Crystal Growth, Surface Phase Transitions and Thermomolecular Pressure

Abstract

The most abundant crystal growth problems in the natural environment involve ice, yet in nearly every setting a crucial roadblock to our progress concerns how microscopic processes influence macroscopic behavior. Recent advances in understanding the surface phase transitions that alter the equilibrium and near equilibrium interfacial structure are central to many pattern formation problems during ice crystal growth, the geometric evolution of polycrystals, and the dynamics of frost heave. This chapter focuses on surface, size and impurity effects in the melting, growth and interfacial evolution of ice crystals. The primary focus in melting and growth concerns how the nature of the microscopic interfacial structure at ice/vapor, ice/gaseous-atmosphere and ice/water interfaces influences the growth shapes. The treatment of impurities in solidification dynamics begins at the molecular level and is traced to the long range transport effects that drive interfacial instabilities. Finally, we examine the dynamical implications of taking subfreezing interfaces into a weakly nonequilibrium regime wherein the underlying causes of frost heave are revealed. Experimental, theoretical and computer simulation techniques all play important roles in our evolving understanding of both the basic phenomena under scrutiny and their environmental implications. The fundamental issues are observed in nearly all materials, but their environmental manifestations are most striking in the case of ice, lying as they do at the heart of the evolving shape of a snowflake, the origin of the forces driving cryoturbation in soils and the microscopic basis of charge transfer inside icy clouds.

Full text

Ice
Physics
and
the
Natural
Environment
Edited by
John
S.
Wettlaufer
Applied Physics Laboratory and
Department of Physics
University of Washington
Seattle, Washington 98105-5640, USA
J.
Gregory
Dash
Department of Physics
University of Washington
Seattle, Washington 98105-1560, USA
Norbert
Untersteiner
Department of Atmospheric Sciences
University of Washington
Seattle, Washington 98105-1640, USA
With 100 Figures and 7 Tables
Springer
Published
in
cooperation with
NATO
Scientific Affairs Division

Introduction
The Advanced Study Institute Ice Physics
in
the Natural and Endangered Environ-
ment was held at Acquafredda di Maratea, Italy, from September 7 to
19,
1997. The
ASI
was designed to study the broad range
of
ice science and technology, and it
brought together an appropriately interdisciplinary group
of
lecturers and students to
study the many facets
of
the subject. The talks and poster presentations explored how
basic molecular physics
of
ice have important environmental consequences, and, con-
versely, how natural phenomena present new questions for fundamental study. The
following sunimary
of
lectures discusses these linkages, in order that overall unity
of
the subject and this volume can be perceived. Not all
of
the lecturers and participants
were able to contribute a written piece, but their active involvement was crucial to the
success
of
the Institute and thereby influenced the content
of
the volume.
We began the Institute by retracing the history
of
the search for a microscopic un-
derstanding
of
melting. Our motivation was straightforward. Nearly every phenome-
non involving ice in the environment
is
influenced by the change
of
phase from solid
to liquid or vice-versa. Hence, a sufficiently deep physical picture
of
the melting tran-
sition enriches our appreciation
of
a vast array
of
geophysical and technical problems.
We found that the theory
of
melting possesses a long and distinguished history in the
study
of
condensed matter, and in particular the interplay between dimensionality and
phase behavior. Crucial in the history
is
the distinction between positional and topo-
logical long-range order; considering only one
is
not sufficient to understand melting.
The search originally focused on a microscopic picture
of
continuous solid breakdown
from within the bulk
of
the crystal, with intermolecular interactions treated
as
har-
monic. Lower dimensional systems seemed to offer the most suitable subjects for
studies
of
continuous melting, but we learned that the best realizations
of
two-
dimensional
matter-adsorbed
monolayer
films-melt
just
as
abruptly
as
three-
dimensional matter. Progress was made by considering the role
of
excitations such
as
vacancies, interstititals, and dislocations, the latter observed using synchrotron x-ray
topography by Ian Baker in ice. But for all the interest continuous melting has cap-
tured, and for all the new physics that has grown out
of
it, it has never been observed
defmitively in ordinary solids. Such effort has thus been important for science but has
left melting in an ambiguous state. By pointing a finger at the surface
of
solids, the
situation has been greatly clarified. All real crystals are finite and their surfaces an-
ticipate their destiny, melting from the outside in.
This process, called surface melting,
is
basic to melting in all materials. It
is
ob-
served in ice at its interfaces with the vapor and other materials and results in the ex-
istence
of
mobile interfacial liquid at temperatures well below the normal freezing
point. It was seen throughout the Institute to be crucial in many varied settings from
frost heave to crystal growth.
Initiated by David Oxtoby's first lecture we turned our attention to a complemen-
tary question: How does solid form in the first place? In other words, within a bulk
fluid or vapor below the normal freezing point, how
is
the solid nucleated? In melting,

2
the
anhannonicity
of
the
outer
atoms
of
the
surface initiates
the
breakdown
of
the
solid
phase.
In
nucleation,
although
the
bulk
solid
phase
is
the
stable
thennodynamic
one,
the
surface
separating
it
from
the
liquid
phase
must
be
created.
The
surface
is
the
antagonist of
the
story,
and
Oxtoby
showed
us
that
there
is
more
to
this
boundary
than
meets
the
eye.
Classical nucleation
theory
fails
to
account
for
the
structural variation
associated
with
moving
from
one
phase
to
the
other.
Plumbing
the
depths
of
the
inter-
face
and
properly
accounting
for
the
thennodynamic
history
of
the
material
are
crucial
for
the
understanding of nucleation
in
general.
Major
roadblocks
in
our
understanding
of
the
nucleation of
ice
impede
our
ability
to
properly treat
the
microscopics
of water
so
as
to
incorporate
the
tenets
of
nonclassical nucleation
theory.
Oxtoby
made
con-
vincing arguments
by
analogy.
For
example,
in
an
"argon-like" material (well
de-
scribed
by
a hard-sphere Lennard-Jones potential)
the
liquid anticipates
the
solid
through
an
ordering
process
which
lowers
the
free
energy
barrier
to
nucleation.
(In
particular, a metastable
bcc
ordering
occurs
at
the
interface
between
a liquid
and
an
fcc
solid.)
By
analogy,
and
reference
to
experimental
data,
Oxtoby
brought
to
discus-
sion
the
question
of
whether
cubic
ice
is
nucleated
fIrst
and
hexagonal
ice
either
grows
from
the
cubic
seed
or
appears
through
a structural
phase
transition.
He
showed
us
that
simulations
demonstrate
the
importance
of electric
fields
in
ice
nucleation
and
illuminated
our
ignorance of
the
solid/liquid interfacial
energy
as
a
function
of tem-
perature, a
quantity
so
important
in
nucleation
and
crystal
growth.
The
experimental
studies
of
ice
nucleation being perfonned
by
Lane
Seeley,
Jerry Seidler
and
Greg
Dash might
shed
light
on
some
of these
issues.
We
learned
about
heterogeneous
nu-
cleation
in
general
and
were
impressed
by
its
importance
in
atmospheric
science
both
in
the
context of
cloud
physics,
by
Marcia
Baker,
and
in
the
context of atmospheric
chemistry,
by
Thomas
Peter.
Nucleation
and
surface
melting
take
us
from
microscopic
to
macroscopic
when
we
consider crystal
growth.
The
length
scale
of
importance
in
the
nucleation
and
growth
of
ice
single crystals
and
polycrystals
ranges
from
molecular
(Angstroms)
to
meters,
and
hence
a proper description requires
an
arsenal
ranging
from
statistical
mechanics
to
transport theory.
On
one
end
of
the
scale
just after nucleation,
when
a crystal
is
smaller
than
any
characteristic
scale
in
the
diffusion
field
of
the
source
material,
its
evolution
depends
solely
on
the
molecular
kinetic
processes
limiting
the
rate
at
which
molecules
attach
themselves
to
the
solid,
the
so-called "interface controlled"
limit.
As
such
a crystal
becomes
larger,
it
begins
to
reach
further
into
the
diffusion
field,
and
the
growth
itself
modifies
the
field,
crossing
into
the
"diffusion
limited"
regime.
Each
regime,
and
the
boundaries
between
them,
presents
us
with
a hierarchy of challenging
and
interesting
problems.
However,
crystal
growth
is
a
history
dependent
process,
and
all
of
these processes conspire
to
create
the
object that motivates
many
of
the
ASI
participants:
the
snowflake.
Many
aspects
of
these phenomena
were
addressed
by
Braslavsky,
Dash,
Furukawa,
Hodgkin,
Maruyama,
Wilen,
and
myself.
Real
ice
surfaces
are
now
being
scrutinized
using
the
tools
of
surface
science
which
we
learned
from
Sam
Fain
and
Victor Petrenko,
who
are
both
employing Atomic
Force Microscopy
in
their laboratories.
The
Institute
was
also
exposed
to
several
types
of "model"
ice
surfaces
and
their
consequences.
Enrique
Batista
and
Y oshinori
Furukawa displayed computer simulations of
ice
surfaces
and
their diffusional
and

3
structural properties. In contrast, Jean Suzanne showed us how similar to ice the
structure
of
water adsorbed on MgO is, and offers this "ice surface" as a potential
model for chemisorption studies
of
atmospheric importance. When water molecules
are deposited ballistically on a very cold substrate one may observe amorphous ices
as
studied by Alexander Kolesnikov and Ji-Chen
Li
using neutron spectroscopy and as
shown by Mayo Greenberg to pervade extraterrestrial environments.
John Nye took us inside an ice polycrystal to observe liquid water present at tem-
peratures below the normal melting point due solely to the curvature and impurity
depressions
of
the equilibrium temperature. The water resides at the tri-grain junc-
tions in microscopic veins (10-100
11m)
and at nodes where four grains come to-
gether. The impurity depression is simply that which we find in normal solution
theory, but the solution resides in the veins and nodes because
of
the strong impurity
segregation properties
of
ice. The consequence
is
that,
if
the temperature
is
suffi-
ciently high, ice poly crystals communicate liquid throughout their volume. Plunging
down between the faces
of
two grains the veins appear to vanish, but does the water
itself disappear? Some participants pointed out that, on a microscopic scale, ice grain
boundaries themselves can interfacially melt, a process facilitated by the presence
of
impurities, so that still more liquid (or quasi-liquid) will
be
present throughout the
polycrystal. The effect might provide a chemical reservoir
in
polycrystalline polar
stratospheric cloud particles (PSC's) allowing the surface to exchange material with
the interior and thus influence chemical reaction rates . Surely many laboratory ex-
periments using vapor deposited ice films experience this phenomenon.
It
had previ-
ously been stressed that these contributions to liquid or quasi-liquid
in
poly crystalline
ice might play an important role
in
facilitating the mobility
of
important chemicals in
ice cores, a kind
of
effective diffusivity enhancing the smearing
of
isochrones. This
diffusivity would thus depend on the crystallinity
of
the ice, and quite clearly the use
of
laboratory values on single crystals might misguide analyzers
of
ice core chemis-
try. The lectures
of
Sigfus Johnson forced the participants to revisit the issue and set
John Nye into action; we expect to see a new publication on the topic soon.
This perspective on water
in
ice also touched on the concept
of
what we mean by
the triple point. How
do
we think about the temperature defming equilibrium between
ice/water/vapor? Such a query has a long history in the thermodynamics
of
phase
equilibrium wherein bulk coexistence concerns a free energy state excluding the con-
tribution
of
interfacial energy. Practically, however, we must construct an ice bath
consisting
of
polycrystalline ice and water, which leads us to consider the question
"should the equilibrium temperature
be
taken as the true triple point?" Or, given the
practical need to use polycrystalline-crushed-ice, should we defme a "practical triple
point" and specify the degree
of
comminution?
The phenomenon also defmes the boundary between "temperate" and "cold" ice
in
polythermal glaciers. The former
is
loosely defmed by glaciologists as ice at a tem-
perature "near"
O°C,
while the latter
is
"cold" and "dry." Both types
of
ice are present
in poly thermal glaciers, and the boundary between them moves in response to
changes in the environment, but it moves slowly, on the order
of
10
meters
in
100
years and slows down with time.

4
The latter study began to bridge the gap from microscopic to macroscopic, taking
the participants up in scale. Nye made a leap in his second lecture on water in ice by
educating
us
on how a particular Icelandic glacier "leaps" periodically
in
a process
called a JOkulhlaup. A complicated sequence
of
events leads to the release
of
a tre-
mendous surge
of
water out from under a glacier, driving disastrous floods
in
the re-
gion. Nye's physically based picture gets the gist
of
the event, and Felix Ng has
showed
us
some recent numerical treatments
of
this impressive piece
of
glaciology.
Basic questions concerning interfacial adhesion and the role
of
surface melting
meet common ground in the important problem
of
the icing
of
structures. Most nota-
ble are ships, which under icing conditions lose their gravitational stability, aircraft,
which lose their aerodynamic integrity, and man-made structures, which buckle under
ice loading. Victor Petrenko showed
us
many examples
of
disasters caused by icing
of
these structures and focused our attention on one aspect
of
the problem that
is
basic
and
unsolved-ice
adhesion. Can we design structures that thwart adhesion?
Some
of
the most important environmental threats are above our heads, in the at-
mosphere. Marcia Baker and Thomas Peter took the Institute on a tour
of
the atmos-
pheric phenomena that sit at the heart
of
climate change. They exposed us to an
impressive range
of
space and time scales from the microphysics
of
cloud formation
and glaciation to the heterogeneous chemistry taking place on polar stratospheric
cloud particles. Baker describes a cloud
as
an extremely complex dynamic multiphase
structure, but one serving as a laboratory for the study
of
nucleation and crystal
growth. A cloud is an amalgam
of
chemistry and physics influencing the global ra-
diation balance and thereby the earth's climate.
We
learned that practitioners study ice
crystal growth in clouds both by flying through them with well-instrumented aircraft
or, as Neil Bacon and Brian Swanson are doing, suspending single ice particles in an
electromagnetic trap. When ice particles collide inside a cloud, Brian Mason and Greg
Dash's laboratory collisions tell us that charge will be exchanged. Perhaps this
is
at
the heart
of
the striking and dangerous phenomenon
of
thunderstorm electrification?
Important clues again implicate the microscopics
of
surfaces.
Moving up into the stratosphere and onto the surface
of
PSC's, which act
as
sub-
strates for chemical reactions, Thomas Peter discussed the reaction HCI + CION02
Cl2 +HN03 regarded by atmospheric chemists to be the most important heterogene-
ous chlorine activation reaction. The reaction can convert a rather inert species into
highly reactive ozone-destroying species within hours under the right PSC formation
conditions. His analysis
of
recent experiments reveals a discrepancy
in
the reaction
rates
of
up to two orders
of
magnitude! Hence, we were exposed to the present and
serious doubts concerning our understanding
of
the most crucial link
in
the chain re-
action leading to stratospheric ozone destruction. The drama will continue to be
played out on model PSC surfaces
in
laboratories, but it raises questions
as
to whether
each laboratory has its "own" PSC surface and how in the future experimental prepa-
rations can best uniformly mimic the surfaces made by nature.
One can consider the glaciation in clouds from another perspective, also
of
impor-
tance in climate change. Snow falls in accordance with meteorological conditions and
it buries clues concerning the deposition conditions
of
past climates. This
is
the basis
of
paleoclimate reconstruction through the analysis
of
ice cores retrieved from the

5
Greenland
and
Antarctic
ice
sheets.
Simply
stated
perhaps
but
certainly not trivially
carried
out.
The
ice
doesn't necessarily
cooperate;
it
flows
under
its
own
weight
and
hence
the
history-depth relationship
requires
an
understanding
of
ice
rheology
and
the
physics
of
ice
flow.
Stan
Paterson provided
us
with
a primer
on
the
latter
field,
and
Kurt
Cuffey
presented current
ideas
concerning
the
rheology.
Sigfus
Johnsen
showed
us
the
time history of climate captured
in
ice
cores,
how
to
read
it
and,
perhaps
the
most difficult,
how
to
retrieve
it-an
immense
logistical enterprise.
An
outstanding
question concerns
the
mechanism
by
which
the
so-called Heinrich
events
occurred.
These
"events"
describe
layers
of sediment
observed
in
the
North
Atlantic sediment
cores
containing
the
geochemical
signature of material originating
in
Hudson
Strait,
Greenland,
and
Iceland.
There
is
evidence
that
an
armada
of
icebergs
made
during
the
retreat
of
the
Laurentide
ice
sheet
may
have
carried
the
debris
composing
the
Heinrich
layers
into
the
ocean
where
melting
released
them.
How
did
the
debris
get
rafted?
Can
such
an
ice
sheet
surge?
How
much
debris
is
required?
Some
of
the
present theories
were
called
into
question,
but
no
new
ones
rose
up
to
replace
them.
Where
there
is
ice
there
is
rock,
and
Krzysztof Birkenmajer
showed
us
that a great
deal
of
information concerning
the
former
is
recorded
in
the
latter.
We
were
given a
guided tour
of
how
the
great global glaciations shaped
high
latitude continents
and
land
forms.
The
information
is
varied
and
difficult
to
obtain,
and
has
taken
him
into
both polar regions
many
times.
Reading
the
climatic
clues
contained
in
rock,
the
re-
shaping of
local
land
forms
and
continental
masses
involves
tying
the
sometimes
dis-
parate techniques of glaciology
and
geology
together.
The
result
is
a
unique
approach
of
viewing
the
earth's
history.
Our
attention
was
drawn
to
the
role
of
ice
in
the
history
of
the
solar
system
and
the
galaxy during
Mayo
Greenberg's lectures. Interstellar
dust
grains
only
a tenth of a
micron
in
size
are
covered
by
water
ice
and
scattered
throughout
the
Milky
Way.
The
striking
fact
is
that
this
dominates
the
galactic
water
mass.
Comets
follow
in
the
mass
hierarchy,
and
Konrad
Kossacki
showed
us
how
the
distribution of
ice
within these
bodies influences their mechanical
and
thermal properties. Following
comets
in
the
extra terrestrial
ice
mass
scale
is
the
ice
found
on
some
planets
and
moons,
and
Jacek
Leliwa-Kopystynski
showed
us
that
water
ice
comprises
half of
the
mass
of
icy
satel-
lites.
The
conditions experienced
by
interstellar
dust
are
recreated
in
the
laboratory,
allowing
ice
formation
and
concomitant
chemical
reactions
to
be
revealed.
Ground-
and
space-based
observations
of
comets
demonstrate
that
they
form
as
an
aggregation
of
interstellar
dust,
and
hence
from
their
early
genesis
consist of
icy
masses.
More
speculative
was
the
issue
of
whether
life
on
earth
arose
from
prebiotic
molecules
un-
der
volatiles
on
interstellar
grains
which
were
deposited
on
our
planet.
The
icy
moon
of
Jupiter,
Europa,
was
also
scrutinized, but
its
shell
will
have
to
await
more
defmi-
tive
observation
before
judgment
can
be
rendered.
Looking back
at
the
earth
from
space
was
Alan
Thorndike
with
an
eye
toward a
minimal description of
ice
and
climate.
By
area,
floating
sea
ice
represents
the
most
dramatic seasonal variation
in
global
ice
cover.
A
minimal
description ofthe balance
of
energy
at
the
surface
of
the
ocean
leads
one
to
the
conclusion that
the
presence
of
ice
is
required.
For
some
it
is
sufficient to
know
that
we
observe
ice
in
the
polar
seas,
but
to
others
addressing
the
question
of
whether
an
ice-free polar
ocean
can
exist
re-

6
quires a fundamental understanding
of
the conditions necessary for the existence
of
the ice-covered state. With this in hand, Thorndike took the view that one can now
model how the low latitudes and the polar regions interact. The important physical
features are that the polar regions are white and hence reflect radiation, and there
is
a
transport
of
heat between low and high latitudes. The temperatures
of
these regions
adjust within the constraint that the system
is
in radiative equilibrium. The concepts
form a minimal model
of
climate. With such a model the issue
of
whether an ice-free
or ice-covered Arctic Ocean describes a stable equilibrium can be addressed. A model
like this can be used
as
a testing ground for global climate simulations run on enor-
mous computers. The advantage is that a minimal model provides relatively simple
solutions in terms
of
the basic properties
of
the materials involved: the ice, the ocean,
and the atmosphere.
Every day one
of
the polar oceans is freezing and,
as
the hemispherical winters
evolve, the implications play an increasingly dramatic role in the lives ofthose.inhab-
iting high-latitude regions.
In
terms
of
population density, the effects are dominant
in
the northern hemisphere, where seas, lakes, and estuaries freeze and modify local
conditions. A minimal model won't tell
us
whether a storm near Helsinki will result in
snowfall over Stockholm. For short-term forecasting and regional predictions in ice-
infested areas, one needs to put
as
much physics into the air/sea/ice model as
is
dic-
tated by the particular environment
of
interest and the observations available. Anders
Omstedt introduced the Institute to this practical "maximalism" beginning from the
simplest situation
of
a shallow lake and developing the necessary tools for dy-
namic/thermodynamic predictions in freezing seas. We were cautioned about the
proper way to assimilate meteorological and oceanographic data in real-time predic-
tions and exposed to the varied problems associated with ice formation in waters near
populations. Most notably, shipping
is
influenced severely, but less well known is the
role that freezing lakes play in the modification
of
regional climate. Areas such as
Scandinavia, with a tremendous areal density
of
lakes, lend a strong impetus to under-
stand these processes.
The motion
of
ice in frozen seas has
an
important influence on large-scale heat and
mass balance, as seen in Ron Kwok's analysis
of
sea ice motion which blends posi-
tion information taken from buoys and satellite data. As the ice moves into warmer
waters it melts, thereby removing heat from the ocean and leaving the surface waters
less saline. The large-scale effect
of
this process can have significant oceanographic
consequences as was shown by Peter Wadhams. As one piece
of
sea ice encounters
another, the weaker one may be deformed irreversibly.
J.
Dempsey showed how this
process scales using concepts
of
fracture mechanics. Many participants are occupied
with various aspects
of
modeling sea ice growth (D. Feltham), observing its motion,
modeling its large-scale mechanical and geometric properties, and making other field
observations on snow and glacial ice (S. Gerland, K. Hietala,
T.
Johnsen,
A.
Paelli,
A. Sinisalo, N. Steiner,
P.
Uotila, L. Smedsrud).
Anyone who has traveled to a marginal ice zone, where sea ice meets the open
ocean, is struck by a particular contrast with the central ice pack: the tremendous
abundance
of
animal life. The food chain, at the crudest level, is obvious. There
is
more light in the water column and biological blooms are frequent. Fish feed on these.

7
Birds and seals feed on fish, and polar bears feed on seals. Thick ice hinders the proc-
ess but, as Ole
L0lUle
showed us, it constitutes an important link in the chain.
L0lUle,
who goes into the Arctic but then ventures under the sea ice, studies this link, which
takes the form
of
three species
of
shrimp-like animals called sympagic macro-fauna.
These creatures cannot swim very well and hence cling to the inner walls
of
brine
channels within sea ice. The wind driven motion
of
sea ice moves a colony long dis-
tances while its population evolves locally on its ice floe home. However, when the
ice meets the open ocean, rapid ablation occurs and the macro-fauna lose their shelter
and are thus suspended throughout the water column, where fish and birds have ready
access to them. Herein lies a very interesting case wherein biological and physical
approaches may blend to solve a puzzle. Most
of
the perennial ice produced in the
Arctic Ocean is exported through Fram Strait and eventually melts.
L0lUle
has ob-
served the macro-fauna population densities on perennial ice. Because ice is continu-
ally lost, there
is
a sink
of
macro-fauna in the marginal seas. How is the population
maintained and transported? Where
is
the source? How long do they live? The Insti-
tute brought together experts on sea ice motion who together with
L0lUle
can address
these and other important biological questions relying on
an
understanding
of
ice
physical science.
This discussion
of
such a pristine substance might have swayed a less educated
group into believing that all ice
is
clean. Many
of
the participants are experts on just
how dirty important natural environmental and technical systems can be. Fiorenzo
Ugolini first emphasized the very large fraction
of
polar regions that consist either
of
ice on land or partially, perennially and permanently frozen soils. Soil science relies
on tools and techniques from geochemistry, geology, botany, geography, and those
that its practitioners must invent themselves. In the polar regions one must fold ice
into the picture and the complete tapestry
is
a rich one.
We
learned that polar soils can
be desert-like because
of
the low precipitation, and have the added exposure to intense
cold and seasons where the sun either never shines or never sets. When humidity
is
adequate a tundra can form, but the paucity
of
liquid water has made agriculture in the
northern regions extremely challenging. The successes and failures were traced by
Ugolini with the main ingredients
of
the former being the choice
of
intrinsically ro-
bust crops, choice
of
location and, interestingly, the sociological factors surrounding
the process. The expense
of
food and other supplies
in
remote polar regions
is
due
mainly to high transportation costs. This must be weighed against the relatively tech-
nical process
of
local production
of
produce and other consumables. Were it substan-
tially simpler to purchase food than to grow it nearby, what would you do?
Refrigeration is never a problem!
The natural thermal fluctuations in saturated soils result
in
the deformation in the
direction
of
the heat flux vector
of
the material known
as
"frost heave."
We
endow the
term with quotations for the simple fact that the phenomenon can be explained by no
single mechanism. There are two very important manifestations
of
frost heave. Per-
haps the most aesthetically pleasing and curious
is
the "cryoturbation"
of
land forms
into impressive features including sorted stone circles and ice wedge polygons. The
motion
of
the soil on this scale has severely deleterious effects on man-made struc-
tures such as pipelines. Although there
is
a contrast between observing naturally oc-

8
curring geomorphological
features
and
engineering pipelines
in
order
to
avoid
the
associated effects
of
differential ground
motion,
understanding
both
requires a
con-
certed
focus
on
the
basic
mechanisms
of
frost
heave.
What
is
certain,
as
was
stressed
in
the
presentations of
Masami
Fukuda,
Peter
Wil-
liams,
Alan
Rempe,
and
myself,
is
that
the
mobility
and
concomitant
pressure
of
wa-
ter
at
temperatures
below
the
bulk
freezing
point
is
important.
I stressed
that
there
are
a
number
of physical
mechanisms
responsible
for
this
water:
interfacial
melting,
cur-
vature,
proximity
of
soil
grains,
and
impurities.
Each
has
unique
temperature
depend-
encies
and
hence each contributes
to
the
volume fraction
of
water with varied
strengths depending
on
the
temperature.
Each
of these
has
been
studied
in
specific
experiments
using
techniques
of
modem
condensed
matter
physics.
Alan
Rempel
dis-
played
the
consequences
of
these
in
a
model
porous
medium.
Bo
Elberling
is
studying
gas
transport
in
partially
saturated,
partially
frozen
soils.
Williams
and
Fukuda
introduced
us
to
the
topic
of
frost
heave
from
an
engineering
perspective,
within
which
we
are
presented
with
two
very
different
challenges.
There
are
two
extensive
oil
pipelines
in
the
northern
hemisphere,
in
Alaska
and
the
Arctic
of
the
Former
Soviet
Union.
Essential
to
environmental
safety
during
their operation
and
to
the
efficiency
of
transport
is
engineering
that
maintains
the
mechanical
integrity
of
the
pipeline
in
the
presence
of differential
ground
motion
that
can
be
caused
by
frost
heave.
The
costs
in
terms
of
environmental
damage
and
the
price
we
pay
for
trans-
portation
are
immense,
and
the
phenomena
driving
frost
heave
are
the
most
important
scientific
problem
coupling
the
two.
This
was
one
of
Peter Williams'
main
messages
.
Masami
Fukuda introduced
us
to
the
second
challenge:
artificial ground
freezing.
A
frozen
soil
has
a
vastly
superior mechanical integrity
than
its
unfrozen counterpart,
making improvements
in
freezing
technology paramount
to
engineering safety
and
structural stability.
We
were
treated
to
a
history
of
the
field
and
shown
the
large-scale
ground
freezing
projects
in
Japan
that
he
has
been
involved
in.
The
main
usage
is
in
stabilization
of
water-saturated
soils
during
pipeline
or
tunnel construction
or
other
excavations.
The
same
techniques
provide
promise
in
the
area
of
waste
containment
because
in
addition
to
mechanical superiority,
the
very
low
diffusivity
and
small
phase
segregation of
impurities
in
ice
spotlight
frozen
soils
as
important
potential
bar-
riers
for
nuclear
and
nonnuclear
materials.
This
is
the
main
focus
of
Greg
Dash's
sec-
ond
chapter.
The
last point,
as
we
learned
from
Lydia
Popova,
should
not
be
taken lightly.
Dumping
of
military
and
nonmilitary radioactive
wastes,
the
testing
of
nuclear
weap-
ons,
the
use
of nuclear
weapons
as
tools
for
excavation,
radioactive fallout
from
mili-
tary
and
nonmilitary
sources,
and
oil
spills
have
all
left their signatures
in
the
Arctic
of
the
Former
Soviet
Union.
Approximately
sixty
percent of
this
region
is
either
per-
ennially
or
seasonally
ice
covered,
which
provided
the
Institute
with
a clear
link
be-
tween a vast array
of
scientific fields, ranging
from
solute trapping
in
growing
contaminated
sea
water
to
the
mobility of
contaminants
in
permafrost,
and
the
ques-
tion
of
long-term
environmental
danger.
Popova
gave
us
an
appreciation
for
the
diffi-
CUlty
in
assessing
the
present
day
state
of
damage
done
by
these
and
other activities,
simply
identifying
the
sites
and
the
extent of
the
varied environmental threats
is
a
monumental
task
requiring a concerted effort
by
both
government
and
industry.
The

9
cost
of
identifying existing
sites
and
preventing
future
ones
is
substantial.
The
cost of
not
doing
so
is
so
immense
it
escapes
assessment.
Future
identification, monitoring,
remediation,
and
containment strategies will build
on
scientific
and
engineering
di-
rected
at
both
fundamental
and
applied
aspects.
These
problems
occurring
in
northern
regions will
rely
heavily
on
knowledge
gained
in
fields
where
the
physics
and
chem-
istry
of
ice
underlie
the
environmental
consequences.
At
the
conclusion of
the
ASI,
most
participants
felt
that
the
meeting
was
successful.
Because
we
came
from
many
different disciplines,
we
were
novices
in
many
of
the
sessions,
and
the
lectures
had
to
be
more
descriptive
and
less
detailed than
they
are
in
more
specialized
conferences.
Some
of
us
came
away
with
a
deep
respect
for
the
dif-
ficulties
and
subtle aspects of another discipline.
In
1.1.
Rabi's
1955
Morris
Loeb
Lecture
he.
stated
his
belief that "the
value
content of
science
or
literary scholarship
lies not
in
the
subject matter
alone,
or
even
in
greater
part.
It
lies
chiefly
in
the
spirit
and
living
tradition
in
which
these
disciplines
are
pursued.
The
spirit
is
almost
always
conditioned
by
the
subject."
It
is
hoped
that
this
collection will
convey
the
spirit of
the
Institute
and
hold
the
interest of
novices
and
professional
ice
scientists.
J.
S.
Wettlaufer

Crystal Growth, Surface Phase Transitions and
Thermomolecular Pressure
I.S.
Wettlaufer
Applied
Physics
Laboratory
and
Department
of
Physics,
University
of
Washington,
Seattle,
Washington,
98lO5-5640,
USA.
Abstract.
The
most
abundant
crystal
growth
problems
in
the
natural environment
involve
ice,
yet
in
nearly
every
setting a crucial roadblock
to
our
progress concerns
how
microscopic
processes
influence macroscopic
behavior.
Recent
advances
in
un-
derstanding
the
surface
phase
transitions that alter
the
equilibrium
and
near equilib-
rium
interfacial structure
are
central
to
many
pattern
formation
problems
during
ice
crystal growth,
the
geometric evolution of
poly
crystals,
and
the
dynamics
of
frost
heave.
This
chapter
focuses
on
surface,
size
and
impurity effects
in
the
melting,
growth
and
interfacial evolution of
ice
crystals.
The
primary
focus
in
melting
and
growth
concerns
how
the
nature
of
the
microscopic interfacial structure
at
ice/vapor,
ice/gaseous-atmosphere
and
ice/water interfaces influences
the
growth
shapes.
The
treatment
of
impurities
in
solidification
dynamics
begins
at
the
molecular
level
and
is
traced
to
the
long
range
transport
effects
that
drive
interfacial instabilities. Finally,
we
examine
the
dynamical implications of taking subfreezing interfaces
into
a weakly
nonequilibrium regime
wherein
the
underlying
causes
of
frost
heave
are
revealed.
Experimental, theoretical
and
computer
simulation
techniques
all
play
important
roles
in
our evolving understanding of
both
the
basic
phenomena
under
scrutiny
and
their
environmental
implications.
The
fundamental
issues
are
observed
in
nearly
all
materi-
als,
but their environmental manifestations
are
most striking
in
the
case
of
ice,
lying
as
they
do
at
the
heart of
the
evolving
shape
of a
snowflake,
the
origin
of
the
forces
driving cryoturbation
in
soils
and
the
microscopic
basis
of
charge
transfer
inside
icy
clouds.
Keywords.
Ice,
Surface
Melting,
Crystal
Growth,
Intermolecular
forces
1 Introduction
Several surface-specific structural
phase
transitions
are
observed
in
ice.
We
will
focus
here
on
complementary
aspects
of
the
basic
phenomena
of
interfacial
melting,
surface
roughening,
and
molecular kinetics
and
their application
to
the
equilibrium
and
growth structures of
ice
crystals. A central issue
is
how
the
interfacial structure
at
ice/vapor, ice/gaseous-atmosphere
and
ice/water interfaces influences
the
adsorption
potential,
the
growth
shapes
and
surface
transport properties.
The
environmental rele-
vance
concerns
the
fact
that
ice
dominates
the
crystal
growth
phenomena
we
observe
NATO
AS! Series, Vol. ! 56
Ice Physics and the Natural Environment
Edited by
John
S.
Wettlaufer,
J.
Gregory Dash and Norbert Untersteiner
© Springer-Verlag Berlin Heidelberg 1999

40
on the surface
of
the earth and
in
the atmosphere. The broad scientific relevance con-
cerns the fact that surface phase transitions occur in a host
of
materials, so that what
we learn in ice may be important in many other settings. Underlying these problems is
the interplay between phase behavior and dimensionality, a topic that developed in the
context
of
quantum materials (Dash et aI., 1994), and a parallel development that
grew out
of
Nakaya's studies
of
ice crystal growth (Nakaya 1954; Furukawa 1997).
The long search for a microscopic theory
of
melting displays how our focus has
moved from the bulk to the surface
of
a crystal. The formation
of
a crystal from a va-
por or a melt phase has its own history in the field
of
nucleation. Both topics, dis-
cussed in complementary chapters by Greg Dash and David Oxtoby, come together in
crystal growth.
2 Surface Structure
and
Growth
2.1
Equilibrium
Crystal
Shapes
It
is
not widely appreciated that the plethora
of
naturally occurring solids we encoun-
ter, ranging from mineral formations to snowflakes, have experienced many
of
the
same basic growth processes during their evolution. Their most striking feature is
their overall shape. Therefore, it
is
important to examine the extent to which this
shape can be understood in terms
of
equilibrium thermodynamics. The concept
of
a
global geometry that arises from an extremum principle is broadly relevant in many
areas
of
condensed matter science and mathematics, and a great deal
of
our contempo-
rary approach to such questions originated in Wulffs goal to understand the shapes
of
naturally occurring crystals (Wulff, 1901). Experiments, exact solutions
of
micro-
scopic models, and mean-field theory have demonstrated that the equilibrium shape
of
a crystal depends on temperature (excellent reviews include Weeks and Gilmer 1979,
Lipson and Polturak 1987, and Balibar et
aI.,
1993). Classical theories have been suf-
ficient to demonstrate the main qualitative features: An ideal dislocation free crystal is
fully faceted at absolute zero, and becomes more rounded, or locally rough, as its
temperature increases.
The surface free energy
of
a crystal,
y(n),
depends on the orientation
of
the unit
normal n
of
the surface relative to the underlying crystalline lattice. The crystal itself
is
built by a replication
of
unit cells. When the dominant intermolecular interactions
in the material are short ranged, one can estimate the bond energies by considering
only nearest neighbors in the lattice.
We
can then think ofy(n)
as
the sum
of
the ener-
gies
of
all the bonds broken per unit area in the creation
of
the surface. The energy
itself depends on where, relative to the underlying crystalline lattice, the surface
is
created
1.
If
one knows a-priori, the surface free energy
of
the crystal as a function
of
1 Note that such a treatment of the surface free energy ignores the role of curvature in local
distortion
of
the surface, an effect with increasing importance
as
a crystal becomes smaller
(e.g.,Oxtoby 1998).

41
all
orientations present, then
the
geometric Wulff construction determines the equilib-
rium crystal shape (e.g.,
Wulff,
1901;
Herring,
1951;
Cahn
and
Carter, 1996). Con-
versely, were
one
to
measure
the
shape
of
a crystal
in
the
absence
of
a growth drive2,
the Wulff construction will provide
the
surface
free
energy
for
all
orientations present
on
that shape (e.g., Heyraud
and
Metois,
1986).
We
suppose that
y(n)
is
known
for
all
crystal
faces,
and
the
construction determines
uniquely the equilibrium crystal
shape
by
minimizing
the
orientation-dependent total
surface
free
energy per unit area
for
the
volume enclosed
by
the
crystal surface:
Is
= f
r(n)dA.
s
(1)
We
can consider Wulffs construction
as
a transform
on
the
surface
free
energy itself,
acting to minimize the
above
integral at constant volume, temperature, and chemical
potential;
ner=/"'y(n).
(2)
Here r
defmes
a radial vector
from
the
origin
to
the
equilibrium crystal surface, which
is
given
by
the inner envelope
of
a set
of
planes perpendicular
to
radial rays, inter-
secting a polar plot
of
y(n)
as
shown
in
Fig.
1.
The
boundary
of
the equilibrium
Wulff
shape
is
given
by
equation
(2)
and
is
geometrically similar
at
any
size determined
by
/..,
above (e.g., Taylor et
aI.,
1992). Dinghas (1944), Herring (1951) and Landau
(1965), among others, have provided proofs that Wulffs construction does indeed
minimize
Is.
An
example with which
we
have
some
familiarity
is
that
of
a soap bub-
ble, wherein
y(n)
is
independent
of
n
and
in
the
absence
of
gravity
the
minimization
provides the spherical equilibrium
shape.
The
effect of gravity
is
only important when
the body under consideration becomes large (e.g., Avron
et
aI.,
1983).
Cusps
in
the
polar plot ofy(n) locate high symmetry crystallographic planes that
are
molecularly smooth.
We
call these facets
or
singular surfaces, the latter referring to
the nonanalyticity
in
the
y(n)
plot. Note that because
some
cusps have a lower
free
energy than others, not all cusps appearing
in
the
y(n)
plot will
be
represented
by
a
facet
on
the Wulff shape. Surfaces
of
a crystal normal
to
which there
is
a gradual
variation over the scale of a number of molecular planes
are
called molecularly rough
or diffuse surfaces.
An
equilibrium crystal shape
may
be
fully
faceted
or
completely
rough,
or
may
have
facets
and
rough regions coexisting.
2The growth drive for crystallization is defined in Equation (3).

42
Crystal
Shape
Fig.
1.
The
boundary
(bold
lines)
of
the
equilibrium
crystal
shape
with
cubic
symmetry
formed
from
the
Wulff construction,
which
is
the
interior
envelope
of
the
set of perpendiculars
to
radial
rays
intersecting
the
polar
plot
of
surface
free
energy
(lighter
lines).
2.2 Surface Roughening
Two thennodynamic interfacial transitions are relevant here. The first involves a
structural rearrangement
of
solid material. A crystal surface
is
characterized by the
free energy
of
the various interfacial configurations: faces, ledges, comers, edges, and
point-defects such as admolecules and vacancies. The energetic cost
of
any state is
reckoned with the sum
of
the fonnation enthalpies
of
the sites, and the benefit toward
lowering the total free energy comes from increasing the configurational entropy
of
the surface. The latter
is
the driving force for the thermodynamic roughening transi-
tion. As the temperature increases, thennal fluctuations give rise to the continual for-
mation
of
steps. Creating a step
is
tantamount to slightly tilting the surface, which
requires doing some work against the periodic potential
of
the underlying crystalline
array. As the temperature Tincreases, thennal agitation facilitates step fonnation and

43
the
free
energy cost of a
step,
cr
s(1),
decreases.
The
coherence
length of
the
surface,
11=')'(n)/cr
s(1),
is
a
measure
of
the
distance
(relative
to
the
mean
orientation
of
the
sur-
face)
over which fluctuations
at
two
points
are
correlated
(e.g.,
Lipson
and
Polturak
1987).
For
an
infmite two-dimensional surface
cr
s(1)
->
0
at
the
roughening tem-
perature, Tr, thermal
agitation
liberates
the
surface
from
the
ordering
influence of
the
underlying crystalline
lattice
and
fluctuations
are
correlated
on
all
length
scales
so
that
the
surface
roughens.
Strictly
speaking,
one-dimensional interfaces
are
always
rough,
and
three-dimensional
(and
higher)
interfaces
are
never
rough.
Any
real crystal
has
facets
of
fmite
size,
and
as
the
temperature increases
11
ap-
proaches
the
facet
size
and
they
roughen
at
T = Tf < T
r.
If
T>
Tf
for
all
facets
present,
an
equilibrium
shape
will
be
completely
rough
and
rounded.
3
There
is
a substantial
literature
in
the
field,
and
while
theoretical treatments
are
in
qualitative agreement,
predictions of
Tr
and
Tf
are
strongly
model
dependent.
Because
the
roughening tran-
sition
is
responsible
for
marked
interfacial restructuring,
the
process
has
a profound
influence
on
the
dynamics
of
molecules
on
the
surface,
and
ultimately
on
near equilib-
rium
crystal growth
shapes.
This
point
reminds
us
of
an
intimately related
phenome-
non
to
be
described
in
a subsequent section; the kinetic
or
dynamic
roughening
transition,
which
occurs
when
the
growth
rate
exceeds
a
critical
value.
There
are
other
distinct, but closely related transitions associated
with
surface
reconstruction that
do
not concern
us
here
but
which
are
fundamentally important
and
often observed
in
other materials
(e.g.,
den
Nijs,
1997).
There
have
been
few
definitive
studies
of
these
transitions
as
they
occur
in
ice.
The
roughening transition of
the
prism
facet
of
ice
Ih
has
been
observed
under
vapor
con-
ditions
by
Elbaum
(1991).
A beautiful series of recent experiments
has
been
able
to
address these transitions
over
a
tremendous
thermodynamic
range
for
ice
in
contact
with water (Maruyama
et
aI.,
1997).
Maruyama,
has
demonstrated
both
the
roughen-
ing
transition
of
the
prism
facet
of
ice
Ih
and
the
equilibrium crystal
shape
by
care-
fully controlling the pressure
along
the
solid-liquid coexistence
line.
Using a high
pressure optical
cell,
he
observed
shape
changes
from
near
the
triple point
to
-21°C
at
200
MPa.
He
observed
the
prism
facets
to
undergo
a
thermodynamic
roughening tran-
sition
at
approximately
-16°C
at
165
MPa.
The
technique
has
the
ability
to
accurately
control the
growth
drive
very
near
equilibrium
and
hence
can
test theoretical predic-
tions
of crystal
shapes
growing
and
melting
near
equilibrium.
2.3 Interfacial and Surface Melting
The
second transition
to
be
discussed
is
a
kind
of
lower
dimensional melting.
The
description
of
the
melting of a solid
as
a first-order
phase
transition
is
a consequence
of
the
discontinuous
change
in
bulk
quantities
at
the
transition
point.
However,
every
crystal
is
fmite,
and
bounded
by
its
own
surface
area
where
the
process of melting
may
actually
be
initiated:
If
there
were
a
layer
of
liquid
at
the
surface,
at
temperatures
below
the
bulk
melting
transition,
then
there
is
little
need
to
activate
the
melting
proc-
3Note
that
11
diverges logarithmically
with
crystal
size.

44
ess. Such a deduction was made by Stranski (1942) on the basis
of
the inability to
superheat crystals. However, since it
is
well known that liquids can be supercooled,
the melting process exhibits an asymmetry about the transition point. The chapters by
Dash and Oxtoby alleviate the need to develop any more detail than is necessary to
insure that this presentation is reasonably self contained.
Recent studies have borne out the intuition
of
Stranski. The main issues are em-
bodied in the effect
of
surface melting, which takes place at the surface
of
a crystal in
contact with its vapor and is therefore extremely important in environmental ice sci-
ence (Dash et al., 1995). Surface melting occurs when a liquid wets its own solid in
the form
of
a stable thin liquid layer disjoining the solid/vapor interface at tempera-
tures below the bulk transition temperature T
m'
(Dash 1998; Oxtoby 1998).
It
is an
equilibrium phenomenon driven by the tendency to reduce the interfacial free energy
of
the system. The process occurs
in
many classes
of
solids, including metals, semi-
conductors, solid rare gases and molecular solids. Typically surface melting begins at
T < 0.9 T
m,
with a film thickness
of
one or two monolayers, which thickens gradually
with increasing
T,
and diverges at T
m'
In a modem framework, we think
of
the prob-
lem
as
a special case
of
wetting (e.g., Schick, 1988). In surface melting the underlying
solid consists
of
the same material
as
the liquid above. Whether or not a layer
of
melt
beads up, or spreads to form a thin film on its own solid depends on a competition
between the attraction
of
the liquid to the solid, adhesion, and the attraction
of
the
liquid to itself, cohesion. The origin
of
the competition may be long range intermo-
lecular dispersion forces, which have unique consequences in ice systems (Elbaum
and Schick, 1991).
If
we replace the vapor phase by a wall
of
a different material, the
process is referred to
as
interfacial melting and the mean field thermodynamic de-
scription is essentially the same. In this case unscreened long ranged surface ioniza-
tion forces are important (Wilen et al., 1995). Depending on the nature
of
the
substrate and the temperature, system specific short ranged electrical interactions may
also be present (Petrenko, 1993).
Most relevant to pattern formation during the growth
of
ice crystals
is
the surface
melting
of
ice at interfaces with vapor and gaseous atmospheres (Furukawa, 1997).
There have been several laboratory experiments, with extremely variable and complex
results.
We
believe that the variability lies in both the great sensitivity to the atmos-
phere and conditions
of
preparation, and in the inherent differences in measurement
technique. The evidence
of
surface melting comes from studies by proton backscat-
tering, ellipsometry, optical reflectometry and interference microscopy, X-ray dif-
fraction and glancing angle X-ray scattering. Because the techniques themselves are
designed to measure different structural aspects
of
the surface there
is
still active de-
bate in the community regarding the interpretation
of
the various resultS throughout
the entire temperature range (Dash et al., 1995). A particularly interesting effect ob-
served by Elbaum et al., (1993)
is
worth noting here.
It
is
called incomplete surface
melting, and is distinguished by the coexistence
of
water droplets with a thin water
film at the ice/vapor interface. The effect
is
due to the attenuation with film thickness
of
the interactions that favor adhesion. Regardless
of
the complexity
of
the results,
what is certain is that sufficiently close to the bulk melting transition, the disordered
surface, whether liquid-like or taking some other structure, the growth kinetics will be

45
substantially altered (Kuroda
and
Lacmann,
1982;
Sei
and
Gonda
1989;
Furukawa
and
Kohata,
1993;
N
ada
and
Furukawa,
1996).
2.4 Near Equilibrium Crystal
Growth
In
addition to the aesthetic appeal of crystal growth shapes, their understanding
in-
forms
many
areas of technology,
from
computer
chip
manufacturing
to
powder met-
allurgy. Furthermore, because growth shapes
are
often cast
in
the
form
of
a free
boundary problem, there
are
a myriad of theoretical analogues
in
hydrodynamic,
chemical
and
biological systems (e.g., Kirkaldy
1992,
Cross
and
Hohenberg
1993).
The problems
of
equilibrium crystal shapes
and
growth shapes merge when
one
be-
gins
to
address nucleation
and
subsequent solidification (Frank
1958;
Chemov
1984;
Elbaum and Wettlaufer
1993;
Bagdassarian
and
Oxtoby
1994;
Wettlaufer et
aI.,
1994).
In
the laboratory, crystal growth
very
close
to
equilibrium
can
be
achieved
(e.g., Heyraud
and
Metois
1983,
1986;
Carmi et
aI.,
1987;
Elbaum et
aI.,
1993;
Maruyama et
aI.,
1997).
When
the growth
drive
is
removed, a crystal growth shape
must eventually relax
to
the
equilibrium
form.
The
relaxation time depends
on
the
mechanism facilitating
the
redistribution of material. For example, when surface dif-
fusion dominates the relaxation
times
are
typically extremely
long
(Cahn
and
Taylor
1994).
Crystals observed
in
Nature
are
often growth shapes that
are
still undergoing
slow
relaxation toward equilibrium. Indeed, nearly
all
the
shapes
we
observe depend
in
some
manner
on
the
history of
the
object.
David Oxtoby (1998) describes
how
a seed of the solid phase
is
nucleated. Subse-
quent
to
nucleation,
the
manner
in
which
growth
proceeds depends
on
the state of
the
crystal surface
and
the
thermodynamic conditions of the nutrient phase. Our main
goal
in
crystal growth
is
to
determine
the
evolution of
the
unknown shape of the
solid.
We
must take care
in
understanding
the
dominant processes through which a mole-
cule
in
the nutrient phase
becomes
part ofthe
solid.
For example,
the
shape
evolution
depends
on
whether the interfacial motion
is
controlled
by
long-range diffusion
Or
by
local interfacial processes.
We
first consider single crystals growing
from
a pure par-
ent phase
and
defme
the
two
common
growth
regimes.
Second,
we
add
impurities
and
discuss a number of the phenomena that control
the
structure
of
sea
ice
and
other
natural
ice
forms.
The
growth drive
for
crystallization
is
the
difference
in
the chemical potential
be-
tween a molecule
in
the parent phase
and
one
in
the
crystal. Molecules
move
in
such
directions
as
will lower chemical potential.
When
bulk
solid
and
liquid phases
are
in
coexistence
at
temperature T m
and
pressure
Pm
the
chemical potentials of both phases
are
equal:
Il sU 'm ' Pm) = Ilf.U'm ' Pm),
where
the
subscripts
denote
the solid (S)
and
liquid
(f)
phases. Hence, there
is
zero growth drive for crystallization:
11
f.
U'm ' Pm) -
11
s U'm ' Pm)
==
dll =
O.
However, were
we
to
adjust
the
pressure
by
dP and/or the temperature
by
d T
to
extend
into
the
solid region of
the
bulk phase
diagram, then the solid becomes
the
stable
phase
and
its
chemical potential
is
reduced
relative
to
the
liquid phase,
and
the
growth
drive
for
crystallization becomes positive.
Expanding the growth
drive
about
the
point (T
m,P
m)
we
fmd

46
(3)
Experimentally it
is
ideal to create either isothermal
or
isobaric control. For
ice
and
water very well controlled growth
can
be
achieved
in
the range 0 m -20°C.
With
either
-AP
=
0.1
MPa
or
-
AT=
O.Ol°C
one
can obtain
AJ!lkbT
=
10-
3 (e.g.,
Maruyama
et
aI.,
1997).
We
do
not
know
in
detail the relationship between a nucleated
shape
and
an
equi-
librium
shape.
However,
it
is
reasonable
to
consider the latter
as
one
that experiences
the least activation barrier,
is
the most likely
to
be
nucleated
and
hence
is
the
initial
shape
for
growth studies.
The
manner
in
which
molecules attach
to
the
existing solid
at a specific orientation
is
loosely referred
to
as
growth kinetics. These kinetics
de-
pend
on
the local interfacial
free
energy
in
a particular crystallographic orientation.
We
can distinguish
two
broad regimes of
growth:
interface-controlled
and
diffusion-
limited.
The
former regime
is
best treated
in
the
context of
geometric
theories wherein
the interfacial growth velocity
is
controlled only
by
local surface parameters that
are
decoupled
from
diffusional or other long-range influences. 4
More
technical descrip-
tions are given
by
Taylor et
aI.,
(1992) Yokoyama and Sekerka, (1992), and
Wettlaufer et
aI.,
(1994).
For interface controlled growth, the goal
is
to
formulate a theory
for
evolution
of
the overall crystal
shape
by
constructing
the
local velocity V(n) normal
to
an
interfa-
cial point.
The
theory should
be
based
on
our knowledge concerning molecular at-
tachment kinetics
in
single interfacial states, faceted or rough, or the transition
between them (Burton et
aI.,
1951;
Cahn
1960;
Cahn
et
aI.,
1964;
Weeks
and
Gilmer
1979;
Chernov
1984).
In
1958
Charles Frank initiated this
type
of study
in
a paper
that
is
highly recommended to
any
student of
the
topic.
The
essential point of the
simplest class of geometric
models
is
that
the
local interfacial velocity will depend
on
local interfacial processes rather than
on
long-range processes
such
as
diffusion
in
the
bulk. A number
of
familiar situations wherein this
is
the
case
include the early stages
of
atmospheric
ice
growth, when the mean
free
path
in
the vapor
is
larger than the
characteristic size of the crystal, the growth of electronic materials using the tech-
niques
of
molecular beam epitaxy,
the
growth of ferromagnetic
or
ferroelectric
do-
main walls, grain growth,
and
stress-driven-zone migration. It
is
important
to
stress
that a number
of
instabilities associated with diffusional growth cannot manifest
themselves
in
this regime. Within a geometric
framework,
we
have
been
able
to
ex-
plain
some
general features of transient anisotropic growth observed
in
metals,
4He,
and
ice.
We
first review basic growth kinetics
and
then
ask
how
a well defmed crys-
talline surface, either faceted
or
rough,
moves
normal
to
itself under
the
imposition of
a
weak
growth
drive
AJ!
that
is
spatially homogeneous
and
steady.
4rhe
diffusion-limited regime can only be treated
by
a geometric theory in special limiting
regimes.

47
2.4.1
Growth
Kinetics
2.4.1.1
Growth
from
the
Melt
Consider a facet in equilibrium
(Lll!
= 0). The addition
of
any integral number
of
identical lattice planes provides the same, minimum free energy, equilibrium configu-
ration The surface itself (except for its size)
is
identical after the completion
of
each
layer. A small growth drive will create some fractional coverage
of
molecules on the
surface, thereby increasing the free energy, until a complete plane
is
added and the
low energy surface is regained. Hence, in analogy with nucleation, a facet
is
metasta-
ble during growth. Continuous growth normal to a facet relies primarily on the lateral
motion
of
steps across the surface (Burton et al., 1951, Cahn 1960). The term step
is
most often reserved for a single molecular layer, but less restrictive defmitions are
appropriate because it
is
observed often
in
semiconductors (Liu et al., 1997), and can
be inferred from optical measurements on ice (Hallett 1961), that steps consisting
of
many molecular layers propagate across a growing facet. I view Cahn's (1960) defmi-
tion to be the clearest: A step describes the transition between two adjacent, parallel,
and structurally identical regions
of
a surface that are separated by an integral number
of
lattice planes. Defect- and dislocation-free facets have available only one step gen-
eration mechanism: the two-dimensional nucleation
of
solid clusters. Other distinct
step generation mechanisms such
as
screw dislocations, or Frank-Read sources (a
ledge separating a pair
of
oppositely signed screw dislocations) are often observed.
Each mechanism has a specific form relating the growth drive to the growth rate, and
which mechanism dominates depends on the material and the principal facets being
studied. Geometric theories can treat a variety
of
step generation mechanisms.
When the primary facets
of
ice are defect free, the relevant step generation mecha-
nism is the formation
of
two-dimensional nuclei. The process
is
driven by thermally
activated fluctuations
of
the liquid phase that create two-dimensional heterophase
clusters with solid-like structure. The most probable cluster geometry
is
that
of
a pill-
box, for which the edge-to-surface free energy ratio will favor spreading at a given
drive. In analogy with three-dimensions, the nucleation frequency I per unit facet area
A is
of
a Maxwell-Boltzmann form, I oc exp( -7t /
Lll!
kb
T), where is the free
energy
of
a critical nucleus on the facet (e.g., Weeks and Gilmer 1979).
If
lattice
plane spacing is given by a then we can write the normal growth rate at faceted ori-
entations as
Vjn)
= a I A. This growth mechanism has been observed on the basal
planes
of
ice growing from water over a large range
of
undercoolings from 0.03 to 0.2
K (Hillig 1958; Wilen and Dash 1995; Hodgkin et al., 1998). The observation
ofthis
step generation mechanism displays the structural perfection
of
the basal plane facets
for pure samples.
It
should be noted that this growth mode can be frustrated by sur-
face active impurities. By contrast, rounded surfaces are molecularly rough in that
steps are already present on the surface so that growth occurs by random incorpora-
tion
of
nutrient molecules onto the surface. Therefore, the entire interface continu-
ously advances normal to itself
in
the presence
of
any fmite driving force, and the
velocity
is
linearly related to the drive, Vr(n)
oc
Lll!
(Burton et
al.
1951, Cahn 1960),
with some orientation-dependent factor accounting for local anisotropy.

48
2.4.1.2 Growth from the Vapor
The linear relationship is most simply shown by analogy with the Wilson-Frenkel
theory
of
growth from the vapor (e.g., Chemov, 1984). The total growth rate V T
is
the
difference between the deposition and evaporation rates. Thus, the actual deposition
rate
v;qis
proportional to the vapor pressure:
V;q
= K
eq
exp(fl./1
/
kbT)
where
Keqis
the equilibrium deposition rate. In a detailed balance V T = 0, but during growth the
simplification is made that the evaporation rate takes the equilibrium value;
=
Keq'
The
Wilson-Frenkel
rate
is
(4)
For
T»T
r. to leading order in
fl.)!,
the rate becomes VWF =
Keq(fl./1
/
kb
T) , which
crudely shows that the interfacial velocity
is
linearly related to the drive.
It
is
physi-
cally plausible that due to the homogeneity
of
surface evaporation and the large sur-
face diffusivity at high temperatures, the linearized
Wilson-Frenkel
expression may be
quantitatively accurate.
It
should be stressed that this
is
a purely illustrative example,
for it is well known that it serves solely
as
an upper bound.
2.4.2 Kinetic Roughening
During kinetic or
dynamic
roughening a large growth drive may eliminate the order-
ing effect
of
the underlying lattice (e.g., Balibar et
aI.,
1993) and thereby change the
nature
of
the interfacial attachment kinetics. Here, the competition
is
between a
"crystallization fluctuation" which may arise from the formation and subsequent
evaporation/melting
of
subcritical surface nuclei, or a growth/melt wave induced by
thermal fluctuations, and a periodic lattice potential
of
the form
<l>cos(21W'ii),
where z
is
the height
of
the surface and the lattice plane has thickness
ii.
Such a fluctuation
carries with it a latent heat which must diffuse. The dissipation occurs over a charac-
teristic time scale 'tF which depends on the size and the diffusivity
of
the fluctuation.
If
the interface
is
moving at
dzl
dt = Vfthen the addition
of
a lattice plane takes a time
iilVf
If
the fluctuation dissipates before a layer
is
added, i.e.,
'tp
<
iilVfi
then the lattice
potential influences the entire history
of
the disturbance. However,
if
growth
is
fast
enough so that
'tF
>
iilVfi
then the influence
of
the lattice potential on the fluctuation
is
time periodic and therefore averages to zero. Hence, the growth process
itself
renders
the anchoring effect
of
the lattice ineffective, and the surface
roughens
dynamically.
Let us be clear in our distinctions: For a zero growth drive, an equilibrium form
is
faceted up to the
thermodynamic
roughening transition Tr. For growth at a fixed tem-
perature and fixed coupling strength to the underlying lattice, a facet will dynamically
roughen within a range
of
growth drives wherein 'tF z iilVf
,.
Roughening is sup-
pressed as the strength
of
the lattice coupling increases, and thermodynamic rough-

49
ening is "smeared out" under nonequilibrium conditions, effectively shifting the tran-
sition to lower temperatures.
A different kinetic realization
of
dynamic roughening is as follows:
If
the rate
of
step-spreading is much greater than the nucleation rate, then a single layer can cover
the facet before the subsequent nucleation event occurs. As the driving force is in-
creased, this condition is more difficult to satisfy, and the facet becomes decorated
with multiple supercritical nuclei simultaneously. Over a time period
'ts
the nuclei
spread and
join
to complete the layer. At still larger growth drives, the increase in the
surface density
of
molecules increases the nucleation rate, and during a time less than
'ts,
secondary nuclei may form on the existing islands. The picture
is
that
of
islands
growing upon islands and effectively roughening the surface. The process also results
in a crossover from activated, or layer by layer growth, to continuous growth (e.g.,
Cahn 1960 and Nozieres and Gallet 1987) and is related to some growth modes ob-
served in molecular beam epitaxy
of
metals (Kunkel et al., 1990). Detailed theories
of
kinetic roughening have been given by Jose et al., (1977), Chui and Weeks (1978),
and Nozieres and Gallet (1987).
2.4.3 An Example: Evolution
of
an Equilibrium Seed
We are now in a position to explore the evolution
of
an equilibrium seed. Different
surface structures coexist on a partially faceted eqUilibrium crystal shape (Fig.
1).
By
considering such an initial shape we are able to perceive a kind
of
intrinsic growth
anisotropy. We impose a small growth drive, so that the shape
is
uniformly bathed by
nutrient molecules. The relaxation rate at rough orientations
is
much more rapid than
that on facets, and
if
the drive itself is less than that necessary to induce kinetic
roughening
on
the facets, then at faceted orientations there will be slow normal
growth by nucleation and spreading
of
mono layers. Relatively fast normal growth
will occur at molecularly rough orientations. Finally, diffusion
of
admolecules on the
facets causes them to serve as catchment areas, and the diffusing material influences
the growth rate in vicinal orientations. 5 .
There are several experimentally realizable limits that are particularly instructive.
We can loosely distinguish these in terms
of
the magnitude
of
the growth drive
L'lll.
The first regime occurs for
L'lll
::;
10-
3, wherein the accretion
of
mass
is
much slower
than all
of
the available rel",xation processes, so that the crystal surface is indistin-
guishable
from
the
equilibrium
crystal
shape. This
regime
is
called
"thermodynamically slow" or "shape preserving" growth and can describe the growth
shapes via a continuous expansion
of
the scale
of
the Wulff diagram (Elbaum and
Wettlaufer, 1993). Growth shapes in this regime have been observed at temperatures
above, below and very near the roughening transition
of
the prism facet growing from
water (Fig .
.3
of
Maruyama et al., 1997). The result is also trivially correct for iso-
5 A vicinal orientation is one with a small angle deviation from a singular orientation, charac-
terized by broad terraces separated by monomolecular steps.

50
tropic surface free energy; a crystal everywhere above its roughening transition, or for
example in a liquid drop. One then simply expands a sphere.
In the second regime
LlJ!
is
larger, but nonetheless smaller than the activation bar-
rier for two-dimensional nucleation on the facets. Therefore, the facets are pinned and
the rough orientations accrete mass to take the Wulff shape
of
an "equilibrium" crys-
tal
of
increasing size (Elbaum and Wettlaufer, 1993). Ultimately the rough orienta-
tions grow themselves out
of
existence leaving a fully faceted growth form behind6.
Theoretical growth shapes are generated by two methods. The fIrst method expands
the polar surface free energy plot uniformly by increasing
A.
of
equation (2), and
re-
drawing the perpendiculars construction subject to the additional constraint that the
facets
do
not move normal to themselves. Hence, because
of
the lack
of
an activation
barrier in the rough orientations, the crystal surface only remains in "contact" with the
polar surface free energy plot at the rough orientations. The latter contribute a de-
creasing fraction
of
the total surface
of
the crystal shape with time (see Fig. 1
of
El-
baum and Wettlaufer, 1993), but the curvature
of
these regions
decreases
with time.
The method is not thermodynamically rigorous, but it is intuitive and qualitatively
reproduces experimental behavior.
It
lacks thermodynamic rigor because we cannot
of
course apply a global extremum principal locally.
It
is intuitive because the rough
orientations can "keep up" with the growth drive and are therefore near equilibrium
(i.e., the free energy polar plot), whereas, the facets fall behind.
The second method
is
a geometric model that predicts the evolution
of
every point
of
the crystal surface. As described above, this class
of
models requires
us
to con-
struct the local velocity
V(n)
normal to
an
interfacial point. In trying to understand the
evolution
of
partially faceted equilibrium shapes, such a function must explicitly ac-
count for the variation in growth kinetics with location on the surface. Our approach
has been to include activated growth on facets, nonactivated growth
in
rough regions
and their modifIcation in the vicinal orientations (Wettlaufer et aI., 1994). An ob-
server moving on the surface
of
the crystal will see activated growth at singular ori-
entations; having to "jump up" as a monomolecular step runs across the surface.
Walking away from singular faces, the observer will see a transition to the rough
growth kinetics, from, in the vicinal regions, surface migration
of
admolecules away
from facets, to the eventual rough terrain where growth occurs via random incorpora-
tion
of
molecules into the surface. Such effects can be modelled as: V(n,LlJ!) =
Vf
(n,LlJ!)
+ Vr(n,LlJ!)
(1
where describes the observer's transition be-
tween facet-like and rough-like growth. The necessary properties
of
are that it
is
periodic
in
2rtln, that °
:::;;
1,
and =
1,
rtln) =
0,
where nfdefmes the
normal at a facet.
We
choose the simplest form
of
the function satisfying these prop-
erties and predict the evolution
of
the same initial shape
as
previously (Fig. 2). We
fmd the same overall evolution
as
in the previous model, but here the facets are al-
lowed to move normal to themselves when the critical supercooling for two-
dimensional nucleation
is
exceeded, albeit the rate
is
rather slow relative to nonsin-
6rhe
growth and relaxation
of
a fully facetted shape is
of
interest in its own right (Carter et aI.,
1995), but here we focus on the details
of
how the shape arises after which a ,host
of
other
effects become important.

51
gular orientations.
We
observe
the
spreading
of
the
facets,
and
the
change
in
charac-
ter,
from
smooth
to
abrupt,
of
the
joint between
facets
and
rounded parts.
These
qualitative
features
have
been
observed
on
crystals of solid
He
growing
into
super-
fluid,
of ordinary
water
ice
growing
into
pure
water
vapor
(Elbaum
et
aI.,
1993)
and
into
liquid
water
(Maruyama,
pers.
comm.
1997),
and
on
small
metal
crystals
growing
into
metal
vapor
(Heyraud
and
Metois
,
1987)
.
0.45
0.4
0.35
0.3
0.25
•.
Fig.
2.
A sequence of growth shapes at dimensionless times t =
0,
0.018,
0.Q38,
0.
068, 0.35
of
the upper right quadrant of a crystal with
cubic
(n =
4)
symmetry.
The
units
are
arbitrary.
The
initial (inner) equilibrium shape
is
that
is
formed
by
the
Wulff construction shown
in
Fig.
1.
The
inset shows
the
full
crystal
shape
at
the
same
times.
Note
that
the
rough orientations
grow
out
of
existence with a decreasing curvature,
and
that there
are
sharp joints where the vicinal
regions join
the
rough
regions.
The
curvatures
at
the
rough
"corners"
where
the
angle 8 defining
n relative
to
the
positive x-axis
is
8 =
rc/4
for
t = 0, 0.
018,
0.038,
0.
068
are
4, 3.7, 3.4,3.1 (the
numerical values agreeing
with
the
exact analytic solutions
to
one
significant figure Wettlaufer
et
aI.,
1994).
As
growth progresses, facet orientations
move
substantially slower than
the
rough
regions,
and
the
crystal surface
loses
orientations
until
it
is
fully
faceted
and
possesses
only
four
orientations.

52
The rough regions
of
the surface take a decreasing curvature
as
growth progresses
between the equilibrium and fully faceted shapes. Recall that equilibrium forms be-
come more faceted as the temperature decreases. This growth process is intrinsically
transient and we refer to it as "global kinetic faceting", in order to emphasize that (a)
it
is
the global (Le., the entire closed surface) effect
of
local dynamics, and (b) it
should be distinguished from equilibrium faceting or local kinetic faceting. The de-
crease in curvature at rough orientations during this transition is consistent with no-
tions
of
the heuristic model above, but has a kinetic basis in that the local normal
growth rate
is
not, as is commonly supposed, a maximum where the density
of
steps is
the largest. Rather, the maximum local growth rate occurs a slight distance away
where the normal motion has contributions from both local attachment
of
molecules
and surface diffusion. This result can be shown by an explicit calculation
of
the sur-
face curvature. Figure 2 shows an example from our own studies, and there are other
forms
of
velocity functions both within the context
of
our model and in the larger
context
of
geometric models
of
interface motion (Langer, 1987; Kessler et
aI.,
1988;
Ben-Jacob and Garik, 1990; Cahn et
aI.,
1991; Goldstein & Petrich, 1991; Taylor et
aI., 1992; Angenent & Gurtin, 1994).
The general features
of
this faceting transition are
of
particular relevance to the
growth
of
atmospheric ice crystals. A study ofNakaya's snowflake catalogue reveals
that the majority
of
platelike snowflakes have evolved from a fully-faceted hexagonal
seed. Such a seed contains no surface which can easily accept accreting material. De-
pending on the size
of
the crystal relative to the diffusion length in the vapor phase,
the faceted state defmes a reasonable lower bound for kinetic roughening
of
the facets
themselves, or the onset
of
either shape instabilities or oscillations (Tersoff et aI.,
1993). Once the crystal size approaches the diffusion length, the comers reach farther
into the nutrient phase and tend to trigger changes in growth morphology and mode.
One likely morphological change
is
discussed presently.
2.5 Diffusion-Limited Crystal Growth
When the rate
of
advance
of
the phase boundary
is
limited by the removal
of
latent
heat or impurities, then the diffusion field modifies the interfacial values
of
the rele-
vant field variables and therefore the growth shape. The physical picture
is
as
follows:
As a phase boundary advances, latent heat and/or impurities are emitted at the front.
Their build up occurs over a characteristic dimension, or boundary layer, that depends
on their respective diffusivities and the rate
of
advance. The motion itself is limited by
the speed with which this excess
is
depleted by diffusion. These diffusion-limited re-
gimes are generally studied using non-geometric formulations. 7 The reason
is
that the
7Rarely can diffusion-limited growth be thought
of
as
local and so be described geometrically.
One exception occurs when the characteristic scale
of
the boundary layer is much less than the
radius
of
curvature
of
the interface, and one can view the rate limiting part
of
the diffusion field
as being "localized". This, so-called boundary layer hypothesis, was invoked to make diffusion-
limited growth problems tractable theoretically (e.g., Langer 1987; Kessler et
aI.,
1988; Ben-
Jacob and
P.
Garik 1990). However, physical realizations consistent with the hypothesis are

53
motion at a point depends
on
the
diffusion field near that location, which itself
de-
pends
on
the history
of
the field both there,
and
at
other points.
More
commonly
we
say
that the growth
is
controlled
by
a non-local retarded
field.
Non-geometric theories
generally treat growth
on
surfaces that
are
everywhere molecularly rough, and the
anisotropy
is
introduced
into
the
interfacial conditions of
the
particular
free
boundary
problem being studied. For example, anisotropy
is
ascribed
to
an
orientation depend-
ence
in
the surface
free
energy
or
some
other interfacial coefficient. Because
of
the
general importance
and
environmental applicability, I describe a morphological insta-
bility characteristic
of
impurity
diffusion limited
growth.
In
nearly every natural setting, phase coexistence will
be
altered
by
the
presence
of
impurities,
and
in
the
case
of water substance
we
most often encounter electrolyte
solutions.
It
is
common
lore
that
ice
rejects salt,
and
it
is
observed that
in
equilibrium
the solubility
of
most impurity species
in
ice
is
much
less
than
that
in
solution.
This
observation
is
commonly represented
by
the
equilibrium distribution coefficient for
that species;
k =
Cs
eq
CL
(5)
where
Cs
(eL)
is
the equilibrium concentration
of
solute (impurity)
in
the solid
(liquid).
In
a subeutectic solution, keq < 1
and
decreases
in
magnitude with the degree
of
lattice mismatch between the solute
and
the
crystal
of
solvent.
In
most aqueous
solutions 10-4 < keq < 10-1 (e.g.,
Gross
1987;
Gross
and
Svec
1997),
so
that ice
grown very slowly
from
solution
may
remain nearly
pure.
In
practice a myriad of
ef-
fects complicate this distillation effect,
one
of which
we
shall describe.
In
the pure
case, especially
in
ice,
growth
is
highly anisotropic (e.g., Elbaum et
aI.,
1993;
Fu-
rukawa and Kohata,
1993),
and
we
expect anisotropy
in
solute segregation which
translates into a facet dependence of k
eq
. However,
in
other materials such
an
effect
has
only been observed
in
semiconductors (e.g., Tiller
1991).
Additional complica-
tions,
some
of which
have
plagued
the
interpretation of experimental
data,
include,
(i)
Dependence of segregation
on
valency (ii)
Samples
are
often modified
by
convection.
(iii) During growth there
is
a possibility of phase boundary instabilities (discussed
below) which
can
result
in
the
trapping of impurities
in
bulk.
(iv)
Some
species
show
a clear concentration dependence of keq
and
others
do
not.
(v)
Measurements near
the
solubility limit
or
the eutectic point
are
difficult
to
interpret.
(vi)
In
almost
all
cases,
above
some
threshold growth rate, the impurity distribution will depend
on
growth
rate (e.g., Aziz
1982).
Therefore, rather than observing
an
equilibrium quantity,
it
is
more common
to
actually measure
an
effective distribution coefficient
kef!'
which
represents the net effect of
some
or
all
of these processes
in
terms
of
a measurement
of
the
bulk concentrations before
and
after
an
experiment.
difficult
to
achieve.
In
the
common
case
of a
free
dendrite
growing
into
an
undercooled
melt,
the
radius
of
curvature
at
the
tip
is
much
smaller
than
the
boundary
layer
unless
very
large,
and
experimentally
awkward,
undercoolings
are
imposed.

54
We consider solutions
as
being
ideal
when the concentration
of
solute molecules
is
low enough that they do not interact. At constant pressure, the impurity-induced shift
in the equilibrium temperature
of
a solution
is
given by Raoult's law, which
is
written
in terms
of
the number
of
moles
of
solute nj dissolved in n moles
of
solvent;
(6)
where R is the gas constant, and
qm
is the heat
of
fusion
of
the solvent. More com-
monly however,
Tc
is related to the solute concentration in the liquid
CD
by an em-
piricalliquidus slope M determined from the appropriate binary phase diagram:
Tc
=
M C
L.
Many salts go into solution as ions, so we expect a deviation from the condi-
tions
of
an ideal solution. Depending on the impurity, the ideal slope may often differ
from the empirical value, so we often rely on the latter. Ion-ion interactions and more
complicated solution chemistry effects complicate theoretical treatments so the de-
tailed experimental values have been tabulated by electrolyte chemists (Partanen
1991) are
of
great utility. The earth's most pervasive and important solution
is
the
ocean, and the physical chemistry
of
seawater a vast field
in
itself (e.g., Millero 1974;
Andersson 1998), and a notable feature
is
that the relative concentration
of
the inor-
ganic components that dominate by weight are constant throughout the world oceans:
NaiCI = 60, KlCI = 228, Mg/CI = 240. The basic physics
of
the ocean freezing that
is
occurring daily in one
of
the polar regions can be understood by vastly simplifying
the detailed chemistry
of
seawater and treating it
as
a simple two component system
described by an empirical phase diagram. More complicated electrolyte effects and
their applications are discussed in the summary.
The freezing point
of
small solid particles convex into their melt phase
is
reduced
from that
of
the bulk. The effect is associated with Gibbs and Thomson and for a
spherical particle
is
written
as
(7)
where
Ps
is
the density
of
the solid,
qm
is
the latent heat
of
fusion, and r
is
the radius
of
the particle.
The three important ingredients for understanding morphological instabilities
driven by diffusion fields are now
in
hand: Solute segregation, and solute and curva-
ture depression
of
the freezing point. Consider a planar ice/solution interface in bulk
two-phase, two-component coexistence. The stationary isothermal system is at the
compositionally determined freezing point given by equation (6), and there
is
ajump
in the solute concentration at the solid/liquid interface determined by equation (5). A
unidirectional constant temperature gradient is instantaneously imposed perpendicular
to the ice/solution interface. It is cold
in
the ice and warm in the solution and the in-

55
terface moves toward the latter at constant speed. Solute, let
us
call it salt, is rejected
into the solution and builds up at the interface to take a value determined by the rate
of
advance (which depends
on
the imposed temperature gradient), the magnitude
of
the segregation coefficient and the value
of
the (negative) liquidus slope
M.
The re-
gion
of
solution adjacent to the interface
is
enriched with salt, and its freezing point
is
depressed relative to the value far into the liquid. The enriched region has a spatial
extent that depends on the diffusivity
of
salt
in
water which
is
several hundred times
slower than that
of
heat. Depending on the magnitude
of
the actual temperature gradi-
ent and the rate
of
motion
of
the interface, the possibility exists that, because
of
the
slow diffusion
of
salt, a region
of
fluid will be constitutionally supercooled. This oc-
curs when the gradient in the liquidus temperature
is
greater than the actual tempera-
ture gradient. Therefore, a fluctuation
of
the planar phase boundary that
is
convex into
the solution will encounter a region
of
metastable fluid and tend to grow, creating a
cellular interfacial geometry that traps regions
of
highly concentrated solution be-
tween regions
of
nearly pure ice. The fluctuation can be stabilized by the curvature
depression
of
the freezing point. This morphological instability
is
prevalent in metal-
lurgy (Rutter and Chalmers, 1953; Mullins and Sekerka 1963), and
is
responsible for
the bulk two-phase properties
of
sea ice which generally grows under unstable condi-
tions (Wettlaufer 1992). Morphological instabilities are
of
the most active present
interest
in
materials science, wherein there
is
wide practical applicability and a long
history (e.g., Tiller, 1991; Sekerka 1993). This instability lies in the molecular diffu-
sion field, but it does not act alone. Rather, it
is
often coupled to fluid mechanical
effects, and the interaction
of
convection and solidification
is
an
important influence
in the phase behavior and structure
of
sea ice (Wettlaufer et
aI.,
1997; Worster and
Wettlaufer, 1997)
3
Thermomolecular
Pressure
and
Frost
Heave
As discussed
in
section 2.3, both surface melting and interfacial melting are treated as
wetting phenomena at the interface between a solid and its vapor phase, or a wall
of
a
different material, at temperatures below the bulk freezing point T
m.
The melting
process may be complete or incomplete. In the former case the film thickness diverges
as T m
is
approached from below, and
in
the latter case the film growth
is
truncated at
finite undercooling, which can be best understood
as
being due to the influence
of
retarded potential effects. Convincing interfacial melting measurements have been
made in ice (Furukawa and Ishikawa, 1993; Beaglehole and Wilson, 1994)
),
and a
primary motivation for understanding the process concerns the important role it can
play in a host
of
environmental problems ranging from polar stratospheric cloud
chemistry to frost heave
in
porous media (Dash et
aI.,
1995).
The dramatic deformation
of
water saturated soils
in
cold climates
is
known generi-
cally as "frost heave" and the chapters by Greg Dash and Peter Williams stress the
important implications
of
the phenomenon from various perspectives. Yet it
is
rather
more appropriate to refer to frost heave
as
the collective action
of
a number
of
phe-
nomena each
of
interest and intense study
in
its own right. Although frost heave is

56
clearly not caused
by
the volume expansion of water during solidification,
for
many
years
it
has
been known to
be
associated with the existence of stable liquid water
at
subfreezing interfaces (Williams
and
Smith
1989).
However,
the
varied
causes
of this
water have obscured attempts
to
extract
the
fundamental mechanisms driving frost
heave.
In
a porous
medium,
curvature, confmement, interfacial roughness
and
disor-
der,
impurities
and
interfacial premelting all contribute
to
the
fmite
liquid fraction
at
subfreezing temperatures. Individually, these effects have distinct temperature
de-
pendendes, but their combined effects
have
thus
far
limited frost heave research
to
semi-empirical treatments. It
is
for
this reason that
we
have focused
on
isolating the
role of interfacial melting
in
the existence
and
mobility of unfrozen interfacial water.
The
intermolecular origin
of
the
films,
the effect of external
forcing
and
the
geometry
of the confming wall
are
the
main features of our studies.
We
test our theories using
experiments
on
isolated crystalline
surfaces.
The
approaches
to
the
problem range
from
microscopic
to
macroscopic (e.g.,
Gil-
pin
1980;
Dash
1988;
LOwen
1994;
Wettlaufer
and
Worster
1995;
Netz
and
Andel-
man
1997;
Oxtoby
1998),
and
here I need only to repeat the aspects necessary
to
study dynamical behavior
and
refer the reader
to
Oxtoby's chapter
for
the
remainder
of
the topic.
We
begin
by
writing
down
the
mean
field grand potential n
for
the
sys-
tem which
is
the natural
free
energy
for
describing interphase pressure differences.
The
total
free
energy
for
a layered wall (w),
film
(f),
solid
(s)
system of area Ai
is
written
as
a combination of bulk
and
surface
terms:
(8)
Here P
and
V denote pressure
and
volume
and
the
liquid
film
has
a thickness
d.
The
interfacial term
3(d)
=
(!1Y
f(d)
+ Ysw)Ai
is
a phenomenological representation
of
the potential between
the
three layers.
By
defmition,
!1y
= Ysf + Yfw -Y
sw
,where
the y's
are
the solid-liquid
(s.e)
liquid-wall
(fw)
and
solid-wall
(sw)
interfacial
free
energies.
We
assume that
the
volume-volume interactions
are
of a power
law
form
represented
by
f(d)
=
1-
(a
/ d)
v-I
where
a
is
on
the order of a molecular diame-
ter,8
and
v depends
on
the
nature of
the
interactions.
The
relevant types include v = 3
(v
= 4)
for
nonretarded (retarded)
van
der
Waals
interactions9,
and
v = 2
(v
= 3/2)
for
8Note that the short range cut off defined by the value
of
cr
is
intimately wed to the effective
nature
of
the solid-wall coefficient Y
sw
which
is
a non-equilibrium interfacial energy (see the
Appendix in Wettlaufer and Worster, 1995).
9various forces arise between charged, uncharged and dipolar molecules. Briefly, intermo-
lecular forces can be Coulombic arising between charges and permanent multi poles. Dipole
moments "induced" in molecules by local electric fields or charges are responsible for polari-
zation
forces between molecules. Chemical bonding itself
is
controlled by
quantum
mechanical
forces.
Van
der
Waals
forces are quantum mechanical in origin and act between all molecules.
Even a neutral molecule has a finite dipole moment due to the fluctuations in the electron posi-
tions, and the dipole creates an electric field that polarizes a molecule in its vicinity. The di-
poles
of
the two molecules interact to give rise
to
a force that depends on the frequency
of
the

57
long range (short range) electrostatic interactions. By electrostatic interfacial interac-
tions we refer to a simple treatment in which the substrate possesses a surface charge
density and the confmement
of
counterions present in the liquid creates a repulsive
force across the layer (Wilen et aI., 1995). The electrical conductivity
of
the ice sur-
face may also be calculated (Petrenko and Ryzhkin 1997) and influence the short
range behavior. This form
of
:3(d) captures the essential features
of
more detailed
treatments
of
complete interfacial melting under the influence
of
dispersion or elec-
trostatic forces, and allows us to develop a dynamical theory for arbitrary power law
interactions and to make experimentally relevant predictions. At fixed temperature
and chemical potential, minimizing n with respect to d provides two very useful, in-
teraction dependent, results. To first order in the reduced temperature
tr
= (T m -
T)IT
m the film thickness
is
d
__
v-I
cr
l1y
-l/v
=
-l/v
(
( )
v-I
)l/V
-. ,
Psqm
(9)
where
Ps
and
qmare
as previously defined. The coefficient
t..v
depends on the expo-
nent v, for example when v = 3, a
'"
l1y
= A /
121t,
where A
is
the Hamaker constant.
The pressure in each phase
is
uniform but the interfacial interactions create a pressure
difference between the film and the bulk solid
(10)
the solid repels the wall through the liquid,
in
analogy with the disjoining pressure
of
wetting. When taken together these results yield a thermodynamic statement that
is
independent
of
the nature
of
the interactions and helps to describe the main ingredient
of
pre melting dynamics:
(11)
We imagine that the pressure in the solid
is
fixed by contact with a reservoir. There-
fore, the pressure in the liquid film increases with temperature and, by imposing a
temperature gradient parallel to the interface, the resulting thermomolecular pressure
gradient will drive premelted liquid from high to low temperature. Thermomolecular
pressure-driven flow as a for frost heave captures our attention because
the maximum pressure in the case
of
water
is
approximately
11
atm per degree The
film thins
as
the temperature decreases, so continuity requires that liquid convert to
solid during its transit toward lower temperatures.
In order to anticipate
the
behavior
of
an ongoing experiment, we made our first cal-
culations in an idealized geometry (Wettlaufer and Worster 1995). Soon thereafter
Larry Wilen and Greg Dash (1995) observed premelting dynamics at ice single crystal
interfaces formed against a flexible polymer membrane with a more complicated ge-
fluctuations and is attenuated with distance. When two molecules are
an
appreciable distance
apart the strength
of
the interaction can be weakened by the time it takes for the field to travel
between the two molecules. The effect is known
as
"retardation".

58
ometry than that in our model.
We
immediately set to work constructing a theory that
accounted for the membrane geometry, and allowed us to back out the relevant types
of
interactions in the system (Wettlaufer et
aI.,
1996). For brevity, the discussion will
be limited to the latter study, but for a more detailed and complete treatment see
Wettlaufer et ai. (1997).
In analogy with the frost heave terminology we call the pressure exerted by the
confming wall the "overburden pressure".
If
it
is
greater than the maximum pressure
in the film then flow will cease, but
if
it
is
less than the maximum, flow will persist.
The thermomolecular pressure gradient
is
written
as
(12)
where Ps is the external pressure exerted on the solid by the confining
wall/membrane. In the observed regime, the radius
of
curvature
of
the deformed
membrane
is
large compared with
d,
so
the membrane exerts a pressure deep within
the solid.
We
assume that any solid deformation
is
slow relative to the film dynamics.
The configuration is shown in Fig.
3.
The film thickness depends solely on the tem-
perature, relaxing to the value determined by equation (9) on a time scale negligible
relative to that for heat conduction. The confming wall evolves in time according to
the spatial variation in the thin film volume flux Q oc _d3
VPI!' Wilen and Dash
(1995) imposed a constant temperature gradient G parallel to the ice/membrane inter-
face.
FLEXIBLE MEMBRA
,I
yT(outer)
Ceo s Section
Grain
Boundary
CRYSTAL DOMAIN
Plan View
Fig.3(a). Cross section and plan view
of
the Wilen
and
Dash (1995) experimental cell. The
temperatures at the cold finger T(inner) and at the outer rim T(outer) are independently con-
trolled.

Cold
59
.....
I----X
HEIGHT
h(x,t)
OF
FLEXIBLE
MEMBRANE
Warm
Fig. 3(b).
The
local
configuration of a radial
frost
heave
cell
treated
in
a one-dimensional
slab
geometry (Wettlaufer
et
aI.,
1996).
The
membrane
is
locally under tension,
and
exerts a restor-
ing
force
proportional
to
its
curvature.
The
temperature gradient G
is
constant
so
tr = G xlT
m,
and
the
film
thickness d
depends
only
onx.
The
thermomolecular pressure gradient drives the
evolution
of
the
membrane
height h(x,t) relative
to
an
initial reference height h(x,tO).
The
three
112
6/11
3/5.
.
power
laws
t
,t
,
and
t
arise
from
the
short
and
long
range
electrostatic
(v
=
3/2
and
v
=
2)
and
nonretarded
Van
der
Waals
(v
=
3)
interactions respectively.
For
each interaction
we
fit
((
v-I)
a
v-I
Llr
r(
4!3),
related
to
AV
as
defined
in
Eq.
(9
).
a,
b,
and
c
are
material
constants,
and
AV=312
=
0.0337
A,
AV=2
= 0.
2101
A,
and
AV=3
=
1.3759
A.
The main qualitative features
of
the steady state temperature gradient experiments
are as follows: The height
of
the membrane, h = h(x,
f),
was measured relative to an
initial reference height
h(xtO).
The bulk solid-liquid interface is located at the fixed
position x. A constant temperature gradient drives a volume flux gradient, and ther-
momolecular pressure continually draws more fluid into the film where it freezes and
deforms the membrane. This response to the flow
of
premelted water occurs in a nar-
row temperature range near T m, with a relative lack
of
membrane deformation at
lower temperatures. The position
of
maximum deformation
Xmax
moves towards
colder temperatures and the maximum displacement hmax increases with time. In
principle, large deformations can be achieved by increasing G and decreasing the
strength
of
the membrane. However, the migration
of
the peak toward lower tem-
peratures will be most dramatic when G is small and the membrane strength
is
large.
The experiment clearly demonstrates the phenomenon
of
interfacial melting, the
viscous nature
of
the premelted liquid and its dynamical consequences. Initially the
observed relative lack
of
membrane deformation at lower temperatures was thought to
be due to an abrupt
jump
to d =
O.
Such an interpretation is consistent with an interfa-
cial free energy that decreases monotonically with d at long range,
but
which pos-
sesses a local minimum at shorter range. The theory has allowed us to show that the

60
data can be described with a monotonically decreasing interfacial free energy over the
entire range
of
film thicknesses. Although the low temperature dynamics are slow,
non-local effects insure that they still
playa
role, and that they will be especially im-
portant in nature where seasonal time scales are very long. The agreement between
theory and experiment (Fig. 4) is best in the case
of
electrostatic interactions which
always dominate van der Waals at long range and, depending on the surface charge
density, may also dominate at short range (Wilen et aI., 1995). Nonetheless, the ob-
servations cannot be fitted by a single interaction over the entire range
of
deformation,
although a simple electrostatic interaction can fit the region
of
maximum deflection.
Of
course other interactions might also
playa
role, and crossovers from one type to
another may occur as the film thins and short range behavior comes into its own.
There
is
much left to explore, especially the role
of
impurities in modifying the inter-
actions that control the film thickness, the effect
of
solid-like ordering due to the
proximity
of
the interfaces and other dynamically significant issues.
2
160h
o
-
400
Ice Water
o
R - R (/lm)
o
400
Fig. 4. Comparison
of
the theoretical predictions and the experimental values described by
Wilen and Dash (1995) for the membrane height h(x,t) at 160 hours. R -Ro
is
the experimental
x-coordinate. At the bulk ice/water interface, R = Ro and G = 0.92
Klcm
. The predictions for
the short and long range electrostatic (v = 3/2 and v = 2), and the nonretarded Van der Waals (v
= 3) interactions are shown by the solid, dashed, and dotted lines respectively. For orientation
"Ice" (R
-Ro
< 0) and "Water" (R -Ro > 0) refer to regions where the bulk phases are stable.
For R -
Ro<
0
an
interfacial water film coexists between the membrane and bulk ice.

61
4 Summary and Future Directions
The
phenomena described
in
this
chapter
centered
on
interfacial disorder, sometimes
liquid
like
disorder, of
both
a
thermodynamic
and
dynamic
origin,
and
the
redistribu-
tion of both
the
disordered material
and
impurities
at
solid-liquid interfaces.
Because
of
the
change
in
the
surface
structure
created
by
both
surface
roughening
and
surface
melting, they have a profound influence
on
the
local
attachment kinetics,
and
ulti-
mately
on
pattern
formation
in
ice
crystals.
We
have
a
clear
picture of several
issues,
namely
(i)
that
roughening
is
a
crucial
aspect of
growth
anisotropy,
and
(ii) that inter-
facially melted water
at
subfreezing interfaces
is
mobile
and
responds
in
a
thermody-
namically consistent
and
predictable
manner.
Less
clear however
are
a number of
other
issues
as
discussed
below.
4.1
Surface Melting
and
Crystal
Growth
The
role
of
surface
melting
in
the
growth
of
crystals
is
a
fertile
area
still
in
its
infancy.
On
isolated surfaces thermodynamic roughening, kinetic roughening
and
surface
melting
have
been
extensively studied, but their coexistence
on
a
single
crystal
sur-
face
results
in
highly
anisotropic
global
growth
shapes
(Furukawa
and
Kohata,
1993;
Wettlaufer et
aI.,
1994;
Nada
and
Furukawa
,
1996).
In
the
case
of
surface
melting,
the
communication with
the
parent phase
is
mediated
by
a
mobile
thin
film
(Sei
and
Gonda,
1989;
Kuroda
and
Lacmann,
1982).
The
difference
in
the
vapor-solid
and
the
vapor-liquid/liquid-solid accretion mechanisms,
and
the
change
in
these with tem-
perature
as
the
film
thickness
changes
will influence
the
growth rate . But what
are
these temperature dependencies?
Some
important
and
telling experiments
have
al-
ready
been
done.
Sei
and
Gonda
(1989)
grew
polyhedral
ice
crystals
at
several percent supersatura-
tion.
From
their observations of
growth
rates
versus
supersaturation,
they
concluded
that,
from
the
triple point
to
-2°C,
the
basal
and
prism
faces
grow
by
a
mechanism
of
vapor/quasi-liquid/solid
attachment
kinetics,
and
from
-2
to
-30°C,
they
grow
by
the
classical kinetic
mechanisms
described
above
which
operate
in
the
absence
of a
quasi-
liquid.
Furukawa
and
Kohata
(1993)
studied negative crystals
formed
by
creating cavities
in
ice
single crystals.
They
observed
receding crystallographic (basal
and
prism)
fac-
ets
in
a
fixed
evaporation
rate.
Below
-2°C,
they
found
that a negative crystal took
the
form
of a hexagonal
prism
, but
changed
habit
to
a
plate
and
back
to
a
prism
ex-
actly
as
does
a positive
ice
crystal
growing
from
the
vapor
or
a
gaseous
atmosphere.
However,
at
temperatures
above
-2°C
the
hexagonal
prism
was
observed
to
change
into
a
sphere
truncated
only
by
basal
faces.
Furukawa
and
Nada
(1997)
have
combined
a
number
of interesting observations
In
his
ellipsometric
studies
of
surface
melting
on
the
basal
and
prism
facets
a crossover
was
observed
in
the
film
thicknesses
at
about
-1°C
.
At
temperatures greater
than
this,
the
basal
plane
film
was
thicker, but
the
prism
plane
film
was
thicker
at
lower
tem
-
peratures.
From
studies
of
the
anisotropic
"growth"
of
negative
crystals
Furukawa
and

62
Kohata (1993) inferred from the growth kinetics that the prism facet undergoes a
roughening transition at approximately -2°C. However, since surface melting
is
initi-
ated at -4°C, this transition occurs between the prism facet and a quasi-liquid layer
tens
of
nanometers thick. Recall that Maruyama et al., (1997) observed the prism fac-
ets to undergo a thermodynamic roughening transition into bulk water at approxi-
mately -16°C at
165
MPa. Finally, using molecular dynamics simulations they found
qualitative agreement with the facet dependence
of
surface melting. However, such
simulations exhibit phase behavior that
is
very sensitive to the details
of
the model
potential, and it is not surprising that the simulated films were substantially thinner
than the experimental films, and the crossover substantially lower. The
qualitative behavior clearly demonstrates the sensitivity
of
the equilibrium and growth
forms to anisotropy in the surface structure. Much caution,
as
noted by Furukawa and
others, must be administered
in
deducing quantitative behavior. Despite the great care
taken by individual groups, there
is
still a substantial spread in the film thickness ver-
sus undercooling curves for the vapor interfacelO.
The question
of
the roughening transition
of
the surface melted prism face
is
an in-
teresting one
in
its own right. Maruyama's experiment in bulk water shows a very low
roughening temperature;
16°C
at
165
MPa. Yet at the surface melted prism facet Fu-
rukawa infers a roughening temperature
of
ice against a quasi-liquid layer
of
ap-
proximately -2°C. Hence, the proximity
of
the quasi-liquid/vapor interface must
influence the fluctuations at the ice/quasi-liquid interface, suggesting a kind
of
"proximity induced roughening transition" but a quantitative explanation
is
still lack-
ing. Simulations are notoriously difficult with water/ice systems, and they are an ex-
cellent intuitive guide, but until they adequately reproduce the bulk phase behavior,
they will remain a tool to drive experiment and theory, much
of
which remains to be
done on this topic.
Were success forthcoming in the area just discussed, an outstanding question
of
general interest concerns the role
of
surface melted films in the morphological stabil-
ity
of
crystals growing in the diffusion limit. Although there
is
no experimental verifi-
cation, Lowen and Lipowsky (1991) predict a dynamical thickening
of
the surface
melt during evaporation. During growth,
if
the solid orientation underlying the surface
melted film is a proper facet, then due to the lack
of
surface attachment impedance at
the film/vapor interface, and activated growth at the solid/film interface, we would
expect a
dynamical
thinning
of
the surface melted film. The actual film thickness
as
a
function
of
growth drive would be exceedingly difficulty to measure directly. How-
ever, it might be best investigated by observing growth anisotropy in a combined ex-
perimental-theoretical study. Once an understanding
of
the film thickness versus
growth drive
is
in hand, its effect on the morphological stability
of
crystals growing
lOsee, for example, Fig. 1
of
Elbaum
et
al. (1993) although it doesn't include
more
recent glancing angle X-ray scattering data taken by Dosch et
al.
(1995). More work
using a combination
of
techniques, including Second-Harmonie-Generation, light
scattering, ellipsometry, X-ray scattering and force microscopy should
be.
embraced to
resolve existing discrepencies.

63
under varying drives and temperatures will shed some light on the pattern formation
problems in the growth
of
snow crystals.
4.2
Electrolyte
Effects
There are a myriad
of
interfacial effects associated with electrolytes that are pervasive
in the natural environment but with which we have only dealt crudely herein.
All
electrostatic fields in any medium containing free charges will become screened by
the polarization
of
the charges. A screened electric field decays approximately expo-
nentially with distance
x;
exJl,.-
KDX),
where
lIKD
is
the Debye screening length.
When charges preferentially adsorb onto
an
interface adjacent to an aqueous solution
they are balanced by counterions creating an electric double layer. For an aqueous
NaCI solution
lIKD
= 30.4 nm at
10--4
M,
0.96 nm at
O.1M,
and
in
pure water
of
pH
7,
lIKD
is
about one micron (e.g., Israelachvili 1992). With this
in
mind, the interfacial
dynamics at an ice/solution interface can become quite complicated, and our studies
of
premelting dynamics might require consideration
of
a continuous variation in the
interaction potential depending on the redistribution
of
ions. Preferential ion incorpo-
ration
is
best demonstrated
in
this context
of
solid/liquid solute distribution introduc-
ing ionic coefficients; A simple example for a monovalent electrolyte solution
is
the electrical imbalance associated with the preferential Na+ rejection during the
growth
of
ice from a dilute aqueous NaCl solution. Within the framework
of
a theory
of
Bronshteyn and Chernov (1991), we imagine the trapping
of
a line
of
negative
charge density parallel to a newly formed planar freezing front. The electric field set
up between the solid and the rejected liquid cations, creates a Debye double layer on
the liquid side
of
the interface. (Of course,
lIKD
will be a dynamical quantity due to
the moving crystallization front.) Charge redistribution
is
driven by a number
of
well
known processes: in the solid it
is
accommodated by the mobility
of
Bjerrum and
ionization defects, and
in
the liquid it
is
determined by the dissociation rates and dif-
fusion coefficients. The higher concentration
of
intrinsic ions
in
the liquid, and their
much larger mobility in the solid, result
in
strongly phase dependent neutralization
rates. Bronshteyn and Chernov (1991) predict that a freezing potential arises only
when the product
of
the impurity concentration
at
the interface and the mean growth
rate exceed a pH dependent critical value. The comparison
of
the theory with experi-
ment is limited by the indetermination
of
the morphological stability
of
the solidifica-
tion front. Chernov (1993) has recently reviewed growth from aqueous solutions.
Electric fields at the ice/solution interface observed during growth have been associ-
ated with Workman and Reynolds (1950) (see also Hobbs 1974, Gross and Svec,
1997 and refs. therein), and combined with the thermoelectric effect (currents driven
by temperature gradients)
in
ice, and ion dependent phase segregation, these phenom-
ena might be central to the issue
of
ice particle collision-driven charge transfer in
thunderstorm electrification (Mason and Dash 1998).

64
4.3 Additional Effects
In
the
field
or
in
a laboratory setting
we
must think of frost
heave
as
the
collective
action
of
a number of
phenomena.
In
a
porous
medium,
the
distinct effects of curva-
ture, confinement, interfacial roughness
and
disorder, impurities
and
interfacial pre-
melting all contribute
to
the
finite liquid fraction
at
subfreezing temperatures.
One
approach
is
to
measure their combined effect
on
the
liquid fraction
and
to
construct a
system specific constitutive relation.
Many
of
these
phenomena
have
been
studied
in
other materials
in
the
context of phase transitions
and
critical phenomena,
and
their
temperature dependencies
can,
in
principle,
be
deconvoluted.
This
suggests that
it
will
fruitful
to
study
these
effects
in
turn,
thereby
building
up
a
fundamental
understanding
of their importance. In this
manner,
one
can
construct a
more
general constitutive
re-
lation between temperature
and
liquid
fraction
that would
be
broadly
applicable.
The
study
of dislocations
is
central
to
understanding
the
mechanical behavior
of
ice,
although their role
in
ice
crystal growth
is
still evolving.
How
and
when
impurities
create
or
pin
dislocations
is
important
in
growth
from
the
vapor or
the
melt
phase.
I
hope
that it
is
clear that
the
microscopic phenomena discussed
here
are
both
inter-
esting
in
their
own
right,
and
have
macroscopic implications
with
real
environmental
consequences. Progress requires serious efforts using well developed methods
from
within
the
disciplines,
and
continuous communication of these
efforts
across
discipli-
nary boundaries.
These
and other
key
questions that
are
impeding progress
can
be
resolved
by
working jointly through direct interaction,
the
sharing
of
different per-
spectives,
data
and
by
visiting
the
respective laboratories.
The
challenges
are
substan-
tial, but
the
rewards
are
great.
It
is
a pleasure
to
acknowledge
extremely
useful
conversations
with
J.W.
Cahn,
J.G.
Dash,
M.
Elbaum,
S.C.
Fain,
Y.
Furukawa,
J.F.
Nye,
M.
Maruyama,
M.
Schick,
L.A.
Wilen
and
M.G
.
Worster,
on
many
aspects
of
the
work
described herein. I
am
fortu-
nate
to
have
as
colleagues
Greg
Dash,
Larry
Wilen
and
Grae
Worster
who
make
the
efforts of research
so
enjoyable.
Our
work
has
been
generously supported
by
the
Na-
tional Science Foundation,
the
Office
of
Naval Research,
and
the
Natural Environ-
ment
Research Council of
the
U.K.
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