Diffusive crystal dissolution

Crystal dissolution may include three component processes: interface reaction, diffusion and complications due to convection. We report here a theoretical and experimental study of crystal dissolution in silicate melt without convection. A reaction-diffusion equation is developed and numerically solved. The results show that during non-convective crystal dissolution in silicate melt, the interface melt composition reaches a constant or stationary "saturation" composition in less than a second, hence interface reaction is not the rate-determining step and crystal dissolution in silicate melt is usually diffusion-controlled. Crystal dissolution experiments (designed to suppress convection) show that the concentration profiles of all components propagate into the melt according to the square root of run duration, and that the dissolution distance is also proportional to the square root of run duration. Thus our experiments confirm that the dissolution is diffusion controlled, which is consistent with our numerical calculations. For some principal equilibrium-determining components, concentration profiles conform approximately to the analytical solution of the diffusion equation with a constant effective binary diffusion coefficient. Diffusive dissolution rates (which are inversely proportional to square root of time) can thus be predicted from the phase equilibria and the effective binary diffusion coefficients. To predict steady-state convective dissolution rates, the thickness of the boundary layer must be known. If the convective compositional boundary layer thickness around a dissolving crystal aggregate or near the wall of a magma chamber during convection is about 2 cm or larger, then convective dissolution would rarely result in any significant alteration of original melt. Our dissolution experiments also illustrate the complexity of the diffusion process. Uphill diffusion is common, especially during olivine dissolution into andesitic melt where a majority of the components show the effect of diffusion up their own concentration gradients. Uphill diffusion has implications to the understanding of crystal zoning, and suggests caution is required in applying least squares mass balance analysis to magmatic rocks affected by processes involving diffusion. * Present address: Division of Geological and Planetary Sciences, 170-25, California Institute of Technology, Pasadena, CA 91125, USA Offprint requests to: D. Walker Introduction An understanding of the processes by which crystals dissolve is necessary for the study of many geological processes, such as the partial melting of mantle and crustal rocks, and the subsequent contamination of these magmas once they leave their source region. Previous studies of crystal dissolution encompass both theoretical and experimental approaches. Kirkpatrick (1975), Dowty (1980), and Kirkpatrick (198l) reviewed the theoretical background of crystal growth, much of which can also be applied to crystal dissolution. Two processes, reaction at the crystal-melt interface and diffusion in the melt (or in the crystal), happen simultaneously during non-convective crystal dissolution. The dissolution rate (or the growth rate) may be controlled by either interface reaction or diffusion. If the reaction at the crystal-melt interface is slow compared to diffusion in the melt (or crystal), or no diffusion is necessary, the process is controlled by the rate of interface reaction; otherwise, if diffusion in the melt (or the crystal) is necessary and is slow compared to the attachment and detachment of atoms at the interface, the process is controlled by diffusion. The coupling of interface reaction and diffusion makes it very difficult to analytically treat even the non-convective crystal dissolution (or growth) problem. Lasaga (1982) was the first to consider the coupling of interface reaction and diffusion through a crystal growth equation in which the growth rate (or interface reaction rate) was related to time or interface concentration. (Prior to Lasaga 1982, analytical treatment of crystal dissolution either assumed constant growth rate or used a Stephan approach, i.e., t t/2 dependence of growth rate.) Lasaga obtained a solution for the interface composition in the form of an integral equation by using a Laplace transform with respect to x (instead of t as in the usual approach). The solution can be applied to any system if the growth rate can be written as a function of melt composition. Experimental studies of crystal dissolution have been conducted by Watson (1982a); Tsuchiyama and Takahashi (1983); Harrison and Watson (1983, 1984); Donaldson (1985); Kuo and Kirkpatrick (1985); Marvin and Walker (1985); Tsuchiyama (1985a, b); Brearley and Scarfe (1986). Among them, Tsuchiyama and Takahashi (1983), Marvin and Walker (1985) and Tsuchiyama (1985a, b) examined the kinetics of partial melting at the mutual boundary of two crystals or of crystal dissolution into finite melt reservoirs. These studies differ significantly from what we will report in this paper and will not be discussed further. Watson (1982a) determined experimental dissolution rates of quartz, feldspar, and a synthetic granite in basaltic melt. He noted uphill diffusion and addressed its significance in the selective contamination of mantle-derived magmas. Donaldson (1985) determined the dissolution rates of olivine, plagioclase and quartz in basaltic melt. Kuo and Kirkpatrick (1985) studied the dissolution rate of forsterite, diopside, enstatite, and quartz in the system D i F o S i O 2 at 1 atmosphere. They all found that the dissolution rate was time-independent (explained by free convection in the experimental charge) and independent of crystallographic orientation. Harrison and Watson (1983, 1984) investigated the kinetics of zircon and apatite dissolution in felsic melt. Brearley and Scarfe (1986) studied dissolution of natural olivine, pyroxene, spinel, and garnet in a natural alkali basaltic melt and found concentration profiles to be time-independent and concluded also that the dissolution rate was time-independent. They also tried to obtain diffusivities from their steady-state concentration profiles and found that the diffusivities thus obtained depend not only on temperature and pressure, but also on the dissolution rate. All the above experimental studies, with the exception of Harrison and Watson (1983, 1984), have examined crystal dissolution where convection operated in the melt. Convection leads to the constant dissolution rates and to the development of steady-state concentration profiles, as obtained by Watson (1982a), Kuo and Kirkpatrick (1985) and Brearley and Scarfe (1986). These dissolution rates can be applied to natural systems only if the convection regimes in natural and experimental systems are similar. We report in this paper studies of crystal dissolution which address non-convective crystal dissolution. We first evaluate the relative importance of interface reaction through numerical solution of the reaction-diffusion equation. We then examine the diffusive aspect of crystal dissolution both by analytical solution to the diffusion equation and by experiments designed to suppress convection. Finally we apply our results by extrapolation to natural magmatic systems, where convection is usually present and complicates dissolution. Interface reaction Two processes occur simultaneously during non-convective crystal dissolution into a melt of different composition. Interface reaction, the attachment and detachment of atoms at the crystal-melt interface, determines the intrinsic maximum dissolution rate. Meanwhile, diffusion, the transport of mass to and from the interface, determines whether this maximum rate can be attained. Because diffusion rates influence interface composition which, in turn, determines the degree of saturation at the interface and thus the interface reaction rate, the two processes are coupled and it is their interplay that controls dissolution. It is important, therefore, to ascertain the conditions under which one or the other of these two processes controls the dissolution rate. In order to examine the relative roles of reaction at the crystal-melt interface and diffusion in the melt, a reaction-diffusion equation is developed which treats both interface reaction and diffusion in the melt simultaneously. The equation is then solved numerically to evaluate the relative importance of interface reaction and diffusion for magmatic 493 conditions. This approach differs from that of Lasaga (1982) in that we relate the reaction rate to diffusion through the degree of saturation at the interface so that we can determine the relative roles of interface reaction and diffusion in a general way. However, mathematically, our numerical solution can be viewed as a special case of Lasaga's master solution. The strategy to develop a reaction-diffusion equation is as follows: First the interface reaction rate equation is expressed in terms of the degree of saturation at the interface. Then the degree of saturation of the melt (including interface melt) is shown to satisfy a diffusion equation if deviation from equilibrium is not great. The interface reaction rate for continuous dissolution can be expressed as (e.g., Kirkpatrick 1981): V~ = f ao v e AG"/RT ( e AGc/RT 1) (1) where V~ is the crystal dissolution rate, AGa and v are the activation energy and vibrational frequency of the activated transition complex, AGe is the difference in partial molar free energies of the dissolving species between liquid and crystalline states, f is the fraction of sites available, ao is the distance between subsequent layers of the crystal, R is the gas constant and T is the temperature in Kelvin. Notice this equation is different from the equation for crystal growth by a negative sign. The crystal dissolution rate V~ in (1) refers to the rate at which a fixed point in the crystal moves relative to the crystal-melt interface. However, when discussing diffusion in the melt it is important to establish the rate at which the interface moves relative to the melt (or a fixed point in the melt). This rate is hereafter referred as the melt growth rate. To dissolve a crystal layer of some thickness, a thicker layer of melt will be produced if the crystal is denser than the melt. As such, the melt growth rate exceeds than the crystal dissolution rate. Specifically the melt growth rate differs from the crystal dissolution rate by a constant equal to the ratio of crystal density over melt density. Multiplying both sides of (1) by this density ratio transforms the crystal dissolution rate into the melt growth rate (denoted as V). The degree of saturation (denoted as w) is defined as: