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3
Gravitational Destabilization of
Stationary States
MECHANICS OF SUSPENSIONS
Study of the effective theological properties of suspensions in
a quantitative sense dates from E~nstem's thesis (1905~. Because
of the industrial and technological importance (slurries, separa-
tions, colloidal dispersions, flocculation, blood flow, and so forth),
suspension mechanics has been intensively studied for the last
80 years. One quantity of interest is the effective shear viscosity
of a suspension of neutrally buoyant particles. Einstein showed
that for dilute suspensions of spheres the shear viscosity depends
linearly on the volume fraction. Data on such systems deviate
from linear behavior, with much scatter at high concentrations;
there is still no convincing theoretical description of such behavior.
More complicated systems (anisotropic particles, emulsions whose
dispersed phase can deform, polymer molecules whose conforma-
tion can change dramatically, and colloidal systems influenced by
electrostatic and London forces) are even more challenging.
Franke} and Acrivos suggested that the viscosity diverges at a
critical concentration Be as (1 - ˘/˘c)-~/3, where above this criti-
cal concentration the suspension can no longer be sheared without
dilation. The uncritical exponent" ~ in reasonable agreement with
experiments, but because of the scatter in the data, both the
71
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72
leading constant and the critical concentration vary considerably.
Recently, large-scale numerical simulations by Brady and Bosses of
sheared, two-dimensional layers of spheres at high concentrations
show that effective properties do diverge, and that this divergence
is a result of the formation of large transient clusters of parti-
cles. The divergence is associated with a percolation threshold
that causes the correlation length of the clusters to diverge. Thus,
as the critical concentration Is approached, the finite size of the
apparatus comes into play, with the accompanying variation from
apparatus to apparatus. In order to probe the region near tic,
it is necessary to use a large apparatus. However, no system Is
neutrally buoyant; small mismatches in density, small tempera-
ture gradients, and so forth, will have large effects as the sample
size increases. We can illustrate this with the following simple
argument.
A measure of the effect variations in density would have on
such a suspension of characteristic dunension is given by the sed-
imentation velocity 9~p`2/p~. Suppose the density variation in
~p/p is controlled to ~ percent and the fluid ~ a viscous solvent
with kinematic viscosity z' ~ 1 cm2/s. Then if the sedimentation
velocity is to be small—less than 1 cm/s, say then 82 << p~/^p9
· ~
In cgs units.
When 9 _ 1, we have ~ << 3 mm, which is far too small a
characteristic scale to probe for percolation behavior near tic. In
practice, greater control of density is possible, but working with
more viscous solvent presents mechanical problems with the high
effective viscosities encountered near ~c.
Obviously, working with smaller gravitational levels allows
~ to be increased without encountering significant sedimentation
problems, thus enabling us to work closer to the critical concen-
tration.
SHEARED SUSPENSIONS OF GRANULAR MATERIALS
Suspension mechanics ~ the description of the theological be-
havior of all sorts of suspensions, but the focus over the past 200
years has been on rigid particles suspended in liquids or gases.
The theory of systems where the interstitial fluid is gaseous dates
to Coulomb's treatment of the static behavior of granular media
in 1776. For solids suspended in liquids, the behavior is heavily
influenced by the nature of the liquid; bulk theological behavior
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73
reflects the mechanical properties of the liquid, modified by par-
ticulate matter. This is in sharp contrast to the situation with
granular materials where the solid Is suspended in a gas. In this
situation the prunary sources of stress involve direct contacts be-
tween particles. One mechanism for producing stresses resembles
molecular momentum transport in gases, where the transport pro-
cess involves the exchange of particles between layers In relative
motion. This mechanism is less important as the volume frac-
tion of solids increases. A second mechanism involves collisions
between particles from layers in relative motion—something akin
to molecular transport in liquids. The third mechanism involves
forces between particles at points of sustained contact. At large
volume fractions, the contact stresses will be dominant for low
rates of deformation, and the collisional stresses will dominate at
high rates. Although the latter two mechanisms resemble momen-
tum transfer In Buids, there is no analog of thermal agitation. De-
formation causes particle movement and the random components
of the particle velocities arise from collisions between particles.
Consequently, these effects die out rapidly as bulk motion ceases.
According to theories now under development, the balance
equations consist of conservation laws for mass, linear momentum,
and a 'pseudo thermal energy." Various versions of the theory
differ in the form of the stress tensor, their treatment of the rate of
conversion of the "pseudo thermal energy" to true thermal energy
by collisions, and the conductivity for Pseudo thermal energy.
Since the particles themselves vary in their density and elastic
properties, have rough surfaces, and need not be spherical, there
are many interesting aspects to any experimental or theoretical
study. It is crucial that the various mechanisms be separated in an
experimentally meaningful way, but in the terrestrial environment
particle behavior is dominated by gravitational effects, making
such separations impossible. The influence of the particle contacts
with any boundary is confounded with that due to particle-particle
contacts, and vice versa. Rheological studies in a m~crogravity
environment offer clear advantages over terrestrial studies since
we could expect to measure and test the tenets of any theory
separately.
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74
GROWTH OF CONDENSATES IN SUPERSATURATED
SYSTEMS
Co~runon examples of the processes referred to here are the
formation of droplets of fog in a supersaturated vapor or the pre-
cipitation of unpurity-rich particles in metallic alloys. Processes
of this kind occur in a wide range of chemical, physical, and bi-
ological situations. These processes have never been adequately
understood from a fundamental point of view, despite having been
known since scientists first looked through microscopes, and de-
spite their importance in determining the structure and properties
of many substances of technological interest.
The conceptual difficulties are twofold. First, there is the
problem of nucleation: the question of how droplets form after
the system has become thermodynamically unstable. This is a
challenging problem in nonequilibrium statistical mechanics that
continues to be of great theoretical interest. It Is in the second
stage of the process, in the growth and coarsening of the precipi-
tate, where microgravity may play a role.
The second stage is characterized by the competition between
several mechanisms. One of these mechanisms is that molecules of
the condensing phase tend to evaporate from the smaller droplets
and recondense onto the larger ones, thus increasing the charac-
termtic size of the objects in the precipitation pattern. Another
competing mechanism is that droplets may merge with one an-
other. In both mechanisms, the driving force is surface tension;
that is, the system evolves toward states of lower surface-to-volume
ratio and thus lower surface energy. At early stages, this coarsen-
ing process is ordinarily controlled by diffusion of material from
one droplet to another. At later stages in fluids, surface tension
can drive hydrodynamic motions in which material flows from one
region to another for example, through channeb formed by merg-
ing droplets. In solid systems, further mechanisms are associated
with elastic forces, interactions with defects, grain boundaries,
and the like. Even the simplest diffusion mechanism presents a
major theoretical challenge, especially when the condensing phase
occupies a nonnegligible fraction of the volume of the system and
the droplets are close enough to each other SO that deformation of
their shapes may be important. At present, the state of our under-
standing of these processes is so rudimentary that a metallurgist,
using the most modern tools of analytic microscopy, cannot tell
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75
whether he is looking at an initial transient or a nearly equili-
brated, late stage of a precipitation pattern. In other words, he
has no fundamental scientific understanding of how to clarify and
control these processes.
One special need in this research area is careful experimenta-
tion using the simplest possible physical systems and observation
of their behavior over the longest possible tunes. It is here that
gravitational effects enter the picture. The simplest mode! systems
are fluids binary mixtures that undergo phase separation when
subjected to changes of temperature or pressure. Fluids can be
made very pure, and automatically have none of the complexity
associated with crystalline anisotropies or defect structures. They
are, however, sub ect to gravity-induced flows when they become
spatially inhomogeneous on sufficiently large length scales. Even
when attempts have been made to use working fluids for which
the two separating phases have nearly the same density, it has
not been possible to prevent precipitation patterns from breaking
down under gravity for long enough times to span the theoretically
interesting range of behavior.
It ~ quite possible that this observational problem could be
solved ~ a microgravity environment; thus, the results would be
of fundamental significance. This class of experiments probably
would require very low average acceleration over periods of the
order of days, but may not be sensitive to small accelerations at
higher frequencies. On the other hand, it is not clear whether
other symmetry-breaking perturbations, such as wall effects or
electrostatic forces, could cause major problems.
FRACTAI AGGllEGATl:S
An exciting area of research ~ the study of systems that can
be adequately investigated only in a low-gravity environment. An
example of such systems is fractal aggregates, objects whose mass
M(~) varies as some power of their size, viz., [D. Here D is the
fractal dimension that lies between 2 and 3 and that depends on
the growth kinetics of the aggregate. Since D < 3, these tenuous
solids become more diffuse as they grow and represent a new state
of matter. Unfortunately, gravitational forces limit the growth of
aggregates in solution because of sedimentation, which depends on
the size of the experimental volume, the viscosity of the medium in
which the aggregates grow, and other factors. While the viscosity,
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76
and hence the sedunentation sane, can be increased, undesirable
physical and chemical ejects occur that fundamentaDy alter the
aggregation process. At present, it has been possible to verify
the fractal growth law M ~ LD only over 2 orders of magnitude
in size; significant questions remain unanswered. If experunents
could be performed ~ a m~crogravity environment, one could in-
vestigate the fractal growth law as well as the scaling dependence
of correlation functions over a considerably wider range of system
parameters. In addition, the complicating influence of hydrody-
namic flow on the aggregation process would be eliminated by
disposing of gravitational effects.
A second major advantage of m~crogravity experiments on ag-
gregate growth is the elimination of mechanical collapse of aggre-
gates due to their own weight. Theory predicts that the max~xnum
size of mechanically stable aggregates La' ~ of order 10 to 100 Em
for clusters of 20~1 gold or silica particles. Thus, even aggregates
led than 0.1 mm in size collapse to Zero volume pancakes," since
the collapsed density vanishes,
M/L3 ~ LD-3 _ O as L
To avoid self-destruction of large aggregates, work must be
done in a nearly weightless environment. Theory predicts that
[M varies inversely with 9: [M(9)—/9. Thus, for gravity
reduced by 105, [M is of order 10 to 100 cm, a desirable range for
experunents.
Once stable aggregates of macroscopic size have been oh
tained, a host of exciting experiments on a single fractal become
possible: elastic, electrical, and thermal transport properties can
be measured. At present, these measurements are impossible be-
cause the medium dominates bulb measurements on aggregate
ensembles.
Another interesting system to study is a fractal gel in which
individual aggregates join together by percolation to form an in-
finite cluster. Such gem are likely to have unique properties. No
doubt, other examples offractal structure occur in nature, possibly
in macromolecular assemblies.
Representative terms from entire chapter:
gravitational effects
