Theoretical Analysis of Calcium Phosphate Precipitation in Simulated Body Fluid

Transcription

1 This is the Pre-Published Version Theoretical Analysis of Calcium Phosphate Precipitation in Simulated Body Fluid Xiong Lu and Yang Leng 1 Department of Mechanical Engineering Hong Kong University of Science and Technology Kowloon, Hong Kong, China Abstract The driving force and nucleation rate of calcium phosphate (Ca-P) precipitation in simulated body fluid (SBF) were analyzed based on the classical crystallization theory. SBF supersaturation with respect to hydroxyapatite (HA), octacalcium phosphate (OCP) and dicalcium phosphate (DCPD) was carefully calculated, considering all the association/dissociation reactions of related ion groups in SBF. The nucleation rates of Ca-P were calculated based on a kinetics model of heterogeneous nucleation. The analysis indicates that the nucleation rate of OCP is substantially higher than that of HA, while HA is most thermodynamically stable in SBF. The difference in nucleation rates between HA and OCP reduces with increasing ph in SBF. The HA nucleation rate is comparable with that of OCP when the ph value approaches 1. DCPD precipitation is thermodynamically impossible in normal SBF, unless calcium and phosphate ion concentrations of SBF increase. In such case, DCPD precipitation is the most likely because of its highest nucleation rates among Ca-P phases. We examined the influences of different SBF recipes, interfacial energies, contact angle and molecular volumes, and found that the parameter variations do not have significant impacts on analysis results. The effects of carbonate incorporation and calcium deficiency in HA were also estimated with available data. Generally, such apatite precipitations are more kinetically favorable than HA. 1 Corresponding author, address: meleng@ust.hk; fax:

2 Keywords: Calcium Phosphate; Simulated Body Fluid; Thermodynamics; Kinetics. Introduction The precipitation of bioactive calcium phosphate (Ca-P) in simulated body fluid (SBF), a solution with ion concentrations and a ph value similar to those of human blood plasma, or in other solutions with supersaturated calcium and phosphate has attracted extensive research interest [1-27]; because such Ca-P precipitation is similar to biological mineralization. Also, the Ca-P precipitation in such biomimetic solutions provides an alternative method of creating Ca-P coatings on titanium implants [15-27]. In addition, Ca-P precipitation in SBF has been widely used to assess the bioactivity of bioactive glass [28, 29], A-W glass-ceramic [1], ceramic α- CaSiO 3 [3], silica gel [31], bioceramics/polymer composites [32-3] and surface treated titanium ([2-, 8-1]. It is well known that precipitated Ca-P phases in aqueous solutions mainly include dicalcium phosphate [CaHPO 2H 2 O, DCPD], octacalcium phosphate [Ca 8 (HPO ) 2 (PO ) 5H 2 O, OCP] and hydroxyapatite [Ca 1 (OH) 2 (PO ) 6, HA]. HA is considered the most thermodynamically stable in physiological environment. OCP and DCPD, however, have been regarded as precursors of HA or the metastable phases of Ca-P because that they are kinetically favorable [35-39]. Experimental studies of Ca-P formation in simulated physiological environments have not clearly indicated the conditions for forming specific Ca-P phases. Although HA or bone-like apatite formation in SBF has frequently been reported, OCP formation in similar environments has also been observed. Marques et al. reported both OCP and HA precipitation on HA/TCP 2

3 biphasic ceramics in a simulated inorganic plasma (CSIP) solution with the same ion concentrations as conventional SBF except for HCO 3 [, 1]. Feng et al. claimed to have found a double layer Ca-P (HA/OCP) deposition by immersing alkali-treated Ti in a supersaturated HA solution [19-21, 2]. Barrere et al. found a pure OCP layer deposited on a non-alkaline treated Ti surface through a two-step SBF immersion procedure [15-16]. Koutsopoulos et al. studied calcification of fibrin and elastin in a simple solution of calcium chloride and potassium dihydrogen phosphate, and they found OCP formation in certain cases even though the driving force of HA formation was larger than that of OCP [2, 3]. Leng et al. examined the crystal structure of Ca-P precipitated in conventional and revised SBF using single crystal electron diffraction and identified exclusive OCP formation [-5]. We believe that the inconsistencies in reported Ca-P formation in supersaturated solutions are partially attributed to misidentification of the Ca-P crystal phases [-6]. On the other hand, we feel that there is a lack of theoretical guidelines for Ca-P precipitation in biomimetic solutions such as SBF. Thus, we are inspired to analyze Ca-P formation in SBF through theories of thermodynamics and kinetics. Thermodynamic analyses of Ca-P precipitation have been reported in the following solution systems: CaCl 2 KH 2 PO KOH [36, 37], Ca(NO 3 ) 2 KH 2 PO NaOH [38, 39], CaCl 2 KH 2 PO NaCl KOH [2-3], CaCl 2 NaH 2 PO NaCl NaOH [7], Ca(OH) 2 H 3 PO KOH HNO 3 CO 2 [8] and supersaturated HA solution [2]. Although analyses of the kinetics of Ca-P precipitation were investigated experimentally [36-39, 5-6], little theoretical analysis of the kinetics has been conducted, except in the work of Boistelle et al. [7]. They proposed a method to calculate a kinetic factor of nucleation rates based on the probability that 3

4 the ions units of calcium phosphates encounter each other to form a nucleus. The theoretical analysis of Ca-P precipitation in SBF is understandably scarce because of the complexity of its chemical composition. The difficulties of analyzing precipitation in SBF arise from the necessary considerations of all the association/dissociation balances between various ions and ion groups. In this report, we present our analysis of the free energy of Ca-P formation in SBF based on the classic theory of crystallization with few assumptions about chemical activities of each ion in SBF in the calculation. Our kinetics analysis of Ca-P heterogeneous nucleation in SBF is based on the kinetic model of Boistelle et al. [7]. The analysis includes Ca-P precipitation in SBF made from the various recipes listed in Table 1, which are conventional SBF (C-SBF), revised SBF (R-SBF), ionized SBF (I-SBF), modified SBF (M-SBF) and synthetic body fluid (S-SBF). Among them, C-SBF is the first SBF recipe being proposed. R-SBF has ion concentrations that are equal to those of blood plasma. The Cl and HCO 3 concentration of C-SBF, M-SBF and S- SBF are not exactly the same with those in blood plasma. Generally speaking, SBF does not include proteins. I-SBF however simulates the influence of proteins by including only the free ions which are not bound to the proteins. Of the magnesium ions in blood plasma (1.5 mmol dm 3 ),.5 mmol dm 3 of the Mg 2 are bound to proteins. Of the calcium ions in blood plasma (2.5 mmol dm 3 ),.9 mmol dm 3 of Ca 2 are bound to proteins [7]. The nominal Ca 2 and Mg 2 concentrations of I-SBF are equal to 1.6 and 1. mmol dm 3, respectively, after subtracting the concentrations of protein-bound ions. We also analyzed the cases of SBF with excessive amounts of calcium and phosphate ions, considering possible dissolution of calcium and phosphate from the substrates containing calcium and phosphor.

5 Analytical Model Driving forces of precipitation The thermodynamic driving forces for Ca-P precipitation were calculated based on the classical equation of free energy change in supersaturated solutions [9]: RT RT G = ln( S) ln( A p / Ksp) n = n, (1) where G is the Gibbs energy per mole of ionic units that compose Ca-P in solution, R is the gas constant (8.31 J K 1 mol 1 ), T is the absolute temperature, n is the number of ion units in a Ca-P molecule, and S is the supersaturation that is defined by the ratio of the activity product of ion units composing precipitates (A p ) to the corresponding solubility product (K sp ). The equations for precipitation in aqueous solutions given below define the ion units of HA, OCP and DCPD: Ca HPO = CaHPO ; (2) 2 2 Ca HPO 2PO = Ca (HPO )(PO ) ; (3) Ca 3PO OH = Ca (PO ) (OH). () Thus, the corresponding supersaturations (S) are defined as follows: 2 2 a(ca ) a(hpo ) S(DCPD) = K (DCPD) ; (5) sp a (Ca ) a(hpo ) a (PO ) S(OCP) = K (OCP) ; (6) sp 5

6 a (Ca ) a (PO ) a(oh ) S(HA) = K (HA), (7) sp where Ksp(DCPD) = [5], Ksp(OCP) = , Ksp(HA)= [51]. To determine the activity of an individual ion unit, all possible association/dissociation reactions in SBF are taken into account. In total, 17 association/dissociation reactions and mass balance equations are used to calculate the activities of all of the ion and ion units in SBF. The association/dissociation reactions are listed in Table 2, while the mass balance equations are given as follows: 2 CHCO = [H CO ] [HCO ] [CO ] [Ca HCO ] [CaCO ] [MgHCO ] [MgCO ] ; ( 8 ) C 2 Ca = [Ca ] [CaOH ] [Ca HCO ] [CaCO ] [CaH PO ] [Ca HPO ] [CaPO ] 2 ; ( 9 ) C 2 Mg = [Mg ] [MgOH ] [MgHCO ] [MgCO ] [MgH PO ] [MgHPO ] [MgPO ] 2 ; ( 1 ) C 2 HPO = [H PO ] [H PO ] [HPO ][PO ] [MgH PO ] [Mg HPO ] [MgPO ] 2 [CaH PO ] [CaHPO ] [CaPO ] 2, ( 11 ) where [ ] is the equilibrium concentration and C, C 2, 2 HCO 3 Ca C C 2 Mg HPO are the nominal concentrations in SBF listed in the recipes (Table 1). The activity coefficients, γ i, of each ion unit were obtained from the modified Debye-Hückel equation proposed by Davies [55] : I 1 I 1/2 2 logγ i = Azi.3 1/2 I, ( 12 ) where A is the Debye-Hückel constant dependent on temperature; A =.5211 at 37 ; z i is the 6

7 charge number of ions; and I is the total ionic strength of the solution and is defined as: 1 2 I = cz i i (13) 2 i in which c i is the molar concentration of each ion unit. A computer program based on the Newton-Raphson iteration method was written to solve numerically the simultaneous equations, including the 17 equations in Table 2 and the mass balance equations (Eqs. 8-11). The ionic strength was obtained by iterative computation, which eliminated possible errors from estimation. The equations for the activity calculations included those with magnesium in SBF, even though it is not an element in the Ca-P composition. Magnesium has been considered the inhibitor of Ca-P nucleation and growth in aqueous solution [15, 56]. This inhibitor role affecting activities of calcium and phosphate ions was taken into account in the analysis. The possible function of magnesium to block the active sites of Ca-P precipitation [56], however, could not be modeled by the classic theories of thermodynamics and kinetics. Nucleation rates The kinetics analysis was focused on the Ca-P nucleation rate (J), which can be estimated based on the classical model of heterogeneous nucleation [57] 2 3 G 16 π v γ f( θ) J = Kexp( ) = Kexp( ) (1) ( ) 3 3 kt 3kT lns in which k is the Boltzmann constant and T is the absolute temperature. The nucleation rate is proportional to a kinetic factor (K), and is exponentially affected by the activation energy of nucleation ( G), which is determined by the interfacial tension (γ ) between Ca-P and the solution, the supersaturation (S) and the contact angle function, f (θ ), for a nucleus on a substrate. 2 7

8 The Ca-P molecular volume (v) is defined by the Ca-P crystal structure. The v values of HA, OCP and DCPD are 263.2, 31.59, Å 3, respectively [7, 6]. The geometrical factor (16π/3) represents spherical nuclei, which should be changed for non-spherical nuclei (e.g., 32 for a cube). In this analysis, 16π/3 was used because our SEM examinations found the nuclei of Ca-P in SBF were in a hemispherical shape. The kinetic factor (K) should be proportional to the probability (P) that the appropriate ion units of Ca-P meet to compose a nucleus in the solution, i.e., K = K P in which K is a constant and the probability depends upon the concentrations of the ion units in the solutions [7]. The P values were calculated using the concentrations of ion units for DCPD, OCP and HA according to the method of Boiselle et al. [7]: 2![Ca ][HPO ] P = ([Ca ] [HPO ]) for DCPD ; (15) 7![Ca ] [HPO ][PO ] P =!2!([Ca ] [HPO ] [PO ]) for OCP; (16) 9![Ca ] [PO ] [OH ] P = 5!3!([Ca ] [PO ] [OH ]) for HA. (17) Boiselle et al. determined the K values from the experimentally measured J with given supersaturation and interfacial energies. According to their report, K should be cm 3 s 1 and this value was adopted in this analysis [7]. Accuracy of the K value does not affect the comparison of nucleation rates of DCPD, OCP and HA. The interfacial energy (γ) is also an uncertain parameter in calculating J in Eq. 1. In our 8

9 calculations, γ values were chosen as 1. mjm -2 for HA,.3 mjm -2 for OCP,. mjm -2 for DCPD based on the studies by Wu et al. [58-6]. Possible effects of the γ variations on the J calculations are addressed in the discussion section. The contact angle (θ) between the nucleus and the substrate surface can be determined experimentally. The globule-like Ca-P nucleus is widely reported [3, 13, 1, 5] regardless of the crystalline phase. We estimated the contact angle (θ) of Ca-P globule-like nuclei on surfaces of the alkali-treated titanium and calcium phosphate using SEM images and found that θ = 9 is a good estimation. The effects of the contact angle variations on the Ca-P nucleation rates are also addressed the discussion section. Results and Discussion The analysis results of the free energy changes of Ca-P precipitation ( G) and the Ca-P nucleation rates (J) in SBF are presented for a range of ph values. Figure 1 shows the comparison of G and J for the HA, OCP and DCPD precipitations in R-SBF, which is the primary SBF recipe in this analysis. The HA precipitation is thermodynamically favorable ( G HA becomes negative) when ph 5., while the OCP precipitation obtains its thermodynamic driving force when ph 6.3. However, there is no thermodynamic driving force for the DCPD precipitation ( G DCPD > ) over the entire ph range. The driving force of HA is always larger than that of OCP, although both increase with increasing ph values. On the other hand, the nucleation rate of OCP (J OCP ) is higher than that of HA (J HA ) by 1 orders of magnitude under physiological conditions (ph = 7.) as shown in Fig. 1b. This difference in nucleation rates reduces with increasing ph values, and J HA approaches J OCP when the ph value approaches 1. The results suggest that the ph value is the critical factor that affects Ca-P nucleation. Note that the DCPD nucleation rate (J CDPD ) is not shown in Fig. 1b because 9

10 nucleation rates can only be calculated when G is less than zero, according to the classical nucleation theory (Eq. 1). We found that Fig. 1 represents the general characteristics of G and J of Ca-P precipitation in SBF. The parameters that possibly affect the G and J calculations are discussed in the following sections. Solution composition Different SBF recipes Variation in the SBF composition results in changes in supersaturation and therefore affects G and J of Ca-P formation. The effects of different SBF recipes on G and J are shown in Fig. 2. The differences in G and J for different SBF recipes are insignificant, particularly when ph values are less than 9. In other words, variations in supersaturation in different SBF recipes are not sufficient to alter the general characteristics of Ca-P precipitation that are shown in Fig. 1. Note that there is no driving force for DCPD precipitation in SBF because G DCPD is always larger than zero (Fig 2e). This might be the reason that no DCPD precipitation has been previously reported in SBF, although DCPD is commonly believed to be a precursor to HA formation in aqueous solutions [37, 61]. According to Fig 2, the SBF recipes however can be divided into three groups: C-SBF and M-SBF (Group 1), R-SBF and S-SBF (Group 2), I-SBF (Group 3). The driving force of precipitation in Group 3 is lower than that of the others because the Ca 2 concentration in the Group 3 is the lowest (Table 1). Note that the higher concentration of HCO 3 in Group 2 than in Group 1 can also reduce the driving force. Since there are only small differences in the G and J of the different SBF recipes, we discuss the cases with R-SBF in the following sections. 1

11 High Ca and P concentrations Although variation in SBF recipe is not sufficient to cause significant G and J changes, it is of interest to calculate G and J of the Ca-P precipitation in SBF with higher than normal concentrations of calcium and phosphate ions. Such solutions with high calcium and phosphate concentrations have been used for biomimetic mineralization [16-21]. A SBF recipe with high calcium and phosphate ion concentrations might represent a local physiological environment in which a calcium phosphate implant is partially dissolved and releases calcium and phosphate ions to the surrounding body fluid. Figure 3 shows the effects of high Ca 2 2 and HPO concentrations in R-SBF on G and J at ph levels, 5, 7., and 1. The x-axis indicates that the Ca 2 and HPO 2 concentrations range from 1 to 5 times higher than those of normal R-SBF. The other ion concentrations remain the same as in normal R-SBF during the analysis. The results show that G HA, G OCP and G DCPD decrease with increasing Ca 2 2 and HPO concentrations at the same rate at ph levels of 5, 7. and 1. Note that G DCPD become negative in solutions when the Ca 2 and HPO 2 concentrations increase slightly at ph = 7.. The significance is that J DCPD is extremely high, even higher than J OCP, when G DCPD becomes negative. Thus, DCPD becomes the most kinetically favorable phase in SBF when the concentrations of calcium and phosphate are higher than the normal level at ph = 7.. Interfacial energy The interfacial energy (γ) is one of the most difficult parameters in Eq. 1 to determine. Note that interfacial energy and interfacial tension are identical and often are used interchangeably in the literature. The interfacial energy data used in our analysis actually are interfacial tension data that were determined experimentally. Note that these data of HA, OCP and DCPD are far from 11

12 certain and were determined by several means including the crystal growth kinetics, dissolution kinetics and contact angle measurements [58-6, 62-63]. Wu and Nancollas reviewed the measurements of the Ca-P interfacial tension and summarized the measured γ HA, γ OCP and γ DCPD as shown in Table 3 [6]. The kinetics methods employ the classical nucleation theory (Eq. 1) to determine the interfacial tension by measuring the nucleation or dissolution rate. Unfortunately, the measurements from different studies based on the kinetics method vary to a large degree. As shown in Table 3, from the kinetics method, γ HA varies from 9.3 to 87 mjm 2, while γ DCPD varies form. to 7 mjm 2. The kinetics method involves a number of variables that are difficult to control or measure experimentally. On the other hand, the contact angle method measures the rate of the rise of a solution in a thin-layer wicking capillary [58-5, 6]. The rising rate in a capillary of a solution relates to the interfacial tension according to the Washburn equation. The contact angle method generates more consistent results than does the kinetic method [6]. Thus, the interfacial tension values obtained with this method were chosen for analysis as listed in Table 3, except for DCPD because γ DCPD is negative from the contact angle method. The negative interfacial energy cannot be used in the calculation of nucleation rates in the classical nucleation model. As an alternative, we chose the minimum value of γ DCPD =. mjm 2 from the kinetics method. Considering the uncertainty of the interfacial tension data, we evaluated the effects of γ HA, γ OCP and γ DCPD on the J calculations. The curves in Fig. represent the J dependence on choosing the γ values listed in Table 3. Fig. a shows the comparison of the γ HA and γ OCP effects on J HA, J OCP in R-SBF, and Fig. b shows the comparison of the γ HA, γ OCP and γ DCPD effects in solutions 12

13 with the Ca 2 and HPO 2 concentrations that are five-fold higher than those of normal R-SBF. Although γ HA, γ OCP and γ DCPD vary to a large degree, J OCP is always larger than J HA as shown in Fig. a. The solution with high Ca 2 and HPO 2 concentrations follow the same trend (Fig b). Thus, we conclude that variations in interfacial energy data from different measurement methods do not change the general trends shown in Figs. 1-3; that is, J OCP is always larger than J HA in SBF, and J DCPD is the highest when DCPD precipitation is thermodynamically possible. Molecular volume Note that the molecular volume (v) used in the nucleation rate calculations (Eq. 1) is defined as the Ca-P crystal volume per molecule. Wu et al. argued that the volume of a crystal unit cell (v c ) should be used in Eq. 1, because the interfacial tension of calcium phosphate in solution is from the crystal/solution interface [6]. The v c values of HA, OCP and DCPD are 582.7, 122., 56.2 Å 3, respectively. This argument was supported by Wu et al. s experimental interfacial tension measurements by substituting v c for v in the kinetics method. When v c is adopted in the kinetics method, Wu et al. were able to obtain γ values comparable with those values obtained by the contact angle method [6]. Thus, we feel that it is necessary to evaluate the effect of replacing v with v c on the Ca-P nucleation rates. We compared the J HA and J OCP calculated with v c with the those calculated with v in Fig. 5. The comparison clearly indicates that we can ignore the variations of nucleation rates by using either v or v c in Eq. 1. Contact angle The contact angle function, f(θ), in Eq. 1 is given by the classical theory of heterogeneous nucleation as [57] : 13

14 2 (2 cos θ)(1 cos θ) f ( θ) =. (18) The f(θ) range corresponding to < θ < 18 is from to 1, while θ = 18 corresponds to homogeneous nucleation. Thus, the effect of the contact angle change on the nucleation rates is minor. Figure 6 shows that J HA and J OCP variations with θ are less than one order of magnitude and therefore can be ignored. This implies that there is little difference in the analysis results using either the homogeneous or the heterogeneous nucleation model. The selection of θ = 9 for the J calculations shown in Figs. 1-3 was simply based on the shape of Ca-P globular nuclei on titanium surfaces. Ion units of OCP formation Notice that the ion units that compose OCP nuclei can be expressed alternatively as in the following equation. Ca 3PO H = Ca (PO ) H. (19) Thus, the corresponding S and P should be a (Ca ) a (PO ) a(h ) S(OCP) = K (OCP) ; (2) sp 8![Ca ] [H ][PO ] P =!3!([Ca ] [H ] [PO ]) , (21) where Ksp(OCP) = [65]. The G OCP and J OCP calculations based on Eqs. 19 and 21 are different from those based on Eqs. 3 and 16. Figure 7a shows that the G OCP calculated based on Eqs. 19 and 21 is only slightly different from the previous calculation. The G difference results from the difference in the number of ion units (n = 7 in Eq. 3 and n = 8 in Eq. 1

15 25) during calculation. The n G is the same in both cases because the value of supersaturation does not change with the expressions. However, the difference in calculating J OCP cannot be 3 ignored because the kinetic factor in Eq. 1 significantly changes when the ion units of PO and H are used for the probability calculation. The probability (P) based on ion units in Eq. 21 is 1 1 times lower than that in Eq. 16. As a consequence, the HA nucleation becomes more kinetically favorable than OCP in SBF as shown in Fig 7b. We do not adopt Eq. 19 in the calculation because it implies that the hydrogen is a separate ion existing in OCP. This is not consistent with the OCP crystal structure in which one hydrogen ion is always associated with one PO 3 to form HPO 2 [66]. In addition, the probability calculation indicates that the chance of ion units in Eq. 19 combining together is too low compared with the chance of the ion units in Eq. 3. Thus, we believe that Eq. 19 should not be used for the OCP nucleation rate analysis. Apatite with carbonate or calcium-deficiency Carbonate can incorporate into apatite and substitute for PO or OH in the apatite crystal structure and subsequently change its properties [67]. Helebrant et al. found that the supersaturation of slightly carbonated apatite (SCHA) was even higher than that of apatite [68] in SBF. Also, calcium-deficient apatite (DOHA) is an important form of Ca-P in biological systems [56]. Comparing the driving force and nucleation rates of such irregular apatites with those of the stoichiometric HA, OCP and DCPD is important. The difficulties in analyzing such apatites come from uncertainties in chemical compositions and the lack of thermodynamic and kinetic data. Based on the available data, we managed to analyze two special cases: 1) Ca 1 (PO ) 6 (CO 3 ).5 (OH) (SCHA) and 2) Ca 9 (HPO ) (PO ) 5 OH (DOHA). The G SCHA and G DOHA were evaluated from 15

16 their supersaturation in SBF: a (Ca ) a (PO ) a (CO 3 ) a(oh ) S(SCHA) = K (SCHA) ; (22) sp a (Ca ) a(hpo ) a (PO ) a(oh ) S(DOHA) = K (DOHA), (23) sp in which Ksp(SCHA) = [68], and Ksp(SCHA) = [56]. There was lack of experimental data, particularly the kinetic factor used in Eq. 1, to calculate nucleation rates for SCHA and DOHA. We however note that the kinetic factor is mainly determined by the probability (P) that the appropriate ion units meet to compose a nucleus in the solution. The probabilities (P) of SCHA and DOHA can be calculated and compared with the stoichiometric HA. Actually, P plays a dominant role in determining the nucleation rates according to the Boistelle s theory [7]. The calculation of P SCHA was based on the concentration equivalency of OH and CO 2 3 : P = 18![Ca ] [PO ] [OH ] eq eq 1!6!([Ca ] [PO ] [OH ] ) (2) in which 2 OH eq = OH 2CO 3 because one CO 2 3 can substitute for two OH in the apatite crystal structure. The calculation of P DOHA is rather straightforward: 16![Ca ] [HPO ] [PO ] [OH ] P = 9!5!([Ca ] [HPO ] [PO ] [OH ]) (25) Fig 8 compares the analysis results of SCHA and DOHA with the other Ca-P. The results indicate that the SCHA precipitation obtains a similar level of thermodynamic driving force as the stoichiometric HA. However, the DOHA precipitation is less thermodynamically favorable and only comparable with OCP (Fig. 8a). The nucleation probabilities of SCHA and DOHA are 16

17 considerably higher than that of stoichiometric HA, but still less than those of OCP and DCPD (Fig. 8b). Thus, higher nucleation rates of SCHA and DOHA than that of stoichiometric HA are expected based on the analysis. We believe that analysis of these special cases indicates a general trend of carbonate-containing and calcium-deficient apatites. The general trend should be that precipitation of carbonate-containing and calcium-deficient apatites in SBF exhibit their kinetic advantages, compared with the stoichiometric HA. The analysis of DOHA indicates that the calcium-deficient apatite exhibits a lower thermodynamic driving force than does stoichiometric HA. Concluding remarks Ca-P formations in SBF were analyzed based on classical crystallization theories of thermodynamics and kinetics. We analyzed the possible effects of data variations and chemistry changes on the nucleation driving force and nucleation rates with best available thermodynamic and kinetic data. The analysis indicates that HA precipitation exhibits a higher thermodynamic driving force than does OCP and DCPD in SBF. OCP precipitation is kinetically favorable in SBF. The HA nucleation rate is significantly affected by the ph value. High ph environment is favorable for HA nucleation and the HA nucleation rate approaches the nucleations rates of OCP when the ph value approaches 1. DCPD does not have a thermodynamic driving force of precipitation in SBF, even though it has kinetic advantages in nucleation. DCPD precipitation becomes possible when the concentrations of calcium and phosphate ions increase to a higher than normal level in SBF. Possible variation of parameter values in the analysis model, such as discrepancies in the interfacial energy data or uncertainty of the contact angle of nuclei on surfaces, does not change these conclusions. However, HA precipitation can be considerably 17

18 affected by containing carbonate or being deficient in calcium. Generally, precipitation of carbonate-containing HA is more kinetically favorable than that of stoichiometric HA and has a same level of thermodynamic driving force; precipitation of calcium-deficient HA is also more kinetically favorable, but its thermodynamic driving force is lower than that of stoichiometric HA. Acknowledgements This project was financially sponsored by the Research Grants Council of Hong Kong (HKUST 637/2E) and the Funds for High Impact Areas at Hong Kong University of Science & Technology. The authors wish to acknowledge the valuable discussion about chemical equilibrium with Mr. Hu Quanyuan in the Department of Chemistry, Hong Kong University of Science and Technology. 18

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24 Ion Table 1 Nominal Concentrations of Different SBF Blood Plasma [6,7] C-SBF [6,7] Concentration (mmol dm 3 ) R-SBF [6,7] i-sbf [7] m-sbf [7] Syn- SBF [22] Na d 12. K Mg Ca Cl a a HCO SO buffer 6.63 b c c c 6.63 b HPO 2 a. Considering mmol dm 3 of extra chloride ion added with the Tris-HCl buffer. b. Tris (hydroxylmethyl) aminomethane (g/l). c. HEPES 2-(-(2-hydroxyethyl)-1-piperazinyl) ethane surfonic acid (g/l) d. Considering 15 mmol dm 3 of extra Na added with the HEPES-NaOH buffer 2

25 Table 2 Reactions in SBF Reaction K Ref. H CO (aq.) H HCO [7] HCO H CO [7] H PO (aq.) H H PO [7] 3 2 HPO H HPO [7] 2 2 HPO H PO [7] 2 3 Ca HCO Ca HCO [7] Ca CO CaCO (aq.) [7] Ca OH CaOH [37] Ca H PO Ca H PO 31.9 [52] [52] 2 Ca HPO Ca HPO (aq.) Ca PO Ca PO [52] Mg HCO 3 MgHCO [53] Mg CO MgCO (aq.) [53] 2 Mg OH MgOH [53] Mg H PO Mg H PO K = 1. [5] Mg HPO Mg HPO (aq.) 2 3 K = [5] Mg PO Mg PO K = [5] Table 3 Summary of Reported Interfacial Tensions[6] Researcher Year HA OCP DCPD Measurement method Nancollas et al Contact angle Crystal growth kinetics Dissolution kinetics Christoffersen et al 88 7 Crystal growth kinetics Crystal growth kinetics 96 6 Dissolution kinetics Lundager et al Crystal growth kinetics The unit of interfacial tension is mjm 2. 25

26 Figure Captions Figure 1. a) Free energy change ( G) of Ca-P precipitation in R-SBF as a function of ph value; b) Nucleation rates (J) of HA and OCP precipitation in R-SBF. Figure 2. Effects of SBF recipes on Ca-P precipitation: a) G HA ; b) J HA ; c) G OCP ; d) J OCP ; and e) G DCPD. Figure 3. Effects of Ca 2 and HPO 2 concentrations in SBF on Ca-P precipitation. The [Ca 2 ] and [HPO 2 ] vary from 1 to 5 times of those in normal SBF. a) G at ph = 5; b) J at ph = 5; c ) G at ph = 7.; d ) J at ph = 7.; e ) G at ph = 1; and f ) J at ph = 1. Figure. Effects of interfacial energy (γ) variations on the calculations of nucleation rates at ph = 7.. The solid line represents J HA ; the dashed line represents J OCP ; and the dotted line represents J DCPD. The triangles mark the data obtained from the contact angle method; the circles mark the data from the crystal growth kinetics method; and the rhombohedra mark the data from the dissolution kinetics method. a) in R-SBF; and b) in R-SBF with high [Ca 2 ] (12.5 mmol dm 3 ) and [HPO 2 ] (5. mmol dm 3 ). Figure 5. Comparison of the nucleation rates calculated from the molecular volume (v) and the volume of the crystal unit cell (v c ) in R-SBF. Figure 6. Effects of the contact angle (θ) on the nucleation rates in R-SBF at ph = 7.: a) J HA ; and b) J OCP. Figure 7. Comparison of using different ion units for OCP formation: a) Small differences in G OCP values using different ion unit; and b) questionable J OCP based on Eqs 19 and 21 compared with J HA. Figure 8. Comparison of precipitation of SCHA and DOHA with other Ca-P: a) G; b) kinetic probability (P). 26

27 G (KJ mol -1 ) 8 - a DCPD OCP HA ph 25 2 b Log J OCP HA ph Figure 1. LU and LENG 1

28 G (KJ mol -1 ) HA a C-SBF R-SBF I-SBF M-SBF S-SBF Log J b HA C-SBF R-SBF I-SBF M-SBF S-SBF ph ph G (KJ mol -1 ) c OCP C-SBF R-SBF I-SBF M-SBF S-SBF Log J d OCP C-SBF R-SBF I-SBF M-SBF S-SBF ph ph 8 e G (KJ mol -1 ) DCPD C-SBF R-SBF I-SBF M-SBF S-SBF ph Figure 2 LU and LENG 2

29 G (KJ mol -1 ) a ph = 5 DCPD OCP HA Log J b ph = 5 DCPD OCP HA Ca_P Ca_P G (KJ mol -1 ) c DCPD OCP HA ph = 7. ph = 7. Log J d DCPD OCP HA Ca_P Ca_P G (KJ mol -1 ) e DCPD OCP HA ph = 1 ph = 1 Log J f DCPD OCP HA Ca_P Ca_P Figure 3 LU and LENG 3

30 a Log J γ ( mjm 2 ). b Log J γ ( mjm 2 ) Figure LU and LENG

31 25 2 Log J OCP v HA v OCP v c HA v c ph Figure 5 LU and LENG 5

32 J (1 7 cm 3 s 1 ) a Contact angle J (1 16 cm 3 s 1 ) b Contact angle Figure 6 LU and LENG 6

33 G (KJ mol -1 ) 8 - a OCP Eqs 3 & 16 OCP Eqs 19 & ph 25 2 b Log J 15 1 OCP HA ph Figure 7 LU and LENG 7

34 G (KJ mol -1 ) a DCPD OCP HA SCHA DOHA ph Log P b DCPD OCP HA SCHA DOHA ph Figure 8 LU and LENG 8