JOURNAL OF COLLOID AND INTERFACE SCIENCE 186, 102–109 (1997)ARTICLE NO. CS964621

The Crystallization of Fluorapatite in the Presence of HydroxyapatiteSeeds and of Hydroxyapatite in the Presence of Fluorapatite Seeds

YUE LIU,* G. SETHURAMAN,* WENJU WU,* G. H. NANCOLLAS,*,1AND M. GRYNPAS†

*Department of Chemistry, State University of New York at Buffalo, Buffalo, New York, 14260; and †Department of Pathology,University of Toronto and Samuel Lunenfeld Research Institute of Mount Sinai Hospital, Toronto, Canada

Received June 10, 1996; accepted October 2, 1996

paper, crystal growth reactions induced by HAP surfacesThe kinetics of growth of crystals induced by hydroxyapatite in FAP-supersaturated solutions and FAP surfaces in HAP-

(HAP) seed crystals in supersaturated solutions of fluorapatite and supersaturated solutions have been investigated using con-of fluorapatite (FAP) seed crystals in supersaturated solutions of stant composition and electron microscopic methods.hydroxyapatite have been studied using the constant composition

Crystal growth mechanisms would be expected to bemethod. The reactions were investigated at relative supersatura-markedly dependent upon the surface energy of the substratations ranging from sFAP Å 0.99 to 12.0 at pH 6.5 and for HAP,on which the reactions are occurring. The surface tension ofsHAP Å 3.6 to 12.6 at pH 7.4. In FAP-supersaturated solutions,solid materials, however, is extremely difficult to measurethis phase was nucleated at the HAP surfaces and underwentdirectly, and most of the surface energy data have beengrowth at a rate more than three times greater than that on FAP

seed crystals of equivalent surface area. Transmission electron obtained by contact angle measurements with the applicationmicroscopy (TEM) and electron diffraction imaging clearly dem- of Young’s equation. Since this approach has serious disad-onstrated that the new needle-like HAP phase originated at the vantages for the calcium phosphate microcrystals used inFAP surface with the 002 planes of HAP growing on FAP. In mineralization studies, a thin-layer wicking technique hascontrast, the mutual orientation of FAP crystallization on HAP been used to determine the contact angles for the calculationseed crystals could not be established. Interfacial energies of FAP, of interfacial energies of FAP, HAP, and OCP against aque-HAP, and octacalcium phosphate (OCP) microcrystals against

ous solutions.aqueous solutions were obtained by using a thin-layer wickingtechnique. The interfacial energy values measured in pure aqueoussolutions were 18.5, 9.0, and 4.3 mJ m02 for FAP, HAP, and OCP, MATERIALS AND METHODSrespectively. The much smaller value for OCP as compared withthe other phases may explain why this phase has been so frequently Crystal Growth Experimentimplicated as a possible precursor to the formation of apatite,especially in biological mineralization reactions. q 1997 Academic Grade A glassware, analytical reagent grade chemicals,Press and triple distilled, deionized carbon dioxide-free water

Key Words: crystal growth kinetics; epitaxy; hydroxyapatite; (TDW) were used for the preparation of solutions whichfluoapatite; surface energy. were subsequently filtered twice through 0.22 mm Millipore

filters before use. Supersaturated solutions were prepared bymixing calcium chloride, potassium dihydrogen phosphate,

INTRODUCTION potassium fluoride, and potassium chloride solutions andadjusting pH to the desired value (6.5 or 7.4) by the addition

The structure-dependent epitaxial growth of one crystal-of potassium hydroxide solution. At higher driving forces,

line material on the surface of another has been of interestthe supersaturated solutions were prepared by simultane-

for many years (1). Examples include a number of thin-ously mixing equal volumes of solutions, one containing

film physics and technology systems, corrosion films, andcalcium and the other containing the base, phosphate, and

corrosion protection layers on crystalline solid materials. Thefluoride in order to avoid spontaneous nucleation induced

intergrowth of apatite and collagen and many pathologicalby local concentration effects.

mineralization events such as the formation of kidney andFAP seed crystals (3, 4) were prepared by the simultane-

bladder stones may also involve epitaxial relationships al-ous dropwise addition of 0.5 liter each of 0.2 M Ca(NO3)2though these are sometimes difficult to verify (2). In thisand 0.12 M KH2PO4 containing 0.04 M KF to 8 liters of acontinuously boiling solution consisting of 0.10 M potassium

1 To whom correspondence should be addressed. acetate buffer at pH 4.7 containing 50 mg of KF. The FAP

1020021-9797/97 $25.00Copyright q 1997 by Academic PressAll rights of reproduction in any form reserved.

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103CRYSTALLIZATION OF APATITES

droxide. These solutions were prepared according to Eqs.[1–4]:

[CaCl2]h Å 2TCa / 5Che, [1]

[NaCl]h Å 2TNaCl 0 10Che, [2]

[KH2PO4]h Å 2TPO4 / 3Che, [3]

[KOH]h Å 2TKOH / 7Che. [4]

In these equations, TCa, TNaCl, TPO4, and TKOH are totalmolar concentrations of the subscripted species in the super-saturated reaction solutions and Che is the effective titrantconcentration with respect to HAP, the phase precipitated.

For FAP growth on HAP seed crystal surfaces, the titrantsolution concentrations were calculated from Eqs. [5]–[9]:

[CaCl2]f Å 2TCa / 5Cfe, [5]

[NaCl]f Å 2TNaCl 0 10Cfe, [6]

[KH2PO4]f Å 2TPO4 / 3Cfe, [7]

[KF]f Å 2TF / Cfe, [8]

[KOH]f Å 2TKOH / 6Cfe. [9]

In these equations, Cfe is the effective titrant concentrationFIG. 1. Scanning electron micrograph of FAP seed crystals.with respect to FAP.

During the reactions, solution samples were withdrawnperiodically and subjected to calcium and phosphate analysiscrystallites were filtered, washed with TDW, and aged atin order to verify concentration constancy. FTIR spectra of377C for several months in polyethylene vessels containingthe solid samples were made using a Perkin Elmer 1760FAP-saturated solution. They were dried at room tempera-research grade FTIR with diffuse reflectance powder acces-ture and analyzed chemically and by FTIR. The molar ratiossory (KBr 1% solutions; spectral resolution 4.0 cm01). Thewere 1.62 { 0.02 for Ca/P and 5.42 { 0.05 for Ca/F. Therate of crystal growth was obtained from the slope of theabsence of other calcium phosphate phases was confirmedtitrant volume vs. time curve normalized to the surface areaby X-ray powder diffraction (Siemens Nicolet/Nic spectrom-of the crystals. Using the constant composition method, iteter, CuKa radiation with Ni filter, position sensitive detectoris possible to obtain the rate of crystal growth with a preci-in the transmission mode and STOE attachments). Crystalsion of approximately 3% (7).morphology is shown in Fig. 1; the specific surface areas

For transmission electron microscope examinations, thedetermined by nitrogen adsorption (5) (BET, 20/80 N2/He,resulting crystals were placed in embedding molds withQuantasorb, Quantachrome) were 1.24 and 21.2 m2 g01 forSpurr epoxy resin without any chemical processing and poly-FAP and HAP, respectively.merized overnight at 707C. Sections of 100 nm thick wereHAP seed crystals were prepared by the dropwise additioncut, placed on Formvar-coated grids, and viewed in a Philipsof 2.0 liters of a solution of 0.15 M in (NH4)2HPO4 andEM 430 at 100 kV. For dark field imaging the field of interest5.7% (1.5 M) in NH4OH to 7.0 liters of a solution 0.08 M Cawas selected and a selected area diffraction pattern was ob-(NO3)2 and 0.8% (0.2 M) in NH4OH. Following the addition,tained. A small objective aperture was inserted and placedwhich was completed in 5 h, 0.06 liter of 28% NH4OH wasover the 002 line of the pattern. When the microscope wasadded and the suspension was refluxed at pH 10 for 12 h.switched from diffraction mode to imaging mode the darkAfter filtration, the solid was washed twice with 2 liters offield appeared. After the dark field image was photographedTDW and dried at 1407C; the Ca/P molar ratio obtained bythe same area was photographed in bright field mode.analysis was 1.66 { 0.02.

For the constant composition growth of HAP (6, 7), in-Interfacial Energy Determination

duced by FAP seed crystals, two titrant solutions were used.The first, containing calcium and sodium chlorides, and the For the determination of the contact angle formed between

a liquid (and air, or vapor) and the finely divided particles,second, potassium hydrogen phosphate and potassium hy-

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104 LIU ET AL.

the capillary rise method using thin-layer wicking was em- TABLE 1Values of the Surface Tension Components and Parametersployed (8). The rate of capillary rise of various liquids,

(mJ m02) and of the Viscosity (poise) of Wicking Liquids Used(h/t), through a packed column of powder supported on afor Contact Angle Measurements (207C)glass microscope slide was recorded. The results were inter-

preted in terms of Washburn’s equation (9):Liquid gL gLW

L g/L g0L h

Octane 21.6 21.6 0 0 0.00542h2 Å tReffgLcos u

2h, [10]

Decane 23.8 23.8 0 0 0.00907Dodecane 25.35 25.35 0 0 0.01493Hexadecane 27.5 27.5 0 0 0.03where Reff is the effective interstitial pore radius, gL thea-Bromonaphthalene 44.4 44.4 É0 É0 0.0489surface tension of the liquid, u the contact angle betweenDiiodomethane 50.8 50.8 É0 É0 0.028

the liquid and the solid, and h the viscosity of the liquid. In Ethylene glycol 48.0 29.0 1.92 47.0 0.199order to evaluate Reff and cos u, separate measurements were Formamide 58.0 39.0 2.28 39.6 0.0455made using low-energy apolar liquids, heptane, octane, dec- Water 72.8 21.8 25.5 25.5 0.010

ane, and dodecane which spread over the solid surface with-Note. From Ref. (8).out forming a finite contact angle. It has been shown (8)

that with such spreading liquids, u is exactly equal to zero,so that cos u Å 1, as the result of the formation of a precursor

s Å [(IP)1/n 0 K1/nso ]K01/n

so , [12]film. Reff was determined for each spreading liquid fromlinear plots of 2hh2/t against gL. These values were used

in which Kso is the solubility product for the calcium phos-with the same particles to solve for cos u for non-spreading

phate phase under consideration, n is the number of ions inliquids (for which cos u õ 1). With the knowledge of Reff, the formula unit, and IP is the ionic activity product, definedcontact angles between the solids and the test liquids a-

by Eqs. [13] and [14],bromonaphthalene, diiodomethane, water, ethylene glycol,and formamide were determined. Finally, the interfacial en-

IP(FAP) Å (Ca2/)5(PO304 )3(F0), [13]ergy was evaluated by means of the Young’s equation

adapted for polar systems (Eq. [11]): IP(HAP) Å (Ca2/)5(PO304 )3(OH0). [14]

gL(1 / cos u) Å 2(√gLW

S gLWL /

√g/Sg

0L /

√g0Sg

/L ), [11] The activities of ionic species, enclosed in parentheses

were calculated using mass balance and electroneutralitywhere gLW

S is the Lifshitz–van der Waals (LW) componentconditions with ionic activity coefficients obtained using the

of the surface tension of the solid, g/S and g0L are the Lewis Davies extended form of the Debye–Huckel equation (10).acid and base components of the surface interaction, respec- Solubility products of FAP and HAP at 377C were 7.1 1tively. 10061 M 9 (3) and 2.35 1 10059 M 9 (11), respectively. The

1.5 wt % suspensions of the HAP, FAP, and OCP crystal- relative supersaturations ranged from 0.99 to 12.0 for FAPlites in distilled water or ethyl alcohol were made with con- growth and from 3.6 to 12.6 for HAP.tinuous magnetic stirring, and 5 ml aliquots were distributed The ability of one crystalline phase to grow on the surfaceon horizontal glass microscope slides (7.5 cm 1 2.5 cm). of another is markedly dependent on their surface character-The slides were allowed to dry overnight in air and for an istics. If there is a lattice misfit of crystallographic parame-additional 24 h at 1057C prior to storing in a vacuum desicca- ters, the initially grown layers may display imperfectionstor. The slides were equilibrated for one hour in stoppered making crystal growth less favorable (12). This is shown inglass containers containing one of the liquids listed in Table Fig. 2 and Table 2 in which it is seen that in supersaturated1. They were then dipped into the liquids to a height of solutions of FAP, the rate of FAP growth is considerablyabout 5 mm, and the vertical movement of the liquid front greater than that of HAP seed crystals.through the layer of powder was observed. With the help of The lattice misfit, d ÅDa/a, where a is the lattice parame-small indentations on the glass slides at 2 mm intervals, ter of the stress-free crystal and Da is the difference in latticeseveral observations were made of the rate of movement of parameters between the substrate and the growing crystal.the liquid front. About 35 slides were used for each sample, The establishment of a definite orientation in the crystallineat least three for each liquid, and the values of u were calcu- overgrowth requires the initial formation of an immobilelated using Eq. [10]. monolayer of regular atomic patterns which serve as the

embryo. It has been suggested that there is a critical misfitRESULTS AND DISCUSSIONvalue in order for the depositing monolayer to attain thesame lattice spacing as the crystalline substrate (13–15). ItRelative supersaturations, s, were calculated using Eq.

[12] is generally considered that a misfit between 10–20% is

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105CRYSTALLIZATION OF APATITES

acceptable for epitaxial growth. Turnbull and Vonnegut (16) TABLE 2Summary of Experiments for FAP Growth Using FAP and HAPdeveloped a formal theory in which the effectiveness of

As Seed Crystals (pH Å 6.5, T Å 377C, IS Å 0.15 (NaCl)) molprimary heterogeneous nucleation is also related to latticeL01 Mmatching, d, between nucleus and substrate and predicted

that nuclei would probably form only for d below aboutFAP on FAP FAP on HAP

0.015.For HAP and FAP surfaces, the basal plane value of d is Slope (ml R (1009 mol Slope (ml R (1009 mol

sFAP min01 m02) s01 m02) min01 m02) s01 m02)0.006 indicating a remarkably close fit between their latticestructures. In addition, crystallographically, they both belong

0.985 — — 0.00976 0.08to the same space group P63/m. Transmission electron micro-1.64 — — 0.0107 0.09

graphs shown in Figs. 3(a) and (b) suggest that HAP growth 2.17 0.198 1.65 — —in the presence of FAP seed crystals was oriented while 3.22 0.287 2.39 0.0667 0.56

4.26 0.34 2.83 0.0784 0.65although FAP crystallization was induced by the presence5.31 0.68 5.67 0.0933 0.78of HAP crystals, their mutual orientation could not be estab-6.34 0.794 6.62 0.136 1.13lished because of their small size. The TEM picture obtained7.38 0.954 7.95 0.222 1.85

by selected area dark field imaging (17) (Fig. 3(b)) indicates 8.41 1.151 9.59 — —that the 002 planes of the crystals of HAP grow on FAP. 9.44 1.378 11.48 0.465 3.88

10.5 1.749 14.58 — —These findings are to be expected from the facts that both12 1.831 15.26 0.567 4.73HAP and FAP grow in predominantly one-dimensional

mode (18, 19) and that their basal planes have a near-perfectlattice matching.

In terms of the phenomenological rate equation which has apparent order of the reaction, n Å 1.4 { 0.1 for FAP growthbeen found to hold for a large number of sparingly soluble on FAP seeds, was in good agreement with that obtainedsalts, the crystallization of FAP and HAP would be expected previously, 1.3 { 0.1, by Amjad et al. (20). For the growthto follow an expression of the form: of FAP on HAP, n was 1.7 { 0.1. In both cases, therefore,

transport processes were unlikely to be involved in the rateR Å kss n [15] determining step (21).

In general, there are two competing factors governingin which k is the precipitation rate constant, s is proportional nucleation: (i) the free energy of the crystallizing systemto the total number of available growth sites on the added decreases when it transforms to solid and (ii) for very smallseed crystals and n is the effective order of the reaction. The particles of solid, the surface area is large relative to thelinearity of the logarithmic plots of Eq. [15] shown in Fig. volume, so that the surface energy of such a particle domi-4 confirms the applicability of this kinetic rate equation for nates. Thus, small particles may decrease the total free en-the growth of FAP on FAP and on HAP seed crystals. The ergy of the system by shrinking and reducing their surface

area while larger particles can achieve the same result bygrowing. A balance between these tendencies describes thecritical nucleus. If a spherical shape is assumed for the nu-cleus, the Gibbs free energy for homogeneous nucleationmay be written as Eq. [16]:

DG Å (4/3)pr3DGy / 4pr2g [16]

where g is the interfacial tension between the nucleus andthe crystallizing medium. DGy is the volume specific freeenergy term associated with the liquid–solid phase changewhich represents the volume free energy and is negative.Primary heterogeneous nucleation may be considered interms of a spherical cap nucleus on a flat nucleating substrateimmersed in fluid matrix as shown in Fig. 5. An importantprediction about nucleation energetics emerges from thismodel. That is the energetics of heterogeneous nucleation,FIG. 2. FAP growth rate on FAP seed crystals and on HAP seed crystalsDGhet, can be described in terms of homogeneous parametersas a function of relative supersaturation with respect to FAP (pH Å 6.5, IS

Å 0.15 M (NaCl), T Å 377C). together with a single additional parameter, u, the contact

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106 LIU ET AL.

FIG. 3. Transmission electron micrographs (16,6501) of (a) FAP growth in the presence of HAP seed crystals and (b) HAP growth on FAP seedcrystals.

angle between the crystalline deposit and the foreign solid an interfacial energy between HAP and OCP, gHAP,OCP, ofsubstrate (Fig. 5). 0.93 mJ m02 was evaluated. In Eq. [19], gLW

i is the Lifshitz-van der Waals (LW) component of the surface tension ofphase i. g/i and g0i are the Lewis acid and base parametersDGhet Å fDG, [17]of the surface tension of phase i, respectively. g12 is theinterfacial tension between phases 1 and 2. The angle ofwherecontact between the HAP deposit and the OCP surface, u,corresponding to the angle of wetting in liquid-solid systems,

f Å (1 0 cos u)2(2 / cos u)4

. [18] was calculated as u Å 67.77 using Eq. [20],

Using the measured surface tension components (Table cos u Å gOCP,WATER 0 gHAP,OCP

gHAP,WATER

. [20]3) together with Eq. [19],

The value of f in Eq. [17] was 0.23, implying that the energyg12 Å (√gLW

1 0√gLW

2 )2 / 2(√g/1 g

01

barrier of nucleation of HAP is reduced by a factor of 0.23on OCP as compared to the homogeneous nucleation of/

√g/2 g

02 0

√g/1 g

02 0

√g01 g

/2 ), [19]

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107CRYSTALLIZATION OF APATITES

FIG. 3—Continued

HAP. Taking into consideration the spreading pressure, for For calcium phosphates for which only microcrystals can bereadily grown, a thin-layer wicking technique described inFAP nucleation on an HAP substrate, the contact angle was

about 1187 while the contact angle for HAP on FAP ap- Materials and Methods was used for contact angle measure-ments and the interfacial tension was calculated usingproached 07 within experimental error. The former system

therefore represented a much higher energy barrier for nucle- Young’s equation. Thus values of the effective pore sizesReff, were 7.5 1 1001 mm for FAP and 8.2 1 1002 mm foration.

In terms of the classical model, the only resistance to OCP particles. The calculated interfacial energies of FAP,HAP, and OCP against water were 18.5, 9.0, and 4.3 mJnucleation is the interfacial energy of the nucleating sub-

stance against the nucleating media (homogeneous nucle- m02, respectively. The lower value for OCP indicates a largereduction in the nucleation barrier, since in terms of classicalation) or against the foreign substrate surface and the nucle-

ating medium (primary heterogeneous nucleation). The sur- nucleation theory, it is proportional to the third power of theinterfacial energy. This important result may provide keyface tension of solid material is very difficult to measure,

and few accurate data are available. Values for some types information to explain why, in many calcium phosphate pre-cipitating systems, OCP appears to form before the thermo-of ionic crystals in a vacuum may be calculated with perhaps

five per cent uncertainty, but the interfacial tensions between dynamically more stable HAP.Another manifestation of the effect of interfacial energysolid and solution are much more important. Most of the

solid surface energy values have been obtained from contact during nucleation, lies in the induction times, ti, for precipita-tion, defined as the time elapsed between the creation of theangle measurements through the use of Young’s equation.

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108 LIU ET AL.

TABLE 3Interfacial Tension g between Water and Various Solids and

Components (mJ m02) Found by Thinlayer Wicking

Solid gLWs g/s g0s g

FAP 32.4 0.64 8.96 18.5HAP 28.5 0.92 16.0 9.0OCP 21.6 2.19 19.7 4.3

ration with respect to FAP increases. When the supersatura-tions are the same, the induction period for FAP growth onFAP seed is always shorter than that for growth on HAPseed.

In Fig. 7, it is seen that HAP growth on FAP seeds occursafter induction periods of about 300 and 275 min at sHAP

FIG. 4. Crystal growth of FAP on FAP and on HAP seed crystals. Plots values of 9.2 and 12.6, respectively. After the inductionof log(rate) against log(relative supersaturation). period, the growth rates were 5.9 1 1006 and 4.9 1 1007

mol min01 m02 for sHAP values of 12.6 and 9.2, respectively.The growth rate of HAP on FAP at sHAP Å 9.2 was foundsupersaturated state and the detection of particles using someto be close to the value for pure HAP growth on HAP seeddevice. Thus we may write:at sHAP É 4.

In Fig. 7, the long delay in HAP growth on FAP seedti Å tn / tg [21]crystals may result from the inhibiting influence of very lowlevels of released fluoride when the FAP seed crystals arewhere tn is the time required for critical nucleus formationfirst introduced into the supersaturated solutions. This inhibi-and tg the time required for the growth of particles havingtion was shown in crystal growth kinetics studies (27) anddetectable dimensions. Although induction times are usuallycontrasts the mineralization acceleration that is observed inobtained from unseeded precipitation experiments (21–24),the presence of higher fluoride concentrations, probably re-the establishment of an induction time for seeded growthflecting the formation of a different phase, namely FAP orreactions has been suggested (25, 26). Due to the arbitrarypartially fluoridated HAP.nature of the ‘‘induction time,’’ this is defined, in the present

In this study, the interfacial energy was obtained by thepaper as the time between seeding and the addition of 20thin layer wicking method. It is interesting to note, however,ml m02 of titrant with an effective titrant concentration ofthat this parameter could also be estimated from the induc-5.0 1 1004 M. In terms of these requirements, for FAPtion time and crystal growth rate measurements. The valueformation, the volume of new phase is approximately 2.36

1 1005 cm3. Following van der Leeden’s formalism, a ÅVmacro /V Å 3.8 1 1003. Here, Vmacro is the macroscopic vol-ume of the new phase formed and V the volume of the seed(usually 20 mg or 6.3 1 1003 cm3). The ratio a is muchless than unity, thus justifying the above arbitrary choice ofinduction times. The induction times for FAP growth in thepresence of FAP and HAP seed crystals as a function ofsupersaturation, plotted in Fig. 6, decrease as the supersatu-

FIG. 5. Heterogeneous nucleation an a phase on a substrate s. The FIG. 6. Dependence of the induction time on relative supersaturationof FAP on FAP and HAP seed crystals.spherical nucleus b makes a contact angle u with the substrate.

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109CRYSTALLIZATION OF APATITES

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12. Matthew, J. W., ‘‘Epitaxial Growth,’’ p. 3. Academic Press, New York,FIG. 7. Titrant addition obtained from HAP constant composition1975.growth experiments.

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57, 325 (1982).to much higher values for the interfacial energy. Thus the19. Liu, Y., and G. H. Nancollas, J. Cryst. Growth 165, 116 (1996).interfacial energy of HAP against aqueous solutions was20. Amjad, Z., Koutsoukos, P., and Nancollas, G. H., J. Colloid Interfacecalculated to be 186 mJ m02 by Koutsopoulos et al. (28).

Sci. 82, 394 (1981).This value is much greater than would be anticipated for a 21. Nielsen, A. E., Croat. Chem. Acta 60, 531 (1987).biocompatible calcium phosphate surface. The reason for 22. Kubota, N., Takadi, T., and Kawakami, T., J. Cryst. Growth 74, 259

(1986).this difference is probably related to the arbitrary assump-23. Wojcieckowski, K., and Kibalczyc, W., J. Cryst. Growth 76, 379tions that are involved in estimation of induction times (25).

(1986).Moreover, the crystallization mechanism may be difficult to24. Price, C. J., and Garside, J., in ‘‘Industrial Crystallization 87’’ (J. Nyvlt

identify or may not correspond to the expressions used in and S. Zacek, Eds.), p. 147. Elsevier, Amsterdam, 1989.the estimation of the interfacial energy (29). Still another 25. Mullin, J. W., ‘‘Crystallization.’’ Butterworth-Heinemann, Oxford,

1993.reason may arise from the question as to the applicability26. van der Leeden, M. C., Verdoes, D., Kashchiev, D., and van Rosmalen,of a thermodynamic quantity to rather small entities.

G. M., in ‘‘Advances in Industrial Crystallization,’’ p. 31. 1991.27. Meyer, J. L., and Nancollas, G. H., J. Dent. Res. 51, 1443 (1972).

ACKNOWLEDGMENT 28. Koutsopoulos, S., Paschalakis, P. C., and Dalas, E., Langmuir 10, 2423(1994).

29. Nielsen, A. E., in ‘‘Crystal Growth’’ (H. S. Peiser, Ed.). Pergamon,We thank the National Institutes of Health (Grant DE03223) for supportof this work. Oxford, 1967.

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