Global Biogeochemical Cycles

Age dependence of mineral dissolution and precipitation rates

Authors

  • Daniel Reeves,

    Corresponding author
    1. Lorenz Center and Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA
    • Corresponding author: D. Reeves, Massachusetts Institute of Technology, EAPS, 77 Massachusetts Avenue, Cambridge, MA 02139, USA. (dreeves@mit.edu)

    Search for more papers by this author
  • Daniel H. Rothman

    1. Lorenz Center and Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA
    Search for more papers by this author

Abstract

[1] Understanding the rates of weathering, and more generally dissolution and precipitation in porous materials, is important for many applications including modeling the global carbon cycle and predicting short-term and long-term behavior in subsurface carbon sequestration sites. However, interpretation of the rates remains elusive as they have been observed to vary with location, measurement procedure, and time. We argue that the mechanisms responsible for the apparent aging in the rates, or gradual decrease over time, can be partially determined by noting which measure of time best characterizes the dependence. If the rate is best described as a function of residence time, then hydraulic and transport limitations are responsible for the variations. If reaction age is a better independent variable, then limitations in the chemical reaction at the fluid-mineral interface are responsible. We discuss several mechanisms in each category and construct mathematical models that demonstrate quantitatively how they affect time variation in reaction rates. These include nonlinear kinetics, disordered kinetics, and a reprecipitation model that accounts for the limited access to the bulk of a dissolving solid. We apply the reprecipitation model to the development of the isotopic composition of porous solids to derive an apparent rate constant that decays with inverse time, similar to that calculated for diagenesis in deep-sea sediments. This paper provides a theoretical framework for understanding the changing and varied dissolution and precipitation rates measured in the laboratory and in nature, and provides testable quantitative models that capture the aging effect.

1 Introduction

[2] Rates of dissolution and precipitation in carbonate and silicate minerals have been studied extensively in many contexts. These rates determine fundamental timescales in the global carbon cycle [Lasaga et al., 1994; Kump et al., 2000; Navarre-Sitchler and Brantley, 2007; Wallmann et al., 2008], during diagenetic processes in marine sediments [Fantle et al., 2010] and in development of soils [Jin et al., 2008]. They are also relevant to the contemporary problem of carbon dioxide sequestration in subsurface reservoirs [Kang et al., 2009; Kelemen and Matter, 2008], where rates determine how dissolution will affect transport through the reservoir in the short term, and how carbonate precipitation will sequester carbon over the long term.

[3] In general, the total rates of dissolution and precipitation are controlled by combinations of transport properties, the chemical environment, material geometry, and surface reaction kinetics. Methods have been developed to probe the effects of each of these types of properties on dissolution and precipitation rates. Techniques utilizing tools such as atomic force microscopy and vertical scanning interferometry have been utilized to directly visualize and probe the rate of material reactions at the molecular level [Teng et al., 1998, 2000, 2001; Arvidson et al., 2003; Yoreo et al., 2009; Xu et al., 2010]. While such microscopic techniques have the clear benefit of illuminating the surface reaction mechanisms at the molecular level, the results do not capture the ensemble of all reactive surfaces. At larger length scales, dissolution and precipitation rates may be dominated by transport limitations induced by limited available solute and flow through porous materials. Reactive transport modeling [Goddérisa et al., 2006; Maher et al., 2006, 2009] is an approach used to capture these complex flows and predict large-scale rates starting from rate laws derived at smaller scales (i.e., upscaling). To understand the rate laws that dominate at the mesoscale, experimental methods have been developed to remove transport limitations and explore systems dominated by surface processes [Weyl, 1958; MacInnis, 1992; Svensson and Dreybrodt, 1992; Zuddas, 1994; Cama et al., 2000; Jeschke et al., 2001].

[4] Several studies of mineral dissolution and precipitation rates have investigated time variability. They find that as the reaction proceeds, the observed rates decrease, a behavior we refer to as “aging.” These observations include investigations of the weathering history of natural soils and rocks [Taylor and Blum, 1995; White et al., 1996; Kump et al., 2000; White et al., 2001, 2009; Maher, 2010]. The observed age dependence and marked disparity between rates measured in the field and those measured in laboratory settings [Stillings and Brantley, 1995; Arvidson et al., 2003] inspired theoretical analyses focusing on varied fluid residence times [Maher, 2010] and further laboratory studies under controlled conditions such as those by White and Brantley [2003]. Those authors measured dissolution rates of silicate minerals in column reactors over a period of several years. They report a power law decay in the dissolution-rate constant and suggest that the loss of reactive surface area in the material and the accrual of secondary precipitates may be the responsible aging mechanisms. Other studies have calculated dissolution and reprecipitation rates by analyzing the isotopic compositions of elements such as calcium, strontium, and uranium in deep-sea sediments [Richter and Liang, 1993; Maher et al., 2004; Fantle and DePaolo, 2006, 2007; Fantle et al., 2010]. They observe aging phenomena similar to that of White and Brantley [2003], but at much longer timescales.

[5] Much of our theoretical understanding of reaction rates is derived from Transition State Theory (TST), as developed over the last century [Evans and Polanyi, 1935; Eyring, 1935; Aagaard and Helgeson, 1982]. TST theory focuses on how the dissolution and precipitation rates of reactants depend upon the energetic and entropic barriers posed at the molecular level. A particle must cross the barrier and change energetic state to free itself from the crystal structure of the mineral and enter solution, and vice versa. We take this as a starting point in understanding reactions at the fluid-mineral interface. However, it is not understood if the perceived aging of reaction rates is due to changes at the fluid-mineral interface or is due to changes in the transport, geometric, and equilibrium properties of the sample and surrounding fluid. Indeed, there is growing evidence from rigorous application of reactive transport models to reactive-flow through field systems [Maher et al., 2009; Navarre-Sitchler et al., 2011] that the perceived correlation between rate and time is an artifact of the way rates are calculated in macroscopic systems. In particular, these approaches suggest solutions to the “lab-field" discrepancy by which laboratory-measured reaction rates differ by orders of magnitude from field-derived rates [Arvidson et al., 2003; White and Brantley, 2003; Fantle and DePaolo, 2006; Navarre-Sitchler and Brantley, 2007; Maher, 2010]. In some cases, this discrepancy can be resolved via consistent definitions of surface area at appropriate length scales [Navarre-Sitchler and Brantley, 2007]. However, appealing only to changes in surface area with time cannot account for the magnitude of the observed discrepancies [Fantle and Depaolo, 2007].

[6] This paper develops a framework for understanding the time dependence of rates and for reconciling differences among various observations. We sense that there are various ideas and observations that deserve being placed in such a cohesive framework, and to that end, this paper is partly a review, partly a reappraisal of published work, and partly novel contribution. In section 2, we identify features that distinguish between perceived aging due to transport properties and aging due to changes at the mineral-fluid interface. The former, discussed further in section 3, includes the complexities of reactive-transport through porous media. The latter, discussed in the final two sections, includes changes in the composition, geometry, and accessibility of the mineral. We introduce several mechanistic and mathematical models that appeal to heterogeneities in the solids to predict variable apparent rates given fixed fundamental rate constants. In other words, although the rate constant determined by TST remains fixed, the calculation or averaging technique used in the measurement gives rise to a time dependence that is essentially an artifact. More detailed descriptions of the system, as with reactive-transport models in the example above, recover a time-independent reaction rate constant.

[7] Although there are many applications for which precise knowledge of the evolution of mineral precipitation and dissolution rates is critical, there is more to be learned than rates and numbers by studying the aging phenomenon. By understanding what physical mechanisms cause the observed aging, we gain insight into the underlying mechanisms of precipitation and dissolution. There is currently no comprehensive understanding of how to upscale measured reaction rates, i.e., how to derive bulk dissolution and precipitation rates from molecular scale rates measured at the crystal surface. The aging phenomenon is one of changing temporal scales, rather than spatial scales. Perhaps, the two are related, and by understanding the origins of long timescale behavior of dissolution and precipitation, we may gain insight into the connections between spatial scales.

2 Transport and Surface Reaction Controls

[8] Precipitation and dissolution rates of geologically relevant porous media are limited by both transport processes and reaction rates at the fluid-mineral interface. The net dissolution of the solid is the difference between the bulk dissolution rate and the bulk precipitation rate. In general, both the forward and reverse bulk rates depend on the local concentration (and activity) [Lasaga and Luttge, 2001, 2004; D. J. DePaolo, Theory of isotopic and trace element fractionation during precipitation of carbonate minerals from aqueous solutions: Surface reaction control limit, submitted to Geochimica et Cosmochimica Acta, 2010] at the fluid-mineral interface. However, concentration measurements are made at the bulk scale, and the local concentration at the mineral surface can differ from the bulk concentration. Transport processes, such as advection and diffusion, must effectively sweep solute away from the dissolving mineral to mix with the bulk; otherwise, the fluid in contact with mineral approaches equilibrium and there is no further net reaction.

[9] This uncertainty of the local concentration at the reacting surface, given only a bulk concentration measurement, gives rise to an apparent rate constant. We define the apparent reaction rate constant, kap, by assuming a first-order reaction, R=−tC=kap(1−Ω), where the relative saturation is Ω=C/Ceq, C is the bulk concentration of the dissolving species, and Ceq is the equilibrium concentration. kap is a composite rate constant where in addition to the fundamental TST relevant rate constant, it includes the effects of catalytic and inhibitory species, the specific surface area of the sample, the surface geometry, and the effects of averaging over heterogeneous samples. If the concentration at the surface is equal to that in the bulk, then kap is controlled by the kinetics of the reaction itself. We refer to this case as “surface-reaction control” because the limiting step is due to the chemistry of the surface reaction. In the limiting case of C=0, the overall apparent rate is equal to the surface-reaction-limited rate.

[10] If the fluid in contact with the surface is near equilibrium, then the net reaction can only proceed as quickly as transport processes remove solute, i.e., the reaction is “transport-limited.” These limitations give rise to gradients in the concentration which could be confined to the microscopic pore scale or span many pores at the mesoscopic scale. In the former case, kap is calculated using bulk average concentrations. In the latter, coarse-grained average concentrations can be estimated at the mesoscale using reactive-transport models [Maher et al., 2009; Maher, 2011]. In either case, the measured concentration is lower than that near the mineral surface, so kap will appear smaller than the surface-reaction-limited rate.

[11] Typically, reaction rate constants are reported as rates normalized by surface area. The value of the surface area depends on the resolution, or ruler size, of the measurement technique. Surface area normalization provides a consistent definition of kinetics, as long as the technique is consistent and the length scale of transport limitations approaches zero (surface-reaction control). However, when transport limitations introduce finite length scales, the relevant measurement of surface area becomes ambiguous. We must take care to measure the surface area with a ruler compatible with the transport length scale when normalizing rate constants.

2.1 Connections Between Rate Limitations and Time Measurements

[12] As discussed above, reaction rates are observed to “age,” or decrease with time. However, the time variable used to compare rates can be associated with either of two distinct measurements. A reacting system, either in the laboratory or in the field, has both a “reaction age” and “fluid age.” The former is the total time the solid has been undergoing material exchange with the fluid phase through dissolution-precipitation reactions. For weathering of soils in a glacial moraine, for example, the reaction age is the time since the soil was deposited and weathering commenced [Bain et al., 1993; Taylor and Blum, 1995; White and Brantley, 2003; White et al., 2005]. In weathering of sediments, such as at the seafloor, it is the time since deposition [Maher et al., 2004; Fantle and Depaolo, 2007; Fantle et al., 2010]. Alternatively, one can use the “fluid age,” or fluid residence time, τof fluid in the reactive system. In laboratory experiments, this is the average time a parcel of fluid spends in a flow-through or batch reactor [Holdren and Speyer, 1985; Burch et al., 1993]. In soils, it is the time rainwater spends in the porous soil column. The time τis measured in many ways such as using discharge rates [Clow and Drever, 1996], groundwater flow models [Kim, 2002], or breakthrough curves of introduced tracers [Jin et al., 2008]. In general, it is estimated using one of two generalized approaches: τ=[Length]/[Flow velocity], or τ=[Fluid volume]/([Length][Infiltration flux]).

[13] When considering how reaction rates depend on time, it is informative to consider which time variable yields a more convincing dependence. If the aging is best characterized as a function of reaction age, then the mechanisms responsible for the aging are related to changes in the reacting surface itself. If, however, the aging is best characterized as a function of residence time, then the aging mechanisms are related to transport processes and the approach to equilibrium.

[14] Therefore, we propose a strategy to determine if the variations in weathering rates both within and between studies are related to transport or surface reaction limitations. The mechanisms responsible for variability can be identified by plotting rates as functions of residence time and reaction age. Figure 1 gives weathering rates that span 4 orders of magnitude from nine laboratory and field studies for which the fluid residence times are available (data as presented in given references or Maher [2010]). Collectively, these data cluster near the power law trend k=4×10−5/τ(appearing as a straight line on the logarithmic axes). Because the dependence on fluid residence time is strong, we conclude that the mechanisms that separate the reaction rates in this subset of studies are related to transport and equilibrium processes. This trend continues across orders of magnitude in residence time and between laboratory and field, where we expect a dramatic difference in the degree of deviation from equilibrium. The clearest exceptions are the data from Burch et al. [1993], which derive from a well-mixed laboratory experiment, and the data of Taylor and Blum[1995]. It should be noted that there are large uncertainties in the fluid residence times of this last study, as we have computed them using the available data of rainfall and soil depths available.

Figure 1.

Weathering rates, or dissolution-rate constants, for silicates from various studies. The rates are plotted as a function of the residence time of the fluid in the experimental or natural system. The dashed line is along k=4×10−5/τ. Some of the data are as presented in Maher [2010].

[15] Systems that are limited by surface reaction mechanisms do not exhibit clustered reaction rates when plotted as a function of residence time. Rather, they cluster when plotted as a function of reaction age as illustrated by Figure 2 which includes reaction rates for nine studies for which the reaction age is available. Collectively, these data cluster near the line k=0.1/t, with the exceptions of Jin et al. [2008] and White et al. [2008], which clustered better with studies plotted as functions of residence time in Figure 1. Note the power law behavior evident in data within some individual studies. For example, data from Taylor and Blum [1995] exhibits more regular behavior that is consistent with other studies (in terms of time dependence) when plotted as a function of reaction age rather than residence time. The dependence on reaction age suggests that the variations observed among the studies are due to surface reaction mechanisms. This distinction leaves one study definitely ambiguous, that of Kim [2002]. In this case, the weathering rates are consistent with the trend observed in either plot, so we can rule out neither transport nor surface reaction mechanisms as relevant to that sample. The same may be true of the other studies for which only one type of age is available.

Figure 2.

Weathering rates for carbonates and silicates from various studies. The rates are plotted as a function of the age of the reaction, or the total time since the solid phase began weathering. The dashed line shows k=0.1/t.

[16] To make a definite statement concerning a single site or group of sites, detailed measurements of both residence time and reaction age must be made. In this paper, we appeal to the collections of studies in Figures 1 and 2 as examples to demonstrate a strategy that helps determine if variations in weathering rates are transport or surface-reaction related.

[17] There are many possible causes of aging, or observed slowing of kinetics, within the two general categories of transport and surface-reaction control. In the following sections, we discuss previously proposed mechanisms and suggest several novel ones. We first discuss mechanisms providing transport limitations in section 3, including nonlinear kinetics and improper use of equilibrium values. We then turn our attention in section 4 to those mechanisms providing surface reaction limitations, including disordered kinetics and slow exchange of reactive surfaces. We employ mathematical models to make the aging mechanisms concrete and quantitative, allowing us to form testable predictions that can distinguish one model from another. Our list of physical processes is by no means an exhaustive list, but the mathematical models we develop are general and can apply to new physical mechanisms that have not yet been considered. Therefore, as we analyze the quantitative behavior of our specific models, we gain insight into the generic behavior of many different physical mechanisms.

3 Transport Limited Aging Processes

[18] As discussed above, the supply of dilute fluid to the reactive surface is one factor that controls the measured reaction rate. As the fluid stays in contact with the mineral, it approaches equilibrium and the measured rate decreases. Thus, the pertinent time over which we measure these changes is the residence time of the fluid in the medium. In this section, we discuss mechanisms that couple time to the approach to equilibrium, thereby varying apparent reaction rates.

3.1 Rate Measurements Near Equilibrium

[19] The first mechanism we discuss that causes time dependence in rates is a trivial result of definitions. One method of measuring total mineral weathering at a field site is to estimate the total influx and efflux of reaction products. The reaction rate is the difference of those concentrations, divided by the average residence time of fluid in the system, τ. However, if at some time within the system, the dissolution/precipitation reaction is arrested by approach to equilibrium or a chemical feedback such as that discussed in Yoshida et al. [2011], then C will remain constant, and the apparent rate will scale as 1/τ. Therefore, this method yields the trivial reaction rate Ceq/τfor any system in which the residence time is long enough that the efflux has reached equilibrium.

[20] There may be a tendency to observe this apparent aging due to sample selection bias, which could be geologically driven or due to constraints in data collection. In soil studies, a natural system size is the entire regolith layer down to solid rock. If the regolith-bedrock interface is generated by the weathering reactions themselves, the interface is naturally located where the weathering reactions cease, i.e., near equilibrium. Rate calculations using concentrations and residence times at the interface will tend to yield a 1/τ dependency.

3.2 Nonlinear Kinetics and Apparent Measurements

[21] Although linear rate laws have proven to be good approximations for silicates and carbonates far from equilibrium [Svensson and Dreybrodt, 1992; Dove et al., 2005], near-equilibrium dissolution and precipitation rates are nonlinear and history dependent [Burch et al., 1993; Teng et al., 2001; Fenter et al., 2003; Dove et al., 2005; Beig and Luttge, 2006; Hellmann and Tisserand, 2006; Arvidson and Luttge, 2010]. We show that as a system with nonlinear kinetics slowly relaxes toward equilibrium, it exhibits a decrease in the apparent reaction rate constant. This provides another mechanism by which kinetics appear to vary with time, while the underlying physical and chemical processes have remained constant.

[22] The body of theory in the literature [Teng et al., 2000; Lasaga and Luttge, 2001, 2004; Vekilov, 2007], supported by visualizations of reacting surfaces [Gautier et al., 1994; Teng et al., 2001; Lasaga and Luttge, 2003; Beig and Luttge, 2006; Hellmann and Tisserand, 2006; Arvidson and Luttge, 2010], provides some understanding to the nonlinear reaction rates by analyzing the energetics and dynamics of various reactive mechanisms on the surface of dissolving and precipitating crystals. These mechanisms include 2-D nucleation and dissolution of etch pits, reactions at crystal edges, and the formation of reactive sites radiating from various types of dislocations. Although the reaction rate at a specific kink site is in general linear with concentration, the density of different reactive features and kink sites also depends upon concentration. The overall reaction is therefore nonlinear in general, which we approximate as

display math(1)

where the rate is in moles per unit time, M is the number of moles of reactive solid, inline image is the putative reaction rate constant in inverse time, and n is the order of the reaction. As written, inline image depends on the surface area per unit mass of the solid. Equation (1) is equivalent to

display math(2)

where w=Cs/Ceqφ, φ is the porosity and Cs is the concentration of dissolving species in the solid phase. The constant w<1 describes the ability of the solvent to dissolve the solid phase. Most systems of interest are more complex, with multicomponent reactions, but we assume that the reaction is limited by the relative saturation of a single species.

[23] The exact form of equation (1) that describes a particular weathering system is typically not available. If we assume the reaction to be first order, as TST would predict for a simple reaction, we define an apparent first-order rate constant

display math(3)

If transport of dissolved solute out of the system is limited, the concentration of reactant near the interface will slowly approach equilibrium. If the reaction is nonlinear (n>1), it turns out that as the relative saturation Ωapproaches 1, kap will decrease. We show this by solving for Ωin the case of n>1:

display math(4)

where m=n−1 and Ω0 is the initial value at τ=0. Applying the definition of kap yields the apparent first-order rate constant

display math(5)

Note that equation (5) is only valid for n>1(i.e., m>0), as for n=1, the assumed first-order rate constant is by definition constant in time. At long times (inline image), the first term in the parentheses in Equation (5) dominates. The initial concentration and the intrinsic rate constant inline image become irrelevant and the apparent rate constant asymptotes to 1/(wmτ). Figure 3 gives kap as a function of time (in temporal units of wτ) for where inline image and three reaction orders. As each nonlinear case approaches equilibrium, the rate constant peels away from the trivial n=1 case and assumes an inverse scaling. This case of inline image predicts the “peeling” away to occur at inline image. Larger values of w or inline image, give an earlier switch to inverse-time behavior. In summary, a system that is assumed to be of lower order than the fundamental chemistry will appear to slow as equilibrium is approached. Specifically, the assumption of first-order kinetics will lead to an artificial inverse-time scaling in the apparent rate constant.

Figure 3.

The apparent rate constant, kap as a function of residence time, τ, for closed reactions of order n=m+1=1, 2, and 3. Time is given in units of w, and the putative reaction rate constant is inline image (see text for definition). The relative saturation, Ω, is initially zero, but as systems for which n>1 approach equilibrium, the apparent rate constant decays. Note the logarithmic axes, and the 1/τ power law scaling for nonlinear reactions beginning near time inline image).

[24] The prefactor A=1/(wm) in the reaction rate relationship, kap(τ)=Aτ−1, reflects the solubility of the reactant. This is a general result for all aging mechanisms that are due to transport limitations leading to approach toward equilibrium. We expect any equilibrium-controlled aging process to yield A≪1 because the solubility is orders of magnitude less than the concentration in the solid phase. This is true even for complex systems that incorporate a wide range of residence times, because the resultant saturation will increase with the mean residence time. This is consistent with the data presented in Figure 1, where A=4×10−5.

4 Reaction-Limited Aging Processes at Surfaces 1: Surface Exchange at Discrete Reactive Sites

[25] The previous section focused on ways that transport limitations can lead to aging kinetics, or apparent variation in time. In this section and the next, we discuss aging mechanisms that are driven by changes at the fluid-solid interface. Several physical mechanisms have been suggested in the literature, including accumulation of clays and impurities [White and Brantley, 2003; Zhu et al., 2004] and changing surface area [Noiriel et al., 2009]. Other suggested mechanisms include changes in grain size and shape, mineral composition, density of reactive sites, and density of reactive surface structures such as dissolution pits and spiral dislocations [Lasaga and Luttge, 2001, 2003; Luttge, 2005]. We supplement these qualitative scenarios with two classes of quantitative models that make scaling predictions similar to those observed in the laboratory and in nature.

[26] In this section, we develop a model for dissolution and precipitation rates of carbonates and silicates derived from interpretations of deep-sea sediment isotopic data. The measurements are isotopic compositions of seawater, pore fluid, and sediment at various depths below the seafloor. Fantle and DePaolo [2006, 2007] and Maher et al. [2004] use an understanding of the isotopic fractionation of major and trace elements during precipitation and dissolution, along with assumptions about the transport of fluid through the subsurface, to infer the rates at which the isotopic composition in the fluid phase mixes with the composition of the solid phase. The rate of mixing between the phases gives the rates of precipitation and dissolution. Across studies and within studies, power law decays are consistent with calculations of the rates (see Figure 2).

[27] Although the concentrations of dissolving molecules are generally out of thermodynamic equilibrium between the solid and fluid phases, in some studies [Fantle and DePaolo, 2006, 2007], they are assumed to be near equilibrium such that the bulk precipitation and bulk dissolution rates are equal. However, the isotopic composition of the two phases are out of equilibrium at the time of burial. After burial, the solid dissolves and dissolved mineral reprecipitates. Over time, the initial fluid and solid phases thereby mix, and the isotopic compositions of the two phases approach an appropriately weighted average.

[28] The sediment studies use sets of reaction-diffusion equations for each isotope and each phase that describe dissolution and precipitation, diffusion, and advection. Such a mean-field (i.e., averaged) model implies well-mixed reservoirs of each phase and isotope. However, the solid phase is not well-mixed because dissolution and precipitation occur only at specific sites on the mineral surfaces. Therefore, the isotopic composition near reactive sites at the solid-fluid interface is accessible to the fluid phase and will quickly evolve to resemble that of the fluid phase with which it is mixing (granted the fractionation factor is zero). In contrast, the composition of the covered solid in the interior of the grains will remain constant. In this sense, the isotopic composition of the solid is poorly mixed and a mean-field, or average, description is inappropriate. Specifically, the composition is spatially heterogeneous in that the interior of grains has a different composition from the accessible exterior, and these heterogeneities are coupled to the dynamics in that the accessible regions react faster.

[29] We propose a pair of models that focus on how poor mixing of the solid phase affects the evolution of the overall isotopic composition of the solid phase. We compute the apparent first-order rate constant which mimics that calculated assuming a well-mixed model. Our models that account for heterogeneous compositions predict aging of the apparent rate constant, despite assumed constant dissolution and precipitation rates. In other words, the apparent aging can be understood as an artifact, similar to what is shown in Maher et al. [2009] and Navarre-Sitchler et al. [2011] using detailed reactive-transport modeling.

4.1 Discrete Model

[30] To avoid the mean-field pitfalls mentioned above, we propose a physical model in which the solid phase is treated explicitly as consisting of individual blocks, or subunits, of mineral (see Figure 4). Dissolution and reprecipitation dominantly occur on the solid phase at a finite set of reactive kink sites. Therefore, our model permits the blocks of mineral, each of which we take to represent a collection of molecules, to be removed and added only when exposed to the fluid phase at these kink sites and not when either buried in the bulk or covered by reprecipitated material.

Figure 4.

Schematic of our reactive site reprecipitation model. Each strip of cubes represents the material accessible via a given reactive kink site. Dissolution and precipitation to and from the well-mixed fluid occurs at the open (left) end of each strip, and describes a random walk in the location of the reactive site. The gray cubes are the α molecules that remain from the initial solid, which are replaced by β molecules from the fluid and α molecules reprecipitating back onto the solid phase. The labels at the right (m, x, ν, and j) demonstrate the indexing scheme described in the main text.

[31] To incorporate these restrictions, we arrange the blocks in stacks where the top of each stack represents a single reactive kink site. Only the topmost block is able to dissolve into solution, and each precipitation event adds one block to the top of the one stack. We designate two isotopic compositions, α and β, which we refer to as block “types.” These represent the initial isotopic ratios within the ocean water and within the sediment at the time of deposition. Each block in the stack is assigned one of these compositions. For simplicity, we assume there is no fractionation in that the dissolution rates for both types are equal and likewise for precipitation.

[32] We initially assign the concentrations of each dissolved type in the fluid phase, inline image and inline image, and the initial fraction of solid blocks of type α, γα. Over time, the blocks that were initially in the solid phase (indicated by dark gray in Figure 4) are replaced by those precipitating from the fluid phase. As blocks dissolve from the solid phase, the concentrations in the fluid gradually change from the initial values. In turn, the population of the reprecipitating blocks changes. This feedback leads to rich dynamics in the overall composition of the solid and fluid.

[33] To quantitatively model the scenario described above, we employ Monte Carlo simulations. We model the solid phase as consisting of M independent step edges, indexed by m. Each step edge consists of a stack of subunits that are added to and removed from the kink site at the top of the stack. Each step edge is described symbolically by the vector vm of length xm where inline image indicates the type of block (αor β) at location j. This finite sample of M step edges is assumed to represent a larger system. Because the step edges are independent, they communicate only through a well-mixed fluid phase which is described by concentrations of the dissolving species. There is no spatial resolution to the model, so there is no explicit diffusion. The system is further defined by setting the fraction, θ, of the solid in the system that is assumed reactive and included in the simulation (the remainder is assumed inaccessible for the duration of our simulation because it is, for example, covered by a very thick layer of bulk solid).

[34] Dissolution and precipitation occur concurrently with rate constants koff and kon, respectively. At each time step, an average of Mkoff kink sites dissolve by one block, and the corresponding step edge lengths are decreased by one. If the departing block is of composition i, then inline image is increased by 1/Vfluid, where Vfluid is the volume of solvent in the system. Also, on each time step, there are on average inline image precipitation events, for which a block of type i is added. Randomly chosen kink sites are advanced by one block of the appropriate species, and the fluid concentrations are adjusted as for dissolution. The initial lengths, xm, are large enough that step edges are not reduced to zero during the simulation.

[35] We run the simulation for one million time steps, then compute an apparent rate constant, kap, as defined by

display math(6)

where K is an equilibrium constant, inline image, and inline image is the total number of reactive and nonreactive blocks of type i. Figure 5 plots kap as a function of time as computed from the averages of roughly 10 identical runs (black curve). After a crossover time tcross, we observe the scaling kap=5×10−10t−1. The crossover time and the prefactor are determined by initial conditions and rate constants which could be tuned to represent a particular physical-chemical system. The main result from this model is the inevitable 1/t scaling in the apparent rate constant which is independent of the chosen parameters. By introducing heterogeneity in the solid phase, a system with invariant dissolution and precipitation rate constants exhibits apparent time dependence if the concentrations are measured assuming homogeneity.

Figure 5.

Effective rate constant, kap as a function of time as predicted by simulations of our discrete reactive site reprecipitation model (black) and the three-reservoir kinetic model (gray). The parameters used in both models are equivalent. Although the initial behaviors are different, at long times, they approach a common asymptotic result, kap=5×10−10t−1 (dashed line).

4.2 Spatially Averaged Model

[36] Although the explicit simulations give predictions for effective reaction rates, the model is not easily extended to include vertical diffusion, advection, or spatially resolved features. Therefore, we construct an analytic three-reservoir box model (see Figure 6). In addition to a well-mixed fluid phase, this model contains a reservoir of bulk solid phase and a reservoir of surficial precipitated solid phase. Each phase is composed of two species, α and β, representing mineral (either dissolved or solid) of two distinct isotopic compositions. The concentration of species i in each phase is inline image, inline image, and inline image, respectively. The surficial interface reservoir begins empty, but fills as molecules precipitate from the fluid. Although each reservoir is well-mixed, which is the problem we cited with the original two-reservoir models, we find that by dividing the solid phase into initial bulk solid and subsequently deposited precipitates, we add sufficient complexity to fully capture the correct dynamics of the system.

Figure 6.

Schematic of our three-reservoir model for reprecipitation. One reservoir represents remaining initial bulk solid (gray boxes) and is quantified by the total concentration of each species inline image (where stars distinguish between species α and β). Another reservoir represents precipitated solid (white boxes), with total concentrations inline image. The probability of a site being in each are Pb and Ps, respectively. Both reservoirs dissolve to the well-mixed fluid reservoir, which has concentration Cf, but precipitation only moves material from the fluid to surface precipitate reservoir.

[37] Over time, the accessibility of the initial bulk solid to the fluid decreases as kink sites are covered by precipitates. We describe the accessibility of the bulk solid using the physical picture of kink sites and stacks. Since we assume the system is at chemical equilibrium, particles are added and removed at the same rate. Therefore, the length of a given stack follows an unbiased 1-D random walk. The site exposes pristine bulk solid (i.e., that which is not a precipitate) only if the step edge is currently dissolved to its greatest extent since initiation. We find the number of such sites by computing the fraction of random walks at time t, Pb(t), that are at their maximum excursion toward the origin since t=0, using both Monte Carlo simulation and the theory of first passage processes (see Appendix A) [Redner, 2001]. We find that for Z>7,

display math(7)

where Z=ρkont is the number of dissolution and precipitation events and ρ is the density of kink sites.

[38] We use this result to write the following system of differential equations that describe the evolution of the concentrations:

display math(8)

where the fraction of a given species in a particular phase is inline image, and the fraction (1−φ)/φ is the ratio of solid to fluid volume in the system. Initially, inline image and inline image, where the ratio of inline image to inline image defines the initial isotopic composition of the solid. The total fluid concentration is inline image, where K is the equilibrium constant, in units where the concentration in the solid phase is unity. We numerically integrate the differential equations in time and calculate the effective first-order rate constant, kap, as defined in equation (6), where Cs=(1−Pb)Cs+PbCb.

[39] Figure 5 gives kap as a function of time (gray curve). The solid is initially composed of species β and the fluid of species α. We observe a crossover time where the fluid approaches isotopic equilibrium with the reprecipitated surface reservoir. After the crossover, we observe a universal kap∝1/t decay for all parameter sets.

[40] The scaling behavior is best understood via the αform of equation (6). Because the solid is dominantly composed of species β, the denominator is dominated by the term inline image,

display math

Over time, dissolved mineral of composition αmixes with the surface layer of the solid. Because the solid phase is much more dense than the fluid phase, the majority of species α will soon be stored within this surficial layer. The thickness of the surficial layer follows a random walk, so over time, its volume scales as t1/2. The concentration of α remaining in the fluid therefore scales as inline image. Because the system is closed,

display math

and therefore, inline image. The apparent rate constant is then kapt−1.

[41] Can this model explain data in deep-sea sediment studies such as Fantle and Depaolo [2007] and Maher et al. [2004]? We have suggested a model, or scenario, in which the time variations calculated in those studies do not reflect physical changes in exchange rate. Rather, they reflect a gradual increase of compositional heterogeneity at the grain scale. In the isotopic composition measurements, the heterogeneities are averaged over, leading to an artificial variability in the apparent rate constant. Specifically, the published rates use solid compositions that include both reprecipitated and initial solid. The solid phase should therefore not be characterized by a single concentration in a reaction-diffusion model. The telltale signature of this mechanism would be heterogeneities in the grains themselves.

[42] Both the analytic model and simulations show that the crossover to kap∝1/t occurs only when the fluid composition is very similar to that of the surficial solid. In general, this requires a closed system whereas the marine burial systems are open to the well-mixed ocean. Nevertheless, at the “diffusive burial depth,” the depth at which the mean time to diffuse to the surface is larger than the burial age, the pore fluid becomes effectively cut off from the ocean reservoir. We expect the inverse-time aging to commence below this depth. Specifically, this depth is given by the ratio D/v where D is the diffusivity in the pore space and v is the sedimentation rate.

5 Reaction-Limited Aging Processes at Surfaces 2: Static and Dynamic Disorder in Reaction Rates

[43] The next class of aging mechanisms appeals to the disorder, or heterogeneity, inherent in any weathering system. These models of disordered kinetics can be applied to various scenarios such as evolving mineral composition or accumulation of impurities. After describing the mathematics, we present a model of shrinking grains and apply it to data reported in Tang et al. [2001].

[44] Models of disordered kinetics involve ensembles of independently reacting elements, which may be definable units of material or unique classes of material. The elements react with rates randomly chosen from a distribution [Plonka, 1986]. In our models, the various component elements react in parallel such that the overall rate is determined by the distribution of rates among the elements. In the “static disorder” case, the rate associated with each element of the ensemble is constant with time. Although each individual rate is constant, the overall rate gradually decreases because the fastest reacting elements equilibrate and cease first, leaving only slower reacting elements. Therefore, the overall behavior is completely determined by the initial distribution. The mathematics of the “random rate model” and the equivalent “random channel model” have previously been explored [Huber, 1985; Vlad et al., 1997; Xie, 2002; Vlad et al., 2005] and have been employed successfully to explain various relaxation processes such as fluorescence [Huber, 1985; Blumen, 1981] and the decay of marine organic carbon [Rothman and Forney, 2007; Boudreau and Ruddick, 1991].

5.1 Theory of the Random Rate Model

[45] In the static disorder, or random rate model, we write the reaction rate R as an integral over all rates krepresented in the ensemble,

display math(9)

where the weight g(k,t) is the total amount of material at time t reacting at rate k, and k has dimensions of inverse time. As mentioned above, the local reaction somehow leads to relaxation by which the reacted elements are removed from the ensemble via first-order kinetics. Therefore, g(k,t)=M0P(k)ekt, where P(k) is the initial density distribution of rates in the material, and M0 is the initial total mass. We find that the total material decreases according to inline image.

[46] We now define the effective first-order rate constant as

display math(10)

Note that if M(t)∼tα, then kapαt−1. It can be shown that this is indeed the case for a wide range of rate distributions. If P(k) can be expressed as a power series for small k, then the total reactive material left as t is M(t)∼a0Γ(α)tα[Rothman, 2008]. Therefore, after a crossover time, the effective rate constant scales as kapαt−1. The time of this crossover depends on the details of the system. If the crossover is early enough in our particular system of interest, then we will observe that kap scales as 1/t at long times. However, the crossover could be so late as to be unobservable, in which case, the inevitable, but unobserved, 1/tbehavior is irrelevant.

[47] In summary, we have briefly outlined the theory for systems exhibiting an ensemble of simultaneous fixed, but random, reaction rates. As elements react at their respective rates, they are removed from the system. For such models, there exists a large class of initial rate distributions that predict 1/t decay of the effective rate constant at long times.

5.2 A Case Study in Dynamic Disorder

[48] We apply this theory to a dissolving and precipitating porous material by noting sources of heterogeneities that could give rise to an ensemble of reaction rates. At the microscopic scale, different types of reactive features, such as crystal edges and etch pits, react at different rates [Arvidson et al., 2003]. At slightly larger scales, grains vary in shape, size, and mineral composition. We expect high curvature regions to have more reactive step edges, leading to higher reaction rates. Furthermore, mineral surfaces may have different histories and may have been exposed to different saturation conditions, leading to varying densities of reactive kink sites [Arvidson and Luttge, 2010]. Laboratory experiments have shown that dissolution rates of some silicate and phosphate crystallites depend on the size of the grains [Tang et al., 2001, 2003, 2004; Wang et al., 2005; Nancollas et al., 2007]. Regardless of the scale at which the heterogeneities occur, kinetic disorder models characterize the entire system by one ensemble. In this way, they are a means to bridge length scales in the system. Although microscopic features dissolve at various rates, together, they form a disordered ensemble at the macroscopic length scale.

[49] An ensemble of reaction rates, by itself, does not give rise to a changing total rate. We require a mechanism by which fast reacting elements in the ensemble are removed preferentially. In the case of grains of different composition, for example, this mechanism is clear; the fastest reacting minerals dissolve first, leaving the slowest reacting minerals. We also suggest other more subtle means. The faster reacting surfaces will accumulate impurities and secondary precipitates, thus stalling their continued reaction. Over time, active kink sites vanish by virtue of the finite size of the grain. In addition, as crystals shrink the density and distribution of dislocation sources on the surface of the crystal can diminish, heavily influencing the reaction rates [Vekilov and Kuznetsov, 1992].

[50] We now provide an example of how disordered kinetics can be employed to model a particular set of experiments. We will employ the model in an inverse fashion to support a suggested aging mechanism, which in this case is grain-size-dependent dissolution. Nancollas and coworkers report that the dissolution rates of various biominerals vary with the size of the grains and crystal faces. Performed under constant composition solute conditions, dissolution experiments with tooth enamel [Tang et al., 2004; Wang et al., 2005], brushite [Tang et al., 2003], β–tricalcium phosphate (β-TCP) [Tang et al., 2001], and hydroxyl-apatite crystals [Tang et al., 2004] have all demonstrated a size dependence on the dissolution-rate constant. By comparing the rate constants of variously sized seed crystals, and by performing prolonged dissolution measurements and comparing the crystal faces before and after dissolution, they have concluded that as the grains decrease in size due to dissolution, the rates decrease.

[51] The authors of those studies suggest that the slowdown can be understood in two ways. They observe that dissolution on the crystal faces is dominated by dissolution stepwaves [Teng et al., 1998; Lasaga and Luttge, 2001; Tang et al., 2003] emanating from dissolution pits. Larger pits were measured to grow faster than smaller pits, which is consistent with theory [Teng et al., 1998; Tang et al., 2004]. Small crystal faces lack large dissolution pits and long step edges, and therefore, the overall dissolution rate is slower. Second, Tang et al. [2004] suggest that small crystallites lack faces large enough to support any stable dissolution pits. In analogy to critical size for crystal nucleation, there is a minimal critical size for dissolution pits to stabilize at a given undersaturation. Proto-pits at dislocation sites fluctuate in size until either closing or spontaneously reaching the critical size; at which point, they expand and effectively dissolve material. Indeed, Tang et al. [2004] observe a saturation-dependent critical grain size, rc, below which crystallites do not dissolve.

[52] We model their experiments using a dynamic disorder model in which the rates that compose the ensemble are permitted to vary according to a prescribed rule. Each mineral face, or feature, in the ensemble dissolves at a rate R(r0,t) that depends on both time and the initial feature size, r0. The total dissolution rate is a sum over the contributions from all feature sizes,

display math(11)

where P(r0) is the probability density of initial feature sizes r0 and M is the total number of features in the sample. R(r0,Ω,t)=ψr(r0,t)2k(r0,t) is the dissolution rate of each feature, where ψ is a unitless shape factor relating surface area to feature size, and k is the size-dependent rate constant. The variables M and φ are not measurable, but they vanish in the total rate ktot, which is normalized by total surface area. Below, we use physical arguments and results from experiments to define rules for how r(r0,t) and k(r0,t) depend on Ω, r0, and t. We then solve the corresponding inverse problem: given observed values of Rtot, find the most likely distribution of feature sizes P(r0).

[53] Tang et al. [2004] cite the dependence of dissolution rate on etch pit size as possibly responsible for the similar relationship at the scale of whole grains. Although this extension is not rigorously defended theoretically, it is consistent with their data [Tang et al., 2003]. Similarly, they suggest that the critical grain size has a similar dependence on saturation state as critical nuclei for etch pit formation. Indeed, the data are again consistent with that assumption, and we adopt those two relationships.

[54] The details of our model, including rules for the evolution of feature sizes and solution methods, are given in Appendix B. Using measured dissolution rates for β-TCP given in Tang et al. [2001], we find the distribution P(r0) that best fits equation (11). Figure 7a gives that solution to P(r0) and Figure 7b gives the corresponding ktot with the experimentally measured values.

Figure 7.

The solution to our dynamic disorder model that best predicts the observed β-TCP dissolution rates in Tang et al. [2001]. Given are (a) the probability density function of initial grain sizes, N(r0), and (b) dissolution rates from both our model (lines) and experimental data (shapes). The rate is given as a function of time for the four activities, Ω, used in the constant composition experiments.

[55] In the theory above, we show that disordered kinetics provide a quantitative and mathematical framework to describe systems with heterogeneity. Such models can quantify systems where heterogeneities in the material are coupled to kinetics, such as through mineral composition or abundance of reactive surfaces. As an example, we hypothesize that the observed time dependence in the grain dissolution data can be captured by a disordered kinetics framework. Consistency of the data with the model provides an initial test of this hypothesis. Figure 8 gives the rates that best fit the experimental data using a simple static disorder model (the details for which are given in Appendix C); the fit is poor, and we conclude that static disorder is not sufficient to adequately describe these data. On the other hand, Figure 7 demonstrates that the dynamic disorder model described above is able to reproduce the observed data adequately with a reasonable P(r0).

Figure 8.

Dissolution rates of hydroxyapatite from experimental data (shapes) and a simple static-disorder model (lines). The poor fit relative to Figure 7 suggests that a model of static disorder fails. The identity of lines and shapes is given in Figure 7.

[56] Disordered kinetics provide a plausible explanation for the observed time variability. We still do not know for certain what the exact underlying physical mechanism or chemistry is that gives rise to the disordered kinetics, but we have been able to rule out mechanisms associated with static-disorder and have proposed an alternative mechanism. We believe that this adaptability and inclusiveness of this modeling strategy lends its strength and utility.

6 Concluding Remarks

[57] In this paper, we have reviewed several examples of time dependence in dissolution and precipitation rates. Such aging has been observed in both natural and laboratory settings and occurs over many orders of magnitude in time. We discuss the variety of physical processes that have been promoted as explanations, and just as the observations span many time and length scales, so do the suggested aging processes. Laboratory experiments and field observations at different length scales, environmental circumstances, and materials likely manifest different aging processes. So identifying the mechanism that causes aging in a given scenario appears to be a daunting task, as the underlying geophysics and chemistry varies from situation to situation.

[58] However, we have argued that the different mechanisms are classifiable and distinguishable based on measurable quantitative behaviors. These include power law relationships, the prefactor of those scaling laws, the timing of the onset of that scaling, and the choice of time variable that yields that scaling. For example, identifying residence time, rather than age, as the relevant independent variable indicates transport processes are causing the time variability. Identifying candidate mechanisms that may be causing aging in a given set of observations helps inform further experiments to distinguish between those physical mechanisms. Such an understanding is critical to making long-term predictions for similar weathering systems.

[59] Furthermore, the diversity of physical processes can be classified by general mathematical models, such as disordered kinetics or nonlinear kinetics. Classification allows us to focus on the similarities between aging mechanisms in terms of quantitative predictions, so that the fine details of the physical process are not required to make long-term predictions. We demonstrate how several of these quantitative models, such as disordered and nonlinear kinetics, predict inverse aging in the rates of dissolution and precipitation. Each of these models, in turn, represent a variety of physical processes that may explain the observed behavior. This flexibility is a strength that can be exploited to gain a deeper understanding of both specific aging systems and time dependence in general.

Appendix A: Finding Pb(t)

[60] In the three-reservoir kinetic reprecipitation model, the rates at which material dissolves from the bulk solid phase and the surficial precipitated phase depend on the parameter Pb(t). In the discrete reprecipitation model, Pb(t) is identified as the probability that the terminal exposed block of a stack is pristine bulk solid (as opposed to precipitated solid). As material is added to and removed from the stack, the length x(t) performs a 1-D random walk. If we define dissolution as positive advance along the x axis, then we define Pb(t):

display math(A1)

[61] After identifying the random walk as a diffusive process, we find the time distribution, Q(x0,t), for a diffusing particle to collide with an absorbing wall at x0, which is equivalent to the time distribution for a stack to first reach a length of x0. We convert the continuous diffusion problem to a discrete random walk and sum over all values of x0 to find the total probability that the stack length is greater than at any previous time. We then relax this strict inequality to include cases of x(t)=x(t), and the result agrees with random walk simulations.

[62] The probability density of a freely diffusing particle on the x axis, released at x=0, develops according to C(x,t)=(4πDt)−1/2exp(−x2/4Dt), where D is defined such that the variance of the distribution is 〈x2〉=2Dt. We identify the diffusion constant corresponding to a random walk of Z steps of length δ and time τ as D=δ2/2τ. Introducing an absorbing wall at x0, such that C(x0,t)=0, the probability density for a diffusing particle can be found using the method of images [Redner, 2001]. The result is

display math(A2)

The distribution of first passage times to x0 is given by the diffusive flux across this wall. The result is the inverse Gaussian distribution:

display math(A3)

The total rate, M(t), at which walkers step into “new” territory that they have not yet traveled, is the sum of Q(x0,t) over all steps where x0>0. This is given by

display math(A4)

To find the probability of entering new territory during a step, P>(t), we integrate K(t) over the duration of the step. For tτ, this integration is equivalent to multiplication by τ. Noting that the number of steps is Z=t/τ, we have the probability,

display math(A5)

[63] To relax the strict inequality and include walks where x(t)=max(x(t)t<t), we make several observations regarding discrete walks. Let N0(t) be the number of walkers in an ensemble that are at their maximum excursion, x0. Also, let N−1(t) be the number that are a distance one less than the maximum excursion, x=x0−1. Note that at the next step, N0/2 walkers move forward past their maximum distance, while N0/2 move back to less than the maximum distance. Simultaneously, N−1/2 walkers advance to the maximum excursion. The total number of walkers that satisfy x(t)≥xmax is N0+N−1. At long times, N0N−1. Therefore, relaxing the strict inequality introduces a factor of 2, giving a result consistent with numerical simulations,

display math(A6)

Appendix B: Details of Dynamic Shrinking Grains Model

[64] Having established equation (11) for the total reaction rate of the ensemble of features, we define the size dependence of the dissolution rate of a single feature,

display math(B1)

where k is the dissolution-rate constant for an infinitely large etch pit. The form of this equation is inspired by the theoretical relationship between etch pit dissolution rate and size [Teng et al., 2000; Lasaga and Luttge, 2001, 2003] and is supported by data in Tang et al. [2001]. We assume that far from equilibrium, the rate is linear in Ω, albeit with a nonzero Ω-intercept. So we let k(Ω)=k00−Ω), where both k0 and Ω0 are undetermined.

[65] We assume that the critical grain size depends on saturation state similarly to critical nuclei in etch pit formation. Therefore, we adopt the relationship

display math(B2)

where the constant rc0 is undetermined. This functional form is supported by data in Tang et al. [2003, 2004].

[66] Finally, we form a model for feature size evolution. Assuming that faces of one size tend to be adjacent to dissolving faces of similar size, the rate at which a face shrinks is roughly proportional to the dissolution rate of that face. We adopt the expression that describes the evolution of faces on a homogeneously dissolving cubic crystal, dr/dt=k/(r2ρmat), where ρmat is the density of the material. Therefore,

display math(B3)

[67] For sizes rrc, the dissolution rate for each feature is R(r,t)=kψr(t)2. We adopt a switch approximation, supported by numerical solutions to equation (B3), in which the grains shrink linearly with time until r=rc at which time the dissolution rate vanishes. Substituting for r, we write,

display math(B4)

where inline image.

[68] We now express the normalized dissolution rate as a function of the known parameters t and Ω, and the unknown parameters, P(r0), k0, Ω0, and rc0. We evaluate the model by finding parameters that best predict the tricalcium phosphate dissolution data given in Tang et al. [2001]. The density of the mineral is ρmat=1.01×10−14mol/μm3, and all critical sizes are restricted to rc<1.5μm, a rough upper bound on the initial diameter of the crystallites. We find best fit parameters in the least squares sense by minimizing the error function,

display math(B5)

where i indexes the measurements and k(ti) are the experimentally measured rates. We apply regularization via the second term, which assures a sufficiently smooth distribution by assigning a roughness penalty, w, and a roughness functional, ΔP(r0) [Lawson and Hanson, 1974; Hansen, 1992].

Appendix C: Comparisons of Static and Dynamic Models

[69] We now compare the above dynamic shrinking grains model to a static-disorder model, in which the dissolution rate of each piece of material is constant. There exists a distribution of rate constants k, with units of mass per area per time. Material is removed from the ensemble at rate Assk, where Ass is the specific surface area. The total rate is the integral over all rate constants and is given by,

display math(C1)

where M is the initial mass of the system and P(k) is the initial distribution of rate constants. Although in general Ass may be vary with k, we assume it is constant and equal to the bulk value. Applying the same Ωdependence to k as in the dynamic model, the normalized rate constant is

display math(C2)

where the integral is now over the saturation independent variable k0, and we use a geometric assumption relating mass to surface area. The optimal solution for P(k0) is found as before, and the resulting rates are given in Figure 8.

Acknowledgments

[70] This material is based upon work supported as part of the Center for Nanoscale Control of Geologic CO2, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under award: DE-AC02-05CH11231, subcontract 6896518.