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Kinetics of Dissolution of β-Tricalcium Phosphate Ruikang Tang, Wenju Wu, Michael Haas, and George H. Nancollas* Department of Chemistry, Natural Sciences Complex, State University of New York at Buffalo, Buffalo, New York, 14260 Received December 11, 2000. In Final Form: March 5, 2001
The dissolution kinetics of β-tricalcium phosphate (β-TCP) was investigated at 37 °C using the constant composition (CC) method over a range of undersaturation. The rates decreased markedly with time despite the sustained driving force and eventually approached zero, indicating a critical condition in the dissolution process. The calculated reaction order with respect to relative undersaturation, 5.5, suggested a surface-pit (polynucleation) mechanism of β-TCP dissolution. The interfacial tension between β-TCP and the solution phase obtained from this model, γSL ) 3.8 mJ m-2, was consistent with its dissolution mechanism and relatively high solubility. The addition of magnesium or zinc ions retarded β-TCP dissolution; the latter completely suppressing the reaction at a concentration as low as 2 × 10-6 mol L-1.
Introduction β-tricalcium phosphate (β-Ca3(PO4)2, β-TCP) is now a widely accepted candidate for synthetic bone grafting, calcium phosphate cements, and surgical implant applications. The effective use of this mineral in biological systems is mostly related to its dissolution behavior, yet few kinetic studies have been performed. For example, the presence of β-TCP in plasma-sprayed hydroxyapatite (HAP) coatings leads to mechanical instability of the implanted coating due to its enhanced in vivo dissolution.1 In other situations, enhanced dissolution may be desired to achieve fast bony adaptation, reduced healing time, and increased tolerance of surgical inaccuracies.2,3 Whitlockite, a mineral related to β-TCP, is a major constituent of human dental calculus,4,5 and it also occurs in carious lesions.6-8 The formation and stability of whitlockite is enhanced by the presence of cations smaller than Ca2+, such as Mg2+ and Fe2+.7 There appears to be a structural difference between material prepared at high temperature (referred to here as β-Ca3(PO4)2) and that formed in an aqueous environment (whitlockite). Although the removal of β-TCP with concomitant calculus demineralization is of considerable interest for improving periodontal health, information on the dissolution kinetics of either form of this salt in aqueous solutions has hitherto been lacking. Zinc, used as an effective anitcalculus ion in mouth rinses, influences the in vitro physical/chemical nature of calcium phosphates usually found in human dental calculus.9 The objective of this work is to use a constant composition technique to study the dissolution kinetics of β-TCP in aqueous solutions in the absence and presence of magnesium and zinc ions as a function of thermodynamic driving force. Experimental Method β-TCP (purity > 96%) was obtained from Clarkson (S. Williamsport, PA) and was characterized by X-ray diffraction, * Corresponding author. (1) Radin, S.; Ducheyne, P. In Characterization and performance of calcium phosphate coatings for implants; Horowitz, E., Parr, J. E., Eds.; ASTM: Philadelphia, 1994. (2) Lemons, L. E. Clin. Orthop. Relat. Res. 1988, 235, 220. (3) Kay, J. F. Biomed. Mater. Res. 1988, 22, 127. (4) Jensen, A. T.; Rowles, S. L. Acta Odontol. Scand. 1957, 16, 121. (5) Leung, S. W.; Jensen, A. T. Int. Dent. J. 1958, 8, 613.
chemical analysis,10 and infrared spectroscopy. The specific surface area was 0.83 m2/g (BET nitrogen adsorption; 30/70 N2/He, Quantasorb II, Quantachrome Corp., Greenvale, NY). Dissolution experiments, initiated by the introduction of known amounts of β-TCP crystals, were conducted in magnetically stirred (450 rpm) double-jacketed vessels thermostated at 37.0 ( 0.1 °C. Undersaturated solutions (ionic strength I ) 0.15 mol L-1, pH ) 6.2) were prepared by slowly mixing filtered (0.22 µm Millipore filter) calcium chloride, potassium dihydrogen phosphate, sodium chloride, and potassium hydroxide solutions. Nitrogen, saturated with water vapor at 37 °C, was purged through the reaction vessels to exclude carbon dioxide. In the constant composition method (CC), a titrant solution containing the lattice ions was simultaneously added to the reaction solutions to compensate for changes due to dissolution. The titrants were prepared having β-TCP stoichiometries:
TNaCl ) WNaCl + WKOH + 2Ceff
(1)
THCl ) 4Ceff - WKOH
(2)
Ceff ) 1/3WCa ) 1/2WP
(3)
W and T are the total concentrations in the reaction solutions and titrants, respectively, and Ceff is the effective titrant concentration. For dissolution experiments made in the presence of magnesium (1 × 10-5 to 1 × 10-4 mol L-1) and zinc (1 × 10-7 to 2 × 10-6 mol L-1) ions, magnesium chloride and zinc chloride were added to both reaction and titrant solutions. Titrant addition was triggered by a potentiometer (Orion 720A, U.K.) incorporating a glass electrode (Orion No. 91-01, U.K.) and a reference electrode (Orion 900100, U.K.). During the dissolution, the output of the potentiometer was constantly compared with a preset value and the difference, or error signal, activated motor-driven titrant burets, thereby maintaining a constant thermodynamic driving force. Chemical analysis of solution samples periodically withdrawn and filtered (0.22 µm Millpore filter) showed that the total calcium (atomic absorption) and phosphate (spectrophotometrically, as (6) Llory, H.; Frank, R. M. Actual Odonto-Stomatol. 1969, 88, 507. (7) LeGeros, R. Z. Calcium Phosphates in Oral Biology and Medicine; Karger: New York, 1991. (8) Takuma, S.; Sunohara, H.; Watababe, H.; Yama, H. Bull. Tokyo Dent. Coll. 1969, 10, 173. (9) Brunski, B. J.; Kim, D. G.; Rubin, M.; Legeros, R. Z.; Pan, P. C. J. Dent. Res. 2000, 79, 258. (10) Tomson, B. M.; Barone, J. B.; Nancollas, G. H. At. Absorpt. Newsl. 1977, 16, 117.
10.1021/la001730n CCC: $20.00 © 2001 American Chemical Society Published on Web 04/10/2001
Kinetics of Dissolution of β-Tricalcium Phosphate
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Figure 1. Plots of titrant volume against time for β-TCP dissolution at different undersaturations. the vanadomolybdate complex) concentrations remained constant to within 1%. The relative undersaturation, σ, was defined by eq 4
σ)
[
]
(Ca2+)3(PO43-)2 Ksp
1/5
-1
(4)
where (Ca2+) and (PO43-) are the ionic activities, calculated using the Davies extended form of the Debye-Hu¨ckel equation using mass balance expressions for total calcium and total phosphate with appropriate equilibrium constants by successive approximation for the ionic strength. Values for the dissociation constants of phosphoric acid were K1 ) 6.22 × 10-3,11 K2 ) 6.58 × 10-8,12 and K3 ) 6.6 × 10-13 mol L-1,13 with the water ionic product 2.40 × 10-14 mol2 L-2.14 The formation constants for the ion pairs CaH2PO4+, CaHPO4, CaPO4-, and CaOH+ were taken as 28.1, 589, 1.40 × 106, and 25 mol-1 L,15 respectively. The solubility activity product of β-TCP, Ksp, was 2.51 × 10-30 mol5 L-5.16
Results and Discussion I. Dissolution Deceleration. The dissolution rates at any instant were determined from plots of titrant volume added as a function of time. An example is shown in Figure 1. The overall dissolution rate, J, is defined by eq 5
J)
Ceff dV AT dt
(5)
where dV/dt is the titrant curve gradient. The value of the surface area during dissolution, AT, was estimated from eq 6 assuming a uniform three-dimensional dissolution
AT ) m0SA(mt/m0)2/3
(6)
in eq 6 m0 and mt are the masses of the crystal initially and at time t, respectively. SA is the specific surface area of the seed crystals. It can be seen in Figure 2 that the corrected rates decreased with time, eventually approaching zero even though only a part of the β-TCP crystals had undergone reaction in the undersaturated solutions. From the total titrant volume added, only about 17%, 26%, 53%, and 69% of the seed crystals dissolved at (11) Bates, R. G. J. Res. Natl. Bur. Stand. 1951, 47, 2236. (12) Bates, R. G.; Acree, S. F. J. Res. Natl. Bur. Stand. 1945, 34, 396. (13) Bjerum, A.; Unmack, A. K. Dans. Vidensk. Selsk. Mater.sFys. Medd. 1929, 9, 1. (14) Harned, H. S.; Owen, B. B. The Physical Chemsitry of Electrolytic Solutions, 3rd ed.; Reinhold: New York, 1958. (15) Zhang, J.; Ebrahimpour, A.; Nancollas, G. H. J. Solution Chem. 1991, 20, 455. (16) Shyu, L. J.; Perez, L.; Zawacki, S. J.; Heughebaert, J. C.; Nancollas, G. H. J. Dent. Res. 1982, 62, 398.
Figure 2. Plots of corrected dissolution fluxes of β-TCP as a function of time at different undersaturations.
σ values of -0.445, -0.582, -0.651, and -0.720, respectively. This interesting dissolution termination has been noted with other sparingly soluble salts such as octacalcium phosphate, dicalcium phosphate dihydrate, and even hydroxyapatite, where it was suggested that the effect was directly related to the formation, movement, and length of the dissolution steps on the crystal surfaces.17 By analogy with the length-dependent step speed equation for crystal growth,18,19 during dissolution, decreases in crystal size and average step length resulted in rate deceleration. Since the dissolution steps do not move spontaneously until the length exceeds the critical limit, the presence of a critical condition can be explained in terms of the increase of edge free energy with the formation of steps/pits on the surface. Thus, when the crystallite reaches a very small size, the formation of critical dissolution steps/pits becomes difficult and existing steps may be too short to undergo dissolution.20 Scanning electron micrographs (SEMs, Hitachi S-800, Japan) of crystallites remaining in the dissolutionterminated solutions showed that they decreased in size with an increase in undersaturation. However, the particle sizes did not change following reaction suppression (Figure 3). It is also interesting to note that although some steps were present on the surfaces, they did not contribute to dissolution. At lower undersaturation, the surfaces of the dissolution-terminated crystals were smooth with few long steps (Figure 3b), while, at higher undersaturations, the surface became rough with multiple short dissolution steps (Figure 3c and d). A previous study demonstrated that the value of the critical length, l*, increased with the decreasing undersaturation or increasing γ, the edge free energy.18,19,21 Thus, higher undersaturations lead to smaller sized dissolution steps, as shown in Figure 3. It should be emphasized that neither chemical analysis nor energy dispersive spectoroscopy (EDS) revealed any phase change during the dissolution process. II. Kinetics and Interfacial Free Energy. Although the interfacial tension, γSL, between a solid crystal phase (S) and the surrounding medium (L) cannot be measured directly, it plays a critical role in nearly all theoretical dissolution kinetics expressions. The mechanisms of dissolution are usually assessed by confronting experimental rate data at different thermodynamic driving forces (17) Tang, R.; Nancollas, G. H. J. Cryst. Growth 2000, 212, 261. (18) Burton, W. K.; Cabrera, N.; Frank, F. C. R. Soc. London Philos. Trans. 1951, A243, 299. (19) Hurle, D. T. J. Handbook of Crystal Growth; North-Holland: Amsterdam-London-New York-Tokyo, 1993. (20) Tang, R.; Nancollas, G. H.; Orme, C. A. The mechanism of dissolution of sparingly soluble electrolytes. Submitted. (21) Chernov, A. A. Sov. Phys. 1961, 4, 116.
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Figure 3. Scanning electron micrographs of β-TCP seeds (a) and dissolution suppressed crystallites at σ of -0.582 (b), -0.651 (c), and -0.720 (d).
and fitting to an empirical rate law such as eq 7:
J ) kd|σ|n
(7)
In eq 7 kd is the rate constant and the effective reaction
order, n ) 5.5, was obtained from the slope of the logarithmic plot shown in Figure 4 of the dissolution rate (a mean value for the first 20 min of reaction) as a function of relative undersaturation. This value of n suggests that dissolution is controlled by the formation and growth of
Kinetics of Dissolution of β-Tricalcium Phosphate
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Figure 6. Plots of titrant volume against time for β-TCP dissolution at σ ) -0.651 in the absence and presence of magnesium ions. Table 1. Experiment for β-TCP Dissolution at Different Relative Uundersaturations, σ
Figure 4. Plot of ln J against ln |σ| for β-TCP dissolution.
103TCaa
103TPa
L-1
L-1
mol
σ -0.445 -0.582 -0.651 -0.720
1.20 0.90 0.75 0.60
mol
0.80 0.60 0.50 0.40
TNaCla mol
L-1
104Ceff mol L-1
0.145 0.146 0.147 0.148
4.0 3.0 2.5 2.0
a T , T , and T Ca P NaCl are total concentrations of CaCl2, KH2PO4, and NaCl in the working solution, respectively.
Table 2. Interfacial Tension γSL between Calcium Phosphates and Solutions Obtained from Dissolution Kinetics
Figure 5. Plot of ln J against 1/ln S for β-TCP dissolution.
surface pits.22 In terms of this model, the interfacial tension and rate of dissolution are given by eq 8
(
)
a4γSL2 J ) A exp β 2 2 k T ln S
(8)
in which A is a constant, a is the mean molecular size (aTCP ) 19.4 × 10-10 m), k and T are the Boltzmann constant and absolute temperature, respectively, and β is a shape factor. A plot of ln J against (ln S)-1 in Figure 5 gives a straight line with slope corresponding to γSL ) 3.8 mJ m-2 between TCP and the undersaturated solution. This value and the β-TCP solubility are reasonable compared with those of other calcium phosphates (Table 2). It has long been recognized that there is a close relationship between solubility and interfacial tension. During dissolution some neighboring ions on the surface are replaced by water molecules to form units that escape into the bulk solution. Higher values of γSL indicate a greater difficulty in forming such an interface between (22) Ohara, M.; Reid, R. C. Modeling Crystal Growth Rates from Solution; Prentice Hall: Englewood Cliffs, NJ, 1973.
phase
γSL/mJ m-2
CaHPO4‚2H2O (DCPD) Ca8H2(PO4)6‚5H2O (OCP) β-Ca3(PO4)2 (β-TCP) Ca10(PO4)6 (OH)2 (HAP) Ca10(PO4)6F2 (FAP)
0.423 2.924 3.8 9.324 17.124
the solid and the aqueous phase. Both HAP and fluorapatite (FAP) have higher values of the interfacial tension and lower solubilities in water while β-TCP, octacalcium phosphate (OCP), and dicalcium phosphate dihydrate (DCPD) have relatively lower interfacial tensions but higher solubilities. III. Influence of Magnesium and Zinc. Typical CC dissolution curves at σ ) -0.651 in the presence and absence of magnesium ion are shown in Figure 6. It can be seen that the reaction rate was significantly retarded at concentrations as low as 1 × 10-5 mol L-1. At a magnesium concentration of 1 × 10-4 mol L-1, the initial rate of β-TCP dissolution (7.4 × 10-6 mol m-2 min-1) was less than one-fifth of that in the absence of this additive. This is in agreement with other findings in our laboratory which showed that whitlockite dissolved less readily than β-TCP even though the chemical compositions were approximately the same. The effectiveness of magnesium as a dissolution inhibitor suggests adsorption of this ion on the crystal surfaces. In terms of the well-known model of Cabrera and Vermileya, where growth is assumed to be accomplished through a barrier of adsorbed inhibitor ions upon the smooth crystal surface,25 a step moving across the crystal surface will be stopped if the inhibitor ions/molecules are separated by a distance less than l*. We suggest that a similar model can be applied to the results of the dissolution experiments. Moreover, the adsorption of impurities such as magnesium ion onto crystal faces may change the relative surface free energies of the faces,
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Figure 8. “Langmuir” kinetic adsorption isotherm for the influence of magnesium ions on β-TCP dissolution.
Figure 9. Plots of titrant volume against time for β-TCP dissolution at σ ) -0.651 in the absence and presence of zinc ions.
Figure 10. “Langmuir” kinetic adsorption isotherm for β-TCP dissolution in the presence of zinc ions.
Figure 7. Scanning electron micrographs of dissolutionterminated β-TCP crystallites in the presence of magnesium at σ ) -0.651. Magnesium concentrations are 3.0 × 10-5 (a) and 1.0 × 10-4 mmol L-1 (b).
lowering the tendency for solute to be released from the crystal lattice. l* may therefore increase with increasing magnesium ion concentration. This suggestion is also supported by Figures 3c and 7, which show SEMs of the
dissolution-terminated crystallites in the absence and presence of magnesium ion, respectively. At a low concentration of Mg2+ (3 × 10-5 mol L-1, Figure 7a), the dissolution steps were similar to those in the absence of this ion (Figure 3c). However, the surfaces of the crystals were much smoother with fewer dissolution steps at a magnesium concentration of 1 × 10-4 mol L-1 (Figure 7b). The adsorption of inhibitor molecules on crystal surfaces may be interpreted in terms of a Langmuir equilibrium adsorption isotherm. The application of this model in interpreting the reduction in the dissolution rates of numerous sparingly soluble salts in the presence of inhibitors has been very successful. In terms of the reaction rate, the Langmuir adsorption isotherm may be written as eq 926 (23) Nancollas, G. H.; Wu, W. J. Dispersion Sci. Technol. 1998, 19, 723.
Kinetics of Dissolution of β-Tricalcium Phosphate
J0 1 1 + ) J0 - Ji 1 - b K(1 - b)C
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(9)
where J0 and Ji are the dissolution rates in the absence and in the presence of inhibitor, respectively, and b is a dimensionless parameter. If b ) 0, the inhibitor is capable of completely inhibiting the rate of dissolution at concentrations approaching infinity. The inhibitor is incapable of completely inhibiting dissolution if 0 < b < 1, while if b < 0, the inhibitor can completely suppress the reaction at concentrations below those corresponding to a monolayer. In Figure 8, values of K ) 7.2 ((0.3) × 103 L mol-1 and b ) -0.82 ((0.4) are obtained from the linear plot of J0(J0-Ji)-1 as a function of [Mg2+]-1. The CC dissolution curves of β-TCP in the presence of zinc ions with corresponding Langmuir parameters K ) 1.8 ((0.1) × 106 L mol-1 and b ) -0.28 ((0.2) are shown in Figures 9 and 10, respectively. It can be seen that zinc (24) Wu, W.; Nancollas, G. H. Adv. Colloid Interface Sci. 1999, 79, 229. (25) Cabrera, N.; Vermileya, D. A. In Growth and Perfection of Crystal; Doremus, R. H., Roberts, B. W., Turnbull, D., Eds.; Wiley: New York, 1958. (26) Nancollas, G. H.; Zawacki, S. J. In Industrial Crystallization ’84; Jancic, S. J., DeLong, E. J., Eds.; Elsevier: Amsterdam, 1984.
ion is a more effective inhibitor of β-TCP dissolution than magnesium, as its adsorption affinity constant is greater by a factor of 250. From the Langmuir representation, the dissolution reaction can be completely suppressed at a very low concentration, [Zn2+] ) 2 × 10-6 mol L-1 (Figure 9). Conclusion Constant composition dissolution results for β-TCP gave an effective reaction order n ) 5.5, suggesting a mechanism involving the formation of surface pits. The rate of β-TCP dissolution was markedly reduced by the presence of magnesium and zinc ions. The latter was the more effective inhibitor, completely suppressing the reaction at a concentration as low as 2 × 10-6 mol L-1. In agreement with the results of previous studies of the dissolution of sparingly soluble salts, dissolution deceleration of β-TCP at constant undersaturation demonstrates the influence of size reduction on the dissolution processes, and the termination suggests that important critical conditions must be taken into account. Acknowledgment. This work was supported by the National Institutes of Health (DE03223) and the Health Care Research Center of Procter & Gamble. LA001730N
