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THE ASTROPHYSICAL JOURNAL, 473 :1104È1113, 1996 December 20
1996. The American Astronomical Society. All rights reserved. Printed in U.S.A.(
CRYSTALLIZATION OF AMORPHOUS WATER ICE IN THE SOLAR SYSTEM
P. JENNISKENS AND D. F. BLAKE
NASA/Ames Research Center, Space Science Microscopy Laboratory, Mail Stop 239-4, Mo†ett Field, CA 94035
Received 1995 September 21; accepted 1996 July 5
ABSTRACT
Electron di†raction studies of vapor-deposited water ice have characterized the dynamical structural
changes during crystallization that a†ect volatile retention in cometary materials. Crystallization is found
to occur by nucleation of small domains, while leaving a signiÐcant part of the amorphous material in a
slightly more relaxed amorphous state that coexists metastably with cubic crystalline ice. The onset of
the amorphous relaxation is prior to crystallization and coincides with the glass transition. Above the
glass transition temperature, the crystallization kinetics are consistent with the amorphous solid becom-
ing a ““ strong ÏÏ viscous liquid. The amorphous component can e†ectively retain volatiles during crys-
tallization if the volatile concentration is D10% or less. For higher initial impurity concentrations, a
signiÐcant amount of impurities is released during crystallization, probably because the impurities are
trapped on the surfaces of micropores. A model for crystallization over long timescales is described that
can be applied to a wide range of impure water ices under typical astrophysical conditions if the fragility
factor D, which describes the viscosity behavior, can be estimated.
Subject headings: comets: general È infrared: ISM: lines and bands È infrared: solar system È
methods: laboratory È molecular data È planets and satellites: general
T he hope is that if we do understand the ice crystal we shall ultimately understand
the glacier.ÈR. P. Feynman (1965)
1. INTRODUCTION
The crystallization of water ice is thought to play an
important role in the phenomena of cometary activity
et al. Weiss-(Smoluchowski 1981; Prialnik 1993; Mumma,
man, & Stern Comets contain water ice that is at low1993).
temperature and pressure and contains up to 20% trapped
impurities. The physical phenomena associated with such
astrophysical ices, i.e., sublimation, outgassing of volatiles,
changes in thermal conductivity, and the like, are controlled
to a large extent by changes in the structure of the water ice
during heating (““ structure ÏÏ in this context refers to the
local ordering of the water molecules in a hydrogen-bonded
network). Hence, by characterizing the dynamic structural
changes that occur within water ice as a function of thermal
history, it may be possible to provide a physical basis for
understanding cometary activity and, perhaps, to predict
the physical properties of comets and other icy bodies on
the basis of their activity or thermal history.
The structure of water-rich ice in astrophysical environ-
ments is usually not that of the familiar thermodynamically
stable hexagonal crystalline polymorph group(I
h
Èspace
found almost exclusively on Earth, where theP6
3
/mmc)
hexagonal symmetry is manifested in the familiar shape of
snowÑakes (Nordenskiold 1893; Schneer 1988; Clopton
Rango, & Erbe Rather, astrophysical1994; Wergin, 1995).
water ice is often observed to be in an amorphous form,
either as a result of the particular conditions of its forma-
tion (e.g., vapor deposition at low temperature) or as a
result of amorphizing processes that a†ect the ices after
formation (e.g., pressure deformation, or processing by
photons or charged particles). When amorphous ice is
warmed to a high enough temperature, the ice crystallizes
into the cubic crystalline polymorph group(I
c
Èspace
Fd3m; before, at higher temperatures, re-Konig 1944)
crystallization to the hexagonal polymorph occurs.
Prior to the onset of cubic crystallization, amorphous
water ice undergoes a weak glass transition that manifests
itself as an endothermic step in di†erential thermal analysis
(DTA) scans & Los(McMillan 1965; Ghormley 1968;
Suga, & Seki Mayer, & JohariSugisaki, 1968; Hallbrucker,
The glass transition is associated with the opening of1989).
pathways that connect di†erent positional conÐgurations in
the ice reached by molecular di†usion or di†usion of
defects, and it results in a sudden change in slope of the
temperature dependence of volume V (T ) and entropy S(T )
& Rice p. 304; In(Sceats 1982; Zallen 1983, Angell 1995).
other words, the solid transforms to a viscous liquid. An
interesting feature of the glass transition is that it is
reversible, although induced structural changes that occur
as a result of the glass transition may not be.
We have investigated the structural properties of vapor-
deposited water ice with a Hitachi H-500H transmission
electron microscope using real-timeÈselected area electron
di†raction and bright-Ðeld imaging. Experimental details
are given in et al. which addresses theJenniskens (1995),
temperature range from 14 to 100 K, in which two structur-
ally distinct amorphous forms of water ice are found: a
low-temperature high-density amorphous form and a(I
a
h)
higher temperature low-density amorphous form At(I
a
l).
temperatures between 100 and 160 K, the low-density
amorphous ice begins to transform to cubic crystalline ice,
but a signiÐcant fraction remains in a third amorphous
form, which has a slightly di†erent structure than I
a
l
& Rinfret & Blake We refer(Dowell 1960; Jenniskens 1994).
to this third amorphous form as the ““ restrained ÏÏ amor-
phous form because its structure apparently inhibits(I
a
r)
crystallization into the cubic polymorph.
Here we study the process of crystallization in detail and
the structural changes that occur in the amorphous com-
ponent when it is warmed above the glass transition tem-
perature. It will be shown that the crystallization of water
ice is well described by classical capillary nucleation theory,
but crystal growth proceeds by rapid crystallization of small
domains conÐned by the restrained amorphous ice matrix.
1104
AMORPHOUS WATER ICE CRYSTALLIZATION 1105
The measured time and temperature dependence of crys-
tallization is found to characterize the viscosity of the
amorphous component in the cubic domain. These results
lead to a more fundamental understanding of the crys-
tallization of (impure) water ice on the very long timescales
of the solar system and the retention of volatiles in water-
rich cometary ices.
2. THE PROCESS OF CRYSTALLIZATION
2.1. Structural Changes
When amorphous water ice crystallizes, the Ðrst amor-
phous di†raction maximum narrows and two sharp peaks
emerge near the second broad di†raction maximum of low-
density amorphous ice (see, for example, &Jenniskens
Blake The latter two peaks are the 220 and 311 dif-1994).
fractions of cubic water ice. The di†raction pattern is typical
of a randomly oriented Ðne powder and does not exhibit
discreet maxima characteristic of single crystals. Amorp-
hous water ice does not crystallize into a single crystal sheet,
as do, for example, annealed Ðlms of methanol et al.(Blake
1991).
The 220 and 311 di†raction peaks do not sharpen during
annealing, and throughout crystal maturation they are
intrinsically wider than the instrumental resolution. This
suggests that the crystalline domains remain small. Domain
sizes can be determined from measurements of the(c
dia
)
width of di†raction maxima using the Scherrer equation
p. 555), where(Cullity 1978, c
dia
\0.9j/[FWHM(radians)
cos (h)]. For cubic ice, the 220 di†raction maximum occurs
at sin (h) \j/2d. With d \2.24 and j \ 0.037 for 100Ó, Ó
keV electrons, the width of the measured di†raction peak is
found to be FWHM \0.183 at 3.1 before correc-Ó~1 Ó~1
tion for instrumental broadening. From this, we calculate
crystal domain sizes of diameter Correcting forc
dia
\68 Ó.
instrumental broadening increases this value to 100È150 Ó.
From X-ray experiments of annealed and probably thicker
vapor deposited water layers, & Rinfret mea-Dowell (1960)
sured a domain size of 400 Ó.
2.2. Crystallization Kinetics
The crystallization process is time and temperature
dependent. When an ice Ðlm is deposited at 14 K and
warmed at a rate of 1È3 K per minute, cubic crystalline
peaks emerge in the temperature range 145È160 K
(typically at 151 ^4 K). Upon continued warming, the
intensities of the crystal di†raction peaks increase until
160È170 K, at which time the pattern appears fully mature.
The integrated intensity of the 220 cubic di†raction peak
increases to at most 30% of the total di†racted intensity. On
the other hand, in ices that are warmed rapidly (10 K per
minute) directly after deposition at 86 K, the onset of crys-
tallization typically occurs at lower temperature, 142 ^ 4K,
the temperature range of crystal growth is smaller, and the
Ðnal di†racted 220 peak intensity is higher. forFigure 1,
example, shows the height of the amorphous component
and the height of the 220 crystalline di†raction peak as a
function of temperature in both experiments from a decom-
position of the di†raction patterns.
In a similar manner, we have measured the growth of the
crystalline fraction during isothermal annealing experi-
ments conducted at various temperatures. In a typical
experiment, ice layers are deposited and held at 86 K for 1
hour, heated to a point 10 K below the annealing tem-
FIG. 1.ÈThe intensity of the amorphous ( Ðlled circles) and 220 cubic
crystalline (open circles) di†raction maxima of vapor-deposited water ice
during gradual warming at 1 K per minute (top) and rapid warming at 10
K per minute (bottom).
perature at a rate of 1 K per minute and then heated rapidly
(D10 K per minute) to the annealing temperature. Figure 2
shows the growth of the 220 cubic di†raction maximum as a
function of time for representative annealing temperatures
in the range 125È143 K. The vertical axis is given in relative
intensity units, since absolute values depend on layer thick-
ness. All intensities are scaled linearly to the mean value for
the intensity of the second di†raction maximum of the
amorphous component, which ranged from four to 12
intensity units. The Ðgure illustrates that there are two
regimes of growth of the crystalline fraction: a regime of
rapid initial increase and a second regime of a more gradual
increase.
FIG. 2.ÈThe intensity of the cubic 220 crystalline di†raction peak
during isothermal annealing at various temperatures. Two regimes of
growth are recognized.
1106 JENNISKENS & BLAKE Vol. 473
The theory of phase transformation by nucleation and
growth yields an equation (the ““ Avrami ÏÏ equation) that
describes the fraction of crystallized material (x) as a func-
tion of time (t) during isothermal annealing at temperature
T . This equation is & Rao(Rao 1978; Doremus 1985)
x(T ) \1 [exp [[k(T )tn] , (1)
in which k(T ) is the rate constant and n is a parameter
whose magnitude is determined by the geometry of the
growing particles and whether the transformation is di†u-
sion or interface controlled; that is, whether the growth
limiting step is the di†usion of molecules to the nucleus or
the ordering of the molecules at the interface.
The parameters n and k can be derived from log-log dia-
grams of di†raction intensity -ln (1 [x) versus time. We
Ðnd that n\2.0 ^0.3 for the Ðrst regime (or n\1.0 ^0.3
if one allows for an induction period). For the second
regime, n \0.8 ^ 0.3 (with or without an induction period).
These values are in good agreement with the n \2.43 and
n \ 0.90, respectively, found by et al. in a recentHage (1994)
infrared study of amorphous ice. These authors annealed a
layer of amorphous ice for 120 minutes at 130 K, warmed
the ice to 144 K, and measured the crystalline fraction by
decomposing the 3.07 km infrared band into crystalline and
amorphous components. In other works by these authors,
three samples studied at 140, 144, and 146 K without prior
annealing did not show a clear distinction between a Ðrst
and second crystallization regime and gave somewhat
higher values of n \1.30, 1.34, and 1.15, respectively,
perhaps because crystallization proceeded rapidly enough
to blend the two regimes.
The rate constants k(T ) are summarized in andFigure 3
are a factor of 2 higher than found by Hage et al. The rate
constant is thermally activated with an activation energy
*H according to et al.(Hage 1994)
k1@n(T ) \k
0
1@n exp ([*H/RT ) , (2)
where R\8.3144 J K~1 mol~1. By plotting the logarithm
of k1@n versus 1/T , we Ðnd an activation enthalpy of
*H \ 39 ^5 kJ mol~1 for the Ðrst regime and
*H \ 58 ^10 kJ per mole for the second regime. By com-
parison, Hage et al. found an activation enthalpy of
*H \ 67 ^3 kJ mol~1 for the second regime (triangles in
Fig. 3). We note that these values are about equivalent to
breaking two and three typical hydrogen bonds in water ice
(Buch 1992).
FIG. 3.ÈTemperature dependence of the growth rate of the crystalline
fraction for the initial fast regime (I) and the later slower regime (II).
2.3. Morphological Changes of the T hin Ice Film
When an amorphous water ice Ðlm is warmed at 1 K per
minute to 160 K, bright Ðeld images reveal that the ice layer
is made up of many isolated islands of material (the
““ crystallites ÏÏ in & Blake This is a thinJenniskens 1994).
Ðlm e†ect driven by the high surface tension at the interface
with the hydrophobic substrate. These ““ droplets ÏÏ stand
out from the surface to a height of 2È6 times the original
Ðlm thickness (0.1È0.3 km). Individual droplets have irregu-
lar shapes and rarely show crystalline morphologies. The
emerging droplets do not grow signiÐcantly after their
initial formation. The droplets do, however, tend to show
increased contrast in bright-Ðeld images as crystallization
proceeds [Pl. 22]). Similar observations were made(Fig. 4
during isothermal annealing experiments of three initially
amorphous ice Ðlms annealed to di†erent stages of the crys-
tallization process. A Ðlm annealed for 48 hr at 122 K had
droplets emerging with a diameter of 0.22 ^0.06 km(1p)
initially at low contrast, while a second Ðlm annealed at 132
K for 26 hours had droplets of diameter 0.27^0.05 km,
and a third ice Ðlm annealed at 143 K for 11 hours had
nearly fully developed droplets of diameter 0.30 ^ 0.06 km.
The mean size of the droplets does not change signiÐcantly
during development, and any di†erences between the three
samples may simply be the result of experiment-to-experi-
ment variations in initial layer thickness.
When electron beam e†ects warp the substrate and
change the orientation of the ice Ðlm relative to the beam,
di†racted intensities are seen to vary within individual
droplets. Hence, many droplets must consist of smaller crys-
talline domains, often less than 100 in diameter, in agree-Ó
ment with the width of the di†raction peaks. A few droplets
appear to be well crystallized with numerous twinning
planes thought to be due to stacking faults. Abundant twin-
ning occurs as a result of fast warm-up ([10 K per minute)
shortly after deposition. The distances between twinning
planes can be less than 50 Ó.
3. THE RESTRAINED AMORPHOUS FORM
3.1. Structural Changes of the Amorphous Form
Vapor-deposited amorphous water ice does not trans-
form completely to crystalline ice when warmed through
the crystallization temperature & Rinfret(Dowell 1960;
& Blake In the cubic domain, an amorp-Jenniskens 1994).
hous component always underlies the cubic crystalline
peaks.
In a gradual 1 K per minute warmup, structural changes
occur in the amorphous component prior to any sign of
crystallization. The intensity of the Ðrst amorphous di†rac-
tion maximum increases before the appearance of cubic dif-
fraction maxima and before di†raction contrast is(Fig. 1)
observed in bright Ðeld images By comparison, the(Fig. 4).
intensity of the second amorphous di†raction maximum
varies little during crystallization (e.g., & BlakeJenniskens
The intensity of the Ðrst amorphous di†raction1994).
maximum continues to increase during crystallization of the
cubic phase and ceases its growth when the crystalline dif-
fraction peaks have matured. The observed changes are
irreversible: cooling to 86 K does not cause a reduction in
intensity or a broadening of the Ðrst di†raction maximum.
The oxygen-oxygen radial distribution function (rdf) of
this new amorphous form cannot be calculated directly
from the di†raction pattern because the subtraction of the
No. 2, 1996 AMORPHOUS WATER ICE CRYSTALLIZATION 1107
crystalline component cannot be performed accurately.
This is true for two principal reasons: Ðrst, the intensity of
the patterns decreases exponentially as a function of radial
distance from the central beam, and second, it is difficult to
obtain in the electron microscope a cubic di†raction pattern
of crystalline ice (with a similar distribution of domain sizes)
that does not have an amorphous component convolved
with it. It is, however, possible to study the di†raction
pattern just prior to the onset of crystallization. The
resulting rdf does not represent fully formed(Fig. 5) I
a
r.
However, a comparison with the rdf of measured at 86 KI
a
l
does reveal di†erences that appear to indicate structural
relaxation: The second nearest neighbor moves to some-
what greater distance (from r \4.46 to 4.59 and a gapÓ),
opens between the Ðrst and second nearest neighbor. These
di†erences suggest that the fraction of bonding angles that
deviate most from the tetrahedral angle decreases, resulting
in a structure closer to that of the thermodynamically
favored crystalline forms.
3.2. T he Glass Transition
The onset of structural relaxation, as deduced from the
increase in intensity of the Ðrst di†raction maximum, occurs
in the range 120È142 K, which coincides with the tem-
perature regime in which water has its glass transition (Fig.
For example, in a 1È2 K per minute warming experiment1).
starting at 14 K, the onset of the transition occursI
a
l ] I
a
r
at 122È135 K, while the glass transition temperature (T
g
)
occurs at K &T
g
\127È133 (Yannas 1968; Rasmussen
MacKenzie A rapid warm-up at 10 K per minute1971).
shortly after deposition postpones the transitionI
a
l ] I
a
r
until 141 ^2 K, just preceding the onset of crystallization.
The glass transition is sensitive to heating rate in a similar
way. Mishima, & Whalley found an onset atHanda, (1986)
about 120 K for a low heating rate of 0.17 K per minute (for
pressure-induced low-density amorphous ice), while in a
fast 10È30 K per minute warm-up, the glass transition has
been reported in the range K &T
g
\135È142 (McMillan
Los & MacKenzie et1965; Rasmussen 1971; Hallbrucker
al. 1989).
We attempted to measure an activation enthalpy from
the time and temperature dependence of the onset of the
FIG. 5.ÈOxygen-oxygen radial distribution function (rdf) of amorp-
hous water ice shortly after the onset of the transition from the low-density
amorphous form to the restrained amorphous form The result is(I
a
l) (I
a
r).
compared to that for low-density amorphous ice at 86 K (dashed line).
transition. The experimental procedure was asI
a
l ] I
a
r
before. The onset was now at a lower temperature, T \122
K, than that found in some of our other experiments. We
found that for experiments with annealing temperatures
above 122 K, i.e., at T \ 125 K and T \ 135 K, the Ðrst
di†raction maximum began to increase in intensity at about
122 ^ 2 K and continued during subsequent annealing. For
Ðlms annealed at T \ 107 K, T \ 110 K, and T \112 K,
however, no increase in peak intensity could be measured
after annealing times of several tens of hours. The resulting
value, *H [ 79 kJ mol~1, is a lower limit to the activation
enthalpy. This value is higher than the value of
*H \ 25 ^6kJmol~1 for the relaxation reportedI
a
l ] I
a
r
in Jenniskens & Blake (1994). This incorrect value resulted
from an incorrect interpretation of electron beam damage
at later stages of the annealing process. Instead, it is likely
that the relaxation follows the viscosity changesI
a
I ] l
a
r
manifested in the glass transition. In comparison,
& McKenzie tentatively derived from theRasmussen (1971)
heating rate dependence of an activation enthalpy thatT
g
limits the kinetics of the glass transition to 80 ^2kJmol~1,
while the activation entropy was estimated at 364 ^ 8J
mol~1 K.
4. DISCUSSION
4.1. Crystal Growth
The steep time dependence of crystal growth in the Ðrst
growth regime of is consistent with di†usion-Figure 2
controlled crystal growth et al. The value of(Hage 1994).
n \ 2.0 ^0.3 for the Ðrst growth regime is between n \ 1.5
expected for di†usion controlled growth of spherical nuclei
in case of rapid nucleation rate and depletion or n\2.5
expected if the nucleation rate is constant & Rao(Rao 1978;
There is no evidence for the alternative thatDoremus 1985).
interface-controlled growth of cylindrical nuclei is
responsible (n \2).
After this initial phase, the crystallization proceeds in a
di†erent manner. Hage et al. suggested that the second
growth regime may be due to di†usion-controlled growth of
cylindrical nuclei growing radially (n \ 1). However,
normal crystal growth should result in a narrowing of dif-
fraction maxima during maturation of the pattern, which is
not observed. Indeed, no well-deÐned large crystals are
observed in bright Ðeld images. Rather, crystallization
seems to occur at many places within individual droplets,
resulting in a gradual increase of di†racted intensity. Our
result that n may be less than 1 also argues against such a
process. For the same reasons, surface-induced crys-
tallization followed by crystal growth in one dimension
(e.g., et al. is not a likely mechanism. In thatZellama 1979)
case also, one would expect a narrowing of the di†raction
peaks with increasing crystal size. Hence, crystal growth
does not control the kinetics in the second crystal growth
regime.
4.2. Nucleation in Small Domains
We propose an alternative mechanism for the kinetics of
crystallization in the second crystal growth regime, follow-
ing earlier suggestions by & Rinfret It isDowell (1960).
assumed that shortly before crystallization, amorphous
water ice consists of stable and unstable zones. Crys-
tallization to the cubic phase can begin spontaneously at a
few sites in the unstable zones, after an induction time that
1108 JENNISKENS & BLAKE Vol. 473
is determined by the nucleation process. After nucleation,
the crystals grow quickly to occupy the domains fully until
the more stable zones surrounding them limit further
growth. After a certain induction period, the rate of nucle-
ation increases to a steady state value, which results in a
nearly linear increase of the fraction of crystallization with
time, i.e., n \1, until all domains have crystallized. An
exponent n slightly less than 1 for this second regime, as
suggested by the observations, is possible if, at the same
time, new domains are formed in the ice.
4.2.1. A Description of the Nucleation Process
Classical (capillary) nucleation theory describes the sta-
tistical process of the formation of a (spherical) critical
nucleus that can crystallize a larger (not necessarily
spherical) domain in the amorphous material. The crys-
tallization of small water domains has been described
recently for the case of the freezing of water clusters and
water droplets by Bartell et al. & Dibble(Bartell 1991;
& Bartell The mean domain size (d)isHuang 1995).
assumed to be of order 50È200 in diameter, similar to theÓ
domains we Ðnd in solid water layers, while the critical
nucleus has a typical size of no more than 5È10 We willÓ.
now apply this model to our data. Note that in our case the
domain size is determined by the size of unstable zones in
the ice, not by the size of the droplets or thickness of the
solid water layer.
During nucleation a small but critical number of unit
cells of the cubic phase self-assemble. The model considers a
steady state rate for the formation of critical nuclei per unit
volume per unit time (J), which is proportional to the prob-
ability of growing a critical nucleus and has the form
p. 342)(Scholze 1977,
J \ A exp ([*G*/kT ) , (3)
where *G* is the free energy barrier to the formation of the
critical nucleus that initiates the transition. The preexpon-
ential factor A is given by
A \ 2(pkT )1@2/[V
m
5@3g(T )] , (4)
where p is the interfacial free energy per unit area of the
boundary between solid and liquid, is the molecularV
m
volume, and g(T ) is the liquid viscosity. uses aEquation (4)
viscous Ñow model for the molecular jumps across the
solid-liquid interface & Dibble The free(Bartell 1991).
energy barrier to the formation of a spherical nucleus
follows from the Gibbs equation, neglecting the work
needed to change density, since the density of the amorp-
hous and crystalline forms are very similar
*G* \ 16np3/(3 *G
v
2) , (5)
where is the Gibbs free energy of the transition from*G
v
the old to the new phase per unit volume, for which Huang
& Bartell give
*G
v
(J per mole) D 1139.5 ]13.016T [ 0.06499T 2
(T
g
\ T \ 226 K) ,
*G
v
(J per mole) D [2007 ]37.163T [ 0.1102T 2
(226 K) \ T \ T
s
) . (6)
We make the assumption that these relationships are an
adequate analytical representation for all relevant tem-
peratures below the melting temperature including(T
s
),
those below the glass transition temperature This(T
g
).
assumption can be made, because these equations for *G
v
have only a weak temperature dependence below HuangT
g
.
& Bartell assumed that because of interfacial entropy, the
free energy of the interface increases with temperature T
according to
p(T ) \p(T
1
)/(T /T
1
)a , (7)
where the cubic polymorph is kinetically favored over the
hexagonal polymorph due to the lower interfacial free
energy of cubic nuclei relative to hexagonal nuclei. For
liquid water clusters frozen at T \ 200 K and somewhat
larger liquid water droplets dispersed in oil that were frozen
at T \235È240 K, Huang & Bartell argue that a \ 0.3 and
p \ 21.6 ^0.1 mJ m~2 at 200 K. These data and the corre-
sponding Ðt are shown in by open squares and aFigure 6
dashed line.
4.2.2. Strong and Fragile L iquids
The temperature dependence in the model that describes
the liquid water clusters (dashed line in is much tooFig. 6)
steep below T \ 150 K to Ðt the nucleation rates for I
a
r.
The data on the growth rate k (s~1) during the initial fast
growth of the pattern convert to nucleation rates of(Fig. 3)
order J \ 4 ] 1018 to 7 ] 1020 m~3 s~1 between 122 K
and 140 K, given a domain size of andc
dia
\100 Ó
assuming that 30% of the ice transforms into cubic crystals.
The growth rates in the second regime are a factor of 2
larger. These values are shown in ( Ðlled circles).Figure 6
We note that there is no continuous curve that connects
these nucleation rates with those found in water clusters at
high temperature.
The source of the steep temperature dependence of the
dashed curve is the viscosity: no other parameter changes
20 orders of magnitude over the measured temperature
interval. In general, the viscosity behavior g(T ) of a liquid
above the glass transition is described by (Angell 1995)
g(T ) \g
0
exp [DT
0
/(T [T
0
)] . (8)
This is an adapted Vogel-Tammann-Fulcher relation. The
parameter D, introduced by Angell, is called the fragility
FIG. 6.ÈNucleation rate in the cubic domain ( Ðlled squares) compared
to nucleation rates in liquid water droplets (open squares) from &Huang
Bartell Model curves are for the same nucleation model, but for(1995).
fragility parameters D \O (short-dashed line), D \200 (dotted line), and
D \ 10 (long-dashed line) respectively.
No. 2, 1996 AMORPHOUS WATER ICE CRYSTALLIZATION 1109
parameter and describes the deviation from an Arrhenius
law for which D \ O and K. A liquid that does notT
0
\0
deviate much from the Arrhenius law, say D [ 100, is called
a ““ strong ÏÏ liquid, while others with much lower values of D
are called ““ fragile.ÏÏ is the temperature at which the con-T
0
Ðgurational entropy of the liquid disappears (the Kauz-
mann temperature). The glass transition temperature is aT
g
kinetic manifestation of whereT
0
, (Angell 1995)
T
g
/T
0
\1 ]D/[ln (10) log (g
g
/g
0
)] , (9)
where log and the viscosity at the glass(g
g
/g
0
) \ 17.0 ^0.5
transition temperature is about Pa s. As pointedg
g
\1012
out by Angell, the use of a canonical value for (the mea-T
g
sured value depends on heating rate) is justiÐed because it
serves merely as a mathematical tool for combining vis-
cosity data for a large number of materials: another choice
of would a†ect the ratio log but would lead toT
g
(g
g
/g
0
)
similar equations. The viscosity behavior of liquid water is
shown in which is adapted fromFigure 7, Angell (1993).
Note that liquid water is a fragile liquid with a small
D \ 10 ^2 (if K).T
g
\136
Our data suggest a viscosity behavior of that is typicalI
a
r
of a strong liquid, with large D. By replacing g(T )in
for the relation given in equations (withequation (4) (8)È(9)
K), we obtain the solid curve in forT
g
\136 Figure 6
D \ O and the dotted curve for D \200. Clearly, any
model with Dº 200 provides a good agreement with the
data. In order for water ice to have a measurable glass
transition, D should be somewhat less than D \ O, but it
does not need to be less than D \ 200. Hence, the restrained
amorphous form above the glass transition is a strong liquid.
It has been argued before that water above the glass tran-
sition (but before crystallizing) is a strong liquid. Angell
made this conclusion based on the delay of crys-(1993)
tallization above the glass transition and the observation
that is more resistant to crystallization than is a range ofI
a
r
aqueous LiCl solutions (dotted lines in This obser-Fig. 7).
vation is subject to experimental uncertainty, however,
because various experiments show a considerable range in
the crystallization temperature of pure water. In fact, pure
water in our experiments does crystallize close to the 150 K
quoted for the LiCl solution of lowest concentration. Our
results for the crystallization kinetics of water ice do,
however, give support to AngellÏs conclusion that amorp-
FIG. 7.ÈViscosity behavior of glasses for a range of fragility parameters
D. Figure adapted from Angell (1993).
hous water ice above the glass transition is a strong liquid,
and we can add to this that the strong liquid persists in the
cubic crystalline domain.
4.2.3. T he Onset of Nucleation
Viscosity also plays a crucial role in delaying or slowing
the onset of nucleation. The induction period *t, the time
until a stable production of nuclei occurs, is approximately
& Gutzow & Dibble(Toschev 1972; Bartell 1991)
*t D p(T )g(T )[T
s
2/(T [T
s
)2]/*G
v
2 , (10)
with the melting temperature. The strongest temperatureT
s
dependence is that of the viscosity. Curves drawn in Figure
show the temperature dependence expected for D \ O8
(solid curve) and D \ 200 (dashed curve), assuming T
g
\136
K. The Ðt is scaled to the data by a constant factor, and it is
the shape of the curve that proves good agreement for
D º 200. Hence, the viscosity of water follows the Arrhenius
law closely. Note that in the limit of D \ O, equation (8)
reduces to
g(T ) \10~5 exp
A
326T
g
RT
B
. (11)
Hence, *t D exp (326 which reduces to a familiarT
g
/RT ),
equation in the interpretation of isothermal annealing data
& Los(McMillan 1965):
*t \ l
0
exp (*H/RT ) , (12)
with *H \ 44 kJ per mole, while is the vibrational time-l
0
scale that limits the kinetics, most likely the vibration or
bending frequency of the water molecules (l
0
D 1 ] 10~14
s). For the onset of crystallization, the measured values are
*H \ 44 ^2 kJ per mole and s.l
0
\3.3 ] 10~15
4.3. T he InÑuence of Impurities
Astrophysical ices usually contain trapped impurities
that will a†ect the crystallization behavior. It is possible,
however to estimate the crystallization behavior of impure
ices in comparison to pure water ice.
Impurities tend to increase the endothermic step in DTA
scans, a characteristic of a reduced D, without a†ecting T
g
very much & MacKenzie In a binary(Rasmussen 1971).
FIG. 8.ÈTime-temperature dependence of the onset of crystallization in
vapor-deposited amorphous water ice ( Ðlled squares). Data are compared
to the time required for full development of the cubic di†raction maxima in
X-ray di†raction experiments of & Rinfret (triangles) andDowell (1960)
electron microscopy studies of et al. (diamonds). ModelDubochet (1982)
curves are as in Fig. 6.
1110 JENNISKENS & BLAKE Vol. 473
solution of an impurity in water, the following equation is
approximately valid & Heush(Jenckel 1958):
T
g
\T
g1
w
1
]T
g2
w
2
]Kw
1
w
2
, (13)
with the weight fraction of (1) water and (2) solute and K a
proportionality constant dependent on the particular com-
pounds involved. For example, K \ 15 ^1 for the water-
methanol system & MacKenzie(Rasmussen 1971),
methanol being an impurity relevant to astrophysical ices.
For K and K, a 6%T
g
(methanol) \ 102 T
g
(water) \ 136
methanol concentration will lower the glass transition tem-
perature by only about 3 K. However, that small decrease
of may underlie a strong change of D (and If solu-T
g
T
0
).
tions of LiCl can be considered typical, a rather strong
change is expected. In a 14.8% solution of LiCl, increasedT
g
to only 140.6 K, while D dropped to 10, and water that is
absorbed up to 42% weight in polymeric materials has T
g
increased similarly by only a few degrees while D has
decreased to 56 ^30 Hence, while is(Angell 1993). T
g
a†ected only slightly (upward or downward), D is lowered
dramatically.
4.4. T he Possible InÑuence of Structural Variation
The presence of impurities in the ice may explain the
somewhat lower value of D (150 ^ 30) derived from X-ray
di†raction data published by & Rinfret ADowell (1960).
progressively steeper time-temperature dependence was
found for temperatures T \ 124 K in comparison to our
data Although the experimental procedure of(Fig. 8).
& Rinfret seems to exclude signiÐcant impu-Dowell (1960)
rities, their timescale for full maturation of the cubic pattern
is systematically shorter than in our experiments, which is
consistent with an impure ice.
An alternative (albeit less likely) explanation for the lower
value of D inferred from Dowell & RinfretÏs work is that
some characteristic of the structure of the water ice itself is
responsible. Let us consider now what that may be.
The restrained amorphous form has a structure close to
that of Liquid water, on the other hand, has a structureI
a
l.
close to that of Johari, & TeixeraI
a
h (Bosio, 1986;
et al. There is no direct thermodynamicJenniskens 1995).
pathway between the two structures Bergren, &(Rice,
Swingle In fact, when1978; Speedy 1992; Angell 1993).
supercooled, water may undergo a high- to low-density
transition similar to going from aH
2
O (l) ] I
a
rI
a
h]I
a
l,
““ fragile ÏÏ to a ““ strong ÏÏ liquid in the process (with a sudden
increase in viscosity, like freezing to a solid). The transition
temperature is thought to be at about 228 K for rapid
changes of temperature (Angell 1983).
A ““ strong ÏÏ liquid is typically characterized by an open
network with strong activation barriers between di†erent
positional conÐgurations Since liquid water is(Angell 1995).
a ““ weak ÏÏ liquid, the suggestion may be put forward that
these activation barriers are lowered for liquid water (and
Angell suggested that this is due to a higher percentageI
a
h).
of Ðve-coordination in liquid water, meaning an interaction
of a water molecule with Ðve instead of four neighbors.
Indeed, Ðve-coordination can act as an intermediate
complex between di†erent water conÐgurations, lowering
the activation enthalpy and catalyzing fast restructuring
Geiger, & Stanley(Sciortino, 1992).
The above considerations suggest to us that any physical
interaction that increases the percentage of Ðve-
coordination or that in some other way can lower the acti-
vation barriers for restructuring, such as pressure increases,
the presence of impurities, or even rapid crystallization, can
render the ice more ““ fragile.ÏÏ
4.5. T he Structure of I
a
r
It is likely that the onset of the transition isI
a
l ] I
a
r
related to the glass transition in water. The observed
changes occur at about the same temperature. We expect
that the glass transition per se has only a minor e†ect on the
ultrastructure of the water and that the molecules remain in
the disordered, nearly fully bonded random network, pref-
erentially with tetrahedral bonding. However, the increased
mobility of water molecules in the structure allows for some
relaxation of the structure and removal of strained bonding
angles, as is observed in the rdf of This structuralFigure 5.
relaxation, once it has occurred, is not reversible (although
the glass transition itself is). Indeed, does not evolveI
a
r
back to when cooled below the glass transition. This isI
a
l
why we introduced the notation for the structure““ I
a
rÏÏ
above the glass transition, which emphasizes the ice I struc-
ture & Blake We note that others have(Jenniskens 1994).
described indirect evidence for a third amorphous form
above the glass transition. From thermodynamic argu-
ments, put forward the hypothesis that aSpeedy (1992)
thermodynamically distinct form of liquid water, ““ Water
II,ÏÏ should exist above the glass transition in order to be
able to link the quite distinct heat capacities of liquid water
and interpreted higher sublimation ratesI
a
l. Kouchi (1990)
above 130 K as due to a third amorphous form
although & Baragiola argued that““ H
2
O
as
(III),ÏÏ Sack (1993)
the reported vapor pressure behavior was a dynamical
e†ect.
The question remains whether one needs to invoke a new
notation for this form of amorphous ice, such as orI
a
r,
merely refer to it as ““ liquid Unlike however, theI
a
l.ÏÏ I
a
l,
restrained amorphous form coexists metastably with cubic
ice And because only part of crystallizes, the remain-I
c
.I
a
l
ing ice must have a structure that either di†ers from or isI
a
l
a subset of conÐgurations. Hence, the real question is asI
a
l
follows: What prevents from transforming fully intoI
a
r
cubic ice? The answer may be that contains a recog-I
a
r
nizable short-range hexagonal stacking order that restrains
the material from crystallizing & Blake(Jenniskens 1994).
To form cubic crystals, the ice has to restructure fully all
layers with ABAB (hexagonal) type stacking into ABCABC
(cubic) type stacking. Domains with hexagonal short-range
order probably resist restructuring into cubic short-range
order and persist metastably until wholesale restructuring
transforms all ice into the hexagonal polymorph at higher
temperatures. Of course, this makes the structural di†erence
between and much more subtle than between andI
a
lI
a
rI
a
l
I
a
h.
5. ASTROPHYSICAL IMPLICATIONS
The crystallization kinetics and the structural properties
of pure water ice above the glass transition underlie and are
responsible for physical phenomena that occur on a much
larger scale in comets, on the icy surfaces of some planets
and satellites, and in the interstellar medium. A knowledge
of the crystallization kinetics described in this paper allows
one to predict the structural state of ice on the timescale of
the age of the solar system for a given thermal history of the
ice, even if the ice contains a signiÐcant amount of impu-
No. 2, 1996 AMORPHOUS WATER ICE CRYSTALLIZATION 1111
rities. Also, the presence of solid amorphous and viscous
liquid forms of water ice at high temperatures (T \ 110È
200 K) has important implications for gas retention and
sublimation properties of the ice.
While our observations relate to the crystallization of
thin ice Ðlms of about 500 thickness (some 100Ó
monolayers), cometary and planetary ices are considered
bulk materials. However, we have no reason to believe that
the process of crystallization as described here will be di†er-
ent in bulk ices. We do note that our mean domain size
(50È200 may be limited by the thickness of the Ðlm.Ó)
& Rinfret found a domain size of 400 forDowell (1960) Ó
much thicker Ðlms. This value may be more characteristic
of bulk ices. Impurities may also cause larger mean domain
size, and some future study of thicker and impure Ðlms is
warranted. For the following discussion, this uncertainty in
domain size is not relevant.
5.1. Comets
Theoretical models of comet sublimation and outgassing
have traditionally concentrated on the sublimation rate of
various pure endmember materials (Delsemme 1982;
Houpis, & Marconi and many current cometMendis, 1985),
models are still based on an oversimpliÐed picture of crys-
talline water ice mixed with pure volatiles (e.g., et al.Orosei
On the other hand, laboratory experiments of gas1995).
trapping and release have shown that gasses are e†ectively
retained in the ice to much higher temperatures than
expected on the basis of their sublimation rate in pure form.
Ðrst showed that the outgassing of impu-Ghormley (1967)
rities from water ice occurs within discrete temperature
intervals, work that has been continued by Bar-Nun,
Herman, & Laufer & Kuroda(1985), Kouchi (1990),
& Donn and others. Many of the tem-Hudson (1991),
perature ranges coincide with structural transitions in the
water matrix that e†ectively trap and release the impurities
in the water lattice. The amorphous to crystalline transition
in particular has been implicated in comet outgassing
Ruprecht, & Schuerman(Patashnick, 1974; Smoluchowski
et al. et al.1981; Klinger 1981; Prialnik 1993; Haruyama
1993).
The construction of a comet model is beyond the scope of
this paper. However, with the data presented here, it is
possible to construct a theoretical model of a cometary
nucleus that incorporates correctly the time-temperature
dependence of the crystallization process. If the fragility
parameter D can be estimated, based on the amount of
impurity present in the ice before crystallization, then the
fraction of amorphous ice transformed into crystalline ice
during annealing at a temperature T for a time t is given by
while the nucleation rate is given by equationsequation (1),
Two regimes of crystal growth should be considered(3)È(9).
that have similar nucleation rate but a di†erent time depen-
dence. In a very gradual crystallization such as occurs when
a comet approaches the Sun from the outer solar system,
the Ðrst regime is the more important one.
Any such model should keep track of the thermal history
of the amorphous component as well. Factors which should
be considered include (1) the total time spent in the induc-
tion period that precedes the stationary production of
nuclei (eqs. and and (2) structural changes[10], [11], [12])
that occur in the ice due to the increased mobility of water
molecules above the glass transition temperature.
At Ðrst glance, the release of volatiles is expected to
follow closely the rate at which ice is crystallized. No impu-
rity is retained in the crystalline fraction, which would
otherwise cause dislocations and other distortions of the
crystalline lattice. However, in laboratory experiments that
consider an initial impurity content of less than 10%, most
outgassing occurs during sublimation of the ice, not during
crystallization & Donn This reÑects merely(Hudson 1991).
that the volatiles can be retained in and that only someI
a
r
30% of the ice crystallizes to cubic ice.
There is an additional mechanism that leads to gas
release at about the crystallization temperature. In fact, for
a high initial impurity content, a much larger fraction of
volatiles is released at this point et al.(Bar-Nun 1987;
& Donn than the 30% suggested by the per-Hudson 1991)
centage of ice that crystallizes. We propose that the release
of gases adsorbed on the surfaces of micropores in amorp-
hous ice is the dominant outgassing mechanism for high
([10%) impurity contents. Fresh amorphous ice has many
free OH groups at these surfaces that can bind impurities,
while in Ðlms that have been crystallized by annealing at
high temperatures, dangling OH groups are not observed
spectroscopically & Roberts & Buch(Scha† 1994; Devlin
Hence, gas release in this situation is the result of1995).
decreasing viscosity associated with the glass transition
rather than crystallization.
The impurities that remain trapped in the ice above the
crystallization temperature are not necessarily in the form
of clathrates. Clathrate formation in impure ices is not only
unlikely but also unnecessary to explain the outgassing data
of et al. and Kochavi, & Bar-NunBar-Nun (1985) Laufer,
Whether or not clathrates are formed in cometary(1987).
ice at high enough temperature will depend mainly on the
abundance of the least volatile impurity, methanol, in com-
etary ice at temperatures above 120 K. Clathrates are an
ordered, crystalline form of ice. Such ordering is imposed by
the impurities, and a high enough impurity content is
needed, at least 7% methanol, for example, for a type II
clathrate hydrate, or 16% for a type I clathrate hydrate
et al. We surmise that the onset of methanol(Blake 1991).
clathrate formation at about 120 K in a 1 K per minute
warmup may well be driven by the glass transition in water,
because both processes occur at about the same tem-
perature.
5.2. Icy Satellites and Planetary Surfaces
The time and temperature dependence of crystallization
has important implications also for planetary ices
Fanale, & Ga†ey(Consolmagno 1983; Klinger 1983; Clark,
although to our knowledge no discussion has yet1986),
emerged on the amorphous or crystalline nature of the ice
bands that are typically observed in the near-infrared reÑec-
tion spectra of these surfaces (e.g., et al. This isCalvin 1995).
perhaps due to a lack of experimental information on the
near-infrared spectra of amorphous water ice (but see
Wells, & WagnerHapke, 1981).
Given the thermal history of these planetary surfaces, one
can predict whether ice can remain in the amorphous state
on the planetÏs surface. For impure ices typical of various
astrophysical environments, there is a temperature below
which di†usive motions are blocked. The exact value of this
““ Kauzmann temperature,ÏÏ which is characterized by the
““ fragility ÏÏ parameter D, depends on the impurity content.
On a timescale of 4.5 ] 109 yr, pure water ice (at low
pressure) will remain in an amorphous state if the tem-
1112 JENNISKENS & BLAKE Vol. 473
perature remains below T \ 72 K for D \ O, T \87 K for
D \ 100, and T D 110 K for D \ 10. Hence, all surfaces
that remained at temperatures below T \ 90 K may have
persisted in an amorphous form over the age of the solar
system, if they were originally deposited in that form or if
they were amorphized by UV photons or solar wind ions at
low enough temperatures (T \ 70 K). Whether amorphous
ice is actually found on the surfaces of planets and satellites
depends on the local thermal history. Permanently shaded
areas can exist in which temperatures remain below the
transition temperature for crystallization. Even on the
surface of shallow craters near the poles of Mercury, water
ice can persist at temperatures less than 90 K Wood,(Paige,
& Vasavada Hence, even on Mercury water ice may1992).
be in an amorphous form.
If amorphous ice is present on planetary surfaces, it can
have interesting physical properties. Long-term annealing
of these ices at temperatures just below the glass transition
can potentially change the structural properties of the ice on
a microscopic scale & Baust et al.(Chang 1991; Hage 1994).
As Hage et al. pointed out, while annealing densiÐes most
glasses (as the crystals of most substances are denser than
their liquids), water is an exception, and annealing issub-T
g
expected to decrease its density instead. However, we note
that annealing will also remove micropores, and thereby it
e†ectively increases the bulk density. Annealing can change
drastically the material properties of amorphous water ice.
For example, the percentage of tetrahedrally bonded water
will increase during annealing, making the material harder.
5.3. Water Ice in Interstellar and Circumstellar Matter
Some infrared spectra of the 3.07 km absorption band of
the water frost on interstellar grains seem to contain a small
crystalline component Sellgren, & Brooke(Smith, 1993).
This is an important indicator of the thermal history of
volatile grain mantles (e.g., Hagen, & GreenbergTielens,
et al. The crystalline component may1983; Kouchi 1994).
have formed during brief moments of heating of the grains
in grain-grain collisions or cosmic-ray impacts. The amount
of crystalline ice reÑects the distribution of heating events.
Such ices will have structural properties very similar to the
warmed amorphous ice in this study, including perhaps the
tendency to form droplets and expel the dust grain impu-
rities. Such an e†ect may also occur at temperatures less
than 70 K, where UV photon irradiation can decrease the
viscosity of high-density amorphous ice &(Jenniskens
Blake 1994).
Crystalline ice has also been observed in circumstellar
matter, where the water is thought to have crystallized
during vapor deposition on warm grains et al.(Omont
Vapor deposition at high temperatures under the1990).
present laboratory conditions always results in both an I
c
and component, although the relative fraction of canI
a
rI
c
be higher. On the other hand, the very slow condensation
rates in space may facilitate the growth of large cubic crys-
tals. This will result in signiÐcantly di†erent infrared
absorption bands than those found in recent laboratory
analog studies. The small di†erence in the position of the
240 cm~1 peak between laboratory measurements and
astronomical observations may be related(Breukers 1991)
to this. We expect the infrared absorption spectra of such
very slowly deposited cubic crystalline water ices to be
similar to those of hexagonal ice, which also forms large
crystals.
6. CONCLUSIONS
During the crystallization of vapor-deposited low-density
amorphous ice, only about 30% of the material is trans-
formed into cubic crystals. Crystalline domains remain
small, on the order of 10~24 m3. The crystallization pro-
ceeds in two steps: the initial growth of crystals in single
domains, and a second steady state growth limited by the
nucleation rate and formation of new domains. The nucle-
ation rates are activated thermally with *H \39 ^ 5 kJ per
mole and 58 ^10 kJ per mole, respectively, equivalent ener-
getically to the breaking of two and three typical hydrogen
bonds in water ice.
Viscosity plays a crucial role in the measured induction
period and the nucleation rate because of its strong tem-
perature dependence. A model that describes nucleation fol-
lowed by a rapid crystallization of small liquid water
droplets can also describe the crystallization of amorphous
water ice, but only if the amorphous water ice above the
glass transition behaves as a ““ strong ÏÏ liquid. A strong
liquid has a temperature-dependent viscosity that is close to
an Arrhenius law, while a ““ fragile ÏÏ liquid such as normal
liquid water, has a much steeper temperature dependence of
viscosity. This is the Ðrst direct evidence that water ice
warmed above the glass transition is a ““ strong ÏÏ liquid.
When low-density amorphous ice is warmed through the
glass transition temperature, ice undergoes a structural
relaxation. The glass transition opens pathways to di†erent
amorphous positional conÐgurations. The Ðrst di†raction
maximum increases in intensity and narrows slightly. The
oxygen-oxygen radial distribution function derived just
before onset of crystallization suggests that bonding angles
that deviate signiÐcantly from the tetrahedral angle are
relaxed. This structural change is irreversible. The slightly
altered amorphous form, which we call the ““ restrained ÏÏ
amorphous form persists metastably with cubic crys-I
a
r,
talline ice, perhaps because it has domains of hexagonal
short-range order that would require restructuring prior to
cubic crystallization. In that sense, is not merely aI
a
r
viscous liquid form of the structure.I
a
l
The temperature dependence of viscosity is a sensitive
function of the impurity content, more so than the glass
transition temperature or the temperature of crystallization.
As a result, impure astrophysical ices can remain in an
amorphous form much longer (or to higher temperatures)
than expected. On the timescale of the solar system, the
maximum temperature for the persistence of amorphous ice
is as high as T D 90 K, and perhaps as high as T \100 K,
as compared to T \72 K for pure water ice. This increases
the likelihood that amorphous ice is preserved on solar
system bodies, even at such unexpected locations as on the
poles of the planet Mercury.
We thank G. Palmer, who is responsible for a number of
modiÐcations to the electron microscope that made this
work possible. Adam Breon automated the procedure for
analyzing di†raction patterns and did a careful analysis of
the di†raction patterns in the study of the tran-I
a
l ] I
a
r
sition. Part of this work was done while P. J. held a Nation-
al Research CouncilÈARC Research Associateship. The
work was continued under a NASA Cooperative Agree-
ment with the SETI Institute.
No. 2, 1996 AMORPHOUS WATER ICE CRYSTALLIZATION 1113
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PLATE 22
FIG. 4.ÈThe process of crystallization as observed in Transmission Electron Microscope images: bright-Ðeld images during gradual warmup and two
normal images of water ice at 155 K in edge-on view at the edge of a curved fragment of broken carbon Ðlm substrate (a) and in a planar view (b).
JENNISKENS &BLAKE (see 473, 1106)
