J. Phys. Chem. B 2001, 105, 1991-1994
1991
Effect of Impurity on Two-Dimensional Nucleation Kinetics: Case Studies of Magnesium and Zinc on Hydroxyapatite (0001) Face Noriko Kanzaki,† Kazuo Onuma,*,‡ Gabin Treboux,‡ Sadao Tsutsumi,† and Atsuo Ito‡ School of Science and Engineering, Waseda UniVersity, 1-6-1 Nishiwaseda, Shinjuku-ku, Tokyo 169-8050, Japan, and Cell Tissue Module Group, National Institute for AdVanced Interdisciplinary Research, 1-1-4 Higashi, Tsukuba-shi, Ibaraki 305-8562, Japan ReceiVed: September 15, 2000; In Final Form: December 7, 2000
The effect of impurity on the two-dimensional (2D) nucleation rate, J, was investigated by estimating the edge free energy, γ, on the (0001) face of a hydroxyapatite crystal in the presence of magnesium and zinc as impurities. It was found that edge free energies, γ, at 0.06 mM magnesium and 0.75 µM zinc concentrations were constant and almost the same as that in the absence of impurities in the supersaturation range of σ ) 9.8-22.0. This indicates that magnesium and zinc did not affect the energy barrier for 2D nucleation at least up to these concentrations, although these impurities inhibited the growth rate of the (0001) face by adsorbing at kink sites of 2D islands. Taking into account the effect of impurities on J, the adsorption constants K of the Langmuir kink model were calculated as KMg ) (1.30 ( 0.2) × 104 L/mol and KZn ) (1.23 ( 0.1) × 106 L/mol at σ ) 22.0 and 25 °C.
Introduction It is important to clarify the effect of impurities on a crystal growth kinetics from industrial and scientific points of view, since impurities control the quality and morphology of a crystal through inhibition or promotion of a crystal growth kinetics.1-12 In the case of a two-dimensional (2D) nucleation growth, the impurity effect is more complicated than that in a spiral growth, since the growth rate, R, is related not only to the step velocity, V, but also to the 2D nucleation rate, J, as
R ) h(V)2/3(J)1/3
to the lack of information about J in the presence of impurities. The aim of the present study is, therefore, to investigate the effects of magnesium and zinc impurities on J by estimating the γ. The edge free energies γ are calculated from growth rates measured by Moire phase shift interferometry in the presence of impurities as a function of supersaturation with respect to HAP, σ. Surface morphology is also observed using atomic force microscopy (AFM). Based on these results, the Langmuir adsorption mechanism of magnesium and zinc on the (0001) face of HAP, which we previously reported, is reanalyzed.16
(1)
where h is the step height.1 The impurity effect on V and J should be separately clarified. Important research related to the impurity effect on J was performed by Malkin et al., who showed that heterogeneous 2D nucleation by unintentional impurities occurred at low supersaturations on the growth of ADP (101) face, thaumatin (101) face, and catalase (001) face.10-12 This phenomenon was theoretically discussed in terms of a promotive effect on 2D nucleation through the increase in J, by taking into account the size and density of the impurity particle adsorbed on the surface, and the contact angle between the impurity particle and the nuclei.13 However, quantitative studies of impurity effects on J in the presence of intentional impurities have rarely been performed.4,5 We investigated the (0001) face of hydroxyapatite, (Ca10(PO4)6(OH)2; HAP), the major inorganic component of human bone and teeth, which grew in a multiple 2D nucleation growth under pseudophysiological conditions.14,15 We reported previously that magnesium and zinc in the solutions reduced the R of the (0001) face by adsorbing at kink sites of 2D islands.16 However, this conclusion was made under the assumption that magnesium and zinc did not have significant effects on J, due * To whom correspondence should be addressed: Tel: +81-298-612557. Fax: +81-298-61-2565. E-mail:
[email protected]. † Waseda University. ‡ National Institute for Advanced Interdisciplinary Research.
Experimental Section 1. Seed Crystals. The seed crystals used were hydrothermally synthetic HAP single crystals bounded by {0001} and {101h0} faces.17 The size of the crystals was 1-4 mm and 20-50 µm along the [0001] and [101h0] directions, respectively. The (0001) faces were used for all growth rate measurements and surface observations. 2. Pseudophysiological Solutions. The pseudophysiological solutions (σ ) 9.8-22.0) used in the present study were prepared using extrapure-grade reagents (Nacalai Tesque, Inc., Tokyo, Japan) and ultrapure CO2-free water. The standard solution (σ ) 22.0) contains 140 mM NaCl, 1.0 mM K2HPO4‚ 3H2O, 2.5 mM CaCl2, and either 0.06 mM MgCl2‚6H2O or 0.75 µM ZnCl2 as impurities, and is buffered at pH 7.4 by tris(hydroxymethyl)aminomethane and 1 N HCl at 25 °C.18 The σ was varied by diluting the standard solution without changing the concentration of magnesium or zinc impurity and the pH. These impurity concentrations were chosen for the following reasons. (1) The growth rates were efficiently decreased and the inhibitory effect was clearly observed at these impurity concentrations. (2) The growth rates were too low to be measured at low supersaturations above these impurity concentrations. 3. Surface Observation by AFM. The effect of magnesium and zinc on the surface morphology of the (0001) face was
10.1021/jp003295b CCC: $20.00 © 2001 American Chemical Society Published on Web 02/15/2001
1992 J. Phys. Chem. B, Vol. 105, No. 10, 2001
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Figure 1. AFM images of (0001) faces after 4 h of growth in the absence of impurity (a), and in the presence of 0.06 mM magnesium (b) and 0.75 µM zinc (c).
observed using a NanoScope III-a AFM (Digital Instruments, USA) in the tapping mode. Seed crystals were grown for 4 h at 25 °C in the pseudophysiological solutions. The solutions at σ ) 22.0 were used without impurity or with either 0.06 mM magnesium or 0.75 µM zinc. 4. Measurement of Growth Rate by Interferometry. To clarify the effect of impurities on the 2D nucleation rate, J, the edge free energies, γ, were estimated from the growth rates of the (0001) face measured by Moire phase shift interferometry at 25 °C. The growth rates were measured at random points around the center of the (0001) face. Common-path two-beam interferometry with a Nomarsky prism was used to eliminate mechanical disturbance. A special signal processor was also employed to analyze the movement of interference fringes with a theoretical height resolution of 0.5 nm. The details of the interferometry and the signal processor can be seen elsewhere.14,15 Results and Discussion 1. Effect of Impurity on Surface Morphology. Figure 1 shows that the (0001) face grows in a multiple 2D nucleation growth regardless of the absence or presence of impurities. Although the size of 2D islands is reduced by the addition of impurities, the effect of impurities on J is not clear in the figure. The size of 2D islands is decreased by 20-40% and 50-70% at 0.06 mM magnesium and 0.75 µM zinc concentrations, respectively. The decrease in size of 2D islands indicates a decrease in the step velocity, V, via a reduction in the step kinetic coefficient, β, caused by impurity adsorption at the kinks or steps of 2D islands. 2. Effect of Impurity on Edge Free Energy. Figure 2 shows the R as a function of σ in the absence and presence of impurities. The growth rates monotonically increase with an increase in σ in all cases. The γ was calculated from these data as follows in order to estimate the effect of impurities on J. According to the heterogeneous 2D nucleation theory by Liu et al., J is given by13
Figure 2. Growth rates of (0001) face as a function of σ in the absence of impurity (4), and in the presence of 0.06 mM magnesium (b) and 0.75 µM zinc (O).
and δ vary as 0 < f e 1 and 0 < δ e 1, and characterize the influence of impurity particles on 2D nucleation growth. With increasing the ratio between radius of the impurity particle and critical radius of nucleus, and decreasing the contact angle between the impurity particle and nucleus, f decreases close to 0. This indicates that heterogeneous 2D nucleation is likely to occur by the impurity particles. Equation 2 covers homogeneous nucleation when f ) δ )1. It was assumed in ref 13 that β is not affected by impurities. However, magnesium and zinc clearly inhibited the lateral growth of 2D nuclei, as seen in AFM images (Figure 1); thus, β is a function of impurity concentration in the present study. Step velocity, V, is proportional to β and σ as
V ) Ceωβσ
(3)
where Ce is equilibrium concentration and ω is molecular volume of HAP. Therefore R is expressed as follows using eqs 1, 2, and 3
J ) C1β(1 + σ){ln(1 + σ)}1/2 exp[-πγ02f/(kT)2 ln(1 + σ)]δ (C1 ) constant) (2)
R ) C2βσ2/3(1 + σ)1/3{ln(1 + σ)}1/6 exp{-πγ02f/3(kT)2
taking into account the promotive effect of an impurity on 2D nucleation, using two factors, f and δ.13 The f is related to the ratio between radius of an impurity particle and critical radius of nucleus and the contact angle between an impurity particle and nucleus. The δ is related to f, radius of the impurity particle, and density of impurity particles on the crystal surface. The f
Equation 4 also covers homogeneous nucleation when f ) δ ) 1. Taking the logarithm of eq 4, we obtain
ln(1 + σ)}δ1/3 (C2 ) constant) (4)
ln[R/[σ 2/3(1 + σ)1/3{ln(1 + σ)}1/6]] ) ln C2 + ln β -{πγ02f/3(kT)2} × 1/ln(1 + σ) + ln δ1/3 (5)
Two-Dimensional Nucleation Kinetics
J. Phys. Chem. B, Vol. 105, No. 10, 2001 1993
Figure 3. Plots of growth rates for calculating γ in the absence of impurity (4), and in the presence of 0.06 mM magnesium (b) and 0.75 µM zinc (O).
TABLE 1: Physical Parameters for the Growth of the (0001) Face Calculated in the Cases of fMg ) 0.8, fMg ) 1.0 and fZg ) 1.0 at pH 7.4 and 25 °C
βi/β0 fMg ) 0.8 0.80 0.70 0.60 0.50 0.40 0.30 fMg ) 1.0 0.6 fZn ) 1.0 0.5
rel diff between y interceptions 1.27 1.27 1.27 1.27 1.27 1.27
δ
Ji/J0 Ri/R0 Ri/R0 (calc) (calc) (act) (σ ) 22) (σ ) 22) (σ ) 22)
0.04 0.07 0.10 0.18 0.35 0.83 1.0 1.0
0.2 0.3 0.4 0.5 0.8 1.4 0.6 0.5
0.5 0.5 0.5 0.5 0.5 0.5
0.6 ( 0.1 0.6 ( 0.1 0.6 ( 0.1 0.6 ( 0.1 0.6 ( 0.1 0.6 ( 0.1 0.6 ( 0.1 0.5 ( 0.1
The edge free energy in the presence of impurity, γi, is described using that in the absence of impurity, γ0, as
γi ) γ0 fi1/2
(6)
Hereafter, we denote parameters in the absence and presence of impurity, and those pertaining to magnesium and zinc by subscripts 0, i, Mg, and Zn, respectively. Figure 3 shows the relationship between σ and R according to eqs 5 and 6. From the slopes of the lines in Figure 3, edge free energies are calculated as γ0 ) 3.3 kT, γMg ) 3.0 kT, and γZn ) 3.4 kT. Therefore, the factor, f, is obtained as fMg ) 0.8 and fZn ) 1.1. It is concluded that zinc does not influence γ and that fZn should be 1.0 in terms of homogeneous 2D nucleation. However, it is unclear whether magnesium influences γ or not, due to the experimental error of γ, which is estimated as (0.3 kT. In the case of fMg ) 0.8, this indicates a slight reduction of the energy barrier for 2D nucleation in terms of heterogeneous 2D nucleation. On the other hand, if the difference between γ0 and γMg is due to experimental error, actual fMg should be 1.0 in terms of homogeneous 2D nucleation. Therefore, we estimated the values of several parameters for the cases of both fMg ) 0.8 and fMg ) 1.0 (Table 1). First, in the case of fMg ) 0.8, subtracting the y interception of line 2 from that of line 1 (Figure 3) is carried out using the least-squares method. The obtained value of 1.27 yields any combination of δ and βi/β0 in the range of 0 < δ < 1 from eq 5. Since β is a function of impurity concentration, δ cannot be determined uniquely. Since J is described as eq 2, the ratio of 2D nucleation rates, Ji/J0, is described as
Ji/J0 ) (βi/β0)δ exp[-γ02π/(kT)2 ln(1 + σ)(f - 1)] (7) As seen in Table 1, Ji/J0 at σ ) 22.0 varies depending on the combination of βi/β0 and δ using eq 7. Since the R is proportional to (J)1/3(β)2/3, by modifying eq 1, the values of
Ri/R0 at σ ) 22.0, using Ji/J0 and βi/β0, are calculated as 0.5 regardless of the combination of Ji/J0 and βi/β0, which is close to the actually measured value of Ri/R0, 0.6. However, the actual value of βi/β0 in the presence of 0.06 mM magnesium is estimated as 0.6-0.8 from the decrease in size of 2D islands observed in the AFM images (Figure 1). Given this range of values, the calculated value of Ji/J0 is from 0.2 to 0.4, which seems to be smaller than that expected from the change in the number of 2D islands (at least Ji/J0 > 0.5) observed in the AFM images (Figure 1). Thus, the assumption of heterogeneous 2D nucleation with fMg ) 0.8 is questionable. Second, in the case of fMg ) 1.0, Ji/J0 is expressed by eq 7 with fi ) δi ) 1 as
Ji/J0 ) βi/β0
(8)
Moreover, since Ri/R0 is proportional to (J)1/3(β)2/3, Ri/R0 is expressed using eq 8 as
Ri/R0 ) βi/β0
(9)
Thus, the values of βi/β0 and Ji/J0 should be 0.6, consistent with AFM observation. Therefore, it is concluded that both magnesium and zinc do not influence γ at 0.06 mM magnesium and 0.75 µM zinc concentrations. This conclusion is also valid at lower concentrations of magnesium and zinc, even though measurements of growth rates were not performed. If magnesium and zinc were to inhibit or promote 2D nucleation, γ, would monotonically increase or decrease depending on the impurity concentration, since γ is affected by the number of impurity particles on the surface. Thus, the γ of the HAP (0001) face does not change, at least up to 0.06 mM magnesium and 0.75 µM zinc concentrations. One reason why magnesium and zinc did not promote 2D nucleation of HAP is the σ with respect to HAP. It is generally believed that heterogeneous nucleation is likely to occur in a low σ range.1,12,19 Heterogeneous nucleation was not dominant in the present study due to the high σ range (σ ) 9.8-22.0). 3. Langmuir Adsorption Model. On the basis of the results presented in the previous section, the Langmuir adsorption isotherms in ref 16 are reanalyzed. In our previous study, the growth rates have been measured in the range of 0-1.5 mM magnesium and 0-7.5 µM zinc concentrations at σ ) 22.0. These data have been analyzed using Langmuir adsorption isotherms for multiple 2D nucleation, as16
R03/2/(R03/2 - (Ji/J0)-1/2Ri3/2) ) Rl-1{1 + 1/KCi} Langmuir kink model (10) {R03/2/(R03/2 - (Ji/J0)-1/2Ri3/2)}2 ) Rs-2{1 + 1/KCi} Langmuir terrace model (11) where Rl and Rs are effectiveness factors as a function of σ, K is the adsorption constant, and Ci is the impurity concentration. The data were well fitted by the Langmuir kink model for both magnesium and zinc. However, these fittings were achieved under the assumption of (Ji/J0)-1/2 ) 1, due to the lack of information about impurity effects on J. Here, when the impurities do not influence edge free energies, eq 12 is obtained from eqs 8 and 9 as
Ji/J0 ) Ri/R0
(12)
Therefore, in the present study, the Langmuir adsorption isotherms are given by eqs 13 and 14 by taking the values of Ji/J0 into consideration. The following equations are obtained
1994 J. Phys. Chem. B, Vol. 105, No. 10, 2001
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Figure 4. Langmuir plots for magnesium (a) and zinc (b). Closed symbols (b) indicate calculated values for kink and terrace models in the range of 0-0.06 mM magnesium and 0-0.75 µM zinc concentrations. Open symbols (O) represent references at higher impurity concentrations.
from eqs 10, 11, and 12 as -1
R0/(R0 - Ri) ) Rl {1 + 1/KCi}
References and Notes
Langmuir kink model (13)
{R0/(R0 - Ri)}2 ) Rs-2{1 + 1/KCi} Langmuir terrace model (14) Equations 13 and 14 are valid in the range of 0-0.06 mM magnesium and 0-0.75 µM zinc concentrations. Figure 4 shows the result of replotting after corrections. Data at higher impurity concentrations than these ranges are also shown in Figure 4 as references (open symbols). The data are fitted by linear functions for the Langmuir kink model, meaning that the inhibitory effect follows the kink model in both cases of magnesium and zinc. Adsorption constants, K, of the Langmuir kink model are recalculated as KMg ) (1.30 ( 0.2) × 104 L/mol and KZn ) (1.23 ( 0.1) × 106 L/mol at σ ) 22.0 and 25 °C. On the other hand, the data cannot be fitted by linear functions for the Langmuir terrace model, indicating that the inhibitory effect does not follow the terrace model in both cases of magnesium and zinc. The values of K in our previous study, KMg ) (1.97 ( 0.3) × 104 L/mol and KZn ) (2.13 ( 0.3) × 106 L/mol, are close to those in the present study.16 The total concentration of zinc in a normal human plasma is 11-19 µM.20 Approximately 98% of zinc in the normal human plasma is protein bound20,21 and is known to promote bone formation by stimulating the activity of osteoblasts.22-25 The rest, 1-2%, 0.1-0.4 µM, is free cation.26 The concentration of free cation under physiological condition is lower than that used in the present study (0.75 µM zinc), and thus, the inhibitory mechanism of zinc on the growth of HAP can be applied to that in vivo. Acknowledgment. This study was supported by JSPS Research Fellowships for Young Scientists.
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