Theory of ice premelting in monosized powders

Journal of Crystal Growth 123 (1992) 101—108 North-Holland

Jo~~o,

CRYSTAL GROWTH

Theory of ice premelting in monosized powders J.W. Cahn

1,

J.G. Dash and Haiying Fu

Department of Physics, University of Washington, Seattle, Washington 98195, USA Received 3 April 1992

The gradual freezing of water in inert porous media is considered to be a consequence of interfacial and grain boundary melting combined with curvature-induced depression of the melting point. The unfrozen fraction is calculated for monosized powders of spherical particles. It is found that the unfrozen fraction contains separate terms associated with interfacial and curvature melting, with distinctive dependencies on temperature and particle size. The calculated fraction is in excellent agreement with measurements on pure water in graphitized carbon black and polystyrene micron-sized spheres. The introduction of dissolved impurities adds a third term, with an intermediate functional dependence.

1. Introduction

• The persistence of unfrozen water in soils and other porous media, to temperatures as low as 40°C,has been known for many years, yet the causes of the phenomenon have not been established [1—3].The controversy has been sustained by the heterogeneity of typical soils and the vanety of their chemical and electrical interactions. New interest in the problem has stemmed from advances in the understanding of surface melting, showing than thin liquid films are thermodynamically stable at planar surfaces of solids below their bulk melting temperatures [4,5]. The effect occurs in various types of solid at one or more common facets of their interfaces with vapor, foreign solids, and grain boundaries. It has been argued that the premelting of ice is an example of such surface melting [6]. Support for that interpretation is provided by a recent study of the freezing characteristics of water in graphitized carbon black by quasi elastic neutron scattering [7], demonstrating the effect in pure H20 at —

chemically and electrically inert interfaces. However, the data show significantly more liquid than 1

Permanent address: National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA.

0022-0248/92/$05.00 © 1992

—

predicted by the standard Van der Waals theory. An explanation was proposed [81,by a curvature correction to the planar theory, but the observed increase is much larger. In this paper we extend the analysis to include the fully developed effects of curvature, together with grain boundary melting, and develop a general and quantitative formalism applicable to compact powders of spherical particles. The model is successful in describing the graphite data and a new study of monosized polystyrene powder. The effects of impurities are also treated in the range of dilute concentrations. 2. Theory We consider an ideal system composed of a pure fusible substance completely filling the pores or interstices of a powder matrix of monosized spherical particles. The solid phase of the substance is assumed to be homogeneous and isotropic: it melts in a first order transition at a bulk melting temperature Tm well below the melting temperature of the spheres. However, the transition is continuous when the substrate is in a powder matrix. This results from a combination of interfacial melting at the surfaces of the pow-

Elsevier Science Publishers By. All rights reserved

J. W Cahn et a!.

102

/

Theory of ice premelting in monosized powders

der spheres and grain boundary melting between differently oriented grains of the solid, and curvature-induced melting point depression in regions of high surface curvature. These are primarily at the contacts between two powder particles and along the curves where grain boundaries meet each other or meet the surface of a particle. The contribution of each to the total fraction melted

where ~T Tm T, p is the solid density, q~is the latent heat of fusion per unit mass, and ~ is a length on the order of a molecular diameter. For premelting to occur at a boundary ~iy has to be positive; it is composed of the free energies of the three interfaces between solid substance (s), partide wall (w), and liquid melt (I): iy

at a given temperature is a function of the detailed geometry of the powder matrix and the grain boundaries in the substance. We assume that before melting begins, the interstices are completely filled with solid. As initially prepared, the solid may be composed of a fine polycrystalline mass, a result of nucleation at many surface sites. The grains coarsen in time, driven by the tendency to reduce their grain boundary area.

grains of the solid substance z~iy where the subscript ss refers to the grain boundary. We neglect any orientational dependence of all these interfacial free energies and assume that values of A averaged over the orientations in the experiments will be similar in magnitude for partide surfaces and grain boundaries.

The process is assumed to continue quickly toward a final state of a single grain randomly oriented in each pore, with the grain boundaries confined to the necks between pores, where their area reaches a local minimum.

—

—

—

‘r~5~.Similarly, for the boundaries between two —

2.2. Melting points shifted by curvature A familiar and important effect of curvature is the depression (or elevation) of the melting point Tm [9]. The shift of melting temperature to T~is given by

2.1. Interfacial melting

(1

1\

r1

r2)

(2) If the two kinds of interfaces, the boundary between the solid and the particles and the boundaries between differently oriented grains of the solid, are wetted by films of the liquid phase, the substance premelts over a range of temperatures. The thickness of the films depends on the temperature T and the nature of the molecular interactions within the solids at the boundaries, We will assume that the interactions are dominated at long range (i.e. at distances greater than aforces, few molecular diameters) Van modified der Waals but the theory may bebyeasily to include other types of interactions. For interfacial melting governed by Van der Waals forces, the thickness at a planar boundary is

with ~

=

where r1 and r2 are the signed principal radii of curvature, taken as positive if the radii point from the solid toward the liquid. With experimental 3, y values qrn 23.33 iO~i/kg, p~ 917 kg/rn (andX nearly isotropic) [10], the cur-5~ 29 mJ/m vature coefficient for the ice—water interface ~ 0.0259 K /.Lm. Pockets of liquid with curvature appropriate to the undercooling given by eq. (2) may appear in a structure at all places of high curvature and of discontinuity in slope, such as =

=

=

=

d 0

=

A ~

(la)

where A is known from experiment (see section 2.3.3 below). An approximate expression for A is given by 3, (ib) A (2~2 ~y T~/pq~)Y =

junctions of surfaces. Two likely places are the contacts between particles and the termination of grain boundaries at particle surfaces. For spherical particles of radius R, that are surrounded by a surface melted layer with thickness d0 ~‘~z R, the terface is —R. —r in the (signed) rradius of Substituting curvature of R thefor ice—water in-

J. W. Cahn et a!.

/

Theory of ice premelting in monosized powders

expression derived by Baker and Dash [8], we obtain

103

2.3.1. Surface melting on the surface of solid spheres and at the grain boundaries

We characterize the powder matrix in terms of d

—

d0 1

—

13\

~‘slm 3p5q~R z.1T

‘

‘

2.3. Powder and crystallite morphology and melting of the solid substance

its average macroscopic its packing fraction f~,defined as thestructure, ratio of total sphere-

occupied volume to total volume of powder and interstices. For sphere radius R and particle number N~per unit volume 3N~, of and powder, the total thesuperpacking is f~ ficialfraction area per unit~rR volume is a~ 47rR2N 0 3) for sc, and V~/(8R3) 3f~/R.Since N~ 1/(8R for fcc, these limits are f~(fcc) ir/3~ 0.7405, and f~(sc) ~/6 0.5236. In random close pack=

=

=

In this section we calculate the amount of

liquid from estimates of particle and grain boundaty areas, the number of particle—particle contacts, and the lengths of the junctions between grain boundaries and particle surfaces. Since these quantities depend on how the particles are packed, we obtain bounds on random close packed powders by calculations for two regular structures, cubic close packing (fcc) and simple cubic packing (sc) of monosized spheres, Because the relaxation time for grain coarsening is expected to be much shorter than typical experimental times, we will assume in the analysis that a final coarsened state has been reached in which there is only one grain in each pore, and that there is a grain boundary in each neck between pores.

=

=

=

=

=

ing the fraction is intermediate between facecentered cubic and simple cubic. In sc packing, there are as many pores as spheres. Each pore has 6 equivalent nearest neighbor pores. Hence there are three grain boundaries per sphere. The necks joining pores are bounded by four touching spheres arrayed in a square; all the grain boundaries spanning the gaps are identical. Each grain boundary has an area of (4 ~-)R2, so the total superficial area contributed by the grain boundary per unit volume is agb(sc) N 2 0 x 1 x 3 x (4 ~)R 0.6148f 0(sc)/R. Therefore, the total superficial area per unit volume is a~(sc) a0 + agb(sc) 3.6148f0(sc)/R 1.893/R. —

=

=

—

=

=

=

individual ice crystals

1i~

grain boundary

r

cross sectional area ae spherical particle

\

\

.~7’~’meIting front

R

water

I p

x

R spherical particle

r

a

spherical particle

b

Fig. 1. Geometry of high curvature regions of ice—water interface in powders of monosized spherical particles: (a) region of contact between two powder particles; (b) region of contact between the edge of a boundary between two ice grains and a powder particle.

104

J. W. Cahn et a!.

/ Theory of ice premelting in monosized powders

In fcc close packing, there are four spheres, eight tetrahedral and four octahedral interstices per unit cell. Octahedral interstices only have

tetrahedral neighbors and vice versa. Each neck is bounded by three touching spheres; all the grain boundaries spanning the gaps are identical. 2. There Each boundary (v~necks ir/2)R are 32grain necks in a unitarea cell is(four for each of the eight tetrahedra or eight necks for each of the four octahedra). So agb (fcc)N 0 >< (2 >< 2 + 1 < 2 8(~/i ~/2)R2N 4) X (v~ ~/2)R 0 0.3080 f0(fcc)/R. And a~(fcc) a0 + agh(fcc) 3.3080f0(fcc)/R 2.450/R. The volume of liquid in the films at the spherical particles and grain boundaries per unit volume of powder is vf aid. If the temperature is —

The liquid volume per crevice is 4~r f0 zx dx, where the upper limit ranges between r p—r

and

p—

_____

Vr2+2Rr. R+r The result, to first order in r/R, is a volume of 2~Rr2per crevice. In sc packing, each sphere touches 6 neighbor-

=

=

—

—

=

=

=

=

=

ing spheres, hence there are 3 crevices per sphere. So the volume of the crevices per unit sample volume is 2 v~(sc) N0 X 3 X 2rrRr r 2 r 2 4.5f 0(sc) 2.356( (5a) =

(~) ~)

=

=

not very close to Tm the film is sufficiently thin to ignore curvature over most of the area of the boundaries, so that d can be approximated by d0 according to eq. (3). Thus, vf(sc)

=

R1.893A ~1T1~3’ vf(fcc)

=

R2.450A ~Tt/3

(4)

2.3.2. Melting at the sharp curvature regions There are two types of region of sharp curvature, where the melting point is appreciably depressed: the crevices between two adjacent spheres and the edge contacts between the grain boundaries and the spheres. In both cases the volume fraction can be expressed in terms of the radius of curvature r of the solid—liquid interface. 2.3.2.1. Melting at the crevices. The crevices between spheres are analyzed with reference to fig. la. We ignore the circumferential radius, which is usually much larger than the radius shown in the cross section. A more detailed analysis without this assumption is available [11]. Inspection shows (R + r)2 R2 + p2 and (R z)2 + x2 R2 solving for p and z in the range r ~ R, =

=

~( x2

—

=

.

In fcc close packing, each sphere touches 12 spheres, hence there are 6 crevices per sphere. So 2 r 2 r 2 v~(fcc)=N0x6x2~-Rr =9f 0(fcc) =6.665(—~. \R)

(~)

2.3.2.2. Melting of the grain boundary edge at the sphere. The edges of the grain boundaries contacting the spheres are shown in cross section, in fig. lb. Neglecting the curvature of the sphere relative to that of the ice—water interface, 2the cross-sectional area These of the regions liquid is contribute ae 2(r a lTr2/4) 0.4292r2. —

=

=

liquid volume equal to ae times the total edge length. In sc packing, each sphere has one pore and three grain boundaries. Each grain boundary has a total edge length of 4 x 2~R/4 2irR, therefore, the grain boundary edge length per sphere /(sc) I x 3 x 2irR. The total edge melting volume per unit sample volume is =

=

v~(sc) N =

0 x /e(5c) X ae 1+~+. 4

4R

‘‘)‘

=9~1 (

—

~)f0(sc) (~) ~

r

2R

+ •~~~~‘‘•‘

2 r~

2

=

z=—

(5b)

~

kR)

(6a)

J. W. Cahn et a!.

/ Theory of ice premelling in monosized powders

In fcc close packing the grain boundary in each x 2~rR irR.

neck has an edge length of 3 x So le(fcc) 8 x irR and

=

=

v(fcc)=N P xl(fcc)Xa e I

=

12(,1

-i-)\ ~T

—

105

temperatures 0.01°< T < 30° [12]. In Gilpin’s analysis the data were fitted to a power law d constant x (4TY~ over the entire experimental range, with an average exponent n 0.42. However, it has been argued that temperatures lowerthe than 10°Cshould beeninduced excluded from fit due to the higher have viscosity by =

=

e

f0(fcc)

r r ~) 1~907L~) (6b) ~2

/

2‘

=

The liquid volumes in the crevices and the

grain boundary edges can now be related to their melting temperature by means of eq. (2). We note that in both regions the two principal radii of curvature are vastly different (as long as the ternperature is not very near Tm), hence we can

‘-~ —

proximity of the solid boundaries. When the low temperature data are excluded, the fit improves, and the empirical exponent becomes consistent with the Van der Waals value 1/3 [6,13]. The parameter corresponding to this fit is A 3.5 X iO~~im K1~3.We adopt this empirical value for the present calculation. Less is known about the values of A for grain boundaries, which depends =

approximate 1/r 1 + 1/r2 1/r. Together with the liquid films at the sphere interfaces and grain boundaries, we obtain expressions for the total volume fraction of unfrozen liquid as a function of the temperature: 1.89A / r \2 fL(sc) 1/3 + 3.367(,_) R ~1T R =

1.89A =

R ZITt/3

2.26 x iO~ +

(7a)

(R ~T)2

2.45A i r \2 fL(fcc)= RL%Tt/3 +8.572~-) 2.45A =

R ~T’~3

5.75 +

=

=

,~

~ ~T

(R ~T)

2.4. Impurity-induced premelting In real soils, impurities play an important part in the freezing process of water. Few impurities

X iO~

(R ~T)2

in addition on the relative orientations of the two crystals. Giving it the same value, the liquid fractions are 6 61 t~3+ 2 26 2 x iO~, (8a) fL(sc) R LlT (R ~T) 8.57 5.75 fL(fcc) D i/3 + 2 X i0~. (8b)

‘

(7b)

where R and A are measured in micrometers (nm). 2.3.3. Determination ofparameter A

A complete quantitative evaluation of the fractions depends on the interfacial melting parameter A. Since the interfacial coefficients y~ and Ylw depend on the wall material, ~ly does not have a unique value. Furthermore, the length 6 is uncertain, since it depends on the structure and dielectric properties of the melted film. At this juncture we can treat A as an adjustable parameter in fitting the theory to experiment. Alternatively, we may assume an estimated based on a series of measurements of wire regelation, which yielded values for the liquid film thickness at

are incorporated in the ice, so that it is valid to assume that the segregation coefficient between ice and water is zero. To explore impurity effects, let us begin by neglecting all surface effects. According to Van ‘t Hoff’s law of freezing point depression for dilute solutions [141, the liquidus

or bulk melting temperature is reduced by a term proportional to the total summed molecular concentration of all dissolved species, including ions,

regardless of the nature of the species and the strength of its interaction with the solvent. For water this freezing point lowering at ambient pressures is 1.86 K for each mole of dissolved species per liter or 1.86 x iO~ K m3/mol. The liquidus slope is denoted by —m. The factors in this coefficient m come entirely from properties of the solvent, such as q~,Tm and the molar density. Liquidus temperatures for water have

106

1 W. Cahn et a!.

/

Theory of ice premelting in monosized powders

long been measured with micro-kelvin accuracy, and deviations from a constant liquidus slope, that come only at non-dilute concentrations from the interactions between the solute molecules, have been extensively cataloged because of their importance in the theory of such interactions [15]. For our purposes the linear law is sufficiently accurate at all concentrations up to the eutectic, which is where the dissolved species have reached their solubility limit for that temperature. For a binary solution (a single dissolved species), below this eutectic temperature there is no liquid (apart from surface melting); both water and the dissolved species will crystallize out as separate solid phases. The eutectic temperature depends on the specific species that is dissolved in the water, Liquidus curves to the individual eutectic temperature are known empirically for many solutes in water. When there is more than one dissolved specie, there will be a hierarchy of saturation temperatures. At each a new solid phase appears, until a final eutectic point, below which there is no liquid. The slope of the liquidus changes discontinuously at each new saturation point [16,171. An aqueous solution, with initial summed solute density p~(and no surface effects), will begin to freeze at ~T mp~.Ice separates out as the temperature is lowered, and the remaining liquid becomes more concentrated. If PT is the concentration of the solution in equilibrium with ice at ~T < ~T, conservation gives fL P/PT’ Since mpT ~tT, we obtain =

=

=

Impurity effects become important only at values of p and R for which the curves of eqs. (8) and (9) would approach each other, or intersect. For low enough concentrations, impurity effects can be neglected at all temperatures. With the approximations that entered into the derivations of eq. (8), a derivation of a single expression for all the capillary and impurity effects is not warranted.

3. Comparisons with experiments Two recent studies of the melting of ice in powders are suitable for testing the model. In the first [7], the powder was graphitized carbon black, a powder composed of compact polyhedral partides with basal plane graphite surfaces. The typical particle diameter estimated by microscopy was 0.24 ~m, compared to 0.19 ~.rmdetermined by gas adsorption. The difference indicates a size distribution which is relatively narrow. The measurements were made by quasi-elastic neutron scattering, a technique particularly well suited to the problem since it can yield a simultaneous measure of the fraction melted and the diffusion coefficient of the liquid. The results detected liquid present down to 30°C.Near the melting point the diffusion coefficient agreed with that of supercooled water. In fig. 2 we show the measured liquid fraction, together with the fraction calculated according to eq. (8). In fitting the data we excluded temperatures above 1°Cas beyond the range of validity of the theory. R was treated as an adjustable parameter, but good agreement —

—

I 9\

—

L

—

mp/

~

.

As mentioned above, for water m 1.86 X iO~ m3 K/mol. This would be the fraction of the interstitial water that is liquid. To obtain the fraction of liquid in the sample, composed of ice, water and particles, these fL expressions have to be multiplied by I —f 0. If the water is spread out as thin sheets of fixed area, d will be inversely proportional to 8T. Because this is intermediate in power to the capillary effects, at low temperatures surface melting will dominate over impurity effects, while near the melting point curvature effects may stay dominant, =

was obtained for the actual range, 0.1
range, the test is not very stringent. In the second study the powder was composed of polystyrene spheres [18[.The typical dimension of the polystyrene spheres under the electron microscopy is 5 ~rm. The water was deionized and degassed before mixing with the polystyrene spheres. The mixture was then put into the space

J.W. Cahn et a!.

_______

\

/

Theory of ice preme!ting in monosized powders

________

Ice! Graphitized Carbon Black

>

5

between ice and water

(—~

3 and

90), the TDR

unfrozen water in the ice/powder mixture [19], and has been used in several investigations of ground oped forfreezing the dielectric [20,21].constant An equation of emulsions [22] develwas employed to analyze the polystyrene—water mix-

fcc sc

0.1

the tremendous difference of dielectric constants technique is sensitive to very small amounts of

0

o

‘~

107

E

ture to deduce its unfrozen water content. We find that the eqs. (8), calculated for the actual value of particle radius, provide close bounds in

0)

I

0.1

1

100

10

Tm-T (K)

Fig. 2. Comparison between measured and calculated liquid fraction of H 20 in graphitized carbon black powder. Average diameter of powder particles is 0.24 ~zm. The computed curves are calculated with independently determined parameters, and no fitting factors.

of a coaxial transmission line, connected to a time domain reflectometry (TDR) spectrometer. The spectrometer sends an electromagnetic pulse into the mixture and detects the reflected wave packet.

By analyzing the combined signal, the dielectric constant ofrange the mixture deduced in the frequency 10 MHzmay to abe few GHz. Due to

plotted versus agreement withtemperature the measurements. and compared The datawith are eq. (8) in fig. 3.

4. Discussion A detailed examination has been made of four premelting effects of pure water in close packed powders; surface melting at the particle—ice interface, grain boundary melting, crevice melting at the contacts between particles, and along the perimeter lines where grain boundaries terminate at particles. These effects combine to give two terms to fL; one that3,and is inversely to anotherproportional inversely proboth R and to 4T~ portional to both R2 and LIT2. The constants of proportionality are given by independent empirical measurements, and geometrical estimates of surface areas, perimeter lengths and number of

is within the rangeInofthe estimation with experiment no adjustable parameters. experi-

01

c/Polystyrene

0)

0.01

are needed, contacts, all per and unit no single volume algebraic ofmicron specimen. term would fit neither ments with term particles overwhelms in the the other; both size The range terms fit the data. A confirmatory experiment with different particle radii would alter the relative im-

0.001

io~ 0.01

portance of the terms. Equating the terms in eq. (8) shows that the effects are comparable when RLI~5~3 0(1), with R measured in micromoters. This relation gives the approximate values of LIT~where the melting curve for a given R bends up from the 1/3 slope in the data. When this =

01

1

10

Tm-T (K) Fig. 3. Comparison between measured and calculated liquid fraction of H 20 in polystyrene powders. Average diameter of powders particles is 10 jzm. The computed curves are calculated with independently determined parameters, and no fitting factors.

—

cross-over condition is used to eliminate LI7~from eq. (8) we is approximately equal to 4”5find at that the fL crossover., As R increases, 102 R

1 W. Cahn et a!.

108

/

Theory of ice preme/ting in monosized powders

both LI7~and fLX at crossover become smaller. These shifts of the crossover region with larger R to lower LIJ~and f~.can be seen by comparing figs. 2 and 3. At larger R than in these experiments, fL at crossover becomes so small that the region where surface melting is dominant may not be detectable from the data. For particle sizes smaller than in these experiments, the crossover LI7~ would move outside the experimental range. For all spheres with R larger larger than 0.05 ~sm, the melting at small LIT is dominated by curvature melting. With parallel planar sheets of graphite, with grains of ice in between, curvature melting is less important [7]. Because the impurity exponent is intermediate between the surface melting and curvature terms, the crossover region is the first to be affected by a rising impurity concentration. This will occur for 7~5 0(1), where temperatures are in 102 mpR K and R is in ~sm. For p less than that, impurity effects can be neglected at all temperatures. For 1 ~sm spheres this concentration would be 5 X i0~ mol/l, a degree of purity that is readily achieved. In real soils, the sizes of solid particles have a broad distribution, typically from nanometers to millimeters. Also the shapes of particles are not necessarily spherical. Nonetheless, if the particles are smooth and the impurity concentration is sufficiently small, we will expect the melting curve to be composed of a sum of terms with the same two undercooling dependencies. The surface areas of the particles and of the grain boundaries will each contribute a LIT 1/3 term, while the number of crevices and the total perimeter lengths each will contribute a LIT2 term. If the particles are jagged, the contributions from the crevices will have a larger negative exponent. The assumption that crevices form where a sharp corner of one particle touches a smooth surface of the other would lead to an exponent of —3 [23]. In addition, the leaching of typical soils will cause appreciable concentrations of solutes, thereby introducing an impurity term as discussed in section 2.4. Because the water may contain many species, and may be saturated with respect to some of them at some ambient temperature above Tm, the impurity effect may be piecewise linear in LIT. =

Acknowledgments This work was supported by NSF Grant DPP9023845 and ONR (J.W. Cahn). We thank S. Arcone, M. Bienfait, P. Black and B. for useful discussions during the course of Hallet the study.

References [1] D.

Taber, J. Geol. 37 (1929) 416. [2] RD. Miller, in: Proc. 2nd Intern. Conf. on Permafrost. Yakutsk, USSR, 1973 (NatI. Acad. Press, Washington. DC, 1973) pp. 344—352. [3] Ice Segregation and Frost Heaving, Polar Research Board (NatI. Acad. Press, Washington, DC, 1984). [4] R. Lipowsky, Phys. Rev. Letters 49 (1982) 1575 [5] H. Ldwen, T. Beier and H. Wagner, Europhys. Letters 9 (1989) 791.

[6] J.G. Dash, in: Proc. NATO Advanced Study Institute, Erice, Sicily, 1990. Phase Transitions in Surface films, NATO, ASI, Vol. B, Eds H Taub, S.C. Fain, Jr., G. Torzo and H.J. Lauer (Plenum. New York, 1991) p. 267. [7] M. Maruyama, M. Bienfait, J.G. Dash and G. Coddens, J. Crystal Growth 118 (1992) 33. [8] MB. Baker and J.G. Dash, J. Crystal Growth 97 (1989) 770. [9] J.W. Gibbs, Scientific Papers, Vol. 1: Thermodynamics (Longmans, Green and Co., London, 1906; republished by Dover, New York. 1961) pp. 219—252. [10] [11] [12] [13] [14]

S.C. Hardy, Phil. Mag. 35 (1977) 471. R.B. Heady and J.W. Cahn, Met. Trans. 1(197(1)185. R.R. Gilpin, J. Colloid Interface Sci. 77 (1980) 435. J.G. Dash, Contemp. Phys 30 (1989) 89. J.H. van ‘t Hoff, Phil. Mag. 26 (1988) 81.

[15] H.S. Harned and B.B. Owen, The Physical Chemistry of Electrolyte Solutions, 3rd ed. (Reinhold, New York, 1958). [16] J.H. van ‘1 Hoff, Zur Bilding der oceanischen Salzablagerungen, 2 Vols. (Brunswick, 1905—1909). [17] D.R.F. West, Ternary Equilibrium Diagrams (MacMil-

lan, New York, 1965). [18] H.Y. Fu, private communication. [19] S.A. Arcone and J. Wills. J. Phys. E (Sci. Instr.) 19 (1985) 448. [20] D.E. Patterson and M W. Smith, Can. Geotech. J. iS (1981) 131. [21] A.J. Delaney and S.A. Arcone, IEEE Trans. Geosci.

Remore Sensing GE-22 (1984) 428. [22] T. Hanai, Electrical properties of emulsions, in: Emulsion Science, Ed. P. Sherman (Academic Press, London, 1968) ~. 353. [23] J.W. Cahn and RB. Heady, J. Am. Ceram. Soc. 53 (1970) 406. ~