Grain growth of pure and saline polycrystalline ice has been studied recently by Reference Jellinek and GoudaJellinek and Gouda (1969). Also the effect of applied tensile stress on the rate of grain growth of pure polycrystalline ice has been investigated (Reference Jellinek, Juznic and OnogiJellinek and Juznic, 1970[b]). Diffusion of radioactive cesium and sodium, respectively, through polycrystalline ice was also studied (Reference Jellinek and ChatterjeeJellinek and Chatterjee, 1971; Reference Jellinek and JuznicJellinek and Juznic, 1970[a]). Two types of diffusion have to be considered for such ice, as is also the case with polycrystalline metals, i.e. volume or lattice diffusion and grain-boundary diffusion. In order to evaluate diffusion coefficients of the latter, the grain-boundary thickness has to be known; this can be calculated accurately for ice of low salinity with the help of the H2O/NaCl phase diagram. Reference VeynbergVeynberg (1940, p. 500-05) made such calculations as early as 1940. However, information concerning growth rates of number-average grain diameters as functions of salinity, time and temperature was not available then. Reference Jellinek and GoudaJellinek and Gouda (1969) have provided experimental data in this respect and it is now possible to calculate grain-boundary thicknesses based on experimental results. The salinity affects profoundly the mechanical and morphological properties of ice (Reference ShumskiShumskiy, 1955; English translation p. 98 Reference PounderPounder, 1965). In polycrystalline ice of low salinity, practically all salt is contained in the grain boundary (see, e.g. Reference De Micheli and IribarneDe Micheli and Iribarne, 1969) which may be a liquid solution above, and a solid one below, the eutectic temperature. Sea ice contains numerous brine pockets and channels and hence docs not contain all salt in the boundaries.
The calculations presented here are in essence similar to those of Reference VeynbergVeynberg (1940, p. 500-05), but the approach is not the same; in addition, as pointed out above, actual grain-boundary thickness, δ, is calculated on the basis of measurements of grain diameter as function of salinity, age and temperature.
Derivation Of Fundamental Equation
The grain boundary in saline polycrystalline ice is given by the amount of substance necessary to form an aqueous saline solution (or a solid one) containing all salt at the particular temperature in question in accordance with the H20/NaCl phase diagram. Let s and p be the salinities of the saline ice or saline melt solution and of the grain boundary, respectively; salinity is defined here as grams of salt per 1 000 grams of saline melt solution or in %0 at the absolute temperature T. The number-average diameter of grains, considered as spheres, were experimentally determined for various conditions by thin-section analysis (Reference Jellinek and GoudaJellinek and Gouda, 1969).
If x is the mass of water plus all salt in the sample needed to form the grain boundary from unit mass of ice of salinity s, then this boundary contains px/1000, or px/ 1000 = s/ 1000 of salt; the grain boundary contains all the salt in the sample. The volume of the grain boundary is 
where pgb, T is the density of the grain boundary at T
abs. The salinity of the liquid (or solid boundary solution) at this temperature is obtained from the H2O/NaCl phase diagram. The latter was determined by Guthrie, Rodebush and Chretien. The data of these three authors agree fairly closely (Reference TimniermansTimmermans, 1960, p. 309-10). The volume oHne ice (all grains) at temperature T given by 
where ρi,T is the density of ice at temperature T.
If the volumeFootnote * average edge-length of the grains, considered as cubes, is 5, then the volume of JV grains is given by
and the volume of the grain boundary is
Eliminating N from Equation (2) gives the grain-boundary thickness δ,
For prisms with average quadratic cross-section b 2, one obtains
If δ ≪
b
and S ≪ р, which is the case here, then 
and р-s ≈ р hence Equations (3) and (4) simplify, respectively, to
and
The grain-boundary thickness is directly proportional to s and b respectively. The case for spherical grains is somewhat more complicated as the grain boundary is not of uniform thickness. It is useful in this case to define a “theoretical" grain-boundary thickness δ/2 here surrounding each ice sphere. The overall geometry is neglected, however the correct total ice-grain and grain-boundary volumes are taken. Each sphere of number-average diameter b is assumed to be enveloped by a grain boundary of thickness δ/2. The total ice-volume is again 11 (I-s/p)/pi,T and the total grain-boundary volume is given by s/ppgb,T as before. Hence Equations (1) and (2) contain a factor π/6 on their left sides. The resulting equation is given by Equation (3), which simplifies for b ≫ δ to Equation (3a). δ in Equation (3a) is here twice the grain boundary thickness. Equation (4a) is valid for cylinders of diameter b ; here again δ is twice the thickness. The number-average grain diameters (including the grain boundary envelopes, which are too thin to be measured separately) were experimentally determined as function of time t for overall salinities s = o%0 and s = i%0 by Reference Jellinek and GoudaJellinek and Gouda (1969). The general functional relationship of b and t (growth law) is given by the expression
where K exp is a rate constant and n is also a constant. Hence for s = I%0
Table I. Ghain Diameters And Thickness Of The Grain Boundaries b And S IN μη\ At Various Growth Ttmf.S Of Ice Obtained From Equations (3a) And (4a), 8(3) And 8(4) Respectively. In The Case Of Spheres 8 Represents Twice Grain-Boundary Thickness
Equation (6) shows that δ, or δ/2, increases directly proportional to the 8th power of time, and to s, respectively.
The temperature dependence of Kexv for the growth of grains in the range from —3°C to —30°C obeyed an Arrhenius equation. Hence.
and log δ decreases linearly as reciprocal absolute temperature increases for any one lime and salinity.
Average values of n for f = 0‰ and s = 1%0 are 0.30 and 0.25 respectively (Reference Jellinek and GoudaJellinek and Gouda, 1969), the Kexp values range from 4.85X10-2 (μm day -n ) at -3-0°C to 1.12 X 10-2 (μm day -n ) at - 36°C for s ‰, and from 5.69 to 0.94 μm day -n ) for the same temperature when s = 1. ‰.
A direct linear relationship was assumed between ƀ and s at any one constant time t and temperature T. Hence ƀ can be expressed by
ƀ o is the average diameter at t = o, which often can be neglected. K exp and n were derived as function of salinity for constant times t from plots of log ƀ versus log t for various salinities. Almost linear relationships were obtained for cither parameter. δ values for spheres, cubes, cylinders, and prisms were calculated from experimental values for ƀ found previously (Reference Jellinek and GoudaJellinck and Gouda, 1969) and are contained in Table I (δ values for spheres represent here twice the boundary thickness). Figure 1 shows plots of δ (spheres) versus absolute temperature for different growth times, obtained from Equation (7). (s = 1‰. A = 3.58 X 108 μm; E = 7.2 kcal/mol).
Fig. 1. δ (spheres) versus Tabs various growth times (s = 1‰). t = 0.5, 1, 2, 4, 7, 12 days. The points in each curve are in the same sequence as the t values above, starting from the left. (δ here is twice the boundary thickness.)
Discussion
It is interesting to note (see Table I) that at temperatures above the eutectic point the grain diameter ƀ increases, while at temperatures below it. ƀ decreases with salinity. It was remarked in the previous paper (Reference Jellinek and GoudaJellinek and Gouda, 1969) that growth in saline ice is slower near or below the eutectic temperature than that in pure (s 0) ice, while the reverse is true at higher temperatures. This is probably due to the fact that the grain boundary is quasi-liquid above and quasi-solid below the eutectic temperature.
In sea ice (c 5‰) or ice of relatively high salinity, there usually exist brine pockets or channels in the grains, especially if rapid freezing has taken place. Hence the calculations presented here will not he accurate enough for such cases. These pockets give an opaque appearance to ice.
Grain thickness obtained on the basis of calculations presented here have been used for evaluating grain-boundary diffusion coefficients for saline polycrystalline ice (Reference Jellinek and ChatterjeeJellinek and Chatterjee, 1971).
Veynberg's calculations are speculative, as some arbitrary values have been taken for grain diameters. Also data were not available on the effect of temperature at that time, and he assumed an approximate coefficient of thermal linear expansion for ice grains in order to compute diameters at different temperatures, which lead to erroneous results. He also was not aware that grain diameter and grain-boundary thickness are functions of the age of ice samples.
Acknowledgement
This work was made possible by a Grant from the U.S. Department of the Interior, Office of Saline Water, No. 14-01-0001-1122.
