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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
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 smallless 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
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 motionsomething 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.
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
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,
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.
