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The significance of wetting in reactor technology
J. Nuclear Enagy. 1955. Vol. 2. pp. 15 to 30.
Peqamon Press Ltd., London
THE SIGNIFICANCE OF WETTING IN REACTOR TECHNOLOGY By J. W. TAYLOR Atomic Energy Research Establishment, Harwell, Rerks (Received 7 IMarch 1955) Abstract-The fundamental forces responsible for the spreading of a liquid metal on a solid (or a liquid) surface are discussed, and a number of interfacial tension and spreading studies are analyzed in terms of the theoretical considerations. In the light of these, methods of modifying the interfacial tension in any given application are suggested. The signiticance of interfacial effects in the following technological processes is outlined: (i) liquid metal corrosion and intergranular penetration; (ii) heat transfer; (iii) mass transfer and zone melting; (iv) liquid metal slurry properties and stability; (v) extraction metallurgy. 1.
INTRODUCTION
Tti proposed use of liquid metals in reactors as coolants, in slurries, and in chemical processing, has focused considerable attention on the role played by interfacial tensions in these several applications. In what follows, the physical parameters responsible for the spreading of a liquid metal on a solid (or a liquid) surface are considered; practical studies of interfacial tensions are analyzed in terms of these factors and methods of modifying intesfacial tensions fundamental parameters; discussed; and the significance of such effects in the principal applications of liquid metals established. 2. (i) Thermodynamics
THEORETICAL
CONSIDERATIONS
ofSpreading
The prim&y forces responsible for the spreading of a liquid metal on a solid (or a second liquid) are the free surface energy components of the two phases, these energies existing as a consequence of unsaturated bonds present at the free surfaces. For a given phase, this surface energy may be expressed (BONDI,1953; HARKINS,1952) thus:
(zaF1p,,=a where F = free energy of the phase (ergs). u = surface area of the phase (cm*). a = surface free energy of the phase (ergs/cma), numerically equal to the surface tension. When a liquid metal (A) is placed in contact with a second phase (B) (which may be solid or liquid), spreading will occur if the total free energy change of the system is negative, i.e., if
dF
(2)
where dF = free energy change on spreading. 15
16
J. W. TAYLOR
Thus, considering the free energy change involved in the formation of one square centimetre of interface, and since aa, = aaAB = -aa, UA +
dF =
ugg
-uB
(4)
where bg = surface energy of the liquid. egg
=
interfacial tension of the interface AB.
oB = surface energy of the solid (or second liquid). Thus spreading of (A) on (B) will occur when: =A
+
OAB
<
aB
(9
Considering a case in which spreading has proceeded to some intermediate stage, short of complete wetting of the solid by the liquid, the equilibrium of a drop of
SOLID B
FIG. 1.
liquid (A) on the phase (B) may be represented as in Fig. 1. The equilibrium of the tensions involved may be expressed as: where 8 = contact angle. Of the three tensions responsible for spreading, only bg is known with any degree of experimental accuracy. bg has been measured for a few solid metals (see page 19), and it is possible, by methods outlined below, to estimate such values at least to an order of magnitude. GAB is peculiar to each system, and there is no ‘way of predicting it from the respective values of the quantities, cg and ug. In addition, the latter surface energies are sensitive to composition and to environment; e.g., cA will be altered by solution of(B) in (A), while uB is affected by the nature of the atmosphere above it; vapour from (A) may alter the value of bg as a result of adsorption. As a consequence of these several possible variations, a number of different contact angle conditions have been recognized (BAILEY and WATKINS, 1951); e.g,, “advancing,” “receding,*’ “initial,” “equilibrium,” etc. While these several angles are of no great significance in the applications to be discussed, nevertheless it is important to consider possible variations in bA and cB in evaluating (TABfrom contact angle conditions. Also, it is quite erroneous to consider all wetting phenomena in terms of two extreme contact angle conditions, i.e., cases of “wetting” where the contact angle is zero, and “nonwetting” cases where the contact angle is approximately 180”. Such a concept was common in the past. The spreading referred to above may be termed “macro-spreading,” as it involves the bulk surface energy of the solid. A second type of “micro-” or “grain boundary spreading” may also be described where a liquid metal is in contact with a grain
The significance of wetting in reactor techn0logy
17
boundary. The respective tensions are depicted in Fig. 2 and the equilibrium may be expressed as : UBB = 2c,*
COS g
where egg = interfacial tension between (A) and (B). a,~ = grain-boundary
tension between two grains of solid (B).
Q = dihedral angle. The dihedral angle (v) is formed as a consequence of the unbalanced tensions existing at the grain boundary/liquid metal interface. Its formation necessitates the surface migration of(B) atoms and it is thus confined to temperatures where this migration is
LIQUID
A
FIG. 2.
appreciable, The magnitude of (I@ is controlled by the ratio cA&BB. discussed in detail later (p. 25).
This will be
(ii) Surface Adsorption A further consequence of surface energy considerations is that of surface adsorption in a two- or higher-component system. By preferential adsorption of the phase of lower surface energy at the surface, the system as a whole attains a state of minimum energy; the extent of this enrichment or depletion, in a binary system, is given by the GIBB’Sadsorption equation: da r, = 2.3RT d bg a, where I’, = surface excess of component (1) relative to a plane of average composition (mol/cn?). da = change in surface energy for a change d log a, in the activity of component ( 1) @g/cm?. If (du/d log aI) is positive, i.e., the surface energy of the solution increases with increasing proportions of component (I), the surface excess (I’d will be negative and the surfaT layers will be enriched in component (2). Use is made of this effect when considering additions to a liquid metal, which will reduce the interfacial tension between it and a solid (or second liquid) metal. Expression (8) is usually applied to liquid solutions, but the underlying principle is general and the same e&ct may well be operative in solid solutions at suitable elevated temperatures. Thus DOBINSKY and JAGIIXSKI(1939) have shown by diffraction studies that surface enrichment of the phase of lower surface energy occurs in the systems Pb-Sb, J.N.E. 2-8
18
J. W. TAYEOR
Zn-Sn, Cu-Ca, Ag-Cd. Thus suitable alloy additions to the solid phase may also be effective in lowering the interfacial tension by means of a similar surface adsorption process, though this has not been considered seriously in the past. 3.
SURFACE
ENERGY
VALUES
Two of the three primary tensions responsible for spreading of a liquid metal on a solid metal are the respective surface tensions of the phases, and it is convenient to summarize the information available on these properties before considering studies that have been made of the interfacial tension (uBB of Fig. 1). The surface energies of a number of liquid metals and alloys have been measured experimentally and a review of these measurements has been presented elsewhere (TAYLOR,1954a). Where experimental values did not exist for a metal, it has proved possible to estimate the surface energy from a number of correlations which exist (TAYLOR, 1954b) between this and other physical parameters of the liquids. The experimental or estimated surface energy values for liquid metals at their respective melting temperatures are presented in Table 1. TABLE~---SURFACE ENERGIES OF LX
-
Metal
METALS AT
THEIR
MELTING
TEMPERATURES
T-
I
Surface energy (ergs/cm*)
914 1134 195* 1620*
!
Metal
Surface energy
Metal
(ergs/m*)
Surface energy (ergs/cm*)
926
Ag Al Au Ba Be Bi Ca 2 ce Co Cr cs CU Fe Ga
D
:z 3030* 608 610* 1740* 1420* 55* 1220 1460 735
Ge Hf Hg In Ir K Li Mg Mn MO Na Ni OS Pb Pd Pt
Pu Re Rb Sb Si Sn Sr Ta Te Ti Tl Th U
644 1510’ 471 630* 2310* 101 398 553 10502 2240’ 191 1570 2450’ 455 1280* 1845
; Zn Zr
580. 2480* 75* 380 860 616 165’ 2330. 300* 1325+ 444 1075* 1070’ 1710’ 2680. 818 1380*
* Estimatedvalues.
The surface energies of solid metals have been determined in only a few cases, these values being presented in Table 2 for the metals, copper, silver, gold, and tin. Attempts to calculate the surface energies of solid metals from fundamental principles (FRICHE, 1948) have not been successful, the calculated values being much larger than the experimental ones. An expression derived by GURNEY(1949) for the difference in free energy between the solid and liquid phases at the melting-point leads to improbable results. A measure of this difference in surface energies may, however, be obtained in two ways. The first is from the supercooling data presented by HOLLOMON and TURNBULL(1951), from which the interfacial tension between the liquid and solid phases can be calculated; this tension may be regarded as the’difference in surface energies of the two phases. The second estimate, obtained by the author, may be
The significance of wetting in reactor technology
19
derived from a consideration of the two factors responsible for this difference in surface energy. The first is the energy difference between the phases as a consequence of TABLE2.-SURFACE ENERGmOF SOLIDMETALS M&al
Temperature (“C)
Reference .
Au Au Ag cu Sn
Surface energy (ergs/cm*) _
SAWAIand NISHIDA,1930 TAMMAN and B~EHME., 1932 SAWAX and NISHJDA,1930 UDIN, SHALER,and WU~FF, 1949 GREENHILL and MCDONALD,1953
650-850 700-850 650-850 950-1050 215
1360 1810 1180 1450 685
melting; the magnitude of this effect (AoH,) may be gauged by considering the surface energy decrease that would result in the liquid state if a quantity of heat equivalent to the heat of fusion (A,) were added, i.e., A*Hf do Aa,, = x C, dT
(9)
where C, = specific heat of the liquid (cal/“/mol). da
-
dT
= temperature coefficient of the surface tension of the liquid (erg/cnP/“C).
The second factor is the difference in the atomic volumes of the liquid and solid phases. The magnitude of this (Aa,) was estimated by considering the surface energy change equivalent to the change in volume on melting, using the approximate relationship which exists between the surface energies of metals and their atomic volumes, Fig. 3. The final difference in surface energy of the liquid and the solid would thus be: EAa = Aa,, + Aa,
(10)
Table 3 compares the difference in surface energies obtained by the two methods with TABLEQ.-DIFFERENCESIN Surface energy (ergs/cm3 Metal
SURFACE
ENERGYBETWEEN
AU A%, (ergs/cm*) (ergs/cm3
Solid
LIQUID
AND
SOLID
MBTAI.8
Aa CA0 Solid-liquid (ergs/cm*) (ergslcma)
Interfacial tension (ergs/cm*)
cu Ag
1400 1180
40 30,
100 40
140 70
180 250
177 126
,Au Sn Na K
1800 1360 685 -
40 25 8 7
80
120
225 665 70 -
132 55 20
Li
-
10
I
-
20
J. W. TAYLOR
the difference existing between the surface energies of solid and liquid gold, tin, copper, and silver. The agreement is reasonable, considering the inaccuracies of surface energy measurements; the two methods of estimation employed are consistent, both in the previous series of metals and in the alkali metals lithium, sodium, and potassium. )‘5-
PO
I 05
I I.0 Loq ATOMIC VOLUME
I.5
I PO
FIG. 3.
4.
SPREADING
AND
INTERFAC,IAL
TENSION
STUDIES
The parameter of primary interest in most interfacial processes is the interfacial tension (cram of Fig. l), and this must be measured experimentally. In view of the difficulty of making such measurements, the number of detailed studies have been few, but these, in conjunction with a number of qualitative observations, serve to indicate the principles governing the magnitude of this tension. BAILEYand WATKINS (1950), using the principles of phase equilibrium established by SMITH (1948), have determined the magnitude of the tensions involved in the system solid copper/liquid lead. By suitable thermal treatment of copper in hydrogen, argon, and in lead vapour, and from contact angle measurements of lead on copper, they evaluated the following tensions: (i) surface energy of clean solid copper; (ii) surface energy of solid-copper in presence of lead vapour ; (ii) interfacial tension between solid copper and liquid lead; (iv) grain boundary tension of copper. These values are contained in Table 4. Values found by SEAIU(1950) for the same system are given for comparison. The following points may be noted. The interfacial tension oc,+r,, = 340 ergs/ems is very much flower than the difference between the two tensions, orb = 450 ergs/cm* and cc,, = 1800 ergs/cm*, and this in a system where there is only a limited degree of interaction between the solid and the liquid components. Secondly, the surface energy of copper in the presence of lead vapour acan = 780 ergs/cm* is less than half
The signiticanceof wetting in reactor technology
21
that of solid copper in the absence of the lead vapour. This feature stresses the need for considering the effect of surface adsorption on the solid when deducing interfacial tensions from contact angle measurements. If this reduction in solid surface energy had been ignored, the interfacial tension deduced would have been 1510 ergs/cm2, as opposed to the true value of 340 ergs/cm 2. It is of interest that the surface energy of solid copper in the presence of lead vapour, 780 ergs/cmS, is intermediate between that of clean solid copper, 1800 ergs/cm%,and the interfacial tension o&,/pb = 340 ergs/cm2. TAPIZA--SURIJA~E ENERGIES OFTHESysIgMSOLIDCDPPER-LIQIJJD LEADSURFACE Surface energy @r&m*)
Boundary Solid copper (uh) Copper with adsorbed lead (a~) Copper/liquid lead interface &,,/p,,) capper grain-lJo@ary (ml/c!“)
1800 780
720 510
E I
-
BAILEY and
Reference
WATKINS
(1950)
The value of 1800 ergs/cm2 for the surface energy of solid copper is larger than UDIN, SHALER,and WULFF’Svalue (1949), but this reflects the experimental error involved in such measurements. Similarly, a number of other interfacial tensions have been measured. VANVLACK (1951) established the liquid copper/solid iron interfacial tension at 430 ergs/cm2. This is of a similar magnitude to the value of 340 ergs/cm2 for the liquid lead/solid copper value. Both systems show a limited degree of solubility of the solid in the liquid phase at the temperatures concerned. The copper/iron tension is much closer to the difference in the respective tensions of the two primary phases, viz. (1460 + 200 - 1220 = 440 ergs/cm2), if 200 ergs/cm2 is allowed for the difference in the surface energy of solid and liquid iron. TABLE ~.SURFACE ENERGIESOF OXIDE-METAL MXRFACES
Surface energy (ergs/cm’)
Interface
Interface
1800 440 905
&O&NW) A1,4(s)/4W) A&O,(s) surface I
ZrCMs)/W) z~,(wJws) zro*(s) Sllrf~
surfaceenergy hgS/UIl’)
970
265 590
I
NORTONand KINGERY(1953) give values for the liquid nickel/alumina interface, 1860 ergs/cm2, and for the liquid nickel/zirconia interface, 970 ergs/cm2; their complete results are given in Table 5. The interfacial tensions between liquid metals and oxides are much higher than values for metal-metal interfaces, and in fact the
J. W. TAYLOR
22
interfacial tensions are higher than the reported free surface energies of the solid phases. These oxide-metal systems are quite inert, and this fact will be referred to subsequently. NORTON and KINGERY’Sother values, Table 5, indicate that the grain boundary tension of oxides is approximately half the corresponding surface energy; a similar ratio of energy to tension for copper is shown in Table 4. A number of qualitative observations of spreading and contact angle relationships serve to establish some general ‘principles. In a study of the spreading of a large number of pure liquid metals on a variety of solid metals, BAILEY,Fox, and WATKINS(1948), and Fox (1948), distinguished the following three conditions : (i) “wetting,” in which a stable liquid coat was left after the bulk of the liquid was drained; (ii) “non-wetting,” TENSIONS TABLE6.-INT~RFAUAL
Solid metal
ci’ SOME LIQUID MIIT~/SOLID
Fe
Ni
1660* 830
1770’ 885
METAL -
T T
. =Lx
caltact
a+
0 tension (e&cm~ wettin b&laviour
Intcrfaci@
??
Sn
Sn
Pb
Cd
620
620
460
610
4; D.W.
2 W
2g: W
72
640
D.W.
1
Esthated v&ea (6). D.W. De-wettingsyystcm. W. Wettingaystsm.
where no spread of the liquid over’the solid took place; and (iii) “de-wetting,” where the liquid Ghn retracted into discrete droplets when the excess liquid was drained away. Systems exhibiting “wetting,” i.e., low’interfacial tension, showed either a tendency to compound formation or extensive solid solution, while “non-wetting” systems, i.e., high interfacial tension, showed no such tendency to interact. In the course of these studies WATKINS(1948) reported contact angle values for a number of systems, and it is instructive to derive therefrom approximate interfacial tension values using equation (6) (p. 16), making the quite arbitrary assumption that the surface .energy of the respective solids in the liquid metal vapour is half that of the metal in an inert atmosphere (cf. copper in lead vapour, Table 4). The interfacial tensions thus derived are compared with the condition of “wetting” of the system in Table 6. This would suggest that the systems showing “wetting” have interfacial tensions below about 300 ergs/cm*, and that those with tensions greater than this exhibit “de-wetting” or “non-wetting” conditions, bearing in mind the assumption made in calculating these tensions. Similar qualitative observations have been made by NORTONand KINGERY(1953) in relation to the’spreading of the transition group of elements on a variety of materials. In general, they established that spreading was facilitated by any interfacial reaction between the solid ceramic and the liquid metal; where this reaction was non-existent or limited, the spreading was restricted, i.i:, the contact angle was high.
23
The significance of wetting in reactor technology
So far only the spreading behaviour of pure metals on solids has been considered, but the case of alloys is particularly interesting. In a study of the contact angle behaviour of a series of lead-tin alloys on solid copper, silver, nickel, and iron, it was clearly demonstrated (Fox, 1948; BAILEY,Fox, and WATKINS, 1948; WATKINS,1948) that where compound formation occurred,between the liquid alloy and the solid, the contact angle was low. Furthermore, it appeared that a definite minimum existed in the contact angle-alloy composition relation at 60‘per cent lead-40 per cent tin, when the alloys were in contact with iron and nickel; this was evident to a less extent with silver. Now it is, apparently, the tin which is effective in lowering the interfacial tension,.as may be seen from the fact that the system iron/liquid tin shows “de-wetting” behaviour, while the corresponding lead system is “non-wetting.” In the liquid
TABLE7.-INTZRFACUL TENsIONS m’rwxx~ Fe-Si AuoY ANDA&O, Composition (A Si)
Surfae$$;;f I
OQ60
liquid 1 Conta$ angle / Interfa$lal
0.265 0.97 3.10’ 8.72 I
I
I
138 132 125 119 115
1450 1400 1425 1440 1425 I
2000 1885 1747 1640 1537 I
lead-tin alloy, the tin will be preferentially desorbed from the surface, and if this surface deficiency is calculated for this system, it is found that the relative effect of this deficiency will be at a minimum round about the 40-50 per cent composition. This is probably responsible for the observed minimumeontact angle values in this region. Similar effects have been observed by NORTONand KINGERYin the case of iron-silicon alloys spreading over alumina. Their results, which are summarized in Table 7, show that the interfacial tension drops from 2009 ergs/cm2 at 0.06 per cent silicon to 1537 ergs/cm2 at 8.72 per cent silicon, while the corresponding surface tension of the liquid remains essentially unaltered. This decrease in interfacial tension is a consequence of adsorption of silicon at the interface, and the relatively strong tendency for silicon-oxygen interaction. A number of other qualitative observations on the spreading of liquid alloys on solid metals may be interpreted in a similar manner. The addition of I.5 per cent silver did not affect the spreading behaviour of lead or of an alloy of 99 per cent lead + 1 per cent tin on nickel. No effect would be anticipated, as silver would not be preferentially adsorbed on the surface, due to its high surface tension; in addition it has little tendency to react with nickel. ,VAN VLACK (1951) found the addition of silver to liquid copper resulted in an increase in interfacial tension between the liquid phase and iron. This is a consequence of the surface adsorption of silver, which has a surface tension lower than that of copper; in addition, silver and iron are immiscible in the liquid and solid states, and therefore the interfacial tension in the presence of silver will be high. An interesting case of additives affecting spreading is that in which 0.1 per cent nickel added to lead markedly reduces the interfacial tension against steel. On the basis of relative surface energy values, nickel would not be adsorbed preferentially,
24
J. W.
TAYLOR
and the lowering of the interfacial tension must be attributed to the strong tendency for the nickel to alloy with the iron. It would appear that, where there is a strong tendency for alloy formation to occur between the solid and a component in the liquid, surface adsorption effects will be overcome and a low interfacial tension will result. MODIFYING INTERFACIAL TENSIONS 5. FACTORS Before considering the several technologically important applications of interfacial effects, it is convenient to consider briefly what factors may modify interfacial tensions and spreading, since the foregoing discussions apply to ideal conditions. Surface lilms are likely to be a signiticant factor under practical conditions. Such films may be present on the liquid or solid phases. Whereas the former can generally be ruptured easily, the latter are likely to prove a more permanent and effective barrier to contact between the solid and the liquid. Two types of film may be present on the solid. The one is an adsorbed gas tim; this may remain effective up to temperatures r-t which desorption occurs, i.e., 300°C. The second type of tilm is a chemical one. The stability of such a film in the liquid medium will be controlled by free energy considerations, and should this film be thermodynamically stable it is likely that the resulting interfacial tension will be considerable, as found by NORTONand KINGERY(1953). The action of a flux in facilitating spreading and lowering interfacial tensions is to dissolve such surface films, although it has been suggested that frequently there is also an electrochemical effect (BAILEY,1948). A second significant method of altering the interfacial tension is by control of the liquid metal composition’; the principles involved have been outlined in the previous paragraph. To reduce the interfacial tension the added element should be such that its surface energy is lower than the value of the solvent liquid, so that it is preferentially adsorbed at the surface. In addition, it should have a strong tendency to interact with the solid, either to form intermetallic compounds or solid solutions. The addition of an element of surface energy higher than the solvent liquid may lower the interfacial tension, provided the interaction effect is large. The interfacial tension will tend to increase if the added element is preferentially adsorbed at the liquid surface, and has no tendency to react with the solid metal, e.g., forms an immiscible system with the solid. Similar considerations apply to interfacial effects between liquid metals and ceramics. Temperature increases will normally lead to decreased interfaciai tensions as a result of the solid-liquid interaction effects being more pronounced. A third factor which may modify the spreading of a liquid on a solid is the roughness of the latter, the interfacial tension of the system’ being independent of roughness. In relation to its effect on the apparent contact angle of the liquid on a surface, roughness has been referred to (MOLLIETand COLLIE,1951a) as “simple,” in which the effective area of solid-liquid interface is increased, and “composite,” in which the surface contains pores tilled with a second fluid. WENZEL(1936) established that simple roughness had the following effect on the contact angle:
cos 8, = f cos 8
(11)
where 8, = contact angle on the rough surface. 8 = true contact angle on a smooth surface. f = roughness factor, i.e., actual area of solid-liquid interface per square centimetre of rough surface.
The sigticance of
wettingin reactor technology
25
“Simple” roughness thus reduces the apparent contact angle when the true angle is less than 90”, and increases it when the true angle is greater than 90”. CASSIEand BAXTER (1944) established the following relation for “composite” roughness. ~0~e,=fc0se-f~.
where 0, = contact angle on the rough surface. 0 = true contact angle on a smooth surface. f = f'
area of solid-liquid interface per square centimetre of rough surface.
= area of solid-pore space interface per square centimetre of rough surface.
Thus “composite” roughness increases the apparent contact angle, and it may well happen that (or.) may be greater than 90” when (0) is actually below this value. Under conditions where “composite” roughness exists, for (0,) to be small, (f3)must be very small, i.e., the interfacial tension must be low, and the pore area of the surface (f ') should be small. 6. TECHNOLOGICAL
SIGNIFICANCE
OF INTERFACIAL
EFFECTS
(i) Corrosion and Intergranular Penetration For liquid metal corrosion in which the solid metal dissolves in the liquid, the role of interfacial tension is obscure due to ignorance concerning the kinetics of the reaction. While, in the solution of a single component from an alloy, the rate of diffusion of the element in the solid may be the rate-controlling process, this cannot be the case for pure metals. In the latter, the rate-controlling process may be diffusion in the liquid phase, or it may be a chemical step at the surface related to the magnitude of the interfacial tension. Other things being equal, the rate of solution may be higher with a low interfacial tension, and vice versa. In the case of liquid-liquid extraction (LEWIS), it appears that a low interfacial tension does not necessarily result in a fast rate of transfer of material, but the two cases are not strictly comparable. Where the liquid metal corrosion is of the intergranular type, the role of interfacial tension is much more clearly defined, as outlined in relation to intergranular penetration, in paragraph 2 (p. 17). Thus the magnitude of the dihedral angle formed, Fig. 2, is given by: OBB P 2- = “‘--’ 2aAB
(13)
and is thus controlled by the ratio of the grain boundary to the interfacial tensions. MORGAN(1954) has recently made a detailed study of the effect of liquid lead and bismuth on the mechanical properties of copper, which may be interpreted in terms of interfacial tension effects. Bismuth was found to embrittle copper severely, whereas the effect of lead was very much less. The dihedral angles formed at the copper grain boundaries in liquid lead and bismuth and in a series of intermediate compositions are listed in Table 8. The results indicate that the low dihedral angle formed in bismuth is not the sole reason for the embrittlement of the copper; in addition, grain-boundary diffusion of bismuth into the copper beyond the end of the dihedral angle results in further embrittlement. Under applied stress, the rate of grain boundary diffusion was considerably increased as a result of the high stress concentration at the base of the sharp grain boundary angles. MORGAN’Sdata give the respective interfacial
26
J. W. TAYLOR
tensions: Cu/Pb = 390 ergs/cm* and Cu/Bi = 280 ergs/cm’, assuming that the copper grain boundary tension remains constant. Thus intergranular genetration is likely to be absent when the ratio of the grain boundary to interfacial tension is small, i.e., when the latter is large. (ii) Heat Transfer In the early stages of the application of liquid metals as heat transfer media it was considered that optimum heat transfer. would be obtained when the .heat transfer tube was “wetted” by the liquid medium. However, much experimental work has failed to verify this, and it has been indicated that heat transfer coefficients may be as high in “wetting” as in “non-wetting” conditions. TABLEIl.-DIHEDFUL ANGLE AND INTERFACLU TENSION VALUES IN Cu-Pb-Bi SYS’TEM
:z
90 87 62
325 292 286 280
;: 20 ??
Assuminggrab boundarytensionof Cu = 550 ergs/d
(iii) Mass Transfer and Zone Melting Interfacial effects wol:!d appear to be of considerable signi&ance in the nonisothermal mass transfer frequently found in liquid-metal corrosion. The overall rate of mass transfer is controlled by three separate rates, viz., that of solution at the higher temperature, that of nucleation in the colder regions, and that of crystal growth on the nuclei. Of the latter two, the rate of nucleation will be the significant factor in mass transfer. Nucleation may occur homogeneously or heterogeneously (HOLLOMON and TURNBULL, 1951; HOLLOMON, 1950; TURNBULL, 1950), depending on whether the process occurs in the absence or presence of nucleating agents such as solid impurities, container walls, etc. While nucleation in most practical applications will be a heterogeneous process, it is instructive to compare the two cases,and establish the significance of interfacial effects. The rate of homogeneous nucleation (IHo) of a pure metal at temperature (T) is given (HOLLOMON and TURNBULL, 1951) as:
bz,=
(F) exp [-(16~TOaa9/3H,*(T -
T,Ja + AF,)/kT]
where n = number of atoms per unit volume of sample. u = interfacial tension between solid and liquid. Hf = heat of fusion of the solid. AF4 = free energy of self-diffusion of the liquid.
(14)
The signif?canceof wettingin reactor technology
27
Thus the larger the interfacial tension (a), the larger the degree of super-cooling to produce a certain rate of nucleation. Under heterogeneous nucleation conditions, where the process starts at any’ solid-liquid interface present in the nucleating zone, the corresponding equation for (I,;) is : exp [- 16~~Voaf(B)/3ZZ,~T - TJa + AF,)/&Tj where n’ = number of liquid atoms in contact with the impurity per unit volume of liquid. j-(e) = (2 + cos e)(1 - cos @a/4. 6 = contact angle formed by the crystallite
on the solid surface of the
nucleating agent. COST=
OLI -
=sr (16)
=Ls
where a,, = interfacial tension between the liquid and the impurity. OSI =
interfacial tension between the crystallite and the impurity.
aLs =
interfacial tension between ‘the crystallite and the liquid.
As (0) appears in equation (15) as a power relation, its effect on the rate of nucleation is correspondingly large. While expressions (14) and (15) are strictly applicable to the solidification of pure metals, they may be applied to predict conditions likely to favour low nucleation rates under mass transfer conditions. The liquid metal should be kept free from solid impurities so that conditions favouring homogeneous nucleation are ap$roached. Under heterogeneous conditions, relative freedom from solid impurities will ensure that (n’) in equation (15) is small. The interfacial tension between the crystallite and the liquid (cLs) should be as large as possible, thus favouring supersaturation conditions. Steps should be taken to make (cos @, equation (16) as small as possible by making (uLs) and (usI) large and (cLI) small. If impurities cannot be avoided, then these are least harmful when they are wetted by the liquid medium. Similarly, if nucleation on the walls of the container is likely, this effect will be minimized if the container walls and the liquid phase have a low interfacial tension. Furthermore, it may be advantageous to choose a container material with a lattice structure differing as widely as possible from that of the precipitating phase, so as to make (asI) large. Roughness of the container wall may also facilitate nucleation (TURNBULL,1950), and it is thus ,desirable to have these surfaces smooth. Similar considerations may well be significant in zone melting. Thus most of the theoretical treatments are based‘on the concentration-temperature relations as given by the equilibrium diagram, but, in practice, undercooling phenomena may cause serious departures from such conditions, especially if the metal is of extreme purity and free from solid impurities. (iv) Slurry Properties and Stability Interfacial tensions are of significance in the application of liquid metal slurries, To form an effective slurry between the particles of a solid phase and a liquid metal, the former must be completely “wetted” by the latter so as to prevent solid-solid contact. The conditions governing spreading and those leading to low interfacial tensions have
28
J. W. TAYLOR
been discussed previously for ideal, flat surfaces, but the conditions for the complete penetration of a particulate aggregate by a liquid do not require a contact angle of zero. WOODROW (1954) has indicated that, under ideal conditions, i.e., a perfectljl smooth surface, free from adsorbed gas, etc., a sufficient condition for penetration of an aggregate is that the contact angle is less than 90”. In practice, the effect of surface roughness must be considered. It has been pointed out that “simple” roughness will reduce the apparent contact angle below the true value provided the latter is less than 90”, whereas “composite*’ roughness may, in fact, increase the apparent contact angle value over the true value on a flat surface. From the point of view of “wetting” a particulate aggregate, it would seem desirable to have a low interfacial tension between the liquid and solid, and to have particles with relatively smooth surfaces. If the surface is rough and of a “composite” nature, i.e., contains pores filled with a second fluid, then steps must be taken to remove this fluid (if air or a gas, by evacuation) so as to facilitate wetting. It is interesting to consider the role of interfacial effects in relation to the escape of fission product gases from slurry particles. If the liquid metal forms a finite contact angle with the solid particle, gas bubbles, released at the particle surface, will conform to this angle. In the settled condition of a slurry, such gas evolution will result in a dilation of the bed and in the generation of hot spots on the particles. If these hot spots develop near the juncture of two or more particles, sufficient heat may be developed to vaporize the liquid in the interstice of the juncture, and under these conditions sintering of the two solid surfaces may take place. The lower the contact angle formed by the liquid on the solid, the more readily will the gas be removed by any de-gassing treatment. In liquid metal slurry systems, it is of importance to control the particle-size distribution, and interfacial effects are involved in this to some extent. Under isothermal conditions in a slurry system with a certain particle-size distribution, the larger particles grow at the expense of the smaller ones, which have a higher solubility in the liquid medium as a result of their larger curvature. The relationship between soluand COLLIE,1951b): bility and particle radius is in fact (MOLLIET (17) where L = solubility. r = particle radius. (T= interfacial tension between solid and liquid. Expression (17) has recently been extended by GREENWOOD to indicate that the rate of particle growth is proportional to the interfacial tension (a) between the liquid and the solid, a high interfacial tension favouring a rapid rate of growth. However, the effect of interfacial tension variations in this case is probably greatly outweighed by changes in the rate of diffusion of the solute in the liquid. In addition to this isothermal effect, a non-isothermal growth factor will also be present where the slurry system is undergoing a temperature cycle, e.g., on circulation through a heat exchanger. Thus material dissolved at the high temperature will be precipitated at the low temperature, and this precipitation will be controlled by the nucleation and growth considerations outlined previously; the effect will obviously
The signiti-
of wetting in reactor technology
29
be greater the larger the differential solubility with respect to temperature of the solid in the liquid phase. Under such conditions, precipitation would undoubtedly take place on the undissolved solid particles, and no separate nuclei would require to form. The ultimate effect of this solution and precipitation cycle might be quite small, the material’precipitated at one stage dissolving in the subsequent one. Initially, however, it might cause the rate of growth of the larger particles at the expense of the smaller ones to be higher than that occurring under isothermal conditions. In &ury systems of the type (liquid A + solid A$,,), it may well be possible to modify the size distribution of the latter by suitable control of nucleation conditions during the formation of the latter compound, i.e., by suitable additions of “wetting” agents and innoculants. lnterfacial tension effects are unlikely to have any pronounced effect on the settling rates of liquid metal slurries due to the relatively large particle sizes used so far. Only when the particle size is such that Brownian effects become sign&ant will interfacial effects become important. (v) Extraction Processes Interfacial effects can be of qualitative value in some metallurgical extraction processes. Thus in liquid-liquid extraction using metals, a low interfacial tension would appear to be necessary. In the first place, a stable dispersion of fine droplets of the one liquid in the other will only be obtained with a low interfacial tension, and secondly, this tension may possibly be of significance in the rate of extraction of the desired material. In extraction processes using slags and electrolytes, interfacial tensions are again of importance. To facilitate rapid separation of slag and metal, a high interfacial tension is desirable, and metal yields may be improved by suitable flux’additions which effectively raise this tension. Similarly, in reduction processes, where the metal is produced by precipitation from a liquid salt bath, a high interfacial tension will favour rapid and complete separation of the two phases. 7.
CONCLUSIONS
While the number of quantitative studies of interfacial tensions between liquid metals and solids is comparatively few, those that have been made, in conjunction with a number of qualitative observations, have indicated which factors control the spreading process. In addition, these observations have suggested suitable methods of modifying the interfacial tension for any particular application. In reactor technology a low interfacial tension appears to be desirable in the application of liquid metals as coolants and slurry media, while a high value appears desirable from the corrosion and mass transfer points of view; in extraction processes both high and low tensions may be desirable, depending on the process in question. ACKNOWLEDGMENTS
The author expresses thanks to Dr. H. M. FINNISTOUNand Mr. B. W. MOTT for helpful discussion during the preparation of this paper, which is published by kind permission of the Director of the Establishment. REFERENCES BAILEY,G. L. J., and WATKINS, H. C. (1951) J. Inst. Met., 80, 66. BAILEY,G. L. J., and WATKINS,H. C. (1950) Proc. Roy. Sot., B63, 350. BAILEY,G. L. J., FOX, M. J., and WATKINS, H. C. (1948) Br. Non-Fer. Met. Res. Assocn. Report, RRA 773.
30
J. W. TAYLOR
BAILEY, G. L. J. (1948) ibid, RRA, 898. A. (1953) Chem. Rev., 52,411. CASSIE,A. B. D., and BARTER, S. (1944) Trans. Faraday Sot., 40,456. DOBINSKI,S., and JAOIEL+%CI, A. (1939) Bull inter. acad. poln. xi., Classe, Sci. math. nat., 193A,423. FOX, M. F. (1948) Br. Non-Fer. Met. Res. Assocn. Report, RRA 771, FRICHE, R. (1948) Z. & Elekirochem., 52,72. GREENHILL,E. B., and MCDONALD, S. R. (1953) Nature, 171,37. GREENWOOD,G. Private communication. GURNEY, C. (1949) hoc. Phys. Sot., A62,639. HARKINS, W. D. (1952) Physical Chemistry of Surfaces, Reinhold Publishing Corp. HOLL~MON,J. H. (1950) Amer. Sot. Metals Seminar, Thermodynamics in Physical Metallurgy, p. 161. HOLLOMON,J. H., and TURNBULL, D. S. (1951) A.I.M.E.E. Symposium, Solidification of Metals and Alloys, p. 1. LEWIS,J. B. Private communication. MOLLY, J. L., and COLLIE, B. (1951a) Surface Actiuity, London, p. 100. MOLLIET, J. L.. and COLLIE, B. (1951b) Surface Actiuity, London, p. 141. MORGAN, W. (1954) Ph.D. Thesis, Cambridge University. NORTON, F. K., KINGERY, W. D., et al. (1953) U.S.A.E.C. Pnbl. NYO-3144. SAWAI, I., and NISHIDA, M. (1930) Z.$ anorg. allgem. Chem., 190,375. BONDI,
G. W. (1950) J. App. Phys., 21,721. C. S. (1948) Met. Tech., Tech. Pub., 2387. TAMMAN, G., and B~EHME,W. (1932) Ann. Phys., 12,820l. TAYLOR, J. W. (1954a) At. Energy. Res. Est. Rep., M/TN 24. TAYLOR, J. W. (1954b) Metallurgia., 50, 161. TURIUBULL,D. (1950) A. S. M. Seminar, Thermodynamics in Physical Metallurgy, p. 282. UDIN, H., SHALER, A. J., and WULFF, J. (1949) J:of Met., 1,186. VAN VLACK, L. H. (1951) ibid, 3,251. WATKINS, H. C. (1948) Br. Non-Fer. Met. Res. Assocn. Report, RRA 772. WENZEL, R. N. (1936) Industrial Eng. Chem., 28,988. WOODROW, J. (1954) At: Energy Res. Est. Rep., ED/M 13. SEARS,
SMITH,
Peqamon Press Ltd., London
THE SIGNIFICANCE OF WETTING IN REACTOR TECHNOLOGY By J. W. TAYLOR Atomic Energy Research Establishment, Harwell, Rerks (Received 7 IMarch 1955) Abstract-The fundamental forces responsible for the spreading of a liquid metal on a solid (or a liquid) surface are discussed, and a number of interfacial tension and spreading studies are analyzed in terms of the theoretical considerations. In the light of these, methods of modifying the interfacial tension in any given application are suggested. The signiticance of interfacial effects in the following technological processes is outlined: (i) liquid metal corrosion and intergranular penetration; (ii) heat transfer; (iii) mass transfer and zone melting; (iv) liquid metal slurry properties and stability; (v) extraction metallurgy. 1.
INTRODUCTION
Tti proposed use of liquid metals in reactors as coolants, in slurries, and in chemical processing, has focused considerable attention on the role played by interfacial tensions in these several applications. In what follows, the physical parameters responsible for the spreading of a liquid metal on a solid (or a liquid) surface are considered; practical studies of interfacial tensions are analyzed in terms of these factors and methods of modifying intesfacial tensions fundamental parameters; discussed; and the significance of such effects in the principal applications of liquid metals established. 2. (i) Thermodynamics
THEORETICAL
CONSIDERATIONS
ofSpreading
The prim&y forces responsible for the spreading of a liquid metal on a solid (or a second liquid) are the free surface energy components of the two phases, these energies existing as a consequence of unsaturated bonds present at the free surfaces. For a given phase, this surface energy may be expressed (BONDI,1953; HARKINS,1952) thus:
(zaF1p,,=a where F = free energy of the phase (ergs). u = surface area of the phase (cm*). a = surface free energy of the phase (ergs/cma), numerically equal to the surface tension. When a liquid metal (A) is placed in contact with a second phase (B) (which may be solid or liquid), spreading will occur if the total free energy change of the system is negative, i.e., if
dF
(2)
where dF = free energy change on spreading. 15
16
J. W. TAYLOR
Thus, considering the free energy change involved in the formation of one square centimetre of interface, and since aa, = aaAB = -aa, UA +
dF =
ugg
-uB
(4)
where bg = surface energy of the liquid. egg
=
interfacial tension of the interface AB.
oB = surface energy of the solid (or second liquid). Thus spreading of (A) on (B) will occur when: =A
+
OAB
<
aB
(9
Considering a case in which spreading has proceeded to some intermediate stage, short of complete wetting of the solid by the liquid, the equilibrium of a drop of
SOLID B
FIG. 1.
liquid (A) on the phase (B) may be represented as in Fig. 1. The equilibrium of the tensions involved may be expressed as: where 8 = contact angle. Of the three tensions responsible for spreading, only bg is known with any degree of experimental accuracy. bg has been measured for a few solid metals (see page 19), and it is possible, by methods outlined below, to estimate such values at least to an order of magnitude. GAB is peculiar to each system, and there is no ‘way of predicting it from the respective values of the quantities, cg and ug. In addition, the latter surface energies are sensitive to composition and to environment; e.g., cA will be altered by solution of(B) in (A), while uB is affected by the nature of the atmosphere above it; vapour from (A) may alter the value of bg as a result of adsorption. As a consequence of these several possible variations, a number of different contact angle conditions have been recognized (BAILEY and WATKINS, 1951); e.g,, “advancing,” “receding,*’ “initial,” “equilibrium,” etc. While these several angles are of no great significance in the applications to be discussed, nevertheless it is important to consider possible variations in bA and cB in evaluating (TABfrom contact angle conditions. Also, it is quite erroneous to consider all wetting phenomena in terms of two extreme contact angle conditions, i.e., cases of “wetting” where the contact angle is zero, and “nonwetting” cases where the contact angle is approximately 180”. Such a concept was common in the past. The spreading referred to above may be termed “macro-spreading,” as it involves the bulk surface energy of the solid. A second type of “micro-” or “grain boundary spreading” may also be described where a liquid metal is in contact with a grain
The significance of wetting in reactor techn0logy
17
boundary. The respective tensions are depicted in Fig. 2 and the equilibrium may be expressed as : UBB = 2c,*
COS g
where egg = interfacial tension between (A) and (B). a,~ = grain-boundary
tension between two grains of solid (B).
Q = dihedral angle. The dihedral angle (v) is formed as a consequence of the unbalanced tensions existing at the grain boundary/liquid metal interface. Its formation necessitates the surface migration of(B) atoms and it is thus confined to temperatures where this migration is
LIQUID
A
FIG. 2.
appreciable, The magnitude of (I@ is controlled by the ratio cA&BB. discussed in detail later (p. 25).
This will be
(ii) Surface Adsorption A further consequence of surface energy considerations is that of surface adsorption in a two- or higher-component system. By preferential adsorption of the phase of lower surface energy at the surface, the system as a whole attains a state of minimum energy; the extent of this enrichment or depletion, in a binary system, is given by the GIBB’Sadsorption equation: da r, = 2.3RT d bg a, where I’, = surface excess of component (1) relative to a plane of average composition (mol/cn?). da = change in surface energy for a change d log a, in the activity of component ( 1) @g/cm?. If (du/d log aI) is positive, i.e., the surface energy of the solution increases with increasing proportions of component (I), the surface excess (I’d will be negative and the surfaT layers will be enriched in component (2). Use is made of this effect when considering additions to a liquid metal, which will reduce the interfacial tension between it and a solid (or second liquid) metal. Expression (8) is usually applied to liquid solutions, but the underlying principle is general and the same e&ct may well be operative in solid solutions at suitable elevated temperatures. Thus DOBINSKY and JAGIIXSKI(1939) have shown by diffraction studies that surface enrichment of the phase of lower surface energy occurs in the systems Pb-Sb, J.N.E. 2-8
18
J. W. TAYEOR
Zn-Sn, Cu-Ca, Ag-Cd. Thus suitable alloy additions to the solid phase may also be effective in lowering the interfacial tension by means of a similar surface adsorption process, though this has not been considered seriously in the past. 3.
SURFACE
ENERGY
VALUES
Two of the three primary tensions responsible for spreading of a liquid metal on a solid metal are the respective surface tensions of the phases, and it is convenient to summarize the information available on these properties before considering studies that have been made of the interfacial tension (uBB of Fig. 1). The surface energies of a number of liquid metals and alloys have been measured experimentally and a review of these measurements has been presented elsewhere (TAYLOR,1954a). Where experimental values did not exist for a metal, it has proved possible to estimate the surface energy from a number of correlations which exist (TAYLOR, 1954b) between this and other physical parameters of the liquids. The experimental or estimated surface energy values for liquid metals at their respective melting temperatures are presented in Table 1. TABLE~---SURFACE ENERGIES OF LX
-
Metal
METALS AT
THEIR
MELTING
TEMPERATURES
T-
I
Surface energy (ergs/cm*)
914 1134 195* 1620*
!
Metal
Surface energy
Metal
(ergs/m*)
Surface energy (ergs/cm*)
926
Ag Al Au Ba Be Bi Ca 2 ce Co Cr cs CU Fe Ga
D
:z 3030* 608 610* 1740* 1420* 55* 1220 1460 735
Ge Hf Hg In Ir K Li Mg Mn MO Na Ni OS Pb Pd Pt
Pu Re Rb Sb Si Sn Sr Ta Te Ti Tl Th U
644 1510’ 471 630* 2310* 101 398 553 10502 2240’ 191 1570 2450’ 455 1280* 1845
; Zn Zr
580. 2480* 75* 380 860 616 165’ 2330. 300* 1325+ 444 1075* 1070’ 1710’ 2680. 818 1380*
* Estimatedvalues.
The surface energies of solid metals have been determined in only a few cases, these values being presented in Table 2 for the metals, copper, silver, gold, and tin. Attempts to calculate the surface energies of solid metals from fundamental principles (FRICHE, 1948) have not been successful, the calculated values being much larger than the experimental ones. An expression derived by GURNEY(1949) for the difference in free energy between the solid and liquid phases at the melting-point leads to improbable results. A measure of this difference in surface energies may, however, be obtained in two ways. The first is from the supercooling data presented by HOLLOMON and TURNBULL(1951), from which the interfacial tension between the liquid and solid phases can be calculated; this tension may be regarded as the’difference in surface energies of the two phases. The second estimate, obtained by the author, may be
The significance of wetting in reactor technology
19
derived from a consideration of the two factors responsible for this difference in surface energy. The first is the energy difference between the phases as a consequence of TABLE2.-SURFACE ENERGmOF SOLIDMETALS M&al
Temperature (“C)
Reference .
Au Au Ag cu Sn
Surface energy (ergs/cm*) _
SAWAIand NISHIDA,1930 TAMMAN and B~EHME., 1932 SAWAX and NISHJDA,1930 UDIN, SHALER,and WU~FF, 1949 GREENHILL and MCDONALD,1953
650-850 700-850 650-850 950-1050 215
1360 1810 1180 1450 685
melting; the magnitude of this effect (AoH,) may be gauged by considering the surface energy decrease that would result in the liquid state if a quantity of heat equivalent to the heat of fusion (A,) were added, i.e., A*Hf do Aa,, = x C, dT
(9)
where C, = specific heat of the liquid (cal/“/mol). da
-
dT
= temperature coefficient of the surface tension of the liquid (erg/cnP/“C).
The second factor is the difference in the atomic volumes of the liquid and solid phases. The magnitude of this (Aa,) was estimated by considering the surface energy change equivalent to the change in volume on melting, using the approximate relationship which exists between the surface energies of metals and their atomic volumes, Fig. 3. The final difference in surface energy of the liquid and the solid would thus be: EAa = Aa,, + Aa,
(10)
Table 3 compares the difference in surface energies obtained by the two methods with TABLEQ.-DIFFERENCESIN Surface energy (ergs/cm3 Metal
SURFACE
ENERGYBETWEEN
AU A%, (ergs/cm*) (ergs/cm3
Solid
LIQUID
AND
SOLID
MBTAI.8
Aa CA0 Solid-liquid (ergs/cm*) (ergslcma)
Interfacial tension (ergs/cm*)
cu Ag
1400 1180
40 30,
100 40
140 70
180 250
177 126
,Au Sn Na K
1800 1360 685 -
40 25 8 7
80
120
225 665 70 -
132 55 20
Li
-
10
I
-
20
J. W. TAYLOR
the difference existing between the surface energies of solid and liquid gold, tin, copper, and silver. The agreement is reasonable, considering the inaccuracies of surface energy measurements; the two methods of estimation employed are consistent, both in the previous series of metals and in the alkali metals lithium, sodium, and potassium. )‘5-
PO
I 05
I I.0 Loq ATOMIC VOLUME
I.5
I PO
FIG. 3.
4.
SPREADING
AND
INTERFAC,IAL
TENSION
STUDIES
The parameter of primary interest in most interfacial processes is the interfacial tension (cram of Fig. l), and this must be measured experimentally. In view of the difficulty of making such measurements, the number of detailed studies have been few, but these, in conjunction with a number of qualitative observations, serve to indicate the principles governing the magnitude of this tension. BAILEYand WATKINS (1950), using the principles of phase equilibrium established by SMITH (1948), have determined the magnitude of the tensions involved in the system solid copper/liquid lead. By suitable thermal treatment of copper in hydrogen, argon, and in lead vapour, and from contact angle measurements of lead on copper, they evaluated the following tensions: (i) surface energy of clean solid copper; (ii) surface energy of solid-copper in presence of lead vapour ; (ii) interfacial tension between solid copper and liquid lead; (iv) grain boundary tension of copper. These values are contained in Table 4. Values found by SEAIU(1950) for the same system are given for comparison. The following points may be noted. The interfacial tension oc,+r,, = 340 ergs/ems is very much flower than the difference between the two tensions, orb = 450 ergs/cm* and cc,, = 1800 ergs/cm*, and this in a system where there is only a limited degree of interaction between the solid and the liquid components. Secondly, the surface energy of copper in the presence of lead vapour acan = 780 ergs/cm* is less than half
The signiticanceof wetting in reactor technology
21
that of solid copper in the absence of the lead vapour. This feature stresses the need for considering the effect of surface adsorption on the solid when deducing interfacial tensions from contact angle measurements. If this reduction in solid surface energy had been ignored, the interfacial tension deduced would have been 1510 ergs/cm2, as opposed to the true value of 340 ergs/cm 2. It is of interest that the surface energy of solid copper in the presence of lead vapour, 780 ergs/cmS, is intermediate between that of clean solid copper, 1800 ergs/cm%,and the interfacial tension o&,/pb = 340 ergs/cm2. TAPIZA--SURIJA~E ENERGIES OFTHESysIgMSOLIDCDPPER-LIQIJJD LEADSURFACE Surface energy @r&m*)
Boundary Solid copper (uh) Copper with adsorbed lead (a~) Copper/liquid lead interface &,,/p,,) capper grain-lJo@ary (ml/c!“)
1800 780
720 510
E I
-
BAILEY and
Reference
WATKINS
(1950)
The value of 1800 ergs/cm2 for the surface energy of solid copper is larger than UDIN, SHALER,and WULFF’Svalue (1949), but this reflects the experimental error involved in such measurements. Similarly, a number of other interfacial tensions have been measured. VANVLACK (1951) established the liquid copper/solid iron interfacial tension at 430 ergs/cm2. This is of a similar magnitude to the value of 340 ergs/cm2 for the liquid lead/solid copper value. Both systems show a limited degree of solubility of the solid in the liquid phase at the temperatures concerned. The copper/iron tension is much closer to the difference in the respective tensions of the two primary phases, viz. (1460 + 200 - 1220 = 440 ergs/cm2), if 200 ergs/cm2 is allowed for the difference in the surface energy of solid and liquid iron. TABLE ~.SURFACE ENERGIESOF OXIDE-METAL MXRFACES
Surface energy (ergs/cm’)
Interface
Interface
1800 440 905
&O&NW) A1,4(s)/4W) A&O,(s) surface I
ZrCMs)/W) z~,(wJws) zro*(s) Sllrf~
surfaceenergy hgS/UIl’)
970
265 590
I
NORTONand KINGERY(1953) give values for the liquid nickel/alumina interface, 1860 ergs/cm2, and for the liquid nickel/zirconia interface, 970 ergs/cm2; their complete results are given in Table 5. The interfacial tensions between liquid metals and oxides are much higher than values for metal-metal interfaces, and in fact the
J. W. TAYLOR
22
interfacial tensions are higher than the reported free surface energies of the solid phases. These oxide-metal systems are quite inert, and this fact will be referred to subsequently. NORTON and KINGERY’Sother values, Table 5, indicate that the grain boundary tension of oxides is approximately half the corresponding surface energy; a similar ratio of energy to tension for copper is shown in Table 4. A number of qualitative observations of spreading and contact angle relationships serve to establish some general ‘principles. In a study of the spreading of a large number of pure liquid metals on a variety of solid metals, BAILEY,Fox, and WATKINS(1948), and Fox (1948), distinguished the following three conditions : (i) “wetting,” in which a stable liquid coat was left after the bulk of the liquid was drained; (ii) “non-wetting,” TENSIONS TABLE6.-INT~RFAUAL
Solid metal
ci’ SOME LIQUID MIIT~/SOLID
Fe
Ni
1660* 830
1770’ 885
METAL -
T T
. =Lx
caltact
a+
0 tension (e&cm~ wettin b&laviour
Intcrfaci@
??
Sn
Sn
Pb
Cd
620
620
460
610
4; D.W.
2 W
2g: W
72
640
D.W.
1
Esthated v&ea (6). D.W. De-wettingsyystcm. W. Wettingaystsm.
where no spread of the liquid over’the solid took place; and (iii) “de-wetting,” where the liquid Ghn retracted into discrete droplets when the excess liquid was drained away. Systems exhibiting “wetting,” i.e., low’interfacial tension, showed either a tendency to compound formation or extensive solid solution, while “non-wetting” systems, i.e., high interfacial tension, showed no such tendency to interact. In the course of these studies WATKINS(1948) reported contact angle values for a number of systems, and it is instructive to derive therefrom approximate interfacial tension values using equation (6) (p. 16), making the quite arbitrary assumption that the surface .energy of the respective solids in the liquid metal vapour is half that of the metal in an inert atmosphere (cf. copper in lead vapour, Table 4). The interfacial tensions thus derived are compared with the condition of “wetting” of the system in Table 6. This would suggest that the systems showing “wetting” have interfacial tensions below about 300 ergs/cm*, and that those with tensions greater than this exhibit “de-wetting” or “non-wetting” conditions, bearing in mind the assumption made in calculating these tensions. Similar qualitative observations have been made by NORTONand KINGERY(1953) in relation to the’spreading of the transition group of elements on a variety of materials. In general, they established that spreading was facilitated by any interfacial reaction between the solid ceramic and the liquid metal; where this reaction was non-existent or limited, the spreading was restricted, i.i:, the contact angle was high.
23
The significance of wetting in reactor technology
So far only the spreading behaviour of pure metals on solids has been considered, but the case of alloys is particularly interesting. In a study of the contact angle behaviour of a series of lead-tin alloys on solid copper, silver, nickel, and iron, it was clearly demonstrated (Fox, 1948; BAILEY,Fox, and WATKINS, 1948; WATKINS,1948) that where compound formation occurred,between the liquid alloy and the solid, the contact angle was low. Furthermore, it appeared that a definite minimum existed in the contact angle-alloy composition relation at 60‘per cent lead-40 per cent tin, when the alloys were in contact with iron and nickel; this was evident to a less extent with silver. Now it is, apparently, the tin which is effective in lowering the interfacial tension,.as may be seen from the fact that the system iron/liquid tin shows “de-wetting” behaviour, while the corresponding lead system is “non-wetting.” In the liquid
TABLE7.-INTZRFACUL TENsIONS m’rwxx~ Fe-Si AuoY ANDA&O, Composition (A Si)
Surfae$$;;f I
OQ60
liquid 1 Conta$ angle / Interfa$lal
0.265 0.97 3.10’ 8.72 I
I
I
138 132 125 119 115
1450 1400 1425 1440 1425 I
2000 1885 1747 1640 1537 I
lead-tin alloy, the tin will be preferentially desorbed from the surface, and if this surface deficiency is calculated for this system, it is found that the relative effect of this deficiency will be at a minimum round about the 40-50 per cent composition. This is probably responsible for the observed minimumeontact angle values in this region. Similar effects have been observed by NORTONand KINGERYin the case of iron-silicon alloys spreading over alumina. Their results, which are summarized in Table 7, show that the interfacial tension drops from 2009 ergs/cm2 at 0.06 per cent silicon to 1537 ergs/cm2 at 8.72 per cent silicon, while the corresponding surface tension of the liquid remains essentially unaltered. This decrease in interfacial tension is a consequence of adsorption of silicon at the interface, and the relatively strong tendency for silicon-oxygen interaction. A number of other qualitative observations on the spreading of liquid alloys on solid metals may be interpreted in a similar manner. The addition of I.5 per cent silver did not affect the spreading behaviour of lead or of an alloy of 99 per cent lead + 1 per cent tin on nickel. No effect would be anticipated, as silver would not be preferentially adsorbed on the surface, due to its high surface tension; in addition it has little tendency to react with nickel. ,VAN VLACK (1951) found the addition of silver to liquid copper resulted in an increase in interfacial tension between the liquid phase and iron. This is a consequence of the surface adsorption of silver, which has a surface tension lower than that of copper; in addition, silver and iron are immiscible in the liquid and solid states, and therefore the interfacial tension in the presence of silver will be high. An interesting case of additives affecting spreading is that in which 0.1 per cent nickel added to lead markedly reduces the interfacial tension against steel. On the basis of relative surface energy values, nickel would not be adsorbed preferentially,
24
J. W.
TAYLOR
and the lowering of the interfacial tension must be attributed to the strong tendency for the nickel to alloy with the iron. It would appear that, where there is a strong tendency for alloy formation to occur between the solid and a component in the liquid, surface adsorption effects will be overcome and a low interfacial tension will result. MODIFYING INTERFACIAL TENSIONS 5. FACTORS Before considering the several technologically important applications of interfacial effects, it is convenient to consider briefly what factors may modify interfacial tensions and spreading, since the foregoing discussions apply to ideal conditions. Surface lilms are likely to be a signiticant factor under practical conditions. Such films may be present on the liquid or solid phases. Whereas the former can generally be ruptured easily, the latter are likely to prove a more permanent and effective barrier to contact between the solid and the liquid. Two types of film may be present on the solid. The one is an adsorbed gas tim; this may remain effective up to temperatures r-t which desorption occurs, i.e., 300°C. The second type of tilm is a chemical one. The stability of such a film in the liquid medium will be controlled by free energy considerations, and should this film be thermodynamically stable it is likely that the resulting interfacial tension will be considerable, as found by NORTONand KINGERY(1953). The action of a flux in facilitating spreading and lowering interfacial tensions is to dissolve such surface films, although it has been suggested that frequently there is also an electrochemical effect (BAILEY,1948). A second significant method of altering the interfacial tension is by control of the liquid metal composition’; the principles involved have been outlined in the previous paragraph. To reduce the interfacial tension the added element should be such that its surface energy is lower than the value of the solvent liquid, so that it is preferentially adsorbed at the surface. In addition, it should have a strong tendency to interact with the solid, either to form intermetallic compounds or solid solutions. The addition of an element of surface energy higher than the solvent liquid may lower the interfacial tension, provided the interaction effect is large. The interfacial tension will tend to increase if the added element is preferentially adsorbed at the liquid surface, and has no tendency to react with the solid metal, e.g., forms an immiscible system with the solid. Similar considerations apply to interfacial effects between liquid metals and ceramics. Temperature increases will normally lead to decreased interfaciai tensions as a result of the solid-liquid interaction effects being more pronounced. A third factor which may modify the spreading of a liquid on a solid is the roughness of the latter, the interfacial tension of the system’ being independent of roughness. In relation to its effect on the apparent contact angle of the liquid on a surface, roughness has been referred to (MOLLIETand COLLIE,1951a) as “simple,” in which the effective area of solid-liquid interface is increased, and “composite,” in which the surface contains pores tilled with a second fluid. WENZEL(1936) established that simple roughness had the following effect on the contact angle:
cos 8, = f cos 8
(11)
where 8, = contact angle on the rough surface. 8 = true contact angle on a smooth surface. f = roughness factor, i.e., actual area of solid-liquid interface per square centimetre of rough surface.
The sigticance of
wettingin reactor technology
25
“Simple” roughness thus reduces the apparent contact angle when the true angle is less than 90”, and increases it when the true angle is greater than 90”. CASSIEand BAXTER (1944) established the following relation for “composite” roughness. ~0~e,=fc0se-f~.
where 0, = contact angle on the rough surface. 0 = true contact angle on a smooth surface. f = f'
area of solid-liquid interface per square centimetre of rough surface.
= area of solid-pore space interface per square centimetre of rough surface.
Thus “composite” roughness increases the apparent contact angle, and it may well happen that (or.) may be greater than 90” when (0) is actually below this value. Under conditions where “composite” roughness exists, for (0,) to be small, (f3)must be very small, i.e., the interfacial tension must be low, and the pore area of the surface (f ') should be small. 6. TECHNOLOGICAL
SIGNIFICANCE
OF INTERFACIAL
EFFECTS
(i) Corrosion and Intergranular Penetration For liquid metal corrosion in which the solid metal dissolves in the liquid, the role of interfacial tension is obscure due to ignorance concerning the kinetics of the reaction. While, in the solution of a single component from an alloy, the rate of diffusion of the element in the solid may be the rate-controlling process, this cannot be the case for pure metals. In the latter, the rate-controlling process may be diffusion in the liquid phase, or it may be a chemical step at the surface related to the magnitude of the interfacial tension. Other things being equal, the rate of solution may be higher with a low interfacial tension, and vice versa. In the case of liquid-liquid extraction (LEWIS), it appears that a low interfacial tension does not necessarily result in a fast rate of transfer of material, but the two cases are not strictly comparable. Where the liquid metal corrosion is of the intergranular type, the role of interfacial tension is much more clearly defined, as outlined in relation to intergranular penetration, in paragraph 2 (p. 17). Thus the magnitude of the dihedral angle formed, Fig. 2, is given by: OBB P 2- = “‘--’ 2aAB
(13)
and is thus controlled by the ratio of the grain boundary to the interfacial tensions. MORGAN(1954) has recently made a detailed study of the effect of liquid lead and bismuth on the mechanical properties of copper, which may be interpreted in terms of interfacial tension effects. Bismuth was found to embrittle copper severely, whereas the effect of lead was very much less. The dihedral angles formed at the copper grain boundaries in liquid lead and bismuth and in a series of intermediate compositions are listed in Table 8. The results indicate that the low dihedral angle formed in bismuth is not the sole reason for the embrittlement of the copper; in addition, grain-boundary diffusion of bismuth into the copper beyond the end of the dihedral angle results in further embrittlement. Under applied stress, the rate of grain boundary diffusion was considerably increased as a result of the high stress concentration at the base of the sharp grain boundary angles. MORGAN’Sdata give the respective interfacial
26
J. W. TAYLOR
tensions: Cu/Pb = 390 ergs/cm* and Cu/Bi = 280 ergs/cm’, assuming that the copper grain boundary tension remains constant. Thus intergranular genetration is likely to be absent when the ratio of the grain boundary to interfacial tension is small, i.e., when the latter is large. (ii) Heat Transfer In the early stages of the application of liquid metals as heat transfer media it was considered that optimum heat transfer. would be obtained when the .heat transfer tube was “wetted” by the liquid medium. However, much experimental work has failed to verify this, and it has been indicated that heat transfer coefficients may be as high in “wetting” as in “non-wetting” conditions. TABLEIl.-DIHEDFUL ANGLE AND INTERFACLU TENSION VALUES IN Cu-Pb-Bi SYS’TEM
:z
90 87 62
325 292 286 280
;: 20 ??
Assuminggrab boundarytensionof Cu = 550 ergs/d
(iii) Mass Transfer and Zone Melting Interfacial effects wol:!d appear to be of considerable signi&ance in the nonisothermal mass transfer frequently found in liquid-metal corrosion. The overall rate of mass transfer is controlled by three separate rates, viz., that of solution at the higher temperature, that of nucleation in the colder regions, and that of crystal growth on the nuclei. Of the latter two, the rate of nucleation will be the significant factor in mass transfer. Nucleation may occur homogeneously or heterogeneously (HOLLOMON and TURNBULL, 1951; HOLLOMON, 1950; TURNBULL, 1950), depending on whether the process occurs in the absence or presence of nucleating agents such as solid impurities, container walls, etc. While nucleation in most practical applications will be a heterogeneous process, it is instructive to compare the two cases,and establish the significance of interfacial effects. The rate of homogeneous nucleation (IHo) of a pure metal at temperature (T) is given (HOLLOMON and TURNBULL, 1951) as:
bz,=
(F) exp [-(16~TOaa9/3H,*(T -
T,Ja + AF,)/kT]
where n = number of atoms per unit volume of sample. u = interfacial tension between solid and liquid. Hf = heat of fusion of the solid. AF4 = free energy of self-diffusion of the liquid.
(14)
The signif?canceof wettingin reactor technology
27
Thus the larger the interfacial tension (a), the larger the degree of super-cooling to produce a certain rate of nucleation. Under heterogeneous nucleation conditions, where the process starts at any’ solid-liquid interface present in the nucleating zone, the corresponding equation for (I,;) is : exp [- 16~~Voaf(B)/3ZZ,~T - TJa + AF,)/&Tj where n’ = number of liquid atoms in contact with the impurity per unit volume of liquid. j-(e) = (2 + cos e)(1 - cos @a/4. 6 = contact angle formed by the crystallite
on the solid surface of the
nucleating agent. COST=
OLI -
=sr (16)
=Ls
where a,, = interfacial tension between the liquid and the impurity. OSI =
interfacial tension between the crystallite and the impurity.
aLs =
interfacial tension between ‘the crystallite and the liquid.
As (0) appears in equation (15) as a power relation, its effect on the rate of nucleation is correspondingly large. While expressions (14) and (15) are strictly applicable to the solidification of pure metals, they may be applied to predict conditions likely to favour low nucleation rates under mass transfer conditions. The liquid metal should be kept free from solid impurities so that conditions favouring homogeneous nucleation are ap$roached. Under heterogeneous conditions, relative freedom from solid impurities will ensure that (n’) in equation (15) is small. The interfacial tension between the crystallite and the liquid (cLs) should be as large as possible, thus favouring supersaturation conditions. Steps should be taken to make (cos @, equation (16) as small as possible by making (uLs) and (usI) large and (cLI) small. If impurities cannot be avoided, then these are least harmful when they are wetted by the liquid medium. Similarly, if nucleation on the walls of the container is likely, this effect will be minimized if the container walls and the liquid phase have a low interfacial tension. Furthermore, it may be advantageous to choose a container material with a lattice structure differing as widely as possible from that of the precipitating phase, so as to make (asI) large. Roughness of the container wall may also facilitate nucleation (TURNBULL,1950), and it is thus ,desirable to have these surfaces smooth. Similar considerations may well be significant in zone melting. Thus most of the theoretical treatments are based‘on the concentration-temperature relations as given by the equilibrium diagram, but, in practice, undercooling phenomena may cause serious departures from such conditions, especially if the metal is of extreme purity and free from solid impurities. (iv) Slurry Properties and Stability Interfacial tensions are of significance in the application of liquid metal slurries, To form an effective slurry between the particles of a solid phase and a liquid metal, the former must be completely “wetted” by the latter so as to prevent solid-solid contact. The conditions governing spreading and those leading to low interfacial tensions have
28
J. W. TAYLOR
been discussed previously for ideal, flat surfaces, but the conditions for the complete penetration of a particulate aggregate by a liquid do not require a contact angle of zero. WOODROW (1954) has indicated that, under ideal conditions, i.e., a perfectljl smooth surface, free from adsorbed gas, etc., a sufficient condition for penetration of an aggregate is that the contact angle is less than 90”. In practice, the effect of surface roughness must be considered. It has been pointed out that “simple” roughness will reduce the apparent contact angle below the true value provided the latter is less than 90”, whereas “composite*’ roughness may, in fact, increase the apparent contact angle value over the true value on a flat surface. From the point of view of “wetting” a particulate aggregate, it would seem desirable to have a low interfacial tension between the liquid and solid, and to have particles with relatively smooth surfaces. If the surface is rough and of a “composite” nature, i.e., contains pores filled with a second fluid, then steps must be taken to remove this fluid (if air or a gas, by evacuation) so as to facilitate wetting. It is interesting to consider the role of interfacial effects in relation to the escape of fission product gases from slurry particles. If the liquid metal forms a finite contact angle with the solid particle, gas bubbles, released at the particle surface, will conform to this angle. In the settled condition of a slurry, such gas evolution will result in a dilation of the bed and in the generation of hot spots on the particles. If these hot spots develop near the juncture of two or more particles, sufficient heat may be developed to vaporize the liquid in the interstice of the juncture, and under these conditions sintering of the two solid surfaces may take place. The lower the contact angle formed by the liquid on the solid, the more readily will the gas be removed by any de-gassing treatment. In liquid metal slurry systems, it is of importance to control the particle-size distribution, and interfacial effects are involved in this to some extent. Under isothermal conditions in a slurry system with a certain particle-size distribution, the larger particles grow at the expense of the smaller ones, which have a higher solubility in the liquid medium as a result of their larger curvature. The relationship between soluand COLLIE,1951b): bility and particle radius is in fact (MOLLIET (17) where L = solubility. r = particle radius. (T= interfacial tension between solid and liquid. Expression (17) has recently been extended by GREENWOOD to indicate that the rate of particle growth is proportional to the interfacial tension (a) between the liquid and the solid, a high interfacial tension favouring a rapid rate of growth. However, the effect of interfacial tension variations in this case is probably greatly outweighed by changes in the rate of diffusion of the solute in the liquid. In addition to this isothermal effect, a non-isothermal growth factor will also be present where the slurry system is undergoing a temperature cycle, e.g., on circulation through a heat exchanger. Thus material dissolved at the high temperature will be precipitated at the low temperature, and this precipitation will be controlled by the nucleation and growth considerations outlined previously; the effect will obviously
The signiti-
of wetting in reactor technology
29
be greater the larger the differential solubility with respect to temperature of the solid in the liquid phase. Under such conditions, precipitation would undoubtedly take place on the undissolved solid particles, and no separate nuclei would require to form. The ultimate effect of this solution and precipitation cycle might be quite small, the material’precipitated at one stage dissolving in the subsequent one. Initially, however, it might cause the rate of growth of the larger particles at the expense of the smaller ones to be higher than that occurring under isothermal conditions. In &ury systems of the type (liquid A + solid A$,,), it may well be possible to modify the size distribution of the latter by suitable control of nucleation conditions during the formation of the latter compound, i.e., by suitable additions of “wetting” agents and innoculants. lnterfacial tension effects are unlikely to have any pronounced effect on the settling rates of liquid metal slurries due to the relatively large particle sizes used so far. Only when the particle size is such that Brownian effects become sign&ant will interfacial effects become important. (v) Extraction Processes Interfacial effects can be of qualitative value in some metallurgical extraction processes. Thus in liquid-liquid extraction using metals, a low interfacial tension would appear to be necessary. In the first place, a stable dispersion of fine droplets of the one liquid in the other will only be obtained with a low interfacial tension, and secondly, this tension may possibly be of significance in the rate of extraction of the desired material. In extraction processes using slags and electrolytes, interfacial tensions are again of importance. To facilitate rapid separation of slag and metal, a high interfacial tension is desirable, and metal yields may be improved by suitable flux’additions which effectively raise this tension. Similarly, in reduction processes, where the metal is produced by precipitation from a liquid salt bath, a high interfacial tension will favour rapid and complete separation of the two phases. 7.
CONCLUSIONS
While the number of quantitative studies of interfacial tensions between liquid metals and solids is comparatively few, those that have been made, in conjunction with a number of qualitative observations, have indicated which factors control the spreading process. In addition, these observations have suggested suitable methods of modifying the interfacial tension for any particular application. In reactor technology a low interfacial tension appears to be desirable in the application of liquid metals as coolants and slurry media, while a high value appears desirable from the corrosion and mass transfer points of view; in extraction processes both high and low tensions may be desirable, depending on the process in question. ACKNOWLEDGMENTS
The author expresses thanks to Dr. H. M. FINNISTOUNand Mr. B. W. MOTT for helpful discussion during the preparation of this paper, which is published by kind permission of the Director of the Establishment. REFERENCES BAILEY,G. L. J., and WATKINS, H. C. (1951) J. Inst. Met., 80, 66. BAILEY,G. L. J., and WATKINS,H. C. (1950) Proc. Roy. Sot., B63, 350. BAILEY,G. L. J., FOX, M. J., and WATKINS, H. C. (1948) Br. Non-Fer. Met. Res. Assocn. Report, RRA 773.
30
J. W. TAYLOR
BAILEY, G. L. J. (1948) ibid, RRA, 898. A. (1953) Chem. Rev., 52,411. CASSIE,A. B. D., and BARTER, S. (1944) Trans. Faraday Sot., 40,456. DOBINSKI,S., and JAOIEL+%CI, A. (1939) Bull inter. acad. poln. xi., Classe, Sci. math. nat., 193A,423. FOX, M. F. (1948) Br. Non-Fer. Met. Res. Assocn. Report, RRA 771, FRICHE, R. (1948) Z. & Elekirochem., 52,72. GREENHILL,E. B., and MCDONALD, S. R. (1953) Nature, 171,37. GREENWOOD,G. Private communication. GURNEY, C. (1949) hoc. Phys. Sot., A62,639. HARKINS, W. D. (1952) Physical Chemistry of Surfaces, Reinhold Publishing Corp. HOLL~MON,J. H. (1950) Amer. Sot. Metals Seminar, Thermodynamics in Physical Metallurgy, p. 161. HOLLOMON,J. H., and TURNBULL, D. S. (1951) A.I.M.E.E. Symposium, Solidification of Metals and Alloys, p. 1. LEWIS,J. B. Private communication. MOLLY, J. L., and COLLIE, B. (1951a) Surface Actiuity, London, p. 100. MOLLIET, J. L.. and COLLIE, B. (1951b) Surface Actiuity, London, p. 141. MORGAN, W. (1954) Ph.D. Thesis, Cambridge University. NORTON, F. K., KINGERY, W. D., et al. (1953) U.S.A.E.C. Pnbl. NYO-3144. SAWAI, I., and NISHIDA, M. (1930) Z.$ anorg. allgem. Chem., 190,375. BONDI,
G. W. (1950) J. App. Phys., 21,721. C. S. (1948) Met. Tech., Tech. Pub., 2387. TAMMAN, G., and B~EHME,W. (1932) Ann. Phys., 12,820l. TAYLOR, J. W. (1954a) At. Energy. Res. Est. Rep., M/TN 24. TAYLOR, J. W. (1954b) Metallurgia., 50, 161. TURIUBULL,D. (1950) A. S. M. Seminar, Thermodynamics in Physical Metallurgy, p. 282. UDIN, H., SHALER, A. J., and WULFF, J. (1949) J:of Met., 1,186. VAN VLACK, L. H. (1951) ibid, 3,251. WATKINS, H. C. (1948) Br. Non-Fer. Met. Res. Assocn. Report, RRA 772. WENZEL, R. N. (1936) Industrial Eng. Chem., 28,988. WOODROW, J. (1954) At: Energy Res. Est. Rep., ED/M 13. SEARS,
SMITH,