1 Determination of the Crystal Growth Rate of Paracetamol As a2 Function of Solvent Composition3 C. T. O’Ciardha,* N. A. Mitchell, K. W. Hutton, and P. J. Frawley
4 Solid State Pharmaceuticals Cluster (SSPC), Materials and Surface Science Institute (MSSI), Department of Mechanical, Aeronautical5 and Biomedical Engineering, University of Limerick, Castletroy, County Limerick, Ireland
6 ABSTRACT: Growth kinetics, growth mechanisms, and the effect of7 solvent composition for the antisolvent crystallization of paracetamol in8 methanol−water mixtures have been determined by means of isothermal9 seeded batch experiments at constant solvent composition. A numerical10 model incorporating the population balance equation based on antisolvent11 free solubility was fitted to the desupersaturation data, and growth rate12 parameters are evaluated. An attenuated total reflectance−Fourier transform13 infrared (ATR-FTIR) probe was employed to measure online solute14 concentration and focused beam reflectance measurement (FBRM) was15 utilized to ensure negligible nucleation occurred. The model is validated by16 the final particle size distributions (PSDs) and online solute concentration17 measurements. Crystal growth rate was found to decrease with increasing18 water mass fractions up to a mass fraction of 0.68 where an increase is19 observed. A method has been introduced linking the effect of solvent20 composition with the growth mechanism and the growth rates. Utilizing the growth mechanism it has been postulated that a21 combination of the solubility gradient, viscosity, selective adsorption, and surface roughening are responsible for the reduction in22 growth rates with solvent composition. Furthermore, the effects of seed mass, size and initial supersaturation on the crystal23 growth rates were investigated to demonstrate the efficacy of the model at predicting these various phenomena.
1. INTRODUCTION24 Crystallization is a widely used technique in solid−liquid25 separation processes and is regarded as one of the most26 important unit operations in the process industries as many27 finished chemical products are in the form of crystalline solids.28 In antisolvent crystallization, supersaturation is generated by29 addition of another solvent or solvent mixture in order to30 reduce the solubility of the compound. Antisolvent crystal-31 lization is an advantageous method where the substance to be32 crystallized is highly soluble, has solubility that is a weak33 function of temperature, is heat sensitive, or unstable in high34 temperatures.1 Antisolvent processes have also been identified35 as a means to produce crystals more efficiently from continuous36 processes due to an ability to generate supersaturations quickly,37 run at low temperatures isothermally, and a low tendency to38 scale reactors which is a significant problem with cooling39 crystallizations. Rigorous determination of an optimal batch40 recipe requires accurate growth and nucleation rate kinetics,41 which can be determined in a series of batch experiments. Once42 the particle formation kinetics are known, they can be used43 together with a population balance model to simulate the44 influence of different process parameters on the final particle45 size distribution (PSD) of the product.46 In precipitation processes, the crystal growth rate is a crucial47 parameter since it determines the final specific properties of48 crystals such as the final particle size distribution. Two methods49 are commonly described in the literature for the estimation of50 crystal growth rate kinetics, namely single crystal growth studies
51and seeded desupersaturation experiments. There are numer-52ous articles describing crystal growth from solutions where the53supersaturation is generated via cooling and precipitation.2−9
54However, very few studies have focused on antisolvent55crystallizations, where the solvent plays a dominant role on56crystal growth. Single crystal studies using transient imagery57were carried out on sodium nitrate in water and isopropox-58yethanol.10 Isothermal seeded batch experiments have been59conducted for paracetamol and acetone−water mixtures, with60nonlinear optimization utilized to evaluate the parameters of a61power law expression to describe the growth rate, as a function62of supersaturation, from the experimental desupersaturation63curves.11 This work does not utilize online measurement64techniques and similar studies have been conducted using65isothermal seeded batch experiments, however employing66ATR-FTIR to track solution concentration online.8,12 This67work collects the previous results, methods, and observations in68the literature referenced above, however expanding them to69improve the methods efficiency while attempting to analyze the70effect of composition on crystal growth rates. This work offers a71novel approach in estimating growth kinetics as a function of72solvent composition from a population balance model solved73utilizing the computationally efficient method of moments and
Received: September 5, 2011Revised: February 20, 2012Accepted: February 20, 2012
74 simultaneously investigating the underlying mechanism of how75 the solvent affects the growth rates. The moments of the76 distribution are reconstructed directly using a simple yet77 accurate method yielding a very fast and effective means of78 estimating growth rates. These reconstructed distributions are79 then compared to distributions obtained experimentally.80 Capturing the effects of solvent composition on the growth81 rate is critical to building a population balance model.82 Neglecting the influence of solvent composition will lead to83 discrepancies between experimental and modeled data. A84 growth rate model that considers its parameters to be functions85 of antisolvent mass fraction has previously been shown to be86 superior to a model without such functionality.13 The87 nucleation kinetics for paracetamol in methanol−water88 mixtures has been evaluated elsewhere and shown to be89 strongly dependent on solvent composition.14 This study aims90 to utilize a population balance to determine growth rates based91 on power law equations, while offering an insight into the92 growth mechanism and the effect of solvent composition. The93 focus of the work is also to determine parameters for power law94 expressions which can be used in predicting and optimizing95 crystal size distributions. To date the literature does not contain96 any work detailing the estimation of growth kinetics of97 antisolvent systems utilizing the method of moments together98 with in situ measurement techniques. In addition to this, a99 greater effort has been made to gauge the effect of solvent100 composition on growth rate kinetics by linking with the101 estimated growth rate mechanism.
2. POPULATION BALANCE MODEL AND PARAMETER102 ESTIMATION PROCEDURE
103 A mathematical model based on population balance equations104 (PBEs) is used in combination with a least-squares105 optimization and the experimental desupersaturation data to106 determine the growth rate parameters of paracetamol in107 methanol/water solutions as a function of solvent composition.108 2.1. Population Balance. In a perfectly mixed batch109 reactor the evolution of the crystal size distribution can be110 described as follows
∂∂
+ ∂∂
=n L tt
G tn L t
L( , )
( )( , )
0(1)
111 where n(L,t) is the population density of the crystals and G(t)112 is the crystal growth rate which is assumed to be independent113 of size. The above equation is solved using the method of114 moments, detailed below.115 The standard method of moments is an efficient method of116 transforming a population balance into its constituent mo-117 ments. The low order moments of the distribution represent118 the total number, length, surface area, and volume of particles119 in the crystallizing system. Using the standard method of120 moments, eq 1 becomes
= −m t
tkG t m t
d ( )d
( ) ( )kk 1 (2)
121 where
∫=∞
m L n L t L( , )dkk
0 (3)
122Equation 2 can be multiplied through by Lk and integrated to123result in an equation in terms of the moments mk:
=m
tB
dd
0(4)
124In the case where the system is sufficiently seeded, negligible125nucleation can be assumed. Ensuring that no nucleation is126present is critical as seeding simplifies the mathematical127treatment of the experimental data. Under this condition the128second moment of the seed particle size distribution can only129be influenced by the growth of the crystal and no other130competing factors.131During the crystallization process, the mass balance of the132solution phase can be described as
∫= − ρ∞c
tk G nL L
dd
3 dv c 02
(5)
133where ρc and kv are the solid density and the volume shape134factor of paracetamol crystals, respectively. In the above135equation, the integral term represents the second moment of136the seed crystals, m2 which is proportional to the total surface137area of crystals present. A value of 1332 kg/m3 will be138employed for the crystal density of form I of paracetamol.15
139The following initial and boundary conditions apply:
=C o C( ) 0 (6)
=n L n L(0, ) ( )0 (7)
=n t( , 0) 0 (8)
140with C0 being the initial concentration of the solute, n0(L) the141initial PSD, and B is the nucleation rate per unit mass. The142supersaturation correlations used in this work for absolute143supersaturation and supersaturation ratio are as follows:
Δ = − *C C C (9)
=*
SC
C (10)
144where C is the solute concentration and C* is the antisolvent145free solubility. The antisolvent free solubility is calculated and146described elsewhere.14 The crystal size distribution was147reconstructed from the moments utilizing a novel technique148developed by Hutton.16 This technique is outlined in more149detail by Mitchell et al.9
1502.2. Optimization. For the estimation of the growth151kinetics parameters the following least-squares problem had to152be solved:
∑ ∑= − θ= =
⎡⎣ ⎤⎦( )R S Smin min ( )i
N
ti
N
t tsim
1
2
1
exp 2d d
(11)
153where θ is the set of parameters to be estimated, Stsim represents
154the predicted supersaturation ratios, Stexp represents the
155measured supersaturation ratios at each time or sampling156interval, and Nd is the number of sampling instances. The157MATLAB optimization algorithm fminsearch which employs a158Nelder−Mead simplex method was utilized to find the optimal159set of parameters.
3. EXPERIMENTAL SECTION1603.1. Materials. The experimental work outlined was161performed on Acetaminophen A7085, Sigma Ultra, ≥99%,
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162 sourced from Sigma Aldrich. The methanol employed in this163 work was gradient grade hiPerSolv CHROMANORM for164 HPLC ≥99%, sourced from VWR.165 3.2. Apparatus. A LabMax reactor system from Mettler-166 Toledo was utilized in this work to estimate the growth kinetics167 of the paracetamol in a methanol/water solution system. The168 reactor was a 1-L round-bottomed borosilicate glass jacketed169 reactor, allowing controlled heating and cooling of solutions.170 All experiments were carried out isothermally at 25 °C and at171 constant solvent composition. Agitation of the solution was172 provided by means of an overhead motor and a glass stirrer,173 with four blades at a pitch of 45°. The system allowed fluid174 dosing and the use of in situ immersion probes. The system175 came with iControl LabMax software enabling real-time176 measurement of vital process parameters and full walk away177 operation. A custom wall baffle described previously9 was178 employed in all experiments to improve the level of mixing in179 the reactor. Antisolvent (water) addition into the solution was180 achieved using a ProMinent beta/4 peristaltic pump, which was181 found to be capable of a maximum addition rate of 30 g/min.182 An electronic balance (Mettler Toledo XS60025 Excellence)183 was used for recording the amount of the antisolvent added to184 the solution.185 3.2.1. FBRM Probe. A Mettler-Toledo Focused Beam186 Reflectance Measurement (FBRM) D600L probe was utilized187 in this work to track the evolution of the PSD and to ensure188 negligible nucleation occurred during desupersaturation experi-189 ments. For all FBRM measurements, the fine detection setting190 was employed, as the detection setting was found to produce a191 significant level of noise due to the agitation of the impeller.192 The instrument provided a chord length distribution evolution193 over time at 10 s intervals, which is useful for indicating the194 presence of nucleating crystals.195 3.3. ATR-FTIR Calibration. ATR-FTIR allows for the196 acquisition of liquid-phase infrared spectra in the presence of197 solid material due to the low penetration depth of the IR beam198 into the solution. An ATR-FTIR ReactIR 4000 system from199 Mettler-Toledo, equipped with a 11.75 ‘‘DiComp’’ immersion200 probe and a diamond ATR crystal, was used to track solution201 concentration. The infrared spectra are known to be affected by202 concentration and solvent composition requiring calibration to203 known experimental conditions. The amide functional group204 contained within paracetamol, which emits a bending frequency205 of 1517−1 in infrared spectroscopy. The calibration procedure206 employed in this work involves tracking the absolute height of207 one solute peak and correlating it to known solution208 concentrations and solvent compositions. This method was209 chosen as it was demonstrated to be capable of predicting210 solute concentration with a relative uncertainty of less than 3%211 for a range of solution systems.17 The procedure involves212 measuring the absorbance of particular peaks and increasing the213 concentration at a set number of intervals until the solubility is214 reached. The calibration points are varied to cover a range of215 concentrations and solvent compositions. The compositions216 and concentrations were varied between 40% and 68% and217 0.010−0.218 kg/kg, respectively. The method requires the218 solubility to be known prior to the calibration in order to219 remain in the undersaturated stable region. The solubility was220 measured via a gravimetric method and detailed in previous221 work.14 The values of absorbance (ABS), composition (Comp)222 and concentration (Conc) were fitted to a second order
223polynomial and the coefficients were computed in matlab with224the regress function as follows:
= + + −
+ +
=
Conc Abs Abs Comp
Comp AbsComp R
0.018 0.244 0.613 0.0009
4.605 0.0005
0.9979
2
2 2
(12)
225The model was found to predict solution concentration with226an average relative error of 0.95% over the concentration and227composition ranges investigated. A typical ATR-FTIR spectra228 f1for a paracetamol and methanol solution is shown in Figure 1,229with the peaks associated with the main functional groups230highlighted.
2313.4.1. Procedure. Measurement of Growth Kinetics.232The measurement procedure for the independent determi-233nation of growth rate kinetics is based on seeded batch234desupersaturation experiments realized at constant solvent235composition. Only the initial PSD of the seed crystals and the236evolution of the solute concentration are needed for the237determination of crystal growth rates. All experiments were238conducted at 25 °C and at an impeller speed of 250 rpm.239Scanning electron microscopy images were taken of the seed240and the final product to ensure no change in crystal241morphology was observed and no polymorphic change242occurred. The mass of solvent in the vessel ranged from2430.365 to 0.45 kg. A saturated solution was created upon mixing244paracetamol in a specific methanol/water mixture at 25 °C. The245solution was then heated 10 °C above the saturation246temperature and held until complete dissolution was observed247by FBRM. The solution was then cooled back to the saturation248temperature. A supersaturated solution was then created via the249addition of a known mass of antisolvent into the reactor. A250range of supersaturations can be induced while avoiding the251solution nucleating with prior knowledge of the MSZW. The252MSZW was determined from the experiments conducted using253the FBRM probe to detect the onset of nucleation outlined254further in previous work.14The masses of solution and255antisolvent were chosen in such a way to obtain the constant256mass fraction of interest. ATR-FTIR and FBRM were employed257to monitor the solute concentration and chord length258distribution during the experiment. At time zero a specific
Figure 1. ATR-FTIR spectra of paracetamol in a methanol−watersolution used for calibration.
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259 mass of dry seeds was charged into the reactor. The experiment260 was monitored until a stable signal was obtained from ATR-261 FTIR, indicating that the solubility had been reached. All
f2 262 desupersaturation experiments were performed twice. Figure 2
263 shows the typical reproducibility of the measured desupersatu-264 ration curves for two repeated runs of experiments at different265 initial conditions. It can be readily observed that the266 repeatability was satisfactory in both cases. The growth kinetics267 were estimated for a range of solvent compositions from 40%268 to 68% mass water, respectively. This range was chosen as269 experiments carried out above 70% result in dilution due to a270 low solubility gradient. The absence of significant nucleation271 was assured by monitoring the CLDs using the FBRM during272 the experiment. Typical time-resolved CLDs are shown in
f3 273 Figure 3. It can be readily observed that no significant increase274 in the counts at small chord lengths occurred, thus indicating275 the absence of significant nucleation. Also the CLD was found
276to maintain a similar shape during the experiment, indicating277that growth of the seed crystals is the dominant supersaturation278consumption mechanism. After each experiment the solution279was filtered, dried, and weighed, and the particle size280distribution was measured using a Horiba L92O particle size281analyzer (PSA). Similar results were obtained for all growth282characterization experiments. A number of experimental283conditions were varied in order to determine the effect of284initial supersaturation, seed mass, and seed size. These results285are discussed in more detail in Section 4. Finally, the measured286growth rate parameters were estimated by comparison of287simulations to experiments at different operating conditions.2883.4.2. Seed Preparation. Seed crystals were prepared by289cooling crystallizations in order to produce a higher mass of290crystals. The crystals were subsequently wet sieved using three291stainless steel, woven wire cloth sieves, with squares apertures292of nominal sizes of 90, 125, and 250 μm, respectively. The293remaining seed crystal fractions in the size ranges of 90−125294μm and 125−250 μm were washed, filtered, and dried. The295PSDs of all the three seed fractions were measured using a296Horiba L92O particle size analyzer (PSA), using saturated297water at room temperature as the dispersal medium. A298dispersant solution saturated with paracetamol and containing299sodium dodecyl sulfate at a concentration of 5 g/L was also300employed to ensure no agglomeration occurred during the301particle size measurement. The particle size distributions were302measured three times in accordance with ISO33320 and all303distributions were found to be less 5%, 3%, and 5% for the d10,304d50, and d90 respectively.
4. RESULTS
305Five seeded batch desupersaturation experiments were306performed at various solvent compositions. An additional307three experiments were performed to investigate the effect of308initial supersaturation, seed size fraction, and seed mass. The309experimental runs were labeled PM1−PM8, and the corre-310 t1sponding experimental conditions are listed in Table 1 where311Minitial and Mfinal are the initial and final percentage of water in312the vessel, respectively. Mw is the mass of solvent added to the
Figure 2. Repeatability of two sets of desupersaturation experimentspresented in Section 2.4.
Figure 3. Measured CLDs from FBRM for typical seeded growthexperiment.
Table 1. Experimental Conditions of Seeded GrowthExperiments
313 vessel to generate the initial supersaturation. The operating314 parameters in Table 1 were chosen to cover the range of315 interest and at the same time to avoid the occurrence of316 nucleation. On the basis of this set of experiments, the growth317 kinetics were evaluated.318 4.1. Estimation of Growth Kinetics. To determine the319 growth kinetics of paracetamol in methanol/water solutions,320 the experimental desupersaturation data were used together321 with the PBE model and the optimization algorithm described322 in Section 2.2. An empirical power law expression was323 employed to express the relationship between supersaturation324 and growth rate, eq 13.
= ΔG k C( )gg
(13)
325 where kg is the growth rate constant, ΔC is absolute326 supersaturation, and g is the growth exponent. The growth327 rate parameters were calculated as a function of solvent
t2 328 composition and are listed in Table 2. The power law eq 13329 provides a good fit of the desupersaturation data in all the
f4 330 growth experiments as can be seen from Figures 4−7.
331 Changing the initial values of the estimated parameters over332 several orders of magnitude in the optimization procedure333 always produced the same results, hence indicating a global334 optimum.335 4.2. Effect of Initial Supersaturation. The effect of initial336 supersaturation on the rate of supersaturation decay is shown in
f5 337 Figure 5. It can be observed that generating a higher initial338 supersaturation results in a faster rate of decay of super-339 saturation. At approximately 400 s the higher initial340 desupersaturation curve cuts across the lower curve. This is341 an expected result as growth rates are a function of342 supersaturation and a higher generation of supersaturation or343 driving force leads to a higher growth rate and desupersatura-344 tion decay.
3454.3. Effect of Seed Mass. The effect of seed mass on the346 f6rate of supersaturation decay is shown in Figure 6. It can be
347seen from the plot that as a result of increasing seed mass, the348supersaturation in solution is consumed at a faster rate. This349increased consumption can be explained by the increase in total350seed surface area available for crystal growth, from the351additional seed. It should be noted that crystal growth rate is352not a function of seed mass. Instead, the effect of seed mass on353the decay of supersaturation is accounted for in eq 5, using the354second moment of the seed crystals present. A larger seed mass355will result in a larger value for the second moment and hence356will result in higher decay of supersaturation.3574.4. Effect of Seed Size. The effect of seed size fraction on358 f7the rate of supersaturation decay is shown in Figure 7. Figure 7359demonstrates that when seeds of a smaller size fraction are360present in solution, a faster rate of desupersaturation decay is
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
361 observed, which can be explained by eq 5. A smaller seed size362 provides a larger specific surface area for the crystal growth363 process. Hence the second moment of the seed crystals, m2 will364 be larger, promoting a faster desupersaturation decay. This365 effect is analogous to the effect of seed mass, although the effect366 of seed size on the desupersaturation decay is not as367 pronounced as the effect of seed mass. This may be due to368 the fact that there is not a substantial difference between the369 size fractions employed to produce significantly different370 results.371 4. 5. Accuracy of Numerical Model. The technique372 employed here provides two methods of verifying the accuracy373 of the numerical model employed in this work. The first is the374 simulated desupersaturation curves shown in Figures 4−7. It375 can be seen that a reasonable fit to the desupersaturation data is376 achieved with a maximum residual calculated using eq 9 of 1.2377 × 10−3. Residuals for all growth rate estimation experiments are378 reported in Table 2. All phenomena associated with the effects379 on crystal growth, such as the effect of initial supersaturation,380 seed mass, and seed size are captured by the numerical model381 as can be seen from Figures 4−7.382 The second method for validating the accuracy of the383 numerical model employed involves comparison of the384 experimental product PSDs with the simulated product PSDs.385 The particle size distributions for PM2, PM3, and PM4 are
f8f9f10 386 plotted in Figures 8, 9, and 10, respectively. It can be readilyf11 387 observed from Figure 11 that the experimental PSD has shifted
388 to larger particle size values. It can be seen that a reasonable389 prediction is obtained from the numerical model of the390 simulated PSD. All other experiments conducted within this391 work were found to be in similar agreement. The numerical392 model captures the particles in the smaller range quiet well,393 however in experiment PM3 the experimental PSD is slightly394 underpredicting the larger particles. Some particle agglomer-395 ation can be seen in the product PSDs, however Figure 11396 shows that this agglomeration originates from the seed and this
f12 397 is supported by both Figure 11 and 12 which show PSDs and398 SEM images of both seed and product PSDs. To some extent399 this agglomeration may be due to the particles agglomerating400 on filtration or on storage as the particles appear to absorb401 moisture quite strongly when present in air. Monitoring the402 particles during the experiments with FBRM also suggests that403 no significant agglomeration occurred.
4044.6. Growth Rate Mechanism. In the previous section
405growth kinetics as a function of solvent composition were
406evaluated via fitting a population balance model to407desupersaturation data. These parameters are essential in
Figure 7. Effect of seed size on the desupersaturation curves. Figure 8. Experimental and simulated final PSD of run PM2.
Figure 9. Experimental and simulated final PSD of run PM3.
Figure 10. Experimental and simulated final PSD of run PM4.
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408 developing a numerical model to optimize an antisolventf13 409 crystallization process. Table 2 and Figure 13 show that solvent
410 composition has a significant impact on the growth rates. In411 this section, we attempt to further investigate the underlying412 mechanisms of the effect of solvent on the growth rates413 observed in Section 4.1.414 Goals of crystal growth theory are to determine the source of415 steps and the rate controlling step for crystal growth. As a416 crystal grows from a supersaturated solution, the solute417 concentration is depleted in the region of the crystal−solution
418interface. Since the concentration of the solute is greater as you419leave the interface, solute will diffuse toward the crystal surface.420If diffusion of solute from the bulk solution to the crystal421surface is rate limiting, growth is diffusion controlled. A model422that focuses on the diffusion of solute through the boundary423layer is known as the diffusion controlled model. If424incorporation into a crystal lattice is the slowest process,425growth is surface-integration controlled. To determine which426growth mechanism is rate determining we use the following427equation:
=ρ
− *G kk M
kc c
3( )d
a
v (14)
428where M is the molar mass of paracetamol (0.15117 kg/mol), c429is the concentration, and c* is the solubility in molar units of430mol/m3. This equation requires the mass transfer coefficient,431which is calculated using the Sherwood correlation.
= + εν
⎛
⎝⎜⎜
⎛⎝⎜⎜
⎞⎠⎟⎟
⎞
⎠⎟⎟k
DL
LSc2 0.8d
4
3
1/51/3
(15)
432where D is the diffusivity or diffusion coefficient, L is the crystal433size, ε is the average power input, υ is the kinematic viscosity,434and Sc is the Schmidt number (Sc = υ/D). Since the solvent435composition in this work is dynamic, the variation in the436density of the solution is considered through
ρ =ρ + − ρ
1mfrac / (1 mfrac )/w w w m
solution(16)
437where Mfracw is the mass fraction of water and ρw and ρm are438the densities of water and methanol, respectively. This results in439dynamic viscosity as a function of solvent composition. The440diffusion coefficient, D, can be evaluated using the Stokes−441Einstein equation as follows:
=πμ
DkT
d3 m (17)
442where k is the Boltzmann constant (1.38065 × 10−23 J/K), T is443the temperature in Kelvin, and μ is the dynamic viscosity of the444fluid. Because all values of viscosity are calculated as a function445of solvent composition a case study of a water mass fraction of
Figure 11. Seed PSD together with the experimental final andsimulated PSD of run PM4.
Figure 12. SEM image of (A) seed and (B) product crystals from runPM4.
Figure 13. Growth rate as a function of supersaturation ratio at varioussolvent compositions.
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446 0.4 will be used for demonstration purposes. The dynamic447 viscosity of the fluid at a water mass fraction of 0.4 is (6.45008448 × 10−4 kg/m s @ 25 °C). The molecular diameter, dm in the449 above expression is evaluated as follows:
=dc N
1m
c A3
(18)
450 where cc is the molar density of the paracetamol (8.8113 kmol/451 m3), and NA is Avogadro’s constant (6.022 × 1026 1/kmol).452 This results in a value of 5.733 × 10−10 m for the molecular453 diameter of paracetamol crystals. For stirred tanks, the average454 power input, ε, can be evaluated using:
ε =N v d
VP s s
3 5
(19)
455 where Np is the Power number of the impeller, vs is the stirrer456 speed (250 rpm), ds is the stirrer diameter (0.06 m), and V is457 the solution volume (0.0004 m3). For the downward-pumping,458 four-blade, 45° pitched blade impeller used in this work, a459 Power number of 1.08 has been estimated previously by460 Chapple et al.18 This results in an average power input of461 0.1493 W/kg for the LabMax reactor. Using the above values,462 for an average particle size, L, of 2 × 10−4 m and a water mass463 fraction of 0.4 at 25 °C, eq 15 yields a value of 1.23 × 10−5 m/s464 for the mass transfer coefficient, kd, for this solution system.465 The values of kd as a function of solvent composition are shown
f14 466 in Figure 14.
467 The knowledge of kd provides the possibility of calculating468 the diffusion growth rates as a function of solvent composition.469 Diffusion limited growth rates calculated from eq 14 are plotted
f15 470 in Figure 15. Over the range of solvent compositions studied,471 the experimental growth rates were found to be lower than the472 diffusion limited growth rates, predicted from eq 14. Therefore,473 surface integration of the solute is deemed to be the rate474 limiting step of the growth mechanism. Figure 15 shows that as475 the mass fraction of water increases, a reduction in the476 experimental growth rates and the diffusion limited growth477 rates is observed. With the exception of water mass fraction of478 0.68, it can be seen that the experimental growth rates reduce at479 the same rate as the diffusion limited growth rates. This480 increase at mass fractions of 0.68 may be due to some481 experimental error as very little supersaturation can be
482generated at this point of the solubility curve, which in turn483can increase the experimental error. It is difficult to obtain484reproducible data in this region of the solubility curve due to a485low driving force. The data in Figure 15 are also calculated for a486constant supersaturation of 1.2 which is outside the super-487saturation range generated in the experiment carried out to488estimate growth rates for a water mass fraction above 0.68. The489reduction in both diffusion limited growth rates and the490experimental growth rates is approximately proportional by a491factor of 2. The slope of both growth rates as a function of492water mass fraction is −1 × 10−7 m/s from water mass fractions493of 0.4−0.6. The discrepancy between the experimental growth494rates and the diffusion limited growths can be attributed to a495reduction in the solubility gradient, higher solution viscosities,496selective adsorption of solvent molecules at specific surface sites497due to strong interactions between solute and solvent498molecules, and the influence of the solvent on the surface499roughening. These mechanisms are discussed in more detail in500the following section.5014.7.1. Effect of Solvent Composition. Selective502Adsorption and Surface Roughening. The role played by503the solvent in enhancing or inhibiting crystal growth is not clear504at present.18 According to the existing literature, the solvent505may contribute to decreasing growth rate due to a selective506adsorption of solvent molecules or may enhance face growth507rate by causing a reduction in the interfacial tension.19−21 The508first mechanism has been attributed to a selective adsorption of509solvent molecules at specific surface sites due to strong510interactions between solute and solvent molecules.20−22 The511second mechanism referred to here as the interfacial energy512effect, is related to the influence of the solvent on the surface513roughening which under certain circumstances may induce a514change in the growth mechanism.23−25 Davey and co-workers515provide a good example of the interfacial effect of the solvent516on crystal interface. They reported on the growth kinetics of517hexamethylene tetramine (HMT) crystallized from different518solvents and solvent mixtures.23−27 It was reported that the519growth rate of the (110) face increased faster when water or520water/acetone mixtures replaced ethanol as the solvent.521Decreasing surface diffusion and a direct integration to the522crystal lattice were connected to a change in the growth523mechanism. The observed effect was attributed to favorable
Figure 14. Mass transfer coefficient kd calculated from eq 15 as afunction of solvent composition.
Figure 15. Diffusion growth rates and experimental growth rates as afunction of solvent composition at a constant supersaturation of 1.2.
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524 interactions between the solute and the solvent at increasing525 solubility.526 4.7.2. Solubility Gradient. It has been shown that at higher527 solubilities more favorable interactions occur between the528 solute and solvent or in the case studied here lower solubilities529 leading to unfavorable interactions. The solubility gradient may530 also have an impact as the gradient is reduced with higher water531 mass fractions leading to a reduced driving force and hence532 reduced crystal growth. Desupersaturation experiments gen-533 erally involve adding a known amount of antisolvent into the534 reactor and generating a specific supersaturation, followed by535 seeding and subsequent growth. However as the water mass536 fraction tends to one the gradient of the solubility with respect
f16 537 to the water mass fraction reduces. Figure 16 illustrates that the
538 driving force is reduced when starting from 0.6 in comparison539 to starting and generating a supersaturation from 0.4. Hence540 the driving force is reduced resulting in a reduced mass transfer,541 solute integration, and subsequent crystal growth rate.542 4.7.3. Viscosity. One other reason could be due to the543 increased viscosity of the fluid inhibiting mass transfer of the544 solute from solution to the crystal face thereby reducing the545 crystal growth rate. Figure 15 shows that at higher water mass546 fractions a reduction in mass transfer is observed. The mass547 transfer coefficient is a function of dynamic viscosity as can be548 seen from eq 15, therefore higher water mass fractions lead to549 higher densities and viscosities leading to inhibited mass550 transfer of solute. This mechanism along with the effect of the551 solubility gradient appears to be the most likely mechanism as552 the decrease in the experimental growth rates is largely553 proportional to the diffusion limited growth rates.
5. CONCLUSIONS
554 Growth kinetics as a function of solvent composition have been555 determined based on seeded isothermal batch desupersatura-556 tion experiments. A population balance model combined with a557 parameter estimation procedure have been utilized to obtain558 growth rate parameters from desupersaturation data. The559 method takes advantage of two online PAT technologies to560 measure solution concentration and to ensure negligible561 nucleation occurs. The method has been shown to predict562 experimental data with good accuracy. The effects of initial
563supersaturation, seed mass, and seed size have been investigated564and the numerical model has been shown to capture these565phenomena with good accuracy. With regard to the effect of566initial supersaturation, faster desupersaturation decay and hence567crystal growth rate was observed with higher supersaturations568due to a larger driving force. A faster desupersaturation decay569was observed for cases where a larger seed mass was used. This570increased consumption can be explained by the increase in total571seed surface area available for crystal growth, from the572additional seed. A similar observation was made when seeds573of a smaller size fraction were used. A smaller seed size provides574a larger specific surface area for the crystal growth process.575Hence the second moment of the seed crystals, m2 will be576larger, promoting a faster desupersaturation decay. Further-577more the role of the solvent has shown to have a significant578impact on the crystal growth rate. Diffusion growth rates have579been calculated in order to provide detail about the growth580mechanism. With the aid of the growth mechanism a detailed581discussion on how solvent composition affects growth rates is582outlined. The effects of solvent composition are split into four583different phenomena and the growth mechanism is utilized to584determine which one is the most probable. The phenomena are585named surface roughing, selective absorption, solubility586gradient, and increasing viscosity due to higher water mass587fractions. This work offers a successful methodology for the588quick determination of crystal growth parameters for use in589modeling and optimizing particulate systems and also highlights590that investigating the crystal growth mechanism can offer new591insights into understanding the role of the solvent in affecting592crystal growth kinetics.
597Notes598The authors declare no competing financial interest.
599■ ACKNOWLEDGMENTS600This research has been conducted as part of the Solid State601Pharmaceuticals Cluster (SSPC) and funded by Science602Foundation Ireland (SFI).
622 dm = Molecular diameter (m)623 ds = Stirrer diameter (m)624 g = Growth order (-)625 ka = Surface area shape factor (-)626 kd = Mass transfer coefficient (m/s)627 kg = Growth rate constant (m/s)628 kv = Volume shape factor (-)629 mfrac = Mass fraction (-)630 mk = kth Moment of particle size distribution (mk)631 n = Population density (no./m4)632 vs = Stirrer speed (no./s)633 ε = Average power input (W/kg)634 μ = Dynamic viscosity (kg/m s)635 π = Pi (-)636 ρc = Crystal density (kg/m3)637 υ = Kinematic viscosity (m2/s)638 ρsoln = Solution density (kg/m3)639 ρw = Water density (kg/m3)640 ρm = Methanol density (kg/m3)641 θ = Parameter set (-)642 o = Initial (-)
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