Upscaling Calcite Growth Rates from the Mesoscale to the Macroscale

May 29, 2013 - Upscaling Calcite Growth Rates from the Mesoscale to the. Macroscale. Jacquelyn N. Bracco,. †,§. Andrew G. Stack,*. ,† and Carl I...
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Upscaling Calcite Growth Rates from the Mesoscale to the Macroscale Jacquelyn N. Bracco,†,§ Andrew G. Stack,*,† and Carl I. Steefel‡ †

Chemical Sciences Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6110, United States Geochemistry Department, Lawrence Berkeley National Laboratory, Berkeley, California 94720, United States

‡

S Supporting Information *

ABSTRACT: Quantitative prediction of mineral reaction rates in the subsurface remains a daunting task partly because a key parameter for macroscopic models, the reactive site density, is poorly constrained. Here we report atomic force microscopy (AFM) measurements on the {101̅4} calcite surface of monomolecular step densities, treated as equivalent to the reactive site density, as a function of aqueous calciumto-carbonate ratio and saturation index. Data for the obtuse step orientation are combined with existing step velocity measurements to generate a model that predicts overall macroscopic calcite growth rates. The model is quantitatively consistent with several published macroscopic rates under a range of alkaline solution conditions, particularly for two of the most comprehensive data sets, without the need for additional fit parameters. The model reproduces peak growth rates, and its functional form is simple enough to be incorporated into reactive transport or other macroscopic models designed for predictions in porous media. However, it currently cannot model equilibrium or pH effects and it may overestimate rates at high aqueous calcium-to-carbonate ratios. The discrepancies in rates at high calcium-to-carbonate ratios may be due to differences in pretreatment, such as exposing the seed material to SI ≥ 1.0 to generate/develop growth hillocks, or other factors.

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have found it necessary to utilize additional fit parameters to capture changing solution conditions6 or surface roughness.10 Ideally a model would be able to predict growth rates without additional fit parameters since this would enhance its transferability across systems and solution compositions. Here we use flow-through atomic force microscopy (AFM) to measure the density of monomolecular steps on the calcite (CaCO3) {101̅4} surface. These measurements are coupled to our previous studies on the rate of advance of those steps to build a model that predicts macroscopic calcite growth rates across a range of aqueous calcium-to-carbonate ratios and saturation indices without the need for additional fit parameters. Calcite was chosen because of its ubiquity in natural systems, the abundance of rate data with which to make comparisons, and its potential role to mitigate numerous environmental problems such as in carbon sequestration, acid mitigation, and toxic metal sequestration.

INTRODUCTION Our ability to predict quantitatively the extent, locale, and rate of mineral reactions in natural environments is limited, as evidenced by the comparatively few geochemical models being used in practical applications (e.g., Records of Decision for Superfund sites only occasionally contain the phrases “geochemical model” or “reactive transport model”).1,2 Specific examples for which an enhanced predictive ability would be useful include the growth of carbonate minerals that determine in part the long-term success of geologic carbon sequestration3,4 and engineered precipitation of carbonate and sulfate minerals, a proposed method to incorporate and thereby sequester toxic metals such as strontium5,6 and radium.7 One reason for our limited predictive ability is that the historically accepted “affinity-based” approaches, where mineral reaction rate is measured relative to the saturation state of the system, are unable to describe reaction rates when the ratio of the aqueous form of the mineral’s constituent ions deviates from that to which the model was calibrated.5,8 An alternative approach are the “process-based” models that are built from reactions thought to occur on the mineral surface itself.9−11 However, the success of process-based models relies on a realistic depiction of both the reaction mechanisms that control growth and dissolution and accurate measurements of the density of the reactive sites on the mineral surface. Thus far models made using a priori assumptions about these processes © XXXX American Chemical Society

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MATERIALS AND METHODS Experimental Methods. Step velocities and the widths of the terraces between steps (i.e., the terrace widths) on the Received: February 13, 2013 Revised: May 23, 2013 Accepted: May 29, 2013

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{1014̅ } calcite surface were measured in situ using an Agilent PicoPlus AFM with a custom designed flow-through fluid cell containing a freshly cleaved crystal of calcite, using procedures previously developed.6,8 Iceland spar crystals were cleaved and blown with compressed nitrogen. Samples were mounted into the fluid cell with the obtuse step orientation facing the solution inlet jets, since in our previous work we determined that the obtuse step velocity was more sensitive to flow direction than the acute.8 Obtuse and acute steps were aligned parallel to the “slow” scan direction. Fluid was injected into the cell using a push−pull syringe pump with a solution flow rate of 120 mL/h, a rate at which step velocities have been shown to be independent of solution flow rate.6 Step velocities (previously reported)6,8 and terrace widths were measured on growth hillocks which were found on the calcite surface after a pretreatment stage using a high saturation index solution (SI = 1.0) lasting 15−30 min prior to the start of the experiment. Since our previous work showed that morphology has a tendency to become irreparably modified at extreme aqueous calcium-to-carbonate ratios,8 each experiment was initiated after pretreatment under a solution with a calcium-to-carbonate close to 1:1. After approximately 40 min exposure to a particular solution, composition was changed to solutions containing successively larger or smaller aqueous calcium-tocarbonate ratios. Saturation indices (SI = log(aCaaCO3/Ksp); Ksp = 10−8.48) were kept fixed at approximately 0.40 and 0.75, but with variable aqueous calcium-to-carbonate ratio. This reduced the amount of time it took to reach a steady state and limited morphological hysteresis (see discussion of Figure 4). The stock solutions (0.1 M CaCl2 and 0.1 M NaHCO3) used to create growth solutions were equilibrated with atmospheric CO2 for at least two weeks or sparged with air overnight to ensure complete equilibration with atmospheric CO2. Chemical compositions of the solutions were calculated using PHREEQC and the pH of the solutions were measured to confirm solution compositions. The methodology for the PHREEQC calculations was designed to mimic the experimental design as closely as possible: The 0.1 M CaCl2 and NaHCO3 stock solutions and water were first equilibrated with atmosphere (in silico). The equilibrated form of each solution was then mixed in a proportionality that matched the liters of each solution to be mixed together at the start of the experiment. For example, if 100 mL of growth solution was needed for SI = 0.75 and [Ca]/ [CO3] = 4, 0.1 M CaCl2, 0.1 M NaHCO3, and H2O, 0.000255, 0.002885, 0.09686 L were added for each, respectively. An example input deck is shown in the Supporting Information (Table S1), along with specific solution compositions, including pH and ionic strength, measured obtuse and acute terrace widths, and overall growth rates (Supporting Information, Table S2). In practice, the much larger water volumes were measured by topping-off a 100 mL volumetric flask after adding growth solutions to ∼50 mL H2O. Terrace widths on calcite under each solution composition were measured by determining the average distance between three different steps in a single image for four to six sequential images (two up scans and two down scans). Step densities were calculated by taking the reciprocal of measured terrace width. Due to the relatively fast rate of advance of the steps, the identity of the individual steps measured in each image were different. Unless otherwise noted, the means and standard deviations for both terrace widths and step velocities are for the variations from image-to-image within a single experiment.

Theory for Net Growth Rate from AFM Measurements. Two measurements must be made in order to obtain a net rate of growth of a mineral surface in units of mols precipitated per unit surface area per unit time from AFM: step velocities and step densities. When step nucleation rate is slow enough to be measured reliably, it can be used to replace these.12 To use the traditional geochemical parlance, we treat the step velocity as the intrinsic reaction rate of the active sites and the step densities as comprising the reactive site density. The expression for step velocity used is one developed previously6,8 where it is assumed to be limited by the rate of kink site nucleation13 by either calcium or carbonate ions: v = aR kn

⎛ ⎞ k Ca[Ca 2 +]k CO3[CO32 −] ⎜ ⎟ = a⎜ − k V kn m − ⎟ 2+ 2− ⎝ Vm(k Ca[Ca ] + k CO3[CO3 ]) ⎠ (1)

where v is the step velocity, a is the row width (0.31 nm for calcite), Rkn is the rate of kink site nucleation, kCa is the apparent rate constant for attachment of a calcium ion to a kink site, kCO3 is the apparent rate constant for carbonate attachment to a kink site, Vm is the molar volume of calcite (32.05 mols/L; used to convert concentration units), k‑kn is a pseudo zerothorder constant for the detachment or back reactions, and [Ca2+] and [CO32‑] are the concentrations of aqueous calcium and carbonate ions. The rate constants in eq 1 can be expanded to reflect attachments that proceed through multiple reactions.14 In fitting this model to saturation states greater than 0.4, the best fit is where detachment reactions are zero.6 Ignoring detachments has the obvious drawback of losing the ability to model equilibrium, but at this time the solution composition where the back reaction becomes important is not known, nor are the details and functional form for convolution of back and forward reactions. For dissolution of calcite steps, the back reaction is only important at SI ≥ −0.7,15 but this observation cannot be extrapolated to growth and dissolution of an initially nucleated kink, since the back reaction for dissolution and the forward reaction for growth are not the same (i.e., the back reaction for growth is detachment from a positive kink whereas the back reaction for dissolution is attachment to a negative kink),16,17 and there are multiple reactions during the attachment/detachment process, whose rate limiting steps are different.14 We speculate that the back reactions for growth are only important when the rate of kink site propagation is slow enough such that ions that form the nucleated kinks have a significant probability of detaching prior to being stabilized by an ion adsorbing next to them so as to continue propagating the kink. Compared to step velocities, there is much less pre-existing data and theory for step densities. In the classical crystal growth theory, the spiral growth hillock with the largest defect that generates the steps (i.e., the Burgers vector) will dominate the growth face of a mineral grain.18 Consistent with this, we observe here that a data set where steps were nucleated from a hillock based on a line defect had a higher step density than other data sets that were generated from point defects. The measured step densities were within the uncertainty of the other measurements, however, so the step densities of all the experiments were weighted equally in our analysis, regardless of defect type that generated the steps. Additionally, flow direction and turbulence have been found to strongly affect step bunching and organization for materials whose kinetics are affected by transport,19,20 with a general trend that uniformity B

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of step densities increases with both flow rate and turbulence. These parameters have been observed to produce fluctuations in step densities of up to 80−100% of the average value and a relaxation time of ∼10 min depending on solution composition and flow conditions.21,22 These fluctuations greatly increase the uncertainty in step density estimates, both since a large number of measurements are required as well as a large amount of time to reach steady state. A fundamental theory, written in terms of individual attachment and detachment of ions or other molecular species to defects, that describes the dependence of both step nucleation rate or step densities, is lacking at this time. Thus, here we have opted to write an empirical description of step densities as a function of saturation state of the system and aqueous cation-to-anion concentration ratio: ρ = mSI + n log(r ) + b

where σavg is the composite standard deviation of the average measurements, Σi is the number of times an experiment was performed for the same solution composition, σi is the standard deviation of the net growth rate from each individual experiment (calculated in eq 4), and μi and μj are the mean net growth rates from the ith and jth experiments of the same solution composition. Fit parameters for the model and their uncertainty were determined using an Levenberg−Marquardt algorithm to minimize the χ2; specifically the automated Curve Fitting routine in the Igor Pro software package was used (Wavemetrics, Lake Oswego, OR). Ninety-five percent confidence intervals for the net growth rate model predictions were estimated stochastically by assigning a normal distribution to each best fit parameter whose uncertainty was assumed to be the standard deviation for the distribution. A Mersenne random number generator24 was used to select random variables weighted to the distribution for each parameter in the model, and a net rate calculated using eqs 1−3. This was repeated one million times for each solution composition. Mean rates from this method were within 2% of those calculated analytically. The mean of these ± two standard deviations was taken as the 95% confidence interval.

(2)

where ρ is the step density (steps/μm), m is the slope with respect to the saturation index (SI), n is the slope with respect to the log of the aqueous [Ca2+]/[CO32‑] ratio (r) and b is the intercept. Although in this work we only report step densities for two saturation indices, the functional form in eq 2 is justified since the slope with respect to saturation index has been demonstrated to be linear at saturation indices greater than ∼0.2.23 The linear slope with respect to the log of the calcium-to-carbonate ratio is justified a posteriori below. Once the step velocities and step densities are obtained, the net growth rate is23 R surface = ρvshVm

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RESULTS AND DISCUSSION Step Densities, Step Velocities, and Net Growth Rate. As shown in Figure 1, measured step densities over the course of an experiment were highly variable and strongly affected by both solution composition and stoppages in flow. Under solutions with the low aqueous calcium-to-carbonate ratios, the

(3)

where Rsurface is the growth rate of the mineral per unit surface area per unit time (mols/m2/s), vs is the step velocity (m/s), ρ is the step density (steps/m), Vm is the molar volume (32.05 mol of calcite/L of step volume), and h is the step height (0.31 nm). The growth rate resulting from this expression is directly comparable to macroscopic measurements, e.g., from a batch reactor. By using this expression, we assume that the rates of growth of the obtuse side of the spiral growth hillocks are representative of all growing crystals regardless of saturation state and pretreatment of the crystal, but this has yet to be tested rigorously. We additionally assume that the {101̅4} surface is representative of all calcite surfaces present in a reactor; its validity is discussed below. Standard deviations for net growth rate measurements calculated in eq 3 were estimated by error propagation methods: σnet ≈ μnet

⎛ σ ⎞2 ⎛ σ ⎞2 σ σ ⎜⎜ vel ⎟⎟ + ⎜⎜ den ⎟⎟ + 2 vel den ρμvel μden μvel μden ⎝ μvel ⎠ ⎝ μden ⎠

(4)

where σnet and μnet are the standard deviation and mean of the net growth rates, respectively, σvel and μvel are the standard deviation and mean of the step velocities, respectively, σden and μden are the standard deviation and mean of the step densities, respectively, and ρμvelμden is the correlation coefficient between mean step velocity and density. The standard deviation of the average net growth rate for a given solution composition was calculated by treating multiple experiments as separate populations: σavg ≈

∑i < j (μi − μj )2 ∑ σi2 + 2 ∑i ∑i

Figure 1. Terrace widths of obtuse step orientations measured during the course of two different AFM experiments. The black solid lines indicate points at which the solution flow was halted to switch to a new solution. Gaps in the data are where images were not being recorded. Often during experiments, brief interruptions in flow, such as when a solution was changed, resulted in spikes in the terrace width. (a) Experiment where solutions were ramped to successively higher calcium-to-carbonate ratios (r). (b) Experiment where solutions were ramped to lower ratios. At low calcium-to-carbonate ratios, the surface does not reach a steady state in the ∼40 min exposure to a solution.

(5) C

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time for the step density to reach a steady state far exceeded the time in which step velocities reached steady state (∼20 min), and it was seemingly independent of step velocity.6 This finding is surprising and suggests the intriguing proposition that a mineral’s morphology (and hence its reactive site density) may not just reflect the current solution conditions to which it is exposed but may contain a history-dependent term based on the previous solutions/flow characteristics the mineral has experienced (some have discussed this type of effect as an inherent unpredictability25). In order to determine the most appropriate step density to use for each experiment, step densities were measured for every image taken over the course of an experiment (Figure 1; Supporting Information, Figure S1) and a representative step density was calculated by taking the average step density of steps where there were no obvious disruptions or systematic changes in step density occurring at that time (discussed above). Under conditions where the system was not at a steady state (e.g., Figure 1b, low calcium-tocarbonate ratios), best estimates were used, but measured step densities are potentially shifted from their steady-state values. Due to the inherent fluctuations in step densities described above as well as other unknown factors, selected data points were necessarily treated as outliers and were not included in the fitting routine for step densities. Step densities where step velocity was zero were also disregarded. Disregarded data points and outliers are noted in the Supporting Information (Table S2). The fluctuations also create a larger amount of scatter in the data than for step velocities, which are wellbehaved with respect to flow velocity and variation due to surface topography.6 Step densities were found to increase with both increasing calcium-to-carbonate ratio and increasing saturation index (Figure 2a; data in the Supporting Information, Table S2), but the functional form is significantly different from that of the step velocities (Figure 2b). As described above, step densities (Figure 2a) were fit to a linear regression as a function of the base-10 logarithm of the calcium-to-carbonate ratio for each saturation index (eq 2). Since there are a limited number of data points at SI = 0.40, the slope of the linear regression with respect to the log of the calcium-to-carbonate ratio for both SI = 0.4 and SI = 0.75 was assumed to be constant for both the SI = 0.4 and 0.75 data sets. Fit parameters and their uncertainties are shown in Table 1. We can speculate that calcium attachment is rate limiting for nucleation of the new steps since step densities are higher at high calcium-to-carbonate ratios and since the step density does not plateau at any of the solution conditions measured here. Presumably at some sufficiently high calcium-to-carbonate ratio, calcium attachment will cease to limit the rate of step nucleation, but this condition is not reached convincingly in these experiments. Measured step densities and the empirical fit (Figure 2a) were combined with the step velocities in the kink site nucleation model reported previously (data in Figure 2b; parameters in Table 1) to derive a model (eq 3) for the overall growth rate as a function of calcium-to-carbonate ratio (Figure 2c). The net growth rate for each measurement of step velocities and densities under a given solution condition was calculated as well. Overall, the model fits the AFM data quite well but it does not capture the cessation of growth at extreme ratios well. We believe this is attributable to the assumption in the step velocity model that kink site nucleation is always limiting the rate of growth. Under extreme ratios, this assumption may not be valid since the rate of kink propagation

Figure 2. Crystal growth data (points) and model (lines). (a) Step densities measured on obtuse calcite steps as a function of calcium-tocarbonate ratio. Step density increases linearly with increasing log of the calcium-to-carbonate ratio for both saturation indices. The step density model is a linear regression for SI = 0.4 and 0.75 (eq 2), assuming the same slope with respect to the log of the calcium-tocarbonate ratio. The SI = 0.97 model is a prediction (see Figure 3b). (b) Step velocities with corresponding model fits/predictions (eq 1).6,8 (c) Net growth rates. Curves are the model response after being calibrated to step densities and velocities in a and b.

Table 1. Summary of Fit Parameters for the Modela constants

obtuse

m n b kCa (s−1) kCO3 (s−1)

1.852 (±0.605) 0.4151 (±0.0716) 1.469 (±0.417) 6.7 (±0.3) × 106 3.6 (±0.2) × 107

k−kn (M s−1)

0.0 (±0.02)

a

The parameters to calculate the step density (in steps/μm; eq 3) are m and n, which are the slopes with respect to the saturation index and calcium-to-carbonate ratio, respectively, and b which is the intercept. The parameters kCa, kCO3, and k‑kn are used to calculate step velocities (in nm/s; eq 1) in the kink nucleation model. Values in parentheses are the standard deviation.

of either the cation or anion which is not in excess will necessarily start to limit the rate of step advance. At this point an accurate, unique solution for the relationship between kink site nucleation and propagation is not known. The data shown in Figure 2 are solely the obtuse step orientation (one of the two commonly found step orientations D

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on calcite, the other being acute). Bunching of acute orientation steps was observed at many calcium-to-carbonate ratios, particularly at ratios close to or greater than one and led to a significant overestimate of acute step densities (Supporting Information, Figure S2). Net growth rates for the obtuse and acute step orientations must be equal since a growth hillock must nucleate steps of both orientations with equal frequency.9 Therefore, solely the net growth rates for the obtuse orientation were used, but these are representative of the entire crystal, even though individual step density and velocity measurements only apply to the obtuse direction. Comparison to Macroscopic Growth Rates. The data and model prediction in Figure 2c are directly comparable to macroscopic rate measurements. A number of bulk experiments have been conducted under conditions close to a saturation index of 0.75. When compared with our derived model fit (Figure 3a), remarkably, our model is consistent within a 95% confidence level with the results of Tang et al.27 and with all but one data point at high calcium-to-carbonate ratio in Nehrke et al.28 and Wolthers et al.10 (Due to an issue in the reported rates in ref 28, data reported in ref 10, are used; Wolthers, M., personal communication.) The rates measured here compare less well with the growth rate measured by Noiriel et al.,29 which is approximately 1.8 orders of magnitude lower than that measured here, similar to the high calcium-to-carbonate ratio data point from Nehrke et al.28 Solution compositions from Reddy et al.30 and Kazmierczak et al.31 at similar saturation indices and high calcium-to-carbonate ratios were estimated from PHREEQC simulations, but growth rates are each approximately an order of magnitude greater than those reported here (not shown). We extrapolated step densities and velocities to SI = 0.97 to compare the predicted model fit with the results obtained by Gebrehiwet et al.5 and Nehrke et al.28 (Figure 3b). (The data point in ref 5 where homogeneous precipitation was observed is omitted since the mechanism of growth changed.) Similar to the lower saturation index data set, our model predicts the Nehrke et al.,28 SI = 1.1−1.3 data set well at low to moderate calcium-to-carbonate ratios, but predicts higher growth rates than were observed at high calcium-to-carbonate ratios. The growth rates predicted by our model alone are approximately 35% less than the measured growth rates of Gebrehiwet et al.5 (solid black line, Figure 3b). To understand this minor discrepancy, we note that quantitative differences in rate arise from the use of different electrolyte composition and ionic strength.32−34 Gebrehiwet et al.5 used a 0.1 M KCl electrolyte whereas our AFM data was collected in ≤0.02 M NaCl. RuizAgudo et al.32 observed that for the same saturation state, obtuse calcite steps advanced at 13.45 nm/s in 0.1 M KCl but at 7.78 nm/s in 0.02 M NaCl. We therefore infer that step velocities should be 1.7 times higher in 0.1 M KCl than 0.02 M NaCl for the same saturation index. Applying this factor of 1.7 to the calculated step velocities results in all of the data points of Gebrehiwet et al.5 falling within the 95% confidence level of our model (Figure 3b) with the exception of a single measurement at low calcium-to-carbonate ratio, discussed below. We note that this outstanding agreement was obtained even though none of our data was measured at SI = 0.97 and no additional fit parameters were included. The agreement is entirely a prediction of growth rates by our model for conditions under which it was not calibrated, indicating a validation of the predictive capability of the model. One possibility that could explain the difference in behavior with

Figure 3. Net growth rates. (a) Growth rates at saturation indices close to 0.75. Macroscopic experimental measurements27−29 fall within the 95% confidence interval, with the exception of a single data point from Nehrke et al.28 and the Noiriel et al.29 measurement at high calcium-to-carbonate ratio. The average of the individual measurements in Figure 2 are shown here, with error bars that are one standard deviation. Solid lines are the mean predicted growth rate from the data in Figure 2, dashed lines are 95% confidence intervals that reflect the uncertainties in the fit parameters in Table 1. (b) Model prediction for SI = 0.97 with (green) and without (black) a correction of the step velocities for the electrolyte composition/ concentration.32 Calcite growth rates from Gebrehiwet et al.5 all fall within the 95% confidence interval except for the low calcium-tocarbonate ratio where no growth was observed. Growth rates for Nehrke et al.28 are consistent at low to moderate calcium-to-carbonate ratio but are smaller at high calcium-to-carbonate ratio.

electrolyte is that the ions comprising the electrolyte may competitively inhibit/accelerate growth through blocking/ facilitating constituent ion adsorption to kink precursor sites on top of the step.6,14 Discrepancies from Macroscopic Data. Given the excellent agreement between the model and much of the macroscopic data, it is still worth discussing the source of discrepancies between data sets and areas where the model does not predict the observed rates. First, as mentioned above our model is unable to predict zero step velocities at present due to uncertainty about the importance of the back reaction and neglecting the kink site propagation reaction. This translates to an inability to predict accurately the data point in Gebrehiwet et al.5 where zero growth rate was observed at low aqueous calcium-to-carbonate ratios in addition to our own E

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by a factor as much as 5 over time in the experiment, presumably due to the fact that their experiments went initially through a discrete nucleation stage, followed later by crystal growth without hillock formation. One is tempted to speculate that since the Tang et al.27 data set and ours model agree in rate, it suggests that the reactive site density on de novo nucleated calcite and seed crystals pretreated to induce growth hillock formation are similar. Another possibility to explain the discrepancies is that variations in measured terrace widths were observed (Figure 4),

AFM measurements where zero growth was observed. Second, our model predicts rates quite well at calcium-to-carbonate ratios less than approximately 10 but, while matching the Tang et al.27 and Gebrehiwet et al.5 data, rates at high calcium-tocarbonate ratios for other data sets fall below the 95% confidence interval for the model.28,29 To isolate whether step velocities, step densities, or both may be creating this discrepancy, we compared four different AFM data sets (SI = 0.4,8 SI = 0.75,6 SI = 0.72,35 SI = 0.6 11) in which calcite growth was measured as a function of the calcium-tocarbonate ratio at a constant saturation index (Supporting Information, Figure S3). Normalized step velocities and etch pit closing rates as a function of aqueous calcium-to-carbonate ratio followed similar trends across four different studies regardless of methodology, thus it is not likely that there is anything systematically wrong with the step velocity measurements presented here and they are not likely to be the source of the discrepancies at high calcium-to-carbonate rate. If it is not due to the step velocities, the discrepancy in growth rates at high calcium-to-carbonate must arise from the step densities. Possible causes include that the pretreatment of the sample for a given experiment is affecting the net rate through the reactive site density, there are inherent differences in rates between calcite samples of different provenance, that solution conditions such as pH are not the same, or that the AFM results are not representative of an entire crystal. To explore the first option, the pretreatment in the sample may become important under high calcium-to-carbonate ratios. To reiterate, here the samples were exposed to high saturation index (SI = 1.0) solutions prior to rate measurements in order to locate spiral growth hillocks. It could be that spiral growth hillock formation/development is limited under lower saturation indices (2 mm from column inlet), evidence for spiral growth hillocks is more difficult to find. The stirred flow-through reactors on which the rates calculated in Noiriel et al.29 are based did not use a pretreatment stage with saturation index >1, so it is likely that the density of growth hillocks was lower in this case. Nehrke et al.28 did not pretreat their samples, but unfortunately this hypothesis is not consistent with their higher saturation index data set (SI = 1.1−1.3; Figure 3b) since this is higher than what we believe is the critical saturation index for growth hillock formation/development. Tang et al.27 similarly did not pretreat their samples, but nucleated calcite de novo. Noiriel et al.29 pointed out that the Tang et al.27 data show changes in the calcite precipitation rate

Figure 4. Effect of morphological hysteresis on measured terrace widths. While measured step velocities were the same within error for both experiments, measured terrace widths for experiment 1 were approximately 1.7 (for r = 1.27) and 1.3 (for r = 0.111) times as large as the measured terrace widths for experiment 2, despite both hillocks appearing to be of the same type of defect at the center of the hillock. Scan sizes are 3 × 3 μm.

even for experiments that had been conducted using the same solution compositions, stock solutions, and crystals cleaved from the same bulk sample. It may be also that the intrinsic defect density of a material can play a role in the growth rate, and this could be further modified if a sample is crushed or cleaved or grown from a powder: In Reddy et al.30 and Kazmierczak et al.31 calcite powders were used in these studies as seed material without crushing with BET surface areas of 0.7 and 0.5−1.0 m2/g, respectively. As mentioned above, Tang et al.27 did not use any seed material all. In Noiriel et al.29 crushed Iceland spar with a surface area of 0.012 m2/g was used (425 μm mesh). Nehrke et al.28 used a cleaved, macroscopic Iceland spar crystal, and Gebrehiwet et al.5 used both powder (1.3 m2/ g) and a single macroscopic Iceland spar seed crystal. In analyzing this list, there does not seem to be any systematic correlation between starting material that can explain the discrepancies in rates or not at high calcium-to-carbonate ratios. Therefore, starting material appears to have little to do with the average steady state step density of the grown calcite. A further area of uncertainty is the extent to which differences in pH between experiments will lead to different growth rates. The pHs used here were in the range 8.4 under high calcium-to-carbonate ratios to 9.1 at low calcium-tocarbonate ratios (Supporting Information, Table S2). These are similar to those in Gebrehiwet et al.5 (pH 8.5) and Noiriel et al.29 (8.2−8.4) whereas Nehrke et al.27 used pH 10.1−10.2. F

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Wolthers et al.10 derived a process based model which incorporates surface complexation modeling to predict calcite growth rates. Based on this model, the step rate of growth depends on the sum of the rates of attachment of bicarbonate and carbonate. However, no substantial changes in X-ray reflectivity (XR) scattering intensity have been reported for calcite−water interfaces under a range of pH and solution compositions.37−39 This implies that the formation of stable inner-sphere complexes (bicarbonate, carbonate, and calcium) on the terrace regions of calcite are unlikely, but attachment is likely infrequent and may occur only on steps, to which the XR would not be sensitive. For step velocities, it has been reported that in general the sum of obtuse and acute step velocities increased linearly with decreasing pH at pH values less than 8.5, but at pH values above that were fairly constant.40 It is therefore difficult to see that changes in pH are strongly affecting step velocities across these experiments. Regarding the possibility that step densities may change with pH, in the dissolution of a similar material, magnesite (MgCO3), step retreat rate during dissolution has been shown to be completely intransient with respect to pH (pH 2−4.2), but at low pH a greater step density was observed which arose from an increase in etch pit formation rate and thus an increase net dissolution rate.12 This implies that the macroscopic changes in reaction rate with pH (e.g., for calcite dissolution41) could result from step density changes as a function of pH. Lastly, here it is assumed that the {101̅4} surface is representative of the entire material, but other surfaces may be more reactive than the naturally occurring cleavage face typically studied with AFM. On the basis of periodically bonded chain theory (PBC), cleavage faces will contain two or more PBC vectors.42 Other planes, such as a fully stepped surface (which will only contain one coplanar PBC vector) or a fully kinked surface (which will not contain any coplanar PBC vectors) will be composed of more reactive sites than the cleavage plane. Since growth of calcite is likely limited by nucleation of kink sites13 under many solution conditions, the fully kinked faces will grow faster than the fully stepped faces and both will grow significantly faster than the cleavage faces. These more reactive faces will not be found in equilibrium crystal morphologies,43,44 but may be found in natural environments due to mechanical and chemical weathering. Due to kink-site nucleation limited growth, the kink sites on these faces will propagate rapidly and other kink sites will be slower to form, leading to the surface eventually reaching a steady-state morphology more similar to a cleavage face than a fully stepped or fully kinked face. This suggests that calcite growth can be generally modeled based solely on growth rates derived from the {101̅4} surface. While not capturing every aspect of every data set, overall we view the work presented here as proof-of-concept that a realistic framework by which mesoscale or even atomic14 scale processes can be incorporated into models that successfully predict macroscopic rates without the need for adjustable fit parameters to describe additional data sets. However, there are many issues remaining before a predictive model built directly from surface processes can be incorporated into a reactive transport or other model that is functional over the full range of solution/substrate conditions expected during, for example, carbon sequestration or engineered precipitation of a mineral in the subsurface. The major outstanding research questions for future study raised in this work are the following: What are the atomic-level reactions for monomolecular step advance/retreat

including kink nucleation versus propagation? What is the effect of adsorption of protonated anions (bicarbonate) and the electrolyte? What is the effect of back-reactions? How does flow rate, direction, and turbulence manifest themselves on growth rates for a given material by affecting step velocities and densities? Lastly, we need to understand what are the criteria that determine the steady-state reactive site density, since comparison of the various data sets indicates that pretreatment at differing saturation states so as to locate spiral growth hillocks may contribute to the observed discrepancies.

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ASSOCIATED CONTENT

S Supporting Information *

An example Phreeqc input file, solution compositions, step densities, pH values, ionic strengths, and a figure of normalized obtuse step velocities from a variety of experiments. This material is available free of charge via the Internet at http:// pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*Phone: (865) 574-8450; fax: (865) 574-4961; e-mail: [email protected]. Present Address §

Jacquelyn N. Bracco: Department of Chemistry, Wright State University, Dayton, Ohio 45435, United States. Author Contributions

J.N.B. performed the experiments, analyzed the data, and wrote substantial portions of the manuscript. A.G.S. developed theory, supervised the work, and wrote substantial portions of the manuscript, and C.I.S. aided in discussion of other data sets, aided theory development, and contributed portions of the manuscript. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Research sponsored by the Division of Chemical Sciences, Geosciences, and Biosciences, Office of Basic Energy Sciences, U.S. Department of Energy (J.N.B.) and 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 Number (DE-AC02-05CH11231) (A.G.S. and C.I.S.). Special thanks to M. Wolthers for her generosity in sharing the growth rate data and to three anonymous reviewers for their comments.

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REFERENCES

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