Ice: Structure and Properties


Contents

  1. The Nature of Ice
  2. The Structure of Ice
  3. Ice In The Sky
  4. References

The Nature of Ice

Ice is an allotrope of water substance, a crystalline solid stable below 0°C, though it sublimes slowly to water vapor at low water vapor pressures. Ice, water and vapor coexist at the triple point at T = 0.01°C and p = 4.579 mmHg. If the temperature and pressure vary from these values, one of the three phases disappears, and the other two reach an equilibrium. Ice and air-saturated water are in equilibrium at 0°C or 273.15K, an important reference point for thermometry. Although ice melts at 0°C, water must be cooled below this temperature to initiate ice formation, after which the mixture warms and freezes as rapidly as heat can be removed. This is called supercooling.

Ice can be used in freezing mixtures to produce temperatures well below the normal melting point of ice. This, and the use of antifreezes, is treated in Phase Rule, since some thermodyamics of mixtures is needed to treat it properly. Salt is the normal substance added for this purpose, as in the traditional making of ice cream.

Ice is a mineral (a naturally-occurring substance of constant chemical composition) and Dana regards it as such, although many mineralogy books do not. It is permanent in polar regions where the average temperature is below 0°C, and of temporary occurrence elsewhere. The greatest accumulations are in the large ice sheets of Greenland and Antarctica. Other permanent accumulations are found in mountain glaciers and snowfields. The flow of glaciers is the best evidence of the flow of matter considered as a solid when long times are involved. The passage of a weighted piano wire through a block of ice without cutting the ice into two pieces is another example of the same thing, often erroneously explained as melting followed by freezing ("regelation").

Pure ice is clear and transparent. In large thicknesses, transmitted light appears blue because of the tail of an infrared absorption due to the protons (hydrogen nuclei) in it. This is not the blue of the sky, Rayleigh-scattering blue, but a greenish absorption blue. Ice is soft, 1.5 on Moh's scale, between talc and gypsum, so cutting it will not dull knives. Ice can easily be scratched by the fingernail. It is brittle, breaking with a conchoidal fracture, and has no cleavage. Its density at 0°C is 0.91671 g/cc, less than that of water, which has many important consequences. Among these are the fact that icebergs float and do not sink to the bottom of the ocean, which was disastrous to the RMS Titanic in 1912. If icebergs were pure ice (which they are not), they would float with a fraction 0.92 submerged and 0.08 visible. Some Antarctic icebergs have an average density of only 0.83, so they float with 0.17 visible. Icebergs usually contain entangled air and rock débris. The surface ice formed on natural water bodies in the winter remains on the surface, hindering the freezing of the water below it by its low thermal conductivity. Life can survive through the winter in the cold water in all but the smallest ponds. The sea itself benefits from this; there is always water beneath the Arctic ice. Water has maximum density at about 4°C, so it is already expanding as it approaches freezing. In lakes, water cooler than this descends and is replaced by warmer water from below, which in turn is cooled. Only when the whole body is at 4° does freezing begin at the surface. Dense 4° water makes an isothermal layer in a lake. The water above this increases rapidly in temperature as the warmer surface is approached, forming a region called the thermocline.

The expansion on freezing also has another wide-ranging effect. Water is attracted by capillary action into fine cracks in rocks and road pavements. When it freezes, the crack is powerfully wedged wider. This makes ice the most powerful weathering agent in geology, in an almost unseen process whose results are soon evident. Water is a powerful agent of chemical weathering, which takes advantage of the mechanical action of the ice. Between them, they create the surface of the planet that we see.

Stones embedded in finer material are forced upwards by alternate freezing and thawing, an example of a general phenomenon called frost heaving. This can produce remarkable structures, such as the gravel circles of Spitzbergen. These are rings of coarse stones about 1 m in diameter that are almost perfect circles, and whose origin has long been a mystery. Recent computer simulations have suggested how these circles can be produced by frost heaving. Stones apparently come to the surface both in regions where the earth is not permanently frozen, and in permafrost areas. Of course, in either case the surface must not be permanently frozen, but in one case the freezing is a surface phenomenon, and in the other it is the melting. At depth, the earth will always be at a constant temperature.

A possible mechanism by which stones are sorted and pushed to the surface is illustrated in the figure. At depth, we presume the earth is at a constant temperature of +5°. The 0° isotherm rises and falls with the surface temperature. When it reaches the surface, the earth is thawed. In arctic permafrost areas, the behavior may be the opposite, with the earth permanently frozen below, and the surface alternately frozen and thawed. On the left, the 0° isotherm is falling, and the stone is plucked upwards since it freezes in at the top, allowing thawed material to come in beneath. On the right, the 0° isotherm is rising. The stone is held up by being frozen in at the top, while the earth around the lower part contracts. This mechanism obviously fails if conditions are reversed, and the earth at depth is constantly frozen, not thawed. The heaving occurs when the 0° isotherm is moving by the stone, and must result in relative motion between the stone and its matrix. It would be good to have a clear explanation of the mechanism, but I have not seen one so far. In making gravel circles, the depression of the 0° isotherm causes the frost heaving to make piles of stones by a feedback effect. The Physics Today article in the References has pictures and further information.

Ice is slippery, with a low coefficient of friction, even with dry, and this is accentuated when the surface is lubricated with water. At 0°C, the static coefficient of friction of ice on ice is 0.05-0.15, while the kinetic coefficient is about 0.02. These numbers increase at lower temperatures, about doubling at -12°C. Brass on ice gives 0.02 at 0°C, and 0.085 at -20°C. Waxed hickory (skis) give 0.09 at 0.1 m/s, and 0.03 at 4.0 m/s. We can conclude that the coefficient of friction decreases with speed and increases with lowered temperature. A coefficient of friction of 0.02 means that sliding will begin (static) or proceed at constant speed (dynamic) on a 1:50 gradient. It is often said that the pressure under the blade of an ice skate melts some ice and lubricates the motion, but the pressure and the magnitude of the effect are insufficient for this to be valid. It is simply that the ice is slippery anyway, especially not far from the melting point. Runners are very satisfactory as a substitute for wheels, especially when wheels would sink into the snow, and their use is traditional in cold climates.

When ice freezes from water, dissolved gases and other substances in the water, such as salt, are rejected and the ice is pure water (practically, of course, there are usually inclusions of the impurities in the ice, but they are not part of the crystal). This is easily seen in ordinary ice cubes, where the last part to freeze in the centre is usually cloudy with these inclusions. If distilled water is boiled and then frozen, the ice will be clear, as can be verified in your kitchen. Sea-water could, in principle, be desalinated by freezing, but the cost of the energy to refrigerate the water is prohibitive, unless natural cold is used. Separating the impurities is also a difficult matter.

Water vapor forms ice crystals in the atmosphere, which then fall as precipitation. This is quite pure water, since it has been purified by evaporation. It is easier for rain to form by the melting of falling ice crystals than by the direct formation of water droplets, which requires a high degree of supersaturation. Snow is formed from hexagonal, dendritic single crystals of remarkable beauty. Although the substance of the snow is clear and transparent, it appears pure white when it falls because of the scattering of the light from the very irregular surface. A similar effect occurs in white paint: titanium dioxide, a white pigment, is actually clear and transparent, but when finely ground appears white. A substance that is not transparent appears black under the same circumstances. White and black are subjective color perceptions; they have no consistent physical meaning.

The main kinds of ice crystals that form in the atmosphere are columns, plates and needles. The beautiful snowflakes with hexagonal symmetry, or stellar crystals, are actually rather rare. It is often said that no two snowflakes are alike, because of the seemingly infinite variety in them, but many snowflakes are really quite similar. Snow rarely consists of perfect crystals, but more often as fragments and clumps and balls of dendritic pieces, with an occasional more perfect plate or star. Columns form at the lowest temperatures, around -40°C, plates and stars at intermediate temperatures, and needles at the lower temperatures, from -8°C to 0°C. Columns falling through lower regions may become capped with plates on one or both ends. The columns formed in high cirrostratus are the ones responsible for the 22° halo. The hexagonal columns and plates adopt a particular attitude when falling because of their aerodynamics. Such bodies turn to an orientation of highest drag, which for plates is horizontal, and for columns sideways. Refraction of the light of the sun or moon by such oriented crystals cause the attractive halo phenomena that will be discussed below.

Many curiosities are associated with snowfall. The wind can blow small lumps of snow that pick up more snow as they roll along, becoming large snow rollers of striking appearance. Icicles grow and melt; hydromites are analogous to stalagmites. Ice freezes on wires and spiderwebs from supercooled moist air. Noise can come from snowbanks, as it comes from sand dunes. Indeed, snow can make drifts analogous to sand dunes, of which the sastrugi (appearently singular) is an example, having special textures of snow. Blowing snow, already fallen, in winds faster than 35 mph is a blizzard. Frost grows in dendritic patterns on windowpanes. Rime is a poetic name for hoarfrost, white frost, caused as explained above ("hoar" is a word for grey). Rime in meteorology always means ice consisting of crystals fused together. Glaze ice is the same, but clear and transparent, or glassy. These two kinds of ice are distinguished in aircraft icing, where they have different effects.

Cold streams in which the water is near freezing can also create ice beneath the surface. Frazil is a spongy ice formed when very cold surface water mixes with lower water and freezes. It can be carried along with the stream and collect where its progress is hindered. Rocks and other bodies beneath the surface may be cooled below 0°C by radiation in quieter waters, and a surface layer of ice may form on them. Supercooled water will also freeze on surfaces it encounters, and this may be a more valid cause than radiative cooling. Because this happened with ships' anchors, this ice is called anchor ice. Spray from turbulent streams and waterfalls can also freeze as rime ice on objects near the water.

Hail is a polycrystalline mass of snow crystals that have clumped together, partially melted, and then have been refrozen in updrafts, so that there may be an onion-like layered structure. It comes in a great range of sizes, but large hail is fortunately rare, because it is very destructive. Most hail is marble-sized or smaller. If the clumps have not been refrozen, the result is the soft masses called graupel. The white color is again due to scattering by the minute surface irregularities. Snow that falls goes through a regular series of changes depending on time and temperature that turn it into grainy névé or firn, and finally into solid ice, unless it melts first. Fresh snow has a density of about 0.1 g/cc (that is, 10" of snow make 1" of water), old snow about 0.4 or 0.5 g/cc.

At -15°C, ice has a coeffient of linear expansion along the c-axis of 46 x 10-6 per K, and perpendicular to the c-axis 63 x 10-6 per K, for a coefficient of volume expansion of β = 156 x 10-6 per K. At 0°C, β is about 200 x 10-6 per K. The latent heat of fusion is 79.72 cal/g, and the latent heat of sublimation is 678 cal/g, at 0°C. The specific heat at constant pressure is 0.505 cal/g/K at 0°C. The Debye temperature is 315K, from which the specific heat at low temperatures can be estimated. The heat conductivity of ice is 0.0057 cal/s/cm/K, about the same as many rocks, and 1/100 of that of aluminium. The thermal conductivity of water at 12°C is 0.00136 for comparison, about a quarter that of ice. New snow with a density of 0.11 g/cc has a conductivity of 0.00256, but old snow with d = 0.45 has 0.000115, a counter-intuitive figure, but that is what the reference (Lange's Handbook of Chemistry) says. Perhaps old snow has voids filled with air that reduce the conductivity. In any case, snow or ice acts as an insulating blanket. This is no secret to the people who live in igloos.

The adiabatic bulk modulus of ice at -13°C is 7.81 x 1010 dyne/cm2. If the Poisson's ratio is assumed to be 1/3, then the shear modulus is 2.93 x 1010 dyne/cm2, and Young's modulus is 7.81 x 1010 dyne/cm2 or 1133 ksi. This means ice is much more springy than steel or aluminium. The speed of longitudinal sound waves is 2919 m/s, and shear waves will travel at 1788 m/s. I have not been able to find information on the compressional or tensile strength of ice on the Internet, or anywhere else for that matter, but these things are not difficult to measure, and must certainly be known. In fact, ice information in general is not as easy to find as one would expect for such an important and familiar substance. Ice has indeed been used as a material of construction, as in the Ice Palace of Leadville, and roads and railways have been built across it, as across the frozen Susquehanna before there was a bridge. Ice sculpture is a winter entertainment.

Structure of Ice

Ice has the chemical composition H2O, molecular weight 18.016 11.2% hydrogen and 88.2% oxygen, by weight. It is a lattice of oxygen atoms, a macromolecule, held together by curious bonds consisting of a proton and two electrons joining O++ ions, arranged tetrahedrally. The bond length is 0.276 nm, with the proton closer to one oxygen than to the other, at distances 0.100 nm and 0.176 nm. This is a hydrogen bond, an interesting feature of water substance in all its forms.

It is fortunate for us that the earth is so massive, since the water molecule is light, and can easily escape the gravitational attraction of a planet. This has happened on Mars to such an extent that water no longer occurs in its atmosphere or surface. Buried ice holds what remains. Although signs of buried ice have been reported, no ice has yet been discovered. The surface of Mars seems to display evidence of earlier water on the planet, however. Without water, weathering is very slow on Mars. Life on Mars probably disappeared with the water. The earth is still covered with a thin sheet of water, water-ice weathering is active, and life continues.

The crystal structure of ice is well-known, and its properties can be calculated from the known structure, in contrast to water, for which the structure is still imperfectly known. (The normal modes of vibration, however, have not been determined.) Bragg inferred the positions of the oxygens by X-ray diffraction in 1922, and neutron scattering later located the protons. The structure known as ice I, with hexagonal symmetry, is the only form of ice existing under usual conditions. At higher pressures, other forms of ice have been found, from ice Ic, which is cubic, to ice VII, which forms at around 22 kbar (22,000 atm). There is also an ice VIII. The high-pressure forms of ice are of great theoretical interest, but are of no practical importance.

If two electrons are taken away from an oxygen atom, the resulting O++ is isoelectronic with C, and its four valence electrons can form four tetrahedral bonds at angles of 109.5°. Of course, this ion does not actually exist and its charges are compensated by other electrons, but it is a good starting point for understanding the structure of ice. From the example of diamond, one might expect the oxygens to form a diamond lattice whose free energy would be less than that of the alternatives. In fact, ice Ic has just this structure, and is still lighter than water. Another possibility is the formation of six-membered rings. With carbon, the great stability of the benzene ring makes graphite the stable form under normal conditions. This is impossible for oxygens and hydrogen bonds, however, so there can be no graphitic ice. However, carbon does form stable, unstrained six-membered rings in cyclohexane, and this oxygens and hydrogen bonds can do. The result is ice I.

A model of the structure of ice I is shown at the right. The model was constructed with a Cochranes of Oxford molecular model kit, which was very satisfactory. The scale is 44 mm = 0.276 nm. The photograph may be confusing and difficult to interpret, which seems to be characteristic of any perspective view of the ice I structure. Even Roger Hayward's excellent and accurate drawing (see References) fails to be clear, although it is much better than most. The trouble is that most of the bonds are slightly inclined in a variety of regular directions, interfering with proper appreciation of the perspective. When making the three-dimensional model, all became clear after about 10 oxygens had been added. If you have continuing trouble in clearly appreciating the ice I structure, I strongly recommend making a model in which the tetrahedral directions are enforced, as in the Cochranes kit's caltrop-shaped carbons. As you can see, I had to use a few tetrahedral nitrogens to eke out the carbons. The structure of ice I is actually very simple and elegant.

The view at the left may help. It shows the hexagonal passages through the crystal parallel to the c-axis, the axis of symmetry. The crystal consists of puckered sheets of oxygens that stack so that the passages are preserved. The model has three such sheets. Alternate sheets are rotated by 60° about the c-axis. In each sheet, every oxygen has its fourth bond, the one that bonds the sheets to each other, either upward or downward, alternately. See if you can recognize this "bed of nails" in the photos. It is easier to do this in the top photo. One bond is alternately up and down in neighboring oxygens. Now note how the vertical bonds align in successive sheets, so that the distances between unbonded oxygens are alternately long and short. The oxygens are arranged like the carbons in the "chair" form of cyclohexane, but the seat of the chair is not horizontal. It would be interesting to discover a carbon allotrope with this structure--it would be a clear, softer form of diamond. Apparently, the urge to collapse into benzene rings is too strong to overcome.

Now we can put in the hydrogen bonds. Between each oxygen ion go a proton and two electrons, which makes the lattice electrically neutral. The electron on each side of the proton teams up with a valence electron on the oxygen ion to pair up in a bonding orbital. Instead of a symmetrical bond with two rather weak single bonds between the proton and each oxygen (hydrogen does not really feel comfortable when divalent), the lowest free energy results when one bond is a strong single bond of length 0.100 nm, while the other is a weaker bond of length 0.176 nm. The proton can pop from one location to the other if it can surmount the low potential barrier. However, for stability every oxygen is surrounded by two short bonds and two long ones. This means that at every oxygen, a water molecule can be identified. However, these water molecules are not permanent, but in a state of constant change as the protons pop back and forth in twos. If there are N oxygens, the number of ways the protons can be arranged is (3/2)N, which makes an observable contribution to the entropy of Nk ln(3/2) = 0.81 cal/mol/K.

The angle between the hydrogens is 105.5° in the water molecule, and the bond length is 0.0965 nm. In ice, the tetrahedral angle of 109.5° and the bond length of 0.100 nm are not much different. Therefore, there can be assumed to be an undistorted water molecule at each oxygen.

Like water, ice has a large polarizability because of the mobile protons. In a single crystal, the dielectric constant parallel to the c-axis is about 105 at 0°C, and perpendicular it is about 92. The dielectric relaxation time at 0°C is about 20 μs, corresponding to a frequency of 7.96 kHz. For comparison, in water the relaxation time is about 1.78 x 10-12 s, and the corresponding frequency is 89 GHz. A microwave oven heats water readily by dipole moment relaxation, but ice remains cold. At higher frequencies, the dielectric constant is about 3.1, with contributions now from molecular vibration. At still higher frequencies, the dielectric constant becomes about 1.7, in line with the optical index of refraction. Ice is a positive uniaxial crystal, with nω = 1.3090 and nε = 1.3104. This is a very small birefringence, and in most cases the index can be taken as 1.310, and the crystal as isotropic. The index of refraction is close to that of water, 1.333, which is seen in the near vanishing of clear ice floating in water. In fact, clear ice is called black ice for this reason. The index varies from 1.307 in the red to 1.317 in the violet. This means a dispersion constant ν = (n - 1)/Δn = 31, about that of a dense flint glass, so ice is strongly dispersive. Unlike flint glass, it is not very refractive.

Ice In The Sky

In the standard atmosphere, 0°C is reached at an altitude of 2300 m or 7550 ft. At the tropopause, 11 km altitude, the temperature is -56°C. Therefore, through much of the lower atmosphere, ice is the stable form of water. It has always been a hazard to aeronautics, from ice forming in the Venturi of a carburetor (now a past hazard in most cases) to ice coating lifting surfaces. Ice is rare at higher altitudes, so it is mostly a mid-altitude problem. Supersaturated air can quickly deposit a coating of ice when encouraged by the presence of the surface. It is difficult to get a water droplet or an ice crystal started unless the supersaturation is large or the temperature very low, so generally they form of condensation nuclei, which are small clusters of charged ions.

Middle-level clouds (3-7 km) may be partly or entirely ice clouds, while high-level clouds (7-11 km) are entirely ice clouds. Observing cloud forms, and the precipitating crystals that fall in streams from them, is a pleasant diversion. Condensation trails from high aircraft are also formed from ice crystals that have condensed from the moisture in the engine exhaust. Optical phenomena are seen mainly in light that has passed through thin, high clouds, usually classed as cirrostratus. Often the cloud itself cannot be seen except for the scattered and refracted light that it has modified.

The commonest phenomenon is the corona of scattered light that surrounds any bright celestial body, usually the moon. If the cloud particles are small, a micron in diameter or smaller, and well-sorted by size, colors due to interference may be seen. This is a result of only the size of the particles, and it does not matter whether they are ice, water or smoke. Usually just a fuzzy ring can be seen, only a degree or so in diameter.

More rarely the particles are hexagonal columns of random orientation, and refraction by these columns causes halo phenomena, which are always impressive. Halo phenomena might be considered common (in some places they occur more than 200 days of the year) if you watched the sky continuously, but random glances for a few minutes are very likely to miss them. However, a look at the moon near full and not too high in the sky will show you lunar halos at least a few times every year, especially when you look at times when high cirrus clouds have been noted during the day. Solar haloes are brighter, and may show rarer phenomena, but seem very much more difficult to see than lunar haloes. Haloes are more easily seen if the direct view of the sun is blocked by a building or similar screen.

The best time to see solar halos is in the late afternoon when cirrostratus is about, and the most frequent phenomenon (at least in Denver) are the sun dogs, mock suns or parhelia a little more than 22° to the right and left of the sun. They correspond to capped columns falling with vertical axes, which is the orientation assumed by such falling crystals because of the cap. The sun dogs are just bright spots to each side the halo. They are actually outside the 22° halo, further away as the sun is higher, sometimes joined to it by a tangential arc, the arc of Lowitz. There are rarer phenomena, such as the 46&dg; halo, the 90° halo, the parhelic circle, the circumscribed halo, and others. Most of these can now be calculated by computer programs, but mysteries and controversies still exist, especially as to causes.

The path of a light ray through a principal plane of a hexagonal crystal is shown at the right. It is much more difficult to handle rays that do not lie in a plane normal to the axis, but it can be done if necessary. We are looking only for approximate results, so this is a reasonable simplification. Snell's Law is applied at the entry and exit points, so sin i = n sin r and n sin i' = sin r. The angles r and i' are connected by A = r + i'. Starting with a certain angle of incidence i, r is first calculated, then i' is found, and finally the angle r. The deviation of the ray is D = i - r + r' - i', or D = i + r' - A. If you carry out the calculation for a few values of i, you will find that the deviation passes through a minimum Dm. This happens when the path of the ray is symmetrical, so that r = i' = A/2. Then, i and r' are equal as well, and i = r' = (Dm + A)/2. By Snell's law, sin i = n sin r, or sin[(Dm + A)/2] = n sin A/2, from which n = sin[(Dm + A)/2] / sin A/2, so that the index of refraction can be found if the angle of minimum deviation has been measured.

In the present case, A = 60° and n = 1.310. If we solve for Dm, we find 21.84°, or about 22°. Now consider refraction from numerous crystals oriented at random. None of these will refract light with a deviation less than the minimum, so we will get no light from the crystals within a cone of 22° about the direction of, say, the moon. Outside of this cone, various crystals at all points will send some light to us. In most cases, the deviation will not be far from the minimum, so the light will concentrate around 22°, and become less at larger angles. This is, in fact, what is observed. The halo has a sharp inner edge and a diffuse outer edge. We will see part of the halo wherever there are crystals in that direction. Since the index of refraction varies with frequency, the angle will be slightly smaller for red light, so there may be a reddish inner border. Outside of this, the colors will fall on one another, and there will be no color. This, also agrees with observation. The rainbow also is a result of refraction, but at the angle of minimum deviation for a refracting sphere there is the interference of two waves that produces peaks in the intensity, and so we see bright colors. In this case, there are no peaks, so colors are absent.

References

D. Eisenberg and W. Kauzmann, The Structure and Properties of Water (Oxford: The Clarendon Press, 1969). Chapter 3, Ice.

A. Holmes, Principles of Physical Geology, 2nd ed. (New York: Ronald, 1965). Chapter XX discusses the geological role of ice.

Cochranes of Oxford Ltd, Leafield, Oxford OX8 5NT, U.K. supplied the molecular model kit, which was ©1973. The price was £11.50 incl. VAT when purchased.

L. Pauling and R. Hayward, The Architecture of Molecules (San Francisco: W. H. Freeman, 1964). Plate No. 41, ice.

R. A. R. Tricker, Introduction to Meteorological Optics (London: Mills & Boon, 1970). Chapter 4, Ice Particle Haloes.

There is a fascinating website on contrails at Contrails.

Bob Fosbury has pages devoted to the solar halo in his website at Halo Site, with some excellent photographs and the results of a program to simulate the halo.

Physics Today, 56(4) April 2003, p. 23. Gravel circles. The reference given there is M. A. Kessler and B. T. Werner, Science 229, 308 (2003).

V. J. Schaefer and J. A. Day, Peterson Field Guides: Atmosphere (Boston: Houghton Mifflin, 1981). Snow and ice, pp. 244-272. On p. 321 are illustrations of 10 types of ice precipitation.


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Composed by J. B. Calvert
Created 6 April 2003
Last revised 7 August 2003