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74 simultaneously investigating the underlying mechanism of how Equation 2 can be multiplied through by Lk and integrated to 122
75 the solvent affects the growth rates. The moments of the result in an equation in terms of the moments mk: 123
76 distribution are reconstructed directly using a simple yet dm 0
77 accurate method yielding a very fast and effective means of =B
dt (4)
78 estimating growth rates. These reconstructed distributions are
79 then compared to distributions obtained experimentally. In the case where the system is sufficiently seeded, negligible 124
80 Capturing the effects of solvent composition on the growth nucleation can be assumed. Ensuring that no nucleation is 125
81 rate is critical to building a population balance model. present is critical as seeding simplifies the mathematical 126
82 Neglecting the influence of solvent composition will lead to treatment of the experimental data. Under this condition the 127
83 discrepancies between experimental and modeled data. A second moment of the seed particle size distribution can only 128
84 growth rate model that considers its parameters to be functions be influenced by the growth of the crystal and no other 129
85 of antisolvent mass fraction has previously been shown to be competing factors. 130
86 superior to a model without such functionality.13 The During the crystallization process, the mass balance of the 131
87 nucleation kinetics for paracetamol in methanol−water solution phase can be described as 132
88 mixtures has been evaluated elsewhere and shown to be dc ∞ 2
89 strongly dependent on solvent composition.14 This study aims
dt
= −3k vρcG
0
∫
nL dL
(5)
90 to utilize a population balance to determine growth rates based
91 on power law equations, while offering an insight into the where ρc and kv are the solid density and the volume shape 133
92 growth mechanism and the effect of solvent composition. The factor of paracetamol crystals, respectively. In the above 134
93 focus of the work is also to determine parameters for power law equation, the integral term represents the second moment of 135
94 expressions which can be used in predicting and optimizing the seed crystals, m2 which is proportional to the total surface 136
95 crystal size distributions. To date the literature does not contain area of crystals present. A value of 1332 kg/m3 will be 137
96 any work detailing the estimation of growth kinetics of employed for the crystal density of form I of paracetamol.15 138
97 antisolvent systems utilizing the method of moments together The following initial and boundary conditions apply: 139
98 with in situ measurement techniques. In addition to this, a C(o) = C0 (6)
99 greater effort has been made to gauge the effect of solvent
100 composition on growth rate kinetics by linking with the n(0, L) = n0(L) (7)
101 estimated growth rate mechanism.
n(t , 0) = 0 (8)
2. POPULATION BALANCE MODEL AND PARAMETER with C0 being the initial concentration of the solute, n0(L) the 140
102 ESTIMATION PROCEDURE initial PSD, and B is the nucleation rate per unit mass. The 141
103 A mathematical model based on population balance equations supersaturation correlations used in this work for absolute 142
104 (PBEs) is used in combination with a least-squares supersaturation and supersaturation ratio are as follows: 143
105 optimization and the experimental desupersaturation data to ΔC = C − C* (9)
106 determine the growth rate parameters of paracetamol in
C
107 methanol/water solutions as a function of solvent composition. S=
108 2.1. Population Balance. In a perfectly mixed batch C* (10)
109 reactor the evolution of the crystal size distribution can be where C is the solute concentration and C* is the antisolvent 144
110 described as follows free solubility. The antisolvent free solubility is calculated and 145
described elsewhere.14 The crystal size distribution was 146
∂n(L , t ) ∂n(L , t ) reconstructed from the moments utilizing a novel technique 147
+ G (t ) =0
∂t ∂L (1) developed by Hutton.16 This technique is outlined in more 148
detail by Mitchell et al.9 149
111 where n(L,t) is the population density of the crystals and G(t) 2.2. Optimization. For the estimation of the growth 150
112 is the crystal growth rate which is assumed to be independent kinetics parameters the following least-squares problem had to 151
113 of size. The above equation is solved using the method of be solved: 152
114 moments, detailed below. Nd Nd 2
115
116
The standard method of moments is an efficient method of
transforming a population balance into its constituent mo-
min ∑ R t2 = min ∑ ⎡(Stexp − Stsim(θ))⎤
⎣ ⎦
i=1 i=1 (11)
117 ments. The low order moments of the distribution represent
118 the total number, length, surface area, and volume of particles where θ is the set of parameters to be estimated, represents Stsim 153
119 in the crystallizing system. Using the standard method of the predicted supersaturation ratios, Stexp represents the 154
120 moments, eq 1 becomes measured supersaturation ratios at each time or sampling 155
interval, and Nd is the number of sampling instances. The 156
dmk (t ) MATLAB optimization algorithm f minsearch which employs a 157
= kG(t )mk − 1(t )
dt (2) Nelder−Mead simplex method was utilized to find the optimal 158
set of parameters. 159
121 where
3. EXPERIMENTAL SECTION
∞ k
mk = ∫0 L n(L , t )dL
(3)
3.1. Materials. The experimental work outlined was
performed on Acetaminophen A7085, Sigma Ultra, ≥99%,
160
161
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162 sourced from Sigma Aldrich. The methanol employed in this polynomial and the coefficients were computed in matlab with 223
163 work was gradient grade hiPerSolv CHROMANORM for the regress function as follows: 224
164 HPLC ≥99%, sourced from VWR.
165 3.2. Apparatus. A LabMax reactor system from Mettler- Conc = 0.018 + 0.244Abs + 0.613Abs 2 − 0.0009Comp
166 Toledo was utilized in this work to estimate the growth kinetics + 4.605Comp2 + 0.0005AbsComp R2
167 of the paracetamol in a methanol/water solution system. The
168 reactor was a 1-L round-bottomed borosilicate glass jacketed = 0.9979 (12)
169 reactor, allowing controlled heating and cooling of solutions. The model was found to predict solution concentration with 225
170 All experiments were carried out isothermally at 25 °C and at an average relative error of 0.95% over the concentration and 226
171 constant solvent composition. Agitation of the solution was composition ranges investigated. A typical ATR-FTIR spectra 227
172 provided by means of an overhead motor and a glass stirrer, for a paracetamol and methanol solution is shown in Figure 1, 228 f1
173 with four blades at a pitch of 45°. The system allowed fluid with the peaks associated with the main functional groups 229
174 dosing and the use of in situ immersion probes. The system highlighted. 230
175 came with iControl LabMax software enabling real-time
176 measurement of vital process parameters and full walk away
177 operation. A custom wall baffle described previously9 was
178 employed in all experiments to improve the level of mixing in
179 the reactor. Antisolvent (water) addition into the solution was
180 achieved using a ProMinent beta/4 peristaltic pump, which was
181 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 to
184 the solution.
185 3.2.1. FBRM Probe. A Mettler-Toledo Focused Beam
186 Reflectance Measurement (FBRM) D600L probe was utilized
187 in this work to track the evolution of the PSD and to ensure
188 negligible nucleation occurred during desupersaturation experi-
189 ments. For all FBRM measurements, the fine detection setting
190 was employed, as the detection setting was found to produce a
191 significant level of noise due to the agitation of the impeller.
192 The instrument provided a chord length distribution evolution Figure 1. ATR-FTIR spectra of paracetamol in a methanol−water
193 over time at 10 s intervals, which is useful for indicating the solution used for calibration.
194 presence of nucleating crystals.
195 3.3. ATR-FTIR Calibration. ATR-FTIR allows for the 3.4.1. Procedure. Measurement of Growth Kinetics. 231
196 acquisition of liquid-phase infrared spectra in the presence of The measurement procedure for the independent determi- 232
197 solid material due to the low penetration depth of the IR beam nation of growth rate kinetics is based on seeded batch 233
198 into the solution. An ATR-FTIR ReactIR 4000 system from desupersaturation experiments realized at constant solvent 234
199 Mettler-Toledo, equipped with a 11.75 ‘‘DiComp’’ immersion composition. Only the initial PSD of the seed crystals and the 235
200 probe and a diamond ATR crystal, was used to track solution evolution of the solute concentration are needed for the 236
201 concentration. The infrared spectra are known to be affected by determination of crystal growth rates. All experiments were 237
202 concentration and solvent composition requiring calibration to conducted at 25 °C and at an impeller speed of 250 rpm. 238
203 known experimental conditions. The amide functional group Scanning electron microscopy images were taken of the seed 239
204 contained within paracetamol, which emits a bending frequency and the final product to ensure no change in crystal 240
205 of 1517−1 in infrared spectroscopy. The calibration procedure morphology was observed and no polymorphic change 241
206 employed in this work involves tracking the absolute height of occurred. The mass of solvent in the vessel ranged from 242
207 one solute peak and correlating it to known solution 0.365 to 0.45 kg. A saturated solution was created upon mixing 243
208 concentrations and solvent compositions. This method was paracetamol in a specific methanol/water mixture at 25 °C. The 244
209 chosen as it was demonstrated to be capable of predicting solution was then heated 10 °C above the saturation 245
210 solute concentration with a relative uncertainty of less than 3% temperature and held until complete dissolution was observed 246
211 for a range of solution systems.17 The procedure involves by FBRM. The solution was then cooled back to the saturation 247
212 measuring the absorbance of particular peaks and increasing the temperature. A supersaturated solution was then created via the 248
213 concentration at a set number of intervals until the solubility is addition of a known mass of antisolvent into the reactor. A 249
214 reached. The calibration points are varied to cover a range of range of supersaturations can be induced while avoiding the 250
215 concentrations and solvent compositions. The compositions solution nucleating with prior knowledge of the MSZW. The 251
216 and concentrations were varied between 40% and 68% and MSZW was determined from the experiments conducted using 252
217 0.010−0.218 kg/kg, respectively. The method requires the the FBRM probe to detect the onset of nucleation outlined 253
218 solubility to be known prior to the calibration in order to further in previous work.14The masses of solution and 254
219 remain in the undersaturated stable region. The solubility was antisolvent were chosen in such a way to obtain the constant 255
220 measured via a gravimetric method and detailed in previous mass fraction of interest. ATR-FTIR and FBRM were employed 256
221 work.14 The values of absorbance (ABS), composition (Comp) to monitor the solute concentration and chord length 257
222 and concentration (Conc) were fitted to a second order distribution during the experiment. At time zero a specific 258
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259 mass of dry seeds was charged into the reactor. The experiment to maintain a similar shape during the experiment, indicating 276
260 was monitored until a stable signal was obtained from ATR- that growth of the seed crystals is the dominant supersaturation 277
261 FTIR, indicating that the solubility had been reached. All consumption mechanism. After each experiment the solution 278
f2 262 desupersaturation experiments were performed twice. Figure 2 was filtered, dried, and weighed, and the particle size 279
distribution was measured using a Horiba L92O particle size 280
analyzer (PSA). Similar results were obtained for all growth 281
characterization experiments. A number of experimental 282
conditions were varied in order to determine the effect of 283
initial supersaturation, seed mass, and seed size. These results 284
are discussed in more detail in Section 4. Finally, the measured 285
growth rate parameters were estimated by comparison of 286
simulations to experiments at different operating conditions. 287
3.4.2. Seed Preparation. Seed crystals were prepared by 288
cooling crystallizations in order to produce a higher mass of 289
crystals. The crystals were subsequently wet sieved using three 290
stainless steel, woven wire cloth sieves, with squares apertures 291
of nominal sizes of 90, 125, and 250 μm, respectively. The 292
remaining seed crystal fractions in the size ranges of 90−125 293
μm and 125−250 μm were washed, filtered, and dried. The 294
PSDs of all the three seed fractions were measured using a 295
Horiba L92O particle size analyzer (PSA), using saturated 296
water at room temperature as the dispersal medium. A 297
Figure 2. Repeatability of two sets of desupersaturation experiments dispersant solution saturated with paracetamol and containing 298
presented in Section 2.4. sodium dodecyl sulfate at a concentration of 5 g/L was also 299
employed to ensure no agglomeration occurred during the 300
263 shows the typical reproducibility of the measured desupersatu- particle size measurement. The particle size distributions were 301
264 ration curves for two repeated runs of experiments at different measured three times in accordance with ISO33320 and all 302
265 initial conditions. It can be readily observed that the distributions were found to be less 5%, 3%, and 5% for the d10, 303
266 repeatability was satisfactory in both cases. The growth kinetics d50, and d90 respectively. 304
267 were estimated for a range of solvent compositions from 40%
268 to 68% mass water, respectively. This range was chosen as 4. RESULTS
269 experiments carried out above 70% result in dilution due to a Five seeded batch desupersaturation experiments were 305
270 low solubility gradient. The absence of significant nucleation performed at various solvent compositions. An additional 306
271 was assured by monitoring the CLDs using the FBRM during three experiments were performed to investigate the effect of 307
272 the experiment. Typical time-resolved CLDs are shown in initial supersaturation, seed size fraction, and seed mass. The 308
f3 273 Figure 3. It can be readily observed that no significant increase experimental runs were labeled PM1−PM8, and the corre- 309
274 in the counts at small chord lengths occurred, thus indicating sponding experimental conditions are listed in Table 1 where 310 t1
275 the absence of significant nucleation. Also the CLD was found Minitial and Mfinal are the initial and final percentage of water in 311
the vessel, respectively. Mw is the mass of solvent added to the 312
Table 1. Experimental Conditions of Seeded Growth
Experiments
seed M initial M final
mass (wt %) (wt %) Mw
exp. no. S0 seed fraction (kg) (water) (water) (kg)
PM1 1.1755 125−250 0.00497 40 50 0.065
μm
PM2 1.3077 125−250 0.00501 40 55 0.11
μm
PM3 1.2330 125−250 0.00497 50 60 0.075
μm
PM4 1.0854 125−250 0.00496 20 40 0.1
μm
PM5 1.2198 125−250 0.00502 60 68 0.075
μm
PM6 1.2160 125−250 0.00993 50 60 0.075
μm
PM7 1.2223 90−125 μm 0.00497 50 60 0.075
PM8 1.4396 125−250 0.00495 40 60 0.150
μm
PM5R 1.2164 125−250 0.00503 60 68 0.075
μm
Figure 3. Measured CLDs from FBRM for typical seeded growth PM8R 1.4396 125−250 0.00502 40 60 0.150
experiment. μm
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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 1.10 × 10−4 3.46 × 10−5 4.93 × 10−5 1.86 × 10−4 8.1232 × 10−5
g 1.7531 1.604 1.895 2.239 1.6083
residual 8.53 × 10−5 1.10 × 10−4 1.20 × 10−3 7.43 × 10−4 8.76 × 10−4
313 vessel to generate the initial supersaturation. The operating
314 parameters in Table 1 were chosen to cover the range of
315 interest and at the same time to avoid the occurrence of
316 nucleation. On the basis of this set of experiments, the growth
317 kinetics were evaluated.
318 4.1. Estimation of Growth Kinetics. To determine the
319 growth kinetics of paracetamol in methanol/water solutions,
320 the experimental desupersaturation data were used together
321 with the PBE model and the optimization algorithm described
322 in Section 2.2. An empirical power law expression was
323 employed to express the relationship between supersaturation
324 and growth rate, eq 13.
G = kg (ΔC) g (13)
325 where kg is the growth rate constant, ΔC is absolute
326 supersaturation, and g is the growth exponent. The growth
327 rate parameters were calculated as a function of solvent
t2 328 composition and are listed in Table 2. The power law eq 13 Figure 5. Effect of initial supersaturation on the desupersaturation
329 provides a good fit of the desupersaturation data in all the curves.
f4 330 growth experiments as can be seen from Figures 4−7.
4.3. Effect of Seed Mass. The effect of seed mass on the 345
rate of supersaturation decay is shown in Figure 6. It can be 346 f6
Figure 4. Desupersaturation experiment PM1: Experimental and
simulated data.
Figure 6. Effect of seed mass on the desupersaturation curves.
331 Changing the initial values of the estimated parameters over
332 several orders of magnitude in the optimization procedure seen from the plot that as a result of increasing seed mass, the 347
333 always produced the same results, hence indicating a global supersaturation in solution is consumed at a faster rate. This 348
334 optimum. increased consumption can be explained by the increase in total 349
4.2. Effect of Initial Supersaturation. The effect of initial seed surface area available for crystal growth, from the 350
335
additional seed. It should be noted that crystal growth rate is 351
336 supersaturation on the rate of supersaturation decay is shown in
not a function of seed mass. Instead, the effect of seed mass on 352
f5 337 Figure 5. It can be observed that generating a higher initial
the decay of supersaturation is accounted for in eq 5, using the 353
338 supersaturation results in a faster rate of decay of super- second moment of the seed crystals present. A larger seed mass 354
339 saturation. At approximately 400 s the higher initial will result in a larger value for the second moment and hence 355
340 desupersaturation curve cuts across the lower curve. This is will result in higher decay of supersaturation. 356
341 an expected result as growth rates are a function of 4.4. Effect of Seed Size. The effect of seed size fraction on 357
342 supersaturation and a higher generation of supersaturation or the rate of supersaturation decay is shown in Figure 7. Figure 7 358 f7
343 driving force leads to a higher growth rate and desupersatura- demonstrates that when seeds of a smaller size fraction are 359
344 tion decay. present in solution, a faster rate of desupersaturation decay is 360
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Figure 7. Effect of seed size on the desupersaturation curves. Figure 8. Experimental and simulated final PSD of run PM2.
361 observed, which can be explained by eq 5. A smaller seed size
362 provides a larger specific surface area for the crystal growth
363 process. Hence the second moment of the seed crystals, m2 will
364 be larger, promoting a faster desupersaturation decay. This
365 effect is analogous to the effect of seed mass, although the effect
366 of seed size on the desupersaturation decay is not as
367 pronounced as the effect of seed mass. This may be due to
368 the fact that there is not a substantial difference between the
369 size fractions employed to produce significantly different
370 results.
371 4. 5. Accuracy of Numerical Model. The technique
372 employed here provides two methods of verifying the accuracy
373 of the numerical model employed in this work. The first is the
374 simulated desupersaturation curves shown in Figures 4−7. It
375 can be seen that a reasonable fit to the desupersaturation data is
376 achieved with a maximum residual calculated using eq 9 of 1.2
377 × 10−3. Residuals for all growth rate estimation experiments are
378 reported in Table 2. All phenomena associated with the effects Figure 9. Experimental and simulated final PSD of run PM3.
379 on crystal growth, such as the effect of initial supersaturation,
380 seed mass, and seed size are captured by the numerical model 4.6. Growth Rate Mechanism. In the previous section 404
381 as can be seen from Figures 4−7. growth kinetics as a function of solvent composition were 405
382 The second method for validating the accuracy of the evaluated via fitting a population balance model to 406
383 numerical model employed involves comparison of the desupersaturation data. These parameters are essential in 407
384 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 readily
f11 387 observed from Figure 11 that the experimental PSD has shifted
388 to larger particle size values. It can be seen that a reasonable
389 prediction is obtained from the numerical model of the
390 simulated PSD. All other experiments conducted within this
391 work were found to be in similar agreement. The numerical
392 model captures the particles in the smaller range quiet well,
393 however in experiment PM3 the experimental PSD is slightly
394 underpredicting the larger particles. Some particle agglomer-
395 ation can be seen in the product PSDs, however Figure 11
396 shows that this agglomeration originates from the seed and this
f12 397 is supported by both Figure 11 and 12 which show PSDs and
398 SEM images of both seed and product PSDs. To some extent
399 this agglomeration may be due to the particles agglomerating
400 on filtration or on storage as the particles appear to absorb
401 moisture quite strongly when present in air. Monitoring the
402 particles during the experiments with FBRM also suggests that Figure 10. Experimental and simulated final PSD of run PM4.
403 no significant agglomeration occurred.
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Figure 13. Growth rate as a function of supersaturation ratio at various
Figure 11. Seed PSD together with the experimental final and solvent compositions.
simulated PSD of run PM4.
interface. Since the concentration of the solute is greater as you 418
leave the interface, solute will diffuse toward the crystal surface. 419
If diffusion of solute from the bulk solution to the crystal 420
surface is rate limiting, growth is diffusion controlled. A model 421
that focuses on the diffusion of solute through the boundary 422
layer is known as the diffusion controlled model. If 423
incorporation into a crystal lattice is the slowest process, 424
growth is surface-integration controlled. To determine which 425
growth mechanism is rate determining we use the following 426
equation: 427
kM
G = kd a (c − c*)
3k vρ (14)
where M is the molar mass of paracetamol (0.15117 kg/mol), c 428
is the concentration, and c* is the solubility in molar units of 429
mol/m3. This equation requires the mass transfer coefficient, 430
which is calculated using the Sherwood correlation. 431
⎛ ⎛ 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 432
size, ε is the average power input, υ is the kinematic viscosity, 433
and Sc is the Schmidt number (Sc = υ/D). Since the solvent 434
composition in this work is dynamic, the variation in the 435
density of the solution is considered through 436
1
ρsolution =
mfrac w /ρw + (1 − mfrac w)/ρm (16)
Figure 12. SEM image of (A) seed and (B) product crystals from run
PM4. where Mfracw is the mass fraction of water and ρw and ρm are 437
the densities of water and methanol, respectively. This results in 438
408 developing a numerical model to optimize an antisolvent dynamic viscosity as a function of solvent composition. The 439
f13 409 crystallization process. Table 2 and Figure 13 show that solvent diffusion coefficient, D, can be evaluated using the Stokes− 440
410 composition has a significant impact on the growth rates. In Einstein equation as follows: 441
411 this section, we attempt to further investigate the underlying kT
412 mechanisms of the effect of solvent on the growth rates D=
3πμdm (17)
413 observed in Section 4.1.
−23
414 Goals of crystal growth theory are to determine the source of where k is the Boltzmann constant (1.38065 × 10 J/K), T is 442
415 steps and the rate controlling step for crystal growth. As a the temperature in Kelvin, and μ is the dynamic viscosity of the 443
416 crystal grows from a supersaturated solution, the solute fluid. Because all values of viscosity are calculated as a function 444
417 concentration is depleted in the region of the crystal−solution of solvent composition a case study of a water mass fraction of 445
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446 0.4 will be used for demonstration purposes. The dynamic
447 viscosity of the fluid at a water mass fraction of 0.4 is (6.45008
448 × 10−4 kg/m s @ 25 °C). The molecular diameter, dm in the
449 above expression is evaluated as follows:
1
dm = 3
ccNA (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 molecular
453 diameter of paracetamol crystals. For stirred tanks, the average
454 power input, ε, can be evaluated using:
NPvs3ds5
ε=
V (19)
455 where Np is the Power number of the impeller, vs is the stirrer
Figure 15. Diffusion growth rates and experimental growth rates as a
456 speed (250 rpm), ds is the stirrer diameter (0.06 m), and V is
function of solvent composition at a constant supersaturation of 1.2.
457 the solution volume (0.0004 m3). For the downward-pumping,
458 four-blade, 45° pitched blade impeller used in this work, a generated at this point of the solubility curve, which in turn 482
459 Power number of 1.08 has been estimated previously by
can increase the experimental error. It is difficult to obtain 483
460 Chapple et al.18 This results in an average power input of
reproducible data in this region of the solubility curve due to a 484
461 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 mass low driving force. The data in Figure 15 are also calculated for a 485
463 fraction of 0.4 at 25 °C, eq 15 yields a value of 1.23 × 10−5 m/s constant supersaturation of 1.2 which is outside the super- 486
464 for the mass transfer coefficient, kd, for this solution system. saturation range generated in the experiment carried out to 487
465 The values of kd as a function of solvent composition are shown estimate growth rates for a water mass fraction above 0.68. The 488
f14 466 in Figure 14. reduction in both diffusion limited growth rates and the 489
experimental growth rates is approximately proportional by a 490
factor of 2. The slope of both growth rates as a function of 491
water mass fraction is −1 × 10−7 m/s from water mass fractions 492
of 0.4−0.6. The discrepancy between the experimental growth 493
rates and the diffusion limited growths can be attributed to a 494
reduction in the solubility gradient, higher solution viscosities, 495
selective adsorption of solvent molecules at specific surface sites 496
due to strong interactions between solute and solvent 497
molecules, and the influence of the solvent on the surface 498
roughening. These mechanisms are discussed in more detail in 499
the following section. 500
4.7.1. Effect of Solvent Composition. Selective 501
Adsorption and Surface Roughening. The role played by 502
the solvent in enhancing or inhibiting crystal growth is not clear 503
at present.18 According to the existing literature, the solvent 504
may contribute to decreasing growth rate due to a selective 505
adsorption of solvent molecules or may enhance face growth 506
Figure 14. Mass transfer coefficient kd calculated from eq 15 as a rate by causing a reduction in the interfacial tension.19−21 The 507
function of solvent composition.
first mechanism has been attributed to a selective adsorption of 508
solvent molecules at specific surface sites due to strong 509
467 The knowledge of kd provides the possibility of calculating
468 the diffusion growth rates as a function of solvent composition. interactions between solute and solvent molecules.20−22 The 510
469 Diffusion limited growth rates calculated from eq 14 are plotted second mechanism referred to here as the interfacial energy 511
f15 470 in Figure 15. Over the range of solvent compositions studied, effect, is related to the influence of the solvent on the surface 512
471 the experimental growth rates were found to be lower than the roughening which under certain circumstances may induce a 513
472 diffusion limited growth rates, predicted from eq 14. Therefore, change in the growth mechanism.23−25 Davey and co-workers 514
473 surface integration of the solute is deemed to be the rate provide a good example of the interfacial effect of the solvent 515
474 limiting step of the growth mechanism. Figure 15 shows that as on crystal interface. They reported on the growth kinetics of 516
475 the mass fraction of water increases, a reduction in the hexamethylene tetramine (HMT) crystallized from different 517
476 experimental growth rates and the diffusion limited growth solvents and solvent mixtures.23−27 It was reported that the 518
477 rates is observed. With the exception of water mass fraction of growth rate of the (110) face increased faster when water or 519
478 0.68, it can be seen that the experimental growth rates reduce at water/acetone mixtures replaced ethanol as the solvent. 520
479 the same rate as the diffusion limited growth rates. This Decreasing surface diffusion and a direct integration to the 521
480 increase at mass fractions of 0.68 may be due to some crystal lattice were connected to a change in the growth 522
481 experimental error as very little supersaturation can be mechanism. The observed effect was attributed to favorable 523
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524 interactions between the solute and the solvent at increasing supersaturation, seed mass, and seed size have been investigated 563
525 solubility. and the numerical model has been shown to capture these 564
526 4.7.2. Solubility Gradient. It has been shown that at higher phenomena with good accuracy. With regard to the effect of 565
527 solubilities more favorable interactions occur between the initial supersaturation, faster desupersaturation decay and hence 566
528 solute and solvent or in the case studied here lower solubilities crystal growth rate was observed with higher supersaturations 567
529 leading to unfavorable interactions. The solubility gradient may due to a larger driving force. A faster desupersaturation decay 568
530 also have an impact as the gradient is reduced with higher water was observed for cases where a larger seed mass was used. This 569
531 mass fractions leading to a reduced driving force and hence increased consumption can be explained by the increase in total 570
532 reduced crystal growth. Desupersaturation experiments gen- seed surface area available for crystal growth, from the 571
533 erally involve adding a known amount of antisolvent into the additional seed. A similar observation was made when seeds 572
534 reactor and generating a specific supersaturation, followed by of a smaller size fraction were used. A smaller seed size provides 573
535 seeding and subsequent growth. However as the water mass a larger specific surface area for the crystal growth process. 574
536 fraction tends to one the gradient of the solubility with respect Hence the second moment of the seed crystals, m2 will be 575
f16 537 to the water mass fraction reduces. Figure 16 illustrates that the larger, promoting a faster desupersaturation decay. Further- 576
more the role of the solvent has shown to have a significant 577
impact on the crystal growth rate. Diffusion growth rates have 578
been calculated in order to provide detail about the growth 579
mechanism. With the aid of the growth mechanism a detailed 580
discussion on how solvent composition affects growth rates is 581
outlined. The effects of solvent composition are split into four 582
different phenomena and the growth mechanism is utilized to 583
determine which one is the most probable. The phenomena are 584
named surface roughing, selective absorption, solubility 585
gradient, and increasing viscosity due to higher water mass 586
fractions. This work offers a successful methodology for the 587
quick determination of crystal growth parameters for use in 588
modeling and optimizing particulate systems and also highlights 589
that investigating the crystal growth mechanism can offer new 590
insights into understanding the role of the solvent in affecting 591
crystal growth kinetics. 592
Figure 16. Antisolvent free solubility gradient as a function of water
■ AUTHOR INFORMATION
Corresponding Author
593
594
mass fraction. *Tel.: 00 353 61 213134. Fax: 00 353 61 202944. E-mail: 595
clifford.ociardha@ul.ie. 596
538 driving force is reduced when starting from 0.6 in comparison
Notes 597
539 to starting and generating a supersaturation from 0.4. Hence
The authors declare no competing financial interest. 598
■
540 the driving force is reduced resulting in a reduced mass transfer,
541 solute integration, and subsequent crystal growth rate. ACKNOWLEDGMENTS 599
542 4.7.3. Viscosity. One other reason could be due to the
543 increased viscosity of the fluid inhibiting mass transfer of the This research has been conducted as part of the Solid State 600
544 solute from solution to the crystal face thereby reducing the Pharmaceuticals Cluster (SSPC) and funded by Science 601
Foundation Ireland (SFI). 602
■
545 crystal growth rate. Figure 15 shows that at higher water mass
546 fractions a reduction in mass transfer is observed. The mass
547 transfer coefficient is a function of dynamic viscosity as can be NOMENCLATURE 603
548 seen from eq 15, therefore higher water mass fractions lead to B = Nucleation rate (no./kg methanol s) 604
549 higher densities and viscosities leading to inhibited mass C = Concentration (kg/kg methanol) 605
550 transfer of solute. This mechanism along with the effect of the C* = Equilibrium concentration (solubility) (kg/kg 606
551 solubility gradient appears to be the most likely mechanism as methanol) 607
552 the decrease in the experimental growth rates is largely D = Diffusivity (m2/s) 608
553 proportional to the diffusion limited growth rates. G = Growth rate (m/s) 609
K = Boltzmann’s constant (J/K) 610
5. CONCLUSIONS L = Particle size (m) 611
M = Molar mass (kg/kmol) 612
554 Growth kinetics as a function of solvent composition have been Np = Power number (-) 613
555 determined based on seeded isothermal batch desupersatura- NA = Avogadro Constant (no./kmol) 614
556 tion experiments. A population balance model combined with a R = Gas constant (J/kmol K) 615
557 parameter estimation procedure have been utilized to obtain Sc = Schmidt number (-) 616
558 growth rate parameters from desupersaturation data. The Stsim = Simulated supersaturation ratio (-) 617
559 method takes advantage of two online PAT technologies to Stexp = Experimental supersaturation ratio 618
560 measure solution concentration and to ensure negligible T = Temperature (K) 619
561 nucleation occurs. The method has been shown to predict V = Solution volume (m3) 620
562 experimental data with good accuracy. The effects of initial cc = Molar density (kmol/m3) 621
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10. Industrial & Engineering Chemistry Research Article
622 dm = Molecular diameter (m) cetamol in methanol/water solutions. J. Cryst. Growth 2011, 328, 50− 687
623 ds = Stirrer diameter (m) 57. 688
624 g = Growth order (-) (15) Brittain, H. G., Ed. Polymorphism in Pharmaceutical Solids; 689
625 ka = Surface area shape factor (-) Marcel Dekker Inc.: New York, 1999. 690
(16) Hutton, K. Mathematical modelling of gibbsite precipitation, 691
626 kd = Mass transfer coefficient (m/s) Ph.D. Thesis, University of Limerick, Ireland, 2009. 692
627 kg = Growth rate constant (m/s) (17) Lewiner, F.; Klein, J. P.; Puel, F.; Févotte, G. On-line ATR-FTIR 693
628 kv = Volume shape factor (-) measurement of supersaturation during solution crystallization 694
629 mf rac = Mass fraction (-) processes. Calibration and applications on three solute/solvent 695
630 mk = kth Moment of particle size distribution (mk) systems. Chem. Eng. Sci. 2001, 56, 2069−2084. 696
631 n = Population density (no./m4) (18) Chapple, D.; Kresta, S. M.; Wall, A.; Afacan, A. The effect of 697
632 vs = Stirrer speed (no./s) impeller and tank geometry on power number for a pitched blade 698
633 ε = Average power input (W/kg) turbine. Trans. IChemE 2002, 80 (A), 364−372. 699
634 μ = Dynamic viscosity (kg/m s) (19) Lahav, M.; Leiserowitz, L. The effect of solvent on crystal 700
635 π = Pi (-) growth and morphology. J. Chem. Eng. Sci. 2001, 56 (7), 2245−2253. 701
(20) Berkovitch-Yellin, Z.; Van Mil, J.; Addadi, L.; Idelson, M.; 702
636 ρc = Crystal density (kg/m3)
Lahav, M.; Leiseroweitz, L Crystal morphology engineering by ″tailor- 703
637 υ = Kinematic viscosity (m2/s) made″ inhibitors; A new probe to fine intermolecular interactions. J. 704
638 ρsoln = Solution density (kg/m3) Am. Chem. Soc. 1985, 107, 3111. 705
639 ρw = Water density (kg/m3) (21) Mersmann, A., Ed. Crystallisation Technology Handbook; 706
640 ρm = Methanol density (kg/m3) Germany, 2001 707
641 θ = Parameter set (-) (22) Meenan, P. A.; Anderson, S. R.; Klug, D. L.; Myerson, A. S., Ed. 708
642 o = Initial (-) Handbook of Industrial Crystallisation; Butterworth-Heinemann: 709
■
Boston, 2002; Ch. 3, vol. 65. 710
(23) Bourne, J. R. The influence of solvent on crystal growth kinetics. 711
643 REFERENCES AIChE Symp. Ser. 1980, 76 (193), 59−64. 712
644 (1) Doki, N.; Kubota, N.; Yokota, M.; Kimura, S.; Sasaki, S. (24) Bourne, J. R.; Davey, R. J. The role of solvent-solute interactions 713
645 Production of sodium chloride crystals of uni-modal size distribution in determining crystal growth mechanism from solution, Part I. The 714
646 by batch dilution crystallization. J. Chem. Eng. Jpn. 2002, 35 (11), surface entropy factor. J. Cryst. Growth 1976, 36, 278−286. 715
647 1099−1104. (25) Bourne, J. R.; Davey, R. J. The role of solvent-solute interactions 716
648 (2) Worlitschek, J, Ph.D. Thesis, Monitoring, Swiss Federal Institute in determining crystal growth mechanism from solution, Part II. The 717
649 of Technology, Zurich, 2003. growth kinetics of hexamethylene tetramine. J. Cryst. Growth 1976, 36, 718
650 (3) Glade, H.; Ilyaskarov, A. M.; Ulrich, J. Determination of crystal 287−296. 719
651 growth kinetics using ultrasonic technique. Chem. Eng. Technol. 2004, (26) Bourne, J. R.; Davey, R. J.; McCulloch, J. The growth kinetics of 720
652 27 (4), 736−740. hexamethylene tetramine crystals from a water/acetone solution. J. 721
653 (4) Finnie, S. D.; Ristic, R. I.; Sherwood, J. N.; Zikic, A. M. Chem. Eng. Sci. 1978, 33, 199. 722
654 Morphological and growth rate distributions of small self-nucleated (27) Davey, R. J. The role of the solvent in crystal growth from 723
655 paracetamol crystals grown from pure aqueous solutions. J. Cryst. solution. J. Cryst. Growth 1986, 76, 637. 724
656 Growth 1999, 207, 308−318.
657 (5) Finnie, S.; Ristic, R. I.; Sherwood, J. N.; Zikic, A. M.
658 Characterization of growth behaviour of small paracetamol crystals
659 grown from pure solution. Trans. IChemE 1996, 74, 835−838.
660 (6) Shekunov, B. Y.; Aulton, M. E.; Adama-Acquah, R. W.; Grant, D.
661 J. W. Effect of temperature on crystal growth and crystal properties of
662 paracetamol. Faraday Trans. 1996, 92 (3), 439−444.
663 (7) Omar, W.; Al-Sayed, S.; Sultan, A.; Ulrich, J. Growth rate of
664 single acetaminophen crystals in supersaturated aqueous solution
665 under different operating conditions. Cryst. Res. Technol. 2008, 43 (1),
666 22−27.
667 (8) Schöll, J.; Lindenberg, C.; Vicum, L.; Brozio, J.; Mazzotti, M.
668 Precipitation of α l-glutamic acid: Determination of growth kinetics.
669 Faraday Discuss. 2007, 136, 247−264.
670 (9) Mitchell, N. A.; Ó ’Ciardhá, C. T.; Frawley, P. J. Estimation of the
671 growth kinetics for the cooling crystallisation of paracetamol and
672 ethanol solutions. J. Cryst. Growth 2011, 328, 39−49.
673 (10) Oosterhof, H.; Geertman, R. M.; Witkamp, G. J.; van Rosmalen,
674 G. M. The growth of sodium nitrate from mixtures of water and
675 isopropoxyethanol. J. Cryst. Growth 1999, 198/199, 754−759.
676 (11) Granberg, R. A.; Bloch, D. G.; Rasmuson, Ǻ .C. Crystallization of
677 paracetamol in acetone-water mixtures. J. Cryst. Growth 1999, 198/
678 199, 1287−1293.
679 (12) Schöll, J.; Lindenberg, C.; Vicum, L.; Brozio, J.; Mazzotti, M.
680 Antisolvent Precipitation of PDI 747: Kinetics of Particle Formation
681 and Growth. Cryst. Growth Des. 2007, 7 (9), 1653−1661.
682 (13) Mostafa Nowee, S.; Abbas, A.; Romagnoli, J. A. Anti solvent
683 crystallization: Model identification, experimental validation and
684 dynamic simulation. J.Chem. Eng Sci. 2008, 63, 5457−5467.
685 (14) Ó ’Ciardhá, C. T.; Mitchell, N. A.; Frawley, P. J. Estimation of
686 the nucleation kinetics for the anti-solvent crystallisation of para-
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