Growth Rate of Calcite Steps As a Function of Aqueous Calcium-to

Feb 4, 2010 - Andrew G. Stack* and Meg C. Grantham. School of Earth ... Once a hillock is located, the slow-scan axis is disabled and the velocity of ...
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DOI: 10.1021/cg901395z

Growth Rate of Calcite Steps As a Function of Aqueous Calcium-to-Carbonate Ratio: Independent Attachment and Detachment of Calcium and Carbonate Ions

2010, Vol. 10 1409–1413

Andrew G. Stack* and Meg C. Grantham School of Earth and Atmospheric Sciences, Georgia Institute of Technology, Atlanta, Georgia 30332 Received November 6, 2009; Revised Manuscript Received January 7, 2010 w This paper contains enhanced objects available on the Internet at http://pubs.acs.org/crystal. n

ABSTRACT: Growth rates of monolayer-height steps on the {1014} calcite surface have been measured as a function of the aqueous calcium-to-carbonate ratio. The maximum growth rates of the two common crystallographic orientations were found to deviate from the ideal stoichiometric ratio of 1:1, and dissolution features were observed under supersaturated solutions containing high calcium-to-carbonate ratios. To explain these phenomena, a theory is applied that treats the rates of attachment and detachment of aqueous calcium and carbonate ions separately. The resultant attachment rate constants are 1-3 orders of magnitude smaller than the water exchange rate of the constituent aqueous ions, suggesting that ligand-exchange processes may directly drive attachment. The broader implication is that the saturation state alone is not adequate to fully describe the rates of the multiple, independent reactions that occur on mineral surfaces under these conditions.

Introduction In geochemical, biological, and other systems, mineral growth and dissolution rates are often modeled as proportional to the bulk saturation state of the system. Under conditions close to equilibrium, crystalline materials often grow and dissolve through the advance and retreat of monomolecular steps that originate from preexisting defects in the crystal structure of the material. Mechanistic models of these processes explicitly assume that the rates of attachment of the constituent ions of a mineral are equal, as are the rates of detachment.1,2 For the ubiquitous (bio)mineral calcite (CaCO3), this implies that growth rates will be a maximum at an aqueous calcium-to-carbonate ratio of 1:1.3,4 There are two commonly found step orientations on calcite, acute and obtuse (Figure 1), that have distinct structures and reactivities.5-7 We measured the growth rate of these two orientations on growth hillocks using an atomic force microscope (AFM) with a custom flow-through fluid cell using established methods.8,9 We find that response of obtuse and acute steps to calcium-to-carbonate ratio is highly variable, and growth becomes kinetically inhibited and dissolution features are observed at high ratios. This result potentially explains the absence of abiotic calcite precipitation in supersaturated, but nonstoichiometric, environments found worldwide such as near-surface ocean water.10-13 This phenomenon may also affect the engineered growth of calcite in the subsurface to sequester contaminants such as strontium.14 Lastly, it is consistent with growth morphologies of calcite seed crystals exposed to high calcium-to-carbonate growth solutions13 and the asymmetric closing rate of etch-pits as a function of the calcium-to-carbonate ratio.4 Methods A calcite crystal is mounted in a custom-made fluid cell such that its acute and obtuse step orientations are perpendicular to the *Corresponding author. E-mail: [email protected]. r 2010 American Chemical Society

slow-scan axis of the microscope (an Agilent, PicoPlus). Step velocities are measured on features of growth known as a spiral hillocks15,16 because the center of the hillock is an easily identified fixed point of reference and shows all of the commonly found step orientations of a crystalline material in a single image. Once a hillock is located, the slow-scan axis is disabled and the velocity of the step is derived trigonometrically from the slope that the trace of the step makes over time. In these experiments, multiple measurements over repeated experiments were averaged. Flowing solutions were created from stock solutions of CaCl2 and NaHCO3 that were well stirred and allowed to equilibrate with atmospheric CO2 for at least two weeks prior to use or sparged with air overnight. Specific solution compositions used are listed in the Supporting Information, Table S1. Saturation state was defined by the saturation index, SI = log(aCa  aCO3/Ksp calcite). Positive SI values are supersaturated, and negative values are undersaturated. Saturation indices, calcium and carbonate concentrations were calculated using PHREEQC17 and the pH of inlet and outlet solutions measured to confirm that the solution pH matched the predicted pH within a few tenths of a pH unit. Measured pHs varied from 8 to 9.3 (Table S1, Supporting Information), but no correlation of pH with step velocity was observed.

Results The startling effects of high calcium-to-carbonate ratios on calcite growth are shown in Figure 1. Sequential images of calcite exposed to a saturation index of 0.64 and a ratio of calcium-to-carbonate concentration of 22:1 in 0.1 M NaCl are shown. Two spiral growth hillocks are observed growing along a preexisting line of defects that trend from the upperright to the lower-left side in the image. The obtuse steps are those that are above and to the left of the center of the hillocks, and acute steps are those that are below and to the right of the center. Growth of the obtuse sides of the hillock is indicated by the position of two relatively wide, adjacent terraces that are close to the center of the upper hillock in Figure 1a but near its left edge in Figure 1b. In contrast, the acute steps are not advancing and numerous features suggestive of dissolution are observed: First, the edges of the growth hillock have become rounded on the acute side, and the high step density Published on Web 02/04/2010

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Figure 2. Step velocities as a function of the aqueous calcium-tocarbonate ratio. Saturation index is fixed at 0.4. The open circles are the average measured velocity; error bars are (1 standard deviation. The acute step orientation grows more rapidly at low calciumto-carbonate ratios, whereas the obtuse step grows more rapidly at high ratios. Curves are model fits for different assumptions about the reactions that control growth: (a) Kink nucleation only. (b) Kink nucleation and propagation.

Figure 1. n w Sequential 15  15 μm AFM images of a calcite surface. (a) Growth morphology immediately following exposure to a solution with a saturation index of 0.64, but a calcium-to-carbonate ratio of 22:1. Acute and obtuse step orientations are as marked. (b) The same surface after ∼30 min exposure, simultaneous growth of obtuse steps, and features likely to be caused by dissolution are observed. A looped image sequence of the data in Figure 1 in Quicktime format is available in the HTML version of this paper. Approximate elapsed time over the loop is 30 min.

suggests that these steps are pinned, or prevented from moving by their proximity to other steps. Second, the curved acute steps bridging the acute faces of the two growth hillocks are retreating. Additional evidence is the terrace-like gap evident in between the hillocks in Figure 1b but not 1a. Lastly, numerous etch-pits have opened along a line (probably a grain boundary), as well as abundant smaller etch-pits decorating the surface. A looped image sequence of the data in Figure 1 is available in the HTML version of the paper. The results in Figure 1 and the looped image sequence could conceivably be produced not by dissolution but by growth inhibition of selected portions of the surface by adsorption of an unknown impurity and the surface imperfectly growing around the pinned area16 to create dissolution-like features. However, a frame-by-frame analysis of the looped image sequence reveals that the features, especially the etch-pits, grow in size over the course of the image loop, and this does not seemingly correlate to rates of advancement of any steps on the calcite. This is especially true for the line of etch-pits

cutting across the acute steps because these steps are nearly completely pinned, but the etch-pits grow in size. The retreat of the steps bridging the acute slopes of the two hillocks would be highly unlikely to be produced by growth around an impurity rather than dissolution without a substantial increase in the size of the base of the hillock on the acute side which is not observed. (Additional evidence that impurities are not responsible for these features is discussed in Figure 3 below.) We thus conclude that the features actually represent true dissolution rather than growth around inhibited portions of the surface. We interpret Figure 1 and the looped image sequence as therefore showing simultaneous growth, step pinning, and dissolution of the calcite surface under conditions where the activities of aqueous calcium and carbonate are four times higher than that necessary for equilibrium with the bulk saturation index. We measured step velocities of the obtuse and acute step orientations on calcite growth hillocks at a fixed saturation index (0.4), but variable calcium-to-carbonate ratio with no additional electrolyte (Figure 2). The growth rates of the obtuse step orientation are higher at high calcium-to-carbonate ratios, whereas the acute step orientation shows the reverse behavior and grows faster at low calcium-to-carbonate ratios. There is substantial asymmetry in the velocity of the two steps as a function of the ratio in that the obtuse steps have a much higher peak velocity than the acute in addition to the difference in the ratio at which the peak occurs. As previous workers have observed,4 this implies that the kinetics of the chemical reactions involved in the movement of these two steps are asym-

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Figure 3. Calcite growth morphology as a function of the calcium-to-carbonate ratio. The orientation of the calcite is the same as in Figure 1. Obtuse and acute step orientations are as marked. (a) At low calcium-to-carbonate ratios (0.003:1), the obtuse steps do not advance. Both step orientations have a high kink density, as evidenced by their roughness. (b) At intermediate ratios (0.4:1), both obtuse and acute step orientations are straight and advance. (c) At high calcium-to-carbonate ratios (20:1), the obtuse step orientation continues to grow, whereas the acute step has become pinned and etch-pits are observed.

metric as well. At calcium-to-carbonate ratios further from 1:1 than the data shown in Figure 2, we found that growth morphology tends to collapse and step velocity measurements become increasingly difficult to make. Coupled to the changes in step velocity are changes in growth morphology (Figure 3). At low calcium-to-carbonate ratios, the obtuse steps become less straight and stop moving (pin) or begin to retreat, whereas the acute steps continue advancing (Figure 3a). As the calcium-to-carbonate ratio is increased (Figure 3b), both steps advance and are straight. At high calcium-to-carbonate ratios (Figure 3c), the acute steps stop advancing all together, pin, and/or begin to dissolve. Figure 3 implies that calcite morphology is affected by the calcium-to-carbonate ratio of solution with which it is in contact, but also that the initial optimum calcium-to-carbonate ratio at which a given crystal will grow depends on its starting morphology. In Figure 3a-c, as the [Ca]/[CO3] ratio is increased, the steps that cease to advance shifts from the obtuse to the acute. This offers additional evidence that impurity pinning

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Figure 4. Kink reactions on a step-edge. For clarity, the calcite has been abstracted to a homogeneous isotropic solid. (a) Nucleation reactions. Both aqueous calcium or carbonate ions can attach to an otherwise smooth step, nucleating a kink. Attachment is treated explicitly for each ion (kCa, kCO3), but only the total detachment is resolved (k-kn). (b) Propagation reactions. Ions from solution add material to an existing step. Attachment reactions are considered as the same as in nucleation, but detachment is considered explicitly (k-kpCa, k-kpCO3) because of the differing number of bonds to the surface.

is not the source of the putative dissolution features, because it would require a potential impurity to change its preference for the acute or obtuse step based on the [Ca]/[CO3] ratio or other system variable. Impurities are known to sometimes have a preference for either an acute or obtuse step orientation,18 but it is difficult to see how an impurity might change its preference of step so dramatically as to create the results shown in Figure 3. To interpret these findings, we adapted modern crystal growth theory16 to allow independent rates of attachment and detachment. The basis of the theory is the following reactions: kCa

Ca2aqþ -F Cakink Rk-

ð1Þ

-Ca

kCO 3

CO23 -aq-F CO3 kink R k

ð2Þ

-CO 3

2where Ca2þ aq is aqueous calcium ion, CO3 aq is aqueous carbonate ion, the kink subscript denotes the ion adsorbed to the

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Table 1. Estimates of Rate Constants for the Two Growth Modelsa nucleation þ propagation

nucleation only constants -1

kCa (s ) kCO3 (s-1) k-kn (M 3 s-1) k-kpCa (M 3 s-1) k-kpCO3 (M 3 s-1) a

acute

obtuse

1.3 (( 0.2)  10 5.0 (( 0.5)  106 8.4 (( 1.0)  10-2 7

acute

1.1 (( 0.1)  10 5.0 (( 0.4)  107 2.0 (( 0.1)  10-1 7

obtuse

3.2 (( 11)  10 2.0 (( 7.9)  107 3.4 (( 13)  10-1 0 (( 0.09) 1.7 (( 18)  102 7

6.5 (( 1.6)  106 3.4 (( 1.0)  107 1.1 (( 0.2)  10-1 0 (( 53) 0 (( 52)

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

step-edge either nucleating or propagating a kink. kCa and kCO3 are the rate constants for attachment, and k-Ca and k-CO3 are rate constants for detachment from the kinks (Figure 4). Rate constants for attachment are assumed to be first-order and the rate depends on the concentration of the ion in solution. Detachment rate constants are also first-order, but depend on the concentration of kink sites along the length of the step. These experiments were conducted on sufficiently long steps such that kinks are consumed by completion of a row as rapidly as they are created through the net attachment rate, so the total concentration of kinks does not change over time; that is, the steps are in the rough limit.16 We assume that the kink density is roughly constant across all of our experiments, and therefore the detachment rate constants are treated as pseudozeroth-order, that is, the product of the average kink site concentration and a first-order rate constant. For any given elementary reaction, the rate of the appearance of products must equal the rate of disappearance of reactants (in this case the product, kinks, can also disappear due to completion of a row of calcite as well as the reverse reaction, detachment). Applying this principle to a heterogeneous reaction yields a problematic issue: rates of different units and dimensions are obtained depending on if the reaction progress is measured by the forward or reverse reaction. Total reaction progress in these experiments is evaluated by solid-phase rates, that is, movement of monomolecular calcite steps in nm/s and converted to rows of calcite added per second (see the a term used in eqs 3 and 4 below), but we wish to compare our rate constants to solution-phase homogeneous reactions. Our tentative solution to this problem is as follows: in eqs 1 and 2, the rate of the forward reaction (i.e., attachment) has the desired homogeneous-phase units, but since the total reaction progress is monitored by the units of the products (nm/s), any rate estimated from a step velocity of the forward reaction has product units and needs to be multiplied by the molar volume of calcite (32.05 mol of calcite/L per unit step volume) to obtain a correct first-order homogeneous rate constant in s-1. For the reverse (detachment) reaction, the units of the reactants (kink sites/ unit step length) are directly comparable to the total reaction progress, but the desired units are solution-phase, so the apparent rate constant has been divided by the molar volume to give a rate constant that predicts solute units and dimensionality of M 3 s-1. The velocity, v, or growth rate of a sufficiently long step is a function of the rates of two separate processes: kink site nucleation and propagation (Rkn and Rkp, respectively):1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ν ¼ a 2Rkn Rkp

ð3Þ

where a is the width of a single molecular row of a calcite step, 0.31 nm. For steps with a high kink density, the propagation rate is most important and nucleation can be neglected.19

Since calcite steps often follow crystallographic directions and have relatively few kinks,20 we initially assume the reverse; that is, that the step velocity is limited solely by the net rate of kink site nucleation (i.e., propagation is fast): ν ¼ aRkn

ð4Þ

Previous treatments of the kink site nucleation rate result in a step velocity that asymptotically approaches a finite positive value for supersaturated solutions at calcium-to-carbonate ratios far from 1:1.1 While some materials do show this behavior,19 the step velocities measured in Figure 2 for calcite abruptly reach zero velocity and can even become negative under supersaturated conditions. We therefore use a different nucleation rate: Rkn ¼

kCa ½CakCO 3 ½CO3  -k -kn kCa ½Ca þ kCO 3 ½CO3 

ð5Þ

The individual rates of detachment for the kink site nucleation reactions cannot be distinguished, so a single rate constant is used, k-kn, that represents their sum (Figure 4a). In order to constrain the fit, the root of eq 5 is taken, that is, solved for where Rkn = 0. If the attachment rates are assumed to be of similar magnitude, when [Ca] . [CO3] the [CO3] in the denominator in eq 5 can be neglected and the expression can be solved for kCO3: kCO 3 ¼ k -kn =½CO3 ν ¼0

ð6Þ

where [CO3]v=0 is the carbonate concentration at the ratio where step velocity goes to zero. Similarly, when [CO3] . [Ca]: kCa ¼ k -kn =½Caν ¼0

ð7Þ

For the obtuse step, [Ca]v=0=1.88  10-5 M at [Ca]/[CO3]= 0.016 and for the acute step, [CO3]v=0 = 1.74  10-5 M at [Ca]/[CO3] = 45.8. Equations 6 and 7 are substituted into eq 5, which is then substituted into eq 4. Fit parameters were then estimated numerically using a Newton-Raphson minimization. The resultant step velocity model fit is shown in Figure 2a, and fit parameters are shown in Table 1. Overall, the fits are adequate but tend to underestimate low step velocities. Since low velocities are also the most likely to have curved steps where the propagation reaction is important (Figure 3a,c), we have fit the data to eq 3 using the standard expression for kink site propagation:1,21 kCa ½CakCO 3 ½CO3  -k -kpCa k -kpCO 3 Rkp ¼ ð8Þ kCa ½Ca þ kCO 3 ½CO3  þ k -kpCa þ k -kpCO 3 We consider the attachment propagation and nucleation reactions for the obtuse and acute steps to be the same. The rationale for this assumption is that the rate of a reaction is not determined by energetically favorable elementary steps such as bond formation, but by unfavorable steps such as bond

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breaking. In this case, attachment is thought to be governed by dehydration reactions of the aqueous ions, diffusion of the ion through the bulk and interfacial water, and dehydration reactions at the mineral surface. These steps are likely to be similar for ions that nucleate or propagate kinks, and thus there may not be much difference in their rates of attachment. In contrast, detachment reactions are thought to be governed by the mineral surface bonding environment and are expected to vary substantially if the detaching site is a nucleated or propagated kink. These are therefore considered explicitly as k-kn for nucleation and k-kp for propagation reactions (Figure 4b). The resultant fit is nonunique, especially in the magnitude of the propagation detachment reactions. Functionally, the primary effect of including the propagation reaction is that step velocity increases from zero more rapidly than when just considering the nucleation reaction, especially for the acute step orientation (Figure 2b). This is consistent with the conditions under which the propagation reaction is the most important, that is, where they are the least straight. The magnitudes of the rate constant estimates for each model are shown in Table 1. The estimated rate constants for attachment for calcium are ∼1-2 orders of magnitude smaller than the rate of water exchange of the aqueous calcium ion22 (kH2O = 6-9  108 s-1), and a computationally estimated rate constant for attachment of an aqueous calcium ion to a planar {1014} calcite surface23 (kattach = 3.3  108 s-1). The smaller rate constant is reasonable if steric hindrances and the different interfacial water-structure at the step relative to an aqueous ion or a planar surface are considered. The water exchange rate for aqueous carbonate is not known, but we estimate an upper limit of 2  109 s-1 based on the diffusion coefficient of the ion (D = 9.2  10-5 cm2 3 s-1) and the Smoluchowski expression.24 This is ∼2-3 orders of magnitude larger than our estimated rate constants for attachment of carbonate. Thus, the fit parameters directly support the concept that ligand-exchange reactions govern the rate of attachment of the ions to the surface and that detachment is generally slower and governed by the rate at which bonds to the surface are broken. As an important check on the validity of the calculated rate constants, an approximate solubility product constant for each step orientation can be calculated. The average detachment rate constant divided by the average attachment rate constant should be similar to the solubility product constant of calcite, Ksp. For the obtuse step we estimate a Ksp of 10-8.33 and for the acute, 10-8.48. The acute direction in particular closely matches the bulk solubility product constant of calcite, 10-8.48. However, this treatment neglects the kink site concentration that is included in the pseudo-zeroth-order rate constants for detachment. As mentioned above, kink density on calcite is considered to be intermediate compared to materials such as lysosyme with one kink every ∼102 unit cells or KDP with a kink density of ∼1.20 Alternately, if the conversion of heterogeneous-homogeneous rates described above is in error, it may mean that these suggest an average kink density might be similar to the reciprocal of the molar volume (one kink per 32 unit cells). This however is probably too low to be accurate.20

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For the more complex model that considers both nucleation and propagation, the best fit for all the rate constants for detachment from propagated kinks were found to be mean of zero except for detachment of carbonate from the obtuse step. The uncertainty in the rate constant estimates increases substantially due to the larger number of parameters in this model, however, is as expected. Future work will focus on reducing the number of unknown parameters in this model, determining the kink site concentration, examination of carbonate/bicarbonate effects, and competition from background electrolyte ions. Acknowledgment. This work was funded by the U.S. NSF Grant EAR-0643139. The authors are grateful for the editorial suggestions of E. Ingall and V. Van Cappellen and to A. A. Chernov for his help with the derivation of equation 8. Supporting Information Available: Table of solution composition summary from data in Figure 2. This material is available free of charge via the Internet at http://pubs.acs.org.

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