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TABLE IV COMPARISON O R CALCULATED TOTAJ., REACTION TIMESFOR AN
Vol. 59
sition but different densities, by means of the absolute reaction rate theory equation
r2c
r2b
(det. head)
r2
79
(noz- (curved zle) front) (rsec.) (psec.)
(isothermal) (psec.)
should give important information on the nature of the equation of state. Thus if the a(v) equation of 70.7/29.3 comp. state were correct using AH* = 38.3 kcal./mole, B-AN 4150 1 0 . 3 1 . 5 0.26 15 one would expect k’(l.53)/k’(l.O) for 50/50 TNT50/50 cast Amatol 3510 20 3 . 4 1.7 30 AN to be somewhat greater than unity. For T250/50 TNT-AN (1.53) - Tz(l.0) = 200 to 500°K., k(1.53)/k’(1.0) 7.4 5.4 110 3300 150 (P 1) should be about 1.4 to 2.5. With the K-W-B equaPure AN 1720 3500 130 100 5000 tion of state in which T2(1.53) - T2(1.0) = -(200 a Computed from a ( v ) g u a t i o n of state. "ram the to 400); however, k’(l.53)/k’(l.O) should be about appro_ximation T S 4 ao/3D. 0 Computed from equations 0.7 t o 0.35. The experimental results obtained in r = R,/Ak’ and IC’ = 4 x 109 Te-”s”o’RT (isothermal de- this study show that the ratio ~2(1.0)/~~(1.53) varcomposition). ied between 7.5 and 2.2 depending on which theory -0.25 corresponding approximately to the Kistia- one assumes to be correct. Part of the ratio ~2(1.0)/~2(1.53) is associated with the larger partikowsky-Wilson-Brinkley equation of state cle size of the AN used in the low density compared pv = nRT(1 + x e s ) ; x = kTc/v (11) with that used in the cast _amatol series. The calin general T2 decreases about 200 to 400°K. over culated value of Rg(1.0)/Rg(1.53)is roughly two. the same increase in density. If, on the other hand, Hence kf(1.53)/k’(l.0) is between 1.2 and 3.5,the c is chosen to be positive (an unlikely value), Tz most probable value beiQg about 2.0. This would increases with density even more rapidly than in require T2(1.53) - Tz(l.0) to be about 300 to the a(u) equation of state. It may also be noted 400°K. which is only slightly higher in this case that while Tt depends rather critically on the form than that computed by the a(v) equation of state. of the equation of state chosen, or on c, all other It is clear, moreover, that k‘(l.53)/k‘(l.O) is greater thermohydrodynamic quantities computed by the than unity showing that the K-W-B equation is hydrodynamic theory are relatively insensitive to inapplicable in this case. I n later papers similar evidence will be presented for other explosives and the form of the equation of state. These considerations suggest that comparisons ingredients also lending strong support to the validof reaction rates for explosives of constant compo- ity of the equation of state(3). Explosive
Tea
(met.)
-
THE STRUCTURE OF ICE’ BY R. E. RUNDLE Conlribztlion No. 384, lnstituie for Atomic Research, and Department of Chemistry, Iowa State College, Ames, I w a Received December 9, 1964
Problems connected with the structure of ice are reviewed, particularly the problem connected with residual entropy and polar DS. non-polar structures. It is shown that if, a t low temperature, ice is truly hexagonal, then there is a good reason to believe it is polar. A polar structure is attractive in explaining some of the physical properties of ice, but i t leaves the residual entropy problem unsolved (though not hopeless), while the non-polar structure leaves some physical properties unexplained, but accounts for the entropy.
There have been many proposals for the structure of ice, the most successful of which has been the random structure of Pauling.2 This structure accounts in a natural way for the residual entropy of ice,3 and is compatible with the neutron diffraction s t ~ d y but , ~ it leaves unexplained other perplexing problems, such as why ice is hexagonal, certain electrical phenomena, and disorder-streaking of X-ray diagrams. Various crystallographic studies have indicated that ice is polar, which is incompatible with the Pauling structure. These have included many (1) Work was performed in the Ames Laboratory of the Atomic Energy Commission. (2) L. Pauling, J. Am. Chem. SOC.,67, 2580 (1935). (3) W. F. Giauque and J. W. Stout, ibid., 6 8 , 1144 (1936). (4) E. 0.Wollan, W. L. Davidson and H. ‘2. 81nill, Phus. Reu., 7 6 , 1348 (1949).
reports of bullet-shaped snow crystals,6 pitting of artificial ice single crystals on a single end of the crystals during sublimation,s and various electrical effect^,^ the most striking of m-hich is the report that under proper conditions ice crystals show a strong piezoelectric effect.s Several investigat,ions have failed to confirm the latter effect,g however, and the other support for the polar ice has been objected t o on several grounds and is not definitive. On the theoretical side, arguments such as BJerrum’s10on (5) U. Nakaya, “Snow Crystals. Natural and Artificial,” Harvard University Press, Cambridge, Mass., 1954. (6) J. M. Adams, Proc. Roy. doc. (London), 11128,588 (1930). (7) See E.S. Csmbell, J. Chem. Phys., 2 0 , 1413 (1952) for a review of these reports. (8) F. Rossmann, Experientia, 6 , 182 (1950). (9) 8. Steinemann, ibid.. B, 135 (1953): J. Mason and P. G. Owclton PhiE. Mag., 43, 911 (1952). (10) N. Bjerrum. Science. 116. 385 (1952).
. m
THESTRUCTURE OF ICE
August, 1955
the nature of dipolar interactions in ice are strong, and lead Do an expectation of polar ice. So far, t,he most compelling reason for believing that ice is nonpolar is that no polar model has yet been found which will explain the residual entropy of ice in the natural manner of the Pauling structure. If one considers that, a t least a t very low temperatures, ice is truly rather than statistically hexagonal, then there is another reason for believing that ice is polar. Let us, for example, require that the true symmetry at each oxygen position be threefold, as is required if ice is hexagonal. Let us also require Ha0 molecules. Now looking a t one puckered net in the ice structure (Fig. 1) we see that there are two kinds of oxygens; osygens of the$& kind lie slightly above the average plane and have a ligand along the C-axis directed upward, while oxygens of the second kind lie slightly below the average plane and have a ligand along the C-axis but directed downward. \
/
68 1
A Fig. 2.-Possible hydrogen positions about an oxygen in the ice structure. There is one set of three equivalent structures (above) with a hydrogen on the C axis, and one set of three (below) with no hydrogen on the C axis.
where U S is totally symmetric and is, in general, of different energy, than the functions U A and UA', which will be degenerate. A similar set of three functions would describe the other three configurations of Fig. 2, with two hydrogens in the three symmetrical positions. Again, the correct description would involve a totally symmetric linear combination plus two other degenerate combinations differing in phase angle. We shall call these wave functions Type TI. The point of interest is that all wave functions of Type I result in a proton density of 1/3 a t each of the three sites 1, 2 and 3, while those of Type I1 result in a density of 2//3 a t each site. Choice of one type of wave function for one oxygen in the plane determines the c,hoicea t every set if we require that there be one hydrogen per 0-13-0 bond. This can be seen in Fig. 3, where one circle represents Type
0000 0000 .0 Fig. 1.-Hexagonal net of oxygena from the ice structure. The hexagonal axis is normal t o the net, and the net is puckered with oxygens lying alternately a t f C above l/16 C are linked to and below the plane. All oxygens at an oxygen in the plane above, those at - '/]E C to an oxygen in the plane below, as noted in the bottom of this figure.
+
Directing attention to one oxygen of the fifirst kind, Fig. 2, we see that there are six mays of putting two hydrogens on the four tetrahedral lines joining the oxygen to its neighbors. These six configurations fall into two classes, three hith a hydrogen on the C-axis and three with no hydrogen along this direction. We could describe the first three configurations in terms of a hydrogen in a parabolic well a t positions 1, 2 or 3 which would result in the usual harmonic oscillator wave functions centered at positions 1, 2 or 3. (Let us call these U1,UZand U3.) These would not, however, he the proper wave functions for describing the system in a symmetrical field. Instead, the correct linear combinations are
00000000000
000000000000
00000000000
OoOooo~oo
Fig. 3.-Arrangement of hydrogen distributions in wells about oxygen in the hexagonal net of the ice structure. A single circle represents a distribution of Type I, with hydrogen in each of the three wells about oxygen, a double circle represents a Type I1 distribution with 2/3 hydrogen in each well about oxygen. To achieve one hydrogen on each 0---0link, distributions must alternate.
I, and two circles represenh Type I1 a t each point, and it is noted that nearest neighbors must alternate between Type I and Type 11. Furthermore all oxygens with Type I distribiition are of the same kind, thus, their 0-H bonds point in the same direction along C. Moreover, it is required that the oxygen directly above in the next plane be of the opposite kind, etc., which determines the entire arrangement. Furthermore, all Type I atoms in all
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682
plmes are of the same kind with their 0-H bands pointing in the same sense along the C-axis, so that the structure is polar. The question of the residual entropy of such a structure, using this non-classical count, is harder to settle. The three wave functions at each site are not all degenerate, even if no interaction among sites is assumed, and if the totally symmetrical state is the lowest in each case, as seems likely, there would be no residual entropy. If, however, the difference in energy is very low, as it would be for a high barrier for rotation about the threefold axis at each hydrogen site, then the three wave functions would become substantially degenerate leading to a residual entropy of R In 3. Furthermore, interactions among sites will undoubtedly alter the simple picture given here, lowering the degeneracy, and all that I feel prepared to say is that the residual entropy for this structure will lie between zero and R In 3. It should be noted that one does not know the real symmetry at an oxygen site at low temperatures. A t temperatures not far below the freezing point of ice, ice shows remarkable disorder scattering, and this is probably best explained if ice has only statistical rather than true hexagonal symmetry. However, this disorder scattering gradually fades as the temperature is lowered, and has substantially disappeared at a temperature above 90OK." The residual entropy, a,s obtained from calorimetric work, persists a t least to 15°K.) the lowest temperature involved in the heat capacity studies. Hence, the disorder causing the diffuse scattering appears to be different from that leading to the residual entropy. The structure above has substantially the same hydrogen positions as one I suggested some time ago.12 It has only partial hydrogens at all sites except those along the C-axis, which are fully ordered, and the neutron diffraction data reported by Wollan, Davidson and Shul14do not distinguish between it and the Pauling structure. Data extending to reflections with larger values of I , the hexagonal index, could, however, distinguish between this polar structure and the Pauling structure, and further work on ice is reportedly in progress at both Oak Ridge and Brookhaven. Because of the disorder scattering at higher temperatures, and large thermal effects, such work will be most definitive if carried out at very low temperatures. (Indeed, neutron diffraction work at high and low temperatures may lead to an understanding of the disorder Scattering.) It should be noted that another polar structure has been found by Owstonl8which satisfies Bjerrum's arguments. It has monoclinic symmetry, and is a fully ordered structure, requiring that the reported (11) (12)
P.G. Owston and K. Lonsdale,
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residual entropy be in error. Because of the close agreement between the residual entropy of DzOL4 and Hz03I have supposed that the residual entropy has been verified, and is an important feature which any correct structure must explain. Owston's work does emphasize anew that the disordered, non-polar structure of Pauling does not seem to be the structure of lowest energy. Moreover, the relaxation time in ice is low enough to make it difficult to see how the full disorder required by Pauling's structure can be frozen in if there are appreciable differences in the energies of various hydrogen configurations. Troubles of this sort do not arise with the polar, hexagonal structure. A tendency for ice to become polar might conceivably help in understanding both the disorder scattering and the slickness of ice. It has long been noted that one can skate on ice, not because the pressure melts the ice, but because even down to fairly lorn temperatures the surface of ice seems wet, that is, the surface energy of ice in air must be lowered by a surface film. If ice were strongly polar, the surface energy would be high due to surface charge unless disorder at the surface led to neutralization. Moreover, a highly polar ice would probably lead to twinning in which there is alternation of t,he direction of the polar axis, much as in the domains of a ferromagnetic single crystal. Such twinning should arise since a parallel arrangement of dipoles is fine when they are placed end to end, but very unfavorable when placed side by side. Submicroscopic twinning of this type would leave the C-axis parallel, or anti-parallel in all twins, and would be difficult to detect optically or crystallographically. Moreover, one might expect somewhat altered oxygen distances at the boundary between twins, leading to disorder scattering. The volume in the boundary layer should decrease with decreasing temperature due to the decrease in importance of the entropy factor which aids the stability of the disordered region. (It is possible that near the melting point the largest part of an ice crystal consists of the disordered regions.) These points are, at least qualitatively, in accord with observations of the disorder scattering, and are quite possibly amenable to experimental check. Finally, it should be stated that the evidence for polar ice seems far from conclusive, and that unless a proper explanation for the residual entropy for polar ice can be given, the Pauling structure must be considered more probable. However, it has seemed worth noting that the evidence for the Pauling structure is also subject to challenge, and, indeed, some points seem easier to understand in terms of a polar structure. It is hoped that this discussion, if it does nothing else, at least makes it clear that there are still problems in the ice structure which deserve further study.
J . Olaciology, 1, 118 (1951).
R. E. Rundle, J . Chem. Phys., 2 1 ,
1311 (1953).
(13) P. G. Owaton, J . chim. p h y s . , S O , C13 (1953).
(14) E. A. Long and J. D. Kemp, J . A m . Chem. Soc., 58, 1829 (1936).
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