Design and Optimization of a Combined Cooling/AntisolventCrystallization Process

Christian Lindenberg, Martin Krattli, Jeroen Cornel, and Marco Mazzotti*

Institute of Process Engineering, ETH Zurich, CH-8092 Zurich, Switzerland

Jorg Brozio

Pharmaceutical & Analytical DeVelopment, NoVartis Pharma AG, CH-4056 Basel, Switzerland

ReceiVed August 25, 2008; ReVised Manuscript ReceiVed October 8, 2008

ABSTRACT: Design and optimization are important steps during the development of crystallization processes. The combinedcooling/antisolvent crystallization of acetylsalicylic acid (ASA) in ethanol-water mixtures is studied by means of experiments andpopulation balance modeling. Model-based approaches require accurate kinetics and thermodynamic data, which are obtained inthis work using in situ process monitoring techniques such as attenuated total reflectance Fourier transform infrared (ATR-FTIR)spectroscopy and focused beam reflectance measurement (FBRM). Solubility is measured in situ as a function of temperature andsolvent composition using a multivariate calibration model for the ATR-FTIR. Nucleation and growth kinetics are determined basedon crystallization experiments by a combination of a population balance model and an integral parameter estimation technique. Themodel is finally used to calculate optimal cooling and antisolvent addition profiles of the combined cooling/antisolvent crystallizationprocess using a multiobjective optimization approach which optimizes the process with respect to product properties, that is, particlesize distribution, and performance, that is, process time. It was found that the solubility exhibits a maximum at 17 wt% water andthat the growth rate correlates well with the solubility. No effect of the solvent composition on the nucleation rate could be identified.The optimized trajectories for cooling and antisolvent could greatly reduce the number of nuclei formed as shown through modelingand experiments. The study shows that combining cooling and antisolvent crystallization allows both improving productivity andreducing the formation of fines, and illustrates how process analytical tools and population balance modeling also are effective incrystallization processes where temperature and solvent composition change.

1. Introduction

Crystallization is a widely used process for the purificationand production of solid particles. In most industrial processesseeded cooling crystallization is applied, in which seed crystalsof the specific compound are added to the reactor and super-saturation is generated by cooling. Cooling crystallization isemployed if the solubility changes significantly with temperatureand if the compound is thermally stable. Alternatively, thesupersaturation can be created by adding an antisolvent, thatis, a solvent with a significantly lower solubility, or by achemical reaction, for example, due to a pH shift. One goal inproduction is the control of the product properties, for example,particle size and particle size distribution, purity, residual solventcontent, crystallinity, and polymorphic, hydrate or solvate form.To a certain extent, the product properties can be manipulatedby changing the process parameters. A crystal size distributionfulfilling assigned specifications, for example, maximum averageparticle size, minimum coefficient of variation, etc., can beobtained by growing given seed crystals and for instance byminimizing nucleation, that is, by keeping the system in themetastable zone, provided agglomeration and breakage can alsobe avoided. The operating conditions to obtain the desiredproduct properties can be selected by a trial and error approachor through a model-based process optimization.

In recent years, the application of in situ monitoring tools incrystallization processes has become more and more popular,also as a consequence of FDA’s process analytical technologyinitiative.1 Attenuated total reflectance Fourier transform infrared(ATR-FTIR) spectroscopy has been employed to measure the

liquid phase concentration in precipitation,2 cooling,3-6 andantisolvent7,8 crystallization processes. The focused beamreflectance method (FBRM)9-11 has been used in crystallizationprocesses to monitor the solid phase, either for tracking theevolution of the chord length distribution12-14 or for monitoringthe particle size distribution by using an appropriate restorationmethod.15-17 Further, it has been used to detect the onset ofparticle formation18,19 or the point of complete dissolution.20

Rohani and co-workers have recently estimated the nucleationand growth kinetics of paracetamol by a combination ofpopulation balance modeling and nonlinear regression with insitu monitoring of supersaturation using ATR-FTIR and mea-suring chord length distribution using FBRM.21 In the samework, optimal antisolvent flow-rate profiles were obtained byapplying nonlinear constrained single- and multiobjectiveoptimization techniques. The multiobjective optimization ap-proach was also applied by the same group to optimize seededbatch cooling22 and reactive crystallization23 processes, and todetermine the optimal cooling profiles and feed flow rates,respectively. The application of the FBRM for the model basedoptimization of the particle size distribution in the batch coolingof paracetamol was also shown earlier.24 While a number ofpublications focus on the offline optimal control, that is, thecalculation of an optimal trajectory for the antisolvent21,25 orthe temperature,22,24,26 Braatz and co-workers have designed apharmaceutical antisolvent crystallization process through feed-back concentration control based on ATR-FTIR.20 A comparisonbetween open-loop temperature control and closed-loop super-saturation control has shown that the latter, though more difficultand complex to implement, is less sensitive to disturbances, suchas changing seed quality or impurity profile, than the former,and hence it is more favorable for industrial applications.27,28

* Author to whom correspondence should be addressed. Phone: +41 44 6322456. Fax: +41 44 632 1141. E-mail: marco.mazzotti@ipe.mavt.ethz.ch.

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2009VOL. 9, NO. 2

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10.1021/cg800934h CCC: $40.75 2009 American Chemical SocietyPublished on Web 12/11/2008

An overview about the different techniques for the measure-ment of nucleation and growth rates can be found elsewhere.29

Regarding antisolvent crystallization, the effect of solventcomposition on the crystallization kinetics of paracetamol inacetone-water mixtures was investigated by Granberg and co-workers, and a good correlation between growth rate andsolubility was found.30 For the same system and an equalsupersaturation, the induction times depended on the solventcomposition and the measured interfacial energies increased withdecreasing solubility.31 Also for paracetamol in isopropanol-water mixtures an effect of the antisolvent concentration on theparameters in the nucleation and growth rate was found.21

It is common practice in industrial cooling crystallization thatan antisolvent is added in order to increase the yield of theprocess. From an engineering point of view the question ariseshow the cooling and the antisolvent addition should becombined. Recently, for the first time such a combination hasbeen considered systematically in the case of crystallization oflovastatin from water-acetone mixtures.32 The authors haveapplied a model-based design approach, that is, equivalent toopen-loop control, to choose the optimal policy in terms ofcombined cooling and antisolvent addition. Using newlymeasured solubility data together with literature expressions fornucleation and growth rates, different single-objective optimiza-tion problems, for example, maximum average particle size,minimum coefficient of variation, etc., have been solved, thusshowing that combining the combination of cooling andantisolvent addition is always better than either one or the otheronly.

Along similar lines, the goal of our study is the developmentand optimization of a combined cooling/antisolvent crystal-lization process of aspirin (acetylsalicylic acid, ASA) inethanol(solvent)-water(antisolvent) solutions. This paper de-scribes first the experimental setup used for solubility measure-ments and for the estimation of the nucleation and growth rates.Then, a process model based on the population balance equation(PBE) is derived. This model is used together with a nonlinearparameter estimation technique to regress the parameters of thenucleation and growth rates from the experimental data. Then,the model is used to calculate optimal temperature and antisol-vent concentration trajectories, which yield a final product withlarge particle size, narrow particle size distribution, and prismaticshape in the shortest process time, that is, by solving thecorresponding multiobjective optimization problem. Such op-timal trajectories are then tested experimentally in an open-loop control mode. Finally, a practical implementation of thetheoretical optimal trajectory is presented and discussed.

2. Experimental Section

2.1. Experimental Setup. The experiments were carried out in ajacketed 500 mL glass reactor equipped with a four-blade glass impellerand 45° inclined blades (LTS, Switzerland). The temperature wascontrolled using a Ministat 230-3 thermostat (Huber, Switzerland) anda Pt 100 temperature sensor. The stirrer had a diameter of 50 mm andwas positioned 10 mm above the reactor bottom. The reactor diameterwas 100 mm. The reactor was equipped with an FBRM and an ATR-FTIR probe. The probe tips were placed in zones of high fluid velocitiesand orientated toward the flow to prevent clogging and to obtainrepresentative measurements by allowing the crystals to impinge onthe probe window. The antisolvent was added using a Kp 2000 pistonpump (Desaga GmbH, Germany) which could deliver flow rates from10 to 5000 mL/h. It has been reported that more consistent results areobtained if the antisolvent is added close to the impeller.33 However,in this study, the antisolvent was fed above the liquid level sinceaddition close to the impeller led to clogging owing to the formation

of incrustations in the inlet pipe. All experiments were carried out ata stirring rate of 250 rpm. A schematic of the setup is shown inFigure 1.

2.2. Materials. Acetylsalicylic acid (>99%, Sigma-Aldrich, Swit-zerland), ethanol (99.9%, Merck KGaA, Germany), and deionized waterwere used in all experiments. ASA has two known polymorphs, whereasform II was recently discovered and is obtained by crystallization ofpure ASA in the presence of levetiracetam from acetonitrile;34 form Iis the well-known drug aspirin. In all our experiments form I wasobtained since form II crystallizes at completely different operatingconditions. Moreover, we employed a RA 400 Raman spectrometer(Mettler-Toledo, Switzerland) to exclude that solvates or hydrates ofASA have been obtained.

Two seed fractions of ASA were produced and are shown in Figure2. The S seeds were obtained by grinding and the L seeds were obtainedby collecting purchased crystals which were sieved in the size rangeof 90-200 µm.

2.3. In Situ Characterization Techniques. ATR-FTIR spectros-copy was employed to measure the liquid phase concentration of ASAand of the antisolvent water. The ATR technique enables one to measureexclusively the liquid phase in the presence of solid material due tothe low penetration depth of the IR beam.3 Spectra were collected usinga ReactIR 4000 system from Mettler-Toledo (Switzerland), equippedwith a 11.75 in. DiComp immersion probe and a diamond ATR crystal.

Figure 1. Schematic of the 0.5 L batch reactor used for the experiments.The setup allows for employing two in situ probes at the same time,namely, FBRM and ATR-FTIR.

Figure 2. Particle size distribution of the two seed fractions, where Sseeds were produced by grinding and L seeds were produced by sieving.

Combined Cooling/Antisolvent Crystallization Process Crystal Growth & Design, Vol. 9, No. 2, 2009 1125

Spectra were collected in the 650-4000 cm-1 region with a resolutionof 2 cm-1, and were recorded over 20 s (16 scans) or 120 s (128 scans),respectively, depending on the required temporal resolution.

The FBRM allows for in situ measurements of the chord lengthdistribution (CLD). In principle, the actual particle size distribution(PSD) can be restored from the obtained CLD.15-17 In this work, alaboratory-scale FBRM 600 L from Lasentec (Redmond, USA) wasapplied to detect the onset of particle formation during the ATR-FTIRcalibration measurements and to verify that no significant nucleationoccurred during the growth experiments.

2.4. Offline Characterization Techniques. Particle size distribu-tions were measured using a Multisizer 3 from Beckman Coulter (Nyon,Switzerland). This device uses the Coulter Principle to measure thenumber and volume particle size distribution with a high resolution inthe range of 0.4-1200 µm. Every measurement consisted of at least50 000 particles and PSDs were smoothed with a moving average filter.A saturated ethanol-water solution with 25 wt% ethanol and 1 wt%sodium chloride (>99%, Fluka, Switzerland) was used as electrolyte.In all experiments the PSD was measured after drying the crystals.

Optical microscopy using a Zeiss Axioplan microscope (Feldbach,Switzerland) was employed to quickly assess qualitative informationabout the particle size and shape of the crystals.

3. Population Balance Model

The PBE is used to model the crystallization process of ASA.The PBE written in terms of length L for a perfectly mixedsemibatch process with size-independent growth reads asfollows:35

∂(Vn)∂t

+VG∂n∂L

) 0 (1)

where n is the particle number density, V is the reactor volume,t is the time, and G is the growth rate. Assuming that the changein reactor volume depends only on the volume flow-rate of theantisolvent, Q, eq 1 can be recast as

∂n∂t

+G∂n∂L

)- nV

Q (2)

The concentration of ASA in solution, c, can be obtained bythe material balance, that is, by the following equation:

dcdt

)- 3M

kvFcG∫0

∞nL2 dL- c

VQ (3)

where M is ASA’s molecular mass, kv is its volume shape factor,and Fc is its crystal density. In this work we have used valuesof Fc ) 1400 g/kg for the crystal density and kv ) π/6 for thevolume shape factor. The following initial and boundaryconditions apply to the PBE and the material balance:

n(L0, t)) J ⁄ G (4)

n(L, 0)) n0 (5)

c(0)) c0 (6)

where J is the nucleation rate, and n0 and c0 are the initial PSDand solute concentration, respectively. The supersaturation S isdefined as follows:

S) c/c∗ (7)

with c* being the solubility of ASA. The solubility c* is not aconstant, but depends on the temperature T and the waterconcentration w as discussed in the following section. Temper-ature and water concentration are functions of time, and theirprofiles, which realize the selected crystallization strategy, areinput to the model. Moreover, nucleation and growth ratesdepend on temperature, concentration, and solvent compositionas discussed in Section 5.

3.1. Solution of the Population Balance Model. The PBEis solved using the moving pivot technique proposed by Kumarand Ramkrishna.36 Thereby, the PBE is discretized and eachsize range is represented by a corresponding length, the pivot,and its lower and upper boundary. The arithmetic mean of thesize of the lower and the upper boundaries was used as the sizeof the pivot. The moving pivot technique greatly reduces theproblem of numerical diffusion and instability, while providingthe applicability to an arbitrary grid to guarantee accuracy andreasonable computational effort.36 The discretized form of eq2 is36

dNi

dt) Jδi,1 -∫bi

bi+1 n(L, t)V

Q dL)Jδi,1 -Ni

VQ (8)

dxi

dt)G, i > 1 (9)

dx1

dt) 1

2G (10)

where Ni represents the total number of particles in the ith sizerange, δi,1 is the Kronecker delta and xi is the size of the ithpivot. Equations 8-10 were solved together with the materialbalance eq 3 using the MATLAB ode45 solver.

4. Multivariate Data Analysis

Liquid properties such as temperature and composition havean influence on the IR absorbance. The pure component IRspectra of ASA and water in ethanol are shown in Figure 3.The spectra overlap in the whole range of wavenumbers ofinterest, thus making a univariate calibration based on peakheight or peak area not feasible. Since the water concentrationchanges in the course of the experiment the ethanol-waterbackground cannot be simply subtracted as often done in coolingcrystallization. Therefore, a multivariate calibration model isused in this study.

4.1. Calibration Set. A calibration set covering the super-saturated as well as the undersaturated region was designed toobtain a calibration model, which is applicable over a wide rangeof process conditions, as described elsewhere.4,37 The experi-mental procedure is described in Figure 4. An undersaturatedsolution was prepared by dissolving a known amount of ASA.After having confirmed complete dissolution by zero FBRMcounts and a constant IR signal, the solution was cooled at 5°C/h and spectra were collected every 2 min. A rapid increase

Figure 3. Pure component spectra of acetylsalicylic acid (ASA) andwater in ethanol indicating overlapping signals in the whole range ofwavenumbers.

1126 Crystal Growth & Design, Vol. 9, No. 2, 2009 Lindenberg et al.

in counts of the FBRM signal indicated nucleation; hencesubsequent samples were discarded since the formation ofparticles decreased the solute concentration by an unknownamount. This procedure was repeated for several different initialconcentrations of ASA and temperatures as shown in Figure 4for a water concentration of 0 wt%, and then also employedfor different water concentrations, namely, 10, 25, 40, 55, and75 wt%, whereas the initial concentration of ASA was decreasedfor larger water concentrations. In total, the calibration setconsists of 740 samples in a temperature range from 20 to 60°C and a concentration range of ASA from 50 to 650 g/kg.

4.2. Calibration Model. In the calibration, a relation betweenthe measured spectra and the independent data, for example,ASA and water concentration, is sought. Since spectroscopicdata are highly correlated, the corresponding regression problemis ill-conditioned, and suitable approaches such as partial least-squares regression (PLSR) have to be applied.38,39 The wholefingerprint region, that is, from 650-1500 cm-1, resulting in443 variables, was used for building the calibration model.

Because of signal drifts induced by external effects, such asnonconstant temperature in the laboratory, a linear baselinecorrection was required to obtain constant results. Moreover,the data were mean-centered to remove unwanted variation inthe data, hence to improve model performance.4

In PLSR a small number (1-10) of variables, which are linearcombinations of the measured spectra, is used to regress theindependent data. These combinations are called latent variablesand their number has a great influence on the model perfor-mance. The optimal number of latent variables is found througha cross-validation procedure during which parts of the calibrationset are left out, a calibration model is built based on theremaining subset, and this model is used to estimate the setthat was left out.4,40 This procedure is repeated for differentsubsetsswe used the leave-one-out methodsand the root-mean-square error of cross-validation (RMSECV) is calculated. TheRMSECV is used to find the optimal number of latent variables,that is, the number where the value of RMSECV reaches aminimum, or, if no minimum exists, to find the number wherethe RMSECV does not change significantly if the number oflatent variables is increased. The best model performance wasfound for a PLSR model using four latent variables. Thecorresponding RMSECV was 5.20 g/kg for the ASA concentra-tion and 0.81 wt% (on a solute free basis) for the waterconcentration, which corresponds to 1.3% and 3.5% of the mean

concentrations as used in the calibration set, respectively. Thesefigures can be assumed to be the average error made inmeasuring the corresponding concentrations using the above-mentioned calibration.

4.3. Solubility Measurement. The calibration model de-scribed in the previous section was used to measure the solubilityof ASA in a temperature range from 25 to 50 °C and anantisolvent concentration from 0 to 75 wt% water (on a solutefree basis). A saturated solution, that is, a solution with an excessof ASA crystals, was slowly heated from 25 to 50 °C at a rateof 1 °C/h. At 37.5 °C the temperature was held constant for 3 hto verify equilibrium conditions during the course of theexperiment. This approach has been applied previously fornonisothermal solubility measurements.41 Figure 5a shows themeasured solubility as a function of time and the correspondingtemperature in pure ethanol. The procedure was repeated at 10,25, 40, 55, and 75 wt% water. The measured solubilities as afunction of solvent composition and temperature are given inFigure 6. It can be observed that the solubility strongly increaseswith temperature and that it exhibits a maximum at about 17wt% water. In Figure 6 a surface described by a 2D polynomialwhich is fitted on the experimental data is shown as well. Thispolynomial is used in the simulations to determine the solubilityat each required combination of solvent composition endtemperature. The coefficients of the polynomial can be foundin the appendix. It must be noted that the polynomial is validonly in the temperature range from 25 to 50 °C and for waterconcentrations between 0 and 75 wt%, and cannot be used forextrapolation.

Figure 4. Temperature and concentration of the samples used for IRcalibration. An undersaturated solution with known concentration iscooled down until nucleation is detected through FBRM monitoring(dashed line).

Figure 5. Measured solubility of acetylsalicylic acid (ASA) andcorresponding temperature in pure ethanol (a), and measured solubilityand corresponding measured water concentration at 25 °C as a functionof time (b).

Combined Cooling/Antisolvent Crystallization Process Crystal Growth & Design, Vol. 9, No. 2, 2009 1127

The solubility can also be measured by slowly adding theantisolvent at constant temperature. The procedure is shown inFigure 5b for an antisolvent addition rate of 20 mL/h and atemperature of 25 °C. Besides the ASA concentration also thewater concentration was measured. It can be observed that inthe very beginning of the experiment and after 2.5 h the waterconcentration stays approximately constant. At this point thepump sucked air, and thus almost no antisolvent was added andalso the ASA concentration changes only slowly. This experi-ment is plotted in Figure 6 as well and a good agreement withthe data measured at constant water content can be observed.The presented approach enables the continuous in situ measure-ment of the solubility for processes with varying temperaturesand solvent composition.

The solubility is also measured gravimetrically and the resultsare shown in Table 1. Each value in the table represents theaverage of three samples. In the gravimetric analysis a solutionwith an excess of ASA crystals was equilibrated in a temper-ature-controlled shaker, after 24 h the solution was filtered off,the solvent was evaporated completely and the mass of theremaining crystals was determined. The agreement with thevalues measured by ATR-FTIR is very good and the relative

error is below 3%. However, for high water content, hence lowASA concentration, the agreement is worse with deviations upto 12.5%. In this case the gravimetrically determined solubilityis always higher, most likely due to residues of water in thecrystal which could not be removed during drying.

5. Growth and Nucleation Kinetics

5.1. Parameter Estimation. The parameters in the kineticexpressions for nucleation and growth were estimated based onbatch desupersaturation experiments using a nonlinear optimiza-tion algorithm that minimizes the sum of squared residuals, SSR,between the experimental and simulated supersaturationvalues:42

SSR)∑i)1

Ne

∑j)1

Nd,i

(Si,jexp - Si,j

sim)2 (11)

where Ne is the number of experiments, Nd,i is the number ofdata points per experiment i, and Sexp and Ssim are theexperimental and simulated supersaturation, respectively. Theoptimization problem was solved using the lsqnonlin algorithmof the MATLAB optimization toolbox.

Approximate confidence intervals of the model parameters bcan be determined by calculating the sensitivity matrix, that is,the derivative of the model predictions with respect to the modelparameters, based on a linearized model in the vicinity of theestimated values.43 The sensitivity matrix can be used tocalculate the approximate covariance matrix of the estimatedparameters. The standard error sk of the kth model parameter isgiven by the square root of the kth diagonal element of thiscovariance matrix.43 The confidence intervals of the kthparameter are given by

bk ( tR,Vsk (12)

where tR,V is the value of the t-distribution for confidence intervalR in the case of V degrees of freedom. We used R ) 0.05 leadingto a 95% confidence interval.43

5.2. Growth Rate. The overall growth rate can be controlledby diffusion or surface integration. There is a wide variety ofgrowth mechanisms which take place in competition and leadto different growth regimes.44 In this work, the followingsemiempirical expression for the growth rate was employed:44

G) kG1 exp(-kG2

RT)(c∗(S- 1))kG3 (13)

where kG1, kG2, and kG3 are empirical parameters that have tobe determined from experiments. The effect of the antisolventconcentration on the growth rate given by eq 13 is implicitlyaccounted for by the dependence of solubility on it.

5.2.1. Seeded Batch Protocol. A series of seeded batchdesupersaturation experiments was performed at different tem-peratures and solvent compositions to determine the growthkinetics of ASA. In each of these experiments a certain amountof ASA crystals was dissolved completely in the ethanol-watersolution by heating it up. The solution was cooled to the desiredset-temperature. Then, the temperature was kept constant andthe seed crystals were added. The experimental method for theestimation of the parameters of the growth rate requires thatonly the added seed grow during the experiment and thatnucleation is absent.2 FBRM was used to ensure that nosignificant nucleation occurred during the experiment. Nucle-ation can be detected by an increasing number of counts in thesmall size range of the FBRM. If nucleation were detected,

Figure 6. Measured ASA solubility (black solid lined) as a functionof solvent composition and temperature. The gray surface is a 2Dpolynomial fitted on the experimental data, whose equation can be foundin the appendix.

Table 1. Solubility of Acetylsalicylic Acid c* in g/kg of SolventDetermined by ATR-FTIR Spectroscopy and Gravimetry for

Different Water Concentrations w and Temperatures T

w [wt%] T [°C] c* (IR) [g/kg] c* (grav) [g/kg] error [%]

0 25 237.9 238.5 0.250 35 378.4 375.4 0.800 50 648.4 663.8 2.3210 25 283.210 35 419.110 50 689.225 25 266.6 271.1 1.6625 35 395.3 404.6 2.3025 50 679.1 663.3 2.3840 25 186.240 35 304.840 50 570.855 25 87.955 35 149.555 50 377.075 25 16.2 17.5 7.4375 35 32.9 37.6 12.5075 50 101.6 110.1 7.72

1128 Crystal Growth & Design, Vol. 9, No. 2, 2009 Lindenberg et al.

which was the case at supersaturation levels higher than 1.4,the experiment would be discarded and not used for parameterestimation. All experiments that were used for parameterestimation are summarized in Table 2. The experiments wereall carried out with the same initial suspension density of ASAseed crystals, that is, 0.2 gseed/kgsolvent of the L seed shown inFigure 2. The value of the suspension density was chosen basedon a trade off between the need of having relatively slowdesupersaturation to allow for monitoring of the process, whichwould require low suspension densities, and that of avoidingnucleation, that would occur if it were too low. Each experimentwas carried out twice, and a good repeatability was observedin all cases, as illustrated in Figure 7a where, as a representativeexample, both experiments at 25 °C in pure ethanol are shownand the two curves overlap. The experimental protocol hasalready been applied successfully to determine the growthkinetics of several other compounds.2,8,45

5.2.2. Results. The parameter estimation technique presentedin Section 5.1 was used together with the population balancemodel and the experimental data of the seeded batch desuper-saturation experiments (Table 2) to determine the growth rateof ASA. The following growth rate parameters in eq 13 wereestimated:

kG1 ) (3.21( 0.18) × 10-4 m s-1 (14)

kG2 ) (2.58( 0.14) × 104 J mol-1 (15)

kG3 ) 1.00( 0.01 (16)

The parameters do not depend on the solvent composition inthis case, unlike that suggested by other authors.21,25,30 It wasfound that the proposed correlation describes the experimentaldata rather well and the use of more parameters was not justified.The R2 values of the growth experiments are given in Table 2.It can be observed that the agreement with the experiment isgood, as indicated by a high value of R2, and that no systematicerror exists, that is, all temperatures and solvent compositionsare represented equally well by the model.

A comparison of experimental and modeled desupersaturationprofiles is reported in Figure 7. The effect of temperature isshown in Figure 7a for pure ethanol. It can be observed thatthe initial supersaturations decrease with increasing temperaturesince the supersaturation had to be adjusted in order to avoidnucleation. Despite the lower initial supersaturations at highertemperatures the desupersaturation curve is steeper, thus indicat-ing a higher growth rate at elevated temperatures. The effect ofsolvent composition is shown in Figure 7b for a constanttemperature of 25 °C. It can be observed that the supersaturationis depleted faster at lower concentrations of water, hence higherconcentrations of ASA. This is only true for water concentrationsup to 17 wt%, that is, the maximum solubility. For lowerconcentrations of water the growth rate decreases again (notshown in the figure). The growth rate as a function of solventcomposition and supersaturation is shown in Figure 8 for atemperature of 45 °C. It can be observed that the maximumgrowth rate at fixed supersaturation is at the maximum of thesolubility, that is, at 17 wt% water. It is noticeable that thegrowth rate scales linearly with the supersaturation, thussuggesting a diffusion limited growth mechanism or a first-

Table 2. Experimental Conditions of the Growth Experimentsa

run T [°C] wH2O [wt%] S0 R2

1 25 0 1.15 0.982 35 0 1.12 0.933 45 0 1.08 0.914 25 10 1.20 0.975 25 25 1.20 0.986 25 40 1.21 0.957 25 55 1.21 0.92

a All experiments were carried out twice.

Figure 7. Growth experiments with ASA in pure ethanol and at differenttemperatures (a; runs 1 - 3), and at a constant temperature of 25 °Cand different solvent compositions (b; runs 5 - 7). Symbols: experi-mental data; Lines: simulation results. The good repeatability is shownexemplarily for pure ethanol and 25 °C (run 1).

Figure 8. Estimated growth rate of ASA as a function of solventcomposition and supersaturation for a temperature of 45 °C (surface).The dotted line below the surface shows the outline of the growth ratesurface at 35 °C.

Combined Cooling/Antisolvent Crystallization Process Crystal Growth & Design, Vol. 9, No. 2, 2009 1129

order reaction mechanism. Below the surface in Figure 8 theoutline of the growth rate surface at 35 °C is plotted as well.This confirms that the growth rate depends strongly on tem-perature as accounted for by the Arrhenius-type temperaturedependence in eq 13.

5.3. Nucleation Rate. There are different nucleation mech-anisms for the formation of crystals in solution: homogeneousand heterogeneous primary nucleation and secondary nucleation.In our experiments it was observed that nuclei form in clearsolution; thus, a primary nucleation mechanism was assumed.Even though secondary nucleation could not be ruled out apriori, there was no experimental evidence that this mechanismcontributed significantly to the formation of new crystals. Thefollowing empirical nucleation rate expression was used in thisstudy:

J) kJ1 exp(- kJ2

RT) exp(- kJ3

ln2 S) (17)

It must be noted that eq 17 does not include any explicit effectof the solvent composition; however, as shown below, it candescribe accurately the experimental data. A more physicallymeaningful expression which accounts for the effect of inter-facial energy on the nucleation kinetics44,46 was also tested, butdid not give satisfactory results.

5.3.1. Unseeded Batch Protocol. A series of unseeded batchdesupersaturation experiments has been performed at differenttemperatures and solvent compositions to determine the nucle-ation kinetics of ASA. In each of these experiments a certainamount of ASA crystals was loaded into an ethanol-watersolution. The solution was heated up till the crystals weredissolved completely. Then, the solution was crash-cooled (ata rate of 50 K/h) to the desired set-temperature and from thattime on the temperature was kept constant. The process wasmonitored using ATR-FTIR and FBRM. The operating condi-tions for the nucleation experiments which were used forparameter estimation are given in Table 3. Each experiment wascarried out twice. The repeatability was satisfactory but worsethan in the growth experiments, most likely due to the stochasticnature of nucleation and to different amounts of microscopicdust, which is difficult to control and might promote heteroge-neous nucleation. The effect of the stirring rate on nucleationkinetics, which was observed elsewhere,33 was not investigatedhere, and the stirring rate was kept constant at 250 rpm in allexperiments to guarantee similar mixing conditions.

5.3.2. Results. The protocol for the estimation of theparameters in the nucleation rate expression is similar to thatfor the growth rate, but it is based on the nucleation experimentssummarized in Table 3. With reference to eq 17, the followingnucleation rate parameters were estimated:

kJ1 ) (1.15( 0.51) × 1021 m-3 s-1 (18)

kJ2 ) (7.67( 0.11) × 104 J mol-1 (19)

kJ3 ) 0.16( 0.01 (20)

The R2 of the nucleation experiments are given in Table 3. Itcan be observed that the agreement between model andexperiment is rather good, and that no systematic error regardingthe effect of temperature and antisolvent concentration isobserved. However, the values of R2 are a bit lower than forthe growth experiments. It is worth noting that although theconfidence interval on the parameter kJ1 looks rather large,namely, ( 44%, the effect of this uncertainty on the course ofnucleation is minor as illustrated in Figure 9a where, as arepresentative example, the experiment at 25 °C in pure ethanolis simulated with the estimated value of the parameter as wellas with the two values at the upper and lower end of theconfidence interval and the three curves do not differ much.

The experimental results of the unseeded batch desupersatu-ration experiments with pure ethanol as solvent are given inFigure 9. In each experiment, first, the supersaturation increasesdue to cooling (Figure 9a). Then, at a certain level, thesupersaturation decreases rapidly because of particle formationand growth. At that point the temperature is typically abovethe final set-point given in Table 3, and the solution is cooledfurther till it reaches the set-temperature (Figure 9b). In someexperiments the supersaturation reaches a plateau after the firstrapid decrease. The reason for this behavior is that the solutionheats up during particle formation and growth, since these areexothermic phenomena. Then, the solution is cooled again

Table 3. Experimental Conditions of the Nucleation Experimentsa

run T [°C] wH2O [wt%] S0 R2

1 25 0 1.52 0.922 35 0 1.51 0.873 45 0 1.22 0.774 25 25 1.40 0.795 25 25 1.75 0.816 25 25 2.00 0.867 35 25 1.48 0.968 25 40 1.50 0.769 35 40 1.50 0.7510 25 55 1.50 0.8111 25 55 1.95 0.8012 35 55 1.51 0.88

a All experiments were carried out twice.

Figure 9. Nucleation experiments with ASA in pure ethanol and atdifferent temperature levels (runs 1-3). The course of the experimental(symbols) and modeled (lines) supersaturation is shown in (a) and ofthe temperature in (b). The dotted lines show exemplarily the effect ofthe uncertainty in the nucleation parameter kJ1 on the simulation resultsfor pure ethanol and 25 °C (run1).

1130 Crystal Growth & Design, Vol. 9, No. 2, 2009 Lindenberg et al.

through the action of the thermostat, whereas some overshootingof the temperature can be observed. This course of thetemperature is also reflected by the experimental course of thesupersaturation. The maximum supersaturation that is reachedin an experiment decreases with increasing temperature due toan increasing nucleation and growth rate. It must be noted thatfor each experiment the temperature profile is measured, thenimported and used in the corresponding simulation as input.The simulation results are plotted in Figure 9a as well and arather good agreement with the experimental data can beobserved. However, there is a discrepancy between simulationresults and experimental data in the phase of the experimentwhen the solution heats up due to nucleation and growth. Thisdiscrepancy is observed in all experiments, sometimes to a largerand sometimes to a smaller extent, and occurs most likely dueto the temperature effect, which is not captured completely bythe model. A study with different initial concentrations of ASA(runs 4-6 in Table 3) revealed that the maximum attainablesupersaturation is not affected thereby. If the initial ASAconcentration is larger, the nucleation just starts at a highertemperature.

A comparison of the experimental and simulated desuper-saturation profiles for different solvent compositions is illustratedin Figure 10. It must be noted that the data of the differentexperiments are shifted in time to allow for a better readability.The course of the supersaturation and the maximum supersatu-ration value are rather similar for the different solvent composi-tions. Thus, as anticipated above, no direct effect of solventcomposition on the nucleation rate was included in the model.Also for these experiments the agreement with the model isrelatively good, whereas the same discrepancy as for theexperiments described in the previous paragraph was observed.

6. Process Optimization

The process model with the kinetic expressions which havebeen presented in sections 3 and 5 is used in this section tooptimize the process, that is, to calculate an optimal trajectoryfor the cooling and antisolvent addition. Such a trajectory isthen implemented experimentally. A crystallization process canbe optimized in different respects, that is, targeting optimalparticle size and particle size distribution, shape, purity, residualsolvent content, yield and process time. In this study, a seededsemibatch process is considered where the supersaturation isgenerated by cooling and antisolvent addition, whereas theantisolvent is added to increase the yield. At which antisolventconcentration the maximum yield is obtained depends on thesolubility behavior of the specific substance; the maximum yieldis in fact obtained at the final conditions where the mass gaindue to a change in solubility is the largest as compared to theloss due to dilution. In this study, the yield was fixed bychoosing a specific and constant value of the final antisolventconcentration. The solution was cooled from 35 to 25 °C, andthe initial and final antisolvent concentrations were 25 and 60wt%, respectively. The theoretical crystal yield obtained bycooling from 35 to 25 °C at a water concentration of 25 wt%was 128.9 g/kg, the yield obtained by changing the waterconcentration from 25 to 60 wt% at a temperature of 25 °Cwas 143.0 g/kg, and hence the total theoretical yield was 271.9g/kg, that is, 110% more than by cooling only.

Particle shape, purity, and residual solvent content were alsonot used as parameters to be optimized, since for the first nocriterion was available, whereas the latter two features did notseem to represent a problem. Therefore, the process wasoptimized with respect to time and particle size distribution.While the multiobjective optimization of cooling22 or antisolventcrystallization21 and the single-objective optimization of thecombined cooling/antisolvent crystallization32 have been re-ported previously, we focus in this study on the multiobjectiveoptimization of the combined cooling/antisolvent crystallization.

6.1. Optimal Trajectory. The objective for the processoptimization was the minimization of process time and of thedifference between the calculated and a specified optimal PSD.The following multiobjective optimization problem was for-mulated where the cooling and the antisolvent flow-rate profileare the manipulated variables:

{ minimize tp

∫0

tp J dt

subject todTdte 30 K/h

Qe 1.29 g/s

(21)

The following initial and final conditions were applied: S0 )1.05, T0 ) 35 °C, w0 ) 25 wt%, Tend ) 25 °C, wend ) 60 wt%.The initial seed mass was 6.25 × 10-2 g per kg of solvent andthe S seeds shown in Figure 2 were used. The cooling and theantisolvent profile are discretized in time in six equidistant steps.This results in 13 parameters to be optimized: two process timesfor cooling and antisolvent addition, respectively, five temper-atures (initial and final temperature were fixed), and sixantisolvent flow-rates. The particle size distribution was notdirectly optimized, but the number of crystals formed bynucleation was minimized as to eq 21. In a seeded process withnegligible nucleation and size independent growth the final PSDis just the initial one shifted to the right along the crystal sizecoordinate of an extent that fulfills the material balance, thatis, it is consistent with the mass being deposited on the seed

Figure 10. Nucleation experiments with ASA at different solventcompositions and at constant final temperature of 25 °C (runs 1, 5,11). The course of the experimental (symbols) and modeled (lines)supersaturation is shown in (a) and of the temperature in (b).

Combined Cooling/Antisolvent Crystallization Process Crystal Growth & Design, Vol. 9, No. 2, 2009 1131

crystals by crystal growth. The constraints in eq 21 aredetermined by the maximum cooling and feed rates in ourexperimental setup.

The multiobjective optimization problem was solved bycombining the two objective functions into a single oneconsisting of a weighted average.47 The optimization problemwas solved using the fmincon algorithm of the MATLABoptimization toolbox. By varying the weights the Pareto-optimalsolutions, which are illustrated in Figure 11, could be calculated.It can be observed that the two objectives are conflicting, thatis, short process time results in a larger nucleation rate and viceversa. The following Pareto-optimal point was chosen and usedfor further studies:

tp ) 1298 s (22)

∫0

tp J dt) 2.60 × 104 m-3 (23)

The optimized trajectory is compared with two other processalternatives, namely, an addition of the antisolvent in thebeginning followed by cooling (case 1) and cooling followedby the addition of the antisolvent in the end (case 2), which arethe extreme cases of the combined cooling/antisolvent crystal-lization. In these cases the cooling and the antisolvent additionare decoupled. These process strategies are interesting as theyare often applied in industry. Initial conditions (concentrationsof ASA and water, temperature, seed size, and mass) and finalconditions (water concentration and temperature) were the sameas for the optimal trajectory. The optimized trajectory (hereaftercalled case 3) and the trajectories of case 1 and 2 are shown inFigure 12. In cases 1 and 2 a maximum cooling rate of 30 K/his applied, either after or before the antisolvent addition (Figure12a). In the optimized case, first, the solution is cooled at amaximum rate, and then the cooling rate is lowered in the secondinterval, till it increases again to the maximum rate. Theantisolvent addition is shown in Figure 12b. In cases 1 and 2,the antisolvent is added at the maximum rate, that is, 77.4 g/min,and for a duration 108.5 s. In the optimized case, the antisolventflow-rate decreases in the second interval and increases after-ward gradually till the end of the process.

The process trajectories as a function of temperature andsolvent composition are plotted together with the solubility inFigure 13. It can be observed that the optimized trajectoryfollows approximately the steepest descent on the solubilitysurface and that the concentration is always close to thesolubility, and thus the supersaturation is low. In case 1 theconcentration is much larger than the solubility in the beginningof the process, while in case 2 it is the largest in the end. Inboth cases the highest supersaturation is attained during the

Figure 11. Pareto-optimal set of the two-objective optimization problem(solid line); the region below the Pareto set is not feasible. The dottedline is the boundary due to the maximum cooling rate. The open circlescorrespond to process alternatives where the antisolvent is added atthe beginning (case 1) or at the end (case 2). The filled circle gives thePareto-optimal set that was implemented (case 3). The dashed linecorresponds to a strategy of industrial relevance in which constantcooling and antisolvent addition rates are applied. The inset showsschematically the corresponding cooling and flow rate profiles.

Figure 12. Cooling profile (a) and antisolvent flow rate (b) as a functionof time. Dashed line: antisolvent addition in the beginning (case 1);dotted line: antisolvent addition in the end (case 2); solid line: optimizedtrajectory (case 3).

Figure 13. Concentration trajectory of the three process alternativesas a function of water concentration and temperature. Dashed line:antisolvent addition in the beginning (case 1); dotted line: antisolventaddition in the end (case 2); solid line: optimized trajectory (case 3);surface: solubility.

1132 Crystal Growth & Design, Vol. 9, No. 2, 2009 Lindenberg et al.

antisolvent addition. The course of the supersaturation as afunction of process time is plotted in Figure 14a. The super-saturation stays at a low value of about 1.1 and is relativelyconstant throughout the whole process in the optimal case 3. Asupersaturation as high as 1.9 is attained if the antisolvent isadded in the beginning (case 1), and a slightly lower maximumsupersaturation of 1.7 is reached if the antisolvent is added inthe end. If the antisolvent is added in the end the available crystalsurface is larger as compared to the case where it is added inthe beginning; thus, the rate of concentration decrease by growthis larger resulting in a lower supersaturation. The nucleationrates corresponding to the three process alternatives are plotted

in Figure 14b. It can be observed that the nucleation rate incases 1 and 2 is by at least 6 orders of magnitude larger than inthe optimized case. The effect of this higher nucleation rate onthe simulated PSDs is illustrated in Figure 15. In case 3 aunimodal distribution with large crystals in the range 350-400µm is obtained. This case is identical to a simulation with growthonly and without nucleation. In case 1 the particles are muchsmaller and the PSD is bimodal. The first peak stems from thenucleation event in the beginning of the process and the secondfrom the grown seed crystals. In case 2 the PSD is also bimodal;however, the grown seed crystals are almost as big as in case3. The first peak on the left-hand side of the plot stems fromthe crystals nucleated at the end of the process during theantisolvent addition.

The process alternatives 1 and 2 are shown as open circlesin Figure 11. It can be observed that for the same process timethe number of nuclei formed in case 1 and 2 is about 5-6 ordersof magnitude larger than in the optimal case 3. However, theoptimal antisolvent addition profile (Figure 12b) is relativelycomplicated and might not be applicable in industrial processes.Therefore, a strategy with constant cooling and antisolventaddition rate was considered as well. In this case cooling andantisolvent feeding time are equal and the same initial and finalconditions in terms of concentration and temperature as for theoptimal case were applied. The corresponding cooling andantisolvent addition rates are plotted schematically in the insetof Figure 11. The number of nuclei formed as a function ofprocess time is shown as well, and it can be observed that forthe same process time as in cases 1-3, that is, around 1300 s,the number of nuclei is 3 orders of magnitude larger than in theoptimized case 3, and about 2-3 orders of magnitude smallerthan in cases 1 and 2. Thus, we conclude that for industrialpurposes an addition of the antisolvent over the whole cooling

Figure 14. Supersaturation (a) and nucleation rate (b) as a function oftime. Dashed line: antisolvent addition in the beginning (case 1); dottedline: antisolvent addition in the end (case 2); solid line: optimizedtrajectory (case 3).

Figure 15. Final simulated particle size distributions of the three processalternatives. Dashed line: antisolvent addition in the beginning (case1); dotted line: antisolvent addition in the end (case 2); solid line:optimized trajectory (case 3).

Figure 16. Experimental and modeled ASA concentration and FBRMcounts as a function of time for the three process alternatives: antisolventaddition in the beginning (a), antisolvent addition in the end (b) andoptimized trajectory (c). Circles: experimental data; black line: simula-tion results; grey line: FBRM data.

Combined Cooling/Antisolvent Crystallization Process Crystal Growth & Design, Vol. 9, No. 2, 2009 1133

(process) time should be favored as compared to processeswhere the antisolvent is added all at once at the beginning orend of the cooling ramp.

6.2. Implementation and Results. A comparison of experi-mental and simulated concentration of ASA is shown in Figure

16. A good agreement is observed, which is remarkable sincein this case the model is used in a fully predictive mode. Besidesthe concentrations, the counts of the FBRM are plotted. It canbe observed that in cases 1 and 2 the addition of the antisolventcoincides with a rapid decrease of the ASA concentration anda sharp increase of the FBRM counts. The comparison ofFigures 14b and 16 shows that the increase in FBRM counts isrelated to the nucleation of ASA crystals. This confirms thatthe FBRM can serve as a valuable tool to detect nucleationevents. In case 3 no significant increase of FBRM counts canbe observed since the nucleation rate is very low. However,there is a slow but steady increase of the FBRM counts due tocrystal growth since it is more likely for the FBRM to detectlarger particles.17 It must also be noted that in cases 1 and 2temperature effects were observed after the fast antisolventaddition owing to the mixing enthalpy and heat of crystallization.However, these effects were accounted for by the model sincethe actual temperature profiles from the experiments weremonitored and used in the simulations.

The measured number distributions of the final particles areshown in Figure 17a. A comparison with Figure 15 shows thatin all cases the experimental PSDs are much broader than thesimulated ones, but that the general trend is well predicted. Ifthe antisolvent is added at the beginning (case 1), the finalparticles are much smaller due to nucleation. Adding theantisolvent at the end (case 2) leads to a bimodal PSD, whereasapplying the optimized trajectory (case 3) yields a unimodaldistribution. The measured mass distributions are bimodal incase 1, whereas unimodal distributions are observed in cases 2and 3 (Figure 17b). However, the mass distribution of case 2 isbroader and shifted toward larger particle sizes as compared tothat of the optimized case 3, mainly due to a larger extent ofagglomeration in case 2.

The broader PSDs in the experiments as compared to thesimulations might be attributed to secondary effects, such asagglomeration, as can be seen on the microscope pictures inFigure 18. The final crystals are elongated in case 1, whereasthey are more compact in cases 2 and 3. The difference mightbe explained by the different solvent compositions. In case 1the solvent composition is about 60 wt% water almost through-

Figure 17. Final measured number (a) and mass densities (b). Greycircles: antisolvent addition in the beginning (case 1); open black circles:antisolvent addition in the end (case 2); filled black circles: optimizedtrajectory (case 3).

Figure 18. Microscope pictures of the final ASA crystals of case 1 (antisolvent addition in the beginning, left), case 2 (antisolvent addition in theend, middle) and case 3 (optimized trajectory, right). All pictures have the same magnification.

Table 4. Parameters of the 2D Polynomial Describing the Solubility of ASA (see appendix)

-7.87 × 102 1.99 × 102 -1.77 × 101 8.15 × 10-1 -1.97 × 10-2 2.41 × 10-4 -1.18 × 10-6

-6.73 × 101 1.13 × 101 -5.71 × 10-1 1.21 × 10-2 -9.85 × 10-5 1.23 × 10-7

-1.30 -4.92 × 10-2 5.63 × 10-3 -1.40 × 10-4 1.08 × 10-6

7.17 × 10-2 -2.50 × 10-3 3.57 × 10-5 -1.23 × 10-7

-1.03 × 10-3 1.60 × 10-5 -1.65 × 10-7

9.41 × 10-6 -1.26 × 10-8

-4.26 × 10-8

1134 Crystal Growth & Design, Vol. 9, No. 2, 2009 Lindenberg et al.

out the whole process. The lowest average water concentrationoccurs in case 2 since the antisolvent is added at the end of theprocess. The optimal case is such that the antisolvent concentra-tion varies continuously during crystallization as shown in Figure13. The comparison of the final crystals from the different runsshows that the solvent composition affects the ratio of the growthrates of different crystal faces, that is, the shape of the crystalchanges and the crystals become more elongated at higher waterconcentrations. This effect was also observed in the nucleationexperiments presented in section 5.3. In case 2 a large numberof small crystals can be observed which have been formed dueto the high supersaturation at the end of the process, and thelarge particles are slightly agglomerated. A mild agglomerationof crystals can also be observed in case 3.

7. Conclusions

The optimized combined cooling/antisolvent crystallizationof acetylsalicylic acid from ethanol-water solutions has beenstudied, showing that process performance can be improved ascompared to conventional process alternatives which are com-monly applied in industry, that is, the addition of the antisolventeither at the beginning or at the end of the cooling process. Inparticular, a unimodal particle size distribution at a minimumprocess time has been obtained. The advantage of the combinedprocess was shown through both modeling and experiments.The model-based optimization required the knowledge ofsolubility, nucleation and growth kinetics of acetylsalicylic acid,which were measured using process analytical technologies, suchas ATR-FTIR and FBRM, and by applying previously develop-ment experimental protocols. A process model based on a PBEwas used for a multiobjective process optimization. Morespecifically, the resulting Pareto-optimal solutions showed thatthe two objectives are conflicting, that is, short process timeresults in a larger nucleation rate and vice versa, and yieldedthe corresponding optimal cooling and antisolvent flow-rateprofiles. However, these profiles are complex and difficult toimplement in industrial processes. Therefore, a strategy withconstant cooling and antisolvent addition, where cooling andfeeding time were equal, was considered. Although this strategyis suboptimal, it outperformed processes where the antisolventis added all at once at the beginning or at the end of the coolingramp, and hence it should be favored. To the best of ourknowledge, for the first time, multiobjective optimization wasapplied to a combined cooling/antisolvent crystallization, andsolubilities as well as nucleation and growth kinetics ofacetylsalicylic acid were determined as a function of temperatureand solvent composition.

AppendixSolubility of Acetylsalicylic Acid. A 2D polynomial of sixth

order giving 28 coefficients was used to fit the experimentalsolubility data of acetylsalicylic acid in a least-squares sense. Thepolynomial is arranged as follows:

c(θ, w)) p0,0θ0w0 + p1,0θ

1w0 + p0,1θ0w1 + p2,0θ

2w0 +

p1,1θ1w1 + p0,2θ

0w2 + · · · + p0,6θ0w6 (24)

where c is the solubility in g per kg on a solute free basis, θ is thetemperature in °C and w is the water concentration in wt% on asolute free basis. The coefficients pi,j are the elements of the ithcolumn and jth row given in Table 4.

Notation

b, vector of parameters [-]c, concentration [mol m-3]

c*, solubility [mol m-3]G, growth rate [m s-1]J, nucleation rate [m-3 s-1]k, empirical parameter [-]kv, volume shape factor [-]L, particle size [m]n, number density [m-4]N, number of particles [m-3]M, molecular mass [g mol-1]Q, antisolvent flow-rate [m3 s-1]R, ideal gas constant [J mol-1 K-1]s, standard error [-]S, supersaturation [-]t, time [s]T, temperature [K]V, reactor volume [m3]w, antisolvent concentration [wt% on a solute free basis]x, size of a pivot [m]R, weighting factor [-]γ, interfacial energy [J m-2]Fc, crystal density [kg m-3]

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