Micronization of Pharmaceutical Substances by Rapid Expansion ofSupercritical Solutions (RESS): Experiments and Modeling

Michael T¸rk*, Britta Helfgen**, Peter Hils*, Ralph Lietzow*, Karlheinz Schaber*(Received: 9 April 2002; accepted: 19 August 2002)

Abstract

An increasing number of newly developed pharma-ceutical substances are poorly soluble in both aqueousand organic media. Thus, the application of oral orinjectable drugs is often limited by its low bioavail-ability. An alternative and promising method toimprove the bioavailability of pharmaceutical agentsis the production of nanoscale particles by the rapidexpansion of supercritical solutions (RESS). Ourresearch is aimed towards an improved understandingof the underlying physical phenomena of the relation-ship between the process conditions and the particlecharacteristics. Therefore, experimental investigationsand numerical simulations were performed. RESS

experiments with the pharmaceutical substances �-sitosterol, griseofulvin, and ibuprofen led to particlesizes in the range of 240� 80 nm. In addition, as onestep towards intravenous application of poorly solubledrugs, �-sitosterol was used to produce aqueoussuspensions of a water-insoluble drug with a particlesize smaller than or equal to those produced by RESSinto air. RESS modeling is focused on the flow throughthe nozzle, the supersonic free jet, the mach shock, andparticle growth in the expansion unit. The comparisonwith experimental results shows a good agreement inthe general trends but does not match exactly themeasured mean particle sizes.

Keywords: bioavailability, griseofulvin, ibuprofen, nanoparticles, RESS, �-sitosterol, supercritical fluids

1 Introduction

An increasing number of newly developed pharmaceut-ical substances are poorly soluble in both aqueous andorganic media. Thus, the application of oral or injectabledrugs is often limited by its insolubility, and thebioavailability of the drug is low compared to the initialdose and its toxicity threshold is close to the therapeuticdosage. The low bioavailability of such drugs can beimproved by the reduction of the particle size [1]. In thepharmaceutical industry, several conventional tech-niques (e.g., milling and grinding, spray drying, freezedrying) havebeenutilized for particle size reduction.Thedisadvantages of using these techniques are thermal andchemical degradation of the product and broad particle

size distributions [2]. An alternative and promisingmethod for the production of drug nanoparticles is theRapid Expansion of Supercritical Solutions (RESS)[3, 4]. This process enables the micronization ofthermally labile materials and the formation of particlesof less than 500 nm in diameter [5 ± 8]. The RESS processutilizes the high solvating power of supercritical fluids[9]. After loading the supercritical fluid with the solute,an extremely fast phase change from the supercritical tothe gas-like state takes place during the expansion in thesupersonic free jet. This leads to high supersaturationand, subsequently, to particle formation. Since thesolvent is a dilute gas after expansion, the RESS processoffers a solvent-free final product [10].The first goal of this work was to explore the processconditions for the formation of nanoscale particles byRESS. Among others, experiments with griseofulvin, �-sitosterol and ibuprofen were performed. Griseofulvin,an oral antifungal drug, is poorly soluble in aqueousmedia [11]. Hence, in gastrointestinal fluids, griseofulvinhas a low dissolution rate resulting in a poor absorptionfrom the gastrointestinal tract [12]. �-Sitosterol ispresent in foods and reduces intestinal cholesterol

¹ 2002 WILEY-VCH Verlag GmbH& Co. KGaA, Weinheim 0934-0866/02/0511-0327 $ 17.50+.50/0

* Priv.-Doz. Dr.-Ing. M. T¸rk (corresponding author), Dipl.-Ing. P. Hils, Dipl.-Ing. R. Lietzow, Prof. Dr.-Ing. K. Schaber,Universit‰t Karlsruhe (TH), Institut f¸r Technische Thermo-dynamik und K‰ltetechnik, Engler-Bunte-Ring 21, 76131Karlsruhe (Germany).E-mail: mt@ciw.uni-karlsruhe.de

** Dr.-Ing. B. Helfgen, Degussa AG, VT-P, RodenbacherChaussee 4, 63457 Hanau-Wolfgang (Germany).

327Part. Part. Syst. Charact. 19 (2002) 327 ± 335

absorption. Structurally similar to cholesterol, �-sitos-terol can be therapeutically useful for cholesterol low-ering in human blood [13]. Ibuprofen is a chiral non-steroidal antiinflammatory drug and is most oftenprescribed to treat fever, arthritis, menstrual symptoms,and pain [14].In addition, as one step towards intravenous applicationof poorly soluble drugs, stable suspensions of nanoscaleparticles of �-sitosterol were produced by rapid expan-sion of a supercritical mixture through a capillary nozzleinto aqueous solutions. In these experiments the non-ionic surfactant Tween-80 was chosen to impede growthand agglomeration of the �-Sitosterol particles anddynamic light-scattering (DLS) measurements wereconducted to determine the size of the stabilizedparticles.Beside the RESS experiments, a model was developedwhich includes the expansion process inside the capillarynozzle as well as the postexpansion conditions withparticle growth inside the expansion chamber. Modelingis focused on the flow through the nozzle, the supersonicfree jet, the mach shock, and particle growth in theexpansion unit. Thus, a useful tool for the understandingand optimization of the RESS process has been created.

2 Fundamentals

Supercritical fluids are gases pressurized and heatedabove their critical pressure and temperature. Thecritical properties of some common supercritical fluidsused as solvents in RESS are listed in Table 1. The mostinteresting applications of supercritical fluids occur inthe range 1� p/pC� 3 and 1�T/TC� 1.2. In the regionslightly above the fluids× critical point, the thermophys-ical properties exhibit a huge variation with respect topressure and/or temperature. Supercritical fluids oftenexhibit a combination of gas-like and liquid-like proper-ties, for example, high transport coefficients and highdensity [15].In Figure 1, the liquid-vapor saturation curve (pS,�L,�V)and the density in the single-phase (p, �, T) region isplotted against the pressure at different temperaturesfor pure CO2. Along the saturation curve liquid and

vapor phase are in equilibrium. The discontinuoustransition from vapor to liquid phase ends at the criticalpoint and the coexisting densities become equal(�L��V��C). Beyond the critical point, a low densitygas can be compressed into the supercritical statewithout a phase transition. Near the critical point, thedensity of the fluid can be manipulated with a smallpressure and/or temperature change. At a temperatureof 308 K, a pressure increase from 7 to 8 MPa leads tomore than a double increase in density (5.2 to 12 mol/dm3). On the other hand, at 8 MPa a temperaturedecrease from 308 to 305 K leads to a density of14.8 mol/dm3.Many applications of supercritical fluids involve dilutemixtures composed of a supercritical fluid and a non-volatile solute. Figure 2 shows the solubility, y*, ofnaphthalene in CO2 at two supercritical temperatures.The mixture shows the typical phase behavior of dilutesolutions of nonvolatile solutes in supercritical solvents.Typically, the solubility of the solute in the solvent has aninitial isothermal decrease, followed by a strong increasein solubility near the solvent×s critical pressure andattainment of a solubility limit in the range above thetwo- to threefold of the solvent×s critical pressure. Atintermediate pressures where the solubility decreaseswith increasing temperature, a retrograde behavior canbe observed. This example shows that in dilute mixtures,solubility is a function of both, the solute×s vaporpressure, (pS), which depends only on temperature, andthe nature of solute-solvent interactions. The latter is thereason for the retrograde behavior near the solvent×scritical point and the various solubility of a solute indifferent solvents.

Table 1: Critical properties of some common supercritical fluidsused as solvents.

Solvent TC

[K]PC

[MPa]�C

[mol ¥ dm�3]

Carbon dioxide (CO2) 304.2 7.38 10.53Trifluoromethane (CHF3) 299.0 4.80 7.42Propane (C3H8) 369.8 4.24 4.93

Fig. 1: Density ± pressure projection of the phase diagram ofpure CO2. The density was calculated by means of the extendedgeneralized Bender equation of state [16].

328 Part. Part. Syst. Charact. 19 (2002) 327 ± 335

3 Experimental Setup

Figure 3 show schematically the apparatus used for theRESS experiments. The RESS apparatus is designed forexperiments in the temperature range from 300 to 600 Kand pressures up to 60 MPa [18, 19].The gaseous solvent is cleaned in a column, condensed,subcooled, and pressurized to the desired pressure with adiaphragmpump, and themass flow rate ismeasuredby amass flowmeter. In a controlled water bath, the super-critical solvent is heated to extraction temperature in apreheater and thenpassed through an extraction column,which is packed with the solute. Then, the saturatedsupercritical solution flows through a heated tube into athermostated high-pressure vessel to reduce the flowvelocity and to adjust thepreexpansion temperature.Thepreexpansion temperature (T0) ismeasuredwith aPt-100thermometer and the preexpansion pressure (p0) with adigital pressure gauge. The supercritical mixture isexpanded through a thermostated capillary nozzle withan inner diameter of 50 �m and a length of 50�m alwaysat atmospheric conditions (300 K, 0.1 MPa). In contrastto the usual off-line examination techniques, the pre-cipitated particles are measured in the expansionchamber (V� 22 dm3, DI� 0.19 m) on-line and in situwith the 3-wavelength-extinction measurement techni-que (3-WEM). In addition to the on-line measurements,samples for SEM examination (scanning electron micro-graph) can be taken at a distance of 300 mm to the nozzleexit directly behind the second 3-WEM probe. A moredetailed description of the apparatus and the measure-ment technique can be found elsewhere [18 ± 20].

4 Formation of Nanoparticles by RESS

First, the influence of the preexpansion conditions (p0

and T0) on the particle size of benzoic acid, cholesterol,ibuprofen, �-sitosterol and griseofulvin was investigated.Due to the relatively low solubility of griseofulvin insupercritical CO2, (y*� 1.6� 10�5 at 323 K and 20 MPa[21]) the polar solvent CHF3 (y*� 7.5� 10�5 at 323 Kand 20 MPa [21]) was used. In case of the other solutes,supercritical CO2 was used as solvent.In the case of ibuprofen, the RESS experiments werecarried out at the preexpansion temperature 308 K and318 K and the preexpansion pressure 20 MPa and13 MPa. All the other experiments were performed atpreexpansion temperatures between 348 K and 418 Kand preexpansion pressures between 20 MPa and30 MPa. These preexpansion conditions were chosen toprevent undesirable particle precipitation inside theRESS apparatus. To avoid clogging, the nozzle temper-ature was equal to or slightly higher (�10 K) than therespective preexpansion temperature. Table 2 showstypical particles sizes (�number volume averageddiameters dNV) obtained in the RESS experiments.

Table 2: Particle size (�number volume averaged diametersdNV) of different solutes obtained from RESS experimentsperformed at various preexpansion conditions.

Solute Solvent Particle size (dNV)[nm]

Benzoic acid CO2 208 ± 461Cholesterol CO2 192 ± 253�-Sitosterol CO2 166 ± 219Ibuprofen CO2 183 ± 326Griseofulvin CHF3 193 ± 323

Fig. 3: A, solvent; B, column with molecular sieve; C, coolingdevice; D, pump; E, thermostated extractor; F, heating; G, cooling;H, thermostated vessel and capillary nozzle; I, expansion cham-ber; J, 3-WEM; K sampling for SEM micrographs; L, vent.

Fig. 2: Solubility of naphthalene in supercritical CO2 at twosupercritical temperatures. The solubility was calculated by meansof a modified Peng-Robinson equation of state [17].

329Part. Part. Syst. Charact. 19 (2002) 327 ± 335

The particle size of benzoic acid varies from 208 to461 nmdepending on the preexpansion conditions.Both,lowering the preexpansion temperature and increasingthe preexpansion pressure lead to smaller particles [7, 8].A detailed discussion of this result will be given in themodeling section of this paper.TheRESSexperiments performedwithCO2/cholesterol,CO2/�-sitosterol, CO2/ibuprofen and CHF3/griseofulvinlead to particle sizes in the range between 166 and326 nm. In contrast to Benzoic acid, no clear dependenceon the different preexpansion conditions on the particlesize of the various solutes was observed. The discrepancybetween benzoic acid, on the one hand, and cholesterol,griseofulvin, ibuprofen and �-sitosterol, on the otherhand, can be explained by the higher solubility of benzoicacid in CO2 in comparison to the other solutes in therespective supercritical solvent [22].At typical extraction conditions (20 MPa and 323 K), thesolubility of griseofulvin inCHF3 aswell as of�-sitosterolin CO2 is about an order of magnitude lower than of

benzoic acid in CO2 [8]. In accordance with the classicalnucleation theory, this lower solubility leads to notice-able lower number concentrations and, therefore, small-er particles [22, 23].As mentioned above, SEM probes can be taken out ofthe expansion chamber. Figure 4 shows typical examplesof the obtained particles. Usually, the agglomeratedparticles consist of primary particles with a particle sizebetween 50 nm and 150 nm. In addition, the upper SEMphotographs show a spongy structure with a high surfacearea. Due to the increase in the surface area thedissolution rate of the drug should be increased, too. Inan earlier investigation, the improvement of the bio-availability of the RESS-produced griseofulvin has beenverified by dissolution experiments according to theStricker model [8, 12]. The dissolution rate of the RESS-produced griseofulvin is about twice as high as thecommon micronized material. A more detailed descrip-tion of the characterization methods can be found in theliterature [12].

Fig. 4: �-Sitosterol (above left), griseofulvin (above right), ibuprofen (below left), and benzoic acid (below right [19]) particlesproduced by RESS.

330 Part. Part. Syst. Charact. 19 (2002) 327 ± 335

5 Stabilization of Nanoparticles

It was shown in previous publications that a higherdilution of the particles in the expansion chamberinhibits postexpansion particle growth [7, 8, 24]. Fromthe theoretical calculations presented in chapter 6follows that it should be possible to form particlessmaller than 50 nm. The difficulty to achieve thetheoretical particle size is likely due to growth andagglomeration during collisions in the supersonic freejet.However, it was shown in sporadic investigations thata promising method to prevent particle growth is to spraythe supercritical solution directly into an aqueoussurfactant solution [8, 24, 25]. To verify the applicabilityof theRESSprocess to produce suspensions of nanoscaleparticles, the RESS apparatus shown in Figure 3 wasused. In contrast to the RESS into-air experimentsdescribed above, the supercritical CO2/�-sitosterol mix-ture is expanded directly into the aqueous solution.Initially, to minimize the unsteadiness of the flow and toaccelerate thermal equilibrium, pure CO2 flows throughthe thermostated bypass section of the extraction unitinto the thermostated high-pressure vessel and throughthe capillary nozzle into the aqueous solution. Afterequilibrium, the bypass section was closed and theexperiments were conducted as described before incase of the RESS into-air experiments. To bring theexpanded solution and, hence, the particles being formedinto rapid contactwith the surrounding area, the nozzle issubmerged approximately 1 cm below the surface of theaqueous surfactant solution. In these experiments thenonionic surfactant Tween-80 (C64H126O26, polyoxyethy-lene sorbitan monooleate, M� 1310 g/mol) was chosento impede growth and agglomeration of the �-sitosterolparticles. Dynamic light-scattering (DLS) measurementswere conducted to determine the size of the stabilizedparticles in terms of number-weighted, mass-weighted, andintensity-weighted size distributions. The aqueous solutionconsistedof either 1.0%or 5.0%(w/w)Tween-80 and theseconcentrations are higher than the critical micelle concen-tration of Tween-80.All of the experiments reported in thispaper were performed at T0�418 K, p0�25 MPa and anaqueous solution temperature of 300 K.First of all as a control, the supercritical CO2/�-sitosterolmixture was sprayed into pure water. In these experi-ments, the aqueous phase became slightly turbid andlarge particles were observed visually which were toolarge for DLS measurements. For comparison, beforeaddition of the supercritical CO2/�-sitosterol mixture,the clear aqueous surfactant solution was analyzed byDLS and at both Tween-80 concentration micelles in therange of 7 to 14 nm were measured (see Figure 5,1st peak). However, it should be noted that particlesizes in the range of 5 to 10 nm correspond to the lower

end of the precision of the DLS measurements. Never-theless, at all operating conditions investigated verysmall �-sitosterol particles were obtained even thoughwith different particle sizes. Figure 5 shows a typicalresult for the size distribution of �-sitosterol particlesstabilized in a 1.0% (w/w) aqueous Tween-80 solution.Table 3 summarizes the results of spraying supercriticalCO2/�-sitosterol mixtures into 1.0% or 5.0% (w/w)Tween-80 solutions. The smallest particles range from10 to 60 nm, however, larger particles in the rangebetween 62 to 200 nm and from 620 to 1600 nm were alsodetermined. The particle size distribution tends to beslightly narrower in the higher surfactant concentrationin comparison to the lower surfactant concentration.Nevertheless, the small particles convincingly demon-strate that the initially formed particles with very smallparticle size can be stabilized within the expanding jetwithout excessive particle growth due to agglomeration.In addition, these experiments show that the rapidexpansion of supercritical solutions into an aqueoussolution is a promising method for the formulation ofwater-insoluble pharmaceuticals.

6 Modeling of the RESS Process

Besides the experimental investigations, the RESSprocess is modeled numerically considering the parts

Table 3: Effect of surfactant concentration on measured particlesize distribution of the stabilized �-sitosterol particles.

Tween-80concentration (w/w)

Particle size distribution [nm]

1.0 % (w/w) 30 ± 60 65 ± 250 420 ± 16005.0 % (w/w) 10 ± 20 80 ± 250 620 ± 1500

Fig. 5: Particle size distribution of the stabilized �-sitosterolparticles measured by DLS (dynamic light scattering). The firstcurve (=aqueous surfactant solution without �-sitosterol particles)holds for both number- and mass- weighted size distribution.

331Part. Part. Syst. Charact. 19 (2002) 327 ± 335

capillary inlet ± capillary ± free jet ± mach shock ±expansion chamber. The one-dimensional flow calcula-tion is based on mass, momentum and energy balancesplus a reliable equation of state (Eqs. (1) ± (3) andEq. (4)) [26].

� ¥A ¥ (dw/dx)�w ¥A (d�/dx)�� � ¥w ¥ (dA/dx) (1)

� ¥w ¥ (dw/dx)� (dp/dx)��2 ¥ f ¥w2 ¥�/D (2)

w ¥ (dw/dx)� (dh/dx)� dq/dx (3)

p� p(�, T). (4)

In Eqs. (1) to (4) � is the fluid density, A (�D2 ¥�/4) theflow area, w the velocity, x the distance along theexpansion device, f the friction factor, h the enthalpy, qthe heat and p the pressure. The extended generalizedBender equation of state was chosen to describe thethermodynamic properties of the pure solvent [16]. Themodel includes nonisentropic flow in the capillary inletarea, heat exchange and friction in the capillary andparticle growth in the expansion chamber. Based on theflow calculations of the pure solvent, the supersatura-tion, S, of the real solute-solvent mixture was calculatedby means of a modified Peng-Robinson equation of state[17].

S� [�(T, p, yE) ¥ yE(TE, pE)]/[�(T, p, y) ¥ y(T, p)]. (5)

In Eq. (5) � is the solute fugacity coefficient in thedilute mixture, yE the equilibrium mole fraction of thesolute in the solvent at extraction conditions and y theequilibrium mole fraction at prevailing expansionconditions. The kinetics of particle formation wasdescribed by the following expression for the nuclea-tion rate J (i.e., the number of nuclei formed per unittime and volume) [23]:

J� � ¥Z ¥�C ¥n2 ¥� ¥ r*2 ¥ c ¥ exp[��G*/(k ¥ T)]. (6)

In Eq. (6) � is the nonisothermal factor (�1 in case ofdilute mixtures),Z the Zeldovich nonequilibrium factor,�C the condensation coefficient, n the number ofcondensable molecules (� �M ¥ yE ¥NA), r* the criticalnucleus size (�2 ¥ � ¥ vS/(k¥T ¥ ln S)) corresponding to themaximum value of the Gibbs energy of critical nucleusformation �G* (� 4³3 ¥� ¥ � ¥ r*2), c the mean thermalvelocity (� (8 ¥k ¥ T/(� ¥m))0.5), k the Boltzmann con-stant, and m the mass of the solute. For the systems ofinterest, solvent dissolution in the incompressible solid isnegligible and a value of 20 mN/m for the solid-fluidinterfacial tension � was assumed. The influence of thephase behavior, solubility and solute properties on the

nucleation rate have been investigated in a previouspaper and, hence, are not subject of this paper [23].For the description of the particulate phase, the generaldynamic equation for simultaneous nucleation, conden-sation, and coagulation is included into the model[27 ± 30]:

�n��t � Jv� � �v� v���G � n���v

� 0�5 ��v

0

� v� v�� v� � � v� v�� t � � n v�� t �dv�

� n v� t � ��

0

v� v� � � n v�� t �dv�� 7�

The term on the left-hand side of Eq. (7) represents thechange of the particle concentration nwithin the limit ofparticle volume v to v� dv. On the right-hand side ofEq. (7), the first term considers the nucleation ofparticles at a given nucleation rate J. The second termtakes into account the variation of the particle volume bycondensation through the condensation rate G. Theparticle growthdue to coagulation is describedby the lasttwo terms. A more detailed description of the hydro-dynamic and aerosol modeling is given in the literature[29, 30]. The results of the calculations show that particleformation occurs mainly in the supersonic free jet. Forthe preexpansion conditions as well as the systemsinvestigated, the calculated particle size ranges from 2to 8 nmat the endof the supersonic free jet [29]. From thesmall particle size, the high particle concentration(�1014 particles/m3) and the very short residence timein the supersonic free jet (�10�7 sec) might be suggestedthat particle growth in the supersonic free jet isunfinished. Hence, the conditions inside the expansionchamber are one key factor to control particle size andthe particles grow inside the expansion chamber to theirfinal size. This result clarifies the influence of twoimportant process parameters on particle size. Both, ashorter residence time and, hence, less time available forparticle growth aswell as a higher dilution of the particlesin the expansion chamber result in smaller particles.Also, this explains the dependency of the particle size ofbenzoic acid on the preexpansion mentioned above. Asshown in Figure 1, increasing the preexpansion pressureat constant preexpansion temperature as well as decreas-ing the preexpansion temperature at constant preexpan-sion pressure leads to a higher fluid density resulting inhigher mass flow rate through the nozzle. For thepreexpansion conditions investigated, the measuredmass flow rate increases from 6 g CO2/min at 389.1 Kand 20 MPa to 11.2 CO2/min at 353.5 K and 30 MPa. Inaccordance with the experimental results for benzoicacid, a higher mass flow rate means shorter residence

332 Part. Part. Syst. Charact. 19 (2002) 327 ± 335

time in the expansion chamber, which results in smallerparticles [8, 22, 29].Figures 6 and 7 show typical results of the hydrodynamicand aerosol modeling of the RESS process describedabove and the comparison with experimental results isalso depicted [29, 30]. In the case of benzoic acid thecalculated particle sizes are in the range of dNV� 750 to800 nm and there is only a weak dependence of theparticle size on the preexpansion temperature. Incontrast to the calculations, the measured particle sizesare about twice to four times smaller than the calculatedones and decrease with lowering the preexpansiontemperature or increasing the preexpansion pressure.The difference between the calculated and measuredparticle size could not be explained in detail until nowand the clarification of these discrepancies is part of ourongoing work.An obviously satisfactory agreement between themeasured und calculated particle size could be observedfor griseofulvin. Both the calculated and the measuredparticle size are in the range of dNV� 200 to 350 nm,whereas the calculated particle sizes are slightly higherthan the measured ones. Neither for the experimentalnor for the calculated values an explicit dependency fromthe pre-expansion condition was found. In summary, thecomparisonbetween experimental and calculated resultsshows a good agreement in the general trends but doesnot match exactly the measured mean particle sizes.

7 Conclusion

The experimental results demonstrate the applicabilityof the RESS process to produce particles smaller than500 nm. The particle size of the pharmaceutical sub-

stances �-sitosterol, griseofulvin, and ibuprofen precipi-tated by RESS was in the range from 160 to 320 nm. As afirst step towards intravenous application of poorlysoluble drugs, stable suspensions of nano-scale particlesof �-sitosterol were produced. In these experiments, thesupercritical mixture was sprayed directly into anaqueous surfactant solution. The particle sizes of �-sitosterol in the aqueous solution were smaller or equalto those produced by RESS into air without thesurfactant solution.Thus, these experiments show that theRESS technique isa promising method to improve the low bioavailability ofinsoluble or poorly soluble pharmaceutical agents aswellas for the formulation of water-insoluble pharmaceuti-cals.The modeling results show that particles are formedsmaller than 10 nm in the supersonic free jet. Hence,particle growth is not completed in the supersonic free jetand the postexpansion conditions are an importantfactor to control particle size. The comparison betweenexperimental and calculated particle size show a goodagreement in the general trends but does not matchexactly the measured mean particle sizes. Nevertheless, auseful tool for the understanding and optimization of theRESS process has been created.

8 Acknowledgements

The authors are grateful to cand. chem. ing. G. Upperwho performed the RESS into the aqueous solutionexperiments, and Prof. C. Posten (Institut f¸r Mechani-sche Verfahrenstechnik und Mechanik, Abteilung Bio-verfahrenstechnik, Universit‰t Karlsruhe (TH)) for theuse of the DLS equipment. Part of this work was

Fig. 6: Comparison of calculated and measured particle numbervolume averaged diameters (dNV) in dependence on variouspreexpansion conditions for benzoic acid [29,30].

Fig. 7: Comparison of calculated and measured particle numbervolume averaged diameters (dNV) in dependence on variouspreexpansion conditions for griseofulvin [29,30].

333Part. Part. Syst. Charact. 19 (2002) 327 ± 335

supported by the State of Baden-W¸rttemberg and bythe grants Scha 524/5-1, 5-2, 5-3, and 5-4 of the DeutscheForschungsgemeinschaft which is gratefully acknowl-edged.

9 Symbols and Abbreviations

A m2 flow area, Eq. (1)c m s�1 mean thermal velocity, Eq. (6)D m diameter of the capillary nozzled m particle diameterf ± fanning friction factor, Eq. (2)G m�3 s�1 condensation rate, Eq. (7)�G* J Gibbs energy of critical nucleus

formation, Eq. (6)h J kg�1 enthalpy, Eq. (3)J m�3 s�1 nucleation rate, Eq. (6)k Nm K�1 Boltzmann constant: 1.38�10�23,

Eq. (6)M kg kmol�1 molecular weightm kg mass of the soluteNA mol�1 Avogadro number: 6.023� 1023

n m�3 number of condensable molecules,Eq. (6)

p MPa pressurepC MPa critical pressurepE MPa extraction pressurep0 MPa pressure at the capillary inlet,

(�preexpansion pressure)pS MPa vapor pressure (depends only on

temperature)q J kg�1 heat, Eq. (3)r* m radius of nucleus of critical size,

Eq. (6)S ± supersaturation, Eq. (5)T K temperatureTC K critical temperatureTE K extraction temperatureT0 K temperature at the capillary inlet

(�preexpansion temperature)Tnozzle K nozzle temperatureV dm3 volume of the expansion chamberv m3 particle volume, Eq. (7)vS m3 solute molecular volume in the

solid phasew m s�1 velocity, Eqs. (1) ± (3)x m actual position, Eqs. (1) ± (3)y* ± solute equilibrium mole fraction in

the fluid phasey ± solute mole fraction in the fluid

phaseyE ± solute mole fraction at extraction

conditions (�TE, pE)

Z ± Zeldovich nonequilibrium factor,Eq. (6)

Greek symbols�C ± condensation coefficient, Eq. (6)� m3 s�1 coagulation coefficient, Eq. (7)� m�3 delta function, Eq. (7)� ± fugacity coefficient of the solute,

Eq. (5)� ± nonisothermal factor, Eq. (6)� mol dm�3 density of the fluid, Eq. (1)�C mol dm�3 critical density of the fluid�L mol dm�3 saturated liquid density�V mol dm�3 saturated vapor density�M mol dm�3 density of the mixture� mN m�1 interfacial tension of the solute

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