Ind. Eng. Chem. Res. 2001, 40, 4821-4827
4821
Crystallization Kinetics of Ampicillin M. Ottens,*,† B. Lebreton,†,‡ M. Zomerdijk,† M. P. W. M. Rijkers,§ O. S. L. Bruinsma,|,⊥ and L. A. M. van der Wielen† Kluyver Laboratory for Biotechnology, Delft University of Technology, Julianalaan 67, 2628 BC Delft, The Netherlands, DSM Research, P.O. Box 18, 6160 MD Geleen, The Netherlands, and Laboratory for Process Equipment, Leeghwaterstraat 44, 2628 CA Delft, The Netherlands
The first results regarding the crystallization kinetics of the pure semisynthetic β-lactam antibiotic (SSA) ampicillin in water are presented. The experimentally obtained results at pH ) 5 and T ) 298 K are described well by the population-based crystallization model presented in this paper. The nucleation rate J (# m-3 s-1) is related to the supersaturation ratio of ampicillin S by J ) 2 × 107 exp[-3.383/(ln(S))2]. The surface energy γ of creating a nucleus of ampicillin molecules therefore equals 5.83 mJ m-2. The growth rate G (m s-1) is determined with a newly developed adapted single-crystal growth rate analysis method and is related to the supersaturation ratio of ampicillin S by G ) 4.57 × 10-8(S - 1)1.72. Introduction Increased product purity demands for pharmaceutical and fine-chemical products as well as legislative and environmental reasons force pharmaceutical industries to investigate new concepts either to optimize existing production operations or to develop new processes for new products. In such new processes, a reduction of the number of process unit operations and waste material streams is paramount. Biotechnological operations, such as enzymatic reactions applied in aqueous environments, are becoming increasingly important for the production of pharmaceutical products such as penicillin derivatives.1 These new synthesis routes imply the application of appropriate separation techniques, which play an important role in the design of cost-effective unit operations. In this respect, crystallization is a suitable technique for the recovery of pharmaceutical products of relatively low solubility, such as β-lactam antibiotics. Crystallization is thus conducted in multicomponent systems where the presence of solutes other than the targeted product might significantly influence the kinetics of crystallization.2 It is known that foreign molecules such as degraded products and byproducts, additives, and other components can interfere with nucleation as well as with the growth process.3 Crystal growth can be disrupted by the incorporation of impurities into the crystal lattice or by their adsorption at the surface of the crystal.4-6 The structure, size, and morphology of the final product might consequently be modified by the presence of impurities.7 The present paper is part of a larger study focused on the identification of possible effects of impurities on the crystallization kinetics of semisynthetic penicillins and on the product quality. The impact of impurities will be monitored herein by * Correspondence concerning this article should be addressed to Marcel Ottens. E-mail:
[email protected]. † Delft University of Technology. ‡ Current address of B. Lebreton: Genentech, 1 DNA way (MS#75), South San Francisco, CA 94080. § DSM Research. | Laboratory for Process Equipment. ⊥ Current address of O. S. L. Bruinsma: SASOL Center for Separation Technology, Private Bag X6001, Potschefstroom, 2520, RSA
investigating the crystallization kinetics of the process, i.e., the induction time, desupersaturation rate, and growth rate.8 In this paper, the crystallization kinetics of pure ampicillin from aqueous solution is determined. Furthermore, a model is developed that is capable of describing the induction time, desupersaturation rate and growth rate. This model provides a framework within which the effects of impurities on semisynthetic antibiotic (SSA) crystallization can be accurately analyzed and reported in future papers. The model presented in this paper is an indispensable tool in further understanding the crystallization of antibiotics and developing realistic mechanistic models for nucleation and growth. As a relevant SSA model compound ampicillin (Ampi, Mw ) 0.349 41 kg mol-1) is considered. The structural formula of this compound is
The solubility of pure ampicillin in water at 298 K is depicted in Figure 1.9 Changing the pH of the aqueous solution to the isoelectric point of ampicillin gives rise to a substantial decrease in the solubility and induces the supersaturation and subsequent crystallization of the semisynthetic antibiotic. The solubility is altered by the introduction of salts.10 Model To describe the crystallization process and to obtain relevant kinetic parameters a crystallization model is used to simulate the solute concentration in the liquid phase as a function of time (the so-called desupersaturation curve) as well as the crystal size distribution (CSD). The model is based on a population balance for crystals in a specific size class11,12
10.1021/ie0101238 CCC: $20.00 © 2001 American Chemical Society Published on Web 10/05/2001
4822
Ind. Eng. Chem. Res., Vol. 40, No. 22, 2001
where kv is the crystal shape factor and Fv is the crystal density in kg m-3. The nucleation rate is described by12
(
J(t) ) J0 exp -
)
B ln(S(t))2
(7)
The growth rate is given by12
G(t) ) kg(S(t) - 1)n
(8)
Both the nucleation rate and the growth rate are dependent on the supersaturation ratio of the crystallizing component
S(t) )
Figure 1. Solubility of ampicillin as function of the pH of the solution at 298 K and 1 bar.
∂Vn(t,x) ∂t
)-
∂Vn(t,x) G(t,x)
+ ∂x Vb(t,x) - Vd(t,x) -
∑k Φknk(t,x)
∂n(t,x) ∂[n(t,x) G(t)] )∂t ∂x
(2)
To solve this PDE, boundary and initial conditions are required. The boundary condition at x ) 0 is given by the commonly applied equation
n(t,0) )
J(t) G(t)
(3)
where J is the nucleation rate in # m-3 s-1. The initial condition at t ) 0 is given by
n(0,x) ) 0
(4)
The desupersaturation is obtained by solving the following component mass balance
MwV
dC(t) dM(t) + )0 dt dt
(5)
where Mw is the molar mass of the crystallizing component. The mass of crystals formed (M) is given by integration over all crystal size classes
M(t) )
∫0∞ kvFvn(t,x)x3 dx
(6)
(9)
Solving the model contained in eqs 2-9 essentially means solving the hyperbolic partial differential equation (PDE). A numerical solution method is used to solve the discretized PDE. A first-order single step upward discretization in space and in time is used, leading to the following Eulerian scheme
(1)
where V is the compartment volume in m3, n is the number of particles per volume per size class in # m-4, t is the time in s, x is the length coordinate in m, G is the growth rate in m s-1, b is the birth function in a certain crystal size class in # m-4 s-1, d is the death function in a certain crystal size class in # m-4 s-1, Φk is the flow rate of the kth stream containing crystals entering or leaving the compartment in m3 s-1. Equation 1 reduces under the constraints of a constantvolume batch operation (no input and output) and under the assumptions of no agglomeration, no breakage, and no death; birth only in the lowest particle class; and sizeindependent growth to the following hyperbolic partial differential equation (PDE)
C(t) CS
) nji nj+1 i
∆t j j j j ni-1 ) (G n - Gi-1 ∆x i i
(10)
with index i indicating the space coordinate and index j indicating the time coordinate. The choice of the values of the time step and space step governs the numerical stability of the system. Care should be taken in selecting these step sizes. In general the so-called CourantFredrich-Levy number should be smaller than 0.5,13 or in this case
Gji∆t < 0.5 ∆x
(11)
To obtain a mechanistic model for the growth rate and the nucleation rate, the comprehensive crystallization model can be used and tested for different relations between the nucleation and growth rates. However, the kinetic parameters can also be determined by plotting the results from the crystallization experiments according to eqs 7 and 8. Nucleation A useful lumped parameter for monitoring the nucleation mechanism is the induction time, which is defined as the period of time that elapses between the achievement of supersaturation and the appearance of crystals having a “detectable” size. The induction time depends not only on the initial supersaturation but also on the detection method. If, for example, a concentration measurement is used, the induction time depends on the conversion, whereas for light reflection detection, it depends instead on the crystal surface area produced. In general, one can simplify this problem as follows: once the detectable value of a moment of the CSD, mi,det, is exceeded, the induction period has elapsed (where i is the order of the moment)
mi,det ) or
∫0G(t)t
ind
J(t) i J(t) x dx ) [G(t)tind]i +1 G(t) (i + 1)G(t)
(12)
Ind. Eng. Chem. Res., Vol. 40, No. 22, 2001 4823
tind )
[
]
(i + 1)mi,det J(t) G(t)i
1/(i+1)
(13)
In terms of initial supersaturation S0, using eqs 7 and 8, this becomes
tind ∝ [(S0 - 1)n]-1/(i+1) exp
[
]
B (i + 1)(ln(S0))2
(14)
Because the exponential term will dominate, a plot of ln(tind) versus [ln(S0)]-2 for different crystallization experiments will yield B/(i + 1) as the slope. Then, the type of detector determines i; in our case, laser reflection is used for detection, so i ) 2. The nucleation in this study is considered to be primary nucleation. The factor B in eq 7 is given by12
B)
16πν2γ3 3(kT)3
(15)
and from B the value for the surface energy γ is obtained. Ultimately, from the desupersaturation experiments, the value of J0 is obtained. The primary nuclei are assumed to be spherical. For anisotropic crystals the surface energy of each face must be different as this is the cause of anisotropy. However, whether a nucleus is already faceted or is spherical is a question that has not been solved. It seems that a certain size is needed before a spherical crystal starts to develop facets. The shape of the crystals becomes needlelike during crystal growth. This is due to the difference in growth velocities for the different crystal faces. No different face-growth velocities are used in this model, but this phenomenon is taken into account by using a shape factor based on the aspect ratio of the crystals. In eq 14, the relation between the induction time and the (initial) supersaturation is given. This relation provides essential information for the nucleation kinetics, i.e., the factor B. To assess the value of B, it is good practice to calculate the associated surface energy γ, ensuring that the rate equation is correct and that the experimental surface energy is of the right order of magnitude. Finally, γ can be used to decide whether the dominant nucleation is homogeneous or heterogeneous. Growth In the case of the growth rate equation, the parameters kg and n in eq 8 can be obtained from growth rate experiments as the abscissa and the slope, respectively, of plots of ln(G) vs ln[S(t) - 1)]. Because ampicillin is a rather large molecule, the crystallization rate is relatively low, and the desupersaturation period turns out to be long enough to determine the growth kinetics from the second part of the batch experiment (see BatchCrystallization Experiments section below). Experimental Setup and Techniques for Determination of Kinetic Parameters Materials. Ampicillin trihydrate (Ampi) was provided by DSM (Geleen, The Netherlands). Batch-Crystallization Experiments. Ampicillin was crystallized from freshly prepared solutions. The experiments were conducted at pH 5.0 and T ) 25 °C
Figure 2. Experimental setup for ampicillin crystallization experiments with the adapted single-crystal growth rate analysis technique (ASCGRA).
with different initial supersaturations S0 ) 2.37 (Xa), 3.29 (Xb), 2.45 (Xc), and 1.72 (Xd). Batch-crystallization experiments were conducted in a 2-L jacketed reactor containing three baffles and two inter-MIG II impellers, with a stirring speed of 300 rpm. The temperature was set to 25 °C. Pure material was first dissolved under acidic conditions at pH 1.9 for 20 min, using HCl as the solvent (2 M). The pH was subsequently raised to a value of 5.0 using ammonia in water (12 M). The final solubility of ampicillin in this solution of 18.9 mol L-1 is higher than the solubility of ampicillin in pure water (see Figure 1) due to the presence of the salt (NH4Cl). This “salting-in” effect is described elsewhere.10 Temperature and pH were kept at their set values. A laser probe (Delft Unversity of Technology, Delft, The Netherlands) was inserted into the reactor to monitor the changes of turbidity by laser light reflection. Temperature, pH, and laser signal were continuously recorded using a Biodacs system (Applikon, Schiedam, The Netherlands); see Figure 2. Samples were carefully extracted at various time intervals. Aliquots of the samples were filtered using nylon membranes (0.2 µm; Gelmann Sciences), and the filtrates were appropriately diluted for subsequent reverse-phase chromatography analysis, while the remaining samples were used for crystal growth rate determination. The final product of each experiment was dried at 40 °C and subsequently analyzed for crystal size distribution. Its crystal size distribution (CSD) was estimated from the image analysis of a minimum of 500 particles observed via light microscopy (Leica Q500IW, Leica, Cambridge, U.K.). Image analysis appeared to be the most suitable and reliable technique for CSD determination because of the needlelike shape of the crystals. Scanning electron microscopy (SEM) images were acquired to characterize the final product of each crystallization using a scanning electron microscope (JSM-5400; JEOL, Tokyo, Japan). Samples were coated with gold for 3 min using an ion sputtering gun (JFC1100E; JEOL, Tokyo, Japan). Reverse-Phase Chromatography. Samples (containing Ampi) were separated and analyzed by reversephase chromatography using a HPLC Waters system comprising a Waters 996 PDA detector, a Waters 910 Wisp injector, and a Waters 590 pump. The reversephase column was a Zorbax SB-C18 column (4.6 × 75 mm with a pore size of 3.5 µm; Hewlett-Packard, Palo Alto, CA). The buffer consisted of 8 mmol L-1 tetrabu-
4824
Ind. Eng. Chem. Res., Vol. 40, No. 22, 2001
Figure 3. SEM image of the needle shaped final product resulting from batch-crystallization of pure ampicillin (experiment Xc). The experiment was conducted with a initial supersaturation ratio of 2.45 (see Experimental Setup and Techniques for Determination of Kinetic Parameters). (Left) Magnification 1 cm ) 100 µm. (Right) Details of a crystal needle tip at magnification 1 cm ) 10 µm.
tylammonium bromide, 10 mmol L-1 Na2HPO4, and 15% (v/v) acetonitrile and was brought to pH 6.6 with H3PO4. The elution profile was isocratic, and the absorbance was measured at 230 nm. In combination with the saturation concentration, the supersaturation could thus be calculated. Crystal Growth Rate Determination via Adapted Single-Crystal Growth Rate Analysis (ASCGRA). A sample was sucked from the batch crystallizer and introduced into a glass cell. The crystal growth was then monitored using a phase contrast microscope (Olympus, Tokyo, Japan) by acquiring images at various time intervals (ranging between 20 and 200 s). A few crystals were selected from the large number of crystals observed per image, and their lengths were measured via image analysis. The growth rate was thus determined. The operation was repeated for a series of samples corresponding to distinct supersaturations, decreasing in time during the batch experiment. Results Crystal Morphology. SEM images of the pure ampicillin crystal obtained from experiment Xc are shown in Figure 3 at two magnifications. It can be seen that the ampicillin crystals formed are boatlike crystals that are tapered and have mixed blunt and sharp ends. Furthermore, they show some limited twinning and limited breakage. The mean aspect ratio AR varies between the experiments performed. For experiments Xa, Xb, and Xc, the AR values were 7.74, 5.44, and 8.17, respectively. The corresponding shape factors kv then become 0.0250, 0.0507, and 0.0225, respectively. Desupersaturation Curve. It can be seen from Figure 4 that the initial supersaturation S0 has a pronounced effect on the crystallization rate. The higher the initial supersaturation S0 the faster the desupersaturation and the crystallization. The inset of Figure 4 also shows that the crystallization actually needs a specific induction time to obtain a substantial rate. The induction times measured with the laser reflection method are indicated in Figure 4. Induction Time Experiments. The plot of ln(tind) versus [ln(S0)]-2 for the different crystallization experiments gives a straight line, as shown in Figure 5. The slope of the curve equals B/3 and has a value of 1.127, so B ) 3.383. From eq 15, the value for the surface energy is calculated. The molar volume of ampicillin is obtained from the UNIQUAC volume parameter R ) 10.587 (R ) V/VCH4, where VCH4 is the reference volume
Figure 4. Desupersaturation curve for amplicillin crystallization at different initial supersaturations. Xa (circles), Xb (squares), Xc (triangles), and Xd (diamonds) pure ampicillin crystallization at pH ) 5 and T ) 298 K. See Experimental Setup and Techniques for Determination of Kinetic Parameters for precise conditions. S based on concentrations; see eq 9.
of methane ) 15.17 cm3 mol-1 9) giving 160.61 cm3 mol-1. The volume of one molecule of Ampi, then, is V divided by Avogadro’s number Na, i.e., ν ) 2.667 × 10-28 m3. The surface energy of creating a nucleus γ is calculated from eq 15 via
x 3
γ)
3B(kT)3 16πν2
with T ) 298 K. γ then is 5.83 mJ m-2, which corresponds to heterogeneous primary nucleation.14 Ex Situ Growth Rate Measurements. From the slope of the curve in Figure 6, the power in the crystal growth relation n is obtained and has an overall value of 1.72. The intercept in Figure 5 gives a value for the growth rate coefficient kg of 4.57 × 10-8 m s-1. It can be seen that, for the different series, a different slope can be identified. It seems that the growth rate is dependent on the initial super saturation S0. Crystal Size Distribution Measurement. The measured CSD can be used to validate the experimentally determined kinetic parameters. Only for experiment Xc was a reliable CSD measured. During the first
Ind. Eng. Chem. Res., Vol. 40, No. 22, 2001 4825 Table 1. Model-Based Validation Results for J0 and Numerical Settings experiment S0 Xa Xb Xc Xd Xec
AR
2.37 7.74a 3.29 5.44a 2.45 8.17a 1.72 10b 2.64 14.9
J0 (# m-3 s-1)
N
M
length (m)
time (s)
∆t (s)
∆x (m)
1 × 107 4 × 107 2 × 107 4 × 107 2 × 107
1000 2000 1000 4000 1000
200 100 200 400 200
1 × 10-3 1 × 10-3 1 × 10-3 2 × 10-3 1 × 10-3
4 × 104 2 × 104 4 × 104 16 × 104 4 × 104
40 10 40 40 40
5 × 10-6 1 × 10-5 5 × 10-6 5 × 10-6 5 × 10-6
a Value based on size classes that are too low. b Not available, interpolated value based on Xa-Xc. c Only S0, AR, and the final CSD are measured (particle class step size ) 50 µm).
Figure 5. Induction time experiments for the determination of the nucleation rate. Xa, Xb, Xc, and Xd pure ampicillin crystallization at pH ) 5 and T ) 298 K. See Experimental Setup and Techniques for Determination of Kinetic Parameters for precise conditions. S based on concentrations; see eq 9.
Figure 7. Model-based desupersaturation curve comparison with experimental values from data set Xc. Initial supersaturation ratio of 2.45. Density of crystals used in calculation Fv ) 1500 kg m-3. Lines are the model; points are experiments. In addition, the modeled amount of crystals formed (M) is given by the increasing line.
Figure 6. Ex situ crystal growth rate determination by the flow cell method. Xa (triangles), Xb (diamonds), Xc (squares), and Xd (circles) pure ampicillin crystallization at pH ) 5 and T ) 298 K. See Experimental Setup and Techniques for Determination of Kinetic Parameters for precise conditions. S based on concentrations; see eq 9.
experiments, i.e., Xa and Xb, because the computer crystal-selecting method for size measurement overselected particles from the lower size classes, these CSDs had crystal size values that were too small. For Xd, no CSD was measured. An extra crystallization experiment Xe was performed with a more accurately determined CSD. This was done to check the model and the obtained kinetic parameters by comparing the model CSD with the experimental CSD. Model-Based Validation. The obtained kinetic data were validated using the model introduced previously, and consequently, the model results are compared with experimental results acquired as described in the previous section. The model is “fed” with the obtained kinetic parameters that are used in the calculations, namely, B ) 3.383, n ) 1.72, and kg ) 4.57 × 10-8 m s-1. By comparing the model and experimental desupersaturation profiles, the value of the preexponential nucleation rate factor J0 can be obtained. The time and space step sizes used in the calculations are mostly ∆t ) 40 s and
∆x ) 5 × 10-6 m, respectively, except for Xb for higher numerical stability. The results and conditions are shown in Table 1. The value of the aspect ratio AR of 14.9 in Xe differs from the previous ones. It was obtained from a manual image analysis and is more accurate than the other values. However, the influence of the AR value on the calculated desupersaturation, nucleation and growth kinetics, and CSD is limited. The optimized values for J0 from the desupersaturation curves vary slightly with S0 (see Table 1). However, the desupersaturation curves and nucleation and growth kinetics can properly be described with a single value for J0, i.e., J0 ) 2.0 107 # m-3 s-1. The results are shown in the Figures 7-10. For the evolution of the crystal size distribution in time, the essential elements of the crystallization are well captured by the model. At S0, the crystal growth is fast, whereas after a while, the decreased supersaturation causes the growth rate to decrease and, finally, to reach zero. Figure 7 shows a comparison between the modeled desupersaturation and experimental data set Xc, and Figure 8 shows the modeled nucleation and growth rates during crystallization, together with the experimental growth rate data. These figures clearly show the ability of the model to describe and predict the crystallization behavior of ampicillin as given by the desupersaturation and the nucleation and growth rates. The induction period is very well captured with the model and the obtained kinetic data. In Figure 9, the calculated and experimental final scaled crystal size distributions are
4826
Ind. Eng. Chem. Res., Vol. 40, No. 22, 2001
Figure 8. Nucleation (dotted line) and growth rate (solid line). Comparison with experimental data set Xc. Conditions are the same as in Figure 7.
Figure 10. Comparison of the modeled final crystal size distribution curves of the product after amplicillin crystallization at an initial supersaturation of 2.64 (Xe) with experimental results (circles, 172 crystals, size classes of 50 µm). Pure ampicillin crystallization at pH ) 5 and T ) 298 K. See Experimental Setup and Techniques for Determination of Kinetic Parameters for precise conditions.
For experiment Xc, with an experimental dp,av of 126.5 µm and a modeled dp,av of 193.4 µm, this yields a ∆ value of 34.6%. For experiment Xe, with an experimental dp,av of 207.2 µm and a modeled dp,av of 227.5 µm, this yields a ∆ value of 8.9%. Thus, the average size is actually predicted quite accurately for Xe. It can be seen that the shapes of the experimental and model curves are similar but with a slight overprediction of the average crystal size. This could be due to several factors, one important one of which is the breakage of the needlelike crystals. Discussion Figure 9. Comparison of the modeled final crystal size distribution curves of the product after amplicillin crystallization at an initial supersaturation of 2.45 (Xc) with experimental results (circles, 525 crystals, size classes of 10 µm). Pure ampicillin crystallization at pH ) 5 and T ) 298 K. See Experimental Setup and Techniques for Determination of Kinetic Parameters for precise conditions. Conditions are the same as in Figure 7.
shown for the data set Xc for final verification of the kinetic parameters. We see that the average crystal size is predicted reasonably well. To validate these kinetic parameters more extensively, we used Xe with an accurately measured final CSD, as shown in Figure 10. The number-average particle sizes from the experimental and modeled CSDs are calculated according to M
dp,ini ∑ i)1
dp,av )
M
ni ∑ i)1 where M is the number of particle size class intervals. The relative difference between the modeled and experimental average particle size values is given by
∆ ) 100%
{
}
mod exp dp,av - dp,av mod dp,av
Different growth mechanisms can be identified by different values for the power number n in the growth rate equation. The birth and spread mechanism results in n ) 5/6; kinetic roughening results in n ) 1, and spiral growth in n ) 2 (e.g., So¨hnel and Garside14). In this work, the power n in the growth rate equation was close to 2 (1.72) for a wide range of experiments. This was also observed for the growth rate of similar molecules, such as aspartame, and indicates a spiral growth mechanism15 (the so-called Burton-Cabrera-Frank (BCF) model16). Thus, surface integration, rather than mass transfer, is the dominant mechanism during crystallization of ampicillin under the conditions reported in this paper. The shape of the experimental CSD in Figure 9 differs from the model CSD. This is largely because of the peak lying at ∼25 µm. This point arises as a result of the selection technique for the crystal size measurements. By shifting to manual selection, a much better picture arises in Figure 10. Furthermore, the front edges of the modeled CSD curves are not completely steep, which can be attributed to numerical diffusion. The average predicted crystal size in Figure 10 is somewhat too large. This can be related to breakage during crystallization and sample preparation. The drying step during this sample preparation process results in breakage of the crystals because of their relative fragility. This gives a smaller average particle diameter. Another issue is the number of crystals in the image analysis needed for
Ind. Eng. Chem. Res., Vol. 40, No. 22, 2001 4827
reliable CSD determination. We chose a minimal number of 500, because of the long analysis time, but this number might actually not be sufficient. Therefore, an extra crystallization experiment was performed, Xe, in which the AR and CSD were accurately measured for comparison with the model and final validation of the kinetic parameters. Overall, it can be stated that the model gives a good prediction of the CSD and the average crystal size, indicating the validity of the obtained kinetic parameters.
V ) volume, m3 x ) length, m
Conclusions
av ) average AMPI ) ampicillin 0 ) initial S ) saturated
The kinetics of pure ampicillin crystallization in an aqueous solution has been determined. These data can be used to design crystallization systems for the separation of ampicillin. The adapted single-crystal growth rate analysis method proves useful in determining growth rates for the relatively slow ampicillin crystallization. Furthermore, a valuable modeling tool is developed for analyzing the experimental crystallization data of ampicillin and more general of SSAs. The model covers the experimental data well, thereby verifying the kinetic relation obtained from experiments. Agglomeration and breakage terms could be inserted, but the need for such a step should be paramount; otherwise, it would only unnecessarily complicate the model. The model can be used to test different relations for growth G and nucleation J and to apply these G and J relations to the entire experimental database. A parametric optimization should be performed, but that falls beyond the scope of the present paper and is work for a future paper. The model can be extended by connecting the crystallization model to an accurate thermodynamic model for a description of the solubilities of the SSAs and their precursors, as developed in another paper.10 This should refine the model predictive capability to capture the specific solubility behavior of the SSAs and their precursors. Acknowledgment The authors thank DSM, Geleen, The Netherlands, for the chemicals and Chemferm and the Dutch Ministry for Economical Affairs for financial support. R. Grimbergen and T. van der Does from DSM are kindly acknowledged for their valuable input. M. Hoeben is thanked for performing experiment Xe and S. H. van Hateren for performing the image analysis. Symbols AR ) aspect ratio b ) birth function, # m-4 s-1 C ) concentration, mol m-3 d ) death function, # m-4 s-1 dp ) crystal length, m G ) growth rate, m s-1 J ) nucleation rate , # m-3 s-1 k ) Boltzmann constant ) 1.83 × 10-23 J K-1 kg ) growth rate coefficient, m s-1 kv ) shape factor M ) mass, kg M ) number of space steps, # n ) number of particles per volume per size class, # m-4 N ) number of time steps, # S ) supersaturation ratio T ) temperature, K t ) time, s
Greek Letters Φ ) flow rate, m3 s-1 Fv ) crystal density, kg m-3 γ ) interfacial free energy, J m-2 ν ) molecular volume, m3 Subscripts
Literature Cited (1) Bruggink, A.; Roos, E. C.; de Vroom, E. Org. Process Res. Dev. 1988, 2, 128-133. (2) McPherson, A. Impurities, defects and crystal quality. In Crystallization of Biological Macromolecules; Cold Spring Harbor Laboratory Press: Woodbury, NY, 1999. (3) Chayen, N. E.; Radcliffe, J. W.; Blow, D. M. Control of nucleation in the crystallization of lysozyme. Protein Sci. 1993, 2, 113-118. (4) Davey, R. J. The effect of impurity adsorption on the kinetics of crystal growth from solution. J. Cryst. Growth 1976, 34, 109119. (5) Kubota, N.; Mullin, J. W. A kinetic model for crystal growth from aqueous solution in the presence of impurity. J. Cryst. Growth 1995, 152, 203-208. (6) Kubota, N.; Yokota, M.; Mullin, J. W. Supersaturation dependence of crystal growth in solutions in the presence of impurity. J. Cryst. Growth 1997, 182, 86-94. (7) Black, S. N.; Davey, R. J. Crystallisation of amino acids. J. Cryst. Growth 1988, 90, 136-144. (8) Lebreton, B.; Zomerdijk, M.; Ottens, M.; Rijkers, M.; van der Wielen, L. A. M. Effects of impurities upon crystallization kinetics of β-lactam antibiotics. Presented at the AICHE Annual Meeting, Dallas, TX, Oct 31-Nov 5, 1999. (9) Rudolph, E. S. J.; Zomerdijk, M.; Ottens, M.; van der Wielen, L. A. M. Solubilities and partition coefficients of semisynthethic antibiotics in water + 1-butanol systems. Ind. Eng. Chem. Res. 2001, 40 (2), 398-406. (10) Rudolph, E. S. J.; Hamelink, J. M.; Zomerdijk, M.; Rijkers, M. P. W. M.; Ottens, M.; Vera, J. H.; van der Wielen, L. A. M. Influence of electrolytes on the phase behaviour of water + β-lactam antibiotics and their precursors, manuscript to be submitted. (11) Randolph, A. D.; Larson, M. A. Theory of Particulate Processes. Analysis and Techniques for Continuous Crystallization; Academic Press: New York, 1971 (12) Tavare, N. S. Industrial Crystallization. Process Simulation Analysis and Design; The Plenum Chemical Engineering Series; Plenum Press: New York, 1995. (13) Press: W. P.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T. Numerical Recipes in Pascal: The Art of Scientific Computing; Cambridge University Press: Cambridge, U.K., 1989. (14) So¨hnel, O.; Garside, J. Precipitation: Basic Principles And Industrial Applications; Butterworth-Heinemann: Oxford, U.K., 1992. (15) Frank, F. C. The influence of dislocations on crystal growth. Discuss. Faraday Soc. 1949, 48-54, 68. (16) Burton, W. K.; Cabrera, N.; Frank, F. C. Philos. Trans. R. Soc. London 1951, 243, 299.
Received for review February 7, 2001 Revised manuscript received July 10, 2001 Accepted July 18, 2001 IE0101238
