1

Theoretical Analysis of Calcium Phosphate Precipitation in Simulated Body Fluid

Xiong Lu and Yang Leng1

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 10. 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, email address: meleng@ust.hk; fax: 852-23581543

This is the Pre-Published Version

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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-

CaSiO3 [30], silica gel [31], bioceramics/polymer composites [32-34] and surface treated

titanium ([2-4, 8-14].

It is well known that precipitated Ca-P phases in aqueous solutions mainly include dicalcium

phosphate [CaHPO4⋅2H2O, DCPD], octacalcium phosphate [Ca8(HPO4)2(PO4)4⋅5H2O, OCP] and

hydroxyapatite [Ca10(OH)2(PO4)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

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biphasic ceramics in a simulated inorganic plasma (CSIP) solution with the same ion

concentrations as conventional SBF except for HCO3− [40, 41]. 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, 24]. 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 [42, 43]. 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 [44-45]. We believe that the

inconsistencies in reported Ca-P formation in supersaturated solutions are partially attributed to

misidentification of the Ca-P crystal phases [44-46]. 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: CaCl2 + KH2PO4 + KOH [36, 37], Ca(NO3)2 + KH2PO4 + NaOH [38, 39], CaCl2 +

KH2PO4 + NaCl + KOH [42-43], CaCl2 + NaH2PO4 + NaCl + NaOH [47], Ca(OH)2 + H3PO4 +

KOH + HNO3 + CO2 [48] and supersaturated HA solution [20]. Although analyses of the

kinetics of Ca-P precipitation were investigated experimentally [36-39, 45-46], little theoretical

analysis of the kinetics has been conducted, except in the work of Boistelle et al. [47]. They

proposed a method to calculate a kinetic factor of nucleation rates based on the probability that

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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. [47]. 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 HCO3− 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), 0.5 mmol dm−3 of the Mg2+ are bound to proteins. Of the calcium ions in blood plasma

(2.5 mmol dm−3), 0.9 mmol dm−3 of Ca2+ are bound to proteins [7]. The nominal Ca2+ and Mg2+

concentrations of I-SBF are equal to 1.6 and 1.0 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.

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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 [49]:

ln( ) ln( / )p spRT RTG S A Kn n

∆ = − = − , (1)

where ∆G is the Gibbs energy per mole of ionic units that compose Ca-P in solution, R is the gas

constant (8.314 J K−1mol−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 (Ap) to the corresponding solubility product (Ksp).

The equations for precipitation in aqueous solutions given below define the ion units of HA,

OCP and DCPD:

2+ 24 4Ca HPO CaHPO−+ = ; (2)

2+ 2 3 2 34 4 4 4 4 24Ca HPO 2PO Ca (HPO )(PO )− − − −+ + = ; (3)

2+ 34 5 4 35Ca 3PO OH Ca (PO ) (OH)− −+ + = . (4)

Thus, the corresponding supersaturations (S) are defined as follows:

2 24(Ca ) (HPO )(DCPD)

(DCPD)sp

a aSK

+ −

= ; (5)

4 2 2 2 34 4(Ca ) (HPO ) (PO )(OCP)

(OCP)sp

a a aSK

+ − −

= ; (6)

6

5 2 3 34(Ca ) (PO ) (OH )(HA)

(HA)sp

a a aSK

+ − −

= , (7)

where Ksp(DCPD) = 10−6.622 [50], Ksp(OCP) = 10−36.48, Ksp(HA)= 2.35 × 10−59 [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 4 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 4 mass balance equations are

given as follows:

3

2 +2 3 3 3 3HCO

+3 3 3

= [H CO ] + [HCO ] + [CO ] + [Ca HCO ]

+ [CaCO ] + [MgHCO ] + [MgCO ]

C −− −

− ; ( 8 )

2+

2+ + +3 3Ca

+2 4 4 4

= [Ca ] + [CaOH ] + [Ca HCO ] + [CaCO ]

+ [CaH PO ] + [Ca HPO ] + [CaPO ]

C−

; ( 9 )

2+

2+ + +3 3Mg

+2 4 4 4

= [Mg ] + [MgOH ] + [MgHCO ] + [MgCO ]

+ [MgH PO ] + [MgHPO ] + [MgPO ]

C−

; ( 10 )

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2 33 4 2 4 4 4HPO

+2 4 4 4

+2 4 4 4

= [H PO ] + [H PO ]+ [HPO ]+[PO ]

+ [MgH PO ] + [Mg HPO ] + [MgPO ]

+ [CaH PO ] + [CaHPO ] + [CaPO ]

C −− − −

−

−

, ( 11 )

where [ ] is the equilibrium concentration and 3HCO

C − , 2+Ca C , 2+Mg

C 24HPOC − are the nominal

concentrations in SBF listed in the recipes (Table 1).

The activity coefficients, γi, of each ion unit were obtained from the modified Debye-Hückel

equation proposed by Davies [55] : 1/ 2

21/ 2log 0.3

1i iIAz II

γ⎡ ⎤

= − −⎢ ⎥+⎣ ⎦ , ( 12 )

where A is the Debye-Hückel constant dependent on temperature; A = 0.5211 at 37°; zi is the

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charge number of ions; and I is the total ionic strength of the solution and is defined as:

212 i i

iI c z= ∑ (13)

in which ci 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

23 3

16 ( )exp( ) exp( )3 ln

G v fJ K KkT k T S

π γ θ∆= − = − (14)

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.

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The Ca-P molecular volume (v) is defined by the Ca-P crystal structure. The v values of HA,

OCP and DCPD are 263.24, 310.59, 126.54 Å3, respectively [47, 60]. 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 [47]. The P values

were calculated using the concentrations of ion units for DCPD, OCP and HA according to the

method of Boiselle et al. [47]:

2 24

2 2 24

2![Ca ][HPO ]([Ca ] [HPO ])

P+ −

+ −=+

for DCPD ; (15)

2 4 2 3 24 4

2 2 3 74 4

7![Ca ] [HPO ][PO ]4!2!([Ca ] [HPO ] [PO ])

P+ − −

+ − −=+ +

for OCP; (16)

2 5 3 34

2 3 94

9![Ca ] [PO ] [OH ]5!3!([Ca ] [PO ] [OH ])

P+ − −

+ − −=+ +

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 13.64×10−24

cm−3s−1 and this value was adopted in this analysis [47]. 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. 14. In our

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calculations, γ values were chosen as 10.4 mJm-2 for HA, 4.3 mJm-2 for OCP, 0.4 mJm-2 for

DCPD based on the studies by Wu et al. [58-60]. 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, 14, 45] 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 = 90° 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

(∆GHA becomes negative) when pH ≥ 5.4, while the OCP precipitation obtains its

thermodynamic driving force when pH ≥ 6.3. However, there is no thermodynamic driving force

for the DCPD precipitation (∆GDCPD > 0) 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 (JOCP) is higher than that of HA (JHA) by 10 orders of

magnitude under physiological conditions (pH = 7.4) as shown in Fig. 1b. This difference in

nucleation rates reduces with increasing pH values, and JHA approaches JOCP when the pH value

approaches 10. The results suggest that the pH value is the critical factor that affects Ca-P

nucleation. Note that the DCPD nucleation rate (JCDPD) is not shown in Fig. 1b because

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nucleation rates can only be calculated when ∆G is less than zero, according to the classical

nucleation theory (Eq. 14). 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 ∆GDCPD 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 Ca2+ concentration in the Group 3 is the lowest (Table 1). Note that the higher

concentration of HCO3− 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.

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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 Ca2+ and HPO42−

concentrations in R-SBF on ∆G and J at pH levels, 5, 7.4, and 10. The x-axis indicates that the

Ca2+ and HPO42− 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 ∆GHA, ∆GOCP and ∆GDCPD decrease with increasing Ca2+ and HPO42−

concentrations at the same rate at pH levels of 5, 7.4 and 10. Note that ∆GDCPD become negative

in solutions when the Ca2+ and HPO42−

concentrations increase slightly at pH = 7.4. The

significance is that JDCPD is extremely high, even higher than JOCP, when ∆GDCPD 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 = 7.4.

Interfacial energy

The interfacial energy (γ) is one of the most difficult parameters in Eq. 14 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

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certain and were determined by several means including the crystal growth kinetics, dissolution

kinetics and contact angle measurements [58-60, 62-63]. Wu and Nancollas reviewed the

measurements of the Ca-P interfacial tension and summarized the measured γHA, γOCP and γDCPD

as shown in Table 3 [60].

The kinetics methods employ the classical nucleation theory (Eq. 14) 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, γHA varies from 9.3 to 87 mJm−2, while γDCPD varies form 0.4 to 70 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 [58-50, 64]. 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 [60]. Thus, the interfacial tension

values obtained with this method were chosen for analysis as listed in Table 3, except for DCPD

because γDCPD 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 γDCPD = 0.4 mJm−2 from the kinetics method.

Considering the uncertainty of the interfacial tension data, we evaluated the effects of γHA, γOCP

and γDCPD on the J calculations. The curves in Fig. 4 represent the J dependence on choosing

the γ values listed in Table 3. Fig. 4a shows the comparison of the γHA and γOCP effects on JHA,

JOCP in R-SBF, and Fig. 4b shows the comparison of the γHA, γOCP and γDCPD effects in solutions

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with the Ca2+ and HPO42−

concentrations that are five-fold higher than those of normal R-SBF.

Although γHA, γOCP and γDCPD vary to a large degree, JOCP is always larger than JHA as shown in

Fig. 4a. The solution with high Ca2+ and HPO42−

concentrations follow the same trend (Fig 4b).

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, JOCP is always larger than JHA in

SBF, and JDCPD is the highest when DCPD precipitation is thermodynamically possible.

Molecular volume

Note that the molecular volume (v) used in the nucleation rate calculations (Eq. 14) is defined as

the Ca-P crystal volume per molecule. Wu et al. argued that the volume of a crystal unit cell (vc)

should be used in Eq. 14, because the interfacial tension of calcium phosphate in solution is from

the crystal/solution interface [60]. The vc values of HA, OCP and DCPD are 582.7, 1242.4, 506.2

Å3, respectively. This argument was supported by Wu et al.’s experimental interfacial tension

measurements by substituting vc for v in the kinetics method. When vc 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 [60]. Thus, we feel that it is necessary to evaluate the effect of replacing v

with vc on the Ca-P nucleation rates. We compared the JHA and JOCP calculated with vc 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 vc in Eq. 14.

Contact angle

The contact angle function, f(θ), in Eq. 14 is given by the classical theory of heterogeneous

nucleation as [57] :

14

2(2 cos )(1 cos )( )4

f θ θθ⎧ ⎫+ −

= ⎨ ⎬⎩ ⎭

. (18)

The f(θ) range corresponding to 0° < θ < 180° is from 0 to 1, while θ = 180° corresponds to

homogeneous nucleation. Thus, the effect of the contact angle change on the nucleation rates is

minor. Figure 6 shows that JHA and JOCP 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 = 90°

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.

2+ 3 +4 4 4 34Ca 3PO H Ca (PO ) H−+ + = . (19)

Thus, the corresponding S and P should be

4 2 3 34(Ca ) (PO ) (H )(OCP)

(OCP)sp

a a aSK

+ − +

= ; (20)

2 4 3 34

2 + 3 84

8![Ca ] [H ][PO ]4!3!([Ca ] [H ] [PO ])

P+ + −

+ −=+ +

, (21)

where Ksp(OCP) = 1.00 × 10−48.7 [65]. The ∆GOCP and JOCP calculations based on Eqs. 19 and

21 are different from those based on Eqs. 3 and 16. Figure 7a shows that the ∆GOCP 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.

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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 JOCP cannot be

ignored because the kinetic factor in Eq. 14 significantly changes when the ion units of PO43−

and H+ are used for the probability calculation. The probability (P) based on ion units in Eq. 21

is 1010 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 PO43− to form HPO4

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 PO4 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) Ca10(PO4)6(CO3)0.5(OH)

(SCHA) and 2) Ca9 (HPO4) (PO4)5OH (DOHA) . The ∆GSCHA and ∆GDOHA were evaluated from

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their supersaturation in SBF:

10 2 6 3 0.5 24 3(Ca ) (PO ) (CO ) (OH )(SCHA)(SCHA)sp

a a a aSK

+ − − −

= ; (22)

9 2 5 34 4(Ca ) (HPO ) (PO ) (OH )(DOHA)

(DOHA)sp

a a a aSK

+ − − −

= , (23)

in which Ksp(SCHA) = 10−115.6 [68], and Ksp(SCHA) = 10−85.1 [56].

There was lack of experimental data, particularly the kinetic factor used in Eq. 14, 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 [47]. The calculation of PSCHA was based on the concentration

equivalency of OH− and CO32− :

2 10 3 6 24

2 3 184

18![Ca ] [PO ] [OH ]10!6!([Ca ] [PO ] [OH ] )

eq

eq

P+ − −

+ − −=+ +

(24)

in which 23OH OH 2COeq

− − −= + because one CO32− can substitute for two OH− in the apatite

crystal structure. The calculation of P DOHA is rather straightforward:

2 9 2 1 3 5 14 4

2 2 3 164 4

16![Ca ] [HPO ] [PO ] [OH ]9!5!([Ca ] [HPO ] [PO ] [OH ])

P+ − − −

+ − − −=+ + +

. (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

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 10. 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

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

6037/02E) 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.

19

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24

Table 1 Nominal Concentrations of Different SBF

Concentration (mmol dm−3) Ion Blood

Plasma [6,7]

C-SBF [6,7]

R-SBF [6,7]

i-SBF [7]

m-SBF [7]

Syn-SBF [22]

Na+ 142.0 142.0 142.0 142.0 157.0d 142.0 K+ 5.0 5.0 5.0 5.0 5.0 5.0

Mg2+ 1.5 1.5 1.5 1.0 1.5 1.5 Ca2+ 2.5 2.5 2.5 1.6 2.5 2.5

Cl− 103.0 187.8 a 103.0 103.0 103.0 165.0a HCO3

− 27.0 4.2 27.0 27.0 4.2 27.0 HPO4

2− 1.0 1.0 1.0 1.0 1.0 1.0 SO4

2− 0.5 0.5 0.5 0.5 0.5 0.5 buffer − 6.063b 11.928c 11.928c 17.892c 6.063b

a. Considering 40 mmol dm−3 of extra chloride ion added with the Tris-HCl buffer. b. Tris (hydroxylmethyl) aminomethane (g/L). c. HEPES 2-(4-(2-hydroxyethyl)-1-piperazinyl) ethane surfonic acid (g/L) d. Considering 15 mmol dm−3 of extra Na+ added with the HEPES-NaOH buffer

25

Table 2 Reactions in SBF

Reaction K Ref.

2 3 3H CO (aq.) H HCO+ −⇔ + 10−6.31 [7] 2

3 3HCO H CO− + −⇔ + 10−10.25 [7] +

3 4 2 4H PO (aq.) H + H PO −⇔ 10−2.196 [7]

+ 22 4 4H PO H + HPO − −⇔ 10−7.185 [7]

2 + 34 4HPO H + PO − −⇔ 10−12.19 [7]

2+ +3 3Ca + HCO Ca HCO− ⇔ 101.16 [7]

2+ 23 3Ca + CO CaCO (aq.)− ⇔ 103.38 [7]

2+ +Ca + OH CaOH − ⇔ 25.12 [37] 2+ +

2 4 2 4Ca + H PO Ca H PO− ⇔ 31.9 [52] 2+

4 4Ca + HPO Ca HPO (aq.)− ⇔ 6.81 × 102 [52] 2+ 3

4 4Ca + PO Ca PO− −⇔ 3.46 × 106 [52]

2+3 3Mg + HCO MgHCO − +⇔ 100.62 [53]

2+ 23 3Mg + CO MgCO (aq.)− ⇔ 101.87 [53]

2+ +Mg + OH MgOH − ⇔ 102.19 [53] 2+ +

2 4 2 4Mg + H PO Mg H PO− ⇔ K = 100.4 [54] 2+

4 4Mg + HPO Mg HPO (aq.)− ⇔ K = 101.8 [54]

2+ 34 4Mg + PO Mg PO− −⇔ K = 103.3 [54]

Table 3 Summary of Reported Interfacial Tensions[60]

Researcher Year HA OCP DCPD Measurement method Nancollas et al 99 10.0 4.3 −4.2 Contact angle 98 17 7 0.4 Crystal growth kinetics 98 9.3 3 0.4 Dissolution kinetics Christoffersen et al 88 − − 70 Crystal growth kinetics 87-92 80-87 − − Crystal growth kinetics 96 46 − − Dissolution kinetics Lundager et al 84 − 45 22 Crystal growth kinetics

The unit of interfacial tension is mJm−2.

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) ∆GHA; b) JHA; c) ∆GOCP; d) JOCP;

and e) ∆GDCPD.

Figure 3. Effects of Ca2+ and HPO42− concentrations in SBF on Ca-P precipitation. The

[Ca2+] and [HPO42−] 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 = 7.4; d ) J at pH = 7.4; e ) ∆G at pH = 10;

and f ) J at pH = 10.

Figure 4. Effects of interfacial energy (γ) variations on the calculations of nucleation rates

at pH = 7.4. The solid line represents JHA; the dashed line represents JOCP; and

the dotted line represents JDCPD. 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 [Ca2+] (12.5 mmol dm−3) and

[HPO42−] (5.0 mmol dm−3).

Figure 5. Comparison of the nucleation rates calculated from the molecular volume (v) and

the volume of the crystal unit cell (vc) in R-SBF.

Figure 6. Effects of the contact angle (θ) on the nucleation rates in R-SBF at pH = 7.4: a)

JHA; and b) JOCP.

Figure 7. Comparison of using different ion units for OCP formation: a) Small differences

in ∆GOCP values using different ion unit; and b) questionable JOCP based on Eqs

19 and 21 compared with JHA.

Figure 8. Comparison of precipitation of SCHA and DOHA with other Ca-P: a) ∆G; b)

kinetic probability (P).

1

4 5 6 7 8 9 10

-12

-8

-4

0

4

8∆G

(KJ

mol

-1)

pH

DCPD OCP HA

Figure 1. LU and LENG

a

b

6 7 8 9 100

5

10

15

20

25

Log

J

pH

OCP HA

2

Figure 2 LU and LENG

6 7 8 9 100

5

10

15

20

25Lo

g J

pH

C-SBF R-SBF I-SBF M-SBF S-SBF

6 7 8 9 100

5

10

15

20

25

Log

J

pH

C-SBF R-SBF I-SBF M-SBF S-SBF

4 5 6 7 8 9 10

-12

-8

-4

0

4

8

∆G (K

J m

ol-1)

pH

C-SBF R-SBF I-SBF M-SBF S-SBF

dc

a

HA

b

HA

OCP OCP

DCPD

4 5 6 7 8 9 10

-12

-8

-4

0

4

8

∆G (K

J m

ol-1)

pH

C-SBF R-SBF I-SBF M-SBF S-SBF

4 5 6 7 8 9 10

-12

-8

-4

0

4

8

∆G (K

J m

ol-1)

pH

C-SBF R-SBF I-SBF M-SBF S-SBF

e

3

Figure 3

LU and LENG

0 1 2 3 4 50

5

10

15

20

25

Log

J

Ca_P

DCPD OCP HA

0 1 2 3 4 5

-12

-8

-4

0

4

∆G (K

J m

ol-1

)

Ca_P

DCPD OCP HA

0 1 2 3 4 50

5

10

15

20

25

Log

J

Ca_P

DCPD OCP HA

0 1 2 3 4 5

-12

-8

-4

0

4

∆G (K

J m

ol-1

)

Ca_P

DCPD OCP HA

0 1 2 3 4 5

-12

-8

-4

0

4

∆G (K

J m

ol-1

)

Ca_P

DCPD OCP HA

0 1 2 3 4 50

5

10

15

20

25Lo

g J

Ca_P

DCPD OCP HA

c

e f

b

d

a

pH = 5 pH = 5

pH = 7.4 pH = 7.4

pH = 10 pH = 10

4

.

Figure 4 LU and LENG

a

b

Log

J

γ ( mJm−2 )

Log

J

γ ( mJm−2 )

5

6 7 8 9 100

5

10

15

20

25Lo

g J

pH

OCP v HA v OCP vc HA vc

Figure 5 LU and LENG

6

Figure 6 LU and LENG

Contact angle

J (

107 cm

−3s−

1 ) J

(1

016cm

−3s−

1 ) a

b

Contact angle

7

4 5 6 7 8 9 10

-12

-8

-4

0

4

8

∆G (K

J m

ol-1)

pH

OCP Eqs 3 & 16 OCP Eqs 19 & 21

6 7 8 9 100

5

10

15

20

25

Log

J

pH

OCP HA

Figure 7

LU and LENG

a

b

8

4 5 6 7 8 9 10

-12

-8

-4

0

4

8

12

∆G (K

J m

ol-1)

pH

DCPD OCP HA SCHA DOHA

4 5 6 7 8 9 10-80

-70

-60

-50

-40

-30

-20

-10

0

Log

P

pH

DCPD OCP HA SCHA DOHA

Figure 8 LU and LENG

a

b