Determination of the Crystal Growth Rate of Paracetamol As a

Article pubs.acs.org/IECR

Determination of the Crystal Growth Rate of Paracetamol As a Function of Solvent Composition C. T. Ó ’Ciardhá,* N. A. Mitchell, K. W. Hutton, and P. J. Frawley Solid State Pharmaceuticals Cluster (SSPC), Materials and Surface Science Institute (MSSI), Department of Mechanical, Aeronautical and Biomedical Engineering, University of Limerick, Castletroy, County Limerick, Ireland ABSTRACT: Growth kinetics, growth mechanisms, and the effect of solvent composition for the antisolvent crystallization of paracetamol in methanol−water mixtures have been determined by means of isothermal seeded batch experiments at constant solvent composition. A numerical model incorporating the population balance equation based on antisolvent free solubility was fitted to the desupersaturation data, and growth rate parameters are evaluated. An attenuated total reflectance−Fourier transform infrared (ATR-FTIR) probe was employed to measure online solute concentration and focused beam reflectance measurement (FBRM) was utilized to ensure negligible nucleation occurred. The model is validated by the final particle size distributions (PSDs) and online solute concentration measurements. Crystal growth rate was found to decrease with increasing water mass fractions up to a mass fraction of 0.68 where an increase is observed. A method has been introduced linking the effect of solvent composition with the growth mechanism and the growth rates. Utilizing the growth mechanism it has been postulated that a combination of the solubility gradient, viscosity, selective adsorption, and surface roughening are responsible for the reduction in growth rates with solvent composition. Furthermore, the effects of seed mass, size and initial supersaturation on the crystal growth rates were investigated to demonstrate the efficacy of the model at predicting these various phenomena. supersaturation is generated via cooling and precipitation.2−9 However, very few studies have focused on antisolvent crystallizations, where the solvent plays a dominant role on crystal growth. Single crystal studies using transient imagery were carried out on sodium nitrate in water and isopropoxyethanol.10 Isothermal seeded batch experiments have been conducted for paracetamol and acetone−water mixtures, with nonlinear optimization utilized to evaluate the parameters of a power law expression to describe the growth rate, as a function of supersaturation, from the experimental desupersaturation curves.11 This work does not utilize online measurement techniques and similar studies have been conducted using isothermal seeded batch experiments, however employing ATR-FTIR to track solution concentration online.8,12 This work collects the previous results, methods, and observations in the literature referenced above, however expanding them to improve the methods efficiency while attempting to analyze the effect of composition on crystal growth rates. This work offers a novel approach in estimating growth kinetics as a function of solvent composition from a population balance model solved utilizing the computationally efficient method of moments and simultaneously investigating the underlying mechanism of how the solvent affects the growth rates. The moments of the distribution are reconstructed directly using a simple yet accurate method yielding a very fast and effective means of estimating growth rates.

1. INTRODUCTION Crystallization is a widely used technique in solid−liquid separation processes and is regarded as one of the most important unit operations in the process industries as many finished chemical products are in the form of crystalline solids. In antisolvent crystallization, supersaturation is generated by addition of another solvent or solvent mixture in order to reduce the solubility of the compound. Antisolvent crystallization is an advantageous method where the substance to be crystallized is highly soluble, has solubility that is a weak function of temperature, is heat sensitive, or unstable in high temperatures.1 Antisolvent processes have also been identified as a means to produce crystals more efficiently from continuous processes due to an ability to generate supersaturations quickly, run at low temperatures isothermally, and a low tendency to scale reactors which is a significant problem with cooling crystallizations. Rigorous determination of an optimal batch recipe requires accurate growth and nucleation rate kinetics, which can be determined in a series of batch experiments. Once the particle formation kinetics are known, they can be used together with a population balance model to simulate the influence of different process parameters on the final particle size distribution (PSD) of the product. In precipitation processes, the crystal growth rate is a crucial parameter since it determines the final specific properties of crystals such as the final particle size distribution. Two methods are commonly described in the literature for the estimation of crystal growth rate kinetics, namely single crystal growth studies and seeded desupersaturation experiments. There are numerous articles describing crystal growth from solutions where the © 2012 American Chemical Society

Received: Revised: Accepted: Published: 4731

September 5, 2011 February 20, 2012 February 20, 2012 February 21, 2012 dx.doi.org/10.1021/ie2020262 | Ind. Eng. Chem. Res. 2012, 51, 4731−4740

Industrial & Engineering Chemistry Research

Article

These reconstructed distributions are then compared to distributions obtained experimentally. Capturing the effects of solvent composition on the growth rate is critical to building a population balance model. Neglecting the influence of solvent composition will lead to discrepancies between experimental and modeled data. A growth rate model that considers its parameters to be functions of antisolvent mass fraction has previously been shown to be superior to a model without such functionality.13 The nucleation kinetics for paracetamol in methanol−water mixtures has been evaluated elsewhere and shown to be strongly dependent on solvent composition.14 This study aims to utilize a population balance to determine growth rates based on power law equations, while offering an insight into the growth mechanism and the effect of solvent composition. The focus of the work is also to determine parameters for power law expressions which can be used in predicting and optimizing crystal size distributions. To date the literature does not contain any work detailing the estimation of growth kinetics of antisolvent systems utilizing the method of moments together with in situ measurement techniques. In addition to this, a greater effort has been made to gauge the effect of solvent composition on growth rate kinetics by linking with the estimated growth rate mechanism.

In the case where the system is sufficiently seeded, negligible nucleation can be assumed. Ensuring that no nucleation is present is critical as seeding simplifies the mathematical treatment of the experimental data. Under this condition the second moment of the seed particle size distribution can only be influenced by the growth of the crystal and no other competing factors. During the crystallization process, the mass balance of the solution phase can be described as ∞ 2 dc nL dL = −3k vρcG dt 0

∫

where ρc and kv are the solid density and the volume shape factor of paracetamol crystals, respectively. In the above equation, the integral term represents the second moment of the seed crystals, m2 which is proportional to the total surface area of crystals present. A value of 1332 kg/m3 will be employed for the crystal density of form I of paracetamol.15 The following initial and boundary conditions apply:

(8)

min

(9)

C C*

∑ i=1

(10)

R t2 = min

Nd

∑ ⎡⎣(Stexp − Stsim(θ))⎤⎦

2 (11)

i=1

where θ is the set of parameters to be estimated, represents the predicted supersaturation ratios, Stexp represents the measured supersaturation ratios at each time or sampling interval, and Nd is the number of sampling instances. The MATLAB optimization algorithm f minsearch which employs a Nelder−Mead simplex method was utilized to find the optimal set of parameters. Stsim

∞ k

(3)

3. EXPERIMENTAL SECTION 3.1. Materials. The experimental work outlined was performed on Acetaminophen A7085, Sigma Ultra, ≥99%, sourced from Sigma Aldrich. The methanol employed in this work was gradient grade hiPerSolv CHROMANORM for HPLC ≥99%, sourced from VWR.

k

Equation 2 can be multiplied through by L and integrated to result in an equation in terms of the moments mk: dm 0 =B dt

n(t , 0) = 0

Nd

(2)

L n(L , t )dL

(7)

where C is the solute concentration and C* is the antisolvent free solubility. The antisolvent free solubility is calculated and described elsewhere.14 The crystal size distribution was reconstructed from the moments utilizing a novel technique developed by Hutton.16 This technique is outlined in more detail by Mitchell et al.9 2.2. Optimization. For the estimation of the growth kinetics parameters the following least-squares problem had to be solved:

where

∫0

n(0, L) = n0(L)

S=

where n(L,t) is the population density of the crystals and G(t) is the crystal growth rate which is assumed to be independent of size. The above equation is solved using the method of moments, detailed below. The standard method of moments is an efficient method of transforming a population balance into its constituent moments. The low order moments of the distribution represent the total number, length, surface area, and volume of particles in the crystallizing system. Using the standard method of moments, eq 1 becomes

mk =

(6)

ΔC = C − C*

(1)

dmk (t ) = kG(t )mk − 1(t ) dt

C(o) = C0

with C0 being the initial concentration of the solute, n0(L) the initial PSD, and B is the nucleation rate per unit mass. The supersaturation correlations used in this work for absolute supersaturation and supersaturation ratio are as follows:

2. POPULATION BALANCE MODEL AND PARAMETER ESTIMATION PROCEDURE A mathematical model based on population balance equations (PBEs) is used in combination with a least-squares optimization and the experimental desupersaturation data to determine the growth rate parameters of paracetamol in methanol/water solutions as a function of solvent composition. 2.1. Population Balance. In a perfectly mixed batch reactor the evolution of the crystal size distribution can be described as follows ∂n(L , t ) ∂n(L , t ) =0 + G (t ) ∂L ∂t

(5)

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dx.doi.org/10.1021/ie2020262 | Ind. Eng. Chem. Res. 2012, 51, 4731−4740


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3.2. Apparatus. A LabMax reactor system from MettlerToledo was utilized in this work to estimate the growth kinetics of the paracetamol in a methanol/water solution system. The reactor was a 1-L round-bottomed borosilicate glass jacketed reactor, allowing controlled heating and cooling of solutions. All experiments were carried out isothermally at 25 °C and at constant solvent composition. Agitation of the solution was provided by means of an overhead motor and a glass stirrer, with four blades at a pitch of 45°. The system allowed fluid dosing and the use of in situ immersion probes. The system came with iControl LabMax software enabling real-time measurement of vital process parameters and full walk away operation. A custom wall baffle described previously9 was employed in all experiments to improve the level of mixing in the reactor. Antisolvent (water) addition into the solution was achieved using a ProMinent beta/4 peristaltic pump, which was found to be capable of a maximum addition rate of 30 g/min. An electronic balance (Mettler Toledo XS60025 Excellence) was used for recording the amount of the antisolvent added to the solution. 3.2.1. FBRM Probe. A Mettler-Toledo Focused Beam Reflectance Measurement (FBRM) D600L probe was utilized in this work to track the evolution of the PSD and to ensure negligible nucleation occurred during desupersaturation experiments. For all FBRM measurements, the fine detection setting was employed, as the detection setting was found to produce a significant level of noise due to the agitation of the impeller. The instrument provided a chord length distribution evolution over time at 10 s intervals, which is useful for indicating the presence of nucleating crystals. 3.3. ATR-FTIR Calibration. ATR-FTIR allows for the acquisition of liquid-phase infrared spectra in the presence of solid material due to the low penetration depth of the IR beam into the solution. An ATR-FTIR ReactIR 4000 system from Mettler-Toledo, equipped with a 11.75 ‘‘DiComp’’ immersion probe and a diamond ATR crystal, was used to track solution concentration. The infrared spectra are known to be affected by concentration and solvent composition requiring calibration to known experimental conditions. The amide functional group contained within paracetamol, which emits a bending frequency of 1517−1 in infrared spectroscopy. The calibration procedure employed in this work involves tracking the absolute height of one solute peak and correlating it to known solution concentrations and solvent compositions. This method was chosen as it was demonstrated to be capable of predicting solute concentration with a relative uncertainty of less than 3% for a range of solution systems.17 The procedure involves measuring the absorbance of particular peaks and increasing the concentration at a set number of intervals until the solubility is reached. The calibration points are varied to cover a range of concentrations and solvent compositions. The compositions and concentrations were varied between 40% and 68% and 0.010−0.218 kg/kg, respectively. The method requires the solubility to be known prior to the calibration in order to remain in the undersaturated stable region. The solubility was measured via a gravimetric method and detailed in previous work.14 The values of absorbance (ABS), composition (Comp) and concentration (Conc) were fitted to a second order

polynomial and the coefficients were computed in matlab with the regress function as follows: Conc = 0.018 + 0.244Abs + 0.613Abs 2 − 0.0009Comp + 4.605Comp2 + 0.0005AbsComp

R2 = 0.9979 (12)

The model was found to predict solution concentration with an average relative error of 0.95% over the concentration and composition ranges investigated. A typical ATR-FTIR spectra for a paracetamol and methanol solution is shown in Figure 1, with the peaks associated with the main functional groups highlighted.

Figure 1. ATR-FTIR spectra of paracetamol in a methanol−water solution used for calibration.

3.4. Procedure. 3.4.1. Measurement of Growth Kinetics. The measurement procedure for the independent determination of growth rate kinetics is based on seeded batch desupersaturation experiments realized at constant solvent composition. Only the initial PSD of the seed crystals and the evolution of the solute concentration are needed for the determination of crystal growth rates. All experiments were conducted at 25 °C and at an impeller speed of 250 rpm. Scanning electron microscopy images were taken of the seed and the final product to ensure no change in crystal morphology was observed and no polymorphic change occurred. The mass of solvent in the vessel ranged from 0.365 to 0.45 kg. A saturated solution was created upon mixing paracetamol in a specific methanol/water mixture at 25 °C. The solution was then heated 10 °C above the saturation temperature and held until complete dissolution was observed by FBRM. The solution was then cooled back to the saturation temperature. A supersaturated solution was then created via the addition of a known mass of antisolvent into the reactor. A range of supersaturations can be induced while avoiding the solution nucleating with prior knowledge of the MSZW. The MSZW was determined from the experiments conducted using the FBRM probe to detect the onset of nucleation outlined further in previous work.14The masses of solution and antisolvent were chosen in such a way to obtain the constant mass fraction of interest. ATR-FTIR and FBRM were employed to monitor the solute concentration and chord length distribution during the experiment. At time zero a specific 4733



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to maintain a similar shape during the experiment, indicating that growth of the seed crystals is the dominant supersaturation consumption mechanism. After each experiment the solution was filtered, dried, and weighed, and the particle size distribution was measured using a Horiba L92O particle size analyzer (PSA). Similar results were obtained for all growth characterization experiments. A number of experimental conditions were varied in order to determine the effect of initial supersaturation, seed mass, and seed size. These results are discussed in more detail in Section 4. Finally, the measured growth rate parameters were estimated by comparison of simulations to experiments at different operating conditions. 3.4.2. Seed Preparation. Seed crystals were prepared by cooling crystallizations in order to produce a higher mass of crystals. The crystals were subsequently wet sieved using three stainless steel, woven wire cloth sieves, with squares apertures of nominal sizes of 90, 125, and 250 μm, respectively. The remaining seed crystal fractions in the size ranges of 90−125 μm and 125−250 μm were washed, filtered, and dried. The PSDs of all the three seed fractions were measured using a Horiba L92O particle size analyzer (PSA), using saturated water at room temperature as the dispersal medium. A dispersant solution saturated with paracetamol and containing sodium dodecyl sulfate at a concentration of 5 g/L was also employed to ensure no agglomeration occurred during the particle size measurement. The particle size distributions were measured three times in accordance with ISO33320 and all distributions were found to be less 5%, 3%, and 5% for the d10, d50, and d90 respectively.

mass of dry seeds was charged into the reactor. The experiment was monitored until a stable signal was obtained from ATRFTIR, indicating that the solubility had been reached. All desupersaturation experiments were performed twice. Figure 2

Figure 2. Repeatability of two sets of desupersaturation experiments presented in Section 2.4.

shows the typical reproducibility of the measured desupersaturation curves for two repeated runs of experiments at different initial conditions. It can be readily observed that the repeatability was satisfactory in both cases. The growth kinetics were estimated for a range of solvent compositions from 40% to 68% mass water, respectively. This range was chosen as experiments carried out above 70% result in dilution due to a low solubility gradient. The absence of significant nucleation was assured by monitoring the CLDs using the FBRM during the experiment. Typical time-resolved CLDs are shown in Figure 3. It can be readily observed that no significant increase in the counts at small chord lengths occurred, thus indicating the absence of significant nucleation. Also the CLD was found

4. RESULTS Five seeded batch desupersaturation experiments were performed at various solvent compositions. An additional three experiments were performed to investigate the effect of initial supersaturation, seed size fraction, and seed mass. The experimental runs were labeled PM1−PM8, and the corresponding experimental conditions are listed in Table 1 where Minitial and Mfinal are the initial and final percentage of water in the vessel, respectively. Mw is the mass of solvent added to the Table 1. Experimental Conditions of Seeded Growth Experiments S0

seed fraction

PM1

1.1755

PM2

1.3077

PM3

1.2330

PM4

1.0854

PM5

1.2198

PM6

1.2160

PM7 PM8

1.2223 1.4396

PM5R

1.2164

PM8R

1.4396

125−250 μm 125−250 μm 125−250 μm 125−250 μm 125−250 μm 125−250 μm 90−125 μm 125−250 μm 125−250 μm 125−250 μm

exp. no.

Figure 3. Measured CLDs from FBRM for typical seeded growth experiment. 4734

seed mass (kg)

M initial (wt %) (water)

M final (wt %) (water)

Mw (kg)

0.00497

40

50

0.065

0.00501

40

55

0.11

0.00497

50

60

0.075

0.00496

20

40

0.1

0.00502

60

68

0.075

0.00993

50

60

0.075

0.00497 0.00495

50 40

60 60

0.075 0.150

0.00503

60

68

0.075

0.00502

40

60

0.150



Article

Table 2. Growth Rate Parameters Estimated from Desupersaturation Data at Varying Compositions parameter

40% mass water

50% mass water

55% mass water

60% mass water

68% mass water

kg g residual

1.10 × 10−4 1.7531 8.53 × 10−5

3.46 × 10−5 1.604 1.10 × 10−4

4.93 × 10−5 1.895 1.20 × 10−3

1.86 × 10−4 2.239 7.43 × 10−4

8.1232 × 10−5 1.6083 8.76 × 10−4

vessel to generate the initial supersaturation. The operating parameters in Table 1 were chosen to cover the range of interest and at the same time to avoid the occurrence of nucleation. On the basis of this set of experiments, the growth kinetics were evaluated. 4.1. Estimation of Growth Kinetics. To determine the growth kinetics of paracetamol in methanol/water solutions, the experimental desupersaturation data were used together with the PBE model and the optimization algorithm described in Section 2.2. An empirical power law expression was employed to express the relationship between supersaturation and growth rate, eq 13. G = kg (ΔC) g

(13)

where kg is the growth rate constant, ΔC is absolute supersaturation, and g is the growth exponent. The growth rate parameters were calculated as a function of solvent composition and are listed in Table 2. The power law eq 13 provides a good fit of the desupersaturation data in all the growth experiments as can be seen from Figures 4−7.

Figure 5. Effect of initial supersaturation on the desupersaturation curves.

4.3. Effect of Seed Mass. The effect of seed mass on the rate of supersaturation decay is shown in Figure 6. It can be

Figure 4. Desupersaturation experiment PM1: Experimental and simulated data. Figure 6. Effect of seed mass on the desupersaturation curves.

Changing the initial values of the estimated parameters over several orders of magnitude in the optimization procedure always produced the same results, hence indicating a global optimum. 4.2. Effect of Initial Supersaturation. The effect of initial supersaturation on the rate of supersaturation decay is shown in Figure 5. It can be observed that generating a higher initial supersaturation results in a faster rate of decay of supersaturation. At approximately 400 s the higher initial desupersaturation curve cuts across the lower curve. This is an expected result as growth rates are a function of supersaturation and a higher generation of supersaturation or driving force leads to a higher growth rate and desupersaturation decay.

seen from the plot that as a result of increasing seed mass, the supersaturation in solution is consumed at a faster rate. This increased consumption can be explained by the increase in total seed surface area available for crystal growth, from the additional seed. It should be noted that crystal growth rate is not a function of seed mass. Instead, the effect of seed mass on the decay of supersaturation is accounted for in eq 5, using the second moment of the seed crystals present. A larger seed mass will result in a larger value for the second moment and hence will result in higher decay of supersaturation. 4.4. Effect of Seed Size. The effect of seed size fraction on the rate of supersaturation decay is shown in Figure 7. Figure 7 demonstrates that when seeds of a smaller size fraction are present in solution, a faster rate of desupersaturation decay is 4735



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Figure 7. Effect of seed size on the desupersaturation curves.

Figure 8. Experimental and simulated final PSD of run PM2.

observed, which can be explained by eq 5. A smaller seed size provides a larger specific surface area for the crystal growth process. Hence the second moment of the seed crystals, m2 will be larger, promoting a faster desupersaturation decay. This effect is analogous to the effect of seed mass, although the effect of seed size on the desupersaturation decay is not as pronounced as the effect of seed mass. This may be due to the fact that there is not a substantial difference between the size fractions employed to produce significantly different results. 4.5. Accuracy of Numerical Model. The technique employed here provides two methods of verifying the accuracy of the numerical model employed in this work. The first is the simulated desupersaturation curves shown in Figures 4−7. It can be seen that a reasonable fit to the desupersaturation data is achieved with a maximum residual calculated using eq 9 of 1.2 × 10−3. Residuals for all growth rate estimation experiments are reported in Table 2. All phenomena associated with the effects on crystal growth, such as the effect of initial supersaturation, seed mass, and seed size are captured by the numerical model as can be seen from Figures 4−7. The second method for validating the accuracy of the numerical model employed involves comparison of the experimental product PSDs with the simulated product PSDs. The particle size distributions for PM2, PM3, and PM4 are plotted in Figures 8, 9, and 10, respectively. It can be readily observed from Figure 11 that the experimental PSD has shifted to larger particle size values. It can be seen that a reasonable prediction is obtained from the numerical model of the simulated PSD. All other experiments conducted within this work were found to be in similar agreement. The numerical model captures the particles in the smaller range quiet well, however in experiment PM3 the experimental PSD is slightly underpredicting the larger particles. Some particle agglomeration can be seen in the product PSDs, however Figure 11 shows that this agglomeration originates from the seed and this is supported by both Figure 11 and 12 which show PSDs and SEM images of both seed and product PSDs. To some extent this agglomeration may be due to the particles agglomerating on filtration or on storage as the particles appear to absorb moisture quite strongly when present in air. Monitoring the particles during the experiments with FBRM also suggests that no significant agglomeration occurred.

Figure 9. Experimental and simulated final PSD of run PM3.

4.6. Growth Rate Mechanism. In the previous section growth kinetics as a function of solvent composition were evaluated via fitting a population balance model to desupersaturation data. These parameters are essential in

Figure 10. Experimental and simulated final PSD of run PM4. 4736



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Figure 13. Growth rate as a function of supersaturation ratio at various solvent compositions.

Figure 11. Seed PSD together with the experimental final and simulated PSD of run PM4.

interface. Since the concentration of the solute is greater as you leave the interface, solute will diffuse toward the crystal surface. If diffusion of solute from the bulk solution to the crystal surface is rate limiting, growth is diffusion controlled. A model that focuses on the diffusion of solute through the boundary layer is known as the diffusion controlled model. If incorporation into a crystal lattice is the slowest process, growth is surface-integration controlled. To determine which growth mechanism is rate determining we use the following equation: kM G = kd a (c − c*) 3k vρ

(14)

where M is the molar mass of paracetamol (0.15117 kg/mol), c is the concentration, and c* is the solubility in molar units of mol/m3. This equation requires the mass transfer coefficient, which is calculated using the Sherwood correlation. ⎛ ⎞ ⎛ εL4 ⎞1/5 D⎜ ⎟ 1/3 kd = ⎜2 + 0.8⎜⎜ 3 ⎟⎟ Sc ⎟ L ⎝ ⎠ ν ⎝ ⎠

(15)

where D is the diffusivity or diffusion coefficient, L is the crystal size, ε is the average power input, υ is the kinematic viscosity, and Sc is the Schmidt number (Sc = υ/D). Since the solvent composition in this work is dynamic, the variation in the density of the solution is considered through ρsolution = Figure 12. SEM image of (A) seed and (B) product crystals from run PM4.

1 mfrac w /ρw + (1 − mfrac w)/ρm

(16)

where Mfracw is the mass fraction of water and ρw and ρm are the densities of water and methanol, respectively. This results in dynamic viscosity as a function of solvent composition. The diffusion coefficient, D, can be evaluated using the Stokes− Einstein equation as follows:

developing a numerical model to optimize an antisolvent crystallization process. Table 2 and Figure 13 show that solvent composition has a significant impact on the growth rates. In this section, we attempt to further investigate the underlying mechanisms of the effect of solvent on the growth rates observed in Section 4.1. Goals of crystal growth theory are to determine the source of steps and the rate controlling step for crystal growth. As a crystal grows from a supersaturated solution, the solute concentration is depleted in the region of the crystal−solution

D=

kT 3πμdm

(17) −23

where k is the Boltzmann constant (1.38065 × 10 J/K), T is the temperature in Kelvin, and μ is the dynamic viscosity of the fluid. Because all values of viscosity are calculated as a function of solvent composition a case study of a water mass fraction of 4737



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0.4 will be used for demonstration purposes. The dynamic viscosity of the fluid at a water mass fraction of 0.4 is (6.45008 × 10−4 kg/m s @ 25 °C). The molecular diameter, dm in the above expression is evaluated as follows: dm = 3

1 ccNA

(18)

where cc is the molar density of the paracetamol (8.8113 kmol/ m3), and NA is Avogadro’s constant (6.022 × 1026 1/kmol). This results in a value of 5.733 × 10−10 m for the molecular diameter of paracetamol crystals. For stirred tanks, the average power input, ε, can be evaluated using: ε=

NPvs3ds5 V

(19) Figure 15. Diffusion growth rates and experimental growth rates as a function of solvent composition at a constant supersaturation of 1.2.

where Np is the Power number of the impeller, vs is the stirrer speed (250 rpm), ds is the stirrer diameter (0.06 m), and V is the solution volume (0.0004 m3). For the downward-pumping, fourblade, 45° pitched blade impeller used in this work, a Power number of 1.08 has been estimated previously by Chapple et al.18 This results in an average power input of 0.1493 W/kg for the LabMax reactor. Using the above values, for an average particle size, L, of 2 × 10−4 m and a water mass fraction of 0.4 at 25 °C, eq 15 yields a value of 1.23 × 10−5 m/s for the mass transfer coefficient, kd, for this solution system. The values of kd as a function of solvent composition are shown in Figure 14.

generated at this point of the solubility curve, which in turn can increase the experimental error. It is difficult to obtain reproducible data in this region of the solubility curve due to a low driving force. The data in Figure 15 are also calculated for a constant supersaturation of 1.2 which is outside the supersaturation range generated in the experiment carried out to estimate growth rates for a water mass fraction above 0.68. The reduction in both diffusion limited growth rates and the experimental growth rates is approximately proportional by a factor of 2. The slope of both growth rates as a function of water mass fraction is −1 × 10−7 m/s from water mass fractions of 0.4−0.6. The discrepancy between the experimental growth rates and the diffusion limited growths can be attributed to a reduction in the solubility gradient, higher solution viscosities, selective adsorption of solvent molecules at specific surface sites due to strong interactions between solute and solvent molecules, and the influence of the solvent on the surface roughening. These mechanisms are discussed in more detail in the following section. 4.7. Effect of Solvent Composition. 4.7.1. Selective Adsorption and Surface Roughening. The role played by the solvent in enhancing or inhibiting crystal growth is not clear at present.18 According to the existing literature, the solvent may contribute to decreasing growth rate due to a selective adsorption of solvent molecules or may enhance face growth rate by causing a reduction in the interfacial tension.19−21 The first mechanism has been attributed to a selective adsorption of solvent molecules at specific surface sites due to strong interactions between solute and solvent molecules.20−22 The second mechanism referred to here as the interfacial energy effect, is related to the influence of the solvent on the surface roughening which under certain circumstances may induce a change in the growth mechanism.23−25 Davey and co-workers provide a good example of the interfacial effect of the solvent on crystal interface. They reported on the growth kinetics of hexamethylene tetramine (HMT) crystallized from different solvents and solvent mixtures.23−27 It was reported that the growth rate of the (110) face increased faster when water or water/acetone mixtures replaced ethanol as the solvent. Decreasing surface diffusion and a direct integration to the crystal lattice were connected to a change in the growth mechanism. The observed effect was attributed to favorable

Figure 14. Mass transfer coefficient kd calculated from eq 15 as a function of solvent composition.

The knowledge of kd provides the possibility of calculating the diffusion growth rates as a function of solvent composition. Diffusion limited growth rates calculated from eq 14 are plotted in Figure 15. Over the range of solvent compositions studied, the experimental growth rates were found to be lower than the diffusion limited growth rates, predicted from eq 14. Therefore, surface integration of the solute is deemed to be the rate limiting step of the growth mechanism. Figure 15 shows that as the mass fraction of water increases, a reduction in the experimental growth rates and the diffusion limited growth rates is observed. With the exception of water mass fraction of 0.68, it can be seen that the experimental growth rates reduce at the same rate as the diffusion limited growth rates. This increase at mass fractions of 0.68 may be due to some experimental error as very little supersaturation can be 4738



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supersaturation, seed mass, and seed size have been investigated and the numerical model has been shown to capture these phenomena with good accuracy. With regard to the effect of initial supersaturation, faster desupersaturation decay and hence crystal growth rate was observed with higher supersaturations due to a larger driving force. A faster desupersaturation decay was observed for cases where a larger seed mass was used. This increased consumption can be explained by the increase in total seed surface area available for crystal growth, from the additional seed. A similar observation was made when seeds of a smaller size fraction were used. A smaller seed size provides a larger specific surface area for the crystal growth process. Hence the second moment of the seed crystals, m2 will be larger, promoting a faster desupersaturation decay. Furthermore the role of the solvent has shown to have a significant impact on the crystal growth rate. Diffusion growth rates have been calculated in order to provide detail about the growth mechanism. With the aid of the growth mechanism a detailed discussion on how solvent composition affects growth rates is outlined. The effects of solvent composition are split into four different phenomena and the growth mechanism is utilized to determine which one is the most probable. The phenomena are named surface roughing, selective absorption, solubility gradient, and increasing viscosity due to higher water mass fractions. This work offers a successful methodology for the quick determination of crystal growth parameters for use in modeling and optimizing particulate systems and also highlights that investigating the crystal growth mechanism can offer new insights into understanding the role of the solvent in affecting crystal growth kinetics.

interactions between the solute and the solvent at increasing solubility. 4.7.2. Solubility Gradient. It has been shown that at higher solubilities more favorable interactions occur between the solute and solvent or in the case studied here lower solubilities leading to unfavorable interactions. The solubility gradient may also have an impact as the gradient is reduced with higher water mass fractions leading to a reduced driving force and hence reduced crystal growth. Desupersaturation experiments generally involve adding a known amount of antisolvent into the reactor and generating a specific supersaturation, followed by seeding and subsequent growth. However as the water mass fraction tends to one the gradient of the solubility with respect to the water mass fraction reduces. Figure 16 illustrates that the

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AUTHOR INFORMATION

Corresponding Author

Figure 16. Antisolvent free solubility gradient as a function of water mass fraction.

*Tel.: 00 353 61 213134. Fax: 00 353 61 202944. E-mail: [email protected].

driving force is reduced when starting from 0.6 in comparison to starting and generating a supersaturation from 0.4. Hence the driving force is reduced resulting in a reduced mass transfer, solute integration, and subsequent crystal growth rate. 4.7.3. Viscosity. One other reason could be due to the increased viscosity of the fluid inhibiting mass transfer of the solute from solution to the crystal face thereby reducing the crystal growth rate. Figure 15 shows that at higher water mass fractions a reduction in mass transfer is observed. The mass transfer coefficient is a function of dynamic viscosity as can be seen from eq 15, therefore higher water mass fractions lead to higher densities and viscosities leading to inhibited mass transfer of solute. This mechanism along with the effect of the solubility gradient appears to be the most likely mechanism as the decrease in the experimental growth rates is largely proportional to the diffusion limited growth rates.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research has been conducted as part of the Solid State Pharmaceuticals Cluster (SSPC) and funded by Science Foundation Ireland (SFI).

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5. CONCLUSIONS Growth kinetics as a function of solvent composition have been determined based on seeded isothermal batch desupersaturation experiments. A population balance model combined with a parameter estimation procedure have been utilized to obtain growth rate parameters from desupersaturation data. The method takes advantage of two online PAT technologies to measure solution concentration and to ensure negligible nucleation occurs. The method has been shown to predict experimental data with good accuracy. The effects of initial 4739

NOMENCLATURE B = Nucleation rate (no./kg methanol s) C = Concentration (kg/kg methanol) C* = Equilibrium concentration (solubility) (kg/kg methanol) D = Diffusivity (m2/s) G = Growth rate (m/s) K = Boltzmann’s constant (J/K) L = Particle size (m) M = Molar mass (kg/kmol) Np = Power number (-) NA = Avogadro Constant (no./kmol) R = Gas constant (J/kmol K) Sc = Schmidt number (-) Stsim = Simulated supersaturation ratio (-) Stexp = Experimental supersaturation ratio T = Temperature (K) V = Solution volume (m3) cc = Molar density (kmol/m3) dx.doi.org/10.1021/ie2020262 | Ind. Eng. Chem. Res. 2012, 51, 4731−4740


Article

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dm = Molecular diameter (m) ds = Stirrer diameter (m) g = Growth order (-) ka = Surface area shape factor (-) kd = Mass transfer coefficient (m/s) kg = Growth rate constant (m/s) kv = Volume shape factor (-) mf rac = Mass fraction (-) mk = kth Moment of particle size distribution (mk) n = Population density (no./m4) vs = Stirrer speed (no./s) ε = Average power input (W/kg) μ = Dynamic viscosity (kg/m s) π = Pi (-) ρc = Crystal density (kg/m3) υ = Kinematic viscosity (m2/s) ρsoln = Solution density (kg/m3) ρw = Water density (kg/m3) ρm = Methanol density (kg/m3) θ = Parameter set (-) o = Initial (-)

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Determination of the Crystal Growth Rate of Paracetamol As a

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