Growth, Dissolution and Pattern Formation in Geosystems

Growth and dissolution processes are of fundamental importance in any geological system, and they occur both near and far away from thermodynamic equilibrium. Familiar examples include the relatively slow growth of faceted mineral crystals, the much more rapid growth of dendritic snow crystals, and the formation of irregular and even spongy mineral grains during weathering near the earth’s surface. Growth and dissolution kinetics are controlled by processes taking place at atomic and molecular scales. However, these processes may affect the patterns that form on a much larger scale in geosysterns through the coupling with transport and/or deformation processes (Barber and Meredith, 1990; Lichtner et al., 1996). Numerous well known patterns that arise through nonlinear couplings between these processes include complex intracrystalline mineral zonation, banded or spotted rocks or polycrystalline aggregates, symplectites and other mineral intergrowths, stylolites and pressure solution seams and reaction front instabilities. These patterns represent the fingerprints of irreversible processes that may leave behind considerable information and memory about important geological processes that took place in the past.

It is a most remarkable phenomenon that most crystals are bounded by flat faces during the growth process. It can be stated that crystals — provided the driving forces for growth are rather low — do indeed grow with flat faces. This phenomenon holds for crystals with very simple structures such as the cubic and hexagonal close-packed crystals of the elements helium, argon and krypton, growing at extremely low temperatures from their melts, and crystals of metals. It also holds for almost all organic crystals of alkanes such as the different paraffin molecules, crystals of aromatic compounds like naphthalene and anthracene, and for much more complex organic molecules like pharmaceutical materials (steroids for example). Even crystals of very complex organic molecules such as proteins that have 20, 000 to 100, 000 atoms per molecule, and crystals of viruses may grow with facets. The occurrence of faceted crystals also holds for almost all inorganic crystals grown in the laboratory and inorganic mineral crystals grown in nature. Examples are the very well known crystals of quartz, gypsum, pyrite, etc. Note that the occurrence of facets on crystals does not depend on the size of the crystals. Crystals with a dimension of one to a few meters in 1, 2 or 3 dimensions, grown in the laboratory (or crystal growth factory) and/or nature may have the same flat faces as tiny crystals of the same compound with dimensions of a few microns or less. In general it can be stated that most crystals show facets (and in principle the same kind of facets) independent of their size, which may vary by up to six orders of magnitude.

It has been observed for a long time that not all individual minerals have a uniform chemical composition. Many of them exhibit complex growth patterns, such as oscillatory zoning (in which the chemical composition fluctuates along a traverse within the crystal) or branching patterns (characterized by a tree-like crystal form) that arise spontaneously as a result of growth kinetics. Liesegang was among the first to investigate the kinetics of a class of growth patterns: periodic precipitation. Until fairly recently (Lasaga and Kirkpatrick, 1981), the importance of kinetics in geochemistry was largely overlooked.

The minor and trace elements present in minerals have long been regarded as retaining important clues to the crystallization and post-crystallization histories of rocks and minerals. Countless studies have examined the presence and concentrations of specific elements as fingerprints of various petrogenetic environments or processes, and their utility is evident in fields as diverse as igneous crystallization and porosity occlusion in shallow aquifers. Whereas many of the factors controlling the distribution of trace elements are well known, there has recently been much attention given to the role of mineral surfaces in influencing element incorporation during crystal growth. Examples of the influence of mineral surfaces on element incorporation are in fact well known to mineralogists in the form of sectoral zoning in such common minerals as augite, staurolite, quartz, calcite, topaz, and zircon (e.g., Hollister, 1970; Dowty, 1976; Reeder and Paquette, 1989; Vavra, 1990). Sectoral zoning arises from differences in the incorporation of minor elements at symmetrically nonequivalent crystal faces during growth, expressed as compositionally distinct growth sectors (Figure 6.1). Reeder (1991) has emphasized the fundamental geometrical and temporal differences of this zoning type in relation to the more common concentric growth zoning; in the familiar case of concentric zoning the surface across which composition changes coincides with prior growth surfaces and hence the zoning pattern records temporal variations in element incorporation. In the case of sectoral zoning, however, the interface across which composition changes cuts across prior growth surfaces and coincides with sector boundaries.

Naturally growing crystals have many different shapes, which are related to their crystal structure, to their surface properties and to the conditions under which they grew. Some crystals are faceted, for example quartz crystals; metal crystals often grow with smooth shapes, and salts and organic materials often grow with needle-like structures. Dendritic crystals, which are the subject of this paper, are also very common. Dendrites are crystals having a tree-like branched structure, with details dictated by their crystallography. The most well-known, and also one of the most beautiful, is the snow-flake. This structure was first noticed by Kepler in 1611 (Kepler, 1611), who drew pictures showing their six-fold symmetry properties.

In the original diffusion-limited aggregated (DLA) model of Witten and Sander (1981) ‘particles’, represented by lattice sites, are added, one at a time, to a growing cluster or aggregate of particles via random walk paths starting from outside of the region occupied by the cluster. In the most simple version of the model (Witten and Sander, 1981), a simulation is started by occupying a site in the center of a square or triangular lattice to represent the ‘seed’, ‘growth site’ or ‘nucleation site’. A site far from the cluster is then selected and a random walk is started from the selected site. If the random walker moves too far from the growing cluster, it is terminated and a new random walk is started. If the random walker eventually reaches a site that is the nearest neighbor to a previously occupied site, the random walk is stopped and the unoccupied perimeter site (the last site occupied by the random walker) is filled to represent the growth process. The process of launching random walkers from outside of the region occupied by the growing cluster and terminating them when they wander too far from the cluster or ‘sticking’ them to the growing cluster when they reach an occupied perimeter site is repeated many times to simulate the cluster growth process.

This contribution focuses upon the predictable aspects of metallurgical microstructures which by inverse logic have a bearing on similar forms from the mainly ceramic geocosmological record. Among the common forms are needles, plates, cells, and dendrites, together with lamellar, wavy, multidisperse and filamentary arrays. Apart from the usual disparity of time and spatial scales, the main distinctions arise from the stronger directionality of bonds in the case of ceramics, with concomitant stronger anisotropy of surface tension and an attendant facetting propensity. In common are the facts that both liquid-solid and solid-solid transformations with control by diffusion or by chemical reaction or both may be involved. The inversion of the logic comes from the fact that metallurgists try to predict micrographic scale from given thermal conditions, whereas geologists would like, among other things, to estimate historical thermal conditions from observed patterning scales.

The patterns on shells of mollusks display an enormous diversity and are frequently of great beauty. Shells consist of calcified material. The animals can increase the size of their shells only by accretion of new material along a marginal zone, the growing edge of the shell. Therefore, the formation of these patterns proceeds in most species in a strictly linear manner. The second dimension is a protocol of what happens as function of time. The shell patterns resemble, so to say, a space-time plot. They provide an unique situation in which the complete history of a highly dynamic process is preserved. They are thus very convenient to study general properties of pattern forming systems. Models for these patterns will be discussed. By computer simulations, it will be demonstrated that even fine details in the pattern can be faithfully reproduced.

Silicon is the second most abundant element in the Earth’s crust and in combination with oxygen the silicates form the largest and most abundant group of minerals. Many silicates are also formed in combination with other elements such as magnesium, aluminium, calcium and iron and as such form many of the minerals in our rocks and soils. Weathering processes over millions of years have also resulted in the occurrence of silica (silicon dioxide) in a variety of largely crystalline forms. The only naturally occurring amorphous silica mineral is opal.

There are numerous motivations for studying the rates and mechanisms of olivine and pyroxene dissolution. The weathering of these minerals are major contributors of Mg, Fe, and Ca to soils and surface waters. In soils, these metals are essential nutrients used by plant communities (Marschner, 1995). The presence of Mg, Fe, and Ca in surficial waters provokes carbonate mineral precipation, which is linked to atmospheric CO₂ content and climate through the carbon cycle (Berner et al., 1983; Brady, 1991; Brady and Carroll, 1994; Berner, 1995; Gislason et al., 1996). The variation of forsterite and enstatite dissolution rates with solution composition and temperature, therefore, provides clues towards the effect of environmental changes on global climate. The dissolution of olivines and pyroxenes are also important buffers controlling surface water pH (Dreyer, 1988; Drever and Clow, 1995).

One of the most rapidly growing research areas in the geosciences is mineral surface science. This is partly because of the recognition of the widely varying role that mineral surfaces play in the cycling of chemicals through the crust. In addition, advances in analytical techniques are making it more and more convenient to gather data pertinent to mineral surface chemistry. In particular, the ability to perform analyses in situ (i.e. with the reactant fluid present) is a key advance. Synchrotron X-ray techniques are proving to be extremely useful for acquiring data from the mineral-fluid interface during reaction. Because of the energy range and high brightness available at a synchrotron, enough signal can be transmitted through a reactant fluid overlayer to perform a variety of measurements without disturbing the system. Experiments detailing the roughening of calcite (Chiarello et al., 1993) and the epitaxial precipitation of otavite (CdCO₃) on calcite (Chiarello and Sturchio, 1994) have been successfully completed in situ. Roughness of the surface is determined at the angstrom-scale and is a root mean square (r.m.s.) measure of the average peak to trough height of features on the surface. Glancing incidence X-ray measurements of the reflected and diffusely scattered components of the beam are particularly useful for describing the structural adjustments that occur during reaction, even if the surface layer becomes amorphous.

The rates of many heterogeneous reactions are dependent upon the mineral-water interfacial area. Examples include release of nutrients from primary minerals, rate of growth of authigenic minerals, adsorption and desorption of metal and organic contaminants on soil and sediment grains, neutralization of acid deposition by weathering reactions, oxidation and reduction of mmetal-containing phases and solutes, clumping of colloids or bacteria by electrostatic attraction, and photocatalytic degradation of organic pollutants at metal oxide surfaces (Davis et al., 1993; White and Brantley, 1995). Several workers have also shown that the surface area is a key parameter in predicting weathering rates using geochemical models (e.g. PROFILE) and soil chemistry under the influence of acid rain (Jönsson et al., 1995; Hodson et al., 1996, 1997a). Along with the permeability, the surface area is one of the most difficult physical parameters to quantify in extrapolating from the laboratory to the soil plot to the watershed (White and Peterson, 1990). Most models of solute transport in aquifers and in soils simply ignore the mineral-water surface area term by combining it with the kinetic rate constant into one fitting parameter, despite the fact that the specific surface area may vary over several orders of magnitude — 10²–10⁶ cm²/g as functions of grain size, mineralogy, oxide coating, weathering history, or biological effects, or as a combination of these factors. Despite the importance of the mineral surface area in many areas of geochemistry, little systematic effort has been expended to understand or predict this term for primary silicates (more work has been completed on the surface area of clays and simple oxides).

Rock strength is profoundly affected by pore fluids through diverse mechanical and chemical interactions (Walder and Nur, 1984; Carter et al., 1990). Conversely, deformation can easily result in large changes of the pore structure (Paterson, 1978; Wong, 1990). Because transport properties of rocks, including permeability and electrical resistivity, are directly determined by the pore structure (Walsh, 1965; Simmons and Richter, 1976; Gangi, 1979; Shankland et al., 1981; Heard and Page, 1982), those properties can also be substantially altered by deformation. As a direct consequence of the relationships among the physical state of pore fluids, fluid transport, and mechanical deformation, rocks may exhibit complex, time-dependent mechanical and transport properties.

Since the earliest work on pressure solution (Sorby, 1863), numerous papers on the subject have appeared in the literature and it is well established that polycrystalline materials containing a solution or partial melt phase within grain boundaries can deform by a process of fluid assisted diffusional creep or intergranular pressure solution creep (Durney, 1972; Stocker and Ashby, 1973; Robin, 1978; Raj, 1982; Pharr and Ashby, 1983; Rutter, 1983; Cooper and Kohlstedt, 1984). This is believed to be distinct from solid state diffusion creep in which grain boundary self diffusion can potentially be (modestly) enhanced by the presence of water defects in the grain boundary zone. In nature, aside from the development of larger-scale solution transfer features such as stylolites and solution cleavages (Kamb, 1959, 1961; Merino and Ortoleva, 1980; Ortoleva et al., 1982; Merino et al., 1983; Dewers and Ortoleva, 1989, 1990; Merino, 1992), pervasive intergranular pressure solution is an important deformation mechanism, manifested by many types of partially dissolved objects such as grains and fossils (Elliot, 1973; Rutter, 1976). It occurs under conditions ranging from diagenetic to greenschist facies (Weyl, 1959; Elliot, 1973; Beach, 1979; Etheridge et al., 1983; Rutter, 1983; Tada et al., 1987; Tada and Siever, 1989). In the diagenetic regime, intergranular pressure solution creep is one of the main mechanisms for compaction and cementation of porous sedimentary rocks (e.g., Tada et al., 1987). In pelitic rocks, it appears to be the dominant deformation mechanism up to medium and even high grade metamorphic conditions (Bell and Cuff, 1989).

In many applications of chemical transport modelling to geological problems, it is very important to take into account the changes to the transport properties of the porous medium that will result from chemical reactions driven by the component fluxes which are being modelled. This is particularly true where the reactions involve breakdowr of carbonate minerals, because they produce very large changes in solid volume, but there are many other fluid-rock reactions, involving both precipitation and dissolution, that are capable of perturbing the pattern of flow that originally triggered the reaction. This paper is concerned with the growth of calc-silicate minerals replacing marbles in metamorphism, which we model through the simplest possible metamorphic reaction: However, our approach is equally appliable a wide range of skarn-forming reactions.

Rocks consist of solid phases and fluid phases. The percentage of fluids by volume is defined as the porosity. At the time of deposition, the solid phases in clastic sedimentary rocks consist of mineral grains, rock fragments, and sometimes also amorphous material. Minerals precipitated after deposition are referred to as authigenic, and they fill the primary pore space or replace detrital grains. The fluid phases may consist of liquid phases like water and oil and gasses like CH₄ and CO₂. Both minerals and water have a low volume compressibility (van Balen and Cloetingh, 1993) while gasses are more compressible and this has important effects on fluid flow.