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Langmuir 2001, 17, 8092-8097

Kinetic Study on the Crystal Growth of Hydroxyapatite S. Koutsopoulos† Department of Chemistry, University of Patras, GR-26500 Patras, Greece Received May 29, 2001. In Final Form: September 26, 2001 The kinetics of hydroxyapatite (HAP) crystallization, which is thermodynamically the most stable calcium phosphate salt, has been investigated at physicochemically defined physiological conditions. The crystal growth process involved metastable supersaturated solutions, and for the study the constant composition method was employed. The kinetic results were tested over the most important crystal growth models. At the experimental conditions with the relatively low supersaturation level, the spiral growth model introduced by the Burton-Cabrera-Frank theory was found to better interpret the kinetic data. With the formulation of the model, thermodynamic and morphological parameters of hydroxyapatite were calculated. The surface kink site density was found to be 4.33 × 1014 kinks/m2, which means that the active growth sites represent less than 1 ‰ of the total number of HAP growth units on the crystal surface. The mean linear growth rate of the HAP crystal surface was found to be 1.34 monolayers per hour.

Introduction The precipitation of calcium phosphates has attracted the interest of many researchers because of its importance in industrial water system scale formation (e.g., heat exchangers, cooling towers, boilers, etc.), in water treatment processes, in catalysis as supporting material, in agriculture as fertilizers, and basically in biomineralization processes.1-9 Depending on the supersaturation level and the solution pH, a number of calcium phosphates may be formed at ambient temperatures and pressure. In aquatic solutions of pH > 4, the order of increasing solubility is as follows: hydroxyapatite (Ca5(PO4)3OH, HAP), tricalcium phosphate (Ca3(PO4)3, TCP), octacalcium phosphate (Ca4H(PO4)3‚2.5H2O, OCP), dicalcium phosphate dihydrate (CaHPO4‚2H2O, DCPD). Hydroxyapatite, thermodynamically the most stable calcium phosphate salt, has been extensively studied because it is the main inorganic compound of hard tissues in vertebrates. Although the precipitation of HAP is of particular importance in the biomineralization processes, very little is known about the crystal growth mechanism. In the present work the kinetics of HAP crystal growth from supersaturated solutions was studied using the constant composition method in which the chemical potentials of the solution species are maintained constant during the reaction. The composition and the supersaturation of the solution are controlled in a way that reactants are added to compensate for those lost during the growth † Present address: Laboratory of Physical Chemistry & Colloid Science, Wageningen University, Dreijenplein 6, P.O. Box 8038, 6700 EK Wageningen, The Netherlands. Tel: +31 317 485603. Fax: +31 317 483777. E-mail: [email protected].

(1) Leckie, J. O.; Stumm, W. In The Changing Chemistry of the Oceans; Dyrssen, D., Janger, D., Eds.; Almqvist and Wiksell: Stockholm, 1972. (2) Stumm, W.; Morgan, J. J. Aquatic Chemistry. An Introduction Emphasizing Chemical Equilibria in Natural Waters; John Wiley and Sons Inc.: New York, 1981. (3) Kibby, C. L.; Hall, W. K. In The Chemistry of Biosurfaces; Hair, M. L., Ed.; Dekker: New York, 1972; Vol. 2, p 663. (4) Joris, S. J.; Amberg, C. H. J. Phys. Chem. 1971, 75, 3167. (5) Ellis, N.; Margaritis, A.; Briens, C. L.; Bergougnou, M. A. AIChE J. 1996, 42, 87. (6) Christoffersen, J.; Christoffersen, M. R. J. Cryst. Growth 1988, 87, 41. (7) Boskey, A. L.; Bullogh, P. G. Scanning Electron Microsc. 1984, 28, 511. (8) Nancollas, G. H. J. Cryst. Growth 1977, 42, 185. (9) Koutsopoulos, S.; Demakopoulos, J.; Argiriou, X.; Dalas, E.; Klouras, N.; Spanos, N. Langmuir 1995, 11 (5), 1831.

process.10-12 Moreover, since the supersaturation level is low and a steady state is established, HAP is formed exclusively on the seed crystals of the same material without the mediation of any precursor phases. The kinetic and thermodynamic data obtained for the crystallization of HAP were examined, and various crystallization theories were applied to interpret the experimental findings and calculate several physicochemical parameters of the precipitating system and the surface characteristics of the crystals. Experimental Section Triply distilled CO2-free water and analytical reagent chemicals were used for the preparation of the stock solutions. All solutions were filtered before use (0.22 µm Millipore). The filters were prewashed to remove residuals and surfactants. Standardization of the phosphate and calcium ion stock solutions was done spectrophotometrically as vanadomolybdate complex13 and by ion chromatography (Metrohm 690-Ion Chromatography Detector, IC Y-521 Shodex), respectively. Potassium hydroxide solutions were prepared from concentrated standards (Titrisol, Merck). In a typical crystal growth experiment the precipitating solutions were prepared in a covered double-walled, watercirculated cell (total volume 250 mL), thermostated at 37 ( 0.1 °C. The working solutions were continuously stirred with a magnetic stirrer (MAG-MIX, Precision Scientific), at a constant rate of 360 rpm (6 Hz). The pH was adjusted to the desired value by the addition of potassium hydroxide solution. To induce precipitation, HAP seed crystals were introduced into the crystallizer containing the phosphate solution at the desired pH, temperature, and ionic strength for 3 h before the experiment was started by the addition of the appropriate volume of calcium chloride solution. This procedure is advantageous over the classical constant composition method (where the experiment is started by the addition of the seed crystals), because of the time lag given to the solid crystal surface to equilibrate with the precipitating liquid environment. This surface wetting process, which can strongly affect the crystal growth process, is eliminated as a variable.14 HAP seed crystals were prepared by a method (10) Tomson, M. B.; Nancollas, G. H. Science 1978, 200, 1059. (11) Heughebaert, J. C.; Nancollas, G. H. J. Phys. Chem. 1984, 88, 2478. (12) Koutsoukos, P. G.; Nancollas, G. H. J. Phys. Chem. 1981, 85, 2403. (13) Tomson, M. B.; Barone, J. P.; Nancollas, G. H. At. Absorpt. Newsl. 1977, 16, 117. (14) Ny´vlt, J. Industrial Crystallization from Solutions; Butterworth: London, 1971.

10.1021/la0107906 CCC: $20.00 © 2001 American Chemical Society Published on Web 11/28/2001

Crystal Growth of Hydroxyapatite

Langmuir, Vol. 17, No. 26, 2001 8093

Table 1. Experimental Conditions, Thermodynamic Parameters, and Kinetic Results, Using the Constant Composition Method for the Crystal Growth of Hydroxyapatite: Total Calcium (Cat):Total Phosphate (Pt) ) 1.67, pH 7.4, Ionic Strength 0.15 M NaCl, 37 °C R/R

Cat (10-4 M)

Pt (10-4 M)

SHAP

σHAP

HAP

S-316 S-307 S-317 S-314 S-318

5.0 4.0 3.5 3.0 2.5

3.0 2.4 2.1 1.8 1.5

5.58 4.60 4.09 3.58 3.05

4.58 3.60 3.09 2.58 2.05

-4.43 × 103 -3.94 × 103 -3.63 × 103 -3.29 × 103 -2.88 × 103

∆G (J/mol) TCP OCP -3.99 × 102 1.63 × 102 5.00 × 102 8.90 × 102 1.35 × 103

described elsewhere17 and had a specific surface area of 34.6 m2 g-1, as determined by the multiple-point BET method (PerkinElmer Sorptometer 212 D). The crystals were analyzed by infrared spectroscopy (KBr pellet method, FT-IR Perkin-Elmer 16-PC) and powder X-ray diffraction (Philips PW 1830/1840, Cu KR radiation, using aluminum as internal standard) and displayed the characteristic X-ray powder diffraction pattern (ASTM card file No. 9-432) and the infrared spectrum of stoichiometric HAP.12 From chemical analysis the experimentally determined ratio of Ca:P was 1.67 ( 0.01, which is identical to the stoichiometric value. Although the precipitating system was closed to the atmosphere, nitrogen gas (99.99%, Linde Hellas) presaturated water at 37 °C was bubbled through the solutions before and during the course of the experiment to exclude carbon dioxide. Solution pH was controlled using a combined glass/Ag/AgCl electrode (Metrohm, 6.0202.100), standardized before and after each experiment with NBS standard buffer solutions.15 Kinetic experiments were performed at constant composition conditions (i.e., a constant molarity of all the reactants was maintained in the supersaturated solution).16 During the crystallization reaction this was accomplished by using a modified pH-stat system incorporating mechanically coupled burets, which delivered the reactants simultaneously into the working solution. The formation of HAP on the HAP seed crystals was monitored by the release of protons according to the following reaction:

5CaCl2 + 3KH2PO4 + KOH a Ca5(PO3)3OH + 6HCl + 4KCl (1) A drop in pH of 0.005 pH units, resulting from the precipitation of HAP, triggered the addition of titrants from the modified pHstat system (Radiometer pH-meter 26, Radiometer titrator 11, Auto-burette ABU1C). The burets contained CaCl2, NaCl, KH2PO4, and KOH; the concentration ratio of calcium and phosphate matched the stoichiometry of the precipitating solid, Ca:P ) 5:3.17 Constancy in solution composition during the course of the experiment was confirmed to be within (3%, by chemical analysis of filtered samples withdrawn randomly from the crystallizer. For the experimental conditions employed in the kinetic experiments, the number of the seed crystal particles into the precipitating solution and the size distribution were measured by a particle size analyzer (Laser Particle Counter ILI-1000).

Results and Discussion The concentration of all ionic species in the supersaturated solutions was calculated from the pH value, the equilibrium constants, and the expressions for the mass and charge balance. Proton dissociation and ion pair association constants were also utilized. This system of equations was solved for the solution species by successive approximations for the ionic strength.18 For this reason computer program software was developed.19 The ionic activity coefficients, f(, of the ionic species with z charge valence were calculated from the modified Debye-Hu¨ckel equation:20,21

I1/2 log f( ) -A|z+ z-| + bI 1 + BReffI1/2

(2)

where I is the ionic strength of the supersaturated solution,

1.28 × 101 5.04 × 102 7.99 × 102 1.14 × 103 1.55 × 103

DCPD

R (10-8 mol‚ min-1‚m-2)

Rlin (10-13 m‚s-1)

3.56 × 103 4.12 × 103 4.45 × 103 4.85 × 103 5.31 × 103

11.30 7.44 6.25 3.67 2.63

3.00 1.97 1.66 0.97 0.70

A and B are constants, which depend on the temperature and the dielectric constant of the solution. At 37 °C A and B are equal to 0.5236 (L/mol)1/2 and 0.3315 (L/cm2‚mol)1/2, respectively (calculated from extrapolation of values given at different temperatures22). The constant b is equal to -0.3 and aeff is the diameter of the ion with its hydration shell (in angstrom units), which depends on the charge density. Typical values for aeff are in the range 3-8 Å for most ions in aquatic solutions.23 The experimental conditions, thermodynamic data, and the kinetic results obtained for the crystallization of HAP are summarized in Table 1, in which the driving force for + the formation of a crystalline phase Muz +Xuz - is expressed as the change in Gibbs free energy, ∆G, of transfer from the supersaturated solution to equilibrium:

∆G ) -RgT ln

( )

(Mz+)u+ (Xz-)uIP 1/u ) -RgT ln ) ks° ks° -RgT ln S (3)

where parentheses denote activities, z+ and z- are the charges on the cation and the anion, respectively, u+ and u- are the corresponding stoichiometric coefficients, u is the number of ions in the molecule, u ) (u+) + (u-) (e.g., for HAP, u is equal to 9), S is the solution supersaturation with respect to the crystallizing compound, IP and ks° are the ionic product and the thermodynamic solubility product of the precipitating salt, respectively (at 37 °C the thermodynamic solubility products of the calcium phosphate salts are k°s,HAP ) 2.35 × 10-59,24 k°s,OCP ) 5.01‚10-50,25 k°s,TCP ) 2.83 × 10-30,26 and k°s,DCPD ) 1.87 × 10-7,27), Rg is the gas constant, and T is the absolute temperature. Positive ∆G values represent solutions undersaturated, and negative ∆G values refer to solutions supersaturated with respect to the crystalline phase under consideration. The Gibbs free energy and the solution supersaturation were calculated from the solution speciation as mentioned before. (15) Bates, R. G. Determination of pH. Theory and Practice, 2nd ed.; John Wiley and Sons: New York, 1973. (16) Koutsoukos, P. G.; Amjad, Z.; Tomson, M. B.; Nancollas, G. H. J. Am. Chem. Soc. 1980, 102, 1553. (17) Amjad, Z.; Koutsoukos, P. G.; Nancollas, G. H. J. Colloid Interface Sci. 1984, 101, 250. (18) Nancollas, G. H. Interactions in Electrolyte Solutions; Elsevier: Amsterdam, 1966. (19) Koutsopoulos, S. Ph.D. Thesis, University of Patras, Greece, 1980. (20) Debye, P.; Hu¨ckel, E. Phys. Z.. 1923, 24, 185. (21) Davies, C. W. Ion Association; Butterworth: London, 1962. (22) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions, 2nd ed.; Butterworth and Co. Ltd.: London, 1959. (23) Kielland, J. J. Am. Chem. Soc. 1937, 59, 1675. (24) McDowel, H.; Gregory, T. M.; Brown, W. E. J. Res. Natl. Bur. Stand. 1977, 81A, 273. (25) Moreno, E. C.; Brown, W. E.; Osborn, G. Soil Surv. Proc. 1960, 99. (26) Gregory, T. M.; Moreno, E. C.; Patel, J. M.; Brown, W. E. J. Res. Natl. Bur. Stand. 1974, 78A, 667. (27) Marshall, R. Ph.D. Thesis, State University of New York at Buffalo, New York, 1970.

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In the kinetic experiments of HAP crystal growth, the rate of the crystallization reaction (i.e., the moles of HAP formed per minute and the surface area of the seed crystals in square meters) was determined from the recorded volume of titrant-reactant consumption per unit time. Crystallization rates were calculated from the first derivative (for t ) 0), and the reproducibility was better than (3%. The kinetics of a growth process of a crystal surface may be expressed by a linear rate perpendicular to the surface. This however results in a complex representation of the growth rate basically because of a tensor-like quantity, which depends on the direction evolved.28 Moreover this method is most applicable to crystallization systems where large and immobilized crystals are subject to microscopic analysis. In the present case it was more useful to express crystallization kinetics in terms of an average overall crystal growth rate which was then transformed to an average linear crystal growth rate. To test the measured growth rate (in units mol‚min-1‚m-2) with the available crystal growth models and their formulation, it had to be transformed into the linear form (in units m‚s-1). The crystal growth experiments were performed at 100 mg/L concentration of well-characterized seed crystals. Experiments were also done at different concentrations of seed crystals in the range of 50-200 mg/L. From these experiments it became evident that the crystal growth rate, normalized per unit surface area of the substrate, was the same in all concentrations of seed crystals. These experiments demonstrated that crystallization of HAP occurs exclusively at active growth sites on the crystal surface, without any secondary nucleation or spontaneous precipitation. Additional evidence to support this correlation comes from the observation that the crystal growth rate was reduced as a function of time.32 In the following section, the experimental kinetic data will be analyzed to elucidate the mechanism of crystal growth, which is followed during the crystallization of HAP from relatively low supersaturated solutions. The first step was to determine whether the crystal growth process was bulk diffusion or surface diffusion controlled. For this reason kinetic experiments were done at different hydrodynamic conditions. Changes in the stirring rate between 100 and 800 rpm revealed that the crystal growth rate was the same in all cases with no observable trend. The stirring rate was determined precisely by the use of a stroboscopic technique (AEG Lichtblitz-Stroboskop LBS 141). This finding means that the rate-determining step in the crystal growth process is surface diffusion of the growth units and not transport of the crystal particles into the solution or diffusion of the crystal growth units from the bulk solution through the stagnant layer to the crystal surface. The independence of the crystal growth rate from the hydrodynamic conditions of the crystallization system may be used as a criterion only if the crystal particles are larger than 5-10 µm.29 If the particles in the crystallization solution are smaller than 5 µm (the size depends on the density difference between solid and liquid phase), then the growth rate is always independent of the system fluid dynamics and the criterion above is not valid.29 Very small particles move so slowly as compared to the bulk fluid and the larger particles therein, that the shape and size of the stagnant layer are practically unaffected by the stirring conditions and a steady-state diffusion field is developed over the crystal surface.29-31 When larger particles are present in the supersaturated solution, the size of the stagnant layer varies inversely with the stirring

Koutsopoulos

rate and its shape changes from a sphere to an ellipse. Hence, the growth units approach the crystal surface more easily since they have to travel a shorter distance, through a thinner layer, as compared to the unstirred. Under the experimental conditions used in the present case, the measured particle size distribution in the precipitating solution was between 9 and 13 µm, (Laser Particle Counter ILI-1000 SPECTREX). Since the dimensions of the crystal particles were large enough so that the crystal growth rate was independent of the hydrodynamics of the system, it can be safely concluded that the rate-determining step was surface diffusion of the growth units. The rate-controlling step of a crystallization reaction can also be determined from the experimental kinetic data. Accordingly, the rate of crystal growth is calculated assuming that bulk diffusion is the rate-determining step, RD. Then this value is compared with the corresponding growth rate obtained from direct experimental data. If the experimentally determined linear crystal growth rate Rlin is greater than RD or of the same order of magnitude as RD, then the rate is strongly influenced by bulk diffusion. If the case is Rlin , RD, then the growth rate is surface diffusion controlled. The bulk-diffusion-controlled crystal growth rate is given by eq 4, derived from Fick’s second law

RD )

DvVM(c - ceq) DvVMceqσ ) rpart rpart

(4)

where Dv is the mean diffusion coefficient of the lattice ions in solution, VM is the molar volume of the crystallizing material, rpart is the mean radius of the crystal particles, c and ceq are the molar concentrations of the crystallizing compound in the supersaturated solution and in equilibrium, respectively, and σ is the relative solution supersaturation equal to σ ) S - 1. Assuming stoichiometric dissolution of the HAP crystal lattice ions in solution, the concentrations c and ceq, may be defined as32,33

cHAP ) ceq,HAP )

(

( )

ksp,HAP

55‚33‚11

IPHAP

)

1/9

55‚33‚11 1/9

(5)

) 8.68 × 10-8 M

The mean radius of the HAP crystal particles in the working solution is rpart. ) 5.5 µm, and a typical value for the diffusion coefficient in an aquatic solution is Dv ≈ 1 × 10-9 m2‚s-1.34,35 The volume of the crystal unit cell of HAP as calculated from X-ray diffraction crystallographic data is Vunitcell ) 5.2871 × 10-28 m3 and the molecular volume is Vm ) 2.6436 × 10-28 m3 (two molecules of HAP per unit cell of the crystal). The molar volume of HAP is (28) van Leeuwen, C.; Blomen, L. J. M. J. J. Cryst. Growth 1979, 46, 96. (29) Nielsen, A. E.; Toft, J. M. J. Cryst. Growth 1984, 67, 278. (30) Nielsen, A. E. In Industrial Crystallization 1978; de Jong, E. J., Jancic, S. J., Eds.; North-Holland Publishing Co.: Amsterdam, 1979; pp 159-168. (31) Nielsen, A. E. J. Phys. Chem. 1961, 65, 46. (32) Hohl, H.; Koutsoukos, P. G.; Nancollas, G. H. J. Cryst. Growth 1982, 57, 325. (33) Koutsoukos, P. G. Ph.D. Thesis, State University of New York at Buffalo, New York, 1980. (34) Dunning, W. J.; Albon, N. In Growth and Perfection of Crystals; Doremus, R. H., Roberts, B. W., Turnbull, D., Eds.; John Wiley: New York, 1958; p 446. (35) Dunning, W. J.; Albon, N. Acta Crystallogr. 1960, 13, 495.

Crystal Growth of Hydroxyapatite

Langmuir, Vol. 17, No. 26, 2001 8095

then calculated as VM ) VmNA ) 1.592 × 10-4 m3, where NA is Avogadro’s number. As may be seen in Table 1, for a typical experiment such as S-316 (Cat ) 5 × 10-4 M, Pt ) 3 × 10-4 M, pH 7.4, ionic strength 0.15 M, 37 °C) the mean linear crystal growth rate is Rlin ) 3.00 × 10-13 m‚s-1, as determined from the overall growth rate. If bulk diffusion is assumed to control the kinetics of the crystallization phenomenon, then the crystal growth rate according to eq 4 is RD ) 1.15 × 10-11 m‚s-1. It is therefore evident that Rlin , RD, and from this it can be safely concluded that the growth rate of the HAP crystals was surface diffusion controlled. A difference of 2 orders of magnitude in the growth rates was also found for the other solution supersaturation levels employed. The thickness of the unstirred stagnant layer of solution which surrounds the crystal particles surface may be calculated from the equation36,37

δ)

[( ) ( ) ] 2 ηl 3 FlDv

1/3

Flvrel ηlx

1/2 -1

(6)

where ηl ) 6.9 × 10-3 P, the viscosity of the solution, Fl ≈ 1 g‚cm-3, the density, x is a characteristic length dimension of the crystal particle (e.g., which depends on the particle size corresponding to the equivalent sphere), which for HAP is equal to 9 × 10-4 cm,33 and vrel ) 3 × 10-2 cm‚s-1,38 the calculated relative rate between solid particles and liquid. Substituting these values in eq 6, the apparent thickness of the unstirred layer was calculated to be δ ≈ 1.8 µm. Therefore, the stagnant layer is negligibly thick and becomes less thick under the stirring conditions employed. Furthermore, the supersaturation on the surface of the crystal is almost the same with the value in the bulk solution (i.e., no concentration gradient is expected along the stagnant layer). The latter means that the resistance for diffusion of the growth units through the stagnant layer to the crystal surface is negligible. Depending on the crystal growth model, a relationship between linear crystal growth rate and supersaturation is proposed and so the first step in the investigation of the crystal growth mechanism is to find the relationship between these two parameters and to compare it with that of candidate models. Surface diffusion controlled kinetics are usually described by linear, parabolic, and exponential rate laws,39,40 regarding the relationship between rate of crystal growth and supersaturation. From Figure 1, which is based on the kinetic data of Table 1, it can be seen that there is a linear dependence of growth rate on solution supersaturation, R ) f(S). Therefore, among the proposed crystal growth mechanisms a few may be excluded. A linear dependence is indicative of the following crystal growth models: (i) spiral growth on the crystal surface when the phenomenon is controlled from surface diffusion, for high supersaturation levels σ . σ1;41 (ii) spiral growth on the crystal surface controlled from bulk diffusion of the crystal growth units, for high supersaturation levels σ . σ1;41 (iii) “birth and spread” of two-dimensional crystal surface nuclei (i.e., polynucleation model), from highly supersaturated solutions, σ . σ1;42 (36) Bennema, P. J. Cryst. Growth 1969, 5, 29. (37) Bennema, P. J. Cryst. Growth 1967, 1, 278. (38) Mullin, J. W. Crystallization, 3rd ed.; Butterworth-Heinemann: Oxford, 1992. (39) Nielsen, A. E. Pure Appl. Chem. 1981, 53, 2025. (40) So¨hnel, O.; Garside, G. Precipitation. Basic Principles and Industrial Applications; Butterworth-Heinemann Ltd.: Oxford, 1992. (41) Burton, W. K.; Cabrera, N.; Frank, F. C. Philos. Trans. R. Soc. London 1951, A243, 299. (42) Hillig, W. B. Acta Metall. 1966, 14, 1868.

Figure 1. Crystal growth rate of hydroxyapatite as a function of the solution supersaturation (pH 7.4, 37 °C).

(iv) crystal growth controlled by transport phenomena of the crystal growth units in the solution.29,43 (v) crystal growth controlled by adsorption of ions or crystal growth units through the stagnant layer which surrounds the crystal particle.43 From the analysis above, cases ii and iv must be ruled out since it has been shown that the crystallization of HAP under the experimental conditions employed is not bulk diffusion controlled. Furthermore, case v cannot be important since the apparent thickness δ of the stagnant layer at the crystal surface has been calculated to be almost zero. Therefore, the possible growth mechanisms are cases i and iii, namely, spiral growth when the phenomenon is surface diffusion controlled (for high supersaturation levels σ . σ1) or crystal growth of two-dimensional surface nuclei (for high supersaturation levels σ . σ1). There is a little point in discussing the mechanism of the crystal growth if one has not determined whether σ . σ1. The value of the constant σ1, can be derived from the equation41

σ1 )

2πγedaσ λskT ln S

(7)

where γed is the edge-free work, k Boltzmann’s constant, T is the absolute temperature, “a” is the distance between two neighboring adsorption positions at equilibrium on the crystal surface, which depends on the temperature. At ambient temperatures, “a” is practically equal to the diameter of the crystal growth unit, the closest distance when two neighboring growth units are in contact. Finally, λs is the mean displacement of the adsorbed growth units on the crystal surface before incorporating in the crystal, given by Einstein’s formula

λs2 ) Dsτs

(8)

In this equation, Ds is the diffusion coefficient on the surface of a growth unit and τs the mean time of an adsorbed growth unit on the crystal surface before it becomes incorporated into the crystal lattice or it returns back to the bulk solution, given by41

Ds ) a2ν′s exp(-Us/kT)

(9)

1/τs ) νs exp(-∆Gs,ads/kT)

(10)

where Us is the activation energy for the displacement between two adjacent sites on the surface and ∆Gs,ads is

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Koutsopoulos

the change in Gibbs free energy for adsorption of a growth unit on the crystal surface. The frequency factors νs and νs′ are of the order of magnitude of the atomic frequency of vibration (∼10-13 s-1). On substitution of eqs 9 and 10 in eq 8, λs is approximated by44

λs ) (4 × 102)a ) 2.57 × 10-7 m

(11)

The edge free energy of eq 7 is equal to γed ) γd2. The interfacial energy, γ, at the interface between crystal surface and supersaturated solution, can be calculated from the semiempirical equation (modified to apply at the working temperature of 37 °C). This equation has been found to apply for many crystallizing compounds,39,43 -2

-3

γ ) d [12.08 - 1.17 ln(ceq/mol m )] × 10 5.58 × 10

-21

-2

2

J/m ) 2

J/m (12)

and therefore, the factor γed/kT ) 5.36 is calculated. This value is in accordance with the literature. For several crystallizing systems the γed/kT was found to vary between 0.2 and 5 for soluble and insoluble salts, respectively.43 HAP is a slightly soluble salt with a very small solubility product (i.e., k°s,HAP ) 2.35 × 10-59,24 as mentioned before). After substitution in eq 7, σ1 is calculated to be equal to σ1 ) 6.85 × 10-2. Thus, for the experimental conditions employed in the crystallization of HAP, the condition of σ . σ1 is fulfilled and so the linear dependence of R ) f(S) may be interpreted in terms of a surface diffusion controlled spiral growth mechanism or by the “birth and spread” of two-dimensional surface nuclei. To distinguish between the two growth mechanisms, the kinetic formulation proposed by the corresponding crystal growth models must be considered. Mathematical expressions describe the crystallization rate as a function of supersaturation.45 Burton, Cabrera, and Frank (BCF theory)41 developed the physical and mathematical background for the case of crystal growth on surface spirals. In the case where surface diffusion controls the kinetics of the crystallization, they demonstrated that the crystal growth rate is defined as

RBCF/s,diff ) C

()

σ1 σ2 tanh σ1 σ

(13)

where C is a constant. This equation may be transformed into the form

()

RBCF/s,diff σ ) Cσ1 σ1

2

()

tanh

σ1 σ

(14)

or

Y ) X2 tanh(1/X)

(15)

where Y ) (RBCF/s.diff/Cσ1) and X ) σ/σ1. The proposed kinetic equation from the “birth and spread” model is42,46,47 (43) Nielsen, A. E. J. Cryst. Growth 1984, 67, 289. (44) Mackenzie, J. K. Ph.D. Thesis, Bristol, U.K., 1950. (45) Garside, J.; Jansen-van Rosmalen, R.; Bennema, P. J. Cryst. Growth 1975, 29, 353. (46) O’Hara, M.; Reid, R. C. Modeling Crystal Growth Rates from Solutions; Prentice-Hall: Englewood Cliffs, NJ, 1973. (47) Gilmer, G. H.; Bennema, P. J. Cryst. Growth 1972, 13/14, 148.

[

RB&S ) kB&S(S - 1)2/3(ln S)1/6 exp -

]

πγ2Vmh

3(kT)2 ln S

(16)

where kB&S is a constant, γ is the interfacial energy between crystal surface of HAP and supersaturated solution, and h is the height of the growing step on the crystal surface (assumed to be equal to the height of the growth unit, i.e., h ) Vm1/3). This equation after approximations is expressed as45

RB&S/kB&SB5/6 ) (σ/B)5/6 exp(-B/σ)

(17)

and takes the form

Y ) X5/6 exp(-1/X)

(18)

where Y ) (RB&S/kB&SB5/6) and X ) σ/B. The constants B and kB&S can be calculated from the slope and the intercept, respectively, of the linear plot of eq 16, in logarithmic form (i.e., ln RB&S as a function of 1/ln S)

[

RB&S ) kB&S(S - 1)2/3(ln S)1/6 exp ln

[

RB&S 2/3

]

1/6

σ (ln S)

πγ2Vmh

]

3(kT)2 ln S

) ln kB&S - B

S

1 (19) ln S

Logarithmic plots according to eqs 14 and 17 are shown in Figure 2, where a better fitting was obtained when the BCF crystal growth model was considered. Best fitting was accomplished by allowing the values of the exponents and the pre-exponential factors of eqs 15 and 18 to change until the correlation factor converges to a minimum. In the case of eq 18, this resulted in a value of 0.62 for the exponent of X (instead of 5/6 ) 0.83 which appears in the equation). Furthermore, the pre-exponential factor should be multiplied by 0.014 (important change compared to unity of the original value). The same procedure followed for eq 15 to give the best fitting and the required changes in the original form were minor: the exponent of X was free to change and best fitting was achieved when it became equal to 1.9 (the original value in eq 15 is 2). No change was necessary for the pre-exponential factor. The fitting improved by the addition of a new term in the right part of the equation, equal to -0.03 (this only alters slightly the intercept from zero of the original equation to the new value). From this analysis it can be concluded that in the experimental conditions employed, the crystal growth of HAP is kinetically controlled by surface diffusion and the crystal growth mechanism is incorporation of growth units into the crystal on spiral dislocations of the surface. In the following, the formulation of the BCF model will be applied for the crystallization of HAP. Assuming one spiral on the crystal face, several morphological and thermodynamic parameters will be calculated. Equilibrium of the growth units between crystal surface and bulk solution must also be acknowledged.41 For supersaturation S ) 5.58 (i.e., experiment S-316), the mean distance between two neighboring kinks of a spiral step is39,41,43,48

( )

xo ) dS-1/2 exp

γed ) 5.77 × 10-8 m kT

(20)

where d is the diameter of the crystal growth unit (taken as d ≈ Vm1/3 ) 6.42 × 10-10 m). The critical radius of the (48) Cabrera, N.; Levine, M. M. Philos. Mag. 1956, 1, 450.

Crystal Growth of Hydroxyapatite

Langmuir, Vol. 17, No. 26, 2001 8097

expression for the rate of crystal growth, is equal to

C ) βBCFVmnoν exp(-∆G/ads/kT)

(25)

where ∆G/ads is the total adsorption energy which is the sum of the adsorption energy factors: from the solution / ) and from the surface to the kink to the surface (∆Gs,ads where the growth unit is incorporated into the crystal / ). Further, no is the number of the molecular (∆Gk,ads positions available for adsorption on the crystal surface, given by

no )

Atot SSAHAP mHAP ) ) Am Am 1.68 × 1018 (positions)·m-2 (26)

Figure 2. Plots of the dimensionless rate of hydroxyapatite crystal growth as a function of the dimensionless supersaturation, according to the transformed equations proposed from the BCF (9) and the polynucleation (b) model (dot and dash graphs represent the theoretical curves of the models, respectively).

spiral according to the BCF model is41

r/BCF )

γeda ) 2.00 × 10-9 m kT ln S

(21)

The distance between two neighbor steps of the spiral is

yo ) 4πr*) 2.51 × 10-8 m

(22)

The kink density on the crystal surface is calculated

S1/2 ln S 1 ) ) xoyo 4πa2(γ /kT) exp(γ /kT) ed ed 6.88 × 1014 kinks/m2 (23) As may be seen from eq 23, the kink density (i.e., the number of the active growth sites) is proportional to the solution supersaturation and it decreases when the interfacial energy, between crystal surface and solution, increases. The height of the spiral step on the crystal surface is assumed to be equal to the height of the crystal unit cell of HAP; thus the vertical mean rate of advance of the crystal surface is 1/3 ) 1.34 (crystal monolayers per hour) Rlin/Vunitcell (24)

where Rlin is the mean linear rate of growth of the crystal surface (at the experimental conditions of experiment S-316) and the volume of the unit cell of HAP, Vunitcell ) 5.287 × 10-28 m3. According to the Burton-CabreraFrank model, the value of the constant C from the

where Atot is the total surface area of the HAP crystals available for crystal growth in the supersaturated solution, SSAHAP is the specific surface area of the HAP crystals, and mHAP is the mass of the seed crystals. The area occupied by one molecule is taken equal to Am ) Vm2/3. From eq 26, after substitution of ν ) kT/h and assuming that βBCF ≈ 1 (which is valid when the reaction is surface diffusion controlled49), the Gibbs free energy for adsorption of a molecule of HAP from the solution to the crystal surface and incorporation into a kink may be calculated from eq 25

∆G/ads ) -kT ln

C ) 97.57 kJ/mol Vmnoν

(27)

The temperature is 37 °C ()310.15 K), and the constant C ) 9.27 × 10-14 m‚s-1 is calculated from the slope of the linear plot of Rlin as a function of the relative solution supersaturation, σ, as shown in Figure 1 (when σ . σ1, then tanh(σ1/σ) ≈ (σ1/σ) and so the crystal growth eq 13 takes a linear form R ) Cσ,41). This value of ∆G/ads is close to 18.50 kcal/mol ()77.33 kJ/mol), which was calculated using a different approach,33 and it is within the range of values obtained for many precipitating salts.50 By use of the crystallographic characteristics of the unit cell of HAP, the moles of HAP on the crystal surface can be calculated assuming that the area occupied by one molecule is equal to its projection on the surface (i.e., Vm2/3 ) 4.12 × 10-19 m2/molecule). Thus, the concentration of HAP molecules on the crystal surface is calculated to be 2.43 × 1018 molecules/m2. As shown in eq 23, when the formulation of the Burton-Cabrera-Frank model is used, the kink density on the crystal surface is 6.88 × 1014 kinks/ m2. After comparison of the surface concentration of molecules with the kink density of HAP, it is concluded that on the crystal surface the number of the active growth sites is 4 orders of magnitude less than the total number of HAP molecules. Therefore, with this mechanistic approach, it is demonstrated that not all of molecules on the surface act as crystallization centers (i.e., active growth sites). Most of them do not contribute in the crystal growth process as they are located on the flat crystal area between the steps of the spiral. LA0107906 (49) Bennema, P. J. Cryst. Growth 1967, 1, 287. (50) Bennema, P.; Boon, J.; van Leeuwen, C.; Gilmer, C. H. Krist. Tech. 1973, 8, 659.