Investigation of Growth Mechanisms of Sodium Chlorate Crystals from

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Investigation of Growth Mechanisms of Sodium Chlorate Crystals from Aqueous Solutions Biljana Z. Radiša, Mićo M. Mitrović,* Branislava M. Misailović, and Andrijana A. Ž ekić Faculty of Physics, University of Belgrade, Studentski trg 12, 11000 Belgrade, Serbia ABSTRACT: Difficulties in determining the crystal growth mechanism from growth rate versus solution supersaturation dependence are discussed. Obtained results indicate that in the supersaturation range of 0.66−1.56%, sodium chlorate crystals grow in accordance with the spiral growth model. It is shown that in the supersaturation range of 0.44−1.32%, the growth mechanism depends on crystal growth history. Namely, the exponent n in the power law R = Kσn depends on the manner of supersaturation changes (varied from 1.3 to 1.9%). This indicates that the overlapping of diffusion fields of neighboring steps depends on growth history.

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INTRODUCTION Depending on supersaturation and temperature, there is a variety of crystal growth mechanisms, which compete and lead to different growth regimes. There exist many difficulties for establishing the mechanism responsible for the substance growth under specific conditions. Analysis of crystal growth rate versus solution supersaturation dependence (R, σ) is widely used to determine this mechanism. Numerous authors used different theoretical equations for fitting (R, σ) dependence. Bennema1 showed that the growth rate of potassium alum crystals, when the relative supersaturation of the solution lies between 3 × 10−5 and 10−2, is proportional to supersaturation and that for supersaturation values greater than 10−2 some points lie above the linear curve. The results obtained for potash alum crystals, single and grown in fluidized bed crystallizer, were fitted to Burton−Cabrera− Frank (BCF), polynuclear, and simple power law equations.2 The data of Botsaris and Denk3 obtained for aluminum potassium sulfate crystals in supersaturation range of 2−16% and data of Alexandru and Antohe,4 obtained for KDP crystals in supersaturation range of 2−10%, were all fitted to the BCF model. Koutsopolous5 found that the dependence of growth rate of hydroxiapatite on supersaturation (2.05−4.58%) is linear. Kim and Myerson6 used a simple semiempirical power law equation to fit experimental data obtained by the other authors. Some authors analyzed (R, σ) dependence for sodium chlorate crystals, grown under different conditions. Bennema1 showed that a nonlinear (nearly parabolic) dependence occurred in the supersaturation range (0.3−5) × 10−2%. However, a linear dependence is found for the range (5−15) × 10−2%. He showed that a theoretical curve resulting from the BCF surface diffusion model7,8 fits measured (R, σ) points in a satisfactory way. Hosoya and Kitamura9 concluded that in the © XXXX American Chemical Society

supersaturation range of 3−5%, the mentioned dependence could be most adequately explained in terms of a twodimensional nucleation mechanism. Ristić et al.10 showed that for sodium chlorate crystals in supersaturation range of 0.1− 1.0% (R, σ) dependence might be parabolic or linear. Surender et al.11 showed that at supersaturations of 3−8% a parabolic trend appeared, suggesting a two-dimensional mechanism. Crystal growth rate dispersion, which occurs for many substances,1,11−25 complicates analysis of (R, σ) dependence. Dispersion of growth rates in one direction (sum of growth rates of opposite faces) is wider than dispersion of face growth rates. Nonequivalent crystal faces can grow by different mechanisms, which additionally complicates analysis. Nevertheless, sometimes conclusions about growth mechanism are drawn from growth rate in some direction, or even from mass changes,1 which depend on growth of more faces. In order to define better (R, σ) dependence, we measured {100} face growth rate of sodium chlorate at different supersaturations, just like the earlier face-by-face growth analysis for paraffin26 and sucrose crystals27,28 was performed. During (R, σ) dependence investigations, authors used different definitions of solution concentration, which resulted in different relative supersaturations. Many papers lack descriptions of the used definition, as in Surender et al.11 Therefore, ranges of used supersaturations are sometimes incomparable. Only two definitions of concentration will result in different relative supersaturations. Our analysis is based on both of them. Received: May 26, 2016 Revised: September 8, 2016 Accepted: September 12, 2016

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DOI: 10.1021/acs.iecr.6b02021 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research

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THEORY Two-Dimensional Nucleation with Surface Diffusion and Two-Dimensional Nucleation with Direct Integration of Growth Units. For the polynuclear model, it can be written R = hACσ 1/2 exp( −ΔGp*/kT )

screw dislocations arranged along the line of length L at equal interdistances, the dependence of crystal growth on supersaturation can be expressed by the formula 19γ Ω σ2 1 2Lσ = + R kT mhΩN0βl* mhΩN0βl*

(1)

where mh is a multiple of the height h of elementary steps (m ≥ 1). Linear (R, σ) Dependence. Beside mentioned equations, our data are fitted by empirical linear dependence

while for the multi-nuclear model, dependence of R on σ can be expressed as R = hBσ 5/6 exp( −ΔGp*/3kT )

(2)

⎛σ ⎞ σ2 tanh⎜ c ⎟ ⎝σ ⎠ σc

Parameters are defined as C* =

where a and b are empirical parameters. Other parameters, defining eqs 1−8, are listed in Notation. Goodness of the Fit. The χ2 (chi-square) test was used in order to test goodness of fit of the proposed equations.37 The existing discrepancy between observed and theoretical values is supplied by statistic χ2: n 2

χ =

and σc =

σ ≫ σc, the face growth rate is given as

9.5γ Ω . kTλs

For

while for σ ≪ σc, it will be

R i ,est

(9)

C1 =

mass of substance mass of solution

(10)

C2 =

mass of substance mass of solvent

(11)

or

σ2 R = C* σc

(5) 3,32−34

Regarding direct integration model and BCF model, (R, σ) dependence is also parabolic: R = Cσ2, where C is the kT constant defined as C = 19γ hc0βl and has the same form as eq

The value of relative supersaturation depends on the concentration definition. Differences between relative supersaturation defined in these two ways are significant, as can be seen below. Using different units for mass (g or kg) or volume (100 mL or 1 L) has no influence in supersaturation calculation. Authors of experimental investigations use both definitions of concentration, but the information about which equation was used is sometimes lacking. Theoretical analyses never consider this fact. Because of that, many results of different authors are not comparable, as occurred in the work of Aquiliano et al.38 Sometimes it is not emphasized if the relative supersaturation is calculated in percentages or not, which presents additional problems. In this paper, concentration is expressed only as σ1 on all graphs. To compare our and other authors’ results, the analysis is performed for both supersaturation definitions.

5.

Simple Power Law. Frequently,35 crystal growth data is expressed by a simple power law such as R = Kσ n

(R i − R i ,est)2

where R̅ is the arithmetic mean of the growth rates; Ri the measured growth rate, corresponding to the crystal size li; Rest the value of R corresponding to the crystal size li; n the number of fitting classes; and k the number of measured growth rates. The function with a smaller value of χ2 describes the observed dependency better than the function with a higher value of χ2. Supersaturation. The relative solution supersaturation is defined as σ = (C − C0)/C0, where C and C0 are the actual and saturated solution concentrations, respectively. The concentration can be defined in different ways. In the crystal growth investigations, concentration is preferably defined as

(4)

R = C*σ

∑ i=1

(3) βΛΩN0 βl* b

(8)

R = a + bσ

The free energy change, corresponding to the formation of a stable circular nucleus of critical radius ρc on the perfect surface, is defined as ΔG*p = πhγΩ/kTσ. For two-dimensional nucleation with surface diffusion, C is defined as C1 = (2/π)n21Ds(Ω/h)1/2 and B is the constant defined as B1 = (2λ/b)2/3β1*2/3C11/3. For two-dimensional nucleation with direct integration of growth units, C is defined as C2 = πhn1c0βl and B is defined as B2 = (Ωc0)2/3C1/3 2 . Concentration of adsorbed growth units on the surface is defined as n1 = hc0 exp(−Ead/kT), kinetic coefficient for steps as β1 = bν exp(−Wa/kT), while the surface diffusion coefficient as Ds ≈ (8kT/πm1)1/2. Both models, polynuclear and multi-nuclear, based on the surface diffusion and direct integration of the growth units, predict an exponential dependence of crystal growth rate on solution supersaturation at low σ, and on σ1/2 or σ5/6 at high σ. Spiral growth: BCF. According to the surface diffusion model,7,29−31 the face growth rate is given as R = C*

(7)

(6)

For n = 1 and n = 2, this equation becomes the same as eqs 4 and 5, corresponding to high and low supersaturations, respectively. Values of n depend on a degree of overlapping neighboring step diffusion fields. For small supersaturations (σ ≪ σc), surface diffusion path is much smaller than terrace width (λs ≪ λ), diffusion fields overlapping, and n = 1; while for high supersaturations (σ ≫ σc), surface diffusion path is much higher than terrace width (λs ≫ λ), diffusion fields are independent, and n = 2. Higher values of n correspond to smaller overlapping of diffusion fields of neighboring steps.36 Growth by Group Cooperating Screw Dislocations: Chernov’s Model.32 When a crystal grows by a group of

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EXPERIMENTAL PROCEDURE The object of all experiments was to investigate the correlation between the growth rate and the solution supersaturation of B


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At the first growth temperature, after its decrease to 28 or 29.5 °C, crystals refaceted for about half an hour. Because of the stabilization of growth conditions, at every growth temperature, crystals were kept for about 15 min before measurements. Also, at each growth temperature, crystals grew within time interval enough to provide growth rate measurement error less than 3%. Depending on the growth temperature, those time intervals were 1−3 h.

small sodium chlorate crystals. Analar grade of this system (99% purity) was used. Crystals grew from aqueous solutions, prepared by equilibrating an excess of crystals with distilled water for 3 days at saturation temperature. The concentrations were calculated on the basis of the following empirical formulas (formula for C1 is from ref 39, and C2 is from ref 40): C1 = 0.226T (°C) + 44.38(g NaClO3 /100 g solution)

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EXPERIMENTAL RESULTS To test dependence of growth rate dispersion on growth temperature, two introductory type 1 experiments were performed: crystals grew at 28 and 30 °C, in the first experiment and in the second experiment, respectively. For both experiments, solutions were saturated at 29.5 and 31.5 °C, respectively, which provides the same solution supercooling of 1.5 °C in both experiments. Significant differences between growth rate distributions do not exist. Mean growth rates and the most probable growth rates are (8.2 ± 0.2) nm/s and (8.1 ± 0.2) nm/s, respectively, in the first experiment and (8.1 ± 0.2) nm/s and (8.0 ± 0.2) nm/s, respectively, in the second experiment. Both growth rates were determined excluding zero growth rates. The first histogram on Figure 1 shows growth rate dispersion in the first experiment, whereas a similar histogram for the second experiment is not presented. Type 1 Experiments. Growth rates of small sodium chlorate crystals in the ⟨100⟩ direction prior to dissolution were measured. After 4 h of growth, crystals reached stabilized growth rates.13,42,43 Wide dispersion of these growth rates similar to results presented earlier was noticed.42,43 The {100} faces growth rates of sodium chlorate crystals after dissolution and refaceting were measured at constant growth temperature, T, in solutions saturated at temperatures TS, corresponding to relative supersaturations (σ1, σ2), Table 1. At the same experimental conditions, growth rate dispersions occurred. Histograms representing these dispersions are presented in Figure 1. These dispersions are described by a simple normal distribution, which is also included in Figure 1. Zero growth rates, whose existence is completely unclear,44 are excluded from the fitting procedure. Several high growth rates at the end of dispersions, which probably pertain to higher activity of dominant dislocation group, are excluded from the fitting procedure, too.43 Number of measured growth rates, N, the most probable growth rate (Rmax) for different supersaturations and standard deviation, s, of the most probable growth rate determination, are presented in Table 1. A scatter diagram of crystal growth rate versus supersaturation is shown in Figure 2a. Data are fitted with functions which describe the different crystal growth mechanisms listed above. Zero growth rates are excluded from the fitting procedure. Figure 2b depicts the most probable growth rate (Rmax) at constant supersaturation versus supersaturation dependence. Goodness of fit is tested with Chi-square test. Values of χ2 for used equations are presented in Table 2 for all observed crystals (a) and for Rmax versus σ dependence (b). Given in parentheses after χ2 values are parameters n corresponding to eq 6 (power law). Colors of the graphs match the colors in the tables. Type 2 Experiments. To obtain large crystals, large enough not to dissolve completely during the dissolution, the first part

C2 = 0.5088 × 10−4T 2 (°C2) + 0.6816 × 10−2T (°C) + 0.8033(kg NaClO3 /kg H2O)

Crystals were nucleated, grown, and observed in a crystallization cell previously described.41 Crystals were nucleated in the cell by introducing air bubbles through a needle into the cell. Dimensions of the observed crystals were measured by a digital optical microscope (Nikon SMZ800) supplied with camera (Luminera, Infinity 1) using transmitted light. During each growth run, 15−33 crystal nuclei, sufficiently distant to avoid intergrowth during the growth, were preselected for growth rate measurements. The experiments had three parts. In the first part, crystals were nucleated and grown for 2−4 h at 28 or 29 °C. In the second part, dissolution and refaceting of crystals were performed. The solution temperature was slowly (heating rate of approximately 0.5 °C/min) increased to (34.0 ± 0.1) °C, remaining at that value for about 25 min. Crystals were partially dissolved, and at the end of the dissolution, crystal sizes in the observed directions were reduced by at least 20%. Some crystals completely dissolved, while the others were used as seeds for further growth in the third part of the experiments. After that, the temperature of the solution was decreased to growth temperature in different ways depending on the type of experiment. Thereby crystals refaceted, which provided clear borders between dissolved and newly growing portions of crystals. Those borders were necessary for the face growth rate measurements. Two Types of Experiments Were Performed. In type 1, crystals grew for about 4 h at 28 °C. In the third part of experiment, after dissolution and refaceting, crystals continued growing 4 h at the same conditions as before dissolution. Sizes of crystals were measured in intervals of about 45 min. Growth rates prior to the dissolution, i.e. stabilized rates,41 were calculated using two experimental data after 3 h of growth. To determine the corresponding average linear face growth rates after refaceting, the face displacement versus time dependence was subjected to the least-squares method. The accuracy of crystal length and face displacement measurements was about ±2 μm. All solution temperatures were set with accuracy of 0.1 °C and remained constant within 0.02 °C. Some of the crystals changed their growth rates during observation (approximately 1%), as described in ref 41. These crystals were excluded from the analysis. Type 2. In the first part of all experiments, crystals grew at 29 °C for about 2 h. After that, the temperature changed in two ways: (a) In experiments with supersaturation decrease, after crystal dissolution, solution temperature was quickly decreased to 28 °C, followed by the increase to 29.5 °C in steps of 0.5 °C. (b) In experiments with supersaturation increase, the temperature was quickly decreased to 29.5 °C, continued with decrease to 28 °C in steps of 0.5 °C. All temperature changes were performed within 5 min. C


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Table 2. Values of χ2 for (a) Scatter Diagram and (b) Rmax vs σ Dependence

of the experiments lasted about 2 h. Growth rates in this period were not measured. After dissolution and refaceting, the {100} face growth rates of sodium chlorate crystals were measured. Growth rate changes of individual crystals due to supersaturation increase and decrease were observed. Experimental conditions are presented in Table 3. During experiments with superstauration decrease, some crystals had intergrown. Growth of these crystals was not further observed. Wide growth rate dispersions occurred at all supersaturations. Histograms representing changes in dispersions with supersaturation decreases (temperature growth increases) are shown in Figure 3, while those with supersaturation increases (temperature growth decreases) are shown in Figure 4. Table 3 lists the total number of observed crystals, ND and NI; most probable growth rates, RmaxD and RmaxI, corresponding to supersaturation decrease and increase; and standard deviation, sD and sI, of the most probable growth rate determination, respectively. Dependence of the growth rates on supersaturation, when values of supersaturation decrease and increase, are shown in Figures 5 and 6, respectively. Panels a and b correspond to all measured crystal growth rates and the most probable growth rates, respectively. Data were fitted with functions listed above. Values of χ2 for used equations are presented in Table 4. Power values, n, in eq 6 are given in parentheses.

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DISCUSSION It was impossible to change supersaturation without changes of growth temperature in type 2 experiments, i.e., it was impossible to grow one particular crystal at constant growth temperature and different saturations in one growth run. Supersaturation depends on supercooling and growth temperature, but as is evident from Tables 1 and 3, supersaturation of sodium chlorate crystals under investigated conditions practically depends only on supercooling. Supersaturations at different temperatures differ in the second decimal place. Also, we have shown that significant difference between growth rates of crystals which were grown at the same supersaturation and different temperatures in our experiments does not exist. In other words, it is possible to neglect supersaturation dependence on growth temperature in the 28.0−31.0 °C range. Therefore, results of type 2 experiments were discussed in the same way as the results of type 1 experiments, even if temperatures were not equal. Many substances exhibit wide face, directional, and mass growth rate dispersion under different growth conditions.12−25 It is shown that dissolution and refaceting narrow these dispersions.44,45 It can be seen from Figures 1, 3, and 4 that for all investigated macroscopic external conditions, a wide range of face growth rates exists after refaceting. These growth rate distributions can be described by a simple normal distribution, as was shown previously.43 It is proposed that in this case {100} face growth rates of sodium chlorate correspond to one

Figure 1. Histograms representing {100} face growth rate dispersions for type 1 experiments.

Table 1. Experimental Results for Type 1 Experiments TS [°C]

T [°C]

σ1 [%]

σ2 [%]

N

29.5 30.0 30.5 31.0 31.5

28.0 28.0 28.0 28.0 28.0

0.66 0.89 1.11 1.34 1.56

1.41 1.89 2.37 2.85 3.33

320 324 337 230 296

Rmax [nm/s]

s [nm/s]

± ± ± ± ±

1.2 2.1 3.8 2.1 2.8

8.1 13.5 18.7 24.9 30.1

0.2 0.5 0.6 0.3 0.7

Figure 2. {100} face growth rate vs supersaturation dependence for (a) all observed crystals and (b) Rmax vs σ.

D


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Industrial & Engineering Chemistry Research Table 3. Experimental Conditions and Results for Type 2 Experiments TS [°C]

T [°C]

σ1 [%]

σ2 [%]

ND

31.0 31.0 31.0 31.0 31.0

28.0 28.5 29.0 29.5 30.0

1.32 1.10 0.88 0.66 0.44

2.85 2.37 1.89 1.42 0.94

113 113 113 102 98

RmaxD [nm/s]

sD [nm/s]

NI

± ± ± ± ±

2.8 1.9 3.0 1.5 0.5

156 156 156 156 156

23.2 15.2 10.9 5.6 3.7

0.7 0.7 0.2 0.7 0.1

RmaxI [nm/s]

sI [nm/s]

± ± ± ± ±

3.5 2.2 1.8 1.9 1.4

24 18.5 13.6 9.4 5.9

2 0.9 0.7 0.9 0.3

Figure 3. Histograms representing {100} face growth rate dispersion for type 2 experiments with supersaturation decreases.

Figure 4. Histograms representing {100} face growth rate dispersion for type 2 experiments with supersaturation increases.

(minimal) activity of the dominant dislocation source. From these figures and Tables 1 and 3 it can be noticed that positions of the distribution maxima are shifted to higher values because of solution supersaturation increase, which is an expected result. From histograms and Table 3 it can be seen that the position of the maxima differs for experiments in which the solution supersaturation decreases, RmaxD (Figure 1) and those in which supersaturation increases, RmaxI (Figure 4). In the first case, after refaceting, crystals started to grow under the higher supersaturation (1.32%, 28 °C). In the second case, crystals started to grow under the lower supersaturation (0.44%, 30 °C). If we compare the values of maxima positions for the growth rate distributions in the case of supersaturation increase, RmaxI,

and decrease, RmaxD, it can be noticed that these values are higher for supersaturation increase. These differences are not in the measurement error limit, so they might be consequences of different growth rate mechanisms. To establish the most suitable correlation between the growth rate and supersaturation and to determine the crystal growth mechanism in the range of investigated experimental conditions, our experimental data were fitted with eqs 1−8. Values of χ2 are listed in Tables 2 and 4. Smaller values of χ2, for the same data set, correspond to equations which better describe the mentioned dependence. High values of χ2 obtained for eqs 1 and 2 show that twodimensional growth practically does not exist in our experiments. Relatively low value of χ2 for eqs 3 and 7, which E


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Type 1 Experiments. From the values of χ2 listed in Table 2, it can be seen that dependence of {100} face growth rate on supersaturation might be best described by eqs 3 and 7. In other words, growth of sodium chlorate crystals for σ1 solution supersaturation range of 0.66−1.56% is in accordance with BCF and Chernov’s dislocations growth models, which was also shown by Bennema1 for the supersaturation range of (0.3−15) × 10−2%. Power obtained by fitting the data with eq 6 takes the value n ≈ 1.5, which suggests that diffusion fields of the steps partially overlap in the investigated experimental conditions, because the surface diffusion path length is comparable to terrace width. This result means that these supersaturations are not extremely small or high compared to σc (σ ≈ σc). Type 2 Experiments. Small values of χ2 (Table 3) correspond to different listed equations for crystal growth when supersaturation decreases and increases. It suggests that growth mechanisms depend on growth history. Influence of crystal and solution history on crystal growth rate was shown earlier.14,15,46,47 In experiments with supersaturation decrease, low values of χ2 (0.47, 0.60, 0.80, and 1.35) correspond to eqs 6, 5, 3, and 7, respectively. On the other hand, in experiments with supersaturation increase low values of χ2 (0.1, 0.49, 0.55, and 2.2) correspond to eqs 6, 7, 3, and 4, respectively. This means that dislocation growth mechanism is dominant, as in type 1 experiments. Values of χ2 suggest that in experiments with σ1 supersaturation decrease between 0.44 and 1.32%, the dependence of the most probable growth rate on supersaturation can be best described by eqs 5 and 6. Power low fitting by eq 6 gives power n ≈ 1.9, which suggests that high interaction between diffusion fields of neighboring steps does not exist. Similar, nearly parabolic (R, σ) dependence was reported by Bennema1 in σ2 supersaturation range (0.3−5) × 10−2% and Ristić et al.10 for smaller values of σ1 supersaturation in the range of 0.1−1.0%. On the other hand, in experiments with supersaturation increase, the best fit is provided by eq 6, for power n ≈ 1.3, which suggests partial overlapping of diffusion fields of neighboring steps. Similar, nearly linear (R, σ) dependence reported by Bennema1 in σ2 supersaturation range (5−15) × 10−2% and Ristić et al.10 for higher values of σ1 supersaturation in the range of 0.1−1.0%. It is considered that interaction of diffusion fields depends on supersaturation. If supersaturation is σ ≫ σc, where σc is the parameter described in eq 3, i.e., if terrace width is much greater than surface diffusion path, diffusion fields of neighboring steps will overlap. In contrast, these diffusion fields are independent. Our experiments show dependence of diffusion fields overlapping on growth history, because experiments with decreasing and increasing supersaturation are realized in the same σ1 supersaturation range of 0.44−1.32%. Dependence of growth rate on crystal and solution history was shown earlier.14,44−46 Our results show that not only the growth rate but also the mechanism of growth depends on history. Existing crystal growth theories predict dependence of growth mechanism on supersaturation and temperature. We have shown that in the same supersaturation range different mechanisms can operate, depending on crystal and solution history. The important parameter which determines growth mechanism is σc. Our results suggest that this parameter depends on growth history. As is evident from Table 3, maxima of the growth rate distributions corresponding to the same supersaturation when

Figure 5. {100} face growth rate vs supersaturation dependence when supersaturation decreases for (a) all observed crystals and (b) the most probable growth rate.

Figure 6. {100} face growth rate vs supersaturation dependence when supersaturation increases for (a) all observed crystals and (b) the most probable growth rate.

Table 4. Values of χ2

correspond to BCF and Chernov’s spiral growth mechanism, respectively, show that {100} faces of sodium chlorate in the σ1 and σ2 supersaturation range 0.44−1.56% and 0.94−3.33%, respectively, grow in accordance with this mechanism. Hosoya and Kitamura9 and Surender et al.11 suggested a twodimensional mechanism for higher supersaturations, 3−5% and 3−8%, respectively (no information about definition of concentration). It can be noticed that a relatively low value of χ2 is obtained for linear equation with intercept (eq 8), i.e., this equation describes well growth rate versus supersaturation dependence; none of the theories predict this type of (R, σ) dependence for spiral growth, which probably exists in our experiments, as is shown below. It can be seen from Tables 2 and 4 that values of χ2 for the dependence of {100} face growth rate on supersaturation are lower for the most probable growth rates than for all observed crystals growth rates, which is expected. Nevertheless, lower χ2 values correspond to the same equations for both dependences. To determine crystal growth mechanism we compare values of χ2 for the most probable growth rate on supersaturation dependences (Figures 5b and 6b). F


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which is better defined for growth after dissolution and refaceting. This is confirmed by the presented results.

supersaturation increases (RmaxI) are higher than when supersaturation decreases (RmaxD). This is in accordance with lattice strain theory,16,17,19,48 whereby the crystal growth rate is inversely proportional to the “overall” lattice strain. Namely, it is expected that crystals which initially grew at smaller supersaturation have fewer defects of structure and have lower chemical potential than crystals which initially grew at higher supersaturation. Consequently, the driving force for the crystal growth should be higher for less defective crystals, resulting in higher growth rates.18 Determination of growth mechanism from growth rate versus supersaturation dependence is difficult because of growth rate dispersion. High scatter of data on diagrams shown at Figures 5 and 6 can be noticed. Practically, graphs of any considered functions can be drawn through experimental data. This means that a small number of growth rate measurements at the same supersaturation may result in a wrong conclusion about the best function describing growth rate versus supersaturation dependence, i.e. about actual growth rate mechanism. It is possible that this is the reason why sometimes different authors do not draw the same conclusions about the growth mechanism for the same substances and experimental conditions. Nonequivalent crystal faces can grow by different mechanisms, especially in the presence of impurities.10,49 It is more difficult to determine the function which describes growth rate versus supersaturation dependence from crystal growth in the ⟨100⟩ direction than from growth of {100} faces. This can be seen from Figure 7 which shows growth rate in

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CONCLUSION We have shown that {100} faces of sodium chlorate crystals grow according to BCF theory in the σ1 supersaturation range of 0.44−1.56%. Overlapping of diffusion fields of neighboring steps depends on the manner of supersaturation change, i.e. the growth history. It is different for experiments in which supersaturation decreases and experiments where it increases in the same range of supersaturation (0.44−1.32%). When supersaturation was reduced, (R, σ) dependence was found to be nearly parabolic and diffusion fields of neighboring steps are independent, whereas for supersaturation increase this dependence was found to be almost linear and diffusion fields of neighboring steps overlap.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: mico@ff.bg.ac.rs. Funding

This work was supported by the Serbian Ministry of Education, Science and Technological Development through Grant No. 171015. Notes

The authors declare no competing financial interest.

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Figure 7. Growth rate of sodium chlorate in ⟨100⟩ direction vs supersaturation.

⟨100⟩ direction versus supersaturation dependence before dissolution. Points are significantly more scattered because the growth rate in the ⟨100⟩ direction is the sum of two opposite face growth rates, which are scattered too. Determination of the growth rate mechanism from mass changes during time is more difficult because it depends on growth rate of all faces.1 If these faces are nonequivalent, conclusions will very probably be wrong. Even though our experimental data, obtained for face growth, are scattered, a clearer conclusion about face growth mechanism can be made. Previously we have shown45 that after dissolution and refaceting of crystals, the correlation between growth rates and the crystal sizes is better defined than before these processes. We presumed that this is not the only correlation

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NOTATION βl = kinetic coefficient for steps γ = surface free energy ΔG = energy necessary for the dehydratation of growth unit during its integration into the crystal ΔG*p = free energy change, corresponding to the formation of a stable circular nucleus on the perfect surface Λ = retardation factor that describes influence of kinks of the density of steps λs = the diffusion distance on the surface ρc = critical radius of nucleus σ = relative solution supersaturation σc = critical value of relative solution supersaturation ν = frequency of the atom vibration on the surface Ω = specific molecular volume of growth units A = surface area a and b = empirical parameters b = the size of the growth unit in the y-direction Ds = surface diffusion coefficient Ead = energy of adsorption of the growth units on the surface h = height of steps of the nuclei K = growth rate constant m1 = mass of the adsorbed growth units n = supersaturation exponent N0 = concentration of growth units at the crystal surface n1 = concentration of adsorbed growth units on the surface Wa = activation energy for the integration of growth units into the kink REFERENCES

(1) Bennema, P. Crystal Growth Measurements on Potassium Aluminium Alum and Sodium Chlorate from Slightly Supersaturated Solutions. Phys. Status Solidi B 1966, 17 (2), 563. (2) Garside, J.; Janssen-van Rosmalen, R.; Bennema, P. Verification of crystal growth rate equations. J. Cryst. Growth 1975, 29, 353.

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