'69 0'05 R 0'1 RAg-1;'b 0'028 6500 0'365 0'47-0'52 0'4 R ,1'1 RAg-Bi 0'027 5250 0'32 0'46-0:52 0'55 R 1'8 RCu-Cd 0'0255 6500 0'36 0'46-0'51 0'4 R 1'1 RZn-Cd 0'015' 3400 0'27 0'40-0'45 0'55'R • 2'0 RAu-Sb 0'01l5 5500 0'24 0'37-0'42 0'6 R 2'4 RAu-Tl 0'009 ' 8000 0'23 0'35-0'40 0'55 R 2'4 RAu-Pb 0'001l 8500 0'15 0'22-0'26 0'5 R 3'3 RAu-Bi 0'0006 9000 0'12 0'20-0'24 0'55 R 4'6 R

,

X _and y in atomic fraction of B; Ima:. in °C; R = gas co~sta~t.-----*) Already predicted by Scheil D)! ,

**) The solid solubility curve of Owen and Roberts shows a kink at 430 °C. No three-phasetemperature has been found here, however, by previous investigators 5).As three-phasetcmperature we take accordingly the eutectic one at 360 °C. • .

A-B ' I Xmax tmax' I ytbeor ' I I Ll dSjdx'

\' - .

..

- I

_'

., ,

": 294 , J, L, lI1EIJERING .

These deviations will he rather small (perhaps even of opposite 'sign)\ . .

~ the systems, Cu-Ag, Pb-Hg.: In-Ga, Sb-Pb (see table 11) and Ag-Hg,otherwise they should have been found to have a retrograde solidus.

In all five systems the value of y 'at the three-phase temperature (andthus .even more so that at a lower temperature where the metastable solu-bility ~aximum would be) is 0'60 or greater. Sb-Pb belongs to the Iowerband in fig: 9, but the other four make it probable that it is no coincidence

, \that Ag-Tl (see table Ill) does' agree with the upper band .•Apparentlythis ba~d should be raised above the theoretical course, but only to the leftof, 'say, y = 0'6. ' '

5. Ana!ysis of systematic entropy' deviations

Fig. 9, with the crosses included, will be aratber good guide in the field'of retrograde solidus curves in binary ahoy systems, but the systematicdeviátions ask for an explanation. This will probably tell u's something'about the thermodynamics of mixtures, which can be of interest, quite apartfrdm retrograde solidus curves.

F~om:the derivation of equation (2) it can be seen that there are two possi-bilities: LJSjLJx may be smaller or dSjdx larger than was calculated. The''sixth column of table III contains the values ijS, which must he subtractedfrom t?-e entropy of the liquid alloy of concentration Ycxp(with the mean'value of o , 1'1) to make LJSjLJx'equal to the theoretical value of dS/dx.If we call the vertical deviations of the crosses in fig. 9 from the middle ofthe upper band LJlog x, then ijS = 2'3 R YcxPLJlog x. Whe~ on the other

, hand Sliq is maintained on the theoretical value, a positive .extra linearterm x LJdSjdx must be added to S.ol to equalize dSjdx and LJSjLJx. ThisLJdSjdx equals ijSjycxpand not ijSj(ycxp_'_:'xmnx)'becausê besides the increasein dSjdx, LJSjLJx is decreased a little: " " , . .

The values of ijS (except that of Ag-Tl of course) are about as great as'the respective Gibbs mixing entropies; in all' four gold alloys the total.mixing entropy of the liquid would even have to be negative to remove the

',discrepanèy, which is very improbable. The' systems Cu-Cd 28) and" Zn-Cd 29) have been examined by measurements of vapou! pressures and,_ electromotive forces respectively. For Cu-Cd the deviation of the 'entropy

from the Gibbs value is somewhat more than 0'1 R; for Zit-Cd we calculatesomewhat less than 0'1 R from Taylor's activities. Both deviations applyto the concentrations in the fourth column of table Ill, and to tempera- •tures about 100° higher than tmnx' _. .

The chance that the form of the mixing entropy curve of the liquid phaseis responsible for u major part of the deviations in the eight cases (belowAg-Tl) in the table thus seems negligihle. We now consider the possi-bility that the melting entropies at tmnx are much less than at the melting

, -'

," ,..

RE_TROGRADE !iOLUBILlTY CURVES IN ALLOY SOLID SOLUTIONS '295

.. . ~. '

, points" of the pure metals, Large' differences in specific heat between liquidand solid would be required to arrive at the discrepancies in the sixthcolumn. Un'dercooled liquid' Ag and Au should have great specific-heat

, ~. ."humps", and still more so liquid Zn. In the system Zn-Üd matters arerelatively simple, because irnax is about the same as the melting point of Cd,and the temperature dependence of th~ melting entropy of this elementcan thus have no great influence. A simple calculation shows that under-cooled liquid Zn, between the jnelting point and 340 °C, should have amean specific heat of about 2t times its value above the melting point,to get rid of the discrepancy 6S. The assumption of such a, not directlycontrollable, specific-heat anomaly would not suffice: the specific heat ofthe Iiquid 27%Cd-~~ alloy between 420° and, 340°C (a wholly stafle _'.range) should also be much greater than the value averaged between solid·. Zn and liquid Cd, to convert the normal entropy of the liquid alloy at I420 0(: into the abnormal one at 340°C. It is all very unlikely. Still, an .'examination of the m~lting entropies of the' right-hand-side (B) elements •is iJ.eeded; not only because their melting points lie well below tmn~ (exceptin Au-Sb and Zn-Cd), a~d specific heats of superheated solids are 'stillmore hypothetical than those of undercooled liquids, but also because face: 'centred cubic Bi, Sb, Cd and Tl *) are not known, 'and their ptelting en-tropies have been assumed to be 1'1 R (see section 3). (Th;" last difficulty'does not apply to Ag-Pb, A~-Pb and Zn-Cd, where both e.".nents have the'same structure). It is easy to see that this value 1'1 R must be diminished'by the values L1dS/dx in the seventh colum~ to explain the discrepancies .by deviating entropies ofB (with A-structure) at tmnx' Nothing would he •left for the melting entropy of B in the case 'of Ag-Pb and Cu-Cd, while in".'the other six éases (not Ag-Tl of course) it would have to he negative.This is not acceptable, even for' superheated hypothetical modifications -.Perhaps one or few öf the eight discrepancies intable III could be thought

: to' be due to' a combination ofthe three factors considered, but to explainall eight of them in this way (and not get one for Ag-Tl) doe~ ,not lookfeasible. Fortunately they can be accounted for by a fourth factor:' theform of the mixing entropy curve of the solid phase (with A-structure) .

• This means that the Gibbs values of dS/dx, namely R ~ Hl-x)/x~, shouldhe raised with the values of L1 dS/dx in the seventh column of table Ill,·while the entropy of pure, B (with structure of A) is held on the samelev~l as that of pure A. If this is really the case, entropy anomalies mustalso b; apparent from' solubility data below the eutectic temperatures,where the liquid phase' does not come into the picture.

. ,

*) .The high température modification of Tl has heen found by Lipson and Stokes 30)to be body-centred cuhic.· ,

296 J. L. lIIEIJERING

, ,

When we plot the logarithm of the solid solubility x ofB in A (the coexist-ing phase having an atomic fraction of B that is fairly constant = x')against ljT, a straight line is mostly found when x remains small. When wewrite log x -- -ajT + (J, it is well known that 2'3 R a = dH/.dx - H'jx',where H is the enthalpy of the A-rich solid s~lution and H' that of thecoexisting phase *). The corresponding relation for the entropies is

dS S'2'3R{J = (-) ,-,'

, dx ex X

where. (dSjdx)ex is the difference between the real dSjdx and the theoreticalGibbs value for it; in contrast with dSjdx, (dSjdx)ex can be treated' as aconstant when x remains small. We now look for which' of the nine cases. in table III this (dSjdx)ex can be deduced, and compared with LldSjdx inthe seventh column. In Cu-Cd, Au-Sb, Au-Pb and Au-Bi there is an' inter-metallic phase coexisting, so that S' is unknowmAg-Bi is also not verysuitable, as S' (i.c, the entropy difference between normal solid Bi and its face-centred cubic modification) can only be guessed to be roughly 1·1...:....2·5= -=1'4times R'(see section 3). For Tl, S' willbe much-less, but no solid solubilitiesbelow the Au-Tl eutectic are known, and the three determinations inAg-Tl 31) ¥e to both sides of the transformation temperature of Tl (be-sides: these solubilities of Tl in Ag are already rather large). In the remain-ing 'two cases, however, Ag-Pb and Zn-Cd (where S' is virtually zero,as the ,coexisting phase is nearly pure B, with the same structure as A),(dSjdx)ex can be found, from the 'X-ray measurements of Chiswik andHultgren 31) and Boas 32) respectively. ~s with these low sol~bilities theerror range of log x is rather large, the straight lines can be shifted some-what without violating the experimental results. A definite positive valuefor (dSNx)ex results, however, which i~ at least some 1'8 Rand 2'0 Rfor Ag-Pb and Zn-Cd respectively. Such values, móstly of the same sign,occur in several other systems.' They (or rather the values of (J) are dis- .cussed' by Scheil 33), who shows that (positive) deviations should arise

'_ when x is not small enough (see espècically his fig. 11; it is to be notedthat there log ~xj(l-x)(is plotted). But in our opinion Scheil goes 'too far,when he regards the matter as settled herewith, and evidently assumes that_th~deviations would disappearwhen the (logx, ljT)-lines could be extended, experimentally to very low x. When the value of x used are only 1%, andf less, as is the case in Ag-Pb and Zn-Cd, less than O'SR can be reasonably'accounted for in this way. So there remain in these two systems v~lues for(dSjdx)ex that are of the same magnitude ~s'those of LldSjdx in table Ill.Thus it looks justified to attribute the systematic discrepancies in fig. 9to deviations of the mixing entropy of the solid solutions from the, Gi~bs*) This can also be easily derived from equation (1) in: section 2.

RETROGRADE SOLUBILITY CÛRVÊS IN ALLOY ~OLlD SOLUTIONS , • 297

, .

., 14~ .....

. - -.'" . .',

mixing' en;tropy. The physical meaning. of these deviations can scarcelybe other than this: ' : , '

When the solid solubility of B in A is small, the substitution of A-atoms by'B-atoms will generally loosen:the lattice considerably,: resulting in an extrarise of the entropy with concentration. . .

This also explains qualitativély why the deviátions LJdS/dx rise with'.. decreasing Xmnx in table Ill, and .seem to .he small in Ag-Tl, Cu-Ag,. Pb-Hg and In-Ga (end ofsection 4), where the solid solubility is rather large .

.when we plot log x against liT for several solutes in the same basis. I ".

.metal A we expect the following: the smaller the solubility is (in theexperimental range), the greater a will be, but also (according tq our pointof view) the greater (Jwill be. Thus on extrapolating the straight lines, they'must cross each other before liT - O.lsreached. Now this is just what.Fink "and Freche 34) found when plorting simultaneously the solubilities o(Ni, .Cr, Mn, Si, Cu and Mg *)in ~luminium.All six lines passed virtually throughthe point with coordinates liT --:-0·35.io-3 and lög x .' + 0'26 (x = atomicfraction). From their values of fJ we c~lcula~e' (dS/_dx)cx- S'/x: to he ,-4'8 R for Al-Ni and 2'3 R for Al-Cu. Admittedly the influence ofS'/x'can be somewhat different in the six cases (in five' of them th~ coexistingphase is an intermetallic one, in AI-Si it is the dianiond-str~ctured Si-richphase], but still the results of Fink a~d Freche gi;'e fUrther support to the'view.that the lattice-loosening effect. can 'give great deviations in dSjdx. ,

- It is only desirable to put this effect on a more quantitative theo~etical basis,' .. :.sothat we can see that the ~ëviations must indeed be ,expe~ted to. he sogreat.. .'If we càu the ch~racterjstic (Debije) temperat~re e, the dependence of .:

the entropy on .e (at temperatures suffi'ciently larger than e) is given ",by S = -:-c3R ln (9. In solid solutions e will be a function of x. The valuesfor pure' A and pure B (with A-structure!) we c~ll el and e2• Putting Sfor x. . 0 and x .' 1 at the same level, we find .') .

... . ~ .- -(dS/dx)ex = 3R (- d Ine/dx~lnel +·ln (2). " '.

-. -,-~.,

.'.. ,

. .-, We now make use of the -Lindemann formula -

. .In J~= const. + rIn Tm - t In Aw ;- t In V"

where Tm, A~ and V ar'e'absolute melting point, atomic weight and a~oinic'volume respe~tively. This for~ula giv~s very satisfactory results for pure <,

metals, and we think for a rough calculation it may be applied for solidsolutions too. While Aw and V may be interpölated, e.g. linearly, and. areof .minor importance in the casès we are interested in, the estimitión ofa melting point for a ·solid solution. presents a difficulty. It is obvious 1;.0

"') in th~.-order of increasing solubility.

,/ !

- ,

298 . J. L. MEIJERING

defin~ it as the temperature where the free enthalpies G of solid.and·li~idare equal; when diffusion were virtually stopped, the solid solution wouldmelt sharply at that point .. ln general it cannot be located between the_solidus .and liquidus, but in dilute solutions \ve shall show it' can.

In fig. 11 the free-enthalpy curves for the solid and the liquid phase aredrawn for a ~emperature just below the melting point of pure A. The

'- atomic fraction of B we call here u; x and y (given by the common t_angent).are the values of u for thc solidus and the liquidus at this temperature, andz is that for the melting "point". For the free enthalpies we can alwayswrite (for binary alloys, 01" otl;cr systems without a dissociating compo-nent B):

Gaol = R T u In u + al + bI u, (3)Gli,!= R T u Ir: u + a2 !f- b2 u, (4)

when u is small enough, so that terms with u2 etc. can he ignored. It fol-lows .that 1- = (à2-fLI)/(bl-b2). When we take the common tangent hori-zontal (which we are allowed to do óf course), we find by differentiatingGa~l-and GHq: .

"

R T (In x+ 1) + bI = R T (lny + 1) + b2= O. (5)

.This gives b~-b2 = RT In (Yjx). After multiplying the first and sé'condmembers of equation '(5) with x and y respectively, and subtracting theresults from (3) '(with U. x) ~md (4) (,Vith u = y), one finds

a2-d1 ='!1- T (y-x) .

_ z (1'::" xjy)- In (yjx)

Thus .(6)

y

-. y " 53650

Fig. Ll , Free-enthalpy curves of solid and liquid phases, at a temperature close below mel- ,. ring point of pure A. Common. tangent chosen as horizontal axis.

- .

. RETROGRADE SOLUiIILIT'Y CURVES IN ALLOY, SOLID SOLUTIONS

\ ""

299 \'" .

Therefore, whim the inti~l slopes of the solid~~ and liquidus curves, at themelting point of pur~ A, in the phase diagram are known, the initial slope.of the "melting curve" can be calculated, and thus also d ln6ljdx and(dSfdx)cx for x = 0, provided that the melting point of pure B with A-.structure is known, /'

Appreciable deviations or' dSjdx must obviously arise when the solid,solubility (or rather xjy) isvery small *). for the five systems in table 'IV;where A and B have reallythe same structurc, (dSfdx)cx is given as calcul-ated with the complete Lindemann formula, with linearly interpolatedvalues for the atomic weight and volume. Aw and V give contributions ofthe, same sign as Tm; those of ,V ~re small (0'2,R or less), bu~ those of Aw ' , '(except in Au-Pb) are quite appreciahle, as can be seen by comparing the ,....second column with the third, where only' the influence of Tm on e istaken into account: \ " , ,

In the last column the values of L1 dSjdx (for the three uppermostsystems, compare table II~) respectively (aSjdx.)c;~S' lx' (for the two,Al-systems, compare p. 297) are given, which should not differ, too' much -:'from (dSfdx)cx' It, seems' that th~ lattice-loosening effect' can be indeed '... -v

largely responsible for the entropy a~omàlies encou'ntered. '

TABLE IV

'~

I _

r '

Lindèmanncomplete only Tm

Ag-PbZn-CdAu-PbAl-CuAl-Ni

1'95 R'2'5 R'

.5'2 R4,'1 R7'~' R

1'35 R2-15 R5'0 R3'3 R6'7 R~ •. r

,,

. -l"lR

. 2'0 R- 3'3 R

2'3 R4'8 R

Recently Lawson 10), in a paper on segregation of solid solutions, also. . ~ ." .\

calculated approximately e as a function of x., The extra entropy can bewritten as 3RHI(Ql + Q2)' H being the mixing en:thalpy of one mole, ..and Ql and Q2 the heats of subliIll~tion of both pure components. Thisformula gives much smaller values; for Ag-Pb and Zn-Cd one gets onlyabout 0'2 R for (dS/dx)clCat x.= 0. The difference in results is due to theloosening of the lattice being viewed with regard to sublimation, by Lawson,and with regard to melting, in this paper.

*) For a rongh orientation we add that, according to the interpolation formula (6), z liesbetween the geometric and the arithmetic means of x and y. '

"Retrograde" *) liquidus curves are universally known (e.g, the solubility_ . " of sèveral salts in water decreases on heating), but from a theoretical

point of view their occurrence is ~uch less natural than that of .retrogradesolidus curves.

In fig. 12 a short piece of the entropy curve of a liquid is drawn to both_.sides of the point representing the concentratien of the liquid in equilibriumwith pure solid A (this is not essential; the solid may just as well be a*) The quotation marks are meant to indicate that here we are not interested in the

" maximum concentration of the Jiquidus being continuous 0': discontinuous (at the -three-phase temperature). -

.-

/ ,

300 J. L. MEIJERING

\

6. Retrograde 'Solidus curves in systems of non-metals

The melting entropy of compounds (per mole) generally increases "withthe number' of atoms in the molecule, although when rotation can occur

' ..in the solid state (e.g. in camphor), it may be small. Consequëntly (seesection 3, equation 2) we can say that, generally speaking, retrogradesolidus curves in systems of compounds are more confined to eases oflow solid solubility than in alloy systems. As the mutual solid soluhilitiesof compounds are often véry small, the number of theoretically retrogradesolidus curves must be very great. Perhaps in some cases they can manifestthemselves in anomalies in the behaviour of slightly impure substancesbeiow the_melting point.

7. Retro~rade allotropie-transformation curves

W é can also apply the considerations of section 2 on the two coexistingcurves which start from an ;llotropic-transformation point in A, when Band A form solid solutions: When the transformátion is lowered in tempe-rature by the ..addition of B, and the boundary of the high-temperaturephase reaches fairly great concentrations of B, the other boundary mayw~ll show a relatively large maximum concentration, because the transfer- ..matión entropy is generally small. The author knows of no other cases thanseveral iron-alloy ~systems, when; such a retrograde transformation curvehas been found. In Fe-Mn 35), F~,;Co36) and Fe-Ni 37) the a-boundarystarring downward from the a -+ y transformation k Fe at 900°C wasfound retrograde. A simple quantitative thermodynamical treatment is not : .. - .' ~feasible, as the differences in specific heat are relatively important' andchange the transformation entropy considerably. As is known, 'in iron itssign is even changed above 900°C, which gives rise to the reversed trans-formation y -+ a at 1400 °C and (in systems where B stahilizes a withrespect to y, e.g. Fe-Si), to the so-called y-Ioops, ,where both 'the a- andthe y-boundary are retrograde. \ .

8. Retrograde solubility in liquids.1

~ " _Jr-_.RETRÇlGRADE SOLUBILITY CURVES IN ALLOY sotro SOLUTIONS '.

, solid solution or e.g. a salt-hydrate}, The straight lin~ gives the entropiesof the -two-phase mixtures. It has to he less steep than the curve of.theliquid, ifthe concentration orA i!}the liquidis to decrease on rising temper-'ature (see section 2). But (metastable] liquid A must have a higher entropythan the solid. Evidently the complete entropy curve of the liquid phasemust have a partthat is convex to the concentration axis (d2Sjdx2 positive)., . .

s

5365/

Fig. 12. Part of entropy-concentration curve ofliquid phase and its heterogeneous mixtureswith solid A, in t.he case that the .soluhility of A in ~he liquid decreases on heating .•

< , i ~

This may partly fall in the stable portion of the curve (to the right infig. 12). The positive curvature' means that there is a rauge ofliquids that'lose entropy by mutual (isothermaljmixing. This is ofcourse quite pos-, sible (hy a stiffening or ordering effect), hut still rather curious.. ,

The same considerations can he applied to miscibility gaps in liquids.When the whole entropy curve of the)iquid phase has normal 'curvature, "and ,ve-draw the tie-line joining the two points representing the coexistingliquids, it is seen that both mutual soluhilities must increase on heating.When one or both soluhilities decrease, the entropy curve must have apart with abnbrmal curvature. Again, without further knowledge of the. .

, . system, we cannot say whether this part is confined to the metastahle andunstable regions óf the liquid. ~,

When a miscibility gap closes in-a lower critical point (e.g. in triethyla-mme-water), we aresure that ther~ is a region of stable liquids with abnor-m:al d2Sjdx2. This follows directly from the fact that à2Gjàx2 is zero at'the critical point, and positive below it. Thus - à3Gjà~2àT = à2Sjàx2 isposi~ive in the neighbourhood of the critical point, . . _. - .

Eindhoven, December 1947

'.

< (~

301. ,

,E . ~ f ,/

.. '

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I I

1.

., .

3p2\

J. I.. MEiJERING I

-, . . _ .REFERENCES

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