Symbols used and Values of Constants
Introduction
Grain-size in glacial ice is of interest because it controls many physical properties of ice and because it may record the age and past history of ice. As yet, however, we do not have a clear understanding of grain growth in glacial ice and some observations remain enigmatic. Here we review relevant theories for grain growth in dry, isothermal firn and glacial ice above the depth where significant flow deformation occurs and strong c-axis textures develop (roughly 10‒1000 m depth in central Greenland or Antarctica); in the succeeding paper (Reference Alley, Alley, Perepezko and BentleyAlley and others, 1986; hereafter referred to as paper II) we use the theories developed here to explain some observations regarding grain growth in natural glacial ice and to identify where further data are needed. We do not consider the more difficult problem of grain growth in deforming ice, which has been addressed by Duval (1984) and Reference Duval and LliboutryDuval and Lliboutry (1985), among others. We also do not consider grain growth in low-density firn in the presence of strong temperature gradients, which has been considered by Reference ColbeckColbeck (1982), among others. In the depth region considered here, grain growth is driven only by grain-boundary curvature. Drag forces opposing grain growth in this region are caused only by inert second-phase particles (microparticles), bubbles, dissolved impurities, and the intrinsic drag of grain boundaries.
Many of the theories describing grain growth have been developed for metallurgical systems. There certainly are many atomistic differences between ice and metals; however, phenomenologie similarities are compelling, including Arrhenius-type temperature dependence of diffusivities, enhanced diffusion along grain boundaries, and segregation of dissolved impurities to grain boundaries. Also, theories for grain growth in metals have found wide application in the study of ceramics, which also show atomistic differences but phenomenologic similarities to metals. Thus, although it is incumbent upon us to exercise caution, we feel justified in applying to ice theories for grain growth in metals.
We follow the metallurgical convention of referring to crystals as grains. The two terms are essentially synonymous in the depth region considered here, although geometrically distinct units composed of two or more crystals each have been reported from shallow firn and have been termed grains (Reference GowGow, 1969).
We consider here the case of cold ice only. At temperatures above about ‒10° C, liquid and pseudo-liquid layers appear in natural ice and the behavior of the system changes. Grain growth in wet snow has been discussed by Reference ColbeckColbeck (1979).
Intrinsic Grain Growth
The general relation describing grain-boundary velocity, v, can be written as
where M is the grain-boundary mobility and P is the driving force for grain-boundary migration (Reference HigginsHiggins, 1974). The “driving force” actually has units of pressure and arises from a gradient in free energy, but we follow conventional usage in referring to it as force. The inverse of the mobility is the drag on grain-boundary migration. In the case where there are no extrinsic drag forces (no microparticles, bubbles, or dissolved impurities in the material), M = M i , the intrinsic grain-boundary mobility, and the driving force for migration of an isotropic, spherically curved section of grain boundary with radius of curvature R’ is given by
where γ is the grain-boundary energy or surface tension. (We use the letter P to denote a driving force for grain growth. The numerical subscripts l, 2, etc. identify the driving force for different conditions, e.g. with or without bubbles, with or without microparticles.) The value of the surface tension between adjacent grains, γ, does not depend on the relative orientation of the grains for most orientations, although it is reduced significantly if there is very low mismatch between lattices of adjacent grains or if adjacent grains are in certain restricted orientations with high densities of coincidence sites (Reference VerhoevenVerhoeven, 1975, p. 208-09). We assume here that γ is the same for all surfaces in a sample, which is a good approximation in the absence of strong deformational fabrics (Reference ColbeckColbeck, 1982).
Grains in fully consolidated, single-phase systems cannot be spherical, so Equation (2) must be modified to account for the actual geometry of materials. Many attempts have been made to do this, using both analytical (Reference HillertHillert, 1965; Reference LouatLouat, 1974; Reference MullinsMullins. 1986) and numerical (Reference Anderson, Anderson, Srolovitz, Grest and SahniAnderson and others, 1984, 1985; Reference Srolovitz, Srolovitz, Anderson, Sahni and GrestSrolovitz and others, 1984) techniques. Analytical techniques suffer from the necessity of oversimplifying the complex geometry of real materials; however, analytical models supply clear, easily interpretable results and allow further calculation. Numerical models offer the possibility of eventually modeling the true complexity of grain growth. Thus far, however, numerical models have beenlimitedtotwo-dimensionalorsimplifiedthree-dimensional systems and the results are not fully independent of the numerical scheme used. Also, numerical models do not yield analytical equations that can be used in further calculations.
We will study grain growth using the analytical model developed by Reference HillertHillert (1965). Empirical, numerical, and analytical studies conducted since 1965 have shown that this model does not provide an exact description of grain growth (see Reference Anderson, Anderson, Srolovitz, Grest and SahniAnderson and others, 1984; Reference Srolovitz, Srolovitz, Anderson, Sahni and GrestSrolovitz and others, 1984), but that it does provide an excellent first approximation. The reader should remember, however, that the theory of grain growth is not complete and that some of our conclusions may require refinement in the light of future advances.
The Hillert model is based on the observation that average grain-size in a material increases because small grains shrink and disappear while large grains grow. This implies that at any instant there must be some critical grain radius, R Cr , such that all grains with longer radii are growing and all grains with shorter radii are shrinking. The length of this critical radius increases with time as the average grain-size increases. (The numerical experiments of Reference Srolovitz, Srolovitz, Anderson, Sahni and GrestSrolovitz and others (1984) confirm this in general but indicate that there is some random noise in the growth of large grains.) The simplest relation that gives growth of grains with radius R j > R Cr and shrinkage of grains with R j < R Cr is
where t is time and K’ is a constant; Reference HillertHillert (1965) based his model on this relation.
Observations of grain growth generally show that, after a short transient period, the grain-size distribution of a sample normalized by the mean grain-size of that sample does not vary with time. By using Equation (3) and requiring that the normalized grain-size distribution be steady, Hillert was able to derive the grain-growth law and steady-state grain-size distribution in terms of the critical radius. He was also able to express the critical radius in terms of the average radius, R. Using these results, the intrinsic driving force for grain growth in Equation (2) can be rewritten as
Hillen also noted that if Equation (1) is applied to bulk material, the velocity can be rewritten as
Combining Equations (1) (with M = M i), (4), and (5),
where
Equation (6) can be solved for R to obtain
where R 0 is the average grain radius at t = 0. A grain-growth equation of this form but with a different value of K was derived by Reference SmithSmith (1948) and was applied to glacial ice first by Reference GowGow (1969) and Reference StephensonStephenson (1967).
Virtually all theoretical and experimental treatments of grain growth conclude that grain-size distributions normalized by their means rapidly approach a steady state, and that in this steady state the growth law is at least approximated by the form
where K” and m are constants (Reference Anderson, Anderson, Srolovitz, Grest and SahniAnderson and others, 1984; Reference Srolovitz, Srolovitz, Anderson, Sahni and GrestSrolovitz and others, 1984). Values reported for m typically range from 2 to 3. Because the data for glacial ice are most consistent with m = 2 (Reference GowGow and Williamson, 1976; Reference Duval and LoriusDuval and Lorius, 1980) and because of the theoretical justification for m = 2, we will use Equation (8) to describe grain growth and will follow previous workers in assuming that deviations from m = 2 arise only from extrinsic effects (Reference Grey and HigginsGrey and Higgins, 1973). The derivation of Equations (7) and (8) involved a number of assumptions and approximations; thus, the constant (16/81) is likely to be somewhat inaccurate. We will use (16/81) here but (1/5) is sufficiently accurate for most purposes.
The intrinsic mobility, M i , is often expressed (Reference VerhoevenVerhoeven, 1975, p. 209)
where D b’is the diffusivity of water molecules across grain boundaries and shows an Arrhenius-type temperature dependence, 2δs is the jump length of diffusing molecules (≈ grain-boundary thickness), Ω is the molecular volume, k is Boltzmann’s constant, and T is absolute temperature. The diffusivity of water molecules across grain boundaries is often identified with the diffusivity along grain boundaries. We discuss the validity of this identification in the succeeding paper.
Extrinsic Effects on Grain Growth
If all ice were pure and fully densified, then Equation (8) would provide a complete description of steady-state grain growth driven by grain-boundary curvature. This is not the case, however, and we must consider the effects of extrinsic materials (microparticles, bubbles, and dissolved impurities) on grain growth. To do this, we will first present a general discussion of extrinsic effects, and then consider microparticles, bubbles, and dissolved impurities in turn, following the approach of Reference HigginsHiggins (1974).
Extrinsic materials can interact with grain boundaries. In almost all cases the interaction energy causes the extrinsic materials to be concentrated on the boundaries, either by diffusion to boundaries or by interaction with migrating boundaries. Microparticles and bubbles replace boundary area, whereas impurities dissolve in boundaries.
The effect of extrinsic materials on the migration rate of grain boundaries is determined by the relative magnitudes of the intrinsic driving force for boundary migration and the force required to separate boundaries from extrinsic materials. If the intrinsic driving force exceeds the separation force, then boundaries will migrate away from extrinsic materials (“high-velocity regime”); after separation, the net driving force will be reduced slightly from its intrinsic value by further encounter with and separation from extrinsic materials (Reference HigginsHiggins, 1974). If the intrinsic driving force is not sufficient to cause separation, then boundaries remain with the extrinsic materials and migrate at a velocity determined by the extrinsic materials (“low-velocity regime”). Depending on the mobility of extrinsic materials, boundaries in the low-velocity regime may be fixed or may migrate rapidly; however, the transition to the high-velocity regime always increases migration rates.
In the high-velocity regime, the intrinsic driving force is reduced to the separation force from some extrinsic material, j, when the grain-size reaches the limiting size R j mDuring grain growth toward this limiting size, the net driving force is given by
where 16γ/(81R) is the intrinsic driving force and l6γ/(8R j m) is the separation force. Most evidence indicates that if several extrinsic materials are present in a sample, their separation forces are additive (Reference HigginsHiggins, 1974). In the presence of i different extrinsic materials, Equation (11) becomes
Then, substituting Equation (12) into the general migration relation, Equation(1), we find
where K is again defined by Equation (7) and dR/dt by Equation (5). The solution of this differential equation is
where R 0 is the value of R at t = 0.
Equation (14) is the general description of grain growth influenced by extrinsic materials, just as Equation (8) is the general description of intrinsic grain growth. In many cases, R >> R 0Then if R m >> R (little extrinsic drag) or if R m is proportional to R (which might occur if the volume fraction of some extrinsic material decreases as grains grow), extrinsic grain growth will mimic intrinsic growth in that grain area will increase linearly with time; however, grain growth will be slower than for intrinsic growth if R m is significant but proportional to R. If extrinsic materials are present, R m is of the same magnitude as R, and R m does not vary directly with R, then Equation (14) shows that linear increase of grain area with time will not occur.
The drag forces discussed above are independent of grain-boundary velocity; however, some drag forces are directly proportional to velocity (Reference Grey and HigginsGrey and Higgins, 1973). If the driving force, P, in Equation (1) is reduced by a drag force proportional to velocity, v, then Equation (1) becomes
where M – e 1 is defined as the velocity-independent coefficient of the velocity-dependent drag force M e -1 v, Then algebraic manipulation of Equation (15) shows that
We thus see that M e -1 is the extrinsic grain-boundary drag just at Mi -1 is the intrinsic grain-boundary drag. Velocity-dependent drag forces thus can be said to reduce boundary mobility. Then Equation (14) again applies, if we replace M i in Equation (7) by (M -1 + Me -1)-1 .
Microparticle drag
The effect on grain growth of inert microparticles with incoherent interfaces was first quantified by Zener (in Reference SmithSmith, 1948) for metallic systems. Zener’s theory has been considered further by Reference Ashby and CentamoreAshby and others (1969), Reference HellmanHellman and Hillert (1975), and Reference NesNes and others (1985). We will present Zener’s derivation and then summarize improvements to it.
Microparticles are assumed to be spherical, uniformly distributed, of only one size which is small compared to grains, and to have zero mobility (Reference Anderson, Anderson, Grest and SrolovitzAshby and Centamore, 1968). If surface energy is independent of orientation of a surface, then the surface-tension balance at a triple junction (the intersection of three surfaces in a line) reduces to a balance of force vectors directed along the surfaces perpendicular to the line of intersection, with magnitudes equal to the surface tensions of the respective surfaces (Fig. 1; see Reference VerhoevenVerhoeven (1975, chapter 7.2) for a more complete development). For a particle of material B resting on a boundary between two grains of A, the particle‒grain surface tensions cause equal and opposite forces (Fig. 1). The force of the boundary on the particle, and thus of the particle on the boundary, F’p, then arises wholly from the A A surface tension, ℓ AA or simply ℓ AA F’ p depends on the angle θ shown in Figure 1, according to
Force balance in microparticle drag. A spherical particle of material B rests on a moving boundary between two grains of A; the boundary is symmetric about the axis shown (dashed line) and moving in the direction indicated. Surface energies γ AA and γ AB act as forces along interfaces in the directions indicated.
where r p is the particle radius. F ’ p is maximized for θ = π/4, at which it has the value F p given by
All particles with centers within r p of a boundary on either side will be contacted by the boundary. If the particles are widely spaced (spacing greater than a few particle diameters), the boundary can bend so that each particle exerts its maximum force. If particles occupy volume fraction V p of the material and the number of particles in contact with unit area of boundary is denoted n p, then
The total drag force caused by microparticles per unit area of grain boundary, P p, is the product of the number of particles per unit area, n p, and the drag force per particle, F p, or
The driving force for grain growth in the presence of microparticles only, P 4 , is the intrinsic driving force, P l in Equation (4), reduced by the microparticle drag forceP pThus
If P 4 is greater than zero, then grain boundaries can separate from microparticles and grain growth occurs in the high-velocity regime shown in Figure 2a. When P 4 falls to zero, grain boundaries cannot escape from microparticles and grain growth ceases for inert microparticles (low-velocity regime; Fig.2a).
We can define R p m to be the grain radius at which the intrinsic driving force equals the microparticle drag force. Equation (21) then can be rewritten as
Equation (18)can be altered to allow for particle type and for enhanced bending of boundaries. The study of Reference Ashby and CentamoreAshby and others (1969) indicates that F p should be multiplied by the factor (1 + cos α’) where α’ is the contact angle between grain boundary and microparticle (taken to be π/2 in Zener’s derivation). This contact angle depends on the type of microparticle and the relative orientation of the particle and grains, and varies between 0 and π/2. For widely spaced particles, Reference HellmanHellman and Hillert (1975) showed that a moving boundary will bend to remain in contact with more particles than assumed by Zener. To allow for this, they proposed that the value of n p be multiplied by the factor β, which is given approximately by
where R’ is again the radius of curvature of the section of boundary under consideration. The factor β will vary between 1 and 2 in most cases.
The drag effect of particles is probably overestimated by these calculations, however, as discussed by Reference HellmanHellman and Hillert (1975). First, some particles may exert less than their maximum effect. Also, a moving boundary will experience an added driving force (negative drag force) upon first encountering a particle. In the light of these factors and the uncertainty involved in the constants in P p , we follow Reference HellmanHellman and Hillert (1975) in taking
and taking β to be identically 1. Then the ratio of the particle drag force to the intrinsic driving force is
The constant 4/9 in Equations (24) and (25) is probably somewhat inaccurate, but it is unlikely to underestimate particle drag. We can also be confident that microparticle drag is directly proportional to the volume fraction of particles and inversely proportional to the particle radius. If particles of different radii are present, then the total particle-drag force is the integral of Equation (24) over all particles in the sample.
Finally, Reference HellmanHellman and Hillert (1975) pointed out that, if the volume fraction of microparticles is near or above 10%, many particles will fall on triple junctions where they exert less drag force on grain-boundary migration. The factor β is thus significantly less than 1 at large particle concentrations, although its exact value has not been calculated.
Bubble drag
Bubbles that are small compared to the grain-size cause drag in the same way as microparticles, except that bubbles are mobile in the low-velocity regime. Continuous porosity and large bubbles cannot separate from boundaries to grain interiors, but can cause drag through the action of thermal grooves and intergranular necks.
We first consider bubble-boundary separation, which is the transition from low-velocity to high-velocity migration relative to bubbles. If the chemical potential of material in contact with a bubble varies across the bubble (owing to differentcurvaturesonoppositesidesofthebubble, imposed temperature gradients, or other factors), then material will diffuse across the bubble and the bubble will migrate. Diffusion may occur through the bubble (vapor diffusion, with diffusivity D v), along the bubble surface (surface diffusion, Ds), or through the material around the bubble (lattice diffusion, D ℓ ). The relative rates of vapor : surface : lattice diffusion are given approximately by (Reference ShewmonShewmon, 1964)
Relation between intrinsic grain-boundary velocity and velocity in the presence of extrinsic materials in ice. Solid lines indicate behavior likely to occur in natural systems; dashed lines indicate behavior unlikely in Nature but theoretically possible: dotted lines indicate unstable behavior: and dot‒dash lines show intrinsic behavior, (a) Behavior in the presence of microparticles only. (b) Behavior in the presence of low-mobility porosity (bubbles in ice) only. (c) Behavior in the presence of high-mobility porosity (small necks and thermal grooves in firn) only. Notice that transition from low-velocity to high-velocity migration occurs at higher velocities than shown in the figure and involves an increase in velocity. (d) Behavior in the presence of dissolved impurities only.
where ρ v and ρi are the densities of water molecules in the vapor and ice lattice, respectively, δS is the thickness of the high-diffusivity surface layer, and r is the bubble radius. (More precisely, r is the radius of the sphere with volume equal to the volume of the bubble in question. Bubbles in ice are sufficiently spherical that this distinction is not significant.) If we take a sample from Byrd Station, Antarctica, with a temperature of ‒28 °C and pore radius of about 0.3 mm (Reference GleiterGow, 1968) as a typical example, and we use the values of physical parameters listed at the beginning of this paper, then the ratio in statement (26) becomes and it is evident that we need worry only about vapor diffusion for the ice‒air system. (Compression of bubbles with depth will decrease D v and r, thus decreasing the vapor-diffusion term and increasing the surface-diffusion term in statement (27); however, the two terms would become equal only if bubbles were subjected to a pressure equivalent to an overburden of more than 50 km of ice and if no air dissolved in the ice lattice.)
Separation of a bubble migrating primarily by vapor diffusion from a boundary between two grains has been considered in some detail by Reference Hsueh and EvansHsueh and Evans (1983). Where a grain boundary intersects a bubble, motion of the boundary causes the bubble to change shape so that the radius of curvature is longer for the leading edge than for the trailing edge (Fig. 3). The equilibrium vapor pressure varies directly with the radius of curvature, so a flux of molecules is established from the leading to the trailing edge, causing the bubble to move in the same direction as the boundary. Reference Hsueh and EvansHsueh and Evans (1983) then calculated the maximum steady-state velocity of the bubble, v b m, assuming that it remains nearly spherical and that the mean free path of diffusing molecules is long compared to the bubble dimension; a boundary moving faster than v b m will separate from an associated bubble.
Geometry used in model of bubble drag, after Reference Hsueh and EvansHsueh and others ( 1982),. Bubble and boundary are symmetric about the axis shown (dashed line) and moving in the direction indicated. The equilibrium dihedral angle is ψ(= ψ t + ψℓ) and ψ t = π/2 to maximize bubble velocity. Vapor pressure across the bubble is shown schematically, and P 0 is the equilibrium vapor pressure over a planar surface. Bubbles in ice are more nearly spherical than shown here.
Bubbles in ice are sufficiently spherical for this analysis, but the mean free path of water molecules diffusing through air at atmospheric pressure is about 10-7 m (Reference Hobbs and MasonHobbs and Mason, 1964), whereas bubble radii typically exceed 10-4 m. We thus must modify the expression of Reference Hsueh and EvansHsueh and Evans (1983) to allow for diffusion of water molecules through air. The resulting expression for v b m then is
where D c is the diffusivity of water molecules in air at pore close-off, Δ x is the characteristic diffusion distance, m’ and Ω are the mass and volume of a water molecule, k: is Boltzmann’s constant and T is absolute temperature, r is bubble radius and r c is its value at pore close-off, P 0 is equilibrium vapor pressure over a planar ice surface, ρ i is the density of ice, γs is the ice-air surface tension, and ψ is the equilibrium dihedral angle where a grain boundary and bubble intersect. The expression in curly brackets is the original relation advanced by Reference Hsueh and EvansHsueh and Evans (1983) and allows for the pressure difference arising from curvature difference across the bubble. The additional terms correct for the effect of air on diffusion and for the variation of diffusivity caused by compression of bubbles below the firn-ice transition; r c should be set equal to r when air pressure is atmospheric. The diffusion distance, ∆x, is on the order of r. (A cylindrical pore with volume and base equal to the volume and maximum cross-section of a pore moving at v b m has an altitude of 1.06r.) The equilibrium dihedral angle of a bubble in ice is about ψ = 0.8π. Substituting these values into Equation (28) and simplifying yields
with the restriction that r c = r above pore close-off. Thus, v b m α 1/r 2 for pores at atmospheric pressure, but v b m α r for closed bubbles undergoing compression.
In the high-velocity regime, the effect of bubbles is entirely analogous to the effect of microparticles. If the bubbles are randomly distributed, then the equations derived for the effect of microparticles in the high-velocity regime also describe the effect of bubbles.
The analysis of bubble-boundary separation presented above does not yield the bubble drag in low-velocity migration easily, so we use the older, phenomenological approximation, which has been discussed by Reference ShewmonShewmon (1964), Reference Hsueh, Hsueh, Evans and CobleKingery and Francois (1965), Reference NicholsNichols (1966, 1968), and Reference Aust and RutterBrook (1969), among others. Using the analysis of Reference ShewmonShewmon (1964), the bubble-drag force in the low-velocity regime, P b , is
where N b is the number of bubbles per grain boundary, 2πR2 is the area per grain boundary (the other 2πR 2 of a spherical grain is allocated to adjacent grains), and F b , the drag force per bubble, is given by
The vapor-diffusion term is dominant in Equation (31). Notice that the grain-boundary and bubble velocities are equal in the low-velocity regime, and that the bubble-drag force is velocity-dependent. For bubbles below the firn‒ice transition, D v varies with r 3 and F b is independent of bubble size; however, if D v is constant, then F b varies with r 3 and small bubbles cause little drag.
We can use Equation (31) to rewrite Equation (30) as
where M b, the bubble mobility, is given by
The general behavior of bubble drag is plotted in Figure 2b and 2c. The slope of the low-velocity regime in these figures is M i -1 /(M -1 + M -1) (see Equation (15)). When M b = 0, bubbles do not migrate and the slope is zero; this is the case for microparticles. If M b is small (Fig. 2b), then the low-velocity regime will exhibit little grain growth; however, if M b is large (Fig. 2c), then low-velocity migration may approach the intrinsic case. In paper II (Reference Alley, Alley, Perepezko and BentleyAlley and others, 1986), we show that Figure 2b applies to bubbles in ice and that Figure 2c applies to thermal grooves and small necks in firn. Remember that, in all cases, transition from low-velocity to high-velocity boundary migration (which occurs off-scale in Figure 2c) involves an increase in boundary velocity.
Figure 2b and c are shown double-valued for some velocities because a boundary in the low-velocity regime collects bubbles as it migrates. Thus, a driving force large enough to cause high-velocity migration through uniformly distributed bubbles may be unable to cause separation from the extra bubbles accumulated during low-velocity migration.
Impurity drag
Dissolved impurities are the most important cause of extrinsic drag forces in many materials. Even a few parts per million of some solutes will reduce grain-boundary mobilities by orders of magnitude (Reference Ashby, Ashby, Harper and LewisAust and Rutter, 1959; Rutter and Aust, 1960).
An impurity atom introduced into a regular lattice will cause strain in the lattice because of misfit in size and/or charge. A grain boundary is disordered relative to a lattice, so an impurity atom causes less strain in a grain boundary than in a regular lattice. There is thus an interaction energy, U, between an impurity atom and a grain boundary that causes the impurity to segregate to the grain boundary. For dilute systems such as glacial ice, the concentration of impurities in the boundary, C
b, is related to the concentration in the lattice, 
, by
where U is positive for attraction between boundaries and impurities, and may be on the order of 10kT for strong interactions (Reference Westengen and RyumWestengen and Ryum, 1978). (A boundary can be viewed crudely as a region transitional between liquid and solid. The limiting case of grain-boundary segregation then is seen to be the commonly observed phenomenon of solute segregation to a melt during solidification.)
For a grain boundary to migrate, it must either escape its associated impurity atmosphere by moving faster than the impurities can diffuse (high-velocity regime in Figure 2d) or it must drag the impurities along (low-velocity regime; Fig. 2d). The plot of actual versus intrinsic velocity for impurity drag is double-valued within a narrow range (Fig. 2d; Reference BrookCahn, 1962), for the same reason that the bubble-drag plot is double-valued (see above). The transition velocity for escape from impurities has not been calculated exactly, but the value estimated by Reference BrookCahn (1962) is several orders of magnitude faster than that observed in glacial ice or in most metallurgical experiments. We thus follow Reference HigginsHiggins (1974) and Reference Grey and HigginsGrey and Higgins (1973) in assuming that, during normal grain growth, the low-velocity regime obtains. The impurity drag force, P s , then is estimated as (Reference BrookCahn, 1962)
where 
is the concentration of impurities in the bulk lattice, v is the grain-boundary velocity, and α is an inter-action parameter between a grain boundary and impurities which depends on the concentration and diffusivity profiles of impurities in the immediate vicinity of the boundary. This parameter can be evaluated experimentally even though it cannot be calculated a priori at present.
Because P s is velocity-dependent, its effect is to reduce the grain-boundary mobility or increase the grain-boundary drag. Just as we did in Equation (16), we can use Equation (35) to rewrite Equation (1) as
where M i is the intrinsic grain-boundary mobility. The term ∞
The value of 
is constant in some systems, but may increase with grain growth under certain restricted circumstances. The variation of 
can be discussed most easily in terms of the grain-boundary partition coefficient, k
0, which we define as
where C b is the impurity concentration in the boundary. Thepartition coefficient in solid‒liquid systems maybe concentration-dependent or concentration-independent, depending on the system and the range of concentrations considered (Reference Gross, Gross, Wu, Bryant and McKeeGross and others, 1975[a], [b]); in the lattice/ grain-boundary system for ice, it is sufficiently accurate to consider k 0 to be a constant.
The total concentration of solute, C
T, is fixed in a sample that is large relative to grain-size. We now make a first-order estimate of the distribution of this solute in the sample. Consider a spherical grain or “unit cell” of radius R and impurity concentration C
T in such a sample. This grain is divided into a lattice region with impurity concentration 
and a surface boundary layer of thickness δS, volume 4πR
2δs, and impurity concentration C
i
Conservation of mass requires that
or, substituting for C b from Equation(37) and re-arranging,
We must now consider two cases. In the first, the surface layer is not saturated with solute and its thickness, δs, is fixed by the intrinsic nature of ice. In the second case, the surface layer is saturated with solute and δS increases as grains grow. (We follow Chaterjee and Jellinek (1971) in assuming that the surface layer is saturated when its composition, C b equals the eutectic composition for the water-impurity system, (C e.)
For δS fixed, Equation (39) shows that 
is nearly equal to C
T and independent of R if k
0 is not extremely large. This case probably applies to most natural glacial ice relative to most impurities. For δs constant and k
0
>> 1, 
varies directly with R and is less than C
T.
If the boundary composition equals the eutectic composition, then Equation (37) gives
Substituting this for 
in Equation (39) and solving for δs as a function of grain radius then leads to
and the grain-boundary thickness is directly proportional to grain radius. (This idea was advanced first for ice by Chaterjee and Jellinek (1971).) This sort of behavior is likely only for highly impure materials, such as first-year sea ice.
For some systems, it has been suggested that vacancies supply an additional drag force analogous to the impurity drag force (Reference Fischmeister and GrimvallGleiter, 1979; Reference Estrin and LückeEstrin and Lücke, 1981; Reference Lücke and GottsteinLücke and Gottstein, 1981). Calculations made following these authors indicate that vacancy drag should not have a significant effect on the slow, high-temperature migration of boundaries in ice, but uncertainties in the theories and in our knowledge of some physical quantities are large enough that significant vacancy drag is possible. If present, vacancy drag would reduce boundary mobility but not cause deviation from linear dependence of grain area on time.
Discussion and Conclusions
We have reviewed how microparticles, bubbles, and dissolved impurities can cause drag on grain growth in both high-velocity and low-velocity regimes and have considered both velocity-dependent and velocity-independent drag forces. Grain growth driven by boundary curvature in natural glacial ice typically occurs in the high-velocity regime relative to bubbles, the high-velocity regime relative to microparticles (see paper II), and the low-velocity regime relative to dissolved impurities. The transition from high-velocity migration to low-velocity migration occurs when the driving force for high-velocity grain growth is reduced to zero at grain radii R p mfor microparticles and for bubbles. Impurity drag in the low-velocity regime is proportional to grain-boundary velocity, so impurities reduce the grain-boundary mobility.
Taking these together, grain growth in ice is described by
where
The value of R p m is given by Equations (21) and (22), and R p m is also given by Equations (21) and (22) in the high-velocity regime.
A number of observations should be made regarding Equation (42). The terms l/R
m and αCℓ are identically zero in pure, fully densified ice, and Equation (42) then predicts linear increase with time of the cross-sectional area of grains. This is also predicted if impurities are present but Cℓ is independent of R, and if R
m is proportional to R. Deviation from linear increase of grain area with time occurs if Cℓ depends on R and α
is significant or if R
m is not linearly dependent on R and 1/R
m is significant.
Finally, we should re-emphasize that our analysis is restricted to cold ice. If the temperature of the ice-impurity system rises above the melting point of the impure grain boundaries, then liquid will form along the boundaries. This will increase diffusivities and allow Ostwald ripening of grains to occur through the liquid matrix (Reference Fischmeister and GrimvallFischmeister and Grimvall, 1973). Under such conditions, an increase in impurity content will increase the amount of liquid present and increase the rate of grain growth; thus, Equation (42) will not apply in high-temperature, impure systems.
Although most of the theories presented here are still under active development, the general ideas seem well established. We are now in a position to assess qualitatively, and often quantitatively, how different factors affect rates of grain growth in natural glacial ice. We do so in the following paper.
Acknowledgements
This work was funded in part by the U.S. National Science Foundation under grant DPP-8315777. We thank J.F. Bolzan, R.H. Dott, jr, P. Duval, A.J. Gow, M. Hillert, J.W. Valley, H.F. Wang, and I.M. Whillans for reading early versions of this manuscript, E. Mosley-Thompson for access to unpublished data, and A.N. Mares and S.H. Smith for manuscript preparation. This is contribution No. 432 of the Geophysical and Polar Research Center, University of Wisconsin‒Madison.
