Premelting at triple grain junctions

Acta metall, mater. Vol. 38, No. 8, pp. 1413 1416, 1990 Printed in Great Britain. All rights reserved

0956-7151/90 $3.00 + 0.00 Copyright © 1990 Pergamon Press plc

P R E M E L T I N G AT TRIPLE G R A I N J U N C T I O N S R. R A J Department of Materials Science and Engineering, Bard Hall, Cornell University, Ithaca, NY 148-1501, U.S.A. (Received 23 October 1989; in revised form 7 February 1990)

Abstract--Recent observations by Hsieh and Balluffi suggest that melting at triple points occurs just slightly below the melting point of aluminum. In this paper the premelting is explained in terms of a macroscopic (rather than an atomistic) analysis, by assuming that the entropy of local melting at triple junctions is equal to the entropy of melting for bulk metal. The results show that melt pockets at three and higher order grain junctions can be thermodynamically stable, their size being inversely proportional to the undercooling. For aluminum this amounts to a melt size of about 4 nm for an undercooling of I K, in fair agreement with the observations by Hsieh and Balluffi. Rrsurar----Des observations r6centes de Hsieh et Balluffi suggrrent que la fusion aux points triples se produit juste fiu-dessous du point de fusion de l'aluminium. Dans cet article, on explique la surfusion partir d'une analyse macroscopique (plutrt qu'atomique) en supposant que l'entropie de la fusion locale aux jonctions triples est 6gale ~ l'entropie de fusion du mbtal massif. Les rrsultats montrent que les poches de fusion sur les jonctions de grains d'ordre 6gal ou sup&ieur fi trois peuvent &re thermodynamiquement stables, leur taille 6rant inversement proportionnelle fi la suffusion. Pour l'aluminium, la dimension de la zone fondue atteint environ 4 nm pour une surfusion de 1 K, ce qui est en bon accord avec les observations de Hsieh et Balluffi.

Zusammenfassung--JfingsteBeobachtungen von Hsieh und Balluifi legen nahe, dab der Schmelzvorgang an Tripelpunkten von Korngrenzen kurz unterhalb des Schmelzpunktes in Aluminium auftritt. In dieser Arbeit wird dieses vorzeitige Schmelzen mit einer makroskopischen Analyse (statt einer atomistischen) erkl~rt. Angenommen wird, dab die Entropie f~r lokales Schmelzen an dem Tripelpunkt gleich der Entropie f/ir Schmelzen des Volummaterials ist. Die Ergebnisse zeigen, dab Schmelztaschen an Tripelpunkten dritter und hrherer Ordnung thermodynamisch stabil sein krnnen, wobei deren Grrl3e umgekehrt proportional zur Unterkfihlung ist. Bei Aluminium ergibt das eine GrrBe yon etwa 4 nm bei einer Unterkfihlung yon 1 K. Dieses Ergebnis stimmt mit den Beobachtungen von Hsieh und Balluffi fiberein.

INTRODUCTION The idea of premelting at grain boundaries may have many origins. It may be rooted in the observation that grain boundaries slide in a liquid-like fashion at temperatures that are substantially below the melting point. Mott's [1] concept of a mixed structure of grain boundaries consisting of regions of good fit and regions of poor fit may also reflect the ambiguity of "solid" and "liquid" properties of interfaces. Indeed considerable experimental and theoretical work on premelting has been reported off and on in the literature. A summary is given in Ref. [2]. Direct experimental evidence of premelting, however, has been difficult to obtain because of the extremely small width of grain boundaries and because of the need for in situ experiments. Hsieh and Balluffi [3] are probably the first to image the melting at grain boundaries below the ideal melting point. Although we cannot rule out the possibility of constitutive melting in their experiments as a result of very minor impurities, we assume this risk in the

interest of further and even more rigorous research on interfaces. Hsieh and Balluffi show the interesting result that the melt accumulates at triple junctions. While the reason for this becomes obvious when we look at their pictures, reproduced in Fig. 1, nearly all the earlier work has concentrated on premelting at two grain junctions. (For a fixed volume fraction of the melt, the free energy is minimized by segregation to the triple junctions because that also minimizes the curvature of the solid-liquid interfaces at the triple junction. The curvature places the liquid under negative pressure and the magnitude of that pressure is minimized by reducing the curvature.) Theoretical studies of melting at two grain junctions, for example by Bolling [4], have had to resort to the assumption that the chemical potential of atoms in the liquid that is trapped between two solid-liquid interfaces depends on the width of the liquid layer in the interface. Without this assumption it is not possible to estimate an equilibrium width of the liquid layer since the contribution of the

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RAJ: PREMELTING AT TRIPLE GRAIN JUNCTIONS melt is equal to zero. A relationship between the size of the melt at a triple junction and the undercooling is derived and the results compared with the measurements by Hsieh and Balluffi. ANALYSIS The phenomenon of premelting in a one-component system is analyzed by considering the difference between the Gibbs flee-energy of the state where there is premeiting and where there is not. The case of three grain junctions is presented in detail; the results for higher order grain junctions can be obtained by extrapolation. The topology of three grain junctions consists of interconnected channels throughout the polycrystal/f the dihedral angle (defined by 0 in Fig. 2) formed by the second phase segregated to the grain boundaries is less than ~/6. This is the range of interest since, as we find later, premelting is possible only if

0 <~16. We now wish to calculate the difference between the free energies of the two states presented in Fig. 2. Both are at the same temperature and pressure, under conditions where the normal state of the metal is a crystalline solid. We consider the general case where 0 can range from 0 to ~/6. For premelting to be possible the free energy of State II must be less than in State I. This difference in free energy, AG, consists of three terms: (i) the increase in free

,h.

State I

State

Tr

// Fig. 1. Disappearance of melts at triple junctions of aluminum at different degrees of undercooling (Hsieh and Balluffi [3]). solid-liquid interface to the total free energy of the system is independent of the width of the melt layer. In the present paper it is shown that melting at triple junctions can be predicted without resorting to the above assumption. The excess energy for premelting at the triple junction is provided by the geometry of the interfaces and can be precisely defined in terms of the dihedral angle and the ideal thermodynamic properties of the material. Premelting is shown to be possible if the dihedral angle is less than n/6, although a natural assumption will be that the wetting angle between a crystal and its

"4"

i/l/l//

Fig. 2. The difference between the Gibbs free energy of States I and II as a function of the radius of curvature of the melt pocket. The minimum in the free energy denotes a stable size of the melt pocket. This stable size depends on the dihedral angle, 0.

RAJ: PREMELTING AT TRIPLE GRAIN JUNCTIONS energy due to formation of a melt below the melting point, (ii) the increase in free energy due to the creation of a solid-liquid interface, and (iii) a decrease in free energy from the elimination of the two grain junction boundaries that are replaced by the melt pocket. If the radius of curvature of the solid-liquid interface is r, then the magnitudes of each of these three components, represented here by the corresponding subscripts, will be given by

AG = rZF~(O)AG~ AGi: = rF~(O)7, and ~Giii =

rFgb(O)Tgb

(1)

where the AG ~ represents quantities for one unit length of the triple junction normal to the plane of the figure. Equation (1) then follows from dimensional arguments since it must be possible to separate the effect of r and 0 on the volume and the surface area given the self similar shapes that would be generated if r were changed while keeping 0 constant. In equation (1), ~.~ is the interface free energy of the crystal-liquid interface and ~gb that of the crystalcrystal interface. In equation (1), AG,. is the increase in the free energy per unit volume when melting occurs below the melting point. If the undercooling is AT then AHmAT AG~ = - --

(2)

VmT~

where T~ is the melting point and AHm is the enthalphy of melting per mole and Vm is the molar volume. The functions F(O) are obtained merely from geometry and are available in the literature [5-7]

An analysis of Fv(0) shows that it is a function that decreases monotonically as 0 increases, going to zero when 0 ~ n/6. Thus the m i n i m u m disappears when 0 ~ n/6. Since the volume of the melt present in one unit length of the triple junctions, V3, is equal to V3 =

r~Fv(O)

(7)

it follows that the melt volume is maximum when 0 = 0 and goes to zero when Fv(O = ~/6) ~ 0. In order to compare the analysis to the measurements by Hsieh and Balluffi [3], we consider the case where 0 = 0 (see their Fig. 4, reproduced here as Fig. 1). We calculate the size of the melt by the diameter of the largest circle that fits interstitially, which leads to the following result z=0.3rc=0.3--

"/1 Vm Tm

(when 0 = 0 ) .

(8)

AHmAT

Equation (8) was applied to aluminum. The result is shown as the solid line in Fig. 3. The following values for the parameters in equation (8) were used: AHm= 321.9kJmol -j, V m = 10-Sm3mo1-1, Tr~ = 933.5 K, and 7~ = 0-42 J m - I (the value for 7a is an estimate: it has been assumed to be one half of the surface energy of aluminum just above the melting point [8]). The comparison between the analysis and the experimental points by Hsieh and Balluffi is shown in Fig. 3. The agreement is qualitative, but a quantitative agreement had not been expected in view of the approximate estimate of the undercooling in the experiments. In fact a leftward shift in the data by 0.12 K would give very good agreement. The trend in the data is certainly comparable with the analysis. APPROXIMATEAT (K) 0.25 0.50 i i

F j O ) = 2 c o s 2 0 s i n ; - ~x- + 3(0 - ~i sin20) E 0 ¢ 500

FJO) = ~ - 60

1415

0.75 i

z

and

400 F-

Fgb(O) = 3 cos O ( t a n 6 - - tan O).

(3) 500

A schematic plot of AG, where AG = AG=I - Gj = AGi + AGii - -

o 0 AGii i

is given in Fig. 2. It gives a m i n i m u m because (AGii - AGiii) is negative when 0 < n/6. The position of the m i n i m u m can be calculated by substituting equations (1~ and (2) into (3), differentiating with respect to r, and equating to zero. This exercise leads to the following simple result

~'~ rc = AGv

(5)

and

IAGcl = r~Fv(O)AGv.

2OO

(4)

(6)

THEORY

~._ ~

.•. ••a'~A~:~HSIEH •.~'~ 8 BALLUFFI

~ tO0 F.-

~ N

o 0

5 I0 15 20 DISTANCE FROM THE MELTAT 955.5K (/a.m)

Fig. 3. The change in the size of the melt pocket at triple junctions (the quantity z in Fig. 2) with undercooling. The solid line was calculated from equation (8). The estimate of the undercooling in the experiments was necessarily approximate [3]. Note that a shift in the data by 0.12 K to the left will give good agreement with theory.

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RAJ: PREMELTING AT TRIPLE GRAIN JUNCTIONS

Melting behavior at higher order grain junctions can be analyzed by simple extrapolation of equations (4) and (6). The critical radius r e is always given by equation (4), with the correction that at four and higher order grain junctions the surfaces enclosing the pocket will have two (equal) radii of curvature which means that r e = 2 × (Tt/AGv). Similarly the volume of the melt pocket in equation (6) will now be given by r~Fv(O) rather than r~Fv(O). The function Fv(0) would have different forms; these are available in the literature [5, 6]. DISCUSSION Both the experimental results and the analysis show that premelting at triple points is possible, even though significant premelting occurs only just below the melting point. Since the melt pocket is stabilized by the curvature of the solid-liquid interface, as given by equation (4), melts can form only at three and higher order grain junctions. An interesting consequence of premelting is that it occurs spontaneously without the necessity

of nucleation and growth. In other words, whereas a melt can be undercooled without freezing, a polycrystal cannot be superheated without melting. Acknowledgements--This research was supported by the Department of Energy at Cornell University under Grant No. DE-FG02-87ER45303. Helpful comments from the anonymous reviewer are gratefully acknowledged. REFERENCES

1. N. F. Mott, Proc. Phys. Soc. 60, 391 (1948). 2. J. K. Kristensen and R. M. J. Cotterill, Phil. Mag. 36, 437 (1977). 3. T. E. Hsieh and R. W. Ballufti, Acta metall. 37, 1637 (1989). 4. G. F. Boiling, Acta metalL 16, 1147 (1968). 5. P. J. Clemm and J. C. Fisher, Acta metall. 3, 70 (1955). 6. P. J. Wray, Acta metall. 24, 125 (1976). 7. R. L. Tsai and R. Raj, Acta metall. 30, 1043 (1982). 8. Aluminum: Properties and Physical Metallurgy (edited by J. E. Hatch), p. 16. Am. Soc. Metals, Metals Park, Ohio (1984).