ICE PHYSICS. Martin Truffer University of Alaska Fairbanks. Spring Semester 2013

Transcription

1 ICE PHYSICS Martin Truffer University of Alaska Fairbanks Spring Semester 2013

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3 Contents Contents 3 List of Figures 6 List of Tables 8 1 Introduction Why study ice and water? Resources The water molecule The atomic constituents The molecular bonds Models of the H 2 O molecule Absorption spectrum of water vapor The hydrogen bond Liquid water The structure of Ice Ih Hexagonal and cubic closed packing Symmetries of Ice Ih The ice rules and zero point entropy The phase diagram of H 2 O Some thermodynamics The phase diagram of H 2 O The high pressure polymorphs of ice The kinetics of phase transitions Clathrates

4 4 CONTENTS 5 Defects in Ice Ih Point defects Line defects: dislocations Planar defects Volumetric defects Thermal properties of ice The heat capacity Latent heat Thermal expansion Thermal conductivity The Stefan problem Electric properties Conductivity Dielectric permittivity Thermo-electric effect Optical properties The electromagnetic spectrum The visible spectrum Optical properties across the EM spectrum Atmospheric halos A The Kramers Kronig relationships Mechanical properties Definitions Elastic properties Brittle failure The creep of ice Visco-elasticity The surface of ice The liquid-like surface layer of ice Surface energy Theoretical models Friction and adhesion of ice Water and Ice in the atmosphere Nucleation of ice Snowflakes

5 CONTENTS 5 12 Snow metamorphism Snow on the ground Snow to ice transformation Firn densification Bubble closure Interpretation of ice cores A Useful Numbers 116 Bibliography 117

6 List of Figures 2.1 sp3 orbital Charge distribution in a free H 2 O molecule Models of the H 2 O molecule Absorption spectrum of water Vibrational modes of the water molecule The orbitals of the H 2 O dimer Lennard-Jones potential The Némethy and Scheraga model of liquid water Translational symmetry of a crystal The 14 Bravais lattices Miller-Bravais indices and the Ice Ih unit cell Geometry of Bragg reflections Illustration of the powder method Ice crystal structure Schematic phase diagram Subglacial regelation Triple point Polymorphs of ice Metastable phase Type I clathrate Clathrate stability diagram Schematic of a D defect Defect potential in electric field Effects of HF doping Dislocations Screw and edge dislocation

7 List of Figures Dislocation motion Coincident lattice sites Hall effect Proton transport and defects Conductivity Dielectric permittivity Snell s Law Ice under cross polarizer Crack propagation Compressional crack Nabarro-Herring creep Creep curve for polycrystalline ice Surface energy Free Energy Nakaya diagram Development of rounded crystals Depth hoar SAR mosaic of Greenland Firn facies Firn densification

8 List of Tables 4.1 Polymorphs of ice The electromagnetic spectrum Elastic properties of ice Seismic speed Viscosities of some materials

9 1 Introduction 1.1 Why study ice and water? Ice is important Ice is very common on Earth and also in the solar system. Among others, it occurs on Mars, our moon, the Jovian moons of Europa and Ganymede and on Saturn s rings and its moon Titan. It is thought that some high-pressure phases of ice that do not occur on Earth can be found on Ganymede. Water is the only common substance that occurs in solid, liquid and gaseous form on Earth. Life evolved in water, and most of our bodies consist of water. Several organisms manage to survive temperatures well below the freezing point of water. Water and ice play an important role in the climate system. Because of the large specific and latent heat, water and ice are substantial heat sinks or sources. This has an important influence on coastal versus continental climates. Ice, and particularly snow, has a high albedo, affecting the radiative budget of the Earth s surface. While the albedo of the open ocean is about 7%, a thin fresh snow cover on sea ice can increase this to more than 87%. The reduction of sea ice in a warming climate is one of the important positive feedbacks in the climate system. Ice sheets build up considerable topography, which affects global weather patterns. Antarctica is the continent with the highest average elevation and Greenland is in the path of the major storm systems coming across the Atlantic. Both of these major topographic features evolve on timescales that are much faster than tectonics. Ice is thought to be important for charge separation in thunderstorms and is thus responsible for lightening. The formation of hail has major economic impacts on farming and is responsible to other damage of human infrastructure. 9

10 10 CHAPTER 1. INTRODUCTION Snow and ice can be important economic factors, both as an asset (ski resorts, glacier visits,...), and by disrupting traffic, electronic transmission lines, etc. Ice storms and snow falls in mid-latitudes can bring large cities and entire region to a virtual standstill. Electrical and optical properties of snow and ice in the atmosphere and on the ground are important for remote sensing studies. Data from satellites are always impacted by the atmosphere, which contains water in solid, liquid and gaseous forms. Ice is interesting While the water molecule is small, it has a surprising amount of complexity that makes it an intriguing target for basic studies of molecules, such as deriving properties from basic physics (quantum mechanics). Water has a number of properties that make it quite unique among many common substances. It has a density maximum at 4 C, rather than at the melting point, which is common. The density of the solid is lower than that of the liquid. Therefore ice floats on water, which has very important consequences for climate, life on Earth, and geologic processes, such as frost heaving, erosion, etc. Water is not a linear molecule and as a consequence has an electric dipole (and higher moments). This has consequences for the structure of water and ice, and their optical, electric and mechanical properties. The ice crystal is anisotropic. That is, its mechanical and optical properties are not the same in all directions. Ice is a disordered crystal. While the oxygen atoms appear in a fixed crystal patterns, the H-atoms (protons) appear at random positions. This gives rise to significant zero point entropy, or configurational disorder. Electric current in ice is transported by protons not electrons. Ice is like a semiconductor. The electric conductivity increases with increasing temperatures. Water can be supercooled substantially (up to -40 C). Ice has a liquid-like surface layer. This has important consequences for the surface properties, such as friction. Ice has many (at least twelve) stable structures at high pressure, some of which have very different physical properties than the ones we are used to. For example, there are high-pressure phases of ice that are stable at temperatures exceeding 100 C.

11 1.2. RESOURCES Resources The best online resource for the physics of water and ice was created and is maintained by Dr. Martin Chaplin of London South Bank University: A good textbook for the physics of ice is Petrenko and Whitworth (1999). Schulson and Duval (2009) is a comprehensive treatment of the mechanical properties of ice. But the out-of-print Hobbs (1974) is still the most comprehensive book on the subject. Another good source is Fletcher (1970), which is unfortunately also out of print. Every few years the International Conference on the Physics and Chemistry of Ice convenes and publishes their proceedings. It is an excellent place to start looking for the newest research developments. This manuscript is based on a course that was taught for many years by Keith Echelmeyer at University of Alaska Fairbanks. It also draws heavily from the books by Hobbs (1974) and Petrenko and Whitworth (1999). The reference list is still evolving and is by no means meant to be complete. Some chapters require background in solid state physics, and a useful reference book is Kittel (2005).

12 2 The water molecule 2.1 The atomic constituents The water molecule consists of two hydrogen atoms (protons) and one oxygen atom. The 2:1 ratio can easily be demonstrated when water is separated into its constituents using electrolysis. The resulting gases (H 2 and O 2 ) clearly occur in that volume ratio, as expected. The hydrogen atom occurs as two stable isotopes. The most stable nucleus consists of only one proton. 2 H has one proton and one neutron and is also known as deuterium (D). 3 H is known as tritium (T) and is radioactive. It has a half-life of 12.5 years. Tritium has been produced in measurable quantities in the atmospheric H-bomb tests in the 1950s and 1960s. Because the date of the first H-bomb test is well-known, tritium has been used as a stratigraphic marker, for example in ice cores. The oxygen atom occurs stably as 16 O, 17 O, and 18 O. Other isotopes are not stable. Naturally occuring water consists of 99.73% 1 H 16 2 O, 0.20% 1 H 18 2 O, 0.03% 1 H 2 D 16 O, and 0.015% 2 D 16 2 O. The ratio of the stable isotopes is an important indicator of past climate. This is because the ratio depends, among other things, on temperatures. For example, at lower temperatures, fewer of the heavy isotopes evaporate from the ocean. 2 D 16 2 O is also regularly used in basic studies of the water molecule or the ice structure. It provides an important second data point for models of the water molecule. 2.2 The molecular bonds Bond strength The energy contained in molecular bonds is often given in electron Volts (ev), as this is a more convenient unit then the SI equivalent Joule (J). For conversion note that 1 ev J. The bond between the hydrogens 12

13 2.2. THE MOLECULAR BONDS 13 and oxygens in the water molecule is strong. It takes 9.6 ev to break up H 2 O into its constituent atoms. This should be compared with the heat of fusion (0.06 ev/molecule) or heat of vaporization (0.39 ev/molecule), which are more than an order of magnitude lower. This means that it is much easier to break up the solid structure of ice, than the molecular bonds of water. It is useful to keep in mind that the energy available depends on the temperature and is given by kt, where k = J K 1 is the Boltzmann constant. kt plays an important role in statistical mechanics and commonly occurs in the form e E/kT, where E is an activation energy. Note that kt =.025 ev at T = 298 K. This means that, near room temperature very few molecular bonds will be found broken. Orbitals According to quantum theory electrons occur in shells around the nucleus. These shells correspond to different energy levels. Electrons fill up the lower energy shells first. Each shell should be thought of as a probability distribution or a cloud within which the electron is most likely to be observed. These probability distributions can be spherical in shape (s orbitals), dumbbell shaped (p orbitals), or more complicated. Each shell can only be occupied by up to two electrons (Pauli exclusion principle). A hydrogen atom has only one electron, which occupies the 1s shell. The oxygen atom has eight electrons, two in the 1s orbital, two in the 2s orbital, and 4 in the 2p orbitals. There are three p orbitals, which can be thought of as mutually perpendicular dumbbells (p x, p y, p z ). They can hold a total of six electrons. So the oxygen needs two more electrons to fill up the 2p orbitals, whereas the hydrogen needs one additional electron to fill up its 1s orbital. In the theory of covalent bonding atoms pair up in a way so that they can share valence electrons (those in the outermost shell) in a way that fills up their half-filled orbitals. In this theory the two hydrogens 1s electrons would be shared with the single 2p oxygen electrons. From the shape of the p-orbitals, one should then expect that H 2 O is a linear molecule or that the H-O-H angle is 90. This turns out to be incorrect. A more refined theory of molecular bonding takes into account the energy of the combined state of one oxygen and two hydrogen. It is then found that the energetically most stable state involves a hybridization of the 2s and 2p orbitals. They form a new kind of orbital, which is called sp 3 orbital (Fig. 2.1). It consists of four lobes that point into the corners of a tetrahedron. Each of these lobes can hold one or two electrons. When oxygen binds with hydrogen, the oxygen will sit in the center of a tetrahedron, the two hydrogens are at

14 14 CHAPTER 2. THE WATER MOLECULE Figure 2.1: Hybridized sp3 orbitals with lobes pointing into the corners of a tetrahedron. From two of the corners, and the other two lobes of the sp 3 orbital point towards the other two corners. These two lobes are also known as the lone pairs. The actual arrangement is somewhat more complicated because of charge inequalities. The lone pairs have an excess negative charge and tend to push themselves apart, squeezing the OH bonds together a bit. The measured O-H- O angle is 104.5, it would be in a perfectly tetrahedral arrangement. The length of the OH bond is 0.96 Å. Figure 2.2: Charge distribution in a free H 2 O molecule. From Combined orbitals are calculated from the quantum mechanical wave functions using the Hartree-Fock approximation. The more detailed the calcula-

15 2.3. MODELS OF THE H 2 O MOLECULE 15 tion, the more complicated the picture gets. For example, there are indications that the theory of sp 3 hybridization and lone pairs is not a very accurate description of a single H 2 O molecule. Instead, the calculated orbitals might be better described as sp 2 hybrids with a p z orbital. Also, orbitals get influenced by neighboring molecules, so the orbital structure of an isolated H 2 O molecule look different for liquid or solid water. 2.3 Models of the H 2 O molecule The mass of the H 2 O molecule is concentrated at the three nuclei, which for most purposes can be considered as point masses, with 16/18 of the mass concentrated at the location of the oxygen atom and 1/18 of the total mass at both of the hydrogens locations. While the molecule is charge neutral, the distribution of protons and lone pairs means that the electric charge is distributed non-uniformly. This results in an electric dipole and higher order multipoles. This is important for bonding between molecules, which is important in liquid and solid water. There are many models of charge distributions that have been used to represent the H 2 O molecule. These models consist of point charges that are distributed in a certain fashion. Some of the well-known models are TIP3P, TIP4P, and TIP5P (Fig. 2.3). TIP3P distributes charge between the site of the oxygen and two hydrogen atoms (three points). TIP4P picks an additional point to better account for the lone pair sites, and TIP5P distributes this negative charge to two sites. There are many such models, and none are completely satisfactory in explaining the properties of water and the phase diagram. For example, TIP5P correctly predicts the density maximum at 4 C of liquid water, but it fails to predict the correct stable structure for ice. Figure 2.3: Water molecule models of increasing complexity in mass and charge distribution. From model

16 16 CHAPTER 2. THE WATER MOLECULE 2.4 Absorption spectrum of water vapor The water molecule can absorb electromagnetic radiation of certain frequencies. The absorption bands correspond to energies that excite rotational or vibrational states of H 2 O. Since one molecule has 3 atoms, it has 3x3 = 9 degrees of freedom (1 per atom and spatial dimension). That is, specifying the Cartesian coordinates of all three atoms completely determines the state of the molecule. The state of the atom can also be expressed by specifying the location of its center of mass (3 degrees of freedom), the rotation around three mutually perpendicular axes (another 3 degrees of freedom) and three vibrational modes. The location of the center of mass is not relevant for the absorption spectrum, so we will now discuss the rotational and vibrational modes. For historic reasons, spectroscopy often uses wave numbers, usually with units of cm 1. The wavenumber k is related to the frequency ν by the speed of light in vacuum c: k = ν/c. The rotational spectrum The rotational spectrum is determined by the moments of inertia of the molecule. The H 2 O molecule has three non-equal moments of inertia. This is in contrast to, for example, a linear molecule, that has only one non-zero moment of inertia and consequently not as broad a spectrum. In fact, the broad far infrared (FIR) absorption spectrum is a clear indication that H 2 O is not a linear molecule. The rotational absorption spectrum is very broad and low energy (k < 600 cm 1 ). Even though the moments of inertia for a rigid rotating molecule can easily be calculated, it is not clear how well this applies for real molecules that are not rigid. This, and the fact that the rotational energy levels are very close to each other are responsible for a broad and spread out spectrum, rather than a series of well-defined peaks. It is possible, however, to use a quantized treatment of the rotational energy to deduce the moments of inertia from the absorption peaks. This can then be used to calculate the O-H distance and the O-H-O angle of the water molecule. The vibrational spectrum The three vibrational modes can be classified into a symmetric stretching mode (ν 1 = cm 1 ), a bending mode (ν 2 = cm 1 ), and an asymmetrical stretching mode (ν 3 = cm 1 ) (Fig. 2.4). The stretch-

17 2.4. ABSORPTION SPECTRUM OF WATER VAPOR 17 Figure 2.4: Absorption spectrum of water. This data is based on the revised 1995 data of Warren (1984). It was revised from the original data in the 1984 paper to account for misinterpretations of data by others. It also includes more recent data. Data from: ing modes depend on the strength of the O-H bond and are very similar. The bending mode involves an oscillation of the H-O-H angle and does not depend on the bond strength. These absorption bands are in the near infrared (NIR) range. The measured absorption spectrum has well-defined absorption peaks, but they are somewhat broadened. This is because the real behavior of molecules is not that of a harmonic oscillator. Rather, it can be shown that by treating water as a anharmonic oscillator, a better fit to measured spectra can be obtained.

18 18 CHAPTER 2. THE WATER MOLECULE Figure 2.5: The three vibrational modes of the H 2 O molecule (not involving translation of the center of mass or rotation). From: Higher energy absorption The H 2 O molecule has a notable absence of absorption bands in the visible spectrum. This is crucial to life as we know it, and it is likely the reason why our eyes evolved to see exactly in that range. In the higher energy range (ultraviolet and higher), absorption becomes stronger again. In this case the energy is absorbed by electrons being excited to higher energy levels. The Greenhouse effect Most of the sun s radiation gets to the surface of the Earth in the visible range. That is the range in which the sun s radiation peaks (due to its surface temperature of about 5000 K) and also the window in which water vapor in the atmosphere does not absorb radiation. The surface of the Earth is much cooler than that of the sun, and its blackbody radiation is peaking in the infrared. Because water in the atmosphere absorbs strongly in those wavelengths, the Earth is not very effective at cooling radiatively. This effect of letting the energy in the visible range pass, but blocking the outgoing radiation is called the Greenhouse effect, even though it has little to do with how a greenhouse works. It has a significant effect on the average temperature on Earth s surface. Without this effect, the average temperature would be well below freezing, and Earth would be a much different place. Water vapor is the strongest Greenhouse gas, but others, in particular carbon dioxide and methane, contribute significantly and increases in their atmospheric concentrations cause a non-negligible radiative forcing that is thought to be responsible for anthropogenically induced global warming.

19 2.5. THE HYDROGEN BOND The hydrogen bond Figure 2.6: The orbitals of the H 2 O dimer obtained from ab-initio calculations, using the Hartree-Fock approximation. From: The hydrogen bond is perhaps the most important concept in ice physics. The nature of this bond arises from the charge separation in the H 2 O molecule, with a slight positive charge surplus near the H-atoms, and a negative charge surplus on the opposite site of the O-atom. The positive charge δ + (for partial charge) and the negative charge δ attract each other and they form a so-called H-bond. Hydrogen bonds occur not only in water, but play a role in many bigger molecules, such as proteins, cell membranes, but also glues etc. They are convenient, because they are reasonably strong, but not as strong as ionic bonds (found in salts) or covalent bonds, so it is easier to separate molecules. Consequently, the boiling point of ionic crystals is considerably higher than that of hydrogen bound substances. The hydrogen bond can be studied in some detail by considering the dimer, which consists of two H 2 O molecules joined by a hydrogen bond. It can be seen that the H does not sit symmetrically between the two O s. Rather, it stays attached to one O, thus leaving the H 2 O molecule intact, except for a small amount of stretch. The O-H bond is about Å longer in a bonded molecule. H-bonds are considerably stronger than Van-der-Waals bonds that are responsible for other non-ionic bonds. For example, the boiling points of the

20 20 CHAPTER 2. THE WATER MOLECULE V LJ repulsive force attractive force Figure 2.7: Lennard- Jones potential O H distance H-bonded NH 3 (-33.3 C), H 2 O (100 C), and HF (19.5 C) are much higher than those of their non-h-bonded neighbors in the periodic table: PH 3 ( C), H 2 S ( 60.7 C), and HCl (-84.9 C). The energy of an H-bond is approximately times higher than that of a Van-der-Waals bond. On the other hand, it is about times lower than that of a covalent bond. The H-bond in ice has an energy of about 0.3 ev. Because two H-bonds per molecule have to be broken to go from solid ice to water vapor, the energy of the H-bond corresponds to half the heat of sublimation. Qualitativly, the H-bond can be modelled using a so-called Lenard-Jones potential (Fig. 2.7): V LJ = A r 12 C r 6 (2.1) A and C are constants to be determined. This potential has no strict physical meaning, but it qualitatively represents a rejection term (due to the Pauli exclusion principle) and an attractive term (the electrostatic force between the partial charges). Models with a Lennard-Jones potential can reproduce density of ice and heat of vaporization quite well, but then they fail miserably at other properties, such as melting point.

21 2.6. LIQUID WATER Liquid water Liquid water is the most difficult of all phases to treat quantitatively. In water vapour most of the molecule interaction can be described classically (little point masses flying around) and a crystal is characterized by long range order. A liquid has some short range structure, so the freedom of motion of a gas is lost. Liquid water contains H 2 O molecules with a certain amount of H-bonding. The amount of intact H-bonds can be estimated by comparing the latent heat of fusion to the latent heat of evaporation. The bonded molecules occur in clusters, which have short range crystal structure. The smallest such cluster is the dimer (H 2 O ) 2, but larger clusters, such as (H 2 O ) 30 could be common. Studies show that the average O-O distance in water is 3 Å, as compared to 2.76 Å for ice. But the coordination number (number of nearest neighbors) is In Ice Ih it is 4. So liquid water has more neighbors than solid ice, but they are separated by a slightly larger distance. A successfull model of liquid water must account for the following properties: Density of the liquid (ρ max at 4 C). Liquid is denser than solid. Large dielectric constant at low frequency (ɛ 100). This implies that the molecular dipoles remain intact and can rotate quite easily. The vibrational spectrum still has peaks at ν 1, ν 2, and ν 3. This implies that the H 2 O molecule remains intact in the liquid phase. An example of a model of liquid water was published by Némethy and Scheraga (1962). It specifies the size of clusters, the percentage of intact H- bonds, and the percentage of unbonded molecules as a function of temperature (Fig. 2.8). This model reproduces the density maximum at 4 C. The reason for the density maximum is that there are two competing mechanisms for density. First, the cluster size, which is a decreasing function of temperature. Large cluster sizes lead to lower density. Second, thermal agitation, which leads to lower density at higher temperature (this is the reason most materials have a decreasing density with increased temperature). Below 4 C, the effect of the cluster size wins, above that thermal agitation is dominant. There are also models suggesting much larger cluster sizes (up to several hundred molecules). Interestingly, constraints on the life time of clusters can

22 22 CHAPTER 2. THE WATER MOLECULE Cluster size % unbroken bonds 40 Figure 2.8: The Némethy and Scheraga model of liquid water 20 % unbonded molecules Temperature ( C) be derived by looking at measurements of electric properties. The dielectric relaxation time defines a time scale, below which molecular dipoles cannot rotate as quickly as the electric field does. This suggests that life times of clusters are shorter than about s. On the other hand, the period of molecular vibrations is about s, and for the concept of a cluster to have physical meaning, it needs to be stable for longer than that. The 4 C density maximum is a property of pure water only. Ocean water, for example, exhibits a density maximum at 3.7 C for a salinity of 35 ppt (Schulson and Duval, 2009). This is lower than the onset of freezing, which occurs at 1.8 C.

23 3 The structure of Ice Ih There are many stable forms of solid H 2 O (14 are currently known). Most of them occur at very high pressures. Only one form occurs naturally on Earth: Ice Ih. The h stands for hexagonal. A similar structure, Ice Ic (cubic), can occur at atmospheric pressures and very low temperatures. It is possible that it forms as a transitional form when atmospheric hexagonal ice is formed (Murray et al., 2005). Some of the characteristics of Ice Ih are: It is stable below 0 C. While liquid water can be supercooled it is not possible to superheat Ice Ih. Ice Ih has tetrahedral coordination (4 nearest neighbors). Ice Ih has an open structure (ρ solid < ρ liquid ). Water molecules remain intact. This is shown, for example, by the IR absorption spectrum. One might expect ice to be an ionic crystal with the proton located half way between two oxygens. This would imply a much higher melting point of ice, because ionic bonds are much stronger than H-bonds (salts have high melting points). It would also imply a different IR absorption spectrum, because the rotational and vibrational characteristics would be different. This is not observed. Finally, it would lead to a small dielectric constant, because the dipole nature of the molecule would be lost. However ɛ ice = 100, which is similar to ɛ water = 80. To discuss the structure of ice Ih we need some background in crystallography. Crystallography: The basics The idea of a crystal is that of translational symmetry: a basic structure (a motive ) repeats itself in 3D space (Fig. 3.6). Formally this means that one 23

24 24 CHAPTER 3. THE STRUCTURE OF ICE IH can find three linearly independent vectors ( a, b, c) such that a translation by u 1 a + u 2 b + u3 c leads to an identical point (u i must be whole numbers). A crystal can thus be defined by a basic motive plus a set of basis vectors. A lattice is a volume defined by the basis vectors. A primitive lattice is the smallest possible cell defined by three linearly independent vectors that fully describes the crystal. The choice of these vectors is generally not unique. The volume of a primitive cell is given by the product of its basis vectors: (( a b) c). Every lattice has translational symmetry (by definition). In addition it can have other symmetries (rotations, mirror planes and screw axes). In crystallography, any crystal can be described in terms of one of 14 so-called Bravais lattices (Fig. 3). The Bravais lattices are not all primitive. Ice Ih has a hexagonal lattice. The unit cell is defined by three vectors, where a 1 = a 2 = a a 3 = c; α = β = 90 ; and γ = 120. The dimensions for ice Ih are a=4.5 Å and c=7.3 Å. This unit cell defines the crystal uniquely (Fig. 3.3). Figure 3.1: A crystal is defined by translational symmetry: A basic pattern (motive) is repeated in all three spatial dimensions. The pattern together with the translation vectors defines the crystal. c b a Miller indices Crystallographers have come up with a system of classifying crystal faces (planes) and directions (edges). Each plane (and edge) is defined by direction only (i.e. parallel faces are equivalent). The way a plane is described is by first determining the points of intersection with the three axes of the lattice. For example, a plane parallel to the a 3 axis that intersects a 1 at 1 and a 2 at 2 has an intersection point of (1, 2, ). The second step is to take reciprocals of each number: (1, 1/2, 0). We then multiply such that we get the lowest possible triplet of whole numbers. In this case: (2 1 0). Faces are written with round brackets, and no commas are used. Negative numbers are indicated by a bar above the number, such as (1 21). Edges are described in normal coordinates and written in square brackets. For example, the a 2 axis is [0 1 0].

25 25 Figure 3.2: The 14 Bravais lattices This way of describing planes and edges is consistent, because the plane and direction descriptions are in reciprocal spaces. It can be shown that the vector product of two edges leads to the Miller indices of the plane that is spanned by these edges. Only in orthogonal systems are the direct and reciprocal spaces identical. The reciprocal space for planes is also highly relevant for X-ray crystallography, and that is mainly why the notation has persisted. The hexagonal system has a peculiarity. Often, faces are written with four indices. The first three indices relate to the three horizontal axes (Fig.3.3). Since these axes are linearly dependent, the third index can be deduced from

26 26 CHAPTER 3. THE STRUCTURE OF ICE IH the first two by the following rule: (hkl) (hk (h + k)l). Discussions about ice crystals routinely use the system with four indices. Figure 3.3 shows an example of how the Miller indices are determined for a particular plane. Figure 3.3: The unit cell is shown in bold, with the three horizontal and the one vertical (c) axes. The Miller indices of the shaded plane are calculated with as described in the text. From: wikipedia/commons/8/82/ Indices miller bra vais.png X-ray diffraction When a crystalline sample is illuminated with X-rays, diffraction patterns are observed. The reason for this is that X-rays are reflected off successive crystal faces within the sample. X-rays are electromagnetic waves and they interact with the periodic electron distribution in a crystal. This series of reflected rays can interact constructively or destructively (Fig. 3.4). If the spacing between planes is given by d and the incidence angle of the X-rays is θ, the difference in path lengths between two reflections is 2d sin θ. A condition for positive interference is that the path length is a multiple of the wavelength. 2d sin θ = nλ (3.1) where n is a positive whole number. This is known as the Bragg condition. This only works when λ < d; that is why X-rays are ideal for crystallography: their wavelength is smaller than the typical spacing of crystal planes. There are several methods for doing X-ray crystallography. Perhaps the simplest is the powder method. The basic idea is to use a sample where any crystal orientation is represented. That means using a powder of lots of randomly oriented crystals. This sample is then surrounded with film, which records diffraction lines (Fig. 3). This guarantees that each plane (hkl) does occur

27 27 ϴ d Figure 3.4: Geometry of Bragg reflections Figure 3.5: Illustration of the powder method. From: matter.org.uk/diffraction/x-ray/powder method.htm in an orientation that will fulfill the Bragg condition (Eqn. 3.1) and therefore produce a reflection and two symmetric lines on the film. For a hexagonal crystal the condition for a band is: ( ) 2 sin θ 2 = (h 2 + hk + k 2 )/a 2 + l 2 /c 2 (3.2) λ where a and c are the lattice constants, and (h, k, l) are whole numbers, which coincide with the Miller indices. Given that h, k, and l are whole numbers, and θ and λ can be measured, it then becomes possible to figure out the lattice constants. It can also be shown that symmetries lead to missing lines in the diffraction pattern, which further helps determine the structure.

28 28 CHAPTER 3. THE STRUCTURE OF ICE IH Figure 3.6: Ice crystal structure. From: cryonics/lessons.html 3.1 Hexagonal and cubic closed packing The ice crystal is made up of hexagonal pluckered rings. Two configurations are possible: boat or chair. The chair configuration is energetically more favorable. A collection of these hexagonal rings can be organized into a basal basal plane. These planes can be stacked in two ways: ABCABC.. (cubic closed packing) or ABABAB... (hexagonal closed packing). The stable form of ice at atmospheric pressure and temperatures just below the freezing point is Ice Ih, which is the hexagonal closed packed structure (Fig. 3.6). Cubic ice (Ic) occurs at lower temperatures. The term hexagonally closed packed is somewhat misleading, since the structure of ice is very open, due to the long distance between the basal planes. 3.2 Symmetries of Ice Ih The space group for Ice Ih is P 6 3 /mmc. P indicates a primitive lattice. 6 3 refers to a hexagonal screw axis. It means that translation along the c-axis by 3c/6 combined with a 2π/6 rotation around the c-axis images the crystal onto itself. The /m indicates a mirror plane normal to the principal axis of symmetry and the remaining mc signifies two symmetry planes parallel to the c-axis: a mirror plane (m) and a glide plane (c). A glide plane is a symmetry that involves slip of c/2 parallel to c followed by a mirror operation. In general, all symmetry operations in 3D can be classified. The result consists of 230 space groups, where the word group is meant in the mathematical sense. These space groups are listed in the International Tables for Crystallography. These symmetries are important, because they manifest themselves in the x-ray diffraction pattern.

29 3.3. THE ICE RULES AND ZERO POINT ENTROPY The ice rules and zero point entropy All the remarks about crystal structure in the preceding sections refer to the positions of the oxygen atoms. The hydrogen atoms (protons) are not ordered in a crystal structure. This leads to a surprising effect: zero point entropy. The reason is that at 0 K (which cannot be reached, although one can get very close), when no thermal agitation exists any longer, there are still a number of possible configurations for a given ice crystal. A number of possible states at a given temperature and pressure leads to the concept of entropy. As a matter of fact, as Boltzmann s tombstone attests: S = k log W (3.3) where S is entropy, k is Boltzmann s constant, and W is the number of possible configurations. Incidentally, Boltzmann never wrote this equation down, although it is attributed to him. Linus Pauling calculated the number of possible configurations for an ice crystal at 0 K. He assumed that a crystal was governed by the following rules: 1. Each oxygen atom has four nearest neighbors (tetrahedral symmetry). 2. Each O-O bond has exactly one proton on it. 3. Each oxygen atom has two protons associated with it. 4. No configuration that satisfies the first three rules is favored. These rules are sometimes also known as the Bernal-Fowler rules. Calculating the number of possible configurations is an exercise in combinatorics, and there are several published approaches. Here is one of them: Take a mole of ice. It contains N A molecules and thus 2N A protons and 2N A hydrogen bonds (each oxygen has four bonds, but they are all shared, so there are effectively two bonds per O). On each bond there are two possible positions for the proton (either by the first O or the second one). So the total of all possible positions for the protons equals 2 2N A. There are 16 possible ways of distributing protons on the four bonds surrounding one O. But, only 6 of these satisfy the Bernal-Fowler rules. So (6/16) N A of the 2 2N A protons fulfill the ice rules. It follows: W = ( ) 6 NA 2 2N A = 16 This results in a zero point entropy of ( ) 3 NA (3.4) 2

30 30 CHAPTER 3. THE STRUCTURE OF ICE IH S o = k log W = kn A log(3/2) = R log(3/2) 3.37 Jmol 1 K 1 (3.5) This is not quite the entire story, because of some long range order that is not accounted for in Pauling s model. However, a much more elaborate analysis yields S o = 3.41 Jmol 1 K 1, which is essentially identical to the experimental value. It might seem counterintuitive at first that long range order increases the configurational entropy. The argument involving long range order goes as follows. The shortest connected sequence of H 2 O molecules is a hexagonal ring. The ring has to form in a way so that the polarities of the first and last molecule match. If we adopt the notation where + stands for a H-occupied bond and - for a lone pair orbital, then a sequence of six molecules (+ -), (+ +), (- +), (- -), (+ -), (+ -) would form a valid hexagonal ring, because the last molecule is of opposite polarity to the first. A correct connection is established if there is an even number of equal signed molecules (+ +) or (- -) in the sequence. The probability of having an equal signed molecule is 1/3 (pick the first polarity, each of the remaining three bonds have an equal probability of having that same polarity) and the probability o having opposite signs is thus 2/3. A ring can therefore contain six, four, two or no equal signed pair. Elementary combinatorics then shows that the probability of matching is 365/729. This is, somewhat surprisingly, more than 1/2, which was the probability assigned by ignoring long range order. The reason S o can be determined experimentally is that entropy is related to heat: and thus δq = T ds (3.6) S(T ) S o = T 0 δq T (3.7) S(T ) can be calculated for the vapor phase from statistical mechanics arguments and the integral contains measurable quantities (temperature and heat).

31 4 The phase diagram of H 2 O 4.1 Some thermodynamics The state of matter is governed by two state variables: pressure p and temperature T. The relevant potential for these two state variables is the Gibbs Free Energy. The Gibbs Free Energy G is one of the thermodynamic (TD) potentials. TD potentials can be transformed into each other by a variable change through a Legendre transform. For example, G can be written in terms of the internal energy U and the state variables S, T, p, V : G = U T S + pv. We will sometimes use specific variables (per mole) and write lower case letters. It can be shown that in differential form we have: dg = ( ) g dt + T p ( ) g dp = sdt + vdp. (4.1) p T We will now use this to calculate the slope of a phase boundary, which is a line in p, T space (phase diagram) that divides two regions where different phases are stable (Fig. 4.1). On the phase boundary two phases co-exist. As one changes into the other, the Gibbs Free Energy remains constant. Formally, for n i moles of phase i and a Gibbs Free Energy of g i per mole for phase i: G tot = n 1 g 1 + n 2 g 2 = n 1 (g 1 g 2 ) + ng 2 (4.2) where n = n 1 + n 2. Because, at the phase boundary, stability does not depend on the fraction of one phase, G n 1 = 0. This implies that g 1 = g 2. This can be used to calculate the slope of the phase boundary. If a point is shifted along the phase boundary (Fig. 4.1), then g 1 + dg 1 = g 2 + dg 2 and hence dg 1 = dg 2. Equation 4.1 then implies: 31

32 32 CHAPTER 4. THE PHASE DIAGRAM OF H 2 O T phase 1 g 2 g 1 dt dp phase 2 p Figure 4.1: Schematic phase diagram used to derive the Clausius-Clapeyron Equation s 1 dt + v 1 dp = s 2 dt + v 2 dp (4.3) (s 2 s 1 )dt = (v 2 v 1 )dp (4.4) dt = v (4.5) dp s This is the slope of the phase boundary and eqn. 4.5 is known as the Clausius-Clapeyron Equation. To calculate this slope in terms of measurable quantities, we need to find s. From thermodynamics we know that δq = T ds, where δq is the heat. The heat that is associated with crossing a phase boundary is the latent heat L f. So, S = L f T. For the liquid-ice Ih boundary this slope turns out to be K bar 1. The fact that the slope is negative is significant. It means that ice has the unusual effect of a freezing point depression under pressure. The effect is not

33 4.2. THE PHASE DIAGRAM OF H 2 O 33 Figure 4.2: Regelation around small bedrock obstacles at the bottom of a glacier large, but it leads to some interesting phenomena, such as regelation, which can be important in subglacial environments. For example, as a temperate glacier (one that is at the pressure melting point throughout) moves over a bump in the bedrock, a higher pressure occurs on the upstream side (Fig. 4.2). This higher pressure results in a lower melting point than on the downstream side. This establishes heat flow through the bedrock. The heat is being used to melt ice and water flows downstream where it refreezes. The overall effect of this is a mass transport that results in basal motion. It is only significant for small obstacles, because the heat flow is neglibible for larger ones. Regelation, however, does NOT play a role in ice friction (such as skating, etc). This is still frequently stated and clearly wrong. The slope of a phase boundary can also be assessed qualitatively. For example, the transition from liquid to solid is one that decreases entropy ( s < 0). For most materials, it also increases density ( v > 0). Consequently, most materials have a positive liquid-solid phase boundary. The fact that the slope has information about the entropy change also means that phase changes from ordered to disordered phases can sometimes be recognized solely because of the slopes in the phase diagram. 4.2 The phase diagram of H 2 O Let us first look at the phase diagram in the solid-liquid-vapor region (Fig.4.3). First, it can be seen that the terminology melting point and boiling point is inaccurate. These phase transitions are in actuality pressure dependent. The only fixed point in the phase diagram is where three phases meet, such as the triple point at K and Pa. At this point vapor, liquid and solid co-exist. It is an excellent point for calibration, and it can be used to define the temperature scale. An interesting fact is that the surface pressure of Mars is Pa, very close to the triple point. That means that with the current atmosphere

34 34 CHAPTER 4. THE PHASE DIAGRAM OF H 2 O the absence of liquid water is not only due to the low temperature. Liquid water, if present, would easily evaporate. 4.3 The high pressure polymorphs of ice Expanding the phase diagram to high pressures and low temperatures reveals at least 12 crystalline phases of ice (Fig. 4.4), with entirely different physical properties. Several phases have ordered proton arrangements. The ordered forms of Ice Ih, VII, and III are Ice XI, VIII, and IX respectively. It is instructive to keep in mind the phase slope relation (Eqn. 4.5). This implies that a vertical phase line implies no change in disorder. In contrast, a horizontal phase boundary implies no change in density. In addition, one can assume that the density increases with increased pressure, so the sign of the slope gives an indication of order-disorder transitions. The fourfold coordination remains for all forms of ice, but the open structure of Ice Ih is lost for the other polymorphs. Ice Ih is the only solid form of water that has a lower density than the liquid form. This manifests itself in the positive slope in the phase diagram. Ice IV has a very interesting interlocking structure, and Ice VI and VII are self clathrates. They consist of two structures that are interlocked, but not directly connected. Ice VII consists of two interpenetrated Ice Ic structures and has thus almost double the density of ordinary ice. The crystal structures of the high pressure polymorphs are vastly different from the familiar Ice Ih (4.3). Cubic ice (Ic) has the same structure as diamond. It has the same configuration of basal planes as Ice Ih, but a different stacking (cubic closed packed, rather than hexagonally closed packed). Carbon also has a hexagonal closed packed structure, named Londsdaleite. It is harder than diamond, but extremely rare, as it only forms under very high pressures, such as meteorite impacts. All ice polymorphs with the exception of Ice X retain an intact H 2 O molecule. In Ice X the protons all move to the middle of the H-bond and the crystal becomes ionic. On Earth, the highest pressure at which ice can be found, is at the bottom of the big ice sheets, where the ice thickness can exceed 4000 m. This is not nearly enough to induce a change to a higher pressure phase of ice. However, it is quite likely that the mantles of the Jovian moons Ganymede and Callisto contain ice at sufficiently high pressures to find some of the high pressure polymorphs.

35 4.4. THE KINETICS OF PHASE TRANSITIONS 35 Figure 4.3: A close-up of the H 2 O phase diagram near the region of the triple point Figure 4.4: The high pressure phases of ice (Petrenko and Whitworth, 1999) The rheology of the other ice phases can be important for the characterization of the evolution of the icy moons of Jupiter and Saturn. It is summarized in Schulson and Duval (2009). 4.4 The kinetics of phase transitions The phase diagram shows the most stable phase for each point in (p, T ) space. That does not necessarily imply that this phase is actually observed. The

36 36 CHAPTER 4. THE PHASE DIAGRAM OF H 2 O Ice Polymorph Density (kg m 3 ) Crystal structure Order Ice Ih 920 Hexagonal disordered Ice Ic 920 Cubic disordered LDA 940 non-crystalline disordered HDA 1170 non-crystalline disordered VHDA 1250 non-crystalline disordered Ice II 1170 Rhombohedral ordered Ice III 1140 Tetragonal disordered Ice IV 1270 Rhombohedral disordered Ice V 1230 Monoclinic disordered Ice VI 1310 Tetragonal disordered Ice VII 1500 Cubic disordered Ice VIII 1460 Tetragonal ordered Ice IX 1160 Tetragonal ordered Ice X 2510 Cubic symmetric Ice XI 920 Orthorhombic ordered Ice XII 1290 Tetragonal disordered Ice XIII 1230 Monoclinic ordered Ice XIV 1290 Orthorhombic mostly ordered Table 4.1: The known polymorphs of ice with their densities crystal structure and proton disorder. LDA, HDA, and VHDA refer to low-density, high-density and very high-density water. Reproduced from reason is that the structure can be trapped in a local minimum of the potential function, and it is separated from the absolute minimum by an energy barrier (Fig. 4.5). If the energy barrier is small, the phase depicted by the local minimum is unstable or metastable, but if the barrier is sufficiently large, both phases are stable. The probablility of a phase change is given by the Boltzmann factor e E barrier/kt. For example, at T = 210 C Ice XI is the most stable phase (global minimum), but E barrier kt, so the phase change from Ice Ic or Ih does not occur. The phase therefore does not only depend on pressure and temperature, but also on the history of pressure and temperature. This is important for laboratory procedures in ice production; often a direct path in the phase diagram will not yield the desired phase.

37 4.5. CLATHRATES 37 Figure 4.5: Schematic diagram of a potential as a function of mean intermolecular distance (a proxy for structure). More than one structure can be stable, but the phase diagram usually only shows the most stable (global minimum). Figure 4.6: Unit cell of a type I clathrate, consisting of 46 water molecules that are organized into 12- and 14-sided cages. From: singer/grenum.html 4.5 Clathrates Clathrates or gas-hydrates are not a form of pure ice. Rather, it is a combination of ice with gasses, such as carbon dioxide or methane. Under favorable conditions the water molecules form a near spherical structure (reminiscent of soccer balls) that holds large gas molecules. There are three different kinds of cages: a 12-sided one consisting of pentagons, a 14-sided one consisting of 12 pentagons and 2 hexagons, and a 16-sided one consisting of 12 pentagons and 4 hexagons. These combine to two forms of clathrates: type I consists of 12 and 14 sided cages (Fig. 4.6), and type II of 12 and 16-sided cages. The type of clathrate depends on the gas, methane forms a type II clathrate, oxygen and nitrogen form type I clathrates. Other common gases in clathrates are CO 2, O 2, SO 2, and H 2 S.

38 38 CHAPTER 4. THE PHASE DIAGRAM OF H 2 O Figure 4.7: The solid line marks the phase boundary between the occurrence of methane clathrates (left side) and the unstable phase. Also shown are typical temperature gradients in the ocean and ocean sediments. From: Kvenvolden (1993) Methane clathrate is stable at 0 C at high pressure. It is found along many of the continental shelves and it is believed to hold vast amounts of methane, likely surpassing all other carbon deposits (oil, coal, natural gas) combined. Some estimates put methane clathrate reserves at more than 50% of all carbon reservoirs and at about 2/3 of the global fossil fuel reservoirs (Maslin, 2003). Clathrates can pack gas molecules to much higher density than the gas would occur in its pure form. For example, 1 m 3 of methane-clathrate holds 168 m 3 of methane. Clathrates are found in the ocean, in permafrost, and deep in ice sheets, where dissolved gas molecules induce a phase change, once the overburden pressure becomes sufficiently high. They can also occur in gas pipelines, where they become a significant problem (because of clogging) and they are discussed as a possibility for transporting natural gas (because of the very dense packing). An example of a clathrate stability diagram for methane is shown in Figure 4.7. There is concern that clathrates could be released in massive amounts under some climate change scenarios. This has the potential effect of massively amplifying any greenhouse effect, because methane is a very powerful greenhouse gas.

39 5 Defects in Ice Ih We have already seen that ice has a number of possible configurations, even at the theoretical limit of 0 K. At any other temperature T > 0 K, thermal agitation creates defects in the crystal. Defects are violations of the ice rules. We distinguish point defects, line defects, planar defects, and gross (volume) defects. The defects are important and they determine many of the physical properties of ice, such as deformation, electrical properties, etc. 5.1 Point defects We distinguish a number of point defects: 1. Impurities: An H 2 O molecule in the crystal structure is replaced by another molecule that fits relatively well into the structure (such as HF, NH 3, etc). This is sometimes done deliberately to study how ice properties change as a result of defects. This process is called doping. 2. Vacancies and interstitials: A vacancy is a missing H 2 O molecule in the ice structure and an interstitial is an extra molecule. 3. Protonic defects: Protonic defects directly violate the ice rules. We distinguish four kinds of defects. Ionic defects occur when an oxygen atom is associated with three protons (+ defect) or with only one proton (- defect). This violates the ice rule that each O has two H s associated with it. Bjerrum defects occur when a given O-O bond has no proton on it (L ( Leer )) defect, or it has two protons (D ( Doppel ) defect). Vacancies A vacancy is defined by an empty molecular site. It takes energy to form a defect. The available energy is the thermal energy kt, where k is the 39

40 40 CHAPTER 5. DEFECTS IN ICE IH Boltzmann constant. This energy is non-zero for all temperatures above 0 K, hence there is always energy available to compensate for the increase in entropy. The number of defects at a given temperature can be calculated from the Free Energy F. The Free Energy (or sometimes Helmholtz Free Energy) is at a minimum for a mechanically isolated and isothermal system. It can be written as F = ne v T S c, where n is the number of defects, and S c is the configurational entropy of defects. S c is related to the number of ways that n vacancies can be arranged among N 0 molecules. E v is the formation energy of forming the defect. For a vacancy it is equal to the heat of sublimation. The configurational entropy can be calculated with similar methods as the zero point entropy : S = k log W, where W is the number of ways to arrange n vacancies among N 0 + n sites: ( ) n + N0 W = n = (N 0 + n)! N 0! n! (5.1) We now use Stirling s approximation, for large n: ln n! n ln n n. With this approximation and using the condition ( ) F n = 0, we obtain ( ) N0 + n E v kt ln n T = 0 (5.2) = n N 0 + n = e Ev/kT (5.3) For vacancies, E v 0.5 ev/molecule. This is related to the latent heat of sublimation, because it signifies the energy needed to remove a molecule n from the crystal lattice. At 263 K this leads to n+n Vacancies are therefore extremely rare. For comparison, metals and alloys have meltingpoint vacancy concentrations of 10 3 to In actuality, an additional factor of e Sv/k enters equation (5.3). This is due to the modification of the configurational entropy introduced by the vacancy. This factor is difficult to estimate and is assumed to be negligible in the case of vacancies. Interstitials An interstitial is formed when an extra water molecule occupies open space in the crystal lattice. The concentration of interstitials is given by C i = e S i/k e E f /kt. (5.4)

41 5.1. POINT DEFECTS 41 Figure 5.1: Schematic of a D defect. The H atoms are not in line, which distorts the H 2 O angles. Also, the H-bond is longer than that of a normal bond. Goto et al. (1986) estimated S i = 4.9k and E f = 0.40eV using X-ray topography observations of dislocation climb. This results in a interstitial concentration of about 10 6 near the melting point. This is significantly higher than the vacancy concentration in ice or the interstitial concentration of metals. That is a direct result of the open structure of ice Ih, which makes the introduction of interstitials relatively easy. Bjerrum defects Ionic defects Ionic defects form when an oxygen has three protons associated with it (+ defect) or just one (- defect). They always form as a pair. The equivalent reaction of forming a +/- defect in liquid water is: H 2 O H 3 O + + OH (5.5) The ph of water is the negative logarithm of the molar [H 3 O + ] concentration. A ph of 7 for neutral water therefore indicates that these ionic defects exist under these conditions. The energy of formation for a +/- defect is relatively high (E + = 1 ev), but once they re formed they move relatively easily, particularly under an applied electric field. We will discuss the role of defects for electric properties later. L/D defects A L/D defect pair happens when a molecule rotates in the ice structure. The formation energy is 0.68 ev, so they are more common than +/- defects. The existence of a D defect distorts the bonds and lengthens the O-O distance from 2.76 Å to about 3 Å (Fig. 5.1). Mobility of a defect A defect has to overcome a potential barrier in order to move. For ionic defects this barrier can be lowered by applying an electric field (Fig. 5.2). Let s look

42 42 CHAPTER 5. DEFECTS IN ICE IH at a non-ionic defect, such as an interstitial. The frequency of successful jumps that leads to motion of a defect is given by an Arrhenius relation: ν j = νe Em/kT (5.6) where ν is the vibrational frequency. To illustrate the law of motion for defects, we restrict ourselves to motion along one dimension x. Assume that the concentration of defects is n(x), such that a n(x) is the number of defects per unit area (normal to x). The number of successful jumps is determined by the jump frequency (Eqn. 5.6) and the number of defects. So a gradient in defects n leads to a net flux j of defects: ( ( a dn j = ν j a n = ν j a 2 dx a )) dn = ν j a 2 dn (5.7) 2 dx dx Generalizing to three dimensions one can write Fick s Law: j = D n (5.8) n Together with a continuity statement: t equation: = j this leads to a diffusion n t = D 2 n (5.9) This is a diffusion equation, which is very common in physics. The most common diffusion equation is the one describing heat diffusion. Note that its derivation follows the same path as the one outlined above. In particular, Fourier s Law of heat conduction replaces Fick s Law (Eqn. 5.8) and conversation of energy is the equivalent of particle conservation. In the case of interstitials, the motion of defects is that of H 2 O molecules within an ice lattice. This is called self-diffusion. It can be measured by radioactively marking the interstitials with tritium. We will return to the mobility of defects later. Suffice it to say at this stage that while L,D defects are relatively easy to form, they are not very mobile. Contrary to that, +/- defects have a high energy of formation, but once formed, they move quite easily. Doping of ice Defects can be created in ice by adding substances that can easily substitute for a H 2 O molecule, but that have excess protons or lacking protons. The most common such substances are HF and NH 3. HF is a strong acid. In liquid water it causes the reaction:

43 5.2. LINE DEFECTS: DISLOCATIONS 43 Figure 5.2: Illustration of how the potential for defects is modified by an electric field (lower panel). Without a field (upper panel) the energy for defect mobilization is larger, and the potential is symmetric. a is the lattice spacing. HF + H 2 O H 3 O + + F (5.10) Hydrofluoric acid thus acts as a proton donor. It creates a + and an L defect (Fig. 5.3). In contrast to that, ammonia acts as a proton acceptor: NH 3 NH OH (5.11) It creates a - and a D defect. Other good dopants are HCl and KOH. But in general, things do not dissolve well in ice, in contrast to water, which is a great solvent. Foreign substances tend to be rejected during the ice formation process and accumulate in liquid or solid inclusions between different ice grains. 5.2 Line defects: dislocations A dislocation is a line defect in a crystal (Fig. 5.4). Depending on the relation of the dislocation line and the direction of displacement, we distinguish edge dislocations and screw dislocations. Most dislocations are in fact mixed dislocations, containing both edge and screw type (Fig. 5.4). The direction of displacement is shown by the Burgers vector. At a screw dislocation the Burgers vector is parallel to the dislocation, whereas at the edge dislocation they are perpendicular to each other (Fig. 5.5). Dislocations will move under an applied stress. Dislocation motion that leads to macroscopic deformation is called glide.

44 44 CHAPTER 5. DEFECTS IN ICE IH Figure 5.3: Effect of HF doping. Replacing a water molecule in the top chain with HF results in a + and L defect (bottom chain). Figure 5.4: Sketch of a dislocation (dotted line). S marks the location of a screw dislocation and E marks the location of an edge dislocation. b shows the direction of the Burgers vector, which shows the direction of displacement. An edge dislocation can have an irregularity, such as a kink or a jog. Jogs join dislocation lines on two different basal planes. The motion of a jog leads to a dislocation climb. A kink is a step in a dislocation line that is within one plane. Dislocations are best observed by X-ray tomography. They show up because the crystal line defect locally alters the Bragg condition, which shows up in the X-ray image. The technique is non-invasive, so the motion of dislocations can be observed. Good observations of dislocations require crystals with very few defects. In the laboratory, it is possible to create ice with fewer than 100 dislocations per cm 2, some of the best quality glacial ice has cm 1. A beautiful reference book for images of dislocations is Higashi (1988). Dislocations play a very important role in the deformation of crystals and we will return to them in Chapter 9. Unlike point defects, dislocations are not a necessary consequence of non-zero thermal energy, because this energy is

45 5.3. PLANAR DEFECTS 45 Figure 5.5: A screw dislocation (left) and an edge dislocation (right). Compare to Figure 5.4. Figure 5.6: When a dislocation travels along a glide plane, it develops kinks (left panel). Sometimes dislocations from different glide planes are joint by joggs (right panel). Moving joggs result in dislocation climb. not compensated for by the increased entropy. Instead, they are the result of flaws that occur during crystal formation. The equilibrium, unstressed crystal would not have any dislocations, but the time scale for removal is so long, that dislocations are very common. 5.3 Planar defects Planar defects can be separated into grain boundaries and free surfaces. We will treat the free surface in a later chapter (10). Grain boundaries are regions within polycrystals that separate two grains and have nanometer scale. A crystal orientation mismatch of more than 10 is called a high-angle boundary and leads to the formation of ledges and facets. Low-angle boundaries are believed to be a result of an accumulation of dislocations. The high-angle boundaries are sometimes interpreted with the Coincident site lattice (CSL) theory. This theory states that a preferred mismatch of crystal orientation is one that preserves some level of lattice order across the boundary. One problem with the CSL theory is that it does not deal with the proton disorder, which would be mismatched across the boundary. It is therefore expected that the grain boundary forms a region of considerable molecular disorder. Any grain boundary comes at an energy cost. It is therefore more favorable

46 46 CHAPTER 5. DEFECTS IN ICE IH Figure 5.7: A grain boundary of two grains rotated relative to each other by 34.1 about the [10 10] axis, which is perpendicular to the diagram. The dotted line shows the grain boundary, which contains lattice sites that are common to both grains. Reproduced from Petrenko and Whitworth (1999). for boundaries to migrate and larger grains to grow at the expense of smaller ones. This is important for glaciers and ice sheets, because the flow properties of polycrystalline ice depend on grain size. Ice is a pure solvent, so impurities concentrate at grain boundaries. Often, they are dissolved in water. Near the melting point, liquid water occurs at the grain boundaries in a network of veins, but the bulk permeability of polycrystalline ice is very small. Grain boundaries are important for mechanical properties. Grain boundary sliding and migration play a role in ice deformation. The grain boundary is also a location of dislocation generation and crack initiation. 5.4 Volumetric defects Volumetric defects occur as pores or as hard particles. Pores form by rejection of insoluble constituents. In glacier ice they are filled with gases, such as oxygen or nitrogen. In temperate ice, they contain water. In sea ice, pores usually contain brine when they are formed, but brine also has a tendency to drain (due to its higher density) and leave behind a network of brine channels. Pores lead to less resistance to creep deformation, and they lower the resistance to crack propagation. Solid particles can originate from the atmosphere (black carbon), through erosional processes, or by the precipitation of salt crystals at sufficiently low temperatures. Depending on the nature and density of these particles, deformation can be enhanced or inhibited.

47 6 Thermal properties of ice 6.1 The heat capacity Definition The heat capacity is a measure of the amount of heat needed to raise the temperature of a substance by 1 K. Commonly, two coefficients are defined, the specific heat under constant pressure and under constant volume: C p = C V = ( ) dq dt ( ) dq dt p V = ( ) de dt V (6.1) (6.2) The reason for the second equality is that under constant volume (dv = 0), the 1st Law of Thermodynamics states: de = dq + pdv = dq (6.3) C p has to be larger than C V, because at constant pressure some of the energy added goes into work for expanding the volume, rather than into raising the temperature. Some observations Note the following observations of specific heat for ice: C V (liquid) C V (ice) C 0 as T 0 C increases with temperature (does not become constant) 47

48 48 CHAPTER 6. THERMAL PROPERTIES OF ICE at very low T (< 5 K): C V T 3 For water vapor at high T: C V 3R Units for C V are JK 1 kg 1 or sometimes we write C V with units of JK 1 mol 1. Theory of specific heat When a solid is heated the crystal lattice absorbs energy through molecular motion (lattice vibrations). This energy is quantized, because the lattice vibrations occur at certain wavelengths corresponding to standing waves. In analogy to electromagnetic waves, which come quantized as photons, one can think of lattice vibrations as particles, the so-called phonons. A phonon is perhaps difficult to visualize, but it s a useful concept, because results can be derived in complete analogy with photons and the electromagnetic field. So, a phonon is a quantized harmonic oscillator with frequency ν and energy: ε n = hν(n ) (6.4) The 1 2 represents zero-point energy. This is not relevant to heat capacity and we will therefore not carry it along in what follows. Phonons turn out to be particles that are indistinguishable from each other (like photons, but unlike electrons). In statistical mechanics there are two different ways of doing statistics, one is known as Bose-Einstein statistics and the other one as Fermi statistics. Fermi statistics applies to particles that are distinguishable and that are defined by unique quantum number. An example are electrons who are subject to the Pauli exclusion principle. On the other hand, bosons are particles that are indistinguishable and are not subject to an exclusion principle. That is more than one can hold a given state. Phonons are bosons and a result from statistical physics is that, for a given frequency ν, the average energy is < E >= εn e εn/kt e ε n/kt (6.5) Now, we apply a number of mathematical tricks to turn this into a useful expression (one that does not contain a sum):

49 6.1. THE HEAT CAPACITY 49 < E > = ( ) ( ) 1 ln e nhν/kt (6.6) kt ( ) = ( ) 1 ln kt = ( ) 1 ln kt = hνe hν/kt = 1 1 e hν/kt (6.7) ( 1 e hν/kt ) (6.8) 1 e hν/kt (6.9) hν e hν/kt 1 (6.10) (6.11) In the above we used a result for infinite geometric series, namely: q n = 1 1 q, (6.12) for 0 < q < 1. Now we simply need to integrate this over all frequencies. For that we need a spectral density function ρ(ν). This function is defined so that ρ(ν)dν yields the number of vibrations excited between ν and ν + dν. The total number of such states (integrated over all frequencies) must be equal to the total degrees of freedom (3N A ): The total energy is then and the specific heat E = 0 0 ρ(ν)dν = 3N A (6.13) hν ρ(ν) e hν/kt dν (6.14) 1 C V = E T (6.15) The spectral density function can be measured (it is essentially the absorption spectrum). It can also be calculated, but that requires a good model for the

50 50 CHAPTER 6. THERMAL PROPERTIES OF ICE lattice. The calculation has been a partial success only so far. In the following we will discuss two simple models: the Einstein model and the Debye model. Both turn out to be poor models for ice, but they serve as an illustration of how such lattice vibration models can be constructed. Note that the derivation of Planck s Law for blackbody radiation proceeds in entirely the same way with a spectral density model that is equivalent to the Debye model discussed below. The Einstein model This was the first quantum mechanical treatment of specific heat. The Einstein theory of specific heat assumes that the spectral density is all contained in one frequency ν E, the Einstein frequency. One could think of ν E being related to the stiffness (elastic modulus) of the crystal (ν E k/m. To fulfill the requirement (6.13), we write: ρ E (ν) = 3N A δ(ν ν E ) (6.16) where δ is the Dirac distribution. One can now derive the total energy quickly from Eqn. 6.14, using the property We find f(x)δ(x x 0 )dx = f(x 0 ) (6.17) and E = 3N Ahν E e hν E/kT 1 (6.18) C V = E ( ) 2 T = 3R hνe e hν E/kT kt (e hνe/kt 1) 2 (6.19) We can define the Einstein temperature Θ E := hν E k and write: C V = 3R ( ΘE T ) 2 e Θ E/T (e Θ E/T 1) 2 (6.20) In particular, one can then show that for T Θ E : C V 3R. This is also known as the Dulong-Petit law. It is not observed for ice. However, water vapor at high temperature does have this property.

51 6.1. THE HEAT CAPACITY 51 The Debye model A somewhat more realistic way of determining the spectral density is to fill up wave space, starting with the lowest frequencies. The wave numbers for the phonons are determined by the lattice spacing L: k = 2π L, 4π L,... (6.21) The number of states up to wavenumber k is proportional to the volume of a sphere of radius k in phase space: n 4 3 πk3 ν 3 (6.22) ρ(ν) = dn dν ν2 (6.23) We still need to worry about condition (6.13). This is satisfied by cutting off ρ(ν) at the appropriate wave number: { Aν 2 ν ν ρ(ν) = D 0 ν > ν D ν D is the Debye frequency. Eqn. (6.13) can be used to find A = 9N A /ν 3 D. We can now find the energy and the specific heat: E = ν D 0 C V = 9N A ν 3 D 9N A ν 3 D If we define the Debye temperature Θ D = hν D k integral, we obtain ν 3 ν 2 ν D 0 hν e hν/kt dν (6.24) 1 hν kt ehν/kt (e hν/kt dν (6.25) 1) 2, and substitute η = hν kt in the ( ) Θ T 3 D /T e η C V = 9N A k Θ D (e η 1) 2 η4 dη = 3Rf 0 ( ΘD T ) (6.26) f is known as the Debye function. Some properties of the Debye function are:

52 52 CHAPTER 6. THERMAL PROPERTIES OF ICE Θ D is related to the wave speed: T Θ D : C V 3R (6.27) T Θ D : C V T 3 (6.28) ν D = v wave 3NA 4πV (6.29) V is the molar volume. For ice at T = 10 C, Θ D 200 K. It turns out that the Debye theory is only good at T < Θ D 50 4K! Some conclusions about specific heat Simple models for the spectral density functions do not work well for ice. The main reason for this is the disorder of the ice lattice with the random arrangement of protons. A more successful assumption is to make Θ D a function of the temperature, or to use an empirical density spectral function. Note that the approach outlined above for the energy and the counting of states in the Debye fashion is exactly analogous to the derivation of Planck s Law of blackbody radiation, which is one of the most fundamental and successful results of quantum statistics. The reason for this similarity is that phonons and photons follow the same statistical rules (those of bosons). 6.2 Latent heat Latent heat is the amount of energy per unit mass that is needed to change the phase of a material. The latent heat of fusion (solid/liquid transition) is L f = 335 kj kg 1 or 0.06 ev/molecule. In contrast, the heat of sublimation (solid/vapor transition) is L s = 2838 kj kg 1 or 0.53 ev/molecule. So, L f = 0.12L s. This can be interpreted in terms of ice-like clusters in liquid water. It is an indication that only 12% of the H-bonds are broken when ice is melted. Finally, note that a phase change that involves latent heat is always associated with a change in entropy: S = Q T = L f T melt (6.30) Entropy (measure of disorder) is higher in the liquid than in the solid.

53 6.3. THERMAL EXPANSION Thermal expansion We commonly define two coefficients of thermal expansion: linear : α = 1 ( ) L K 1 (6.31) L T p volume : β = 3α = 1 ( ) V K 1 (6.32) V T L is the length and V the volume of a sample. The linear coefficient of expansion varies with temperature. A peculiar property of ice is that below 70 K the coefficient becomes negative. That is, an increase in temperature results in contraction of the sample! This property occurs for other tetrahedrally bound crystals. This can be explained as follows: The thermal expansion is determined by lattice vibrations, just as the specific heat. There are different lattice modes, namely compressional modes and transverse modes. The compressional modes expand when excited. The transverse modes contract somewhat when excited. They become important at low temperatures. p 6.4 Thermal conductivity The thermal conductivity K is a measure of how well heat is conducted through a material: q = K T (6.33) This is Fourier s law of heat conduction. q is the heat flux and T the temperature gradient. The negative sign is a direct consequence of the Second Law of Thermodynamics, which states that, without work being performed on the system, heat must flow from high to low temperature. The units of K are W m 1 K 1. For ice at 0 C it is about 2.1 W m 1 K 1. If a material is anisotropic (not the same properties along all directions), K is a second order tensor. For ice it might make sense to write: K = K a K a K c (6.34)

54 54 CHAPTER 6. THERMAL PROPERTIES OF ICE where K a is the conductivity in the basal plane and K c that along the c- axis. There are, however, no conclusive measurements of a difference in those conductivities, even though many other properties are measurably anisotropic. Thermal conduction is a process that also happens through lattice vibrations. The energy transfer occurs via phonon-phonon interaction and phonongeometry (such as boundaries) interaction. Phonon-phonon interaction is due to anharmonic oscillations. Remember that harmonic oscillations do not interact. Kinetic theory shows that the conductivity due to phonon-phonon interaction can be written as: K = 1 3 C V v w λ (6.35) Here v w is the wave velocity and λ is the mean free path for the phonons. We can use this relationship to look at the temperature dependence of K. At temperatures above the Debye temperature (about 200 K for ice), the number of excited phonons is approximately proportional to temperature. The mean free path is inversely proportional to the number of phonons (and thus T). Because, at these temperatures, the temperature dependence of C V and v w can be ignored, we have K 1/T. Indeed, experimentally it is found: K = 651W m 1 T (6.36) At lower temperatures, the temperature dependence of C V becomes important. Very close to 0 K, the mean free path will essentially be the dimension of the crystal D c and Eqn (6.35) becomes: K = C V v w D c (6.37) and the temperature dependence of C V dominates. Thus, K T 3. The mathematical treatment of heat conduction requires Fourier s Law plus a description of heat flow in terms of temperatures. This is given through energy conservation. The internal energy u is given by ρc p T (more accurately it is T 0 ρc pdt ). Energy conservation, in the absence of heat sources, is a statement expressing that the change in internal energy has to equal the divergence of the heat flux. Mathematically: ρc p T t If K is constant this equation can be written as: = q = (K T ) (6.38)

55 6.5. THE STEFAN PROBLEM 55 T t = k ρc p 2 T = κ 2 T (6.39) Here κ is the thermal diffusivity ( 36 m 2 yr 1 ). A mistake that keeps reappearing in the literature is that a treatment of non-constant K is applied to Eqn. (6.39) rather than eqn The Stefan problem A classic application of the heat diffusion equation is to solve for the rate of freezing of a lake that is at freezing temperature T m (this condition can be relaxed) and is exposed to a below freezing surface temperature T s < T m. Assume that the surface is at z = 0, with z positive downward. The freezing front is at a position z s, which is a function of time. The equation describing the temperature of the ice is: T t = κ 2 T z 2 (6.40) The boundary condition has to be applied at a moving boundary z s (t). It is an energy balance that states that the heat carried away through the temperature gradient in the ice is equal to the heat released by freezing, and the temperature is at the freezing point T m : k T z = ρl dz s z=zs dt (6.41) T = T m (6.42) It is of advantage to use the variable transformation Θ = T T s, so that the surface boundary condition becomes Θ = 0. It can be shown that the error function is a solution to the diffusion equation that fulfills this boundary condition: ( ) z Θ(t, z) = Berf 2 (6.43) κt where erf(x) = 2 π x 0 e ξ2 dξ (6.44)

56 56 CHAPTER 6. THERMAL PROPERTIES OF ICE The boundary conditions at the freezing front (6.42) require ( ) z Berf 2 = Θ m = T m T s (6.45) κt This condition must be fulfilled for all times, so the argument of the error function cannot be time dependent: z s 2 κt = const = λ = z s = 2λ κt (6.46) This is a nice and simple enough result. To recover the exact value for λ and ultimately B, the other boundary condition (6.42) has to be applied. This results in an equation for λ that cannot be solved analytically. The Stefan Problem is one of the classical problems in geophysics, and one of the few free boundary problems, which can be addressed analytically.

57 7 Electric properties The chapter on electric properties contains a treatment of the electric permittivity and the electric conductivity and their dependence on frequency. In more general terms, it is a description of how ice reacts to the application of an electric field. We will discuss several topics. Specifically, an electric field results in the polarization of individual molecules by shifting electron clouds (fast process) the rotation of dipoles (slow process) the conduction of an electric current The first two items pertain to permittivity, the third to conductivity. When we discuss frequency dependence we will pay particular attention to the low frequency limits (ɛ s, σ s ) and the high frequency limits (ɛ, σ ) The next chapter (Optical properties) is really the very high frequency limit of the properties discussed here. 7.1 Conductivity The conductivity σ is a measure of how well a material conducts electricity. One often defines the resistivity ρ = 1/σ. The resistivity can be related to the electric resistance: R = ρl A (7.1) where L is the length and A the cross sectional area of the material. The resistance thus depends on the dimensions of a sample of the material, whereas 57

58 58 CHAPTER 7. ELECTRIC PROPERTIES the conductivity and resistivity are simply material properties. The SI units for conductivity are Ω 1 m 1. The conductivity relates the current density J to the electric field: J = σ E (7.2) This is one form of Ohm s Law. A perfect conductor (superconductor) has infinite conductivity (or zero resistivity) and a perfect electric insulator has zero conductivity. Typical conductivities for metals are σ 10 8 Ω 1 m 1. For ice it is σ 10 7 Ω 1 m 1. Ice is not a good conductor. In fact, it is better described as a semiconductor. An additional similarity to semiconductors is that electricity is conducted by protons. In metals, electricity is conducted by electrons, which exist in the form of a Fermi gas. A neat demonstration to discover the nature of the charge carrier is based on the Hall effect. This effect is based on the Lorentz force, which results in a side-ways force when charge carriers move along an electric field with a velocity v and a magnetic field B applied perpendicular to the electric field. This is also known as E B drift. The Lorentz force is F = q v B (7.3) In a given electric field E, the product q v is always in the same direction as E, because negative charge carriers have opposite signs from positive charge carriers for both q and v. Consequently, either charge carrier would be drifting in E B direction. By measuring the induced field, the sign of the charge carrier can be determined. For ice it turns out that the charge carrier is positive, and the most likely candidate is the proton. It is immediately clear that the transport of protons across the ice crystal must be connected to the formation and motion of defects. Figure 7.1 shows how protons move through the ice crystal lattice. The combination of a + and a D-defect results in resetting the lattice to its original configuration. The net result of the two defects is the displacement of one unit charge by one lattice spacing. A treatment of electric conductivity assigns an effective charge to each defect. The effective charges for the + and D defects should thus add up to the unit charge e = q + + q D. The + defect moves the proton across a longer distance, so q + > q D. It can be shown that q + = 0.6e and q D = 0.4e. A treatment of conductivity must consider the contribution of all defects to the total current density: J = i j i = σ E (7.4)

59 7.1. CONDUCTIVITY 59 Figure 7.1: The Hall effect. A current is conducted through a material with a magnetic field applied in a perpendicular direction. The induced voltage perpendicular to the magnetic field and the current, depends on the nature of charge carriers j i are the contributions of each of the four defects (+,-.L,D). Each contribution can be written as: j i = n i q i µ i (7.5) where n i is the concentration of defect i, q i is its effective charge (q +, = ±0.6e, q L,D = ±0.4e), and µ i is the mobility of the charge. The contribution of each defect to the total current density thus depends on its concentration as well as its mobility. Defects are formed in pairs, and, as we have seen previously, L and D defects form much easier than +/- defects: It can be shown that the mobilities are related as n L = n D n + = n (7.6) µ + > µ µ L > µ D (7.7) So, while + defects are difficult to form, they are very mobile. The inverse is true for D-defects. But, a detailed treatment of current densities also needs to take into account the fact that defects interact with each other: j i = n i q i µ i n i q i Ω( j + j + j L j D ) (7.8) Ω is a configuration parameter that describes the effects of one defect needing another one before continued passage (such as the +/D combination shown in Fig. 7.1).

60 60 CHAPTER 7. ELECTRIC PROPERTIES Figure 7.2: Proton transport and defects. Panel a shows how the addition of a proton leads to a + defect, which then propagates through the lattice. A subsequent addition of a proton creates a D-defect, which again has to propagate through the lattice (b). This resets the lattice to its original state. The electric field is from left to right. The interaction of defects depends strongly on frequency. frequency limit, defects move independently of each other and At the high σ = σ + + σ + σ D + σ L (7.9) The largest conductivity, σ L, is thus the most important. On the other hand, in the static limit, the defects do not move independently and we have e 2 = q2 + + q2 LD (7.10) σ s σ + + σ σ L + σ D The conductivity in the static limit is determined by the smaller conductivities, σ + + σ, because the rate limiting factor for charge transport is the unblocking of the chain by the passage of a + defect, as illustrated in Fig The transition from σ s to σ defines a relaxation frequency and a typical time scale (Fig. 7.3). The conductivity is also an important measure to determine how far an EM wave travels into a material. A measure for penetration depth is the skin depth :

61 7.2. DIELECTRIC PERMITTIVITY 61 (µ S m 1 ) σ 10 0 σ 10 2 s f (Hz) Figure 7.3: Conductivity. The point of inflection defines a resonance frequency and a relaxation time. Figure is only illustrative and does not represent actual measurements. d = c 2πωµσ (7.11) µ is the magnetic permeability, which is close to 1 for ice. The conductivity of ice (and thus the resistivity) is temperature dependent. In metals, the resistivity increases with temperature, because the increased lattice vibration impedes the motion of electrons. The resistivity of ice, however, decreases with temperature. This is typical of semiconductors. It can be explained by the fact that the concentration of defects increases with temperature, which means that more charge carriers are available. 7.2 Dielectric permittivity The dielectric permittivity ɛ is a material property that describes how a material, such as ice, reacts to an electric field. On the one hand, an electric field induces a current and this reaction is described by the conductivity. On the other hand, an electric field induces a polarization P in the material, which is then added to the electric field E. The resulting field is known as the displacement field D: D = E + P (7.12) A simple assumption is that the polarization is linearly dependent on the applied field P = χe, (7.13) where χ is known as the electrical susceptibility. We thus have D = (1 + χ)e = ɛe (7.14)

62 62 CHAPTER 7. ELECTRIC PROPERTIES The electric permittivity is a function of frequency. We write: D(ω) = ɛ(ω)ɛ 0 E(ω) (7.15) where ɛ 0 is the permittivity of free space, and ɛ the relative permittivity. Physically, the frequency dependence of ɛ can be interpreted as follows: At low ω, the displacement field D is in phase with E. As the frequency increases, D will start to lag behind E, because rotating molecules are trying to catch up with the field. At some intermediate frequency there is resonance between the molecules rotation frequency and that of the electric field. In general terms, for any linear medium, the electric permittivity should be a second rank tensor. This tensor only has two independent components: the permittivity along the c-axis and perpendicular to it. Even though there is a difference in those permittivities, in what follows, we will only look at the simplified case of an isotropic medium, where ɛ reduces to a scalar. Let s look at a simple relaxation model for polarization. We assume a polarization perturbation P. In a linear model we can write: dp dt = P τ D (7.16) The Debye time τ D defines a typical time scale for the rotation of the dipoles in ice. Eqn. (7.16) can be easily solved: P (t) = P 0 e t/τ D (7.17) We can explore the functional form of the permittivity s frequency dependence by taking the Fourier transform of Eqn. (7.17). Here we will not worry about factors of 2π, as we are only interested in the frequency dependence. P (ω) 0 e t/τ D e iωt dt = 1 1/τ D iω (7.18) This functional dependence of P on ω translates directly to the permittivity ɛ. We can write ɛ(ω) = a + b 1/τ D iω (7.19) where a and b represent constants that account for the magnitude of polarization and the effect of electron cloud shifting (the high frequency behavior). Written in terms of ɛ s and ɛ, we then have:

63 7.2. DIELECTRIC PERMITTIVITY 63 ɛ(ω) = ɛ + ɛ s ɛ 1 iωτ D (7.20) The high frequency limit of the permittivity is not 1, even though dipoles cannot rotate at such high frequency. In that case, the permittivity results from shifting electron clouds. We can now find the real and imaginary parts of ɛ: ɛ R = ɛ + ɛ s ɛ 1 + ω 2 τ 2 D ɛ I = (ɛ s ɛ )ωτ D 1 + ω 2 τ 2 D (7.21) (7.22) The Debye time also implies a typical frequency ν D = ω D /(2π) = 1/(2πτ D ). For ice at -10 C, τ D = s, and ν D = 3.1 khz. This should be compared to water where τ D = s, and ν D = Hz at 10 C. The much shorter Debye time for water is a result of molecules that can rotate much easier. The Debye time for water is an indication of the life time of clusters. At times shorter than the Debye time, clusters are stable, and they are too big to adjust to the electric field quickly. The real part of the permittivity varies as a function of ω from ɛ s to ɛ with an inflection point at ωτ D = 1. At that point the imaginary part has a maximum. This can be explained by a resonance phenomenon; At ωτ D = 1 the molecular dipoles rotate at a natural frequency, and the energy of the oscillating magnetic field is absorbed very effectively. This energy absorption is often described by the loss tangent: tan δ = ɛ I ɛ R (7.23) The loss tangent defines the phase lag between the displacement field and the electric field. Note that in this model, the real and imaginary parts of ɛ are related. In particular, it follows from Eqns (7.21) and (7.22) that ( ɛ R ɛ ) s + ɛ 2 + ɛ 2 I = 2 ( ) ɛ ɛ 2 s (7.24) This equation defines a circle in (ɛ R, ɛ I ) space. A plot of the real versus the imaginary parts of the dielectric permittivity is called a Cole-Cole plot. In the range where the measurements fall onto a circle, this simple oscillator model is applicable. 2

64 64 CHAPTER 7. ELECTRIC PROPERTIES 100 ε R Figure 7.4: The real and imaginary parts of the dielectric permittivity for a simple oscillator model ε I ωτ D The static electric permittivity ɛ s is temperature dependent and follows a Curie-Weiss law: ɛ s 1 T T c (7.25) T c = 50 K is the Curie temperature. It is the temperature at which ice would become ferro-electric. That is, the dipoles would be aligned and Ice XI is formed. Note that the same temperature dependence is typical for the magnetic permeability of a ferro-magnetic material (one that has a permanent magnetic field below the Curie temperature). There is a small jump of the permittivity across the phase boundary with liquid water, but ɛ s,water ɛ s,ice, because, given enough time, equally many dipoles rotate into the electric field in ice and water. The general decrease of ɛ with T is due to the additional agitation of molecules at higher temperatures. However, ɛ,water = 5.5 > ɛ,ice = 3.2. This is because there are some free molecules left in liquid water (not tied up in clusters) that can rotate with the EM field at high frequency. Finally, it is interesting to compare the values of dielectric permittivity for the different phases of ice. The disordered phases (such as Ices III, V, VI, VII) have a slightly higher static permittivity than Ice Ih. This is entirely due to their higher density, and thus the availability of more dipoles. The high frequency permittivity is similar for all disordered phases. The ordered phases (such as II, VIII, IX, and XI) however have their molecules in a fixed arrangement. The dipoles are therefore fixed, and the static permittivity is very similar to the high frequency permittivity.

65 7.3. THERMO-ELECTRIC EFFECT Thermo-electric effect Another interesting phenomenon is the thermoelectric effect: Applying a temperature gradient across a sample of ice results in a electric potential difference of about 2 mv K 1. This is again an effect that is similarly observed in semiconductors. The reason for the thermoelectric effect is that the number of +/- defects depends on temperature ( e E f /kt ). A positive temperature gradient thus implies a positive gradient of defect concentration. The defects will migrate against this gradient (remember Fick s Law). But the + defects are more mobile than the - defects, and in that way charge separation will occur.

66 8 Optical properties 8.1 The electromagnetic spectrum The optical properties are really a subset of all electrical properties, but we will discuss them here separately. Here we concentrate on the interaction of an ice crystal with an electromagnetic wave. The relevant electric property is the dielectric permittivity ɛ. Optical properties of ice are relevant for remote sensing of snow and ice, as all remote sensing products are a result of electromagnetic waves travelling through the atmosphere (which contains water and ice) and being emitted from (passive) or interacting with (actively) a snow and ice cover. Moreover, ice in the atmosphere leads to interesting optical effects, such as halos. Some terminology is common in connection with EM waves. The wave speed v is the speed of wave propagation, which depends on the medium the wave travels in. The wave speed in vacuum c ms 1 is one of the fundamental constants of nature. The frequency ν (or f) is related to the wavelength λ by ν = c/λ. Units of frequency are s 1, which is also written as Hz (Hertz). The angular frequency is ω = 2πν. The wave number k is generally 2π/λ in physics, but in spectroscopy it is 1/λ. As an EM wave impacts a crystal, parts of the wave is reflected, the wave that continues in the crystal is refracted. In the crystal part of the wave is absorbed, and the rest is transmitted. 8.2 The visible spectrum Refraction The index of refraction n is defined as the ratio between the speed of light in vacuum c and the wave speed in ice v: n = c/v. It depends on the wavelength (n = n(λ)), ranging from for violet to for red light. The refractive 66

67 8.2. THE VISIBLE SPECTRUM 67 ν λ k radio khz (AM) - GHz (radar) m microwaves GHz 1 mm - 10 m cm 1 FIR 10 µm - 1 mm cm 1 NIR 1-10 µm cm 1 visible Hz nm UV nm X-rays Hz nm γ-rays Table 8.1: The electromagnetic spectrum with commonly used names, frequencies, wave lengths, and wave numbers. air θ air ice θ ice Figure 8.1: Refraction: A light ray is bent towards the vertical when it traverses from a fast to a slow material index determines the bending of light, and the wavelength dependence therefore results in different amounts of bending for different wavelengths. This is the cause of the prism effect, i.e. the splitting of white light into its components as it travels through ice. The refractive index increases further for longer wavelengths; it is 1.8 in the NIR and 10 in radio frequencies. A refractive index greater than one results in bending of light. If the angle between the incoming wave and the normal to the interface is Θ air and that between the refracted wave and the surface normal is Θ ice, we have (Snell s Law): n ice n air = sin Θ air sin Θ ice (8.1) n air is very close to one, which is equivalent to stating that the speed of light in air is essentially the same as that in vacuum. Snell s Law is illustrated in Fig. 8.1.

68 68 CHAPTER 8. OPTICAL PROPERTIES In electromagnetic theory the refractive index is treated as a complex number: n = n R in I. This should be seen in the following context. If a wave is described by the complex function e iω(t x/v) and v = c/n, we can write: e iω(t x/v) = e iωt e iωx( n R c i n I c ) (8.2) = e iω(t n R c x) e ωn I c x (8.3) This shows how the real part n R is associated with the wave propagation speed, while the imaginary part n I describes absorption. The amplitude of the wave (the factor that is not time-dependent) decreases as e αx/2, where α = 2 ωn I c. It turns out that n I,red n I,blue, which simply states that ice absorbs red light much stronger than blue light. This is one of the reasons why ice appears blue at depth. The other reason is that ice also scatters blue light preferentially. The refractive index is related to dielectric permittivity ɛ. We have n = ɛ, and ɛ is also complex. Ice is optically anisotropic: n c-axis n a-axis = (8.4) The c-axis is sometimes also called the optical axis. The difference in refractive index (and thus wave speed) results in an effect called birefringence. The wave traveling along the c-axis is called the ordinary wave, and the one traveling in the basal plane is the extraordinary wave. If an incident wave hits a crystal at an angle, the wave has an ordinary and an extraordinary component. Because they travel at different speeds that creates a phase lag and the polarization changes. If the incident wave is parallel or perpendicular to the c-axis, the polarization will not change. For a sample between cross-polarizers this means that no light will be transmitted. This property is used to determine c-axis orientations and grain sizes (Fig. 8.2). Relationship between refraction and absorption We have seen that the refractive index is related to the dielectric permittivity: n = ɛ. The dielectric permittivity results from the effect of an electric field on a material. It is defined as the relationship between the electric displacement field D and the electric field E: D = ɛ E. Since the electric permittivity is dependent on frequency, we can write:

69 8.2. THE VISIBLE SPECTRUM 69 Figure 8.2: An ice thin section viewed through cross polarized filters. The different colors appear because of the wavelength dependence of the refractive index. From: pditlev/annual report/projects.html D(ω) = ɛ(ω)e(ω) (8.5) The frequency dependent field can be transferred to the time domain using the Fourier transform: E(t) = 1 2π E(ω)e iωt dω (8.6) An important result for Fourier transforms is the convolution theorem, which results in D(t) = = 1 2π ɛ(t )E(t t )dt (8.7) 1 ɛ(t )E(t t )dt (8.8) 2π 0 The second of these equations is a consequence of causality: The electric field at a future time cannot determine the displacement field now. Since E(t) and D(t) are real valued, so must ɛ(t). One can thus show that ɛ(ω) = 1 2π ɛ(t)eiωt dt is analytic, and so is n = ɛ. Furthermore, n(ω) 1 tends to zero, and therefore all conditions for the applicability of the Kramers-Kronig relationships are fulfilled (see appendix 8.A). This leads to

70 70 CHAPTER 8. OPTICAL PROPERTIES Re n(ν) = π P 0 0 ν Im n(ν ) ν 2 ν 2 dν (8.9) Im n(ν) = 2ν π P Re n(ν ) 1 ν 2 ν 2 dν (8.10) where P designates the Cauchy Principal Value (see appendix 8.A). The interesting thing about the Kramers-Kronig relationships is that they can be used to infer the wave speed (related to n R ) from absorption measurements (related to n I ), or vice versa. This is a direct consequence of causality, which leads to n being an analytic function. Reflection The intensity I of a wave propagating through a material can be written as a function of the propagation depth x I(x) = I 0 (1 R)e αx (8.11) I 0 is the intensity of the incident wave, R is the reflection coefficient, and α is the absorption coefficient. The reflection coefficient R is a function of the incident angle, the polarization and the frequency of the incident wave. The reflection coefficent is given by the Fresnel equations: R s = R p = [ ] n1 cos Θ i n 2 cos Θ 2 t (8.12) n 1 cos Θ i + n 2 cos Θ t [ ] n1 cos Θ t n 2 cos Θ 2 i (8.13) n 1 cos Θ t + n 2 cos Θ i n i are the real parts of the refractive indices of the two materials, Θ t is the refracted angle (which is given by Snell s law (8.1), and the two reflection coefficients are for different polarizations: s refers to a wave with an EM field perpendicular to the plane defined by the incoming and reflected rays, and p refers to an EM polarization in that plane. Instead of a reflection coefficient, we often talk of albedo, which is the ratio of reflected to total incident energy. It is often designated by the Greek letter ρ. The albedo is a function of frequency; ρ(ν) is referred to as the spectral albedo. We also define the emissivity e = 1 ρ. An emissivity of e = 0 defines

71 8.2. THE VISIBLE SPECTRUM 71 a perfect mirror and e = 1 defines a perfect black body. In remote sensing the concept of brightness temperature T B is defined: T B = et S, where T S is the sensitive temperature. T S is the temperature measured on the ground, while T B is the temperature inferred from measuring the outgoing radiation. It is an important concept in remote sensing for satellites that measure thermal emissions (such as AVHRR, Modis, etc.), and determining the emissivity of a material is the main difficulty with estimating temperatures from spaceborne measurements. Transmission As shown above, α is related to the imaginary index of refraction n I. The factor of 2 difference in the exponent (eqn. 8.3) arises because intensity is the square of the wave amplitude. If the wave amplitude decreases as e ωn I/cx, then we have α = 2ωn I c = 4πn I λ = 4πkn I (8.14) where k = 1/λ is the wavenumber as used in spectroscopy. Values for α range from cm 1 in the blue to cm 1 in the red. The absorptivity is defined as the ratio of absorbed energy to total energy. Scattering A detailed treatment of the interaction of an EM wave with a material leads to a theory of scattering. A complete analytical treatment can be made if the individual particles are smaller than the wavelength of the radiation. This is treated by Rayleigh scattering. Rayleigh scattering is highly frequency dependent (proportional to λ 4 ), so that blue light is scattered more strongly than red light. For most molecules in the air, Rayleigh scattering is a good approximation, and it can explain, for example, why the sky appears blue. If this approximation cannot be made, various assumptions are needed. Scattering models are an important topic for models of radiative transfer in the atmosphere. Ice in the atmosphere is often treated by making some assumption about the shape of the ice particles. A treatment of scattering of individual ice particles in a snow pack leads to the concept of the Bidirectional Reflectance Functions (BDRFs). These functions describe the amount of radiation scattered into a unit solid angle as a function of incident angle and observed angle. In that way they provide a fully three-dimensional picture of the radiation pattern.

72 72 CHAPTER 8. OPTICAL PROPERTIES 8.3 Optical properties across the EM spectrum An excellent summary of optical properties across many spectral bands is given by Warren (1984). What follows is a short summary: Ultraviolet The interaction with these high energy waves mostly results in electronic transitions. UV radiation is thus highly absorbed (α = 50, 000cm 1 ), and little energy is transmitted. Visible The absorption coefficient in the visible spectrum varies over an order of magnitude from cm 1 in the blue to cm 1 in the red. Scattering off individual H 2 O molecules is described by Rayleigh scattering, which is proportional to 1/λ 4, so shorter (blue) wavelengths are scattered preferentially. Bubbly ice contains air pockets that are often much larger than the wavelength of light. In that case a better description of scattering is that of Mie scattering, which is independent of wavelength. Bubbly ice therefore appears white. Bubbly ice is also much more absorptive: While only 1.8% of light is absorbed in a passage through 10 cm of pure ice, about 26% of light is absorbed in the same thickness of bubbly ice. The small absorption coefficient in the visible range for clear ice makes it possible to use ice as a neutrino observatory. This is currently done at the South Pole, where the Icecube project consist of a large array of more than 1000 m deep boreholes that are equipped with very sensitive photodetectors. These detectors record light pulses that result from the very few reactions of neutrinos with the protons of the water molecules. Infrared The absorption coefficient in the IR is again large ( 0.34 cm 1 ). Absorption peaks are caused by the vibrations of the H 2 O molecules. At these peaks the absorption coefficient reaches 1400 cm 1, which means that 1 µm of ice absorbs essentially all energy. The peaks of the absorption spectra are broadened due to the disordered nature of the ice crystal. The ordered phases of ice, such as Ices II, VIII, IX, and XI, show more peaked spectra. The same is true for water vapor. Because the absorption of IR radiation is so strong, the longwave radiation balance is important for ice. It can account for a large part of surface melt on a glacier or on sea ice.

73 8.4. ATMOSPHERIC HALOS 73 Radio waves (khz - MHz) Ice is very transparent to radio waves. Radio echo sounding can be done on ice that is over 4000 m thick. Temperate ice poses more of a challenge, because it contains inclusions of liquid water that can be cm in diameter. This leads to increased scattering. Lower frequencies (1-10 MHz) are less susceptible to that (they are not sensitive to that range of inclusion sizes). However, the larger wavelengths result in longer dipole antennas, and the imaging resolution decreases. The index of refraction is large for radio waves, and the wave speed is only in the range mµs 1 for temperate ice. The wave speed is temperature dependent, and this property could, in theory, be exploited for the determination of ice sheet temperatures with radar surveys. 8.4 Atmospheric halos One of the perks of living at high latitudes is the occasional display of halos, which are caused by ice particles in the air. They are the ice equivalents of a rainbow. Good resources for halos are the books by Tape (1994) and Tape and Moilanen (2006). There are also some nice online resources, such as and the links provided there. Halos can be explained by refraction of light in a multitude of crystals of a certain shape. Light entering one crystal face and exiting another one at an angle A R to the first experiences a fixed deflection D which is given by solving: n = sin( 1 2 (A R + D)) sin( 1 2 A R) (8.15) The most common halo is created by sunlight entering vertically aligned columnar crystals through one prismatic face and exiting it through another. The faces are in 60 orientation towards each other. This results in a deflection angle of These halos are known as the sun dogs. The dependence of the refractive index on wavelength results in a splitting of the colors. Another halo from columnar crystals results from horizonally aligned crystals, when sunlight enters the basal face and exits a prismatic face, which have a relative angle of 90. This results in the second arc at 46. Because the angles at which sun dogs can be observed is a direct function of the angles between crystalline faces, the kinds of crystals in the air can be

74 74 CHAPTER 8. OPTICAL PROPERTIES identified by noting the angles between the sun and halos. There are several reports of halos that indicate the presence of cubic ice (Ic). Some atmospheric halos result from reflections of light off crystal faces. The most commonly observed is the reflection of the basal face of platelets. This results in pillars that extend vertically from the light source. During a Fairbanks winter night, one can sometimes see a pillar above every light in the city. 8.A The Kramers Kronig relationships The Kramers-Kronig relationships establish a relationship between the real and imaginary part of certain complex valued functions. Consider a complexvalued function χ(ω) with the following properties: χ = χ r + iχ i is analytic in the upper half plane lim ω χ(ω) = 0 For such a function, we can choose a real-valued ω and then integrate χ(ω ) ω ω dω (8.16) where the integral path goes along the real axis, avoiding the point ω = ω by a half circle in the upper half plane. The path is completed by a half circle at infinity. Via the residue theorem, we can then write χ(ω ) ω ω dω = P χ(ω ) ω ω dω iπχ(ω) = 0. (8.17) Here, P is the principal value, which is defined for a function f(x) with a singularity at x = a, as P f(x)dx = lim ε 0 a ε f(x)dx + provided this limit exists. We can then write χ(ω) = i π P a+ε f(x)dx, (8.18) χ(ω ) ω ω dω. (8.19)

75 8.A. THE KRAMERS KRONIG RELATIONSHIPS 75 It is now possible to take the real and imaginary part of this equation to obtain the following equalities: Reχ(ω) = 1 π P Imχ(ω) = 1 π P Imχ(ω ) ω ω dω (8.20) Reχ(ω ) ω ω dω (8.21) These relationships allow a determination of the real part of the function from measurements of the imaginary part, and vice versa. These equations are also known as Hilbert Transforms. If the function also has the property χ( ω) = χ (ω), (8.22) where is the complex conjugate, then one can show that Reχ(ω) = 2 π P 0 ω Imχ(ω ) ω 2 ω 2 dω (8.23) Imχ(ω) = 2ω π P Reχ(ω ) ω 2 ω 2 dω (8.24) 0

76 9 Mechanical properties The question to be answered in this chapter is: How does ice react to stress? The answer depends not only on the magnitude of the applied stress, but also the time scale over which it is applied. Over short time scales, ice behaves like a solid: it reacts elastically and it can transmit compressional as well as shear waves. If the stress magnitude is high, it can react with brittle failure. Over longer time scales, however, ice acts like a fluid und undergoes solid creep. The deformation behavior is linearly viscous at low stresses (constant viscosity, Newtonian flow), but it becomes non-linear at higher stresses. The deformational mechanisms differ significantly between single crystals and polycrystalline ice. 9.1 Definitions Stress All mechanical properties involve the concept of stress. Simply speaking, stress is the force per unit area. As forces have directions and faces have different orientations, a variety of stresses can be defined (shear stresses and normal stresses). This concept of different stresses can be neatly accommodated by defining the second order stress tensor, σ ij, which is the i-component of the force vector acting on the j-th face (as defined by the normal). One can thus define nine components of the stress tensor. But conservation of angular momentum demands that σ ij = σ ji. The stress tensor is thus a complete description of the stress state. The force on any plane with normal vector n and area da is then df = σnda. (9.1) It can be shown that a symmetric matrix can be transformed to a diagonal matrix, containing the three eigenvalues of the stress. The stress eigenvalues 76

77 9.1. DEFINITIONS 77 are called principal stresses. In that sense, normal or shear stresses are relative concepts, that is, they depend on the frame of reference. For example, if one looks at a map view of a glacier with x pointing downglacier, and y pointing perpendicular to the margin, the stress field can be described by: σ = ( 0 τ τ 0 ) where τ is the shear stress at the margin. rotated by π/4, the stress tensor becomes: σ = ( τ 0 0 τ If the system of coordinates is ) In this system the stresses are principal stresses, which are tensional (positive) along x and compressional (negative) along the y direction. Crevasses form under tension and they are thus typically oriented at 45 upglacier. They are sometimes referred to as shear crevasses, which is not a correct term, because they actually form under tension. Strain and strain rate Strain is the relative change in length. In one dimension: ε = l l. In general, strain can be defined as a second rank tensor: ε ij = 1 2 ( ui + u ) j x j x i (9.2) where u i is the displacement in the i-direction. Similarly, one can define a strain rate: ε ij = 1 2 ( vi + v ) j x j x i (9.3) where v i is the ith velocity component. Note that ε d dt ε. Constitutive (or material) relations relate stress to strain (for elastic materials) or strain rate (for plastic materials) or both (for visco-elastic materials). In the case of perfect elasticity, strain is proportional to stress and it is fully recoverable. In a plastic material, strain rate is a function of stress. When stress is reduced to zero, the strain rate also becomes zero, and the total strain is not recoverable.

78 78 CHAPTER 9. MECHANICAL PROPERTIES 9.2 Elastic properties Isotropic materials In one dimension ideal elasticity is described by Hooke s Law: F = kx, where F is a force, k a spring constant, and x the displacement from equilibrium. In continuum mechanics the concept of force is replaced by stresses and displacements by strains. For normal stresses we write: ε = σ E (9.4) E is known as the elastic modulus or Young s modulus. For shear stresses we write: ε = σ (9.5) 2G where G is the shear modulus. In general when a material is compressed in one direction, it expands in the other directions. The magnitude of this effect is described by Poisson s ratio ν. Assume that a stress σ x is applied that results in a strain ε x. The Poisson s ratio is now defined by the strain in the other directions: ε y = νε x. For an incompressible material ν = 1 2, because V V = ε x + ε y + ε z = ε x 1 2 ε x 1 2 ε x = 0 (9.6) The fractional volume change is also called the bulk modulus κ. For ice, ν = 1 3. So ice is not incompressible during elastic deformation. Nonetheless, to a very good approximation it can be treated as an incompressible material for viscous deformation. Any isotropic material can be defined by two elastic constants, such as (E, G), (κ, ν), (λ, µ), etc. λ and µ are called the Lamé parameters, where µ equals the shear modulus G. Even though ice is highly anisotropic, polycrystalline ice can be described isotropically, if the c-axes are randomly distributed. Table 9.1 shows a comparison of elastic constants for ice and other materials. The elastic constants increase at lower temperatures. If rocks and ice are compared at similar homologous temperature (T/T m ), where T m is the melting temperature, then the elastic and the shear moduli are quite similar between rocks and ice. Anisotropic materials For anisotropic materials the stress tensor is related to the strain tensor in a more general way. The most general linear relationship requires the introduc-

79 9.2. ELASTIC PROPERTIES 79 polycrystalline ice granite sandstone Young s Modulus 9.5 GPa 50 GPa 30 GPa Shear Modulus 3.5 GPa 25 GPa 10 GPa Poisson s ratio 1/3 1/ /3 Table 9.1: Elastic properties of ice at -5 C and granite and sandstone at room temperature tion of a fourth order stiffness tensor called the compliance tensor C: σ = Cɛ, or in component form: σ ij = C ijkl ε kl. (9.7) Alternatively, it is often more convenient to describe strains in terms of stresses, in which case we can define a stiffness tensor: ε ij = S ijkl σ kl. (9.8) A general fourth order tensor has 81 components. But using various symmetries, such as the symmetry of σ and ɛ, as well as the hexagonal symmetry of the ice crystal, this can be reduced to no more than five independent components, which are sometimes named (C 11, C 33, C 12, C 13, C 44 ), and correspondingly for S. The components have the following meaning: S 11 relates normal strain perpendicular to the c-axis to normal stress perpendicular to the c-axis S 33 relates normal strain along the c-axis to normal stress along the c-axis S 12 relates normal strain along the basal plane to a normal stress also in the basal plane but in a direction perpendicular to the strain S 13 relates the normal strain along the c-axis to a normal stress in the basal plane S 44 relates the shear strain parallel to the c-axis to a shear stress in the same plane The stiffness along the c-axis (C 33 ) exceeds that in the basal plane (C 11 ) by about 8%. An elastic constant can be calculated for any direction, but this requires a return to the full tensor notation. For example, if the elastic constants are

80 80 CHAPTER 9. MECHANICAL PROPERTIES desired in a system that is obtained through a transformation a ij from the system that is aligned with the c-axis, then the stiffness matrix components are given by S ijkl = a ima jn a kp a lq S mnpq (9.9) Elasticity is determined by lattice parameters. The theory of phonon interaction is therefore relevant here, and the fundamental physical processes for a treatment of thermal capacity and conductivity is very similar to that of elasticity. Anisotropy in polycrystalline ice Often, polycrystalline ice can be treated as isotropic. Elastic constants can be determined with models of various complexity, by assuming that individual grains in the polycrystal are either subject to the same strain or to the same stress, or by modeling the complete stress-strain distribution. The difference between these treatments turns out to be small. However, some polycrystals are anisotropic. In floating ice, some c-axis orientations are preferred during ice formation. In glacier ice, anisotropy can develop during ice deformation. These cases can be treated properly by introducing stiffness tensor components, similarly to the treatment of single ice crystals. Additional anisotropy is introduced when fractures occur in the ice. In compression, cracks develop preferentially along the direction of compression. In tension, cracks are preferentially oriented perpendicular to the direction of extension. Elastic waves The measurements of the elastic constants is not trivial. If static stresses are applied, non-elastic effects can quickly be important, and lead to wrong values of the elastic constants. Instead, it is better to determine dynamic values through measurements of the speed of elastic waves. Each component of the stiffness matrix has a wave velocity associated with it: v ij = C ij ρ (9.10) where ρ is the ice density. In general we distinguish between two kinds of waves: compressional (p) waves and shear (s) waves. p and s stand for primary and secondary, because the compressional wave speed is much higher than the shear wave speed. For compressional waves the displacement at a given

81 9.2. ELASTIC PROPERTIES 81 p-waves s-waves v c = 4040m s 1 v c = 3892m s 1 v c = 1810m s 1 v c = 1930m s 1 v c = 1810m s 1 polarization c polarization c Table 9.2: Measurements of seismic speeds for single crystals. Source: Gammon et al. (1983) position is along the line of wave propagation, whereas for shear waves the particle displacement is perpendicular to the direction of wave propagation. The shear wave can thus be composed into two components of polarization. The five components of the stiffness tensor lead to the five wave speeds listed in Table 9.2. These different wave speeds lead to a similar phenomenon as birefringence for optical waves. Polycrystalline ice can be treated isotropically and only two wave speeds need to be considered. At 0 C they are approximately 3700 ms 1 for the p wave and 1900 ms 1 for s waves. The wave speed is temperature dependent and for T < -5 C the following empirical relations hold: v p [ms 1 ] = T [K] (9.11) v s [ms 1 ] = T [K] (9.12) A good reference for seismic work on glaciers is Roethlisberger (1972). Applications in glaciology Active seismics has been used to find the depth of the ice for glaciers with deep channels, where radio echo sounding often fails. Examples include the discovery of a 1500 m deep channel on the Ruth Glacier, 1500 m of ice at Taku Glacier, 1300 m at LeConte Glacier (all in Alaska), and 2700 m at Greenland s Jakobshavns Isbrae (all collected by K. Echelmeyer, UAF). Another application is to use active and passive seismicity to detect the nature of the bed. For example, this is how it was discovered that there were water-saturated soft sediments underneath the West Antarctic ice streams of the Ross Sea (e.g. Blankenship et al., 1987). It can also be used to determine changing conditions at the glacier bed when surface water reaches the base (ex. Black Rapids (Nolan and Echelmeyer, 1999)). In West Antarctica, passive seismics has been used to find sticky spots, places at the bed with higher

82 82 CHAPTER 9. MECHANICAL PROPERTIES resistance to flow that generate seismicity (e.g. Anandakrishnan and Alley, 1994). More recently, glacial seismic sources have raised some interest, as it was discovered that glacial sources (mostly in Greenland) generate very low frequency earthquakes ( Hz) that can be detected on the global seismic network (Ekström et al., 2003). The mechanism behind these large ( M5) events is still unclear, but many of them seem to be related to large calving events (Amundson et al., 2008). In principle it might be possible to find the vertical temperature distribution of an icesheet using detailed seismic surveys. Due to the anisotropy of ice it is possible that seismic measurements can also reflect changes in crystal orientation that could be due to large accumulated strain (Horgan et al., 2008). Anelasticity In a theory of ideal elasticity no energy is lost as the waves travel through a medium. In reality seismic waves are attenuated due to energy loss. This can be due to effects such as the re-orientation of molecules, during which energy is lost, which will turn into heat. These anelastic effects lead to a damping of the seismic wave. In the theory of electricity it was the phase lag between the displacement field D and the electric field E that let to a loss tangent and to attenuation of the EM wave. Similarly, in elasticity, anharmonic behavior leads to a phase lag between stress and strain. Just like the energy contained in an electromagnetic wave is given by D E(+H B), the energy of the elastic wave is given by σε, and a phase lag between σ and ε leads to a elastic loss tangent tan δ. δ is also known as the angle of internal friction. The energy loss rate σ ε divided by the total energy integrated over a cycle equals ω tan δ. Seismologists define a quality factor: Q = 1 tan δ If we consider the absorption of an elastic wave of amplitude A 0 : (9.13) A(x, t) = A 0 e αx = A 0 e αvwavet (9.14) where α is an absorption coefficient and v wave the wave speed. We can therefore write:

83 9.3. BRITTLE FAILURE 83 de/dt E = αv wave = ω tan δ (9.15) α = ω tan δ v wave (9.16) The peak absorption occurs at ω khz. This corresponds to a time scale of τ 10 4 s τ D. It turns out that the processes leading to absorption of elastic waves is similar to that which determines dielectric permittivity (rotation of molecules). The absorption of these waves is therefore also impacted by L,D defects. The absorption of seismic energy is highly temperature dependent and measuring absorption as a function of depth in an ice sheet should allow a determination of the temperature profile. 9.3 Brittle failure Brittle failure occurs when stress magnitude is high enough that the stress cannot be accommodated elastically, or it is applied so quickly that it cannot be accommodated plastically. The path to failure depends on the strain rate. At high strain rates stress and strain increase monotonically until failure. At lower strain rate the stress reaches a maximum when microcracks are starting to appear. The strain continues to increase, but stress is decreasing again until the point of failure is reached. Failure under tension Under high strain rates the relevant process for brittle failure is crack nucleation. The crack nucleation stress can be written as σ N T = σ 0 + kd 1/2 (9.17) where σ 0 and k are constants and d is the grain size. At low strain rates the limiting process is crack propagation. The stress necessary for crack propagation is: π σt P = 2 K Ica 1/2 (9.18) where a is the radius of fracture, and K Ic is the fracture toughness for mode I failure (failure under tension). It can also be shown that a = αd, where α is

84 84 CHAPTER 9. MECHANICAL PROPERTIES of order unity. Because the slope of the crack propagation stress as a function of d 1/2 is steeper than that of the nucleation stress, there is a cross-over point between the two stresses. At small grain sizes, crack propagation is rate limiting, and at larger grain sizes it is crack nucleation (Fig. 9.1). Figure 9.1: Crack propagation and crack nucleation as a function of grain size d 1/2. Compression Ice can also fail under compression. Figure 9.2 shows a sketch of how a crack develops. The propagation of the wing cracks is described by: σ P c = ZK Icd 1/2 1 µ (9.19) K Ic appears again, because the wing crack is actually opening under tension. d is the grain size (as above) and µ is the frictional coefficient for the AB crack. Z is a numerical constant that does not depend on strain rate or temperature. The strength of ice is temperature dependent through the fracture toughness and the frictional coefficient. Ice gets stronger as it gets colder. A consequence of this is that floating ice tongues and ice shelves exist for cold ice, but they are quite rare for temperate ice. At a critical strain rate ɛ B/D a transition from brittle to ductile deformation occurs. This strain rate is a function of many of the parameters defined above: ɛ B/D = fct(a, K Ic, Z, µ, d, p) (9.20) where A is the flow rate factor for ice creep (defined in the next section) and p is the pressure. The dependence on p limits the depth of crevasse propagation to about 30 m in temperate ice, and somewhat more in cold ice.

85 9.4. THE CREEP OF ICE 85 Glacier ice Water Oil Mantle rocks Pa s 10 2 Pa s 10 3 Pa s Pa s Table 9.3: Viscosities of some materials In glaciers crevasses usually occur under tension, but shear cracks can occur. Examples have been observed during surges. Very prominent shear cracks formed after the 2002 Denali Fault M7.9 earthquake. Also, thrusting has been observed on several glaciers, particularly polythermal ones. Figure 9.2: Sketch of a compressional crack. At A and B so-called wing cracks develop. 9.4 The creep of ice Ice deforms under stress like a fluid. The simplest fluids deform under constant viscosity η: σ = 2η ɛ. Table 9.3 lists some viscosities for comparison. A material of constant viscosity is called a Newtonian material. The viscosity of ice is stress dependent for σ > 10 kpa. To properly treat the deformation of ice it is important to consider several deformation mechanisms and find which one is rate-limiting. Also, we need to distinguish between deformation mechanisms of single crystal ice versus polycrystalline ice.

86 86 CHAPTER 9. MECHANICAL PROPERTIES Diffusional creep Diffusional creep is also known under the name Nabarro-Herring, or N-H creep. It results from a diffusion of interstitials and vacancies that results from an applied stress field (Fig. 9.3). Recall (Chapter 5) that the number of defects is given by a Boltzmann factor: n V n 0 = e E/kT (9.21) where n V is the number of vacancies (for example), n 0 is the total molecule number, and E is the formation energy for a vacancy. When a stress is applied, this energy can be written as E = E 0 + pdv = E 0 + σr 3 m (9.22) where r m is a typical spatial scale, in this case the size of a vacancy. We can therefore write: c = n V n 0 = e E 0/kT e σr3 m/kt = c 0 e σr3 m/kt (9.23) If we now compare the concentration of vacancies at a location of stress σ to that at +σ (Fig. 9.3), we obtain a difference in defect concentration: c + c = c 0 (e σr3 /kt e σr3 /kt ) (9.24) According to Fick s Law this concentration gradient c will lead to a flux j of defects: j = D SD c (9.25) D SD is the coefficient of self-diffusion (see section 5.1). The flux j is the number of vacancies moving through a unit area in a unit time. It results in motion of mass and therefore in a strain rate ɛ. We can write an expression for strain rate per unit area by considering the size of a vacancy r and the crystal size d: Using Fick s Law (9.25) and ε Jr d (9.26)

87 9.4. THE CREEP OF ICE 87 So: c c + c d = c 0 /kt d (eσr3 e σr3 /kt ) (9.27) = 2c ( ) 0 σr 3 d sinh (9.28) kt 2c 0 d σr 3 kt (9.29) ε Jr d 2c 0r 4 D SD d 2 σ (9.30) kt This shows several interesting relationships. First, the strain rate is proportional to the stress. Diffusional creep is therefore Newtonian or linearly viscous. The strain rate depends on the inverse square of the grain size, and the temperature dependence is: ε e E SD/kT kt (9.31) where the exponential factor describes the temperature dependence of the coefficient for self-diffusion. Diffusion creep is only observed to be dominant at low stresses (σ 10 kpa). It is difficult to study experimentally, because such low stresses result in very low strain rates and experiments have to be carried out for a very long time. For polycrystalline ice, diffusion can also occur in a boundary layer along grain boundaries. This mechanism is also known as Cobble creep. A similar expression to Eqn (9.30) can be derived: ε cobble = α δ σ D GB d 2 kt (9.32) Here, α is a geometrical factor that depends on the crystal shape, D GB is a grain boundary diffusion coefficient, and δ is the thickness of the boundary layer along which diffusion occurs. Dislocation creep A dislocation is a line defect in a crystal (Chapter 5). The motion of a dislocation causes bulk deformation. Here we will estimate the amount of shear deformation that is caused by a dislocation moving through an ice sample of

88 88 CHAPTER 9. MECHANICAL PROPERTIES Figure 9.3: The movement of interstitials (extra H 2 O ) and vacancies (missing H 2 O ) under an applied stress field. length L. The length b of the Burgers vector describes the displacement by one kink. The strain caused by the motion of the dislocation is: ε = b h (9.33) where h is the height of the sample above the dislocation. This expression is correct if the dislocation has moved through the entire sample length L. If it has only moved a distance L, the strain is: For N dislocations we can write: ε = b L h L (9.34) ε = N b L h L = Nl b L = ρb L (9.35) V Here, l is the width of the sample and ρ is the dislocation density. If we now take a time derivative, we have: ( ε = b L dρ ) dt + ρd L dt (9.36) In steady state the dislocation density is constant, and the previous equation simplifies to ε = ρbv (9.37)

89 9.4. THE CREEP OF ICE 89 where v is the dislocation speed. Equation (9.37) is known as Orowan s equation. It can be adjusted by a geometrical factor Φ to account for non-parallel dislocations: ε = Φρbv (9.38) We know that dislocations exist in the unstressed state, but the application of a stress leads to the creation of more dislocation as well as to dislocation motion. It can be shown that the dislocation density ρ depends quadratically on the stress σ: ρ σ 2 (9.39) Dislocations move by developing a kink, which then travels along the dislocation in an effort to smooth it out. The motion of the dislocation is resisted by an energy barrier known as Peierls barrier. These barriers are separated by a distance h, which is related to the lattice spacing. A result from the theory of dislocations for the dislocation velocity is v = ν 2σbah2 kt { exp F } k + F m. (9.40) kt Here, ν is a jump frequency (similar to that encountered in chapter 5), a is the lattice parameter, and b the magnitude of the Burgers vector. F m and F k are free energies required to form an isolated kink and to move it. Dislocation mobility is low compared other materials if they are compared at the same normalized stress (σ/g) and at the same homologous temperature (T/T f ). The above relations lead to a deformation law: ε = Ae Ecreep/kT σ 3 (9.41) Note, however, that there are different models for the dislocation density, which result in different stress exponents. All of these models leads to a nonlinear deformation law: ε σ n, but the exponent n varies between models. For example, dislocation density is proportional to stress for basal glide in single crystals, leading to a flow law exponent of 2 (Schulson and Duval, 2009). It is important to note that the different values of n are based on specific physical models, rather than empirical fits to deformation tests. Dislocations are preferentially located on the basal plane, which is also known as the glide plane. If the c-axis is oriented vertically, the crystal is in an ideal configuration for vertical shear deformation. This mode of deformation is referred to as easy glide. In easy glide there are two possibilities for the

90 90 CHAPTER 9. MECHANICAL PROPERTIES location of the glide plane: It can be the basal plane itself (this is known as the glide set) or in between (but parallel to) two basal planes. The latter is known as the shuffle set. The glide set is believed to be the dominant set, but good observations are missing. Slip parallel to the c-axis is very difficult to achieve and might in fact never have been observed. Deformation tests are often carried out an angle to the c-axis. If the applied stress is transformed to the basal plane, the observed strain rates are consistent with a model of single slip in the basal plane. Note that the motion of dislocations will lead to the creation of (L,D) defects, or alternatively requires the existence of pre-existing defects. An analysis shows that stresses which lead to ice deformation are not high enough to lead to the formation of defects, so existing defects should be important. This would also imply that doped ice should deform easier, because it has more defects. There are some indications that HF doping reduces ice viscosity, but direct observations of dislocations have, so far, failed to show a strong effect of doping. The dependence on defects also leads to the expectation that there should be a relation between viscosity and the dielectric relaxation time (because both are related to number of defects). Indeed, Ice II, which is ordered and has a very long relaxation time, is more viscous than Ice Ih. On the other hand, Ice III has a relaxation time about 100 times smaller than Ice Ih, and also a lower viscosity. Detailed investigations of dislocation motion show that it is not steady, as suggested by Orowan s Equation (eqn. 9.37). Rather, at the small scale, dislocations move in bursts, so-called dislocation avalanches. The strain produced by such avalanches is inversely related to grain size, so for larger grain sizes deformation becomes smooth. Also, in polycrystalline ice, dislocation avalanches are stopped by grain boundaries and plasticity appears smooth. Polycrystalline ice In a treatment of the deformation of polycrystalline ice it is often assumed that the crystals are oriented randomly. This implies that the deformation law will be isotropic (i.e. not depend on direction). It also implies that not all crystals are oriented for easy glide, but they must deform in unison. In addition to diffusional creep (Nabarro-Herring and Cobble) and dislocation creep, we will also have to consider grain boundary sliding, grain boundary migration, and dynamic recrystallization. Grain boundary sliding is the process of grains sliding past each other; grain boundary migration is the process of one grain growing at the expense

91 9.4. THE CREEP OF ICE 91 of another; and finally, dynamic recrystallization results in the continuous formation of new crystals that are oriented for basal slip. Grain boundaries are planar defects. They have a surface energy associated with it. This surface energy can be kept small if the molecules on the boundary fit into the lattice of either one of the crystals. Such models are called coincident site lattice models, and they require specific relative orientations of the two neighboring crystals. In temperate ice, the grain boundaries often contain water veins, even though the overall permeability of ice is very low. Also, most impurities are concentrated at grain boundaries, because ice is a very bad solvent (unlike water). Measurements of ice deformation show four distinct areas of deformational behavior (Fig. 9.4): elastic strain primary creep secondary creep, where the minimum strain rate occurs tertiary creep For reasons that are not always entirely clear, the minimum strain rate obtained during secondary creep is often regarded as the steady state creep rate. This can be compared against strain rate, and that is how power laws are empirically derived. Figure 9.4: The creep of polycrystalline ice, showing elastic (fully recoverable) strain, primary creep, secondary creep (where the minimal strain rate is reached), and tertiary creep.

92 92 CHAPTER 9. MECHANICAL PROPERTIES Primary creep Primary creep occurs early in the deformation history. It is a stage that is characterized by initially high strain rates that decrease with time. Sometimes, primary creep is described by the following equation: ε(t) = βt 1/3 + κt (9.42) This leads to a decreasing strain rate with time. One physical explanation for primary creep is that it is due to a build-up of internal stress. Some of this primary creep is recoverable, possibly because dislocations can glide backwards when the internal stress is released. Because of this partial recoverability, the deformation is sometimes called delayed elastic creep. Secondary creep Early experiments on the deformation of ice by Glen (1955) and Steinemann (1958) were looking for the minimum strain rate obtained in secondary creep. It was shown that data can be fit to a Glen-Steinemann flow law: ɛ = A(T )σ n = A 0 e Q/kT σ 3 (9.43) where Q= 60 kj mol 1 for T < 263 K, and Q= 135 kj mol 1 for T > 263 K. A 0 also changes value at 263 K, so that the strain rate is a continuous function of temperature. A is known as the flow rate factor. Tertiary creep Tertiary creep takes years to decades to establish. It is therefore difficult to measure in the lab, but it should be relevant to ice flow in glaciers and ice sheets. During tertiary creep, strain rates increase again. They can be as much as eight times the rates during secondary creep. This effect is due to dynamic recrystallization: Under stress, crystals will align themselves so that they are oriented for easy glide. This occurs after the accumulation of large strains and can occur at the bottom of ice sheets, or perhaps in the margins of ice streams. The crystal alignment during tertiary creep poses a big problem: Ice becomes anisotropic again. A law of the form (9.43) is therefore not sufficient to describe the flow. An anisotropic flow law would involve a 4th order viscosity tensor relating stress to strain rate. This tensor would be stress dependent. There are no truly anisotropic flow laws in use now. Instead the effect of anisotropy is taken into account through an enhancement factor E:

93 9.4. THE CREEP OF ICE 93 ε = EA(T )σ n (9.44) Eqn is still an isotropic equation, however. It can only apply when one mode of deformation is strongly dominant. Multiple deformation processes Goldsby and Kohlstedt (2001) attempted to derive a flow law that accounts for all the relevant deformation processes. They reviewed existing deformation data and conducted their own experiments. Based on that they suggest the following flow law: ( 1 ε = ε diff ) 1 + ε disl (9.45) ε basal ε GBS The first term is due to diffusional flow (Nabarro-Herring and Cobble), the second term is grain-boundary-sliding accomodated basal slip, and the third term is due to dislocation motion. For ice-sheet relevant stresses, the second term, and in particular grain boundary sliding, turns out to be most important. Grain boundary sliding is the rate-limiting term: it leads to smaller strain rates than basal slip and therefore dominates the second term. Basal slip and grain boundary sliding appear in parallel, because these mechanisms only occur in mutual dependence. Goldsby and Kohlstedt (2001) fitted each of the strain rates in equation (9.45) to a power law of the form: ε = A σn Q+pV e kt (9.46) dp where d is the grain size. There are still some problems with this flow law. First, the application to icesheets requires the knowledge of grain size, and typical ice sheet grain sizes are two orders of magnitude or more larger than the data used in the derivation of the flow law. So, a huge amount of extrapolation is required. This is done so that low stress experiments can be completed in a reasonable amount of time (lower grain sizes lead to higher strain rates). Also, the result that grain boundary sliding is a dominant deformation mechanism would not lead to preferred crystal orientation, which is observed in nature.

94 94 CHAPTER 9. MECHANICAL PROPERTIES Effects on flow rate parameter We already looked at the temperature dependence of the flow rate factor A, which follows an Arrhenius relation. Other variables can affect the flow. It should be expected that there is a pressure dependency of A of the form e pv/kt. It is found, however, that the activation volume V is very small (V = mm 3 mol 1 was found in one study). So the effect of pressure is usually ignored. Work by Duval (1977) has shown that water content in temperate ice can have an important effect on deformation. He found that A = ( W ) Pa 3 s 1 (9.47) where W is the percentage water content. Water contents of about 0.5% to 1% are typical for glacier ice. Water in ice facilitates processes such as grain boundary sliding and recrystallization (because water molecules can move around easier). A proper treatment of water and ice requires mixture theory. There are various levels of mixture theory, depending on whether the mass balance, the momentum balance, and the energy balance are treated independently for the two phases. Mixture theories are rarely applied to glaciological problems. The effect of density on A is generally ignored. It is important for snow and firn, but ice can be treated as incompressible for flow problems, even though in elasticity, it is not incompressible at all! The tensor form of the flow law The Glen-Steinemann flow law (eqn 9.43) relates one component of the stress tensor to the corresponding component of the strain rate tensor. Nye (1957) generalized this flow law to tensor form. The first thing to recognize is that the mean hydrostatic stress does not lead to deformation. We therefore need to define the deviatoric stress tensor σ through a decomposition of the stress tensor σ: σ ij = pδ ij + σ ij (9.48) p = 1 3 σ ii is the mean hydrostatic stress. The negative sign is chosen so that compressive stresses, which are negative by convention, result in a positive number for p. The non-linearity of the flow law is expressed by a dependence on the mean deviatoric stress σ eff. This is also known as the octahedral stress or the second invariant of the deviatoric stress tensor:

95 9.4. THE CREEP OF ICE 95 σ 2 eff = 1 2 tr(σ 2 ) = 1 2 (σ 2 xx + σ 2 yy + σ 2 zz) + σ 2 xy + σ 2 xz + σ 2 yz (9.49) where tr stands for the trace of a tensor. The generalized Glen-Nye flow law then takes the form: ε ij = A(T )σ n 1 eff σ ij (9.50) The effect of incorporating the second stress invariant means that the existence of one stress component can soften the ice when it is sheared in any other direction. It is possible to write down a more generalized flow law for an isotropic incompressible fluid. The property of isotropy means that coefficients can only depend on the three invariants of the deviatoric stress tensor. The first invariant is the trace, which is zero by definition. The second invariant is the octahedral stress, as defined above, and the third invariant is the determinant of the tensor. The dependency on this invariant is often considered negligible, although some deformation tests do show a weak dependence. A general flow law then takes the form: ε = Aid + Bσ + Cσ 2 (9.51) where id is the identity tensor and A, B, and C are functions of the second and third invariant and the temperature. Higher order terms do not appear here, because the Cayley-Hamilton Theorem of linear algebra implies that any higher order term can be written as a sum of the lower order ones. Incompressibility implies that tr( ε) = 0, and thus 3A + 2CII σ = 0 (9.52) where II σ is the second invariant. This leads to the equation of a Reiner-Rivlin fluid: ε = 2 3 CII σ id + Bσ + Cσ 2 (9.53) Note that colinearity between ε and σ implies that C = 0, and we therefore recover Glen s flow law. In the case where C 0, a stress in one direction can result in deformation in another. For example, simple shear with σ xz 0, but all other stress components equal to zero, results in a non-zero deformation component ε xx. If this effect exists in glaciers, it is likely too small to be observable, although Schoof and Clarke (2008) speculated that it might explain flutes in subglacial tills.

96 96 CHAPTER 9. MECHANICAL PROPERTIES Finite viscosity laws A property of Glen s flow law is that the viscosity tends towards infinity for very small strain rates. This can pose numerical problems, but there are also physical reasons why this should not be so. In particular, at low stresses, diffusional creep with a linear flow law becomes dominant. This behavior is sometimes incorporated by introducing a finite viscosity parameter k: ε ij = A(σ eff + k)n 1 σ ij (9.54) Note that the Goldsby-Kohlstedt law does incorporate a linear flow at low stresses. 9.5 Visco-elasticity At sufficiently short time scales, both viscous and elastic deformation can be important. This time scale is known as the Maxwell time and is found by comparing viscous and elastic strains: ɛ el = σ 2µ (9.55) ɛ visc = ε t = σ 2η t (9.56) The Maxwell time τ M is obtained by equating these two strains: τ M = η µ (9.57) The viscosity of ice is stress dependent, and so is therefore the Maxwell time. For glaciologically relevant situations, the Maxwell time is on the order of minutes to hours. Viscoelasticity can be relevant to describe processes such as the tidal flexure of an ice tongue, or the sudden drainage of a lake through a glacier.

97 10 The surface of ice 10.1 The liquid-like surface layer of ice Everybody knows that ice is slippery. But this should come as a bit of a surprise. Ice is a solid and most solids have high coefficients of friction when sliding against other solids. What makes ice peculiar is that most of the time it is quite close to the melting point. Near the melting point ice, and other solids, show a behavior of a liquid-like surface layer. This layer can be explained by the fact that molecules near the surface are not as tightly integrated into the crystal structure of ice. Instead, they are in a disordered state that is somewhat more like a liquid. The thickness of this layer can be as small as a few atoms, and it disappears entirely at temperatures below about 40 C, although estimates vary greatly. People living in the Arctic know that at these temperatures the frictional coefficient of ice becomes much higher, and ice starts behaving more like a rock, or any other solid. Excellent review articles about the surface properties of ice are Dash et al. (1995) and Dash et al. (2006). Observations of the surface layer differ greatly depending on the method used. Enhanced electric conductivity only occurs to about 1 C, indicating that below that temperature the layer is not mobile enough to play a role in conducting electric charge. X-ray diffraction shows the effect of the surface layer to about 10 C, but proton scattering, which probes the open structure of the crystal, shows an effect to 60 C. Optical ellipsometry observes the different index of refraction for ice and water and shows an effect of an liquidlike layer to about 5 C. Finally, nuclear magnetic resonance is sensitive to the location and motion of protons. It is sensitive to a single layer of disorganized molecules, and that effect is measurable to 100 C. It can be concluded that different techniques lead to vastly different numbers. This is likely due to the fact that liquid-like has different meanings, depending on the physical property that is being investigated. 97

98 98 CHAPTER 10. THE SURFACE OF ICE 10.2 Surface energy Surfaces have surface energy associated with it. It is the work required to create a particular surface, and it is given as a free energy per unit area. The surface energy depends on the phases on either side of the surface. A familiar example is the surface tension between liquid water and air. Figure 10.1: The angle of a liquid drop on a solid at the triple point has an angle θ that depends on the respective surface energies vapor solid liquid At the triple point, solid, liquid and vapor phases coexist. Where the three phases meet (Fig. 10.1), a force balance between the different surface tensions requires γ sv = γ sl + γ vl cos θ. (10.1) This condition can only be fulfilled if γ sv < γ sl + γ vl, otherwise the surface will be completely wetted. To date, observations of water at the triple point have not yielded solid conclusions on the surface energies Theoretical models Following Dash et al. (1995), a model for the thickness of a liquid-like layer considers the reduction in the total free energy by covering the solid with a thin liquid layer. The change in free energy per unit area due to the formation of a liquid layer of thickness d can be written as G = ρ l µd + γf(d) (10.2) µ is a difference in free energy of the liquid and the solid: µ = L m(t m T ) T m, (10.3) where L m is the latent heat of melting and T m the melting temperature. γ = γ lv + γ sl γ sv is a difference in surface energy between a solidliquid-vapor transition and one without the liquid phase. f(d) is a function that varies between 0 and 1, as d goes from 0 to. An empirical model for it can be derived from Van der Waals interactions:

99 10.4. FRICTION AND ADHESION OF ICE 99 f(d) = d2 d 2 + σ 2, (10.4) where σ is a typical molecular distance. Minimizing the Gibbs Free Energy results in a liquid-like layer thickness of ( 2σ 2 ) 1/3 γ T m d = (10.5) ρ l L m T m T This model has been disputed, because there is some evidence that γ > 0 and hence no liquid layer would exist. The reason for this apparent failure to predict a liquid layer is that the boundary layer is not truly liquid and the above thermodynamic treatment not applicable. The layer is therefore often referred to as liquid-like or quasi-liquid Friction and adhesion of ice When water is frozen to another solid, the forces of adhesion can be very high. On the other hand, ice in contact with many solids is characterized by very low coefficients of friction. Both of these facts can pose many engineering challenges. Adhesion is a problem for icing of man-made structure, such as power lines and airplane wings. The lack of friction is desirable for skiing, but poses a considerable challenge in the design of car tires. Some of the early experiment on ice were conducted by Faraday, who studied the effects of adhesion between two ice crystals by bringing them into close contact with minimal force. Modern experiments show that at temperatures down to 25 C the two particles will stick together if the experiment is carried out in a wet atmosphere. If the atmosphere is dry, the particles will only adhere to each other down to 5 C. This has been interpreted in terms of a liquid-like layer evaporating in the dry atmosphere, but a more likely consideration is that the wet atmosphere helps molecule transport for adhesion. This adhesion of two ice particles is an important mechanism of snow metamorphism. Adhesion of ice to stainless steel shows a temperature dependence. The shear stress needed to separate ice from steel increases linearly to about 1.5 MPa at 13 C. Below that it remains constant, because failure will occur in the ice, rather than between ice and steel. Separating ice from other solids can be achieved by mechanical breaking (deicing boots on airplane wings), heating and melting (sometimes done on

100 100 CHAPTER 10. THE SURFACE OF ICE power lines) or by rapid cooling to induce stress, because ice has a higher thermal compressibility than many other solids. It is a popular misconception that the slipperiness of ice is due to the pressure melting depression. Supposedly, because of the lower melting point at higher pressure, ice melts when the pressure of a skate or a ski is applied and one can glide on the small layer of water. This cannot be correct for several reasons. For example, a back of the envelope calculation shows that pressures could not possibly be high enough to depress the freezing point by more than a fraction of a degree. Also, at -20 the slope of the liquid-solid phase transition reverses sign, so according to that theory, skating would not be possible below that temperature. It is now believed that two effects are important for gliding. The first one is the presence of a pre-melted liquid-like layer. The second factor is the generation of heat by friction, which leads to additional melting and a liquid layer to slide on.

101 Water and Ice in the atmosphere Nucleation of ice Ice can form either from the liquid or the vapor phase. In this first part we will treat the formation of ice from the liquid phase. A recent summary of nucleation processes in ice is given by Koop (2004). In general we distinguish homogenous from inhomogenous nucleation. Note that Koop (2004) is very sparse on inhomogenous nucleation, which reflects the poor knowledge of the fundamental physics in that area. Homogenous nucleation Homogenous nucleation refers to the freezing of pure water without the presence of seeds. The freezing of water releases latent heat, so a certain amount of supercooling (cooling below the melting point) is necessary to initiate freezing. Freezing does not necessarily start at very small amounts of supercooling, because there is an energetic price to pay: the formation of an ice crystal introduces surface tension σ LS. If H 2 O molecules come together to form a volume v of ice with surface A, the free energy of the system is increased by an amount: G LS = n s (µ S µ L )v + σ LS A (11.1) Here, µ L and µ S are the chemical potentials of the liquid and solid phase, respectively, and n s is the number of molecules per unit volume of ice. It can be shown that µ S µ L = kt ln(p L /p S ) (11.2) where p L and p S are the saturation vapor pressures over the liquid and solid phases respectively. Recognizing that the volume of ice increases linearly with 101

102 102 CHAPTER 11. WATER AND ICE IN THE ATMOSPHERE Figure 11.1: Change in free energy when supercooled water is frozen as a function of cluster size Number of molecules n i the number of ice molecules n i and the surface area increases with ns 2/3, we can rewrite eqn G LS kt = An i + Bn 2/3 i (11.3) where A and B are factors that depend on surface energy and chemical potential, as well as some geometrical factors. Figure 11.1 illustrates the effect of the embryo size. At some critical size, G LS reaches a maximum. Below that maximum, increasing the crystal size is not energetically favorable, because the necessary surface energy outweighs the energy released due to freezing. Above the maximum, however, the volumetric term is dominant and ice crystals can grow rapidly. The right hand side of eqn is temperature dependent, so the position of the maximum in free energy is as well. At lower temperature the critical embryo size decreases. Depending on drop size, water can be supercooled substantially: 1 kg of H 2 O can be supercooled to -10 C, while a water droplet in the air ( 1µm) can be supercooled to -40 C. At that temperature a critical embryo size is about 11 Å, which corresponds to about 190 molecules. This is close to the size of clusters in liquid water, so ice formation should always occur at that temperature. After attaining critical embryo size, the crystal can grow quickly and essentially adiabatically (no heat exchange with the

103 11.2. SNOWFLAKES 103 environment, δq = 0). The growth speed is proportional to some power of the degree of supercooling: v growth Tsupercooling m, where m 2. An open question about homogenous nucleation includes the location of freezing initiation (at the surface or in the interior of a droplet). Heterogenous nucleation The more common mode of crystal formation is that of heterogenous nucleation. In this case a foreign particle acts as a nucleus on which ice grows. These seed crystals are relatively large, and do not have to have similar lattice structure to Ice Ih. Curiously, the controlled growth of ice onto another crystal structure (epitaxy) is difficult to establish. Seeding of clouds with AgI has been tried with some success. AgI has a c-axis that differs by about 2% and an a-axis that differs by about 1% from that of Ice Ih. There seems to be some success at initiating ice growth at temperatures of -4 to -10 C. It is not clear, however, whether this is an effect of the closely matched lattice. The nucleation process is of great importance for animals living in the far north. Uncontrolled freezing is dangerous, because ice crystals can damage cell membranes. Two strategies are commonly used: prevent or control freezing. Freezing prevention happens with the help of proteins with antifreeze properties, which lower the freezing point. Controlled freezing occurs when ice crystal growth is controlled to produce small crystals only, that cannot damage cells. This often leads to vitrification, the formation of a low-temperature ultraviscous water. Pure H 2 O cannot simply be vitrified: Below 235 K liquid water does not exist, and ultraviscous water and glassy water only occurs below 150 K. The region between 150 K and 235 K is therefore referred to as No man s land Snowflakes Snowflakes are ice crystals that grow directly from the vapour phase. Three processes have to be considered for crystal growth: Water molecule transport to the freezing front Incorporation of H 2 O into the lattice Transport of latent heat away from the freezing front The crystal growth is determined by a chemical potential:

104 104 CHAPTER 11. WATER AND ICE IN THE ATMOSPHERE Figure 11.2: The Nakaya diagram shows the shape of snow flakes as a function of temperature and supersaturation. From Kobayashi (1967) µ vapor µ crystal [ ln p ] n i (11.4) p sat where n 2. p i is the vapor pressure over ice and p sat is the saturation pressure. According to eqn. (11.4) growth can only occur when p i > p sat, that is when the vapor pressure exceeds saturation pressure. This is known as supersaturation and it can be shown that the growth rate is proportional to the square of the degree of supersaturation. The crystal growth speed depends on temperature. The temperature dependence is different for growth along the basal plane and growth in the c-axis direction. Depending on the temperature, growth along c-axis is dominant and the snow crystals have the form of columns or needles. At other temperatures growth in the basal plane dominates and plate-like crystals form. A diagram that shows crystal shape as a function of temperature and supersaturation is known as a Nakaya diagram (Fig. 11.2). Snow crystals come in an amazing variety, and it has often been claimed that no two of them are alike. Bentley and Humphreys (1962) compiled a beautiful book with over 2000 photographs of unique snow crystals. To resolve the apparent paradox why two snow flakes from the same storm are different from each other, even though they can have a high degree of symmetry, Libbrecht (2005) proposed the No two alike conjecture: Each snow flake

105 11.2. SNOWFLAKES 105 experiences slightly different atmospheric conditions during growth, leading to a unique shape. However, on the scale of one snow flake, conditions are uniform, so that all sides of the crystal are exposed to identical growth conditions. This leads to a high degree of symmetry. A snow flake does faithfully records the history of supersaturation and temperature on its way from crystallization to meeting the ground.

106 Snow metamorphism Snow on the ground Snow consists of ice, liquid water, water vapor and air. The ice crystals in the snow pack can be atmospheric in nature, or they can be a result of recrystallization. The density of snow varies greatly, from 50 kg m 3 for very light powder snow to almost the density of water for slushy snow. Old seasonal snow often has a density of about 500 kg m 3, whereas the density of firn (one year old snow) is about 830 kg m 3. There are two kinds of snow metamorphism: destructive and constructive. Destructive metamorphism includes: Development of rounded crystals, which reduces the free surface energy. Saltation: Breaking of crystals by tumbling through wind action. Melting and refreezing Constructive metamorphism includes the following processes: Depth hoar formation: Large temperature gradients lead to water vapor pressure gradients, which lead to the formation of new cup-shaped crystals. This depth hoar has little strength and relatively low density. It is also known as sugar snow. Surface hoar: During cold and clear nights the snow surface can cool effectively and this can lead to the formation of new crystals on the snow surface. Crystal growth: Large crystals reduce the free surface energy and they grow at the expense of smaller crystals. This is a very slow process. 106

107 12.1. SNOW ON THE GROUND 107 Figure 12.1: Water molecules move from areas of high vapor pressure (convex) to low vapor pressure (concave). This leads to the formation of rounded crystals. Development of rounded crystals Snow is a material with connected pore spaces. The ice therefore has a large surface area, which is energetically unfavorable. The vapor pressure is higher over convex surfaces than over concave ones, because a water molecule is better integrated into the ice crystal structure at a concave surface. This leads to a transport of molecules from convex to concave areas and to a rounding of crystal (Fig. 12.1). Some of this molecular transport also happens along a quasi-liquid or premelted layer on the surface of the ice crystal. Molecules in that layer are more mobile than those in the ice crystal. Sintering Faraday (1933) already recognized that two snow crystals, when brought together without significant application of force, will adhere to each other very strongly. He attributed this process to a quasi-liquid surface layer that enables water molecules to move in a way to join the two crystals into one solid. Faraday (1933) called this process regelation, but it does not refer to the same process that is nowadays known by this name. Instead, in today s language this is called sintering and is known to occur for other crystals near their melting points. Sintering is an important process in snow metamorphism. It leads to the formation of larger crystals and a binding of snow packs, generally increasing its stability. Depth hoar Depth hoar and snowed in surface hoar form weak layers that can lead to avalanches. They are both a form of constructive metamorphism. Depth hoar forms under large temperature gradients (15-10 Km 1 ). This large gradients typically occur in more continental climates with dry (thin snow packs) and

108 108 CHAPTER 12. SNOW METAMORPHISM Figure 12.2: Optical and Scanning electron microscope images of depth hoar crystals. From: Eric Erbe, USDA, and LT-SEM/default.html cold (large temperature difference) conditions. The large temperature gradients lead to large vapor pressure gradients, and water molecules migrate upwards in the snow pack, building crystals in the process that can reach several cm s in size (Fig. 12.1) Snow to ice transformation When snow has survived at least one full accumulation/ablation cycle it becomes known as firn (from the old German word Ferner, which means last year ). The transformation from firn to ice is very temperature dependent. It was first described by Benson (1962). He recognized several distinct zones of snow to ice transformation in Greenland (which he called facies, borrowing from Geology). It is interesting that many of the facies described by Benson

109 12.2. SNOW TO ICE TRANSFORMATION 109 Figure 12.3: This SAR mosaic of Greenland clearly shows the different snow facies. Particularly obvious is the dark signature of the dry facies. Courtesy: M. Fahnestock, UNH (1962) were found again 30 years later in satellite synthetic aperture radar (SAR) imagery (Fig. 12.3; Fahnestock et al., 1993). A recognition of the different facies and a clear understanding of the process of densification is very important when interpreting ice sheet altimetry data. For example, it is possible that the process of densification accelerates in a warming climate, leading to higher average density of the firn layer. This could lead to a surface lowering, which might erroneously be interpreted as a mass loss. Dry facies The dry facies occurs in places where the mean annual temperature is below -25 C. Little or no melting occurs and the transformation to ice occurs by densification due to the pressure of overlying firn. The dry facies is observed almost everywhere on the Antarctic Ice Sheet, in central Greenland, and on high altitude subpolar mountain glaciers (such as Mt. Wrangell in Alaska). The dry facies is one of the darkest radar surfaces that occur naturally, because radar penetrates into the dry snow/firn with little scattering. Occasional surface melting can occur in the dry facies, but water will refreeze very close to the surface, and not much penetration and disturbance of stratigraphy occurs in that process. Nevertheless, the frequency of such melt layers can serve as an important proxy for a warm climate in ice cores.