Determination of the activation energies of phase transition for calcium orthophosphates based on powder X-ray diffraction data

Kinetic studies of the transformation of calcium orthophosphates metastable precipitates were performed under different synthesis conditions. Phase composition and degree of crystallinity were investigated by X-ray powder diffraction analysis. In acidic solution, precipitates of CaHPO 4  2H 2 O (DCPD) and CaHPO 4 (DCPA) are formed at the early stage of precipitation, with the degree of crystallinity at the range of 17-35%. Specifically, DCPD precipitates at 30  C and anhydrous DCPA at 50°C. In alkaline solution (pH 8-10), only amorphous forms of calcium orthophosphate is precipitated, which is explained by the high degree of supersaturation (i.e., the high rate of precipitation compared to that in acidic media). The diffraction peaks of DCPD and DCPA are found to be 0.3-0.45 degrees lower relative to their reference data, which is caused by decrease of lattice strain during early stage of crystallization. Furthermore, the initial molar ratio of Ca/P in reagent mixture was found to play subordinary role in determining the composition of final calcium phosphate precipitates. The effect of pH on the composition of precipitates is illustrated by the solubility isotherms of pure calcium orthophosphates. Given that the intensities of diffraction peaks are proportional to planar density of the material in the given plane, we propose, for the first time, to determine activation energy of phase transformation of calcium orthophosphate from X-ray powder diffraction patterns. Based on this relationship developed, the activation energy for the recrystallization DCPD and DCPA are 10.2 and 13.1 kJ/mol, respectively and for the phase transition of DCPD to DCPA – 36.7 kJ/mol. Further recrystallization to most thermodynamically stable Ca 10 (PO 4 ) 6 ( OН) 2 hydroxyapatite (HA) occurs at the activation energy of 5.2 kJ/mol. These findings are critical on phase transition and transformation of calcium phosphate minerals. recrystallization 10.2 kJ/mol. Similarly, the energy for the recrystallization of anhydrous DCPA – 13.1 kJ/mol, transition DCPD to DCPA phase – 36.7 kJ/mol, and for recrystallization to HA – 5.2 kJ/mol. These results suggest that the reconstruction of the crystal lattice requires much more energy than the crystal growth or hydrolysis of certain kJ/mol, for recrystallization of anhydrous DCPA is – 13.1 kJ/mol, and phase transition of DCPD to DCPA is – 36.7 kJ/mol, and for recrystallization of hydroxyapatite Ca 10 (PO 4 ) 6 (OН) 2 to be – 5.2 kJ/mol. These results, overall, show that the physical model of calcium phosphate transformation where reconstruction of the crystal lattice requires more energy than that in the crystal growth or dissolution.


Introduction
Among inorganic compounds of calcium and phosphorus, calcium orthophosphates are of particular interest. They are the mineral basis of bone tissue and are therefore considered as promising biomaterials. They are actively used in modern plastic surgery and medical cosmetology. [1][2][3] For example, one of the most common procedures in cosmetology today is contouring, the filler of which includes calcium hydroxyapatite (HA). [4][5][6] Calcium phosphates have various other applications including acidity regulator, emulsifier, leavening agent, anticoagulant and moisture retaining agent, plant tissue sealant, and fertilizer.
The changes in physical and chemical properties, including crystallography and thermodynamic parameters in calcium phosphates such as due to ion substitution and phase transformations are well studied in the 20 th century 1 . However, it is unclear so far why certain forms of calcium phosphates are precipitated under specific pH of the solution, Ca/P molar ratio, or precipitation method. Specifically, conditions of precipitation and ranges of phase transitions of metastable states of calcium phosphates are also not fully investigated.
This gap in fundamental science has limited accurate behavier of calcium phosphates in various applications such as to medical and cosmetic applications and novel phosphate fertilizers for agricultural applications 4 . Therefore a detailed understanding of the mechanism of phase transformation during synthesis as well as during dissolution is needed, Ca-deficient hydroxyapatite Ca10-x(HPO4)x(PO4)6-x(OH)2-x (Ca/P = 1.50-1.67) 7,8 . A separate group of calcium orthophosphates are formed from ion-substitution: fluoroapatites Ca10(PO4)6F2 and Ca10(PO4)6F(ОН), carbonate-containing apatites, apatites with the addition of cations of various metals 4,9,10 . Among these groups, two-, three-and even multi-phase calcium orthophosphates, in which the individual components cannot be separated from each other, are known 1 . Transformational or dry synthesis routes at high temperature generate another group: tetracalcium phosphate Ca4(PO4)2O, oxyapatite Ca10(PO4)6O, tricalcium phosphate Ca3(PO4)2. For example, tricalcium phosphate is formed by heat treatment above 800°C from calcium and phosphorus precursors at a Ca/P ratio of 1.5. This phosphate is stable at room temperature but hydrolyzed in an aqueous solution, which indicates its metastable nature.
As outlined above, the possibility of the formation of metastable phases and additional phase transformations significantly complicates the choice of optimal conditions for the synthesis of calcium orthophosphates 8,11,12 . The complexity of the calcium orthophosphates precipitation is well illustrated by Ferreira et al 13 by highlighting the equimolar Ca/P ratio of Ca(OH)2 and phosphoric acid forms DCPD and five stages of intermediate stages before the formation of hydroxyapatite (HA). According to Briak-Ben 14 , hydroxyapatite precipitation, when calcium salts and phosphate ions are mixed even at a strict stoichiometric Ca/P ratio of 1.67, occurs due to the formation of amorphous or Ca-deficient calcium phosphate. It should also be noted that the chemical composition of the hydroxyapatite surface does not necessarily coincide with its unit cell composition 15 . This discrepancy could be explained by the processes of hydrolysis and the metastable nature of the material. These lines of evidence from past studies allude that processes of chemical precipitation of calcium orthophosphates, specially metatable and intermediate phases, is by far from fully understood. In this communication, the phase transformations of metastable states calcium orthophosphates are studied. For the first time, we show that the kinetics of the reaction is controlled by distinct differences in activation energies, which form different metastable phases and limit phase transformations.

Materials and methods
Various calcium phosphate materials were synthesized at different conditions. In all cases, 1.0 M solutions of CaCl2 and H3PO4 of chemically pure or reagent grade (Reachim) were used as precursors of the synthesis. The calcium phosphate precipitation was performed by varying the pH of the solutions in the range of 3 to11 and the molar ratio of Ca/P 0.5 to 1.67. 500 mL of distilled water was placed in the reaction flask and the pH was adjusted to the target value using solutions of hydrochloric acid or ammonia. The solutions of CaCl2 and H3PO4 prepared with a given ratio of Ca/P. The solution was continuously stirred using a blade stirrer at a speed of 300 rpm and 1.0 M ammonia solution was simultaneously added to the solution using peristaltic Micro-Dosers at a flow rate of 0.05-1 mL/min. The flow rate of the reagent solutions was selected based on the stoichiometry of precipitation reactions and based on pilot experiments so that the pH of the solution remained at a given level with an accuracy of 0.1 pH using a pH meter Adwa AD8000 (accuracy of 0.05 pH units) throughout the precipitation process. After precipitation, the products were kept under stirring in the reactor for additional 6 h. This process of "maturation" contributes to the crystallization of precipitates, and allows proper identification by powdered X-ray diffraction.
The resulting suspensions were filtered under vacuum, washed with small portions of cold distilled water and then freeze dried to constant weight before analysis.
To determine the Ca/P molar ratio, synthesized precipitates were dried to constant weight and homogenized by grinding in an agate mortar. A weighed quantity of the precipitates were dissolved in a concentrated HCl in volumetric flasks. Concentration of orthophosphate was performed in separate aliquots by using colorimetric method in which reduction reaction of heteropolyacid formed by the reaction of orthophosphate ions with ammonium molybdate in acidic condition. Concentration of calcium ion was determined by inductively coupled plasma optical emission spectroscopy (ICP-OES) using an Optima 8000 spectrometer (PerkinElmer, Norwalk, CT, USA). X-ray phase analysis was performed using a DRON-2 diffractometer in monochromatic Co-K 1 radiation. To identify the phase composition from the PCPDFWIN database, the diffraction patterns were recalculated on Cu-K 1 radiation. The degree of crystallinity (XC), corresponding to the fraction of the crystalline phase in the test volume, was evaluated by the equation: where K is a constant equal to 0.24, used for different hydroxyapatite in powder diffraction, and Bhkl is the width (hkl) of the reflex at its half-height 16 .
The solubility of calcium phosphate precipitates was calculated by using the formula for the solubility product, as: The following values of solubility products were used in the calculations (negative

Results and discussion
The summary of experimental conditions and compositions of the synthesized products are reported in Table 1. At the early stage of precipitation, the maturation of precipitates at lower pH and at 15-30C lead to the formation of well crystalline forms of DCPD and DCPA. The degree of their crystallinity varies in the range of 17-35% and can be considered as a result of chemical precipitation. With a decrease in pH of the reactant mixture, a progressive decrease in the degree of crystallinity of the precipitates was found.
In acidic solutions at room temperature, crystal hydrate of DCPD (PCPDFWIN No. 72-0713) is formed. However, its anhydrous form, DCPA (PCPDFWIN No. 77-0128), is crystallized directly in the solution during precipitation at 50C and above. The conversion of DCPD to DCPA has been reported earlier 17 . This process proceeds at a noticeable rate already at a temperature of 37С. The dehydration of DCPD is reported to occur at temperature above 80C 1 , which, however, has been found at a much lower temperature in this study.  17 . It should be noted that this effect is only observed in precipitates that have not been thermally treatment after precipitation, such as reference calcium phosphates presented in the PCPDFWIN. These differences are better explained based on the solid-state chemistry principle that lattice strain decreases with an increase in temperature or time of maturation. During thermal treatment of crystalline or partially crystalline precipitate in hydrated media, crystallites continue to grow, which in turn reduces the lattice strain of the crystal structure and consequent decrease in interplanar spacing. Contrarily to acidic synthesis, the precipitation in an alkaline medium showed stable amorphous precipitate, which practically does not crystallize at 15-30C, even if they are kept in solution for several days (Fig. SI 1-3). They did not show any shift in peak position on the diffraction patterns.
Given that the structure of the precipitate is determined by the precipitation rate (r), which is a function of the degree of supersaturation of the product in the solution, the relationship could be written as follows: , where k is the rate constant, n is the order of precipitation (as a rule of thumb, n  10),  is the degree of supersaturation (the ratio of the ion activities product in the supersaturated solution to the thermodynamic solubility product of the precipitate 18 . In acidic solutions, the degree of supersaturation of the reagent mixture is relatively small due to excess of protonated H2PO4over HPO4 2species. This reduces the growth rate of DCPD nuclei and thus creates the conditions for the formation of the most thermodynamically stable crystal lattice at given pH. As the degree of supersaturation increases, the precipitation rate increases and the precipitate is formed in a less energetically favorable crystal lattice. With a further increase in pH, the size of the DCPD particles decreases with the eventual formation of amorphous products in alkaline solutions. As illustrated from data reported in Table 1, the phase compositions of calcium phosphate precipitates do not depend on the starting Ca/P molar ratio. For example, mixing aqueous solutions of reagents with molar ratios of Ca/P = 0.5 and 1.5 at pH = 3-4 and pH = 6, formed the solid precipitate of DCPD with Ca/P ratio of 1.000.05. Similarly, mixing aqueous reagents with a molar ratio of Ca/P = 1.0 and 1.67 at pH 8, amorphous precipitates with a molar ratio of Ca/P ca. 1.50-1.61 were obtained. An increase of the maturation temperature to 50°C improved the degree of crystallinity of the precipitates and formed Cadeficient hydroxyapatite (Fig. SI 4). A further increase in synthesis pH to 11 with an initial molar ratio of reagents of Ca/P = 0.5, an amorphous precipitate was formed with Ca/P = 1.670.05. However, PXRD diffraction peaks confirmed the presence of some long-range atomic order of hydroxyapatite (Fig. SI 3).
The role of differences in the Ca/P ratios in the reagent mixture in the products have been repeatedly noted in the literature. For example, Zyman at al. 12 prepared phosphate products with Ca/P ratio 1.5 at pH = 10 from a reagent mixture with an initial ratio of Ca/P = 1: 1. They explained this discrepancy by the hydrolysis of the fraction of DCPD during washing of precipitates after synthesis. These results indicate that when mixing solutions of calcium and orthophosphate salts with the ratio Ca/P = 1, the formation of hydroxyapatite with Ca/P ratio = 1.5 is possible. In our study, mixing solutions with Ca/P ratio as low as 0.5 in alkaline solutions formed an amorphous precipitate with a ratio of Ca/P = 1.670.05 (Table  1). This amorphous phase is a precursor for the crystallization of apatite, although the rate of crystal growth is very low at 20C. Thus, in contrast to the pH of the reaction medium-a master variable, the molar Ca/P ratio of initial reagents does not play a critical role in determining the final composition of calcium phosphate precipitates. The differences in Ca/P ratio between the initial reagent mixture and resulted precipitate at the end of crystallization should be only induced by minor presence of CaCO3 from atmospheric air at the impurities level because the magnitude of coprecipitation of certain ions depends on their concentration in solution and the order of mixing of solutions of synthesis precursors. It is noteworthy to mention that even if Ca/P ratio of initial reagents used for synthesis is equal to the stoichiometric hydroxyapatite (i.e., 1.667), non-stoichiometric precipitates, such as amorphous phosphates, are still formed as a side product. 1 .
The effect of pH on the composition of chemically pure calcium orthophosphate precipitates can be illustrated in a solubility isotherm (Fig. 2). As shown, the optimal pH for the synthesis of calcium phosphates can be chosen based on the need of solubility and stability of crystalline phases, which, however, depends on the precipitation method, synthesis temperature, and aging condition. According to thermodynamic calculation on phase precipitation, hydroxyapatite is the least soluble in the entire range of pH values from 0 to 14 (Fig. 2).
The variable range of solubility of precipitates are shown by shaded zones (4 and 7 in However, other authors 19 have shown that DCPA is fairly resistant to hydration and does not transform to DCPD at a temperature of 4-50 °C for as long as seven months. The solubility isotherms in Fig. 2 explain these patterns well. The change of thermodynamic state from isotherm 6 to 8 is energetically favorable. However, activation energy could be a barrier to this transformation. It is obvious that an increase in temperature is required to overcome this barrier, which is confirmed by our experimental results and literature data. For example, Cadeficient hydroxyapatite can be obtained by simultaneous addition of calcium and phosphoric acid salts in the ratio Ca/P = 1.50-1.67 in boiling water at pH = 6.5-9.5, followed by boiling the suspension for several hours. Similarly, stoichiometric hydroxyapatite can be synthesized at a temperature of 90 at the ratio of Ca/P = 1.67 and pH = 9.5-12 20 .
Considering that the intensity of diffraction peaks is proportional to the planar density of all crystallites formed under specific crystallization conditions, activation energies After integrating the Arrhenius equation in a narrow temperature range T1 and T2 (assuming that the activation energy (Ea) does not depend on the temperature), the revised expression of the equation: The above equation can be expressed using the logarithms of the peak intensities of the diffraction patterns instead of the rate constants for precipitates that are present in the supernatant for the same time but at different temperatures. Given that peak intensity can increase or decrease, the expression above can be expressed in a modulus form: This formula allows to calculate the activation energies at any given value of 2 diffraction patterns of precipitates that undergo phase transition after precipitation. The results of calculated values of Ea for diffractograms in Fig. 1  Based on the results obtained from calculations using equations above and presented in Fig. 3, we conclude that the change of each interplanar distance in the crystal lattice is characterized by its activation energy. The appearance and disappearance of certain peaks on the diffraction pattern require overcoming a specific activation energy barrier. Fig. 3 show the upper limit of the activation energy, which is a more appropriate estimate of the crystallization process than the average values. Crystallization (growth of long-range atomic order in particles due to diffusion of ions from solution), recrystallization (growth of longrange atomic order in particles due to diffusion of ions within the solid phase of particles), or phase transformation are interconnected and complementary and cannot exist, in most cases, without each other. Fig. 3 show a notable difference in activation energies for the recrystallization process, which in turn, will determine the overall rate of this reaction. Thus, the transformation of the interplanar distances in the crystal lattice with the largest Ea value will be the limiting factor for all reactions.
The maximum activation energy calculated using the above relationship for DCPD Ostwald ripening on the surface and recrystallization in the volume of precipitate particles occur side by side. Comparable and low activation energies (10.8 kJ/mol for DCPD and 11.2 kJ/mol for DCPA) suggest that the limiting reaction is likely the diffusion of ions. For the phase transition of DCPD to DCPA, the activation energy of 36.7 kJ/mol indicates the kinetic nature of reaction, which induces a conversion of the monoclinic crystal lattice of DCPD into a triclinic lattice of anhydrous DCPA. Overall, these results conclude that a caution needed to compare the literature data 21,22 with current results on activation energy. To make a distinction, we used the term "recrystallization", to represent the growth of crystallites of polycrystalline material at the expense of others.

Conclusion
The study on kinetics of crystallization and transformation of metastable calcium orthophosphates under different synthesis conditions (pH, temperature, and molar ratio Ca/P), specifically the phase composition and degree of crystallinity, generated insightful outcomes.
In acidic solutions (pH = 3-6, t=15-90°C), DCPD and DCPA precipitates are formed in the crystalline state ( degree of crystallinity =17-35%). In alkaline solutions (pH = 8-10, t=15-30°C) calcium orthophosphates are precipitated only in amorphous state. This is due to a high degree of supersaturation of solutions, and consequently, the higher precipitation rate compared to that in acidic media. A shift of all diffraction peaks of DCPD and DCPA by 0.3-0.45 2 degrees relative to the reference data was found to be related to time for maturation.
Our data shows that the magnitude of the shift decreases with the increase in time of maturation of precipitates in the supernatant, analogous to commonly described solid state chemistry principle in which decrease of lattice strain as a result of growth of long-range atomic order.
We conclude that, unlike pH as the master variable of the form, the initial molar ratio of Ca/P reagents in the media has a minor role in determining the fundamental composition of calcium phosphate precipitates. Differences should be expected only at level of minor impurities, because the magnitude of coprecipitation of certain ions depends on their concentration in solution and the order of mixing of solutions of synthesis precursors.
The effect of pH on the composition of precipitates is illustrated by the solubility isotherms of known calcium orthophosphate phases. Based on this calculations, for the entire range of pH (0-14), hydroxyapatite is the least soluble phosphate. The solubility isotherms allowed to establish energetically possible phase transitions between metastable states of calcium phosphates with activation energy being a barrier of transition. An increase in temperature is often the factor to overcome this activation barrier. Since the intensity of diffraction peaks is proportional to planar density of given plane in the material, we proposed an equation to calculate the values of activation energy of phase transformation, for the first time, from X-ray diffraction patterns. The calculated activation energy for DCPD recrystallization is 10.2 kJ/mol, for recrystallization of anhydrous DCPA is -13.1 kJ/mol, and phase transition of DCPD to DCPA is-36.7 kJ/mol, and for recrystallization of hydroxyapatite Ca10(PO4)6(OН)2 to be -5.2 kJ/mol. These results, overall, show that the physical model of calcium phosphate transformation where reconstruction of the crystal lattice requires more energy than that in the crystal growth or dissolution.