Determination of Metastable Zone Widths and the Primary Nucleation

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Determination of Metastable Zone Widths and the Primary Nucleation and Growth Mechanisms for the Crystallization of Disodium Guanosine 5′-Monophosphate from a Water−Ethanol System Fengxia Zou,†,§ Wei Zhuang,†,§ Jinglan Wu,† Jingwei Zhou,† PengPeng Yang,† Qiyan Liu,† Yong Chen,† and Hanjie Ying*,†,‡ †

College of Biotechnology and Pharmaceutical Engineering and ‡State Key Laboratory of Materials-Oriented Chemical Engineering, Nanjing Tech University, Nanjing 210009, China S Supporting Information *

ABSTRACT: Metastable zone widths (MSZWs) and induction times (tind’s) were measured by turbidity techniques during the antisolvent crystallization of disodium guanosine 5′-monophosphate (5′-GMPNa2). Measured MSZWs can be affected by numerous process parameters, including temperature, the agitation rate, and the antisolvent addition rate. An exponential equation was used to correlate the supersolubility and solubility data for different conditions, and to afford predictions of the MSZW values. Values of tind at different temperatures and mole fractions of water were assessed to determine the primary nucleation and growth mechanisms of 5′-GMPNa2 crystals in the water−ethanol system. The measured tind’s were then correlated using mononuclear and polynuclear mechanistic models. The fitting results identified the primary nucleation mechanism for 5′-GMPNa2 as polynuclear, which relates tind and the supersaturation for various growth mechanisms. The growth mechanism of 5′-GMPNa2 was found to be diffusion-controlled at all experimental temperatures.

1. INTRODUCTION Crystallization from solution is a common but important process for the purification and separation of many different chemical species, including pharmaceuticals, biological macromolecules in foods, food additives, and other fine chemicals. The process impacts the control of quality with respect to particle size distribution, crystal shape, and purity of the desired product. The determination of nucleation and growth mechanisms is important for optimizing industrial crystallization processes. Numerous methods have been used to determine these nucleation and growth mechanisms, such as induction time experiments,1 metastable zone width (MSZW) methods, mixed-suspension, mixed-product removal experiments, and combined population balance modeling and particle size distribution measurements. Myriad detection methods have also been developed, which include measurements of electrical conductivity,2,3 ultrasound velocity,4,5 focused beam reflectance,6−9 and turbidity.3,10 In this study, induction time experiments based on turbidity techniques were selected to predict nucleation and growth mechanisms. To date, there does not exist any theoretical model that can completely and satisfactorily explain crystallization kinetics. Most models for primary nucleation and growth mechanisms are expressed as empirical power-law equations with supersaturation as the independent variable.8,11,12 The parameters in these expressions can be obtained by applying optimization techniques to fit model predictions with experimental data.13,14 General theoretical expressions have been derived for the dependence of induction time on supersaturation for different crystal growth mechanisms.15,16 On the basis of these expressions, the © XXXX American Chemical Society

underlying crystal growth mechanism for a particular compound can be identified by measuring the induction time over a range of supersaturated states. In industrial-scale processes, crystallization occurs at high partial supersaturation. This is because it is difficult to avoid a sharp nucleation point, and the formation of amorphous precipitates leads to particle aggregation and impurities. Therefore, there is a need to focus on controlling solution supersaturation and the metastable zone width. The MSZWs and tind for 5′-GMPNa2 crystallization can be affected by a number of parameters, including the initial supersaturation, temperature, pH, agitation speed, and impurity levels.12,17−19 Disodium guanosine 5′-monophosphate (5′-GMPNa2; the structure is shown in Figure 1) is a commercially available product that is widely used in the food, pharmacological, and health products industries. Very few studies have focused on

Figure 1. Molecular structure of disodium guanosine 5′-monophosphate (5′-GMPNa2). Received: October 13, 2014 Revised: December 2, 2014 Accepted: December 2, 2014

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dx.doi.org/10.1021/ie504035j | Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Industrial & Engineering Chemistry Research

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supersaturated but spontaneous nucleation does not occur in a sufficiently short time.13 That is, the crystal nucleus existing within the MSZW can be well controlled for quality and particle size; if the supersaturation is over the MSZW, the crystallization process is not controlled, affording unsatisfactory results. The MSZW is calculated from the mole fraction of water at which the linear parts intersect the x-axis, as shown in eq 1:

the separation and crystallization of 5′-GMPNa2, although the crystallographic and solubility data for 5′-GMPNa2 have been reported previously.20−22 Studies of the as-yet-unreported metastable zone, the induction time, and the primary nucleation and growth mechanisms for 5′-GMPNa2 are necessary to control its crystallization. In the present work, we adopt the induction-based method to determine the primary nucleation and growth mechanism of 5′-GMPNa2. In addition, the MSZWs are predicted using an exponential equation for different temperatures, agitation rates, and antisolvent addition rates in the water−ethanol system.

MSZW = C ss(x) − C s(x)

(1)

3.2. Induction Time. The period of time that usually elapses between the attainment of supersaturation and the appearance of crystals is called the induction time. According to Mullin,12 the induction period may be considered to consist of several parts or stages: tr, tn, and tg. Nielsen12 pointed out that the nucleation time lag (tr) is often negligible; therefore, this will not be considered in this paper. Usually, the induction time can be determined by both the rate of primary nucleation and the growth rate of crystallites. A general expression proposed by Kashchiev et al.,23 including the nucleation and growth mechanisms, can be written as follows:24,25

2. EXPERIMENTAL SECTION 2.1. Materials. 5′-GMPNa2, with a purity of ≥99%, was obtained by recrystallization. Reagent grade ethanol (≥99.7%) was used as an antisolvent, and was supplied by the Shanghai Chemistry Reagent Co. (China). Deionized water was generated by an ultrapure water system (YPYD Co., China). 2.2. Metastable Zone Width Measurements. A solution of 5′-GMPNa2 was prepared in a multiport flask, with stirring by an agitator (IKA 40) and heating by a thermostatic bath at temperatures ranging from 293.15 to 303.15 K. The uncertainty in the temperature is ±0.05 K. Ethanol was added to saturate the system, and the supersolubility was calculated based on the solubility data and the amount of added ethanol. The crystallization process was monitored by turbidity changes using a Trb8000 turbidity transmitter and InPro8200 turbidity probe (Mettler Toledo, USA). We also measured the supersolubility at different temperatures and mole fractions of water. Each experiment was repeated at least three times, and a mean was calculated. The estimated uncertainty of the solubility and supersolubility based on the error analysis was less than 2%. 2.3. Measurement of Induction Time. The values of tind were determined by an isothermal method and a turbidity method with the inline probe.10 A suitable amount of 5′GMPNa2 was added into the prepared ethanol−water mixture with a water mole fraction of 80%. The system was heated to 333.15 K in a prepared thermostatic bath and then rapidly cooled in another prepared thermostatic bath, set at temperatures ranging from 288.15 to 298.15 K, which correspond to the temperatures needed to achieve the required supersaturation. The uncertainty in the temperature is ±0.05 K. In addition, we also measured tind values at different mole fractions of water (70, 80, and 90%) at constant temperature. The uncertainty in supersaturation is about ±2%. The supersaturation was maintained until the first crystals were observed. We measured the time elapsed from placing the flask in the bath of desired temperature to the formation of a new phase; this was regarded as the induction time for crystallization. During the experimental process, the turbidity was used to detect any variation in the system. Each experiment was carried out at least three times, and the average values were used as tind. The uncertainty in tind is ±0.1 s. The estimated uncertainty, including the system error and experimental error, in the induction time for repeat experiments carried out under the same condition was less than 2%.

t ind =

⎡ α ⎤1/ n 1 ⎥ +⎢ JV ⎣ anJGn − 1 ⎦

(2)

Equation 2 contains two terms: the first term represents a mononuclear mechanism, whereas the second originates from a polynuclear mechanism. The first part of eq 2 assumes that the appearance of the first nucleus brings the system out of its metastable state. The loss of the metastable equilibrium is then due to a mononuclear nucleation mechanism.24 Therefore, the induction time for the appearance of the first nucleus is given by 1 t ind = JV (3) In our system, classical theory is not suitable because the recovered 5′-GMPNa2 crystals are sometimes circular or platelike in shape (as shown in the Supporting Information, Figure S2), depending on the crystallization process. Consequently, the equation must be reformulated to include other shapes:1,12 ⎛ −4f 3 γ 3ν 2 ⎞ ⎛ ⎞ S ⎟ = A exp⎜ B ⎟ J = A exp⎜⎜ 2 3 3 2 ⎟ 2 ⎝ ln S ⎠ ⎝ 27fV k T ln S ⎠

(4)

From eqs 3 and 4 in conjunction with the logarithm of the induction time, we obtain ln t ind = ln

1 B + 2 AV ln S

(5)

This model is based on the mononuclear mechanism, which is suitable for characterizing the tind required for critical nuclei formation; therefore, eq 4 is only valid in this case, as Mullin has indicated.1,12 tind in our experiment includes the time for the nucleus to grow to a detectable size, as defined by tg, and therefore, the polynuclear mechanism must be introduced to explain the nucleation. The second term in eq 2 supposes that metasaturation is lost by the nucleation and growth of a large number of nuclei. The mononuclear mechanism is often negligible in comparison to the polynuclear mechanism. Hence, the induction time for the

3. THEORY 3.1. MSZW. The MSZW (shown in the Supporting Information, Figure S1) is a region bounded by the solubility curve and the metastable limit, where the solution is B



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Figure 2. MSZWs of 5′-GMPNa2 obtained in water−ethanol solvent mixtures at (a) 293.15 and (b) 303.15 K. The symbols represent (○) solubility and (□) supersolubility.

Figure 3. MSZWs for 5′-GMPNa2 obtained in water−ethanol solvent mixtures at 298.15 K under different operating conditions. (a) Ethanol addition rate: (○) solubility, (△) supersolubility at 0.28 mL/min, and (□) supersolubility at 4.48 mL/min. (b) Agitation rate: (○) solubility, (□) supersolubility at 150 rpm, and (△)supersolubility at 300 rpm.

The different forms of ln tind for the four different growth mechanisms are listed in Table S2 in the Supporting Information. Many investigations have employed the foregoing methods to determine the crystal growth mechanisms for different materials.1,24,26 The crystal growth mechanism can be identified by another expression of the crystal growth rate, which is represented by a general empirical power-law relationship.12 The expression is shown as follows:

formation of a new phase by the polynuclear mechanism is shown as follows: ⎡ α ⎤1/ n ⎥ t ind = ⎢ ⎣ anJGn − 1 ⎦

(6)

Generally, the relationship between G and S is of the form G(S) = KGf (S)

(7)

G = KG(S − 1)g

The expressions corresponding to f(S) are listed in Table S1 in the Supporting Information for normal, spiral, volumediffusion-controlled, and two-dimensional (2D) nucleationmediated growth.24 By combining eq 6, eq 7, and nucleation rate J = KJS exp(B/ (ln2 S)), we obtain ⎛ B ⎞ ⎟ t ind = A ind [f (S)]−(n − 1)/ n S −1/ n exp⎜ ⎝ n ln 2 S ⎠

where Aind = (α/anKJKG rearranged as follows: ln t ind = ln A ind −

(10)

By combining eqs 6 and 7 and taking the logarithm of the induction time, a new relationship between tind and S can be established: ln t ind = ln A −

(n − 1)g 1 B ln(S − 1) − ln S + n n n ln 2 S (11)

(8)

The underlying nucleation and growth mechanism can thus be identified by comparing the correlation indexes R2 between eqs 9 and 11.

n−1 1/n

) . Therefore, eq 8 can be

4. RESULTS AND DISCUSSION 4.1. Metastable Zone Width Measurements. Metastable zone widths were determined for the crystallization of solutions

1 n−1 B ln S − ln f (S) + n n n ln 2 S (9) C



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Figure 7. Plots of ln tind versus S for 5′-GMPNa2 at different temperatures. Data were fitted using eq 10 (polynuclear mechanism).

Figure 4. Plots of tind versus S for 5′-GMPNa2 at different temperatures. Data were fitted using eq 21.

of 5′-GMPNa2 in ethanol−water systems with different mole fractions of water at 293.15 and 303.15 K. Solubility and supersolubility are critical parameters that define the operational limits for the crystallization process and provide information on the maximum theoretical yield for a given system. The obtained results are shown in Figure 2. The solubility and supersolubility curves in the investigated ranges of water mole fractions x can be calculated from the following equations. When T = 293.15 K: C s = 2.078 × 10−5e15.56x C ss = 0.0069e11.01x

R2 = 0.9967

R2 = 0.9924

(12) (13)

When T = 303.15 K:

Figure 5. Plots of ln tind versus 1/ln2 S for 5′-GMPNa2 at different temperatures. Data were fitted using eq 4.

C s = 1.374 × 10−5e13.36x

R2 = 0.9938

(14)

Figure 6. Plots of ln tind versus S for 5′-GMPNa2 crystallization data at different temperatures fitted to four growth mechanisms: (a) normal growth, (b) spiral growth, (c) diffusion-controlled growth, and (d) 2D nucleation-mediated growth. D



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Figure 9. Comparison of the influence of g in eq 11 on predicted ln tind values at different temperatures: (a) 288.15, (b) 293.15, and (c) 298.15 K.

Figure 8. Comparison of the influence of B in eq 11 on predicted ln tind values at different temperatures: (a) 288.15, (b) 293.15, and (c) 298.15 K.

C ss = 0.4412e6.530x

R2 = 0.9953

(15)

of the MSZW with water mole fraction between 0.4 and 0.85 is considerable. It is preferable to control the initial mole fraction of water between 0.7 and 0.85. This is because, under such conditions, supersaturation occurs, which forces the nuclei to grow. Certainly, the specific amount of water and antisolvent should depend on the experimental process. The MSZW in this study is shown as a reference for an industrial crystallization process; in this case, the product particle size can be controlled, and nucleation must occur in the MSZW. Solubility and supersolubility are generally known to be a function of temperature, so temperature exerts a direct influence on the MSZW. Considering the process constraints of industrial crystallization, the studied temperatures ranged from 293.15 to 303.15 K. From Figure 2, it is evident that the MSZWs increase with temperature. Thus, elevated temperatures during the entire crystallization process can help control the quality of the products. Higher temperature can accelerate the movement

As shown in Figure 2a, the MSZW increases with x, indicating that, for a system with a smaller amount of antisolvent, the crystallization process can be well controlled. However, in Figure 2b, the MSZW is not as wide as it was expected to be. The MSZW in Figure 2b is more narrow than that in Figure 2a on the top of the figure, which is not a normal phenomenon. During the course of the experiment, the sample temperature may be a bit higher than the room temperature. Owing to this, at higher temperatures, in particular, a small amount of solute might precipitate, so the result should be smaller than the real value, which can be seen in Figure 2b. However, this kind of an error is allowed in our system, because the water mole fraction needs to be under 0.85. This is because a water mole fraction higher than 0.85 is not suitable for industry crystallization processes. In an addition, the tendency E



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Figure 10. Plots of ln tind versus 1/ln2 S for 5′-GMPNa2 crystallizations at different mole fractions of water (70, 80, and 90%) at (a) 288.15 and (b) 293.15 K. The lines are drawn as visual aids.

that, when nucleation occurs, the rate of antisolvent addition must be slow enough. Of course, in terms of process economy, a slow rate will increase time and labor costs; thus, we must ensure that the entire process of crystallization occurs within the MSZW. In addition, eqs 16−20 can offer useful reference information in this respect. 4.2. Induction Time Measurements. Induction times were determined experimentally for 5′-GMPNa2 at different supersaturation states under various conditions. Plots of tind versus S at different temperatures and ethanol−water ratios are shown in Figure 4. The induction time increases with decreasing supersaturation and temperature. The relationship between the induction time and supersaturation can be expressed as28

of molecules within a system and reduce the degree of supersaturation in the metastable region, thus making it easier to control the crystallization process. In industrial crystallization processes, the agitation speed and the rate of antisolvent addition can also influence the final crystal quality. The MSZWs at different agitation speeds and antisolvent flow rates were also measured to determine the extent of their impact. The results are shown in Figure 3. The fitted equations at T = 298.15 K are shown as follows: C s = 1.119 × 10−5e16.57x

R2 = 0.9937

C ss(0.28 mL/min) = 0.07504e8.454x

(16)

R2 = 0.9986 (17)

C ss(4.48 mL/min) = 0.01884e9.884x

R2 = 0.9928

t ind =

(18) ss

10.23x

C (150 rpm) = 0.0154e ss

9.314x

C (300 rpm) = 0.03615e

2

R = 0.9971 2

R = 0.9987

(19)

K Sr

(21)

where K and r are constants. The values of K and r, as well as the correlation indexes for 5′-GMPNa2, are illustrated in the Supporting Information, Table S3, for different temperatures. It is apparent that all the correlation indexes are above 0.99, which implies the validity of the model. 4.3. Nucleation and Growth Mechanisms. The induction time data for different supersaturations at different temperatures are presented in the Supporting Information, Table S3, and the standard deviation (α) and relative standard deviation (RSD) were calculated. From the Supporting Information, Table S4, it can be seen that for all cases the values of α and RSD were below 2.0 and 1.2%, respectively, implying that the experimental induction time data exhibit a small variation around the mean. Plots of ln tind versus 1/ln2 S at different temperatures are shown in Figure 5. Clearly, the measured induction times for 5′-GMPNa2 follow the linear relationship given by eq 4. However, there are regions of higher slope at higher supersaturation and lower slope at lower supersaturation. Similar results have been reported in other studies.1,24,29 These changes illustrate that the nucleation mechanism for 5′GMPNa2 is not a simple homogeneous process. The inflections in Figure 5 may reflect a homogeneous process that is gradually transformed to heterogeneous. The reason for this change is that homogeneous nucleation occurs with much more difficulty in the interior of a uniform solution, whereas heterogeneous nucleation induced by external particles occurs at preferential

(20)

From Figure 3, the MSZW increases with decreasing antisolvent addition rate and increasing agitation rate. When the temperature is fixed, a high rate of ethanol addition and a higher agitation rate can lead to partial supersaturation. The system nucleates easily under this condition. It is widely believed that a higher agitation rate leads to a narrower MSZW. In addition, because a higher agitation rate can lead to an increased rate of mass transfer, the probability of collision nucleation of 5′-GMPNa2 molecules increases, too. At the same time, an increase in the heat transfer rate promotes the diffusion of heat, so as to reduce the degree of supersaturation. Therefore, crystallization occurs sooner and the MSZW is narrower. However, it is known that only when a crystal particle grows to a certain critical particle size can it form a stable nucleus for further crystal growth. According to crystal survival theory,27 in a supersaturated solution, only when the particle size is greater than the critical size can the crystal particle survive and grow; particles with a size lower than the critical size are dissolved. Therefore, as the supersolubility increases, the MSZW becomes wider with an increasing agitation rate. Thus, to control the quality of the final product, we must choose an appropriate agitation rate and antisolvent flow rate. However, the rate of antisolvent addition exerts a greater impact on the MSZW than the agitation rate, which indicates F



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atures ranging from 288.15 to 298.15 K. As can be seen from parts a, b, and c of Figure 8, when the value of B was taken to be 1−4% greater or less than that of the experimental value, it was found to have an influence on the predicted value of ln tind with errors of 4, 1, and 2% at 288.15, 293.15, and 298.15 K, respectively. When the value of g was taken to be 1−4% greater or less than that of the experimental value, it was found to have little influence on the predicted value of ln tind with an error of 1% at 288.15, 293.15, and 298.15 K (Figure 9). Consequently, eq 11 was deemed to be correct, with the results obtained from it deemed reliable. Several investigators have studied the influence of solvent composition on primary nucleation in different antisolvent crystallization systems such as water−ethanol,7 water− acetone,30 and water−methanol.2,6 However, their focus was to study primary nucleation with different antisolvents, and they did not examine whether the antisolvent changed the nucleation mechanism. To determine whether the antisolvent effects change the nucleation mechanism, we also measured the induction times in the presence of different mole fractions of water (x = 70, 80, and 90%) at 288.15 and 293.15 K. The reason for selecting this condition is that the mole fraction of antisolvent required for saturation during crystallization is between 0.15 and 0.25, and the temperature range for obtaining the best quality products is 288.15−298.15 K. The experimental results are shown in Figure10. In the foregoing figures, the relationship between ln tind and 1/ln2 S is nonlinear, suggesting a heterogeneous or even a secondary mode of nucleation. On comparing Figures 5−10, we found evidence for the existence of a different mechanism. According to the experimental data, it may be possible that the antisolvent changes the nucleation mode. However, there are no reports in the literature supporting the idea that antisolvents influence the nucleation modes and mechanisms of crystallization.

sites such as phase boundaries or impurities such as dust. Heterogeneous nucleation also takes place much more often than homogeneous nucleation. It is impossible to remove all dust particles, and as a result, the longer the time lapse without homogeneous nucleation in a supersaturated system, the greater the chance that dust particles, as well as the walls of the crystallization vessel, can act as heteronuclei for crystal formation. The slopes (B) and intercepts (Am) of these two regions, in addition to the correlation indexes (R2) at different temperatures, are shown in the Supporting Information, Table S5. The lowest correlation index is 0.923, which indicates poor correlation between the induction time and eq 4. In other words, the mononuclear mechanism was not well-suited to explaining the results obtained with this system. Therefore, polynuclear mechanistic models must be considered to fit the results of induction time and supersaturation. As mentioned before, the different forms of f(S) can be found in the literature and the relationship between the induction time and the supersaturation for different growth mechanisms of 5′-GMPNa2 are listed in the Supporting Information, Table S1. The plots for the data fit to the different crystal growth mechanisms of 5′-GMPNa2 at different temperatures (288.15−298.15 K) are shown in Figure 6. The parameters and the correlation indexes for ln tind versus S for the different growth mechanisms at different temperatures are listed in the Supporting Information, Table S6. The plots in Figure 6a show the 5′-GMPNa2 crystallization data fitted to the normal growth mechanism. In the following plots (Figure 6b), the data are fitted to the spiral growth mechanism; these fits are obviously poor, as shown by the low correlation coefficients and the systematic deviations of the data points from the fitted lines at 293.15 and 298.15 K. Thus, the spiral growth mechanism was not found to be suitable to explain the results obtained with this system. The plots in Figure 6c and Figure 6d show the data fitted to the diffusioncontrolled and the 2D nucleation-mediated growth mechanisms. On the basis of the fitting results, the 2D nucleationmediated growth mechanism is a superior model to the diffusion-controlled growth mechanism. Only on the basis of correlation indexes, the preferred growth mechanism for 5′GMPNa2 at 288.15 K is the spiral one, whereas at 293.15 and 298.15 K, the mechanism changes to that of a 2D nucleationmediated growth. However, the values of B are negative at 288.15 K for spiral growth and at 293.15 and 298.15 K for 2D nucleation-mediated growth. However, according to the expression for B in eq 4, its value must be positive. Therefore, the growth mechanism for crystals of 5′-GMPNa2 must be diffusion-controlled. As mentioned in section 3.2, another method can also be used to identify the growth mechanism and confirm our results. Values of g were obtained by correlating the induction time with the supersaturation through eq 10, as shown in Figure 7 and the Supporting Information, Table S6. As shown in the Supporting Information, Table S7, all the correlation indexes are above 0.99 and g = 1 at the three different temperatures. These values indicate that the crystal growth mechanism for this compound is diffusion-controlled, which confirms the conclusion obtained using eq 9, and confirms the validity of eq 9. For parameters B and g in eq 11, Figures 8 and 9 show a comparison of the influences of B and g as a factor of scatter at different experimental supersaturations at different temper-

5. CONCLUSION The MSZW of 5′-GMPNa2 in an ethanol−water system increases with temperature, mole fraction of water, and agitation speed, and with a decreasing rate of ethanol addition. All the supersolubility data were fitted using exponential equations. The induction times for the crystallization of 5′GMPNa2 in ethanol−water were determined over a range of supersaturation states by turbidity measurements. If the supersaturation was caused by temperature, two different nucleation mechanisms were found to operate: homogeneous nucleation at a high supersaturation and heterogeneous nucleation at a low supersaturation. However, if the supersaturation was caused by antisolvent, the nucleation mechanism must be either heterogeneous nucleation or secondary nucleation. The growth of 5′-GMPNa2 crystals is governed by a polynuclear mechanism: the metastability is lost by the nucleation and growth of nuclei. Four different models of growth were correlated with the experimental data, and the best fit was obtained with the diffusion-controlled growth mechanism, consistent with the result from the other model (g = 1). Thus, the growth of 5′-GMPNa2 crystals occurs through a diffusion-controlled mechanism, as suggested by the fitting of two different equations with different parameters. G



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ASSOCIATED CONTENT

S Supporting Information *

Some expressions corresponding to f(S), functions of ln tind versus 1/ln2 S for different growth mechanisms, and parameters fitted with different growth mechanisms. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] or [email protected]. Tel.:+86 25 86990001. Fax: +86 25 58139389. Author Contributions §

F.X. and W.Z.: These authors contributed equally to this work.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Basic Research Program of China (No. 2011 CBA00806), Natural Science Foundation of Jiangsu Grants (No. BK20130929, No. BK2011031), Jiangsu Postdoctoral Science Foundation (1301038B), National Outstanding Youth Foundation of China (No. 21025625), National High-Tech Research and Development Plan of China (2012AA021202), and Natural Science Foundation of China Grants (No. 21106070).

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NOTATION A = pre-exponential factor, m−3·s−1 an = shape factor B = constant Cs = concentration solubility of 5′-GMPNa2, g·L−1 Css = concentration supersolubility of 5′-GMPNa2, g·L−1 f S = surface shape factors f V = volume shape factors G = growth rate, m−3·L−1 g = parameter of growth mechanism J = nucleation rate, m−3·L−1 k = Boltzmann constant, 1.38 × 1023 J·K−1 KJ = nucleation constant KG = growth rate constant K = constant MSZW = metastable zone width m = dimensionality of growth n = nucleation exponent r = constant R2 = correlation index S = supersaturation tind = induction time, s tg = time needed for growth of nuclei, s tn = time needed for formation of nuclei, s tr = relaxation time, s V = volume of solution, m3

Greek Symbols

α = volume of new phase formed ν = molecular volume

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REFERENCES

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Determination of Metastable Zone Widths and the Primary Nucleation

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