Review and Recalculation of Growth and Nucleation Kinetics for Calcite, Vaterite and Amorphous Calcium Carbonate

Round 1

Reviewer 1 Report

This paper aims to put together data of kinetics of calcium carbonate nucleation and growth scattered in large quantities of literature.  

Even though I do appreciate this brave efforts to collect data to compare, to build a unified method and it could be very useful to some extent,  I think we as scientists, need to accept the fact that individual papers have their own objectives and applications, allowing to choose suitable conditions for their own works.   

The introduction could be much improved if the part  of “sources of inconsistency” is significantly shortened.   I do understand where it is coming from.  However not only “the sources of inconsistency” doesn’t really do justice and redundant,  but also even though it is partially true, it is rather genetic issues of all scientific literatures, not only CaCO3 kinetics. (If I can say, it is very naive)  therefore it would be much useful if the introduction provides more relevant  background and implications for this specific review, which lack in the current version.   

1. 

“The first source is inconsistent definitions. For example, the definition of saturation, which is the core of any theory on mineral precipitation kinetics, is not consistent and is often confused with a driving force (e.g. ln(S) or (√S-1 )). “

The authors must be confused with supersaturation and supersaturation index. Definition of supersaturation is always the same, but how to express it is varied (often termed as “supersaturation index” rather than supersaturation” depending on their own applications or research field by using square root S- 1, or log or ln.  

I found that the authors had the same dilemma shown in figures where mixed uses of supersaturation index (log x or square root S-1 ) in this paper. If the authors can present absolute “IAP” along with supersaturation as double x-axis, it would be much easier to compare figure by figure. 

One such example leading to errors is the ambiguous usage of the mathematical operator ‘log’. It is not always clear when results are plotted on a logarithmic scale whether 'log10’ or ‘ln’ is meant (e.g. [3-5]). This kind of inconsistency can easily lead to incorrect results and make it difficult to check or reproduce the results. 

I was taught in high school that Log(x) means the base 10 logarithm, in this case no need to label 10, this is the mathematical promise, no confusion there. 

2. Aragonite, one of CaCO3 polymorphs, is entirely omitted in discussions. It should be justified. 

3. ACC is not one of polymorphs of calcium carbonate. Polymorph by definition should be a different crystalline structure with the same composition. ACC is an amorphous phase and has undefined compositions. ACC is the phase that forms through an aggregation mechanism, rather than a classical growth mechanism. this has not been considered in the calculation. 

4. How were the equations 11-13 deduced ? It is difficult to understand “the update” without detailed explanation of intermediate steps. 

It was only mentioned as “We have updated these equations from Kaschiev [6] to incorporate the shape factors of equations 5 and 6”  

5. table 3, the values for calcite are all indicated as N/A, why ?

6. Page 5 . 

“For calcite reported growth rate constants have a broad range of values and units. The reported values are 4.0 – 8.8 L mol-1 s-1 (gseed L-1)-1 [1,11], 1.43 10-10 – 1.08 10-9 mol m-2 s- 1 [12,13], 5.37 10-7 – 2.34 10-6 mol m-2 s-1 [14], 2.91 10-9 – 1.39 10-6 mol m-2 s-1 [3], 1.64 10-7 – 2.81 10-7 mol m-2 s-1 [15], 4.05 10-11 – 1.98 10-10 m s-1 [16], 5.0 10-10 m s-1 [17], 1.1 10-10 m s-1 [18], 8.56 10-8 – 2.65 10-7 mol m-2 s-1 kgw-1 [19], 2.21 10-8 – 7.48 10-4 mol m-2 s-1 [20], 6.48 104 L2 mol- 1 m-2 s-1 [21], 3.42 10-7 mol m-2 s-1 [22], 6.64 10-8 – 7.80 10-7 mol m-2 s-1 [23], 1.13 10-10 – 9.2 10- 10 m s-1 [2], 5.8 10-7 – 3.35 10-6 mol m-2 s-1 [24], and 8.08 10-26 m3 s-1 [25]. The growth order, g, varies between 0.91 and 4.71 in these publications. Additionally, several parameters for alternative growth models [26-28] are reported. “

These dont provide any valid information, so would only occupy the valuable space of the paper.  it will of course be widely varied because the values must be varied depending on the solution conditions and precipitation methods. If the authors really want to add these data, it would be much helpful to make a table with solution conditions in SI or main paper. 

6. As a review article,  the number of cited references is substantially low,  especially given the author's vast amount of literature on this topic. 

This must be typo, * indicate Leon van Paassen as corresponding author, while the * Correspondence email is  lukebergwerff@gmail.com; 

Author Response

This paper aims to put together data of kinetics of calcium carbonate nucleation and growth scattered in large quantities of literature.  

Even though I do appreciate this brave efforts to collect data to compare, to build a unified method and it could be very useful to some extent, I think we as scientists, need to accept the fact that individual papers have their own objectives and applications, allowing to choose suitable conditions for their own works.   

We don’t deny the scientists’ freedom to select a method most suitable for the purpose. However, as we demonstrated in this paper different approaches lead to inconsistencies, which make it difficult to reproduce and compare different data sets. Hence, we defined the objective of this review paper to collect and interpret the available data sets and use a unified approach, which allows to compare the data and recalculate the kinetic constants.

The introduction could be much improved if the part  of “sources of inconsistency” is significantly shortened.   I do understand where it is coming from.  However not only “the sources of inconsistency” doesn’t really do justice and redundant,  but also even though it is partially true, it is rather genetic issues of all scientific literatures, not only CaCO3 kinetics. (If I can say, it is very naive)  therefore it would be much useful if the introduction provides more relevant  background and implications for this specific review, which lack in the current version. 

We agree that the identified sources of inconsistency have a generic character which may also be relevant for other scientific fields. However, the specific examples we included are all related to the growth and nucleation kinetics of calcium carbonate and as we demonstrate in the review the fact that these inconsistencies exist is one of the main reasons for the observed variability in kinetic parameters, and the motivation to write this review paper. We have revised the paragraph slightly to remove some redundancy.

“The first source is inconsistent definitions. For example, the definition of saturation, which is the core of any theory on mineral precipitation kinetics, is not consistent and is often confused with a driving force (e.g. ln(S) or (√S-1 )). “

The authors must be confused with supersaturation and supersaturation index. Definition of supersaturation is always the same, but how to express it is varied (often termed as “supersaturation index” rather than supersaturation” depending on their own applications or research field by using square root S- 1, or log or ln.  

We agree with the reviewer that the definition of (super-)saturation, S, is in most cases the same, but the way in which it is expressed or used varies. We revised the paragraph to address this issue.

I found that the authors had the same dilemma shown in figures where mixed uses of supersaturation index (log x or square root S-1 ) in this paper. If the authors can present absolute “IAP” along with supersaturation as double x-axis, it would be much easier to compare figure by figure. 

We selected the x-axis in the various figures for a reason and used definitions consistently. Figure 1 we used IAP as suggested by the reviewer to compare the obtained growth kinetics for the different polymorphs. In Figures 2-4 we used √S_cal-1, √S_vat-1 and √S_acc-1 respectively, as these are the variables used to fit the polynomial trendline and obtain the kinetic growth rate parameters. In figures 5 and 6, S and (ln S)² were used using the reported values for S from literature. Figure 7 contains the recalculated values for S_cal, where the subscript ^“cal” refers to the polymorph calcite, like the subscript vat and acc in figures 8 and 9 refer to vaterite and acc. Similar to figures 2-4 the terms (ln S_cal)², (ln S_vat)², and (ln S_acc)² were used on the x-axis since these were the variables used to fit equation 15.

One such example leading to errors is the ambiguous usage of the mathematical operator ‘log’. It is not always clear when results are plotted on a logarithmic scale whether 'log10’ or ‘ln’ is meant (e.g. [3-5]). This kind of inconsistency can easily lead to incorrect results and make it difficult to check or reproduce the results. 

I was taught in high school that Log(x) means the base 10 logarithm, in this case no need to label 10, this is the mathematical promise, no confusion there. 

We know that log means log10, however, it was not always clear from literature whether log or ln was used.

2. Aragonite, one of CaCO3 polymorphs, is entirely omitted in discussions. It should be justified.

Aragonite was not included as this polymorph was not observed in the experimental observations which motivated this study. A small paragraph is added to the introduction to explain the context of this study:

“The study was performed based on observations during experimental studies on Microbially Induced Carbonate Precipitation (MICP) by urea hydrolysis, which showed that size, shape and type of the calcium carbonate crystals varied significantly depending on the hydrolysis rate and available surface area for crystal growth [6,7] Based on Scanning Electron Microscope (SEM) and X-ray Diffraction (XRD) analysis three different polymorphs were identified: calcite, vaterite and amorphous calcium carbonate (ACC). Other polymorphs (aragonite, ikaite and calcium carbonate monohydrate) were not included in the study, as these were not observed in the experiments and are considered uncommon in freshwater environments or at the relevant temperature range [8,9]”.  

3. ACC is not one of polymorphs of calcium carbonate. Polymorph by definition should be a different crystalline structure with the same composition. ACC is an amorphous phase and has undefined compositions. ACC is the phase that forms through an aggregation mechanism, rather than a classical growth mechanism. this has not been considered in the calculation.

Although ACC may not have a crystalline structure it is a (metastable) ‘form’ of calcium carbonate with a specific solubility product and ‘growth’ and nucleation kinetics, and can therefor be considered one of the polymorphs

4. How were the equations 11-13 deduced ? It is difficult to understand “the update” without detailed explanation of intermediate steps.It was only mentioned as “We have updated these equations from Kaschiev [6] to incorporate the shape factors of equations 5 and 6”  

We were considering to include the derivation of the equations in the supplementary information, but instead refer to Kashiev [10] for the derivation. 

5. table 3, the values for calcite are all indicated as N/A, why ?

The analysis shown in figure 7, 8 and 9 shows that the considered nucleation of calcite is in fact nucleation of vaterite or ACC.

6. Page 5 . 

“For calcite reported growth rate constants have a broad range of values and units. The reported values are 4.0 – 8.8 L mol-1 s-1 (gseed L-1)-1 [1,11], 1.43 10-10 – 1.08 10-9 mol m-2 s- 1 [12,13], 5.37 10-7 – 2.34 10-6 mol m-2 s-1 [14], 2.91 10-9 – 1.39 10-6 mol m-2 s-1 [3], 1.64 10-7 – 2.81 10-7 mol m-2 s-1 [15], 4.05 10-11 – 1.98 10-10 m s-1 [16], 5.0 10-10 m s-1 [17], 1.1 10-10 m s-1 [18], 8.56 10-8 – 2.65 10-7 mol m-2 s-1 kgw-1 [19], 2.21 10-8 – 7.48 10-4 mol m-2 s-1 [20], 6.48 104 L2 mol- 1 m-2 s-1 [21], 3.42 10-7 mol m-2 s-1 [22], 6.64 10-8 – 7.80 10-7 mol m-2 s-1 [23], 1.13 10-10 – 9.2 10- 10 m s-1 [2], 5.8 10-7 – 3.35 10-6 mol m-2 s-1 [24], and 8.08 10-26 m3 s-1 [25]. The growth order, g, varies between 0.91 and 4.71 in these publications. Additionally, several parameters for alternative growth models [26-28] are reported. “

These don’t provide any valid information, so would only occupy the valuable space of the paper.  it will of course be widely varied because the values must be varied depending on the solution conditions and precipitation methods. If the authors really want to add these data, it would be much helpful to make a table with solution conditions in SI or main paper. 

We have transferred all kinetic constants into a table, which in case the paper exceeds the length limits can be transferred to the supplementary information.

6. As a review article, the number of cited references is substantially low,especially given the author's vast amount of literature on this topic. 

We only included those references we considered relevant for the evaluation and recalculation of the kinetic parameters

This must be typo, * indicate Leon van Paassen as corresponding author, while the * Correspondence email is  lukebergwerff@gmail.com; 

The email address for the corresponding author is corrected

Reviewer 2 Report

This manuscript attempts to correlate all the publicly available data on calcium carbonate and determine important parameters such as growth rates, nucleation rates etc. in order to obtain universal constants for interfacial energies of the solids.

Overall, this paper is well written and with some minor modifications can be published. My suggestions are below:

In the methods section, I believe yi should be gamma for the unitless activity coefficient.

Given that the rate is always mass/time in eqn 3 can the units of k really be defined if the order g is unknown?

There appear to be two equations labelled 3, the second of these is an empirically derived equation, no? So is it appropriate to use this rather than one derived from fundamental principles?

Do the shape factors Af and Vf include the 4pi and 4/3pi factors required to convert in eqns 5 and 6?

Ostwald ripening is the dissolution of smaller particles at the expense of growing larger particles (but the particles are the same form). If the particles are transforming into some other polymorph, probably Ostwald's rule of stages is more appropriate.

The calcite growth rates should be presented in a table, writing them down in a paragraph is difficult to follow.

Please indicate in the caption or legend what the red lines refer to in Figure 1.

In figure 2, are there really two regimes or is it one regime with some error at low supersaturations?

Given that the vaterite interfacial energy has been quoted at between 15-94 mJ m-2, obtaining a value of 16 mJm-2 is hardly a different order of magnitude. Maybe the wording in this paragraph can be improved.

Overall, however, I can recommend publication once these points have been considered.

Author Response

This manuscript attempts to correlate all the publicly available data on calcium carbonate and determine important parameters such as growth rates, nucleation rates etc. in order to obtain universal constants for interfacial energies of the solids.

Overall, this paper is well written and with some minor modifications can be published. My suggestions are below:

In the methods section, I believe yi should be gamma for the unitless activity coefficient.

We are not sure where the error is, as we have used the symbol ‘gamma’ for γ_i

Given that the rate is always mass/time in eqn 3 can the units of k really be defined if the order g is unknown?

We believe this is correct, since  is unitless

There appear to be two equations labelled 3, the second of these is an empirically derived equation, no?

Indeed we made an error labeling the equations and revised all the labels accordingly

So is it appropriate to use this rather than one derived from fundamental principles?

Both equations 3 and 4 are derived from fundamental principles, but are different ways to express the growth rate equation. Equation (3) is in mol s^-1, whereas equation (4) is in m s^-1. The two equation are related through equation (5) 

Do the shape factors Af and Vf include the 4pi and 4/3pi factors required to convert in eqns 5 and 6?

The unitless shape factors Af and Vf depend on the crystal shape. Assuming spherical crystals (which is most common), Af is 4π and Vf is 4/3 π. We added that sentence to make it more explicit., , 

Ostwald ripening is the dissolution of smaller particles at the expense of growing larger particles (but the particles are the same form). If the particles are transforming into some other polymorph, probably Ostwald's rule of stages is more appropriate.

At the end of page 4 we mention the process of Ostwald ripening: “For a polymorphic crystal such as calcium carbonate, the balance of growth and dissolution leads to crystal mass eventually transforming a less stable polymorph into the most stable phase. This process is called Ostwald ripening and was placed outside the scope of this study”. However, we focused the paper on growth and nucleation kinetics only and left Ostwald ripening out of the scope.

The calcite growth rates should be presented in a table, writing them down in a paragraph is difficult to follow.

We have listed the reported growth rate constants for calcite and vaterite in a table

Please indicate in the caption or legend what the red lines refer to in Figure 1.

The red lines are the trend lines, which were fitted using the selected data for each growth mechanism or mineral type as shown in figures 2 to 4. We added this in the caption.

In figure 2, are there really two regimes or is it one regime with some error at low supersaturations?

We agree with the suggestion that describing the growth rate of calcite assuming only one growth rate constant and one kinetic order will result in a limited error. However, we follow here the observations by Teng et al and others who identified that there are different growth mechanisms for calcite with different kinetics.

Given that the vaterite interfacial energy has been quoted at between 15-94 mJ m-2, obtaining a value of 16 mJm-2 is hardly a different order of magnitude. Maybe the wording in this paragraph can be improved.

We agree that the specific surface energy of vaterite is indeed within the reported range, however the values for A_n differ several orders of magnitude from the reported values, which is the result of using the proposed method to calculate the saturation values as well as using nucleation rates instead of induction times.

Overall, however, I can recommend publication once these points have been considered.