New Insights into the Mechanisms Controlling the Rate of Crystal

Oct 22, 2014 - One is the effect of solution stoichiometry on the crystal growth rate of ionic ...... processes and relationships to macroscopic rate ...
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New Insights into the Mechanisms Controlling the Rate of Crystal Growth Helge Hellevang,* Rohaldin Miri, and Beyene G. Haile Department of Geosciences, University of Oslo, Pb. 1047, Blindern, Oslo, Norway 0316 ABSTRACT: Our understanding of the fundamental mechanisms controlling the rate of crystal growth is largely based on the groundbreaking works done in the first half of the 20th century. There are however still aspects of crystal growth that are not satisfactorily explained with the present state-of-the-art models. One is the effect of solution stoichiometry on the crystal growth rate of ionic compounds, with rates peaking at or close to stoichiometric solutions. The most updated models are all based on the mechanistic model proposed by Zhang and Nancollas in 1998 (ZN98) (J. Colloid Interface Sci. 1998, 200, 131−145). We have done a review of experimental growth rate data and evaluated ZN98 and the more recent Wolthers et al. model (W12) (Geochim. Cosmochim. Acta 2012, 77, 121−134). In order to quantify the variation in the experimental data, we constructed an empirical model, and it was found that two adjustable parameters (α and β reflecting the width and position of a Gaussian respectively) were sufficient to satisfactorily reproduce all data. On the basis of the data analysis, it was found that the ZN98 and W12 models cannot generally reproduce the experimental data and that critical assumptions of the models must be reevaluated. Potential relations between β and saturation state and α and ionic strength and saturation state were found, and a possible relation between α and the aqueous ionic charge density z2/r may be indicated in the data. One possible explanation for the large variation in α is statistical in nature, where the number of possible incorporation sites for the A and B units determines the width of the Gaussian. If there are few incorporation possibilities, growth will be strongly limited by the deficient growth unit, whereas if the number of incorporation possibilities at steady-state is large, both for the deficient and surplus growth units, the Gaussian will be less pronounced.

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INTRODUCTION Our understanding of mineral growth rate mechanisms has steadily progressed since groundbreaking works such as by Kossel,3 Stranski,4 and Burton et al.5 (see reviews by Bennema,6 Chernov,7 and De Yoreo and Vekilov8). In the last decades, direct observations by atomic force microscopy (AFM) and increasingly complex computational models have provided new insight into the processes occurring at the water−mineral interface.9−13 Growth in accordance with the Burton− Cabrera−Frank (BCF) model has been demonstrated for ionic compounds such as calcite10 and gypsum,14 and the limitations of the model have been identified.7,10 Despite large progress in the field of crystal growth, there are still experimental observations that cannot be satisfactorily explained by today’s growth models. One example is the dependency of growth rates of ionic minerals on the solution stoichiometry. Experimental data show that growth rates for ionic AB type minerals (A being the cation and B being the anion) such as calcite,12,15−17 Barite,13 gypsum,18 Mg- and Caoxalate,19 and AgCl20 peaks at or close to the stoichiometric [A]/[B] activity or concentration ratio. Such behavior has been explained in the mechanistic models of Zhang and Nancollas1 and Wolthers et al.2 (hereafter referred to as the ZN98 and W12 models). The main framework of these models was developed in Zhang and Nancollas.1 In 2010, Larsen et al.15 demonstrated in an AFM study that the ZN98 model cannot © XXXX American Chemical Society

generally reproduce their experimental data, and they suggested that some key assumptions of the model should be reconsidered. Wolthers et al. improved the ZN98 model further two years later to take into account surface speciation and the pH dependency of attachment and detachment frequencies of growth units and allowed peak rates at nonstoichiometric solutions.2 They did however not demonstrate that the new model could generally reproduce experimental data. The principal aim of this study was to provide a review of experimental crystal growth rates and their dependencies on the solution stoichiometry and to evaluate how well existing mechanistic models can reproduce these data and be used as general predictive tools. For the latter we choose to evaluate the ZN98 and W12 models as they share the same mechanistic basis, and these models must be considered as state-of-the-art within this research field. In order to evaluate the experimental data, we first constructed a two-parameter empirical model that could reproduce all experimental data, and the values of the two adjustable fitting parameters were then used for the data analysis and model comparisons. Received: August 30, 2014 Revised: October 13, 2014

A

dx.doi.org/10.1021/cg501294w | Cryst. Growth Des. XXXX, XXX, XXX−XXX

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Article

R = k′(Ω − 1)2

MODEL DEVELOPMENT The growth of ionic compounds can be written as a general reaction between aqueous ions A and B (cation and anion respectively) forming the solid compound AB. The stoichiometry of the reaction depends on the number of cations and anions incorporated in the ionic compound: m A + n B = A mBn (1)

where Ω denotes aqueous saturation state with respect to an ionic compound: Ω=

n[A] m[B]

(2)

where [ ] may denote activity or concentration. A perfectly symmetric Gaussian dependence of growth rates on the solution stoichiometry r is then given by g (r ) =

−1 ⎛1 r⎞ ⎜ + ⎟ ⎝ 2r 2⎠

REVIEW OF THE EFFECT OF SOLUTION STOICHIOMETRY ON THE EXPERIMENTAL GROWTH RATES OF IONIC COMPOUNDS In order to see if there is any systematic trend in the values of the α and β parameters for different compounds and at different conditions (aqueous supersaturation and background electrolyte concentration), we performed a review of published growth rate data and fitted our model to these data. Experimental growth rates for various ionic compounds (CaSO4·2H2O, BaSO4, CaC2O4·H2O, MgC2O4·H2O, AgCl, CaCO3) are compared to modeled values in Figure 1. Table 1 provides a summary of the experimental conditions and best fit values found for α and β. Experimental data obtained from changes in measured water concentrations or seed crystal weight gain are hereafter referred to as bulk rates, whereas AFM rates are reported as step or surface advancement rates and crystal face and step orientations are provided when available. The units of the rates plotted on the ordinate vary from experiment to experiment and the original reported values are used here. Zhang and Nancollas18 provided bulk growth rates of CaSO4· 2H2O (gypsum) at various aqueous Ca2+/SO42− ratios and at 0.5 KCl and 1 N (mol/Kgw) NaCl background electrolyte concentrations (Figure 1a). The experiments were done at low aqueous supersaturations (1.6 < Ω < 1.7), implying that the rates are in the spiral-growth regime.10 The best fit between model and experimental data was found using α and β pairs (α, β) of (0.35, 100) and (0.6, 11) respectively for the 0.5 and 1.0N experiments. β values larger than 1.0 show that the peak Gaussian is skewed to the left of the 1:1 aqueous Ca2+/SO42− ratio. The Gaussian peak is not observed in the data, but the model indicates that the position should be at r ≈ 0.01 and 0.1 respectively for the 0.5 and 1.0 N solutions. Data for another sulfate, BaSO4, were provided by Kowacz et al. using AFM to obtain growth rates for the (001) surface.13 Two sets of experiments were done at aqueous supersaturations of Ω = 7.2 and 12.6. The fit of the model curves to the data is poorer than for the other data series, especially at some distance from the stoichiometric r. The best fit was found using α and β pairs of (0.15, 0.2) and (0.12, 0.1) respectively for the Ω = 7.2 and 12.6 data (Figure 1b). Values of β smaller than 1.0 indicate in this case that the peak Gaussian is skewed to the right of the 1:1 Ba/SO4 ratio. In Chernov et al. the growth rates of Ca and Mg oxalate was examined using the AFM technique.19 For Ca-oxalate, the

(4)

A value of α = 0 gives g(r, 0) = 1 and no dependency of growth rates on the solution stoichiometry. At this stage we will treat eq 4 as a purely empirical model, but possible mechanistic explanations for α will be discussed below. Earlier the calcite growth rate data from Nehrke et al.22 were used as an example of a Gaussian peaking at the stoichiometric 1:1 [Ca2+]/[CO32−] ratio. Other calcite growth rate data, such as provided by van der Weijden,17,23 may however suggest that this peak at the stoichiometric m/n ratio is not always the case. In order to shift the peak to other values we further modified eq 4 adding the empirical β parameter: ⎛1 ⎞−α g ″(r ) = 2α⎜ + rβ ⎟ ⎝ rβ ⎠

(7)

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(3)

where function g is a correction function for the effect of solution stoichiometry. The height (or depth) of the Gaussian will hereafter be defined as the relative difference between g at the maximum value (for eq 3 at r = 1) and at a fixed distance in r from this maximum. The maximum growth rates are not always at r = 1,12,15,21 and the choice of representative r values to provide depth values may therefore be system dependent. Using the exponent of −1 in eq 3 provides a Gaussian depth that is very similar to what has been observed for calcite growth, and the fit is quite good over a large range of r.22 Other experimental data from other ionic compounds, such as AgCl,20 suggests however a much more narrow (deeper) Gaussian, whereas yet other compounds, such as CaC2O4·H2O,19 may suggest a broader (more shallow) Gaussian. The fixed exponent of −1 is therefore replaced by α giving: ⎛1 ⎞−α g ′(r , α) = 2α⎜ + r ⎟ ⎝r ⎠

q ⎛ ΔG ⎞ ⎟ ≡ exp⎜ ⎝ RT ⎠ K

where q is the aqueous ion activity product, K is the thermodynamic equilibrium constant, ΔG is the Gibbs free energy, R is the gas constant, and T is absolute temperature. The coefficients α and β can then found from finding the best fit between adjusted experimental (k′) and modeled (g″(r)) rates. Below we provide a review of estimated α and β values for a range of ionic solid compounds (CaCO3, CaC2O4·H2O, MgC2O4·H2O, CaSO4·2H2O, BaSO4, AgCl), before we discuss possible mechanistic models that can explain the variations observed for the different compounds.

where the stoichiometric ratio r of A and B in a bulk aqueous solution or in the diffuse boundary layer can be defined as r=

(6)

(5)

As will be demonstrated in the next section, eq 5 can satisfactorily reproduce all experimental data within uncertainty limits. To obtain α and β coefficients for the various experimental data sets, we used a Trust Region quasi-Newton algorithm. Because some data series were obtained at various aqueous saturation states, we used the approximated BCF growth model (eq 6) together with the reported rates to estimate growth rate coefficients k′ and thereby removed the impact of saturation state on growth rates. The simplified BCF model valid for low aqueous supersaturations can be written as B

dx.doi.org/10.1021/cg501294w | Cryst. Growth Des. XXXX, XXX, XXX−XXX

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were measured for Mg-oxalate at a constant aqueous supersaturation of Ω = 2.0. For both minerals β values of 1.0 were found, indicating a symmetric Gaussian at the stoichiometric Me2+/C2O42− value (Figure 1c,d). The α parameters varied both for the different step orientations for Ca-oxalate (α = 1.0 and 0.5 for the [001] and [021] steps respectively) and also between Ca- and Mg-oxalate (the latter with an α value of 1.4). Compared to the sulfates, generally larger α values for the oxalates correspond to the narrower (deeper) Gaussian for the latter. The oldest data were obtained from Davies and Jones, providing the dependency of AgCl bulk growth rates on the solution stoichiometry.20 The experiments were done at low aqueous supersaturation (Ω = 2.1) and at low background electrolyte concentrations. The data suggest a symmetric Gaussian at the stoichiometric 1:1 Ag+/Cl− ratio, and therefore β = 1 (Figure 1e). The data suggest a very narrow (deep) Gaussian, and more defined than for any of the other minerals in the study, and a best-fit of the model results in α = 5. Calcite is the only ionic compound with several independent studies performed with different methods (bulk and AFM) and at varying aqueous supersaturations and background electrolyte concentrations (Figure 1f−j).12,15,17,22 The first we notice is that there is quite a range in both α and β for calcite. While the experiments by Nehrke et al.22 suggest a peak Gaussian close to the stoichiometric 1:1 Ca2+:CO32− ratio (Figure 1f), experiments by Larsen et al.15 (Figure 1g), van der Weijden17,23 (Figure 1h), and Bracco et al.12 (Figure 1i, j) suggest that this is not always the case. One reason for this may be differences in the experimental conditions. The background electrolyte concentrations ranges from 1) and Bracco et al.12 (β < 1) data. We may expect that the values of α and β at least partly depend on the ionic compound in question. Moreover, the experiments shown in Figures 1 have been done with different methods and at a range of conditions. It is therefore not trivial to recognize general patterns of α and β as a function of the type of ionic compound and growth conditions. Nevertheless, some systematics may be observed. Table 1 provides a summary of parameter values and experimental conditions for the various compounds. From Figure 1 it is clear that α shows a large variation from ionic compound to ionic compound. For calcite the variation is much more limited, with a variation from 0.2 to 0.95. If we plot α as a function of saturation ratio (Ω) (Figure 2a) and electrolyte concentration (Figure 2b), we see that α is to some degree positively correlated with the ionic strength (background electrolyte concentration), whereas there may be a general negative correlation between α and aqueous saturation state. A plot of α versus pH reveals no apparent trends, neither for the whole data set or for calcite isolated (Figure 2c). The β parameter varies over a large range not only across different ionic compounds but also for the same compound at different experimental conditions. We therefore plotted β as a function of electrolyte concentration and saturation state for all experimental data in this study (Figure 3). From this figure

Figure 1. Experimental growth rates of ionic compounds as a function of solution ion stoichiometry. Curves are modeled rates using R = kg″(r), where k is an empirical rate constant, r is the solution stoichiometry defined in eq 2, and the correction function g″ is defined in eq 5. Optimal values of the empirical fitting parameters k, α and β were found by a Trust-Region quasi-Newton algorithm. The ionic compounds that were analyzed were (a) CaSO4·2H2O; (b) BaSO4; (c) CaC2O4·H2O; (d) MgC2O4·H2O; (e) AgCl; (f−j) CaCO3.

advancement rates of the [001] and [021] steps were measured on the (001) face at a constant aqueous supersaturation of Ω = 5.6, whereas the advancement rates of steps at the (010) face C

dx.doi.org/10.1021/cg501294w | Cryst. Growth Des. XXXX, XXX, XXX−XXX

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Table 1. Estimated α and β Values for the Reviewed Ionic Compounds Together with Type of Experiment (Batch/Bulk Solution or AFM) and Experimental Conditions (Saturation State Ω, Ionic Strength, pH)a ionic compound CaCO3 I22 CaCO3 II22 CaCO3 III12 CaCO3 IV12 CaCO3 V12 CaCO3 V12 CaCO3 VI17,23 CaCO3 VI15 CaCO3 VI15 BaSO413 BaSO413 AgCl20 CaC2O4·H2O I19 CaC2O4·H2O II19 MgC2O4·H2O19 CaSO4·2H2O I18 CaSO4·2H2O II18 a

exp type bulk bulk AFM AFM AFM AFM bulk AFM AFM AFM AFM bulk AFM AFM AFM bulk bulk

{101̅4} {101̅4} {101̅4} {1014̅ }

acute obtuse acute obtuse

{101̅4} acute {101̅4} obtuse

(100)[001] (100)[021] (010)

molarity

pH

Ω

α

β

0.1 0.1