Calcite Growth Rates As a Function of Aqueous Calcium-to-Carbonate

Jun 6, 2012 - Growth rates of monomolecular steps on calcite surfaces were measured using atomic force microscopy. It is found that the amount of stro...
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Calcite Growth Rates As a Function of Aqueous Calcium-toCarbonate Ratio, Saturation Index, and Inhibitor Concentration: Insight into the Mechanism of Reaction and Poisoning by Strontium Jacquelyn N. Bracco,†,‡ Meg C. Grantham,† and Andrew G. Stack*,‡ †

School of Earth and Atmospheric Sciences, Georgia Institute of Technology, Atlanta, Georgia 30332, United States Chemical Sciences Division, Oak Ridge National Laboratory, Oak Ridge, Tennesee 37831, United States

‡

S Supporting Information *

ABSTRACT: Using in situ atomic force microscopy, the growth rates of the obtuse and acute step orientations on the {101̅4} calcite surface were measured at two saturation indices as a function of the aqueous calcium-to-carbonate ratio and aqueous strontium concentration. The amount of strontium required to inhibit growth was found to correlate with the aqueous calcium concentration, but did not correlate with carbonate, suggesting that strontium inhibits attachment of calcium ions to reactive sites on the calcite surface. Strontium/ calcium cation exchange selectivity coefficients, Kex, are estimated at 1.09 ± 0.09 and 1.44 ± 0.19 for reactive sites on the obtuse and acute step orientations, respectively. The implication of this work is that, to avoid poisoning calcite growth, the concentration of calcium should be higher than the quotient of the strontium concentration and Kex, regardless of the saturation index. Previous analytical models of nucleation of kink sites on steps are expanded to include growth rates at multiple saturation indices and the effect of strontium. The rate constants for calcium attachment are found to be similar for the two step orientations, but those of carbonate vary significantly. This work will have implications for natural or engineered calcite growth, such as to sequester subsurface strontium contamination. between the fluid and calcite. At the mesoscale, preferential incorporation of strontium into the obtuse (as opposed to acute) monomolecular step orientations on {101̅4} surfaces has also been observed.12 The same study also found that low strontium concentrations modestly enhances calcite growth rates, but as strontium is increased, a critical concentration is reached that causes growth rates to collapse, suggesting a complex mechanism of inhibition of calcite growth by strontium. A second important factor affecting growth of minerals is the ratio of the aqueous form of the mineral’s constituent ions in solution. It has been shown that the aqueous cation-to-anion ratio affects the growth rates of calcite6,7,13−16 and barite.17 Often, net growth rates are maximized at cation-to-anion ratios greater than that dictated by stoichiometry,6,13,14,16,17 implying that the reactions of attachment and detachment of cations and anions are not uniform and that cation attachment may be rate limiting under stoichiometric solutions. The peak velocities of the obtuse and acute step orientations have been found to occur at different calcium-to-carbonate ratios as well,13,15 implying different mechanisms and/or rates of attachment for

1. INTRODUCTION The growth kinetics of the mineral calcite are of particular interest to geochemists because calcite is the most stable form of calcium carbonate at conditions found on and near the Earth’s surface. The extent to which trace metals incorporate into the calcite crystal lattice is affected by variables such as precipitation rates, temperature, and water chemistry.1−7 For example, the effect of temperature on incorporation rate enables one to measure strontium-to-calcium ratios in biogenic calcite and aragonite and using these ratios extrapolate paleoocean temperatures from the recent past and many thousands of years ago.8 Additionally, strontium itself is of environmental interest because legacy strontium-90 contamination associated with nuclear weapons and energy production is present at some U.S. Department of Energy sites.9 Sequestration of the strontium-90 via coprecipitation into calcite is a potential method to immobilize the contaminant for a sufficiently long time that it decays.6,10 Bulk precipitation experiments have shown that the rate of incorporation of strontium into calcite increases as the precipitation rate of the calcite increases.1,2,5,7 A model was recently developed by DePaolo11 to predict metal/calcium ratios in calcite as a function of the precipitation rate which established that the strontium-to-calcium ratio in the mineral is a result of the competition between precipitation and exchanges © 2012 American Chemical Society

Received: March 14, 2012 Revised: June 5, 2012 Published: June 6, 2012 3540

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ppm. While some trace strontium was observed in the calcite samples, the concentrations found were negligible compared to the amounts added to the growth solutions. 2.2. In Situ AFM Experiments. In the experiments show here, two atomic force microscopes were used to measure step velocities on freshly cleaved calcite samples, an Agilent PicoPlus SPM and an Asylum Research MFP-3D. These were found to give similar step velocities under similar conditions. Calcite surfaces were exposed to continuously flowing growth solutions at a rate of 120 mL/h with the obtuse step orientation facing the inlet solution jet, a flow rate and direction under which growth was found to be not limited by mass transport (data in Figure S2, Supporting Information). Prior to mounting samples, calcite dust produced during cleaving was removed using compressed nitrogen. Step velocities were measured on growth features known as spiral growth hillocks, primarily by calculating the slope of the trace of the obtuse and acute steps over time. However, as the curvature and irregularity of the steps increased with increasing strontium concentration or extreme calcium-to-carbonate ratios, velocities were then measured by the displacement of individual points on steps in sequential images. Velocities were corrected for thermal scanner drift and/or misalignment of the calcite crystal by taking the average of both an up and a down scan for each data point, and two to four replicates were performed per data point (the velocities reported in this work are the arithmetic mean of the replicates and error bars are plus or minus one standard deviation).

the two step orientations. Regarding the shape of the curve for growth rate as a function of calcium-to-carbonate ratio, both asymmetric and symmetric peaks have been observed. Under high ionic strength (I.S. = 0.1), growth rates decrease symmetrically about the peak.6,7,15 At low ionic strength (I.S. ≤ 0.025), growth rates decrease asymmetrically in that growth decreases more rapidly at low calcium-to-carbonate ratios.13,14,16 Coupled to this is that the concentration and identity of the background electrolyte can affect the magnitude of growth rates under a fixed calcium-to-carbonate ratio by a factor of 2 or more.18 Both findings indicate that the presence of the background electrolyte affects the mechanism of growth and that care is therefore needed when comparing different data sets. In the present work we use in situ, flow-through atomic force microscopy kinetic measurements19,20 to examine the effect of strontium on calcite growth as a function of aqueous calciumto-carbonate ratio and saturation index. Along with significant model development, we find that the concentration of strontium required to inhibit growth is correlated with the aqueous concentration of calcium but is not correlated or inversely correlated with the aqueous carbonate concentration, indicating strontium is affecting attachment of calcium to the calcite surface.

3. RESULTS AND DISCUSSION 3.1. Observations in the Absence of Strontium. Step velocities of the obtuse and acute step orientations on calcite growth hillocks at a fixed saturation index (SI = 0.75), but variable aqueous calcium-to-carbonate ratio, are shown in Figure 1a. The step velocities measured here follow a trend similar to step velocities previously reported by Stack and Grantham,13 who worked at a lower saturation index (SI = 0.40) (Figure 1b). The peak velocity of the obtuse steps occurs under solutions with calcium-to-carbonate ratios greater than 1, whereas the acute steps advance fastest at ratios close to 1 or slightly less than 1. As expected, the maximum step velocity is greater at the higher saturation index used here than in Stack and Grantham,13 particularly for the obtuse step orientation. However, steps were observed to continue advancing under solutions containing ratios that were extremely far from one with the saturation index of 0.75. This contrasts to the behavior observed at a saturation index of 0.40, where growth abruptly ceased at a critical aqueous calcium-to-carbonate ratio for either the obtuse (zero velocity at low calcium-to-carbonate) or acute (zero velocity at high calcium-to-carbonate) steps. Morphologies were similar between the two saturation indices, where obtuse steps were rounded at low ratios and acute steps became bunched at high ratios. This raises the question of whether the zero and even slightly negative step velocities observed in Stack and Grantham13 at extreme ratios are indicative of an intrinsic process or whether they resulted from an experimental artifact. To address this, we performed the substantial supporting analyses and verification/ validation discussed in the Methods section. In none of these could we find anything extraordinary suggesting that the observations of Stack and Grantham are due to artifacts. Additionally, the velocities measured here are consistent within error with those measured by Ruiz-Agudo and co-workers18 for similar solutions, suggesting that there is not an isolated systematic error in our measurements. Moreover, there are an increasing number of studies where growth rates inconsistent with the nominal supersaturation of the solution have been observed: Gebrehiwet et al.6 observed calcite growth only

2. METHODS 2.1. Sample and Growth Solution Preparation. Stock solutions used to prepare growth solutions were created from ACS reagent grade CaCl2, NaHCO3, and SrCl2 dissolved in distilled deionized water (18.2 MΩ-cm) and equilibrated with the atmosphere for at least two weeks or sparged with air overnight prior to use. We found that use of solutions not equilibrated with the atmosphere resulted in increased acute step velocities at low calcium-to-carbonate ratios, likely due to incomplete equilibration with carbon dioxide. The results of these measurements are shown in the Supporting Information (Figure S1). The chemical compositions of the growth solutions were calculated using PHREEQC.21 The system was treated as equilibrated with the atmosphere, and the solubility product constant for calcite used was Ksp = 10−8.48. The pH of inlet and outlet solutions was measured to confirm that the growth solution pH matched that called for by PHREEQC to within a few tenths of a pH unit. Depending on the solution composition, measured pHs varied from 8.4 (at high calciumto-carbonate ratios) to 9.2 (at low calcium-to-carbonate ratios). The calcium-to-carbonate ratio of selected solutions was measured using inductively coupled plasma optical emission spectrometry (ICP-OES), alkalinity titrations, and were confirmed to be similar to the to be similar to the compositions predicted by PHREEQC. Solution compositions are shown in Table S1a,b, Supporting Information for experiments conducted in the absence of added strontium and in Table S2, Supporting Information for experiments conducted with added strontium. Saturation indices (SI) were kept fixed at 0.40 and 0.75 (SI = log(aCaaCO3/Ksp); positive values correspond to supersaturation, negative to undersaturation), and the aqueous calcium-tocarbonate ratio was varied from 0.0143 to 381 for SI = 0.75 and 0.0156 to 43.7 for SI = 0.40. Strontium concentrations used fell in the range of 0 to 9.07 × 10−3 M (Table S2, Supporting Information). The ionic strength of the growth solutions were all less than 0.02 M, with the exception of the [Ca]/[CO3] = 381 with added strontium, which was I = 0.04 M. Since large deviations in step velocity were observed with increasing ionic strength using this solution, these results were not used in model development. Calcite samples (Wards Scientific) were cleaved along the {101̅4} surface immediately prior to use in each experiment. A representative calcite sample was analyzed using ICP-OES to determine trace divalent cation impurity content (Table S3, Supporting Information). Of the different impurities found, magnesium was the most common at 588 3541

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within error and was determined to be not statistically significant using a heteroscedastic two-tailed t-test. This test is used to determine if the means of two samples are statistically the same (in this case, the samples are the fit parameters calculated using the doubled data set and the fit parameters calculated using the not doubled data set). A heteroscedastic test was chosen because the variances were not found to be statistically equal using an F-test, which in turn compares the variances of the fit parameters. The combined SI = 0.75 and SI = 0.40 data sets were fit simultaneously to the kinetic ionic ratio model,24 and fit parameters were estimated using a Newton−Raphson minimization (χ2 = 1740 and 321 for the obtuse and acute orientations, respectively). Information on the analytical expressions used for our implementation of this model is summarized in the Supporting Information and model fits are shown in Figure S3. Importantly, we were unable to use this model to predict step velocities of both saturation indices simultaneously (i.e., using the same fit parameters for both saturation indices), nor can it replicate the shapes of the curve well for obtuse step orientation at a saturation index of 0.75, resulting instead in an overbroad peak. Other recent models built from the kinetic ionic ratio model are expected to behave in a similar fashion.25,26 To obtain a model that is more suitable to describe growth at multiple saturation states, the kink-nucleation limited model previously derived in Stack and Grantham13 was used. This model is distinct in that it does not explicitly include saturation state as a master variable but treats attachment and detachment of calcium and carbonate ions independently. Here, kink site propagation will be neglected in order to obtain a unique fit and since this process is assumed to be rapid on calcite steps due to kink-site nucleation limited growth.27,28 Briefly, the velocity, v, or growth rate of a step is defined as

Figure 1. Step velocity as a function of calcium-to-carbonate ratio at saturation indices of (a) 0.75 and (b) 0.40. Obtuse steps advance at a maximum rate at high ratios, whereas the acute step grows at a maximum rate at ratios close to 1, or slightly less than 1. Curves are model fits of the combined data set (SI = 0.40 and 0.75) using the kink site nucleation model from Stack and Grantham.13

v = aR kn

within a limited range of calcium-to-carbonate activities and outside of this observed either an immediate cessation of calcite growth or homogeneous precipitation of calcium carbonate from the growth solution. The peak growth rate6 occurred at a calcium-to-carbonate ratio of 3.3,6 similar to our peak net growth rate:13 3.8 (taken from the average of the calcium-tocarbonate ratio of the peak velocities of the obtuse and acute steps, weighted by the peak velocity). Furthermore, simultaneous growth and dissolution of the acute and obtuse steps, respectively, has also been observed in stoichiometric calciumto-carbonate solutions but close to equilibrium.22 One possibility that could explain a sudden cessation of growth is that one or more back-reactions (i.e., detachment of ions from the surface) only become significant at a critical saturation or cation-to-anion ratio. This was recently demonstrated in the case of calcite dissolution in the presence of magnesium.23 Specifically, magnesium suddenly inhibited dissolution at SI ≥ −0.7, but below this saturation index the back reaction (ion attachment in this case) was not significant, implying that the importance of the back reaction(s) could be nonlinear with solution composition. Since there was approximately twice as much data for SI = 0.75 as SI = 0.40, prior to fitting the combined SI = 0.4 and SI = 0.75 data sets of the current work and that of Stack and Grantham,13 each data point in the SI = 0.40 data was doubled to give equal weighting of each data set in the fits that are described below. The difference between fit parameters using a doubled SI = 0.40 data set and original data set was the same

(1)

where a is the width of a molecular row of a calcite step, 0.31 nm, and Rkn is the rate of kink site nucleation. This rate is defined as R kn =

k Ca[Ca]k CO3[CO3] k Ca[Ca] + k CO3[CO3]

− k −kn (2)

where kCa is the first-order rate constant for attachment of calcium ions, kCO3 is the first-order rate constant for attachment of carbonate ions, and k−kn is a pseudo-zeroth order combined detachment rate constant and kink site density. In Stack and Grantham,13 the roots of eq 2 were used to help constrain the fits at the calcium and carbonate ratios where the step velocities were observed to be zero (or slightly negative). Since no step velocities of magnitude zero were observed at SI = 0.75, here we have opted initially to fit the model without forcing it to zero velocity at a given ratio. Figure 1 shows the step velocity model fit to the two saturation indices simultaneously; fit parameters are shown in Table 1. The fit is much more appropriate for both step orientations than the kinetic ionic ratio model, and the nonlinear goodness-of-fit statistics are correspondingly lower (χ2 = 417 and 195 for the obtuse and acute orientations, respectively). However, the fit does slightly underestimate the peak obtuse step velocity at a saturation index of 0.40. It also predicts nonzero step velocities at ratios where step advancement was not observed and underestimates some step velocities 3542

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This finding suggests that it is the difference in the attachment of carbonate ions that is the determining factor in the anisotropy of the growth rates of each of the acute and obtuse steps. This anisotropy will in turn manifest itself in the morphologies of the crystals crystal grown under solutions with different calcium-to-carbonate ratios (discussed below). The anisotropy is rationalized by recognizing that the rates of attachment/detachment of the trigonal-planar carbonate ion may be much more susceptible to steric effects induced by the geometry of the step than the more spherically symmetric aqueous calcium ion. Since the acute step has a smaller peak velocity and is less open geometrically, it could be that carbonate attachment to this step is sterically hindered and is limiting the rate of advance. Since the obtuse step is more open geometrically and has a greater peak velocity, it could also be that this steric hindrance for attachment of the carbonate is not present on this step orientation. The larger peak velocity of the obtuse step in this scenario is consistent with the relative favorability of the hydration free energy of the aqueous ions29 as well as the water exchange rate of the ions.13,18 Thus, once the steric effects that limit carbonate attachment on the acute step are removed, the more favorable hydration free energy and/or slower water exchange rate requires more calcium in solution in order to maximize growth rate. Ruiz-Agudo and coworkers18 also postulated that obtuse step growth rates may be controlled by the availability of calcium ions in solution (as per above), whereas acute steps may be limited by kink site nucleation and the dehydration of surface sites. While uncertainties still remain in the molecular mechanism of attachment and detachment, our model suggests that the comment in the introduction that cation attachment limits the rate of growth under stoichiometric solutions is oversimplified. The fit parameters in Table 1 suggest that cation attachment is rate limiting for the obtuse step under stoichiometric solutions, but acute step advance appears to be controlled more or less equally by attachment of both ions since the rate constants for attachment are of similar magnitude (perhaps a slight control by carbonate). Taken all together, the data and model fits reinforce the observations that affinity-based rate models are not applicable to describe mineral growth kinetics when the ratio of the mineral’s constituent ions in solution varies.6 3.2. Observations in the Presence of Strontium. Velocities of both the obtuse and acute step orientations were measured at variable strontium concentrations for five different calcium-to-carbonate ratios (Figure 2), three of which were at SI = 0.40 ([Ca]/[CO3] = 0.100, 2.72, and 12.1) and two of which were at SI = 0.75 ([Ca]/[CO3] = 0.0143 and 27.2). Consistent with previous observations,12 the step advancement rate remained relatively constant as strontium was increased until a critical concentration was reached whereupon the growth rate collapsed and became almost completely inhibited. We find that the amount of strontium necessary to inhibit growth can vary by more than an order of magnitude under the same saturation index but differing aqueous calcium-to-carbonate ratios (Figure 2). At low calciumto-carbonate ratios, relatively low concentrations of strontium are necessary to inhibit growth; however as the calcium-tocarbonate ratio is increased, progressively higher concentrations of strontium are necessary to produce similar degrees of growth inhibition. In Figure 2, step velocities were fit using sigmoid curves to ease visualization and to provide a quantitative measure of the reduction in step velocity for a given strontium concentration. The choice of this particular functional form is

Table 1. Estimates of Rate Constants for the Nucleation Modela constants kCa (s−1) kCO3 (s−1) k−kn (M·s−1) KS−Sr KS−Ca kCa (s−1) kCO3 (s−1) k−kn (M·s−1)

acute

obtuse

Present Study 6.4 (±0.5) × 106 4.7 (±0.4) × 106

6.7 (±0.3) × 106 3.6 (±0.2) × 107

0.0 (±0.02) 2.9 (±0.6) × 104 1.1 (±1.2) × 103 Stack and Grantham (2010)13 1.3 (±0.2) × 107 5.0 (±0.5) × 106 8.4 (±1.0) × 10−2

0.0 (±0.02) 1.8 (±0.3) × 104 0.0 (±2.2) × 102 1.1 (±0.1) × 107 5.0 (±0.4) × 107 2.0 (±0.1) × 10−1

a

Values are the best estimates, while standard deviations estimated from residuals are shown in parentheses.

at extreme ratios. As shown in Stack and Grantham,13 some of these issues may arise from improperly neglecting the propagation reaction at extreme ratios. Forcing the model to zero velocity at the extreme ratios for SI = 0.40 yielded fit parameters within error of those parameters reported in Stack and Grantham13 (i.e., within one standard deviation of the uncertainty of the parameters). However, the resulting goodness-of-fit statistics were larger (χ2 = 1030 and 297 for the obtuse and acute orientations, respectively), indicating the overall fit is worse than the model that is not constrained to zero velocity at a certain ratio. Not tying the model fit to zero velocity results in attachment rate constants for calcium ions for both step orientations that are within a factor of 2 of the previous fit at solely the saturation index of 0.40 and the magnitude of the attachment rate constants for carbonate ions are similar (Table 1). The decision to tie the rates to zero at a certain ratio (again, forcing the back reaction to be significant) or not rests in part on whether the observations of zero step velocity are accurate. If those observations are indeed accurate, it is clear that our treatment here does not account for this phenomenon properly, nor does any other existing crystal growth theory to our knowledge. Since a sufficient, independent understanding of the magnitude of the attachment and detachment reactions at the crystal surface is lacking, we have left the model as written and not tied it to a zero step velocity, which produces the best fit. These and other reactions may be important but including more than the three parameters used in this model is not justified statistically since the data to which it is fit only include two independent variables (concentrations of calcium and carbonate) and one dependent variable (step velocity). The magnitude of the rate constants for attachment of calcium ion in the best fit are similar for both the obtuse and acute step orientations; however, the rate constant for attachment of carbonate ion is an order of magnitude slower for the acute step than for the obtuse step (this was also observed in Stack and Grantham13). Using a heteroscedastic two-tailed t-test, it was determined that the difference between the rates of attachment for a calcium ion here are not statistically significant (p value of 0.4), but differences between the rates of attachment for a carbonate ion are extremely statistically significant (p value of 0.0008). Since the attachment rate constants for a calcium ion are the same within error, and the observed difference in the mean is not statistically significant, we will assume that the attachment rate constants for a calcium ion to either step orientation are in fact the same. 3543

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concentrations of both calcium and carbonate increase, whereas as the calcium-to-carbonate ratio is increased, the concentration of calcium increases but the concentration of carbonate decreases. This suggests that the concentration of strontium required to inhibit step advancement may correlate with the concentration of aqueous calcium, but not the concentration of aqueous carbonate. To test this hypothesis, the data in Figure 2 were plotted in terms of step velocity as a function of the aqueous strontium-to-calcium ratio (Figure 5). Distinct trends for the obtuse and acute step velocity as a function of the strontium-to-calcium ratio are readily observed. At high strontium-to-calcium ratios, growth is mostly inhibited for both step orientations, even under calcium-to-carbonate ratios where the peak velocity occurs in the absence of strontium. To confirm that there is no correlation or inverse correlation between the amount of strontium necessary to inhibit growth and the aqueous carbonate concentration, the step velocities were also plotted as a function of the strontium-to-carbonate ratio (Figure S4, Supporting Information) and the product of the strontium and carbonate concentrations (Figure S5, Supporting Information). Our initial hypothesis was that strontium inhibited growth by labilizing carbonate ions adsorbed next to it, or by interfering with their attachment. However, if that mechanism was correct, the amount of strontium necessary to poison growth would inversely correlate with the carbonate concentration (i.e., loading the system with carbonate would overcome the inhibition). Since no correlation exists, inverse or otherwise, this mechanism cannot be possible. To derive an alternate hypothesis for the mechanism by which strontium can inhibit attachment of calcium, but not carbonate, we turn to recent rare event theory simulations for barium detachment/attachment from a barite [120] step edge.33 It was found that barium detachment/attachment from the step edge passes through multiple intermediate adsorption states through a process resembling an inverse-Schwöbel barrier but without surface diffusion. These states include outer-sphere, innersphere, and bidentate species. If calcium attachment during calcite growth occurs via a similar mechanism, strontium may compete with one or more of these precursor adsorption sites and limit or block attachment of calcium ions to the step edge. Simultaneous inner-, outer-, and extended outer-sphere adsorption of strontium has been shown to occur on mica surfaces.34 In contrast, for calcite, X-ray reflectivity (XRR) studies35,36 and a recent molecular dynamics study29 show little or no binding of inner-sphere adsorbed species onto planar calcite surfaces as a function of saturation index. The mechanism of attachment Stack et al.33 may still be favorable however since it allows an oppositely charged ion embedded in the step edge to assist in desolvation of the attaching ion. This process lowers the free energy of attachment37 but also involves multiple bond formation events which may result in intermediate species. Adsorption onto steps would not be detected by the XRR. If this conceptual picture is accurate, the choice of a sigmoidal curve was prescient, since this allows us to calculate the relative affinities of strontium and calcium sorption to the precursor site(s) (also sigmoidal curves are appropriate to model equilibrium adsorption speciation). If it is assumed that the step velocity in excess strontium is zero, and that the step velocity in the absence of strontium or limited strontium is vmax, the sigmoidal curves in Figure 5 can be defined as

Figure 2. Step velocities as a function of the aqueous strontium concentration at a saturation index of 0.75 (triangles) and 0.40 (circles) for the (a) obtuse and (b) acute step orientations at five different aqueous calcium-to-carbonate ratios. Substantially higher concentrations of strontium are required to inhibit growth at high calcium-to-carbonate ratios (green circles and red triangles) than near stoichiometric conditions (blue circles) or low calcium-to-carbonate ratios (black circles and orange triangles).

justified a posteriori below. The form is not completely ideal, however, since step velocities of the acute step orientation were sometimes observed to increase slightly with increasing strontium at low strontium concentrations, a finding also consistent with previous work.12 Changes in step velocity were accompanied by morphological changes. In Waslyenki et al.,12 rounding and elongation of the steps were observed perpendicular to the c-glide plane, which contrasts to step elongation parallel to the c-glide plane observed in the presence of magnesium,30 silicic acid,31 and hexavalent chromium.32 While elongation perpendicular to the c-glide plane was observed at all calcium-to-carbonate ratios, the degree of rounding was found to be strongly dependent on ratio. Under solutions with low calcium-to-carbonate ratios ([Ca]/[CO3] = 0.0143) and high strontium concentrations, substantial rounding and bunching of the obtuse steps were observed (Figure 3). Significantly less rounding occurred for a given strontium concentration at high calcium-to-carbonate ratios ([Ca]/[CO3] = 381) (Figure 4) and step bunching was exclusive to the acute step orientation. (That is, the opposite in the case of the low calcium-to-carbonate ratios.) To apply these measurements to the understanding of the mechanism of growth inhibition by strontium, we noted that the strontium concentration required to inhibit growth increased with both saturation index and solution calcium-tocarbonate ratio. As saturation index is increased, the 3544

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Figure 3. Calcite growth morphology at low calcium-to-carbonate ratio ([Ca]/[CO3] = 0.0143) as a function of the aqueous strontium concentration. (a) Step velocity as a function of strontium concentration. The dashed lines denote the strontium concentrations used for b−d. (b) [Sr] = 8.65 × 10−6 M, acute and obtuse steps advance at similar rates. Obtuse and acute step orientations are as marked. (c) [Sr] = 2.00 × 10−5 M, step advancement for the obtuse orientation is approximately half of the rate of growth of the strontium-free system and obtuse steps begin to round. (d) When [Sr] = 5.00 × 10−5 M, obtuse steps become extremely rounded and bunched, and steps elongate perpendicular to the c-glide plane. The scan size in (b) and (c) is 5 μm and 7.5 μm in (d).

v=

vmax ⎛ [Sr] / [Ca]v1/2 − [Sr] / [Ca] ⎞ ⎟ 1 + exp⎜ α ⎝ ⎠

orientation) indicates that the amount of strontium necessary to inhibit growth correlates with the calcium concentration and supports the concept that strontium acts via competitive inhibition with calcium for some precursor site(s) at the step edge. We attempted to perform the same transform on the previous work on this system12,38 but were unable to obtain a similar correlation. Since there is the known effect of the background electrolyte amount and composition on growth rate,18,39 we hypothesize that the 0.1 M NaCl background electrolyte is interfering with the correlation by affecting the mechanism of attachment and detachment of the constituent ions to and from the surface. To include the hypothetical “multiple intermediate-state” mechanism into the nucleation-limited model discussed above, we need to reconsider the attachment of ions onto the step edge. In eq 2, attachment and detachment of calcium ions are assumed to occur via a single reaction:

(3)

where [Sr]/[Ca]v1/2 is the strontium-to-calcium ratio at which v = vmax/2, and α is a parameter which affects the steepness of the curve. At this time we do not know of an immediate physical interpretation for the α parameter. For a given strontium-tocalcium ratio and saturation index, the aqueous strontium-tocalcium ratio at which the velocity is one-half the maximum velocity then indicates where the concentrations of strontium and calcium adsorbed into a precursor state on the step edge are equal. This may be written as: 2+ 2+ step‐Sr + Ca aq ⇌ step‐Ca + Sr aq

(4)

where step-Sr and step-Ca correspond to strontium or calcium adsorbed in a precursor site as described above. Ignoring differences between activity and concentration and solving for the cation exchange selectivity coefficient yields: [step‐Ca][Sr] Kex = [step‐Sr][Ca]

k Ca

2+ step + Ca aq HoooI kink

(6)

k −Ca

where kCa and k−Ca are the attachment and detachment rate constants for calcium. However, if calcium first adsorbs at some sort of precursor site (step-Ca) prior to forming a kink site (kink), the mechanism should be written as a two reaction process:

(5)

so that under conditions when v = vmax/2, [step-Ca] equals [step-Sr] and the [Sr]/[Ca] ratio is equal to the Kex. The curves in Figure 5 yield a mean strontium-to-calcium ratio of 1.09 ± 0.09 for Kex of the obtuse the step orientation (Figure 5a) and 1.44 ± 0.19 for the acute step orientation (Figure 5b). The convergence of the curves (especially on the obtuse step

KS − Ca

k 2Ca

2+ step + Ca aq HoooooI step‐Ca HooooI kink k −2Ca

3545

(7)

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Figure 4. Calcite growth morphology at high calcium-to-carbonate ratio ([Ca]/[CO3] = 381) as a function of the aqueous strontium concentration. (a) Step velocity as a function of strontium concentration. The dashed lines denote the strontium concentrations used in (b−d). (b) When [Sr] = 5.05 × 10−4 M, acute steps advance more slowly than obtuse steps. (c) When [Sr] = 5.79 × 10−3 M, step advancement of the acute step orientation has slowed substantially. (d) When [Sr] = 9.07 × 10−3 M, acute steps have become pinned and rounding of obtuse steps is observed. The scan size in (b) is 7.5 μm and 10 μm in (c) and (d).

where step is the empty precursor site. We assume that the kink site formation reaction is rate limiting33 and that its back reaction (k−2Ca) is negligible (supported by the fit parameters in Table 1), and that the first reaction can be treated as a local equilibrium and can be described by an equilibrium constant (KS−Ca). In this case the apparent kCa estimated in eq 2 (Table 1) is actually:

[step] =

(8)

and the terms in the right-hand side of this expression are assumed to be relatively constant across experimental conditions (in the absence of strontium). We hypothesize that strontium competes for the precursor site, but in contrast strontium does not proceed to a kink site with a sufficient frequency to contribute to the rate (Sr incorporation rate is on the order of 1−2%6,12):

cf =

1 + KS − Ca[Ca] 1 + KS − Ca[Ca] + KS − Ca[Sr]

(12)

where kCa apparent (with Sr) = kCa apparent (without Sr) × cf. This is substituted into eq 2 and the entire data set of Figure 2 is fit using the fit parameters of Table 1, but the kCa is adjusted by cf and KS−Ca and KS−Sr are treated as adjustable fit parameters. Equation 12 is functionally similar to those derived to explain the Gibbs-Thompson effect,40,41 with a key difference in that here we are treating the impurity as blocking a precursor site rather than the kink itself. The resultant fits are shown in Figure 6 (χ2 = 1609 and 358 for the obtuse and acute steps, respectively). Overall the quality of the fit is quite good considering that it is fit across all strontium concentrations, aqueous calcium-to-carbonate ratios and saturation indices and that inaccuracies in the zero strontium model will propagate to the fit with strontium. The fit qualitatively captures the trend of nonzero step velocities only being observed under high calcium-to-carbonate ratios. Unfortunately, the simple interpretation of the Kex from the sigmoidal empirical fits does not

KS − Sr

(9)

where step-Sr strontium bound to the precursor site and KS−Sr is the equilibrium constant for precursor formation. Combining the first step of eq 7 with eq 9 and simplifying results yields eq 4 above. We can express the total concentration of precursor site as [site]total = [step] + [step‐Ca] + [step‐Sr]

(11)

To fit this, we have substituted eq 11 into eq 8 and taken the ratio of apparent rate constants for with and without strontium, thus deriving a correction factor (cf) to be applied to the apparent rate constant for calcium attachment in the absence of strontium to account for the effect of the inhibitor (thus avoiding refitting the model for the case without strontium):

k Ca apparent = k Ca intrinsic[step‐Ca] = k Ca intrinsicKS − Ca[step]

2+ step + Sr aq HooooI step‐Sr

[step]total 1 + KS − Ca[Ca] + KS − Sr[Sr]

(10)

Substituting expressions for [step-Ca] and [step-Sr] from eqs 7 and 8 and solving for [step] yields: 3546

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Figure 5. Step velocities as a function of the strontium-to-calcium ratio at a saturation index of 0.75 (triangles) and 0.40 (circles) for the (a) obtuse and (b) acute step orientations at five different calcium-tocarbonate ratios. Growth is inhibited at a similar strontium-to-calcium ratio, regardless of the calcium-to-carbonate ratio or saturation index of the solution.

Figure 6. Obtuse step velocities as a function of calcium-to-carbonate ratios at a saturation index of (a) 0.75 and (b) 0.40 at constant strontium concentrations of 0 M (black), 1 × 10−4 M (red), 4 × 10−4 M (blue), and 1 × 10−3 M (green). Curves are model fits of the combined data set (SI = 0.40 and 0.75) using the modified kink site nucleation model (eqs 1, 2, 12). Data and model fits in the absence of strontium are the same as in Figure 1, but for clarity, only those data points whose calcium-to-carbonate ratio matches the strontium experiments are shown. Fits and data for the acute step orientation are shown in Figure S6.

correspond to the quotient of KS−Sr and KS−Ca derived from this analytical expression since the functional form for the step velocity is more complicated. The quotient of KS−Ca and KS−Sr can be forced to the Kex from the sigmoid fit, but the fit quality is significantly degraded (χ2 = 3772 and 430 for the obtuse and acute steps, respectively). The model additionally has difficulty predicting a zero step velocity where a zero velocity is actually observed, similar to the case in the absence of strontium. Again, we attribute this to improperly neglecting the propagation reaction, a nonlinear back reaction, or assuming that certain quantities remain constant across experimental conditions when in fact they do not (e.g., the concentration of precursor site in eq 8). Overall, we view this model is as good as it reasonably can be without independent confirmation of the reactions that actually occur at the step edges and their rates. That is, a unique analytical solution for crystal growth rate is unlikely to be derived solely from step velocities alone. The results of the current study will have some implications for the areas of interest listed in the introduction. For the case of the strontium remediation via incorporation into calcite growth, our results suggest that in order to avoid strontium unduly poisoning the growth of calcite during remediation, calcium concentration should be kept above the quotient of the cation exchange selectivity coefficient and the strontium concentration. In turn, the carbonate concentration should be tied to the calcium concentration such that the aqueous calcium-to-carbonate ratio produces the optimum growth rate

since, as noted above, the strontium incorporation rate is correlated with growth rate.1,2,5,7 This study may also aid in the interpretation of paleo-ocean temperatures records based on strontium-to-calcium ratios. Specifically, foraminifera have been shown to incorporate higher concentrations of strontium when the aqueous calcium concentration is increased at constant supersaturation (increasing the calcium-to-carbonate ratio), but incorporation is not increased when the aqueous carbonate concentration is increased (decreasing the calcium-to-carbonate ratio) at constant supersaturation.42,43 Lastly, understanding the mechanism by which growth inhibition occurs may contribute to the design of compounds and strategies to modify morphologies and other properties of engineered crystals, such those used in nanodevices44 and for industrial purposes such as frequency doublers for lasers.45

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ASSOCIATED CONTENT

S Supporting Information *

Solution compositions and measured step velocity values, impurity contents, the kinetic ion ratio model fit, effect of step velocity on solution equilibration time, flow rate and reactor response, additional analyses of step velocities and discussion of 3547

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(26) Nielsen, L. C.; DePaolo, D. J.; DeYoreo, J. J. Geochim. Cosmochim. Acta 2012, http://dx.doi.org/10.1016/j.gca.2012.02.009. (27) Rashkovich, L. N.; De Yoreo, J. J.; Orme, C. A.; Chernov, A. A. Crystallogr. Rep. 2006, 51, 1063. (28) De Yoreo, J. J.; Zepeda-Ruiz, L. A.; Friddle, R. W.; Qiu, S. R.; Wasylenki, L. E.; Chernov, A. A.; Gilmer, G. H.; Dove, P. M. Cryst. Growth Des. 2009, 9, 5135. (29) Raiteri, P.; Gale, J. D.; Quigley, D.; Rodger, P. M. J. Phys. Chem. C 2010, 114, 5997. (30) Wasylenki, L. E.; Dove, P. M.; De Yoreo, J. J. Geochim. Cosmochim. Acta 2005, 69, 4227. (31) Pina, C. M.; Merkel, C.; Jordan, G. Cryst. Growth Des. 2009, 9, 4084. (32) Sanchez-Pastor, N; Gigler, A. M.; Cruz, J. A.; Park, S. H.; Jordan, G.; Fernandez-Diaz, L. Cryst. Growth Des. 2011, 11, 3081. (33) Stack, A. G.; Raiteri, P.; Gale, J. D. J. Am. Chem. Soc. 2012, 134, 11. (34) Lee, S. S.; Fenter, P.; Park, C.; Sturchio, N. C.; Nagy, K. L. Langmuir 2010, 26, 16647. (35) Fenter, P.; Geissbuhler, P.; DiMasi, E.; Srajer, G.; Sorensen, L. B.; Sturchio, N. C. Geochim. Cosmochim. Acta 2000, 64, 1221. (36) Heberling, F.; Trainor, T. P.; Lutzenkirchen, J.; Eng, P.; Denecke, M. A.; Bosbach, D. J. Colloid Interface Sci. 2011, 354, 843. (37) Piana, S.; Jones, F.; Gale, J. D. J. Am. Chem. Soc. 2006, 128, 13568. (38) Wilson, D. S. Master’s Thesis, Virginia Tech, 2003. (39) Becker, U.; Risthaus, P.; Bosbach, D.; Putnis, A. Mol. Simul. 2002, 28, 607. (40) Weaver, M. L.; Qiu, S. R.; Friddle, R. W.; Casey, W. H.; De Yoreo, J. J. Cryst. Growth Des. 2010, 10, 2954. (41) Pina, C. M. Surf. Sci. 2011, 605, 545. (42) Duenas-Bohorquez, A.; Raitzsch, M.; de Nooijer, L. J.; Reichart, G. J. Mar. Micropaleontol. 2011, 81, 122. (43) Raitzsch, M.; Duenas-Bohorquez, A.; Reichart, G. J.; De Nooijer, L. J.; Bickert, T. Biogeosciences 2010, 7, 869. (44) Aizenberg, J. Adv. Mater. 2004, 16, 1295. (45) Stack, A. G.; Rustad, J. R.; DeYoreo, J. J.; Land, T. A.; Casey, W. H. J. Phys. Chem. B 2004, 108, 18284.

the above. This information is available free of charge via the Internet at http://pubs.acs.org/.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We would like to thank three anonymous reviewers whose helpful comments have improved this manuscript. Research sponsored by the Division of Chemical Sciences, Geosciences, and Biosciences, Office of Basic Energy Sciences, U.S. Department of Energy.

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