Mathematical modeling of nucleation and growth of particles formed by the rapid expansion of a supercritical solution under subsonic conditions

Transcription

1 Journal of Supercritical Fluids 23 (2002) Mathematical modeling of nucleation and growth of particles formed by the rapid expansion of a supercritical solution under subsonic conditions Markus Weber 1, Lynn M. Russell, Pablo G. Debenedetti * Department of Chemical Engineering, Princeton Uni ersity, Princeton, NJ 08544, USA Received 22 May 2000; received in revised form 29 August 2001; accepted 20 September 2001 Abstract The size distribution of fine powders formed during the rapid expansion of supercritical solutions (RESS) depends on the operating conditions, as well as on the geometry of the expansion device. In order to meet product specifications and improve process control, a fundamental understanding of the interplay between nucleation, condensation, and coagulation during this type of expansion is needed. In this work, we model the particle dynamics resulting from homogeneous nucleation, condensation and coagulation during the subsonic expansion of a nonvolatile solute in a supercritical fluid inside a cylindrical capillary. The calculations show that subsonic RESS is a very effective technique for producing particles in the nm diameter range. The particle formation process is characterized by delayed nucleation, low particle number concentrations, precipitation of a comparatively small fraction of the total solute mass, and by a narrow size distribution. In a few cases where the expansion trajectory enters the fluid s vapor liquid coexistence region, the particle formation exhibits early nucleation, strong coagulation, and higher particle number concentrations. In order to explain and describe quantitatively the much larger particle diameters found in actual RESS experiments, additional condensation and coagulation processes that occur in the transonic flow field outside the expansion device, and their interaction with this complex flow field, would also need to be incorporated Elsevier Science B.V. All rights reserved. Keywords: Supercritical carbon dioxide; Rapid expansion; Particle formation; Aerosol dynamics; Modeling 1. Introduction * Corresponding author. Tel.: ; fax address: pdebene@princeton.edu (P.G. Debenedetti). 1 Present address: Degussa, VT-F, Gebaude 1024, Rodenbacher Chaussee 4, D Hanau-Wolfgang, Germany. Fine powders are of interest in many industrial applications, such as pharmaceutical aerosols, inks, pigments, and filtration media. The use of supercritical fluids for the formation of fine particles is receiving increased attention as a possible alternative to grinding and spray-drying with liquid solvents, especially in cases involving high /02/$ - see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S (01)

2 66 M. Weber et al. / J. of Supercritical Fluids 23 (2002) value-added products [1 3]. The relatively mild operating conditions and the dryness and purity of the resulting particles produced by precipitation from a supercritical solution makes this approach particularly attractive for pharmaceutical applications. The desired particle characteristics in this case often include a carefully controlled size distribution. In pulmonary delivery applications [4], this is important in order to minimize losses due to exhalation of small ( 2 m) particles and early deposition of large ones ( 5 m). Size uniformity is also important in order to optimize the efficiency of controlled drug delivery by injection of composite microspheres into the blood stream [5 7]. The rapid expansion of a supercritical solution (RESS) with CO 2 as the solvent can produce particles with small mean diameters and narrow size distributions requiring only moderate processing temperatures [8 12]. The latter helps to prevent the processed substance from being decomposed or losing therapeutic activity. The practical implementation of a RESS unit for industrial applications requires predictability and constancy of the product quality throughout a number of different batches and over a long operating period. Therefore, understanding the influence of all relevant process parameters on the size and shape of the particles formed by RESS is important. The crucial step in the process, the rapid expansion, leads to a pronounced drop of the solute equilibrium mole fraction in the homogeneous mixture via a simultaneous drop in pressure and temperature. Due to the large compressibility of supercritical fluids, modest changes in pressure are accompanied by large changes in density and, therefore, in solvation strength. Consequently, supersaturation increases, giving rise to a sharp increase in the homogeneous nucleation rate. The resulting initial generation of new and very small particles (nuclei) is gradually replaced by the addition of single molecules at the particle surface by the mechanism of condensation. Growth of particles can also occur by events in which two particles collide and stick together forming bigger particles, which is called coagulation. A comprehensive model of the RESS process must account for all three mechanisms of particle formation and growth and must cover flow in both the subsonic and supersonic regimes. The supersonic region cannot be neglected a priori. Attempts to model the supersonic part of the process have been based on fundamental investigations and visualizations of high-pressureratio, free-expanding supersonic jets, as reported by Bier and Schmidt [13] and Ashkenas and Sherman [14]. Most authors assumed axisymmetric, one-dimensional flow and calculated the pressure and temperature profile throughout the first shock cell. In addition, Berends [15] estimated the time of flight from the nozzle exit to the Mach disc while Türk and co-workers [16,17] derive complete supersaturation and critical nucleus size profiles under the simplifying assumption of negligible precipitation. Reverchon and Pallado [18] incorporated the heat transfer through the transonic shear layer and found the calculated temperature profiles to be in reasonable agreement with measurements. Finally, Shaub et al. [19] modeled a spherically symmetric, one-dimensional supersonic flow field and simultaneously solved the aerosol dynamic equation, while Ksibi [20] studied numerically a two-dimensional flow field including shock waves, coupled with particle population balances but neglecting coagulation. In this work, we compute the particle size distribution accounting for nucleation, condensation, and coagulation by using a sectional method to solve the aerosol dynamic equation along an expansion device with given pressure, temperature, and density profiles. These profiles are the result of a one-dimensional single-phase calculation of the subsonic flow inside expansion devices like the one shown in Fig. 1. We quantify the role that several process parameters, such as the expansion device aspect ratio, the pre-expansion pressure, temperature, and supersaturation play in manufacturing submicron particles by subsonic expansion. Based on extrapolations of our results to supersonic conditions, we provide a plausible explanation of experimental results.

3 M. Weber et al. / J. of Supercritical Fluids 23 (2002) Governing equations and integration scheme The cylindrical expansion device, henceforth referred to simply as capillary, is preceded by a converging inlet section (Fig. 1). The flow through this device (including the conical inlet section) is modeled as one-dimensional, compressible and adiabatic. The balance equations [10,12] allow for changes in the cross-sectional area along the axis, and account for friction through a constant Fanning friction factor. It is assumed that the fluid thermodynamic properties are represented accurately by stagnation conditions at inlet velocities less than 3 ms 1. For the conditions studied here, this occurs when the inlet diameter is between 3 and 10 times larger than the capillary diameter. Conservation equations of mass, momentum, and energy were written for the pure solvent, an excellent approximation given the dilute ( 1 mole%) mixtures considered here and in practical RESS applications. With and denoting the device s Fig. 1. General schematic of an expansion device used for the RESS process. A tapered inlet section (a common drilling angle is 118 ) is followed by a cylindrical section with an inner diameter d and the length L. p 0, T 0, S 0, y 0 e and u 0 denote stagnation pressure, temperature, supersaturation, solute equilibrium mole fraction and velocity, respectively (the latter is sufficiently close to zero). The corresponding values along the capillary s axis are denoted by p(z), T(z), S(z), y e (z) andu(z). cross sectional area and perimeter, respectively, and u,, h, and f denoting the velocity, mass density, enthalpy per unit mass, and Fanning friction factor, the conservation equations read: 1 d +1 du u +1 d =0 (1) u du +dp = 2 u 2 f a (2) u du +dh =0 (3) We chose the Carnahan Starling van der Waals equation of state [21,22] to describe the thermodynamic properties of the solvent including the partial derivatives with respect to and T needed to compute the axial derivatives of p and h. dp = p d + p dt (4a) T dh = h T T d + h T dt (4b) Integrating the system of three linear differential equations and one algebraic equation (the equation of state), yields values for pressure, temperature, density, and velocity at each location along the z axis. The precipitated solid material is represented by a discretized particle size distribution (PSD) at every location z. Each component of this vector function gives the number N p, i and the total mass M p, i of particles per unit volume that are contained in the corresponding section i on a logarithmically-spaced diameter scale. The two values for number and mass are related by the reference diameter d p, i as given by M p, i (z)=n p, i (z) m p, i (z) (5) where m p;i (z)=( /6)d p, 3 i S is the particle mass and S is the solid density. Each of the log-normally spaced sections (bins) is assigned a particle diameter d p, i with the index i ranging from 1 to n b.this discretization scheme implies that material precipitating by homogeneous nucleation is always allocated to only one of n b sections at a time. Accordingly, condensation leads to an increase in

4 68 M. Weber et al. / J. of Supercritical Fluids 23 (2002) mass in a section but does not affect the section particle number. The effects of coagulation on the number and mass distributions contrast those of condensation in that they include changes in particle number in some sections but do not change the overall mass. Following Russell and Seinfeld [23], and neglecting deposition of the precipitated material on the capillary walls, the aerosol dynamics equations read: 0 dn p, i 0 dm p, i 0, =dn p, nuc i =dm p, 0, i nuc + dn 0, p, cgl i + dm 0, p, cnd i + dn 0, p, grw i + dm 0, p, cgl i (6a) + dm 0, p, grw i (6b) In Eqs. (6a) and (6b) N p, i and M p, i denote the number and the mass concentration of particles belonging to section i. The superscript 0 indicates that these concentrations are normalized by the density of the surrounding gas and the superscripts nuc, cnd, and cgl denote contributions due to nucleation, condensation, and coagulation, respectively. The growth contribution (grw) is a numerical term arising from re-allocation of mass and particle numbers among the various bins after each integration step, to account for discretization. Particles are grown by adding mass and allocating the resulting particles into the size bins using the constraints of number and mass conservation [23]. We used the same formulae describing the condensation rate as were used in previous work [24]. The interfacial tension that enters the nucleation term, taken from Debenedetti [25], was estimated using the Macleod Sugden correlation [26]. Although this parachor-based method was developed for liquid mixtures, it yields interfacial tension values of O (10 2 N m 1 ) (e.g., Nm 1 for solid phenanthrene in contact with supercritical carbon dioxide at 260 bar and 343K) which are typical of solid gas interfacial tensions obtained by fitting to a nucleation model experimental particle size distributions obtained from the vapor phase. Coagulation was calculated based on the collision frequency given by Seinfeld and Pandis [27] accounting for Brownian motion as well as for laminar [28] and turbulent [29] shear forces. A sticking factor of unity ( =1) was assumed. Details on the solution of the aerosol equation using the sectional method are given in the original reference [23]. The simplification, that all thermodynamic properties can be derived under the assumption of a pure single-phase fluid allows us to decouple the fluid dynamics calculation from the aerosol dynamics equation. This approach differs from earlier work [24] and facilitates the iterations required to determine the critical mass throughput from the capillary. For a given capillary geometry (L/D ratio) and stagnation conditions (temperature, pressure, initial supersaturation), the inlet velocity is varied until the Mach number at the exit is unity (choked flow). The steps in each integration are allowed to be bigger by a factor of 10 or more than the steps required for the calculation of the aerosol evolution. 3. Results 3.1. An example of particle formation and growth Fig. 2 shows the pressure, temperature, density, and Mach number for carbon dioxide expanding along the capillary s axis. The capillary is 100 mm long and has an inner diameter of 100 m. Its friction factor is taken to be f= The stagnation parameters are 260 bar and 383 K. Under these conditions, the major part of the pressure and density drop is due to friction rather than acceleration. This effect can be seen from the very small vertical drop of the pressure profile at z=0, which represents the acceleration-driven expansion in the inlet section. The velocity at the capillary exit is 193 ms 1, which corresponds to sonic (choked) conditions at the exit temperature and density. The equilibrium mole fraction of the solute in the supercritical fluid is a function of pressure and temperature. For given inlet conditions, the actual solute mole fraction can be calculated at each location by integrating the aerosol dynamic equations referenced above. The difference between the actual and equilibrium mole fractions provides the driving force for precipitation.

5 M. Weber et al. / J. of Supercritical Fluids 23 (2002) Fig. 2. Pressure, temperature, dimensionless density, and Mach number of pure carbon dioxide when expanded through a capillary of 100 mm length and 100 m inner diameter as functions of the position z. The Fanning friction factor was assumed f=0.005, the inlet conditions are T 0 =383 K, p 0 =260 bar. Fig. 3 depicts the actual and equilibrium solute mole fraction for the system CO 2 /phenanthrene predicted from the fluid and aerosol dynamics Eqs. (1) (3), (4a), (4b), (5), (6a) and (6b) for the profiles presented in Fig. 2. It can be seen that the actual mole fraction remains nearly unchanged up to the middle of the capillary, where homogeneous nucleation starts to produce nanometersized particles at a significant rate. Subsequently, condensation becomes the prevailing precipitation mechanism up to the capillary exit. Finally, just prior to the exit, pressure and density drop abruptly (Fig. 2) and a second burst of nucleation occurs. This can be seen in Fig. 4, which shows the nucleation rate along the capillary. Fig. 3 shows that this second burst of nucleation results from the slowing of condensation due to the decreasing driving force (difference between actual and equilibrium mole fraction) and specific surface area as more mass precipitates, while the ratio of actual to equilibrium mole fraction remains large because of the continued solvent expansion and the consequent large drop in solvent power. The fact that the supersaturation ratio is large, while the difference between actual and equilibrium mole fraction is small (because the solvent mole fractions themselves are small) provides a strong driving force of homogeneous nucleation. The number of molecules needed for a critical nucleus, n cr, is also plotted against the axial position along the capillary s axis in Fig. 4. This quantity is calculated from classical nucleation theory, with due account for fluid-phase non-ideality, following Debenedetti [30] and Kwauk and Debenedetti [24]. Particle size distributions (PSD) corresponding to this specific example are shown in Fig. 5a c. Comparison with Fig. 4 shows that even in the

6 70 M. Weber et al. / J. of Supercritical Fluids 23 (2002) very early stages of precipitation the mean particle diameter is larger than the diameter of the critical nucleus at the same location. This illustrates the fact that condensation is present at the very early stages of the process. The number concentration of particles is increased by several orders of magnitude within a small fraction of the capillary s length (Fig. 5a). Growth processes not only shift the mean particle diameter towards larger values but also narrow the spectra (Fig. 5b). At the second burst of nucleation, the spectrum becomes bimodal (Fig. 5c), the variance of the distribution increases markedly, and the number-based mean diameter drops. This example shows quite clearly the way in which different microphysical processes compete Influence of capillary geometry on particle size Since a 100-mm capillary produces particles in the nm size range, it is of interest to inquire what length would be needed to produce particles of the size that is typically encountered in experiments. In experiments, the capillaries studied had diameters between 1 and 200 m and lengths usually less than 15 mm [11,12,31 38], with the exception of the restrictors used by Berger [39], which had higher L/D ratios and lengths up to several centimeters. Fig. 6a shows the mean particle diameter at the capillary exit as a function of the aspect ratio (L/D) and the inner diameter. The aspect ratio was varied over three orders of magnitude, and the calcula- Fig. 3. Actual and equilibrium mole fractions of phenanthrene dissolved in carbon dioxide as functions of the position z in a 100 mm capillary. The equilibrium mole fraction is calculated from the pressure, temperature, and density given in Fig. 2. The actual mole fraction is the result of a simulation accounting for homogeneous nucleation, condensation, and coagulation.

7 M. Weber et al. / J. of Supercritical Fluids 23 (2002) Fig. 4. The number n crit of phenanthrene molecules forming a critical nucleus and the nucleation J, rate as functions of the position z along the capillary axis. Same conditions as in Fig. 2. tions were carried out for four different inner diameters ranging from 1 m (representative of a channel in a porous plate) to 1 mm (representative of a laboratory-scale high-pressure tube). For a given tube diameter, the mean particle size is insensitive to length for aspect ratios less than 10. As the aspect ratio is increased beyond 10, particles grow appreciably in all capillaries. While those in the 1- m-channel exceed the size calculated for plain orifices only at aspect ratios greater than 100, growth in the 1-mm capillary increases much faster with L/D, but saturates at aspect ratios of more than 200. Particle growth is commonly represented by an exponential law correlating the mean particle diameter d p and the growth time, sothatd p3. For growth processes that are controlled by coagulation, Lesniewski and Koch [40] give values between 0.8 and 1.2 for the exponent based on theory and measurements. Growth profiles based on the actual residence time approach the exponent =1.2 close to the capillary s exit. From this analysis, we conclude that over the broad range of capillary geometries explored here (1 m D 1 mm, 1 L/D 1000) particle growth by condensation and coagulation will not lead to number-based average diameters appreciable larger than 100 nm in subsonic RESS expansions. Expansion trajectories starting from lower T 0 values can lead to somewhat larger particle sizes, but solvent condensation then becomes a key factor that needs to be taken into consideration. As L/D increases, the major part of the pressure drop is due to friction and is shifted closer to the exit (in normalized distances). Thus, expansions in long capillaries are closer to isenthalpic paths, in contrast to the virtually isentropic paths followed by expansions in short devices. Size distribution of particles generated in long capillaries are generally broader. Fig. 6b shows the resulting mass median diameter (the particle diameter above which 50% of the precipitated

8 72 M. Weber et al. / J. of Supercritical Fluids 23 (2002) mass occurs) for the same set of aspect ratios and capillary diameters as in Fig. 6a. This figure thus indicates that roughly half of the mass produced by RESS in a 1 mm 1m capillary consists of particles having diameters 800 nm. The production of particles with a number-based mean diameter greater than one micrometer appears unattainable in capillaries with aspect ratios less than In expansion devices with high aspect ratios, bimodal particle size distributions can occur because of the second burst of nucleation Influence of process conditions on particle size To examine the influence of inlet conditions on the mean particle diameter we chose a capillary of the same geometry as used by Berends [15] namely an inner diameter of 40 m and a length of 200 m. A Fanning friction factor of f= was assumed for all simulations. Fig. 7 shows a contour map of the mean particle diameter achieved at the capillary exit as a Fig. 5. Particle size distributions (PSD) representing the number density of phenanthrene particles at three different positions (indicated by the distance z from the inlet) at early stages of homogeneous nucleation inside a capillary of 100 m inner diameter and 100 mm length. Same conditions as in Fig. 2. The distributions correspond to the initial (a), middle (b), and exit (c) sections of the capillary. Note the bimodal distribution near the exit due to the second burst of nucleation (c).

9 M. Weber et al. / J. of Supercritical Fluids 23 (2002) Fig. 5. (Continued) function of stagnation pressure and temperature (p 0, T 0 ). The supersaturation ratio at stagnation conditions, S 0, was set to unity in this calculation. The locus of maximum particle size along isobars coincides virtually identically with the locus of solubility maxima along isobars. High initial pressures combined with low initial temperatures are often avoided in practice because the expansion trajectories come closer to (or enter) carbon dioxide s two-phase region. Supersaturation ratios of more than 1.2 are hard to maintain at stagnation conditions. Therefore, the region of experimental and practical relevance corresponds to high stagnation temperatures and pressures and low initial supersaturations. In this region, initial pressure and temperature have a negligible influence on the mean diameter of the resulting particles. The size of the particles is determined primarily by the initial solute mole fraction, although in practice nucleation limits the extent to which this variable can be manipulated.

10 74 M. Weber et al. / J. of Supercritical Fluids 23 (2002) Fig. 5. (Continued) 4. Discussion 4.1. Analysis of different growth characteristics In order to study the relative importance of three microphysical processes (nucleation, condensation, and coagulation) on the evolution of the particle size distribution (PSD) in a RESS process, we correlate the number-based mean diameter and the fraction of precipitated material, along the capillary axis. The resulting growth pattern can be interpreted as the superposition of the characteristic paths for nucleation, condensation and coagulation [27]. Fig. 8 shows the versus d p trajectories for stagnation conditions included in Fig. 7. The stagnation pressure was changed in steps of 30 bar at a constant stagnation temperature T 0 =383 K and constant supersaturation ratio S 0 =1. While it is less than 20% in cases where p 0 =230 bar or lower, the relative contribution of coagulation to the total growth can be as high as 85% in expansions where p bar. The relative contributions of coagulation and condensation processes to this growth can also be seen graphically in the final part of the expansion trajectories illustrated in the insert of Fig. 8. Dur-

11 M. Weber et al. / J. of Supercritical Fluids 23 (2002) Fig. 6. Mean (a) and mass median (b) diameter of phenanthrene particles at the exit of expansion devices with varying inner diameter and length. Inlet conditions are p 0 =200 bar, T 0 =403 K, S 0 =1. ing condensation, increases while the total number of particles remains unchanged. In this idealized case the relation between the cube of the mean diameter and the precipitated mass fraction is linear [41]. In practice, the condensation rate on each particle is proportional to its surface area, favoring smaller particles. Coagulation reduces the number and increases the size of existing particles, but has no direct influence on precipitation. The path representing this process is always horizontal. For most of the expansion conditions that are possible, the slope of the growth path never falls below the condensation limit. In these cases, nucleation occurs farther downstream in the expansion device (even though this is not shown explicitly in a representation, such as Fig. 8), and condensation becomes the dominant precipitation process at relatively low number concentrations. Although the growth of a smaller number of particles by condensation never stops entirely, the mean diameter stagnates or even decreases during the last part of the precipitation in a subsonic expansion. This decrease is due to the second nucleation burst.

12 76 M. Weber et al. / J. of Supercritical Fluids 23 (2002) For expansions terminating either inside or very close to the supercritical fluid s vapor-liquid coexistence region, nucleation occurs comparatively early and condensation is a relatively weak contributor to the overall precipitation process. This allows for a very large number of particles to be formed before growth processes become relevant. Due to the resulting high number density, particles grow by coagulation as well as condensation. Therefore, the slope of the growth paths for 290 bar and for 320 bar in Fig. 8 decreases continuously along the upper branch and finally falls below the condensation limit given by: log(d p ) log( ) From this analysis of particle growth, we conclude that generally in subsonic RESS, expansions are characterized by the almost vertical, nucleation-dominated precipitation trajectory ending abruptly when choked conditions are reached at the nozzle exit. In a few exceptional expansion conditions, particle formation processes include not only the initial nucleation-dominated trajectory, but also the subsequent shift towards condensation and coagulation. This shift is shown in Fig. 7. Mean diameter of phenanthrene particles formed by homogeneous nucleation and growth inside a capillary of 200 m length and 40 m inner diameter. Stagnation conditions of the supercritical carbon dioxide/ phenanthrene mixture before the adiabatic expansion are, 140 bar p bar; 333 K T K; and S 0 1. Fig. 8 by the decrease of the slope of trajectories close to equal unity (cases with T 0 =383 K and 260 bar p bar). Fig. 8 not only reveals important information about growth processes inside of the capillary but also illustrates the discrepancy between mean particle diameters found by subsonic simulation and those found in experiments. The assumption of a low fraction of precipitated substance at the capillary outlet (typically less than 10 2 ) is supported by some of our simulations as well as by Berends et al. [11]. After exiting the capillary, the fluid will undergo further expansion in two distinct steps: (1) the fluid expands almost isentropically before reaching the Mach disc at a distance of about 10 capillary diameters (L/D 10, and pressure ratio 200 according to Ashkenas and Sherman [14]). (2) Recompression and expansion occur in shock cells, together with mixing with the surrounding gas at atmospheric pressure. The shock cell region typically covers a distance of not more than 100D [20]. The short extent of this region is one reason why the size distributions of particles produced by RESS are only analyzed after the particles exit the transonic mixing zone. This result shows the importance of developing a comprehensive model that includes the evolution of the particle size distribution across the transonic region. While we do not have an explicit model of the transonic region, we can speculate from the comparison of our results to experimental measurements which mechanisms dominate the microphysical behavior in this regime. Berends et al. [11] produced phenanthrene particles with mean diameters between 2 and 8 m in their RESS experiments. Since all of the solute eventually precipitates ( =1), we can draw a line from the locations representing choked conditions at the nozzle exit to those representing experimental values. These lines are shown as arrows in Fig. 8. Some factors that contribute to the formation of the much larger particles found in experiments include shear forces generated in the turbulent transonic flow field, and slip due to deceleration associated with shock waves. A detailed analysis of this complex problem is clearly required if supersonic RESS is to be used as a means of

13 M. Weber et al. / J. of Supercritical Fluids 23 (2002) Fig. 8. Mean diameter of particle size distributions evolving during the expansion of a carbon dioxide/phenanthrene mixture in a capillary of 200 m length and 40 m i.d. plotted against the mass fraction of precipitated material. Growth paths (full lines) are shown for seven different inlet pressures p 0 ranging from 140 to 320 bar at constant T 0 =383 K and S 0 =1.0. The dark line depicts the locus of all nozzle exit conditions. The horizontal bar at the top represents particle sizes typically found in RESS experiments. The insert shows characteristic paths that are followed by each of the three microphysical processes. forming larger particles. Of course, another possibility, which we cannot rule out is that the discrepancy between our calculations and experimental observations reflects a limitation of the model Sensiti ity to experimental and model uncertainties Our simulations rely both on idealizations of experimental conditions, and on parameterizations of empirical data to estimate physical properties. In order to quantify the impact of these idealizations on our calculations, we have varied some of the less accurately known parameters for nucleation and growth and quantified their effects on the model predictions. Table 1 shows the mean particle diameter and the initial mole fraction of phenanthrene for typical expansion conditions, along with the predicted deviation in the mean particle diameter with respect to the base-case simulation resulting from changing each parameter. As the interfacial tension between solid phenanthrene and carbon dioxide has not been measured and a liquid-phase estimation method was used here [26], a large, two-fold variation in this quantity was imposed. This corresponds to the calculated change in along a typical expansion trajectory. It can be seen from Table 1 that in-

14 78 M. Weber et al. / J. of Supercritical Fluids 23 (2002) creasing (decreasing) the interfacial tension by a factor of 2 results in the formation of nanoscale particles that are 12% smaller (19% larger) in diameter than for the base case. This modest sensitivity of particle size to changes in interfacial tension indicates robustness of the main trends identified by the model to uncertainties in the estimation of physical properties. The condensation rate is proportional to the accommodation coefficient, which is a finite value between zero and unity (Fuchs [42]) in the transition regime, where the length of the free mean path is comparable to the diameter of the particles [27]. In the continuum regime, where the ratio of mean free path to particle length scale (i.e. the Knudsen number, Kn) is very small, can be greater than 1 depending on the particle size and the relative velocity between particle and bulk flow [43]. We accounted for incomplete accommodation by setting to a value 100 times smaller than calculated at each location. Similarly, extreme conditions of energy dissipation in a turbulent flow were accounted for by increasing by a factor of Conclusion In this work, we have modeled nucleation and growth of particles formed by the rapid expansion Table 1 Deviations of the number-based particle mean diameter induced by the change of different parameters Base case Interfacial tension Pre-expansion conditions 200 bar, 403 K y e (0) (exit) D p nm 11.9% : % Transport coefficient : The deviations referring to the base case on top are either given in percent or as factors, by which the mean diameter in the base must be increased, or multiplied, respectively. of a supercritical solution during one-dimensional sub-sonic flow. We have studied the influence of stagnation conditions, initial supersaturation, and capillary geometry on the size and size distribution of the precipitated particles. In order to understand the mechanisms controlling particle production under different conditions, we have identified the role of nucleation, condensation, and coagulation in determining the size distribution of particles at the capillary exit. In most of the cases investigated, the expansions are controlled by late nucleation, giving rise to smaller particles, and to a smaller precipitated fraction. In a few cases in which the temperature is lower and the solvent vapor-liquid coexistence is approached, the expansions show appreciable coagulation and produce larger particles and higher fractions of precipitated material. Dramatic differences in particle morphology between expansions that cross the pure solvent two-phase region and those that do not have been reported by experimentalists (Petersen et al. [31]) earlier. Phenanthrene particles larger than 100 nm (number-based mean diameter) cannot be formed during subsonic expansions through capillaries shorter than 1 meter. Micron-size particles, such as are commonly observed in RESS experiments with short nozzles, must therefore, grow in the transonic part of the expanding free jet. For practical stagnation conditions, the observed experimental trends correlating initial pressure and temperature with the resulting particle mean diameter (Tom et al. [8]; Shaub et al. [19]) were reproduced. In addition, we have shown that these trends reverse as soon the relative maximum in particle diameter is crossed. Relative maxima of the mean particle diameter with varying stagnation temperature, T 0 or saturation ratio, S 0 were found experimentally by Griscik et al. [44] and by Domingo et al. [37], respectively. The results of our calculations strongly support the notion that (subsonic) RESS is an excellent method for the formation of nanoparticles. Our calculations suggest that the much larger particles found in experiments are the result of strong growth processes in the transonic flow field outside of the nozzle. These subsequent growth processes are condensation and coagulation. While

15 M. Weber et al. / J. of Supercritical Fluids 23 (2002) condensation plays an appreciable role in the transonic region, coagulation may dominate free jet processes in expansions sufficiently close to the liquid-vapor coexistence dome. It appears, therefore that the production of nanoparticles in a practical RESS application depends on the proper control of particle growth in the transonic flow field. It is also possible that the discrepancy between predicted and observed particle sizes reflects a limitation of our model, but it is not obvious what this limitation should be. Acknowledgements This work was supported in part by the ERC program of the National Science Foundation under Award number EEC The first year of Markus Weber s contribution to this project was financed by the Schweizerischer Nationalfonds (Swiss National Science Foundation). Studies of the aerosol dynamics fundamental to this work were made possible by funding from the Merck Foundation to Lynn M. Russell. References [1] C.A. Eckert, B.L. Knutson, P.G. Debenedetti, Supercritical fluids as solvents for chemical and materials processing, Nature 383 (26) (1996) [2] B. Bungert, G. Sadowski, W. Arlt, Separations and material processing in solutions with dense gases, Industrial and Engineering Chemistry Research 37 (8) (1998) [3] K.A. Larson, M.L. King, Evaluation of supercritical fluid extraction in the pharmaceutical industry, Biotechnology Progress 2 (2) (1986) [4] W.C. Hinds, Aerosol Technology: Properties, Behaviour and Measurement of Airborne Particles, Wiley, New York, [5] S.-D. Yeo, G.-B. Lim, P.G. Debenedetti, H. Bernstein, Formation of microparticulate protein powders using a supercritical fluid antisolvent, Biotechnology and Bioengineering 41 (1993) [6] R.S. Langer, New methods of drug delivery, Science 249 (1990) [7] L.M. Sanders, Controlled delivery systems for peptides, in: V.H.L. Lee (Ed.), Peptide and Protein Drug Delivery, Marcel Dekker, New York, [8] C.J. Chang, A.D. Randolph, Precipitation of microsize organic particles from supercritical fluids, The American Institute of Chemical Engineers Journal 35 (11) (1989) [9] R.S. Mohamed, P.G. Debenedetti, R.K. Prud homme, Effects of process conditions on crystals obtained from supercritical mixtures, The American Institute of Chemical Engineers Journal 35 (2) (1989) [10] A.K. Lele, A.D. Shine, Morphology of polymers precipitated from a supercritical solvent, The American Institute of Chemical Engineers Journal 38 (5) (1992) [11] E.M. Berends, O.S.L. Bruinsma, G.M. van Rosmalen, Nucleation and growth of fine crystals from supercritical carbon dioxide, Journal of Crystal Growth 128 (1993) [12] J.W. Tom, P.G. Debenedetti, R. Jerome, Precipitation of poly(l-lactic acid) and composite poly(l-lactic acid)- pyrene particles by rapid expansion of supercritical solutions, The Journal of Supercritical Fluids 7 (1994) [13] K. Bier and B. Schmidt, Zur Form der Verdichtungsstöße in frei expandierenden Gasstrahlen, Zeitschrift für angewandte Physik, Band 13, 1961, Heft 11, pp [14] H. Ashkenas, F.S. Sherman, The Structure and Utilization of Supersonic Free Jets in Low Density Wind Tunnels, in: H.L. Dryden, Th. von Kármán (Eds.), Advances in Applied Mechanics, Suplement 3: J.H. de Leeuw (Ed.), Rarefied Gas Dynamics, 1965, vol. II, pp [15] E.M. Berends, Supercritical Crystallization The RESS Process and the GAS Process, Proefschrift, Universiteitsdrukkerij Technische Universiteit Delft, NL, [16] M. Türk, Formation of small organic particles by RESS: experimental and theoretical investigations, The Journal of Supercritical Fluids 15 (1999) [17] B. Helfgen, M. Türk, K. Schaber, Micronization by rapid expansion of supercritical solutions: theoretical and experimental investigations, Advanced Technologies for Particle Processing AIChE 1 (1998) [18] E. Reverchon, P. Pallado, Hydrodynamic modeling of the RESS process, The Journal of Supercritical Fluids 9 (1996) [19] G.R. Shaub, J.F. Brennecke, M.J. McCready, Radial model for particle formation from the rapid expansion of supercritical solutions, The Journal of Supercritical Fluids 8 (1995) [20] H. Ksibi, Mise en forme des matériaux par détente d une solution supercritique: Simulation et experimentation, Thèse, Université de Paris XIII, Institut Galilée, Laboratoire d Ingénierie des Matériaux et des Hautes Pressions (CNRS), [21] K.P. Johnston and C.A. Eckert, An analytical Carnahan Starling van der Waals model for solubility of hydrocarbon solids in supercritical fluids, The American Institute of Chemical Engineers Journal 27(5) (1981) [22] N.F. Carnahan, K.E. Starling, Intermolecular repulsions and the equation of state for fluids, The American Institute of Chemical Engineers Journal 18 (6) (1972)

16 80 M. Weber et al. / J. of Supercritical Fluids 23 (2002) [23] L.M. Russell, J.H. Seinfeld, Size- and composition-resolved externally mixed aerosol model, Aerosol Science and Technology 28 (1998) [24] X. Kwauk, P.G. Debenedetti, Mathematical modeling of aerosol formation by rapid expansion of supercritical solutions in a converging nozzle, Journal of Aerosol Science 24 (4) (1993) [25] P.G. Debenedetti, Metastable Liquids: Concepts and Principles, Princeton University Press, Princeton, NJ, [26] R.C. Reid, J.M. Prausnitz, B.E. Poling, The Properties of Gases and Liquids, fourth ed., McGraw-Hill, New York, 1987, p. 642ff. [27] J.H. Seinfeld, S.N. Pandis, Atmospheric Chemistry and Physics-From Air Pollution to Climate Change, Wiley, New York, 1998, p [28] M. von Smoluchowski, Versuch einer mathematischen theorie der koagulationskinetik kolloider lösungen, Zeitschrift für physikalische Chemie 92 (1917) [29] P.G. Safmann, J.S. Turner, On the collision of drops in turbulent clouds, Journal of Fluid Mechanics 1 (1956) [30] P.G. Debenedetti, Homogeneous nucleation in supercritical fluids, The American Institute of Chemical Engineers Journal 36 (9) (1990) [31] R.C. Petersen, D.W. Matson, R.D. Smith, Rapid precipitation of low vapor pressure solids from supercritical solutions: the formation of thin films and powders, Journal of the American Chemical Society 108 (1986) [32] R.D. Smith, J.L. Fulton, R.C. Petersen, A.J. Kopriva, B.W. Wright, Performance of capillary restrictors in supercritical fluid chromatography, Analytical Chemistry 58 (1986) [33] D.W. Matson, J.L. Fulton, R.C. Petersen, R.D. Smith, Rapid expansion of supercritical fluid solutions: solute formation of powders, thin films and fibers, Industrial and Engineering Chemistry Research 26 (1987) [34] D.W. Matson, R.C. Petersen, R.D. Smith, Formation of silica powders from the rapid expansion of supercritical solutions, Advanced Ceramic Materials 1 (3) (1986) [35] D.W. Matson, R.C. Petersen, R.D. Smith, The preparation of polycarbosilane powders and fibers during rapid expansion of supercritical fluid solutions, Materials Letters 4 (10) (1986) [36] D.S. Halverson, Precipitation from Supercritical Fluids- Effects of Process Conditions on the Morphology and Particle Size of Precipitation Products, Masters Thesis, Princeton University, [37] C. Domingo, E. Berends, G.M. van Rosmalen, Precipitation of ultrafine organic crystals from the rapid expansion of supercritical solutions over a capillary and a frit nozzle, The Journal of Supercritical Fluids 10 (1997) [38] C. Domingo, E.M. Berends, G.M. van Rosmalen, Precipitation of ultrafine benzoic acid by expansion of a supercritical carbon dioxide solution through a porous plate nozzle, Journal of Crystal Growth 166 (1996) [39] T.A. Berger, Modeling linear and tapered restrictors in capillary supercritical fluid chromatography, Analytical Chemistry 61 (4) (1989) [40] T.K. Lesniewski, W. Koch, Production of rounded Tiand Al-hydroxide particles in a turbulent jet by coagulation-controlled growth followed by rapid coalescence, Journal of Aerosol Science 29 (1/2) (1998) [41] S.K. Friedlander, Smoke, Dust and Haze Fundamentals of Aerosol Behavior, Wiley, New York, [42] N.A. Fuchs, The Mechanics of Aerosol, Translated by R.E. Daisley, Marina Fuchs from Russian Original Mekhanika aerozolei, Pergamon Press, [43] R.H. Perry, D.W. Green, J.O. Maloney, Perry s Chemical Engineers Handbook, seventh ed., McGraw-Hill, New York, 1997, pp. 5 42ff. [44] G.J. Griscik, R.W. Rousseau, A.S. Teja, Crystallization of n-octacosane by the rapid expansion of supercritical solutions, Journal of Crystal Growth 155 (1995)