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A Review of Sinteringin Seasonal SnowSamuel C. Colbeck December 1997

Abstract: Strength and electrical pathways develop insnow as bonds grow among grains. Strong ice-to-icebonds form in wet snow at low liquid contents but notin highly saturated wet snow. In freely draining wetsnow, grain clusters form, and these require a certainconfiguration among the three phases of water. Thisdepends somewhat on the number of grains in thecluster, but always leads to bonding. In dry snow,bonds form more slowly, but considerable strength

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Cover: Two sets of sintered particles: the upper photo is of sintered glass beads, and the lower is of sintered snowgrains, showing a grain boundary and grain boundary groove.

can develop as long as rounded grains develop. Therate of bond growth is probably controlled by thetemperature gradient, because both grains and bondsare observed to grow very slowly in dry snow in theabsence of a temperature gradient. The basic shapeof the bonds is dictated by the geometrical require-ments of grain-boundary grooves and is not a simpleconcave neck. In dry snow, this shape, and possiblythe processes, have been misunderstood.

CRREL Report 97-10

A Review of Sinteringin Seasonal SnowSamuel C. Colbeck December 1997

Prepared for

OFFICE OF THE CHIEF OF ENGINEERS

Approved for public release; distribution is unlimited.

US Army Corpsof EngineersCold Regions Research &Engineering Laboratory

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US Army Corpsof Engineers®

Cold Regions Research &Engineering Laboratory

PREFACE

This report was prepared by Dr. Samuel C. Colbeck, Senior Research Scientist, U.S.Army Cold Regions Research and Engineering Laboratory, Hanover, New Hamp-shire.

This work was supported at CRREL by DA Project 4A161102AT24, Work Package127, Snow Properties and Processes, Work Unit SC-S01, Physical Properties of Snow Covers.The author benefited from the review of Dr. Jerome Johnson. The author thanks Dr.Edward Arons for pointing out the Zhang and Schneibel reference.

The contents of this report are not to be used for advertising or promotionalpurposes. Citation of brand names does not constitute an official endorsement orapproval of the use of such commercial products.

ii

CONTENTSPage

Preface ................................................................................................................... iiIntroduction .......................................................................................................... 1Wet snow .............................................................................................................. 2

Grain clusters ................................................................................................... 2Slush .................................................................................................................. 3

Dry snow ............................................................................................................... 4Rounded grains ............................................................................................... 4Faceted grains .................................................................................................. 9

Discussion and conclusions ............................................................................... 9Literature cited ..................................................................................................... 10Abstract ................................................................................................................. 13

ILLUSTRATIONS

Figure1. Cluster of ice grains in wet snow at a low liquid content .................... 22. Cross section of a three-grain cluster in wet snow................................ 33. Slush, which consists of well-rounded grains 0.5 to 1 mm in size

immersed in water ................................................................................ 3 4. Geometry long assumed to describe rounded grains with necks

in dry snow ............................................................................................ 35. Depth hoar, the extreme case of faceted crystals growing in dry

snow at high growth rates due to large temperature gradients ..... 46. Rounded grains, which grow at low growth rates in snow ................ 47. Snow grains stored at –24°F for several years showing distinct

grain-boundary grooves at the bond .................................................. 68. Fresh, dry snow with newly formed bonds showing a grain-

boundary with a grain boundary groove instead of reversecurvature ................................................................................................ 6

9. Equilibrium form of two grains of ice consisting of singlecrystals where the grain-boundary groove angle is 145° ................ 7

10. Two idealized particles shown at an early and the final stagesof sintering ............................................................................................. 8

11. End grain pointing downwards into the upward-moving streamof vapor ................................................................................................... 9

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INTRODUCTION

Snow on the ground consists of three basiccomponents: air, ice grains, and liquid water whenthe snow is wet. Ice bonds form between the grainsin most, but not all, types of snow. Much atten-tion has been paid to the size, shape, and growthof the grains, but the bonds are of equal impor-tance to the grains themselves, and relatively littleattention has been paid to them. This probablyoccurs for two reasons. First, the bonds are smaller,partly hidden by the grains, and much harder tosee with a hand lens. Second, the physics of theirgrowth is not well understood, partly becausetheir basic geometry is usually misunderstood. Infact, the literature describes sintering in snow asif the snow were a noncrystalline material withno imposed temperature gradient. The widelyused approach to sintering in dry snow could beapplied to glass beads held in an adiabatic cell,but not to ice grains in a seasonal snow cover.

When snow pits are dug to look for weaklayers in the snow profile, the grains are gener-ally examined rather than the bonds. For ex-ample, depth hoar is known to be weak due topoor bonding, so the existence of these highlyfaceted crystals is taken as evidence for the pres-ence of a weak layer. In the International Classifi-cation System for Seasonal Snow on the Ground(Colbeck et al. 1990), there are photographs of thegrains, but information about the bonds is onlyinferred. The size and shape of the grains arerecorded in snow-pit logs, even when it is onlythe bonding that is of concern. Instead of directexaminations of the bonds, stereological methodshave sometimes been used to infer informationabout the bonds, including their size and an as-sumed shape (e.g., Keeler 1969, Alley et al. 1982).Direct information about the degree of bondingcomes from strength tests (e.g., Keeler 1969, Gow1975), but these tests do not provide information

about the geometry of the bonds nor about theprocesses that form them. Models describing thebehavior of snow are often based on an assumedgeometry or observations of the bonds from sur-face sections (e.g., Brown and Edens 1991.)

The strength of snow is not the only propertythat is controlled by the size of the bonds. Someproperties depend primarily on the grains, suchas the optical properties, where scattering andabsorption can depend on grain size and shape.However, some properties are most sensitive tothe narrow constrictions between grains wherestresses are larger but heat and electrical flowpaths are reduced. Thus the size, shape, and fre-quency of grain bonds greatly affects many of themost important properties of snow. Sintering isthe process by which these bonds form and thestudy of their size, shape, and number density.

To study the bonds in snow, it must be recog-nized immediately that wet snow and dry snoware basically two different materials. Whilechanges do occur more rapidly in wet snow be-cause of the higher temperature, the fundamentaldifference is that the introduction of a third phase,liquid water, causes major reconfigurations of bothgrains and bonds. The geometries of wet and drysnow are markedly different, and their propertiesdiffer for several important reasons. Wet snow isactive thermodynamically because of the hightemperature and presence of the liquid phase, butvapor flow due to a macroscopic temperature gra-dient can only occur in dry snow.

Within each of these categories there are alsotwo important divisions: wet snow at low andhigh liquid contents and dry snow at low andhigh growth rates. Wet snow is cohesionless andslushy at high liquid contents, but well-bonded atlow liquid contents. Rapidly growing grains indry snow lack bonding, whereas strong bondsform when the grains grow slowly. To under-stand the formation of bonds in snow, it is first

A Review of Sintering in Seasonal Snow

SAMUEL C. COLBECK

necessary to understand the growth of the grainsbetween which the bonds develop. Thus, for eachcategory of snow, the growth of the grains is re-viewed first to put the growth of the bonds incontext. More details of the growth of grains insnow are given in an earlier review of the physicsof snow metamorphism (Colbeck 1987a) and amore recent review from a more practical point ofview (Colbeck, in press).

WET SNOW

Wet snow contains an observable quantity ofliquid water in one of two basic modes of satura-tion. First, at low liquid contents where air iscontinuous throughout the pore space, the liquidis held by “grain clusters” in a mode of liquidsaturation known as the “pendular regime.” Sec-ond, at high liquid contents where the liquid iscontinuous throughout the pore space, the airoccurs only in isolated bubbles trapped in thepores. This is “slush,” where the liquid is in the“funicular regime” of saturation. The sintering ofthese two modes is markedly different becausegrain clusters develop strength quickly whereasslush is cohesionless. The first mode occurs whenthe snow is free to drain, while the second modeoccurs in snow overlying a surface that impedeswater flow.

Grain clustersAt low liquid contents, all of the liquid is held

by capillarity in the crevices, veins, and junctions

of the clusters (see Fig. 1), and the remaining porespace is filled with air. The basic unit of a clusteris the well-rounded single crystal of ice: the grain,or the minimum observable unit. These singlecrystals join in groups of two or more and aretightly bonded by ice-to-ice contacts, not by capil-larity as is often supposed; the ice-to-ice grainboundaries are depicted in Figure 2 for a three-grain cluster. Their large size gives the snow con-siderable strength. The liquid-filled veins form atthe junctions of three crystals, and more water isheld at the junctions of four veins. The geometryof these veins and junctions can be most easilyvisualized by examining the lines joining soapbubbles that have had time to grow to a size ofabout 10 mm.

While the growth of individual grains at lowliquid contents is not as rapid as grain growth inslush, the clusters do form rapidly by the collect-ing together of existing grains into clusters. Fullydeveloped clusters arise from drained slush inabout 24 hours, which is remarkably fast com-pared with the growth of particles by any otherprocess in snow. This happens in part because theclusters are at the melting temperature and thustransport through the liquid phase is possible. Asa result, vapor diffusion is probably not the rate-limiting process that it is in grain growth in drysnow. Of course, these clusters are multicrystal-line collections, so their growth processes aredifferent from the processes that lead to the growthof grains of single crystals.

Clusters form in this manner because this con-figuration of the vapor/ice, vapor/water, and ice/

water interfaces minimizes the to-tal surface free energy (Colbeck1979a). Neighboring clusters arewell bonded to each other, withice-to-ice bonds forming betweentwo grains, one ice grain from eachcluster. These ice-to-ice bonds arestrong enough to give this form ofsnow some considerable strength,both within the clusters and withinthe snow cover as a whole. In factthe strength is well-known to varysignificantly with the liquid-water content of snow (Kinosita1963, Colbeck 1979b).

Another form of well-bondedsnow is also common, one thatforms a transition between the cat-egories of wet and dry snow.Amorphous, multicrystalline par-

2

Figure 1. Cluster of ice grains in wet snow at a low liquid content. Theindividual ice grains are single crystals, usually 0.5 to 1.0 mm in size.

ticles arise from melt–freeze cycles, simply by thefreezing together of individual grains. When thishappens, it destroys the granular geometry of thegrain cluster but probably increases the strengthof the snow cover. These particles are also ice-bonded to their neighbors.

SlushAt higher liquid contents, the air is no longer

continuous throughout the pore space, but is lim-ited to isolated air bubbles trapped by constric-tions in the pores. Since these bubbles occupy thelargest part of the pore space, the volumetric aircontent can still be higher than the volumetricliquid content, but only the liquid phase is mo-bile. In fact, the permeability to the liquid in-creases with liquid content, and slush is highlycapable of conducting liquid water. In addition,

since the ice grains are surrounded by water (seeFig. 3), grain growth in slush is very rapid as firstmeasured by Wakahama (1968) and later ex-plained by Colbeck (1987b). Slush lacks inter-granular bonding as do rapidly growing grainsin dry snow, but for very different reasons. Inslush, the bonds are unstable because, whenstressed, they melt away by pressure melting,whereas with clusters, the ice-to-ice bonds arestable against pressure melting even though thesnow contains liquid water (Colbeck 1979a). Thisis a very fundamental difference between thependular and funicular regimes of water con-tents since it leads directly to high strength atlow liquid contents and low strength at highliquid contents. This fundamental difference inthe thermodynamics is due to the basic differ-ences in the geometry.

IceGrain

IceGrain

IceGrain

GrainBoundary

Water

Figure 2. Cross section of a three-grain clus-ter in wet snow. Liquid is held in the crevicesbetween two grains, the veins among threegrains, and the junctions that join four veins.Air fills the remaining pore space.

Figure 3. Slush, which consists of well-roundedgrains 0.5 to 1 mm in size immersed in water.These do not bond, and therefore slush lackscohesion.

Figure 4. Geometry long assumed to de-scribe rounded grains with necks in dry snow.

Actually, these grains are glass beads: ice grainsform a neck with a grain-boundary groove and, given

enough time, an equilibrium grain-boundary grooveangle of about 145°.

3

growth of snow grains, although Sturm (1991)has observed the various stages of the growth ofdepth hoar.

Rounded grainsRounded grains grow in dry snow at high den-

sities and/or low temperature gradients in thesnow cover. Figure 6 shows a bond between tworounded grains that are the “equilibrium form” ofthe ice crystal (Colbeck 1983a). This photographwas taken to show the grains, not the bond, andthus the bond is not fully visible in the photo-graph. However, the grain-boundary groove isvisible along with a feature that I have observedin all of my laboratory observations of the growthof bonds between grains: one of the ice grainsreshapes itself to form an elongated neck at thecontact but the other grain retains a much moresphere-like shape adjacent to the contact. Thus, adistinct dissymmetry tends to develop at thecontact, probably due to the different crystallo-graphic orientations of the two crystals.

The growth process for all types of snow grainsin dry snow was described by Yosida et al. (1955)as “the hand-to-hand delivery of water vapor”because water vapor migrates through the snowcover from the warmer part to the colder part,which is usually from the lower layers to theupper. It does so by the step-by-step conveyanceof water from each ice grain to its coldest neigh-bor. This process enhances the rate of vapor diffu-sion for two reasons. First, the vapor diffusesacross the pores only, so the flow path is short-

DRY SNOW

The most studied case of sintering in snow isthat of well-rounded grains in dry snow wherethe grains grow slowly while they build inter-granular bonds (e.g., Kingery 1960, Kuroiwa 1962,Hobbs and Mason 1964). The bonds have longbeen described as necks with a concave geometry(see Fig. 4) where the growth of the bonds isdriven by vapor pressure differences over the con-vex (grain) and concave (bond) surfaces. Althoughthis assumption about the geometry is supportedby few observations, it is common to assume thisgeometry in the sintering of many materials (e.g.,Swinkels and Ashby 1981, Moya et al. 1987, Lenel1992). Figure 4 shows that it is the appropriategeometry for a noncrystalline material, but thisgeometry would not seem possible for a crystal-line material because, at least at slow rates ofgrowth, the equilibrium form of the crystal mustevolve, and that requires the presence of a grain-boundary groove at the crystalline boundary.

The rapid growth of faceted grains, depth hoarbeing the extreme form (see Fig. 5), is of equalinterest to the case of the slower-growing roundedgrains. When rapid growth occurs, the roundedgrains are consumed, leaving poorly bonded, fac-eted grains that do not sinter rapidly because oftheir large size and because their rapid growthleaves little time for sintering. They do sinter oncethe rapid growth slows, but they sinter slowlybecause of their large size. I know of no directobservations of the bonds formed during rapid

Figure 6. Rounded grains, which grow at lowgrowth rates in snow. They sinter, giving thesnow a slab strength. This shows a newly formedbond with a sharp grain boundary groove angle.The neck on one grain only is not uncommon.

Figure 5. Depth hoar, the extreme caseof faceted crystals growing in dry snowat high growth rates due to large tem-perature gradients. These are poorlysintered because their formation con-sumes the well-sintered, roundedgrains, and large grains sinter slowly.

4

ened. Second, the temperature difference acrossthe pores is increased, since the thermal conduc-tivity is much higher for ice than for air. For thesereasons the coefficient of diffusion of water vaporin snow is much higher than it is in air. This grain-to-grain movement of water vapor was describedby Colbeck (1983b) using temperature differencesamong the grains, and this theory was furtherdeveloped by Gubler (1985) to include the dy-namics of a population of grains.

A “slab” strength develops simultaneously withthe growth of rounded grains, especially whendeposition of the snow layer is accompanied byhigh winds. Ideas from other materials have beenapplied to describe the formation of the bondsthat provide that strength. It has long been as-sumed that the bonds, or necks, have a reverse orconcave geometry that causes the migration ofwater molecules to the neck by sublimation, sur-face diffusion, surface flow, volume diffusion,plastic flow, and/or grain boundary diffusion.Kuczynski (1949) pioneered the classical approachto the physics of sintering. His basic idea was thatdifferent mechanisms occurred at different charac-teristic rates and that the dominant mechanismcould be determined from the rate. This result isoften summarized by the equation

xR

f TR

n

m

=

( )(1)

where x is the radius of the neck, R is the radius ofthe grain, f(T) is a function of temperature (T),and t is time. The constants, m and n, assumedifferent values for different processes and aredetermined from the appropriate experimentalobservations of sintering.

This approach was promoted and extended byKingery (1960), who first applied it to ice. Kingeryconcluded that the welding together of pieces ofice at subfreezing temperatures was due to sur-face diffusion, an idea that has not received widesupport. By direct observation, he found that therate of neck growth, when normalized to grainsize, was proportional to t1/6.9. The rate of sinter-ing was much more rapid for smaller grains, inaccordance with eq 1. Unfortunately, his obser-vations do not allow a close examination of thegeometry of the neck, which could provide someinsight into the processes. Thus, his conclusionabout the role of surface diffusion was based onthe time-dependence of the experimental resultsand the fact that theory shows that sublimationcould not happen fast enough to account for the

observed rate. However, the coefficient of surfacediffusion required was very high and the tem-perature dependence was large, but this might bedue to changing surface structure as the meltingtemperature is approached.

It is tempting to account for rapid surface dif-fusion by assuming a liquid-like or a liquid layeron the surface of ice, at least at higher tempera-tures (Dash et al. 1995, Petrenko 1994), but thisidea must still be put in a convincing, quantita-tive form. Gubler (1982) has laid out this problembut left the following concerns: First, he assumedthe usual reverse curvature for the geometry ofthe bond, but there is very little evidence that thisis the correct geometry. Second, the viscosity andthickness of the surface layer are critical, but con-troversial. Much remains to be learned about thesurface of ice before this can be resolved. Forexample, the rate is very sensitive to the humidity(Hosler et al. 1957), which is not surprising sincethe structure of an ice surface can change visiblywith changes in humidity.

Kuroiwa (1962) examined the grain bonds us-ing Kuczynski’s (1949) basic ideas about the pro-cesses. He concluded that volume diffusion wasthe dominant process when air filled the porespace and found that the rate of sintering wasmuch lower when the air was displaced by kero-sene. It is disappointing that Kuroiwa missed anapparent conclusion from his own figures, even ifthey were made from thin sections: they showgrain-boundary grooves, not the concave geom-etry normally assumed (Fig. 4). Even most of themore recent thinking about sintering still as-sumes this concave geometry (Swinkels andAshby 1981), but sintering with grain-boundarygrooves has been at least partly described (Zhangand Schneibel 1995). It is important to realize that,since a grain-boundary groove is present, replac-ing air with another fluid will change the dihe-dral angle at the groove. It could also change thesurface structure of ice and its surface energy.Thus, we should expect the rate of sintering tochange, even if the dominant process stays thesame.

Hobbs and Mason (1964) rejected the approachof the metallurgists and ceramists, stating that itcould not be applied to ice. Instead, they believedthat sublimation, transfer through the vapor phase,had to be the dominant mechanism, and Ramseierand Keeler (1967) supported this conclusion withstrength tests of snow with either air or oil in thepore space. Hobbs and Mason did leave open thequestion of the mechanism(s) during the initial

5

period of the formation of a bond when surfacediffusion, or even pressure melting–regelation,could occur. In fact, it makes a lot of sense to thinkabout different mechanisms dominating duringdifferent phases of bond growth, just as Alley etal. (1982) and Wilkinson (1988) did for densifica-tion. Ideas based on a concave curvature (Fig. 4)could dominate during the early phases whensintering is most rapid, and microphotographsare not yet available to disprove the notion of asimple concave bond. Figure 6 shows that onegrain can be purely convex while the other hasmixed curvature, and thus different processes mayoperate on adjacent grains, or even on differentparts of one grain. Figure 7 shows the nature ofgrain bonds in snow stored at a low temperaturefor several years: these bonds are crystal bound-aries with grain-boundary grooves and the ap-propriate grain-boundary groove angle for ice andwater vapor, about 145° ±2° (Ketcham and Hobbs1969). Because the grains are not packed in a regu-lar manner, part of the grain surface can be still beconcave, even at this late stage of sintering.

The formation of grain-boundary grooves isnot limited to old snow that has had a long time toreach its equilibrium form. A fresh snow bond ofa similar nature is shown in Figure 8. In freshsnow, the grain-boundary groove angle appearsto be much less than the equilibrium value of 145°because the bonds have just formed. The bondgrows rapidly at first as the stress imbalance atthe junction is reduced, probably with an expo-nential decay. Kuroiwa’s (1962) photographs andmy own microscopic observations suggest to methat the dihedral angle increases with time ofsintering. However, his photographs can be mis-

leading in this regard since they where made fromsections and not from the whole grains. As dis-cussed later, it is important to know if the dihe-dral angle of about 145° is always maintainedthroughout sintering, or if the angle is small ini-tially and then increases as sintering proceeds.This question should be answered with more mi-croscopic observations, since its answer will de-termine how we think about the processes andhow they are modeled. For example, if an angle ofabout 145° is always maintained, there will neces-sarily be concave curvature adjacent to the grain-boundary groove, at least during the initial stageof sintering. This will affect all of the processes,regardless of what they are.

Figure 7. Snow grains stored at –24°F for several years showing distinct grain-boundary grooves at thebonds.

Figure 8. Fresh, dry snow with newly formed bondsshowing a grain boundary with a grain-boundarygroove instead of reverse curvature.

6

Keeler (1969) found that the rate of bond growthwas much greater in natural snow than predictedby sintering theory. Although he attributed thisto stresses in the snow, the effect of stress on thethermodynamics is very small and thus it seemslikely to me that this is due to the presence of amacroscopic temperature gradient. These are al-ways imposed on the snow cover by environmen-tal factors, and thus water vapor is driven throughthe snow. This enhanced vapor movement shouldbe much greater than vapor movement due todifferences in curvature or stress. A much greaterrate of vapor flow should cause faster bondgrowth, just as it causes faster grain growth(Colbeck 1983b). While the bonds must move to-ward their equilibrium shape, the rate at whichthe bonds grow is probably controlled by vapor-density gradients caused by macroscopic tempera-ture gradients, not by microscopic curvature orstress differences. Without an imposed tempera-ture gradient, depth hoar could not grow at all insnow, and rounded crystals would grow muchmore slowly than observed (de Quervain 1958).Given that the rate-limiting factor in mass flow insnow is the vapor density gradient, which is con-trolled by the temperature profile, the classicaltheory of sintering may have little to do with therate of formation of bonds in dry snow. This prob-ably explains why the rate of sintering has oftenbeen found to occur faster than is described bymodels or laboratory experiments.

The basic geometries of rounded grains andtheir bonds are controlled by phase equilibrium.Thus, the shape of the bond, but not its rate ofgrowth, can only be understood by examinationof the equilibrium condition at the bond. Ice grainsin dry snow generally consist of single crystals, so

the bonds must be simple grain boundaries withgrain-boundary grooves. Kuroiwa’s (1962) Figure9 shows bonding between polycrystalline grains,but the actual bond appears to connect only twoof the crystals, one within each grain.

When two ice grains consisting of a single crys-tal each are joined by a bond, the equilibriumform of the arrangement should be as shown inFigure 9. For the old grains shown in Figure 7b,the grain-boundary groove angles are about 138°and 148°, close to the equilibrium value observedby Ketcham and Hobbs (1969). For the fresh snowshown in Figure 8, the grain-boundary grooveangle appears to be much smaller because thebond has not yet had sufficient time to approachthe equilibrium condition. However, a closer ex-amination of this bond could prove that the actualangle is closer to the equilibrium value.

Zhang and Schneibel (1995) described the sin-tering of grains joined by grain-boundary grooves.They explained the imbalance of forces that oc-curs in the grain-boundary groove before the equi-librium condition has been established; theyassumed that the dihedral angle was fixedthroughout the process. Sintering in their modelwas limited to grain boundary and surface diffu-sion and expressing their results in terms of thediffusivity ratio of these two processes. Unfortu-nately, the values for the diffusivities are stilluncertain for ice, so it is not possible to calculatemeaningful rates of bond growth based on theseprocesses. Furthermore, in seasonal snow coversit seems likely that the shape is determined by therequirement for equilibrium, but the rate is deter-mined by vapor flux due to the macroscopic tem-perature gradient. In spite of these limitations,the ideas expressed by Zhang and Schneibel (1995)are clearly applicable to sintering in snow and, iffact, are vital to understanding what is actuallyobserved in snow. For this reason their work onsurface and grain-boundary diffusion is summa-rized here.

Surface diffusion is proportional to the gradi-ent of the chemical potential, or the gradient ofcurvature along the surface. As the resulting fluxreconfigures the surface, the growth or decay ofthe surface is given by the gradient of the flux, so

∂∂

∂∂

rt

BK

sa =

2

2 (2)

where ra is the normal vector to the surface, K isthe surface curvature, which is positive for con-vex surfaces, s is length along the surface, and

Figure 9. Equilibriumform of two grainsof ice consistingof single crystalswhere the grain-boundary grooveangle is 145°. Ac-tual bonds maynot have had timeto achieve thisconfiguration.

7

B

DT

=δ γs s

kΩ

(3)

where δ s = the surface diffusive width,Ds = the coefficient of surface diffusion

γ = the surface free energy for the solid–vapor surface

Ω = the atomic volumek = Boltzmann’s constant.

The surface flux is then taken as

J

DT

Kss

s sk

= δ γ ∂∂

. (4)

Zhang and Schneibel (1995) assumed a straightgrain boundary from which parallel layers of mat-ter are removed during sintering. These moleculesdiffuse away by flux along the grain boundary, aflux that arises from the stress gradient along theboundary. The flux is given by

JD

T

d

dybb bk

= δ γ σ(5)

where the subscript b refers to the grain bound-ary, σ is the normal stress acting across the grainboundary, and y is the radial distance across theboundary. The normal stress at the base of thegrain-boundary groove due to the pull of thevapor–ice surfaces is –γ K0, where K0 is the curva-

ture at the base of the grain-boundary groove.The stress along the grain boundary is easily de-rived with that end condition and requires thatKY equal sin(A/2) at the base of the groove whenequilibrium is established and the flux disappears.Thus, the theory requires that equilibrium canonly be achieved when the curvature is the sameeverywhere. This does ignore crystallographic dif-ferences between the two grains, differences thatclearly arise when two ice grains are observed tosinter, but for now we will have to accept thislimitation.

When material is removed from the grainboundary by grain-boundary diffusion and movesonto the free surfaces of the two grains by surfacediffusion, the fluxes are in balance when, at thebase of the groove,

J Jb s= 2 . (6)

Zhang and Schneibel (1995) used this and twoother conditions to describe the changing geom-etry as two particles grow in size and move closer.They do so until the two grains achieve the finalcondition, capped spheres, as shown in Figure 10.The two other conditions are that the dihedralangle remain constant and that the base of thegroove move outward, as shown in Figure 10.

In their numerical calculations they used theratio of the grain boundary and surface diffusiv-ities, Γ, where

Figure 10. Two idealized particlesshown at an early and the finalstages of sintering. The dihedralangle is maintained at a constantvalue throughout sintering in thisexample. However, it appears thatthe dihedral angle changes duringsintering of ice grains. (After Zhangand Schneibel 1995.)

EarlyShage

FinalShape

Dihedral Angle

EarlyShape

8

Γ =

DD

b b

s s

δδ . (7)

They found, for example, for a dihedral angle of150°, that the time to reach 50% of the final neckshape decreased as log (Γ) increased. Thus, for eachorder-of-magnitude increase in grain-boundarydiffusivity, there is an increase of a factor of twoto five in the rate of sintering due to the removalof material from the grain boundary and deposi-tion of that material on the free surfaces.

While this theory ignores sublimation, crystal-lographic differences, and the role of the macro-scopic temperature gradient in determining therate of sintering, at least it includes the role of thedihedral angle and grain-boundary diffusion. Per-haps its greatest limitations are the assumptionsof a constant dihedral angle and an adiabatic en-vironment.

Faceted grainsFaceted grains grow rapidly due to high tem-

perature gradients and low densities. They havelong been of interest because they are associatedwith low strength and avalanche release. In 1973,de Quervain proposed that grains situated at spe-cific sites would preferentially grow more rap-idly. This includes “end grains” that are notconnected at their lower end and thus point down-ward into the upward-moving stream of vaporbeing driven by the temperature gradient (Fig.11). Without being connected at their lower ends,they grow rapidly, especially if there is a largedistance between the end grain and the grain be-low it. Furthermore, because they are not con-nected at the bottom, they fail to form a bondthere. This is one reason why, during a majorrecrystallization, where the rounded grains arereplaced by faceted grains, the bond density ofthe new grains is low compared with that of the

Cold

Warm

Figure 11. End grainpointing downwardsinto the upward-movingstream of vapor as sug-gested by de Quervain(1973).

old grains. Direct observations of bond growthrate and geometry are needed during their growth.

DISCUSSION AND CONCLUSIONS

There is a fundamental difference between wetand dry snow since liquid water causes majorreconfigurations of both grains and bonds. Withinthe wet and dry snow categories there are alsotwo important divisions: wet snow at low andhigh liquid contents and dry snow at low andhigh growth rates. Wet snow is cohesionless andslushy at high liquid contents because the grainboundaries are unstable against pressure melt-ing. However, wet snow is well-bonded at lowliquid contents where ice-bonded clusters form.A transitional form of snow, melt–freeze grains,can be either wet or dry. These amorphous,multicrystalline particles arise from melt–freezecycles. They are solid within and well-bonded totheir neighbors.

Rapidly growing grains in dry snow lack bond-ing because they consume the existing grains, theyare large, and they grow rapidly. However, strongbonds form among rounded grains, and they growslowly. Their growth processes and geometry haveprobably been misunderstood, even though thisis the most studied case of sintering in snow. Thebonds are usually described as necks with a con-cave geometry as in most studies of sintering ofother materials. However, this geometry wouldnot seem possible for a crystalline material be-cause the equilibrium form of the crystal requiresthe presence of a grain-boundary groove at thecrystalline boundary.

In the past it has been assumed that the reversegeometry causes the migration of water moleculesto the neck by one of many possible processes; thedominant mechanisms were identified by the ob-served dependence on time. Kingery (1960) firstapplied this approach to ice; he concluded thatthe bonding was due to surface diffusion. How-ever, the coefficient of surface diffusion requiredwas very high, and although this might be ex-plained by a highly mobile surface layer, this idearemains to be convincingly demonstrated.

Kuroiwa (1962) concluded that volume diffu-sion was the dominant process, but Hobbs andMason (1964) believed that sublimation, transferthrough the vapor phase, had to be the dominantmechanism. The vapor transfer mechanism hassince received wide support and was supportedby the strength tests of Ramseier and Keeler (1967).

9

In fact, various mechanisms may dominate underdifferent conditions or even during different stagesof sintering under given conditions. Mechanismsbased on the concave curvature could dominateduring the early phases when sintering is mostrapid, because microphotographs are not yetavailable to disprove the notion of a concavebond for all stages of sintering. However, theclassical concepts based on concave curvature can-not dominate after the grain-boundary groove isestablished, which could be at the instant thatcontact is made. The grain-boundary groove anglein the bond at equilibrium is about 145° but, infresh snow the angle appears to be much smallerbecause the bonds have just formed. The bondgrows rapidly at first due to the stress imbalanceat the junction, but the growth rate decreasesrapidly with time.

Keeler (1969) found a higher rate of bondgrowth in natural snow than expected from labo-ratory experiments. This is almost certainly dueto the temperature gradients that occur in naturebut were absent in the laboratory experiments.These gradients cause vapor movement at amuch greater rate than could occur just due todifferences in curvature or stress. Since the rate-limiting factor in mass flow in snow is probablythe vapor density gradient, which is controlled bythe temperature profile, the classical theory ofsintering may have little to do with the rate offormation of bonds in dry snow.

Zhang and Schneibel (1995) described the sin-tering of grains joined by grain-boundary groovesbased on the imbalance of forces that occurs inthe grain-boundary groove before the equilib-rium condition has been established. If new bondsassume a very small angle between the ice grains,this would cause a large grain-boundary drag,which would lead to continuous reconfigurationuntil the equilibrium condition is established.Zhang and Schneibel (1995) modeled grain-bound-ary growth due to grain boundary and surfacediffusion, expressing their results in terms of thediffusivity ratio of these two processes. Unfortu-nately, the values for the diffusivities are still un-certain for ice, so it is not possible to calculatemeaningful rates of bond growth based on theseprocesses. Furthermore, in seasonal snow coversit seems likely that the shape is determined by therequirement for equilibrium, but the rate is deter-mined by vapor flux due to the macroscopic tem-perature gradient.

LITERATURE CITED

Alley, R.B., J.F. Bolzan, and I.W. Whillans (1982)Polar firn densification and grain growth. Annalsof Glaciology, 3: 7–11.Brown, R.L., and M.Q. Edens (1991) On the rela-tionship between neck length and bond radiusduring compression of snow. Journal of Glaciology,37: 203–208.Colbeck, S.C. (1979a) Grain clusters in wet snow.Journal of Colloid and Interface Science, 72, 371–384.Colbeck, S.C. (1979b) Sintering and compactionof wet snow. Philosophical Magazine, 39: 13–32.Colbeck, S.C. (1983a) Ice crystal morphology andgrowth rates at low supersaturations and hightemperatures. Journal of Applied Physics, 54: 2677–2682.Colbeck, S.C. (1983b) Theory of metamorphismof dry snow. Journal of Geophysical Research, 88:5475–5482.Colbeck, S.C. (1987a) A review of the metamor-phism and classification of seasonal snow covercrystals. In Avalanche Formation, Movement and Ef-fects (S.C. Colbeck, Ed.). International Associationof Hydrological Science, vol. 162, p. 1–34.Colbeck, S.C. (1987b) Theory of particle coarsen-ing with a log-normal distribution. Acta Metal-lurgica, 35(7): 1583–1588.Colbeck, S.C. (in press) The basic ideas behindsnow metamorphism. In Snow as a Physical, Eco-logical and Economic Factor, Davos, Switzerland, 1996.Colbeck, S., E. Akitaya, R. Armstrong, H. Gubler,J. Lafeuille, K. Lied, D. McClung, and E. Morris(1990) The International Classification for SeasonalSnow on the Ground. The International Commis-sion on Snow and Ice of the International Associa-tion of Scientific Hydrology (available from WorldData Center, University of Colorado, Boulder, Colo.).Dash, J.G., H. Fu, and J.S. Wettlaufer (1995) Thepremelting of ice and its environmental conse-quences. Reports on Progress in Physics, 58: 115–167.de Quervain, M.R. (1958) On metamorphism andhardening of snow under constant pressure andtemperature gradient. In International Associationof Scientific Hydrology (M.R. de Quervain, Ed.).Publication 46, p. 225–239.de Quervain, M.R. (1973) Snow structure, heatand mass flux through snow. International Asso-ciation of Scientific Hydrology, p. 203–226.Gow, A.J. (1975) Time–temperature dependenceof sintering in perennial isothermal snowpacks.In Snow Mechanics. International Association of

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Hydrological Sciences, IAHS–AISH, vol. 114, p.25–41.Gubler, H. (1982) Strength of bonds between icegrains after short contact times. Journal of Glaciol-ogy, 28: 457–473.Gubler, H. (1985) Model of dry snow metamor-phism by interparticle vapor fluxes. Journal of Geo-physical Research, 90: 8081–8092.Hobbs, P.V., and B.J. Mason (1964) The sinteringand adhesion of ice. Philosophical Magazine, 9:181–197.Hosler, C.L., D.C. Jensen, and L. Goldshlak (1957)On the aggregation of ice crystals to form snow.Journal of Meteorology, 14: 415–420.Keeler, C.M. (1969) The growth of bonds and theincrease of mechanical strength in a dry seasonalsnow-pack. Journal of Glaciology, 8: 441–450.Ketcham, W.M., and P.V. Hobbs (1969) An ex-perimental determination of the surface energiesof ice. Philosophical Magazine, 19: 1161–1173.Kingery, W.D. (1960) Regelation, surface diffu-sion, and ice sintering. Journal of Applied Physics,31: 833–838.Kinosita, S. (1963) Compression of snow im-mersed in water of 0°C. Low Temperature Science,A21: 13–22.Kuczynski, G.C. (1949) Self-diffusion in sinteringof metallic particles. Journal of Metals, 1: 169–178.Kuroiwa, D. (1962) A study of ice sintering. USACold Regions Research and Engineering Labora-tory, Research Report 86.Lenel, F.V. (1992) Sintering. In McGraw-Hill Ency-clopedia of Science and Technology (F.V. Lenel, Ed.).New York: McGraw-Hill, vol. 16, p. 461–462.

Moya, J. S., C. Baudin, and P. Miranzo (1987)Sintering. In Encyclopedia of Physical Science andTechnology (J.S. Moya, C. Baudin, and P. Miranzo,Eds.). Orlando, Florida: Academic Press, vol. 12,p. 699–712.Petrenko, V.F. (1994) The surface of ice. USA ColdRegions Research and Engineering Laboratory,Special Report 94-22.Ramseier, R.O., and C.M. Keeler (1967) The sin-tering process in snow. USA Cold Regions Re-search and Engineering Laboratory, ResearchReport 226.Sturm, M.A. (1991) The role of thermal convec-tion in heat and mass transport in the subarcticsnow. USA Cold Regions Research and Engineer-ing Laboratory, CRREL Report 91-19.Swinkels, F.B., and M.F. Ashby (1981) A secondreport on sinter diagrams. Acta Metallurgica, 29:259–281.Wakahama, G. (1968) The metamorphism of wetsnow. IUGG General Assembly of Bern, Septem-ber–October 1967. International Association of Sci-entific Hydrology, p. 370–379.Wilkinson, D.S. (1988) Pressure-sintering modelfor the densification of polar firn and glacier ice.Journal of Glaciology, 34: 40–45.Yosida, Z., and colleagues (1955) Physical stud-ies on deposited snow. I. Thermal properties. LowTemperature Science, 7: 19–74.Zhang, W., and J. H. Schneibel (1995) The sinter-ing of two particles by surface and grain bound-ary diffusion—a two-dimensional numericalstudy. Acta Metallurgica and Materiallia, 43: 4377–4386.

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December 1997

A Review of Sintering in Seasonal Snow

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18Dry snow Snow

UNCLASSIFIED UNCLASSIFIED UNCLASSIFIED UL

Strength and electrical pathways develop in snow as bonds grow among grains. Strong ice-to-ice bonds form inwet snow at low liquid contents but not in highly saturated wet snow. In freely draining wet snow, grainclusters form, and these require a certain configuration among the three phases of water. This dependssomewhat on the number of grains in the cluster, but always leads to bonding. In dry snow, bonds form moreslowly, but considerable strength can develop as long as rounded grains develop. The rate of bond growth isprobably controlled by the temperature gradient, because both grains and bonds are observed to grow veryslowly in dry snow in the absence of a temperature gradient. The basic shape of the bonds is dictated by thegeometrical requirements of grain-boundary grooves and is not a simple concave neck. In dry snow, this shape,and possibly the processes, have been misunderstood.