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Comments on “Heterogeneous Nucleation Rate of Calcium Carbonate Derived from Induction Period” Lie-Ding Shiau* Green Technology Research Center, Pollution PreVention Group, Department of Chemical and Materials Engineering, Chang Gung UniVersity, Taoyuan 33302, Taiwan, R.O.C In a recent paper, Chien et al.1 proposed a novel mechanism for the formation of critical nuclei by heterogeneous nucleation based on the Michaelis-Menten equation.2 By fitting the derived heterogeneous nucleation rate with the induction period data of calcium carbonate, they concluded that the Michaelis-Menten mechanism can be applied to describe the heterogeneous nucleation kinetics. I would like to bring to attention that their derivation of the Michaelis-Menten equation may involve error and subsequently their conclusion is questionable. The mechanism of heterogeneous nucleation proposed by Chien et al.1 was described by the following scheme: k1
k2
CaCO3 + FP 798 complex 98 nuclei + FP
(1)
k-1
In their derivation, it is assumed that the calcium carbonate cluster is first adsorbed onto foreign particles to form a CaCO3-FP complex and then converted into the nuclei when the cluster reaches the critical size. The nucleation rate is written as JP )
d[nuclei] ) k2[complex] dt
(2)
(4)
In their derivation, Chien et al.1 assumed the following relationship: [CaCO3]T ) [CaCO3] + [complex]
(5)
Subsequently, they derived the nucleation rate caused by the effect of foreign particles as JP )
Jp,max[FP] k2[CaCO3]T[FP] ) KM + [FP] KM + [FP]
(0.00782Sa - 0.02895)[FP]
JP )
1.505 × 103 + [FP]
for diatomaceous earth
(0.00680Sa - 0.02630)[FP] 1.563 × 104 + [FP]
(8) for zirconium oxide
(6)
(9)
We note from the reaction sequence that eq 5 is questionable since some calcium carbonate is consumed by the reaction to form nuclei. Instead, the foreign particle, which acts as the catalyst in the proposed heterogeneous nucleation kinetics based on eq 1, is not consumed by the reaction. Thus, eq 5 should be replaced by
(10)
By substituting eq 10 into eq 2, the nucleation rate originated from the surface of the foreign particles can be derived as
(3)
As the production and consumption rates of the complex are equal, k1[CaCO3][FP] ) (k-1 + k2)[complex]
JP )
[FP]T ) [FP] + [complex]
The pseudo-steady-state assumption is postulated for the formation of the complex. Thus, d[complex] )0 dt
By fitting the experimental induction period data with eq 7, the positive values of KM and JP,max with high correlation coefficients are obtained. Thus, they concluded that the Michaelis-Menten mechanism was satisfactory to describe the heterogeneous nucleation kinetics of calcium carbonate. The nucleation rate originated from the surface of the foreign particles can be expressed as
JP )
k2[FP]T[CaCO3] KM + [CaCO3]
(11)
Where, KM ) (k-1 + k2)/k1. If we let JP,max represent the maximum nucleation rate for a given [FP]T, JP,max ) k2[FP]T
(12)
eq 11 reduces to JP )
JP,max[CaCO3] KM + [CaCO3]
(13)
Thus, instead of eq 6 derived by Chien et al.,1 eq 13 is obtained based on the proposed Michaelis-Menten mechanism. In terms of supersaturation ratio, eq 13 can be expressed as JP )
Where, KM ) (k-1 + k2)/k1 and JP,max ) k2[CaCO3]T. Inverting eq 6 yields
JP,maxSa KS + Sa
(14)
Inverting eq 14 yields KM 1 1 1 + ) JP JP,max [FP] JP,max
(7)
* To whom all correspondence should be addressed. Tel.: 011-8863-2118800ext 5291. Fax: 011-886-3-2118700. E-mail: shiau@ mail.cgu.edu.tw.
KS 1 1 1 ) + JP JP,max Sa JP,max
(15)
Where, JP ) 1/tind is assumed for simplicity by Chien et al.1 In general, the nucleation rate is considered to be inversely
10.1021/ie901938d 2010 American Chemical Society Published on Web 02/26/2010
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Table 1. Obtained Values of KS and JP,max/N by Equation 18
JP )
Foreign Particle: Diatomaceous Earth [FP]T (no./cm3)
KS (-)
JP,max/N (1/s)
correlation coefficient (-)
224 335 559 894 1230 1678
–8.69 –8.73 –7.75 –8.75 –8.71 –8.70
–0.00093 –0.00132 –0.00130 –0.00278 –0.00323 –0.00373
0.881 0.910 0.965 0.915 0.908 0.948
Foreign Particle: Zirconium Oxide [FP]T (no./cm3)
KS (–)
JP,max/N (1/s)
correlation coefficient (–)
1326 1988 3311 5291 7283 9901
–7.99 –7.49 –7.58 –8.27 –7.84 –8.08
–0.00032 –0.00036 –0.00060 –0.00117 –0.00123 –0.00158
0.853 0.910 0.927 0.890 0.899 0.898
-1.73 × 10 N[FP]TSa -7.87 + Sa
for zirconium oxide
(19) Figures 1 and 2 illustrate the curves of the induction period determined from eqs 18 and 19. Although the calculated curves fit the induction period data well, the negative values of KS and JP,max/N provide no physical significance. Thus, eqs 18 or 19 can only be used as an empirical equations to calculate the heterogeneous nucleation rate as a function of supersaturation ratio and concentration of foreign particles. In essence, the mechanism of the heterogeneous nucleation in the presence of foreign particles is much more complicated than that of the enzymatically catalyzed reaction described by the Michaelis-Menten equation. For the typical enzymatically catalyzed reaction, the substrate first binds with the enzyme to
Table 2. Average Value of KS and the Linear Dependence of JP,max/N on [FP]T Calculated from Table 1 foreign particle: diatomaceous earth the average value of KS
the linear dependence of JP,max/N on [FP]T
correlation coefficient (–)
–8.56
JP,max/N ) –2.52 × 10–6[FP]T
0.868
foreign particle: zirconium oxide the average value of KS
the linear dependence of JP,max/N on [FP]T
correlation coefficient (–)
–7.87
JP,max/N ) –1.73 × 10–7[FP]T
0.932
proportional to the induction period.3,4 Thus, the nucleation rate originated from the surface of the foreign particles is written as JP ) Jh - Jb ) N/tind|[FP]*0 - N/tind|[FP])0
(16)
Where N is a nucleation rate constant. For example, N ) 1012-1015 no./cm3 for SrMoO4 at Sa ) 66-172.3 Substituting eq 16 into eq 15 yields 1/tind|[FP]*0
Figure 1. Induction period versus [FP]T compared between the experimental data and the calculated curves using eq 18 for diatomaceous earth as the foreign particles.
1 1 ) - 1/tind|[FP])0 (JP /N) KS 1 1 ) + (JP,max /N) Sa (JP,max /N)
(17) A plot of 1/(1/tind|[FP]*0 - 1/tind|[FP])0) versus 1/Sa at a given [FP]T gives a straight line with an intercept 1/(JP,max/N) and slope KS/ (JP,max/N). Thus, KS and JP,max/N can be recovered. By fitting the induction period data from Chien et al.1 with eq 17, the obtained values of KS and JP,max/N at various [FP]T are listed in Table 1. As it results in the negative values of KS and JP,max/N at various [FP]T for both diatomaceous earth and zirconium oxide, the proposed Michaelis-Menten equation fails to describe the heterogeneous nucleation kinetics of calcium carbonate. In Table 1, the values of KS obtained at various [FP]T are close to their average value and the absolute value of JP,max/N increases linearly with increasing [FP]T, as described by eq 12. By substituting the average value of KS and the linear dependence of JP,max/N on [FP]T in Table 2, eq 14 can be expressed as -2.52 × 10-6N[FP]TSa JP ) -8.56 + Sa
for diatomaceous earth
(18)
Figure 2. Induction period versus [FP]T compared between the experimental data and the calculated curves using eq 19 for zirconium oxide as the foreign particles.
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form an enzyme-substrate complex, which then dissociates into product and free enzyme. However, for the heterogeneous nucleation in the presence of foreign particles, the calcium carbonate cluster is first adsorbed onto foreign particles to form a CaCO3-FP complex. Then, the calcium carbonate cluster on the complex continues to grow by aggregating with other clusters. The foreign particle acts as the catalyst to reduce the energy barrier for the further incorporation of the solute clusters into the complex.3-5 Once the size of the calcium carbonate cluster on the complex reaches the critical size, the complex is desorbed from the foreign particles and becomes the nuclei. Such complicated heterogeneous nucleation kinetics can not be described by the Michaelis-Menten equation.
Literature Cited (1) Chien, W. C.; Lee, C. C.; Tai, C. Y. Heterogeneous nucleation rate of calcium carbonate derived from induction period. Ind. Eng. Chem. Res. 2007, 46, 6435. (2) Fogler, H. S. Elements of Chemical Reaction Engineering; PTR Prentice-Hall Inc.: New York, 1999. (3) Sohnel, O.; Garside, J. Precipitation; Butterworth-Heinemann Ltd.: Woburn, MA, 1992. (4) Mullin, J. W. Crystallization; Butterworth-Heinemann Ltd.: Woburn, MA, 1993. (5) Liu, X. Y. Heterogeneous nucleation or homogeneous nucleation. J. Chem. Phys. 2000, 112, 9949.
IE901938D
