Kinetics of Nucleation and Growth of Calcium Carbonate and Calcium Sulphate Crystals from Aqueous Solutions

Abstract

The dependence of the induction period on supersaturation during the nucleation of CaCO₃ and CaSO₄ ∙ 2H₂O is experimentally determined and the interfacial tension values are measured. Using the constant concentration method, the kinetics of crystal growth of both salts is studied, the dependences of the crystal growth rate on solution supersaturation are obtained, and the kinetic order of the growth reaction is determined. Possible mechanisms of nucleation and growth are considered. The kinetics of transformation of the solid phase during the entire process of reagent precipitation is studied. The evolutionary curves are compared with the calculation results for the previously proposed models. It is shown that the model dependences can be used to verify and adjust experimental data.

The previous study [1] was devoted to simulating the kinetics of reagent precipitation of poorly soluble calcium salts. This article presents the results of an experimental study of the kinetics of nucleation and growth of calcium carbonate and calcium sulfate dihydrate crystals and their comparison with the theoretical kinetic curves.

Numerous publications on calcium carbonates and sulfates can be found in the scientific literature, since they are extremely widespread in nature and are used on a large scale in industry [2–5]. This includes quite a rather large number of publications on the kinetics of nucleation and growth of their crystals from aqueous solutions. However, there are few among them in which both the stages and the kinetics of eliminating supersaturation under the same conditions during the entire process would be studied.

Reagent precipitation is characterized by great supersaturations and is more difficult to study compared to classical crystallization, since there are several competing stages with their own kinetics (mixing of reagent flows, chemical reaction, and nucleation) at the beginning of the process. They proceed simultaneously and quickly enough, so it is difficult to isolate the limiting stage. In addition, homogeneous (spontaneous) crystallization at high supersaturations is extremely sensitive to the processing conditions (up to the influence of microimpurities and uncontrolled random factors). Therefore, the nucleation process is difficult to control, and the obtained data are barely suitable for calculating a crystallization process that is carried out in another apparatus and under other conditions.

It should also be noted that the trajectory of the crystallization process can cross several zones with different crystal growth mechanisms (from a diffusion mechanism to a mononuclear one) with initial supersaturations S above 8–12 and, accordingly, different kinetic dependences of the growth rate in time.

As a result of reagent precipitation peculiarities, the scatter of the available experimental data on the nucleation and crystal growth rates and their energy characteristics is extremely wide even for a single salt.

In the published sources, the surface-tension energy for calcium carbonate varies over a wide range—namely, 19.5 [6], 37.3 [7], 64 [8], 86 [9], 90 [10], 108 [7], and 280 [11] mJ/m² for homogeneous crystallization and 11 [7], 31.1 [8], and 40.7 [12] mJ/m² for heterogeneous crystallization. The surface tension for calcium sulfate dihydrate obtained by different authors was 4 [13], 12 [14], 37 [15], 40 [16], 95 [17], and 117 [18] mJ/m² for homogeneous crystallization and 7 [19] and 14 [16] mJ/m² for heterogeneous crystallization. The values of the exponent g in the equation of the crystal growth rate vary from 1.29 to 3.11 [2, 20] for calcium carbonate and from 1.5 [21] to 3.4 [20, 22, 23] for calcium sulfate dehydrate.

The reasons for this spread can be associated with the difference in many factors that have an effect on the process, such as the design of the crystallizers and the hydrodynamic conditions in them; the composition of the solution and, accordingly, its ionic strength; the degree of solution supersaturation and, accordingly, the different crystal growth mechanism of the precipitated salt; polymorphic composition (calcite, vaterite, and aragonite for CaCO₃) and the geometric shape of the crystals themselves; and the presence of soluble and insoluble impurities.

Therefore, to verify the operability of the proposed models, it is necessary to carry out the following series of experiments in the same apparatus and on the same systems: (1) the kinetics of crystal nucleation and its dependence on the degree of solution supersaturation, (2) the kinetics of crystal growth on the nucleus of the same name and without it at various degrees of supersaturation, and (3) the kinetics of evolution of the solid phase and the corresponding removal of supersaturation during homogeneous and heterogeneous crystallization of CaCO₃ and CaSO₄ ∙ 2H₂O.

EXPERIMENTAL

Experiments on determining the induction period, crystallization rates, and supersaturation elimination kinetics were carried out on the model solutions with a given degree of supersaturation, which were prepared using reagents СaCl₂, Na₂CO₃, NaHCO₃, Na₂SO₄, and KCl of the reagent and analytical purity grades and their 0.1 and 0.2 M aqueous solutions. To crystallize calcium carbonate and calcium sulfate, the nuclei with the corresponding compositions were used as a seed material, which were specially prepared using the method of reagent precipitation.

For the experiments, we used a laboratory glass crystallization reactor with a capacity of up to 0.8 L equipped with a thermostating jacket. The setup layout is shown in Fig. 1.

Fig. 1.

Experimental setup for studying the crystallization process: (1) magnetic stirrer, (2) anchor for magnetic stirrer, (3) crystallization reactor equipped with jacket, (4) electrode of pH/ion meter, and (5) two-channel autotitrator.

A model solution with preset calcium carbonate supersaturation and ionic strength values was obtained by mixing working 0.1 M solutions of CaCl₂, Na₂CO₃, and NaHCO₃, as well as a weighed sample of dry KCl. For each component, concentrations were determined using the SequentiX WinIAP software package.

A model solution with a given degree of calcium sulfate supersaturation was obtained by mixing CaCl₂ and Na₂SO₄ solutions obtained by dissolving the required amount of the corresponding salt in distilled water. The concentrations of each component were calculated by the following formulas:

$${{K}_{{{\text{sp}}}}} = \left[ {{\text{C}}{{{\text{a}}}^{{2 + }}}} \right]{{{\gamma }}_{{{\text{C}}{{{\text{a}}}^{{2 + }}}}}}\left[ {{\text{SO}}_{4}^{{2 - }}} \right]{{{\gamma }}_{{{\text{SO}}_{4}^{{2 - }}}}},$$

(1)

where K_sp is the product of activities of ions of the dissolved substance and γ is the activity coefficient.

$$\log w{{{\gamma }}_{ \pm }} = - \frac{{Az_{i}^{2}\sqrt I }}{{1 - \sqrt I }},$$

(2)

where I is the ionic strength of the solution, z_i is the charge of the corresponding ion, and A is the temperature coefficient.

$$I = \frac{1}{2}\sum\limits_{i = 1}^k {{{c}_{i}}z_{i}^{2}} ,$$

(3)

$$\begin{gathered} I = \frac{1}{2}([{\text{C}}{{{\text{a}}}^{{2 + }}}]z_{{{\text{C}}{{{\text{a}}}^{{2 + }}}}}^{2} + [{\text{C}}{{{\text{l}}}^{ - }}]z_{{{\text{C}}{{{\text{l}}}^{ - }}}}^{2} \\ + \,\,[{\text{SO}}_{4}^{{2 - }}]z_{{{\text{SO}}_{4}^{{2 - }}}}^{2} + [{\text{N}}{{{\text{a}}}^{ + }}]z_{{{\text{N}}{{{\text{a}}}^{ + }}}}^{2}), \\ \end{gathered} $$

(4)

$$S = \sqrt {\frac{{{{K}_{{{\text{sp}}}}}}}{{K_{{{\text{sp}}}}^{0}}}} ,$$

(5)

where S is the value of solution supersaturation, $K_{{{\text{sp}}}}^{0}$ is the product of activities of ions of the dissolved substance in the saturated solution, and c_i is the concentration of the corresponding ion.

The $K_{{{\text{sp}}}}^{0}$ for calcium sulfate dihydrate was determined from the temperature dependence obtained in [24] as follows:

$$\begin{gathered} \lg K_{{{\text{sp}}}}^{0} = 390.9619 - 152.6246\log T \hfill \\ - \,\,\frac{{12545.62}}{T} + 0.0818493T. \hfill \\ \end{gathered} $$

(6)

The concentration of Ca²⁺ ions was determined using the standard techniques of complexometric titration by Trilon B and comparison with the results obtained with an EKSPERT-001 pH/ion meter. The change in the optical density of a solution, which indicates the end of the induction period, was measured on a KFK-2MP photocolorimeter. This period was determined visually from the beginning of solution turbidity due to the formation of crystals of sparingly soluble salts.

RESULTS AND DISCUSSION

Determining the Crystal Nucleation Rate

The experiments on determining induction period t_ind were carried out in the following fashion. Two solutions with a volume of 150 mL were prepared; the first was a solution of calcium chloride ions and the second was a solution of sodium carbonate or sodium sulfate ions. Both solutions were mixed in a crystallization reactor under constant stirring with a magnetic stirrer at a speed of 240–400 rpm. Changes in the pH and pCa indices were recorded in real time. To determine optical density D_opt of a solution, the samples were taken into a cuvette of the photocolorimeter at regular intervals. The end of the induction time was detected visually, or from рН = f(t), рCa = f(t), and D_opt = f(t) dependences (Fig. 2).

Fig. 2.

Time dependences of (1) pH and (2) the optical density of the solution for determining the induction period of crystallization of calcium carbonate.

In Fig. 2, the vertical line marks the end of the induction period at t_ind = 4 min determined visually, which indicates the formation of calcium carbonate nuclei and the transition to the active phase of crystal growth; the initial degree of solution supersaturation is 14. The induction time is inversely proportional to the nucleation rate; therefore, these are interchangeable values. In a similar way, D. Verdoes, S. Halevi, and others determined the nucleation rates in [7, 19].

To confirm the power-law dependence of the nucleation rate of particles, it is necessary to construct, using the results of experiments, the dependence of the induction period on the degree of solution supersaturation, t_ind = f(S). For calcium carbonate, the t_ind value was determined at different S values in the range from 10 to 25. For calcium sulfate dihydrate, t_ind was determined at supersaturation degrees in the range from 1.35 to 2.0. Figure 3 shows the results in the logarithmic coordinates, logt_ind = f(logS), which are well described by linear dependences 1 and 2 for CaCO₃ and CaSO₄ ∙ 2H₂O, respectively. Using the line slopes, we determined the values of the exponent in the dependences of the nucleation rates of calcium carbonate and calcium sulfate dihydrate crystals (t_ind = 8.9 × 10⁷S^–4.8 for CaCO₃ and t_ind = 3.2 × 10⁴S^–9.0 for CaSO₄ ∙ 2H₂O).

Fig. 3.

Dependence of the induction periods of (1) calcium carbonate and (2) calcium sulfate dihydrate on the degree of supersaturation of the solution.

Since the induction period is inversely proportional to the nucleation rate, i.e., t_ind ~ 1/v_n, and the removal of supersaturation in a solution is determined by a change in the concentration of calcium, f(С – С₀), the results confirm the following power-law dependences of the nucleation rate of particles:

$${{{v}}_{n}} = {{k}_{n}}{{\left( {C - {{C}_{0}}} \right)}^{n}}$$

(7)

or, in the general case,

$${{{v}}_{n}} = {{k}_{n}}{{S}^{n}},$$

(8)

where С is the mass concentration of a salt in a solution, С₀ is the mass concentration of a salt in the saturated solution, n is the exponent in the equation of nucleation rate, and k_n is the rate constant of nucleation in a solution.

In Fig. 4, the dependence of the induction period on the degree of supersaturation is expressed by the following equation:

$$\log {{t}_{{{\text{ind}}}}} = a + \frac{b}{{{{{(\log S)}}^{2}}}},$$

(9)

Fig. 4.

Dependences of the induction period duration on the degree of solution supersaturation for (1) homogeneous and (2) heterogeneous crystallizations of CaCO₃ and (3) homogeneous and (4) heterogeneous crystallizations of CaSO₄ ∙ 2H₂O.

in which a is an empirical constant and b can be defined as

$$b = \frac{{{\beta }{{{\sigma }}^{3}}V_{m}^{2}{{N}_{{\text{A}}}}f\left( \phi \right)}}{{{{{\left( {2.3R} \right)}}^{3}}{{{v}}^{2}}{{T}^{3}}}},$$

(10)

where β is the geometric shape factor, σ is the interfacial tension, N_A is the Avogadro number (6.022 × 10²³ mol^–1), f(ϕ) is the correction factor, R is the universal gas constant (8.31 J/(mol K)), V_m is the molar volume of a solid body, v is the number of ions that comprise the solid body, and T is the temperature.

The experimental data on the induction periods of calcium carbonate and calcium sulfate dihydrate are described well by linear dependences in the logarithmic coordinates, logt_ind = f(logS)^–2. The slope allows one to determine the magnitude of interfacial tension σ for homogeneous and heterogeneous nucleation, which corresponds to the diffusion model of crystal growth. For homogeneous crystallization of CaCO₃, σ = 70.1 mJ/m² with a shape factor of β = 32 for a cube (calcite) and σ = 87 mJ/m² with a shape factor of β = 16π/3 for a sphere (vaterite) at T = 293 K; the molar volume of solid calcium carbonate is V_m = 36.9 cm³/mol and f(ϕ) = 1. For heterogeneous crystallization, σ = 56.2 mJ/m² for the cubic shape and σ = 69.7 mJ/m² for the spherical shape. The interfacial tension values for homogeneous crystallization do not contradict the data from the other published sources [8–10, 12].

For CaSO₄ ∙ 2H₂O at T = 293 K with β = 32 (cubic) and β = 50 (parallelepiped), V_m = 74.69 cm³/mol , and for S in the range from 1.35 to 2.0, the calculated interfacial tension value for homogeneous crystallization is 9.2 mJ/m² for the cubic shape and 8 mJ/m² for the parallelepiped shape. In the resulting suspension, needle-shaped crystals are also present, but their mass fraction is quite small. For heterogeneous crystallization, the corresponding σ values are 6.9 and 5.9 mJ/m², respectively. Such low values of interfacial tension indicate that the crystal growth mechanism is different from a diffusion mechanism. At relatively small supersaturations, the nucleation of CaSO₄ ∙ 2H₂O is more likely accompanied by polynucleation growth. In this case, the equation for calculating the σ value takes the following form [25]:

$$\begin{gathered} {{t}_{{{\text{ind}}}}} = {{\left( {\frac{3}{{2{\pi }}}} \right)}^{{\frac{1}{4}}}}{{\left( {\frac{{V_{{\text{m}}}^{{\frac{5}{3}}}}}{{N_{{\text{A}}}^{{\frac{8}{3}}}{{D}^{4}}{{C}_{0}}}}} \right)}^{{\frac{1}{4}}}}{{\left( {\frac{S}{{{{{\left( {S - 1} \right)}}^{2}}}}} \right)}^{{\frac{1}{4}}}} \\ \times \,\,{\text{exp}}\left[ {\frac{{{\beta }{{{\sigma }}^{3}}V_{{\text{m}}}^{2}{{N}_{{\text{A}}}}}}{{4{{{\left( {RT} \right)}}^{3}}{{{v}}^{2}}{{{(\ln S)}}^{2}}}} + \frac{{{\beta }{\kern 1pt} {\text{'}}{{{\sigma }}^{2}}V_{{\text{m}}}^{{\frac{4}{3}}}{{N}_{{\text{A}}}}^{{\frac{2}{3}}}}}{{4{{{\left( {RT} \right)}}^{2}}{v}\ln S}}} \right], \\ \end{gathered} $$

(11)

where D is the coefficient of diffusion of the component in a diluted solution and β' is the geometric factor (β' = 4).

For the homogeneous crystallization of CaSO₄ ∙ 2H₂O with D = 5 × 10^–9 m²/s and β = 32, the interfacial tension is in the range from 21.1 to 29 mJ/m². For heterogeneous crystallization, the interfacial tension ranges from 17.5 to 19.9 mJ/m². With a shape factor of β = 50, the σ values were in the ranges from 18.3 to 25.1 mJ/m² and from 15.2 to 17.3 mJ/m² for homogeneous and heterogeneous crystallizations, respectively. The values obtained for homogeneous crystallization are in the range from 15 to 30 mJ/m². These data do not contradict the interfacial tension values obtained by most authors [5, 14–16].

Determining the Crystal Growth Rate

The experiments on determining the crystallization rates were carried out using the constant concentration method so that the driving force of the process remained unchanged throughout the entire experiment. To implement this, 0.1 M solutions of CaCl₂ and Na₂CO₃ and 0.2 M solutions of CaCl₂ and Na₂SO₄ were supplied into the crystallization reactor by means of a two-channel autotitrator for the crystallization of calcium carbonate and calcium sulfate, respectively. The cation-to-anion ratio of sparingly soluble salts in the solution was maintained at a one-to-one level. The concentration of calcium ions in the solution was recorded using a Ca²⁺ ionmeter. The crystal growth rate with constant supersaturation was determined from the volume of a calcium chloride solution added over a certain period of time.

Kinetic experiments with calcium carbonate were performed for heterogeneous crystallization. To accomplish this, 0.2 g of CaCO₃ seed particles serving as crystallization centers were added to the model solution with a given degree of supersaturation after 30 s. In such a way, we tried to exclude the effect of primary nucleation and, hence, to avoid an increase in the number of particles capable of growing. The ionic strength increases as the precipitate forms and the titration solutions are constantly added to the mother liquor, which has a direct effect on the activity of ions and the degree of supersaturation. To prevent this effect, the ionic strength in the model solution was initially adjusted to 0.1 M by dissolving KCl. The amount of chloride and sodium ions that accumulated from titration solutions throughout the experiment is much less than the amount of potassium and chloride ions present in a solution at the initial point in time; therefore, the ionic strength over the entire period of time changes insignificantly.

A seed and a constant ionic strength of the solution were also used to study the crystallization process in [7], and the authors obtained linear crystal growth rates. However, the amount of titrant added over time increases nonlinearly and dV_t/dt = f(t) ≠ const, since the volumetric growth rate is constantly increasing in proportion to the surface area of growing crystals. In [13, 20], it was suggested to use the value of adjusted titrant volume V_tc, which corresponds to a constant effective area of the crystal growth surface. For this purpose, Eq. (12) was used; spatial crystal growth orientation d allows one to adjust the V_tc = f(t) dependence so that equality dV_tс/dt = const is satisfied. The adjusted volume allows one to determine the linear dependence of the volumetric crystal growth rate. The equation for determining the V_tс value with a known real volume of titrant at time point t is represented by the following expression:

$${{V}_{{{\text{tc}}}}} = \frac{{{{{\left( {1 + \frac{{M{{C}_{{\text{t}}}}{{V}_{{\text{t}}}}}}{{{{m}_{0}}}}} \right)}}^{{\left( {1 - d} \right)}}} - 1}}{{\frac{{\left( {1 - d} \right)M{{C}_{{\text{t}}}}}}{{{{m}_{0}}}}}}\,\,\,\,(d \ne 1),$$

(12)

where M is the molecular weight of the growing crystals, C_t is the titrant concentration, V_t is the titrant volume, and m₀ is the initial weight of the seed particles.

Line 1 in Fig. 5 shows the time dependence of the titrant volume, which is obtained experimentally with a solution supersaturation of S = 3.16 and a seed particle weight of 0.2 g. Lines 2–4 show the dependences of the adjusted volume for d values of 0.1, 0.4, and 2/3, respectively. Since a mixture of two solutions, namely, solutions of CaCl₂ and Na₂CO₃ with concentrations of 0.1 M, serves as a titrant, concentration C_t should be taken equal to 0.05 M when calculating the volume of the titrating solution from the concentration of Ca²⁺ ions. To determine the dependence of the growth rate on supersaturation, the concept of adjusted volume should be used. For our experimental conditions, the curvatures of the V_tc = f(t) dependences vary insignificantly and a strong deviation from the real titrant volumes is observed in the region, in which this dependence becomes linear. This is associated with use of a large amount of seed and a substantially smaller change in the total weight of crystals during crystallization in comparison with [13, 20]. Therefore, experimentally obtained V_t values were used in calculating the growth rates.

Fig. 5.

Changes in the titrant volume over time: (1) experimental values, (2) corrected values of the volume at p = 0.1, (3) at p = 0.4, and (4) at p = 2/3.

We obtained the volumetric growth rates of calcium carbonate for supersaturation values of 2.24, 3.16, 3.87, 5, and 7. Figure 6 shows the linearized dependence of titrant feed rate v_t on the driving force in a solution, which is expressed as –logv_t = f[log(S – 1)]. The power-law dependence of the growth rate of calcium carbonate has the following form:

$${{{v}}_{{\text{g}}}} = {{k}_{{\text{g}}}}{{\left( {C - {{C}_{0}}} \right)}^{g}},$$

(13)

where k_g is the rate constant of crystal growth and g = 2.3 is the exponent in the equation of the CaCO₃ growth rate.

Fig. 6.

Dependence of the rates of growth of (1) CaCO₃ and (2) CaSO₄ ∙ 2H₂O crystals on the degree of solution supersaturation.

The experiments on determining the growth rates of calcium sulfate dihydrate crystals are different from the analogous experiments for calcium carbonate. First, the process of homogeneous crystallization without seed particles was considered. Second, the ionic strength in the solution was larger than 0.1 M at the start because of the larger solubility of calcium sulfate; therefore, it is difficult to create the same conditions as those used in the case of calcium carbonate. The feed rate of the titration solution was recorded in the time interval between the end of the induction period and the beginning of uncontrolled crystal growth. The volumetric crystal growth rates were obtained for supersaturation values of 1.3, 1.55, 1.8, and 2.3. Figure 6 shows the experimental results expressed by a linear dependence, from which the value of the exponent in the equation of the crystal growth rate of CaSO₄ ∙ 2H₂O can be determined. In such a way, an exponent value of g = 1.6 was determined for a 0.2 M CaCl₂ titration solution. The exponent values for calcium carbonate and calcium sulfate dihydrate correspond to the polynucleation model with spiral and/or two-dimensional layer-by-layer crystal growth [20, 22, 26–29].

Kinetics of the Crystallization Process

In the previous study [1], the following generalized model describing the crystallization of sparingly soluble salts from their aqueous solutions is given:

$$\frac{{{\text{d}}P}}{{{\text{d}}t}} = {{k}_{{\text{n}}}}{{\left( {{{P}_{{\text{m}}}} - P} \right)}^{n}} + {{k}_{{\text{g}}}}{{\left( {{{P}_{{\text{m}}}} - P} \right)}^{g}}{{P}^{d}},$$

(14)

where P is the mass concentration of a crystalline substance in a suspension and P_m is the maximum mass concentration of a crystalline substance in a suspension.

To verify the adequacy of the model, experiments based on the removal of supersaturation from a solution were performed. The experiments were carried out in a crystallization reactor (Fig. 1). Solution samples were periodically taken and the content of calcium ions was determined in them by complexometric titration with preliminary filtration of the test sample from salt crystals.

Points 1 in Fig. 7 show the experimental data on changes in the fraction of crystallized calcium carbonate over time. The initial parameters of the solution are as follows: the degree of supersaturation is 7, the ionic strength is 0.1 mol/L (adjusted by dissolving KCl), the temperature is 21°C, pH is 9.5, the initial concentration of calcium is 1.14 × 10^–3 mol/L, and the concentration ratio of ions is [Ca²⁺]/$\left[ {{\text{CO}}_{3}^{{2 - }}} \right]$ = 1. The process of crystallization of calcium sulfate dihydrate is shown by points 2. The initial solution parameters are as follows: the degree of supersaturation is 1.6, the concentration of calcium is 6.1 × 10^–2 mol/L, the ionic strength is 3.66 × 10^–1 mol/L, the temperature is 25°C, and the concentration ratio of ions is [Ca²⁺]/$\left[ {{\text{SO}}_{4}^{{2 - }}} \right]$ = 1. The dependences indicated by points 3, 4, and 5 are taken from the studies published by S. Halevy [19], T. Rabizadeh [30], and P. A. Kekin [31], respectively.

Fig. 7.

Changes in the degree of crystallization over time during homogeneous crystallization: (1, 2) original data for CaCO₃ and CaSO₄ ∙ 2H₂O, respectively; (3, 4) data for CaSO₄ ∙ 2H₂O published in [19] and [30], respectively; and (5) published data for CaCO₃ [31].

Differential equation (14) can be solved analytically in some cases. For example, if one takes n = 1, g = 1, and d = 1, then Eq. (14) is simplified and its solution takes the following form:

$$\begin{gathered} P = {{P}_{{\text{m}}}} - \left( {{{P}_{{\text{m}}}} + \frac{{{{k}_{{\text{n}}}}}}{{{{k}_{{\text{g}}}}}}} \right) \\ \times \,\,\frac{{{{P}_{{\text{m}}}}}}{{{{P}_{{\text{m}}}} + \frac{{{{k}_{{\text{n}}}}}}{{{{k}_{{\text{g}}}}}}{\text{exp}}\left[ {{{k}_{{\text{g}}}}t\left( {{{P}_{{\text{m}}}} + \frac{{{{k}_{{\text{n}}}}}}{{{{k}_{{\text{g}}}}}}} \right)} \right]}}. \\ \end{gathered} $$

(15)

The degree of crystallization, η, is expressed as follows:

$${\eta } = \frac{P}{{{{P}_{{\text{m}}}}}}.$$

(16)

The lines in Fig. 7 show theoretical S-shaped dependences of the fraction of crystallized calcium carbonate and calcium sulfate dehydrate, which are obtained using Eqs. (15) and (16) under the condition that k_g k_n. The k_g and k_n coefficients of calcium carbonate in Eq. (15) are 70 and 5 × 10^–2 and 18 and 2 × 10^–3 for our experimental data and the data published in [31], respectively. For the time dependence of the degree of crystallization of calcium sulfate dihydrate, the k_g and k_n values are 1.4 and 9 × 10^–3, 1.9 and 1.9 × 10^–5, and 3 × 10^–3 and 1 × 10^–3 for our experimental data and the data published in [19] and [30], respectively. Since all experiments were carried out under different conditions and concentrations, a significant discrepancy is observed in the kinetic coefficients, taking into account the rates of nucleation and crystal growth. As can be seen from comparison of the experimental data on the degree of crystallization of CaSO₄ ∙ 2H₂O given in Fig. 7, the nucleation rate of crystals was the slowest in [19] and the fastest in our experiments, which is reflected, in turn, in the values of the k_n constants in the theoretically obtained dependences.

Experimental data on the crystallization process are compared with theoretically calculated dependences in Fig. 8. Lines 1 and 2 are constructed by calculation on the basis of experimentally obtained data indicated by dots. They take into account dependence (13) for the crystal growth rate, the duration of the crystallization process, and the initial concentration of calcium ions in the solution. The calculation principle is as follows.

Fig. 8.

Changes in the degree of crystallization of CaCO₃ over time: points show the experimental data; (1, 2) are theoretical lines that take into account experimentally obtained crystallization rate v_t with exponents of (1) g = 2.3 and (2) g = 2.1.

(1) We find the weight of the formed nuclei in a solution.

(2) We subtract the amount of calcium required for nucleation during the induction period from the initial amount of calcium. If this change is very small, then it can be neglected in the calculation.

(3) We divide the time of the entire crystallization process into equal time intervals.

(4) Knowing the degree of solution supersaturation at the beginning and at the end of each considered time interval, we calculate the crystal growth rate by Eq. (13).

(5) Taking into account the crystal growth rate, we find the amount of calcium required for crystal growth in a given time interval. For this purpose, we take the mean rate between the beginning and the end of the considered time interval.

(6) We subtract the amount of calcium consumed in crystal growth from the remaining amount of calcium in the solution at the beginning of the considered time interval.

(7) We find the degree of crystallization in the current period of time.

(8) After that, we repeat the calculation procedure, considering the next time interval.

The nucleation and growth rates included in the kinetic equation of crystallization are obtained under different conditions. The nucleation proceeds under homogeneous crystallization, and the crystal growth proceeds under heterogeneous crystallization. Line 1 takes into account the dependence of the rate of crystallization of CaCO₃ on the degree of supersaturation with an exponent of g = 2.3. This value was obtained in experiments with a constant degree of supersaturation and sufficiently large seed particles with a rough surface on which there are many active sites (dislocations) for the formation of nuclei. Such a case is characterized by polynucleation and/or spiral growth models [7, 20, 26, 28].

Constructed line 1 is positioned significantly higher than the experimental points. In experiments with the removal of supersaturation, the nuclei are much smaller than the seed particles and have a smoother and more structured surface. Therefore, the g value may be different and must be adjusted. The theoretical line obtained using the adjusted growth rate with g = 2.1 (line 2) satisfactorily describes the crystallization process and the removal of supersaturation in a solution.

CONCLUSIONS

Experimental studies on the kinetics of nucleation and crystal growth of sparingly soluble calcium salts during their precipitation from aqueous solutions are analyzed. A very large scatter of published data on the energy and kinetic characteristics of the precipitated salts is shown. The reasons for these inconsistencies are considered. Many of them are explained by the peculiarities of reagent precipitation and are associated with a high initial degree of supersaturation and direct mixing of reagents in the crystallization zone. Supersaturation has a crucial effect not only on the nucleation and crystal growth kinetics, but also on the mechanism of these processes (the presence of impurities also has a strong effect). In the process of eliminating supersaturation and transforming the solid phase, the trajectory of the process can intersect several zones with different crystallization mechanisms and crystal shapes, as well as, consequently, different kinetic characteristics.

When salts with low solubility are precipitated under mild stirring, the crystallization process can begin in the region controlled by bulk diffusion with S > 10 and then move into the kinetic region controlled by the surface reaction. A parabolic kinetic law (the theoretical order of reaction is g = 2, and the experimentally determined order of reaction is close to 2) with dislocation (spiral) and/or polynucleation growth (g > 2) often operates in the latter region. At the last stage of transformation of the solid phase, the layered two-dimensional growth with g < 2 and so on up to the formation of a mononuclear layer upon approaching to S = 1 is possible with a decrease in the degree of supersaturation (or in the presence of impurities).

In this study, the surface energy and the orders of reactions in the equations for the nucleation and crystal growth rates were determined for the precipitated salts from the experimental data. The values fall into the range of the most reliable values and are consistent with the data obtained by other researchers. Using the constant concentration method, we have obtained power-law dependences of the growth rates of calcium carbonate and calcium sulfate dihydrate crystals with exponents g = 2.3 and 1.6, respectively. Such values are typical of combined growth mechanisms. Spiral growth with polynucleation is the most probable mechanism for CaCO₃, and spiral growth in combination with layered growth is the most probable mechanism for CaSO₄ ∙ 2H₂O.

Using the previously obtained model dependence, the experimental g value for calcium carbonate is corrected. Calculation results obtained with use of the new value g = 2.1 satisfactorily describe the kinetics of the entire crystallization process. The model S-shaped dependences can be used to verify, approximate, and adjust experimental data on the kinetics of reagent precipitation. Comparisons of our results with the experimental data published by different authors show good agreement and confirm the efficiency of the models. If the kinetic characteristics of the precipitated compound are available, then the models make it possible to predict the entire crystallization process and, in particular, the time of crystallization for a given value of initial supersaturation.

NOTATION

A	temperature coefficient
C	mass concentration of salt in solution, kg/m3
C 0	mass concentration of salt in saturated solution, kg/m3
C t	titrant concentration, mol/L
c i	molar concentration of corresponding ions, mol/L
D	diffusion coefficient, m2/s
D opt	optical density of solution
d	exponent characterizing spatial orientation of crystal growth
f(ϕ)	correction factor
g	exponent in equation of crystal growth rate (13)
I	ionic strength of solution, mol/L
K sp	product of activities of dissolved substance ions
$K_{{{\text{sp}}}}^{0}$	product of activities of ions in saturated solution
k n	rate constant of nucleation in solution
k g	rate constant of crystal growth
M	molecular weight of growing crystals, kg/mol
m 0	initial mass of seed particles, kg
N A	Avogadro number, mol–1
n	exponent in equation of nucleation rate (7)
P	mass concentration of crystalline substance in suspension, kg/m3
P m	maximum mass concentration of crystalline substance in suspension, kg/m3
R	universal gas constant, J/(mol K)
S	degree of supersaturation of solution
T	temperature, K
t	time of crystallization process, s
t ind	induction period of crystal nucleation, s
V m	molar volume of solid body, m3/mol
V t	titrant volume, m3
V tc	corrected titrant volume, m3
v	number of ions that comprise a solid body
v n	nucleation rate, s–1
v g	crystal growth rate, kg/s
v t	feed rate of titration solution, m3/s
z i	charge of corresponding ion
β, β'	geometric shape factor
γ	activity coefficient
η	degree of crystallization
σ	interfacial tension, mJ/m2

SUBSCRIPTS AND SUPERSCRIPTS

0	value in saturated solution
i	sequential number of ions of dissolved substances