Estimation of the Characteristic Time Scales in the Supercritical

This difference in the characteristic times indicates that only a small amount of the supercritical fluid has had time to diffuse into the liquid jet ...
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Estimation of the Characteristic Time Scales in the Supercritical Antisolvent Process Francisco Cha´ vez and Pablo G. Debenedetti* Department of Chemical Engineering, Princeton University, Princeton, New Jersey 08544

J. J. Luo, Rajesh N. Dave, and Robert Pfeffer New Jersey Institute of Technology, Newark, New Jersey 07102

Simple engineering correlations and a linear jet breakup model are used to estimate the orders of magnitude of characteristic times of the various processes involved in the supercritical antisolvent method (SAS) of particle formation. The characteristic times of jet breakup, mass transfer, and nucleation are studied under typical conditions of pressure and temperature in the two-phase regime for a mixture of carbon dioxide and ethanol. The results of the calculations suggest that the jet breakup phenomenon is much faster than the mass transfer process. This difference in the characteristic times indicates that only a small amount of the supercritical fluid has had time to diffuse into the liquid jet stream by the time the breakup takes place. The characteristic nucleation times can vary appreciably with process conditions (e.g., supersaturation) and the specific properties of the mixture where precipitation occurs (e.g., liquid-solid interfacial tension). On the basis of the present calculations, two different regimes can be anticipated: a diffusion-limited regime leading to a precipitation front and a nucleation-limited, well-mixed regime. Introduction

Table 1. Critical Properties and Acentric Factor Used with the Soave-Redlich-Kwong Equation of Statea

The supercritical antisolvent technique (SAS) is a method currently used for producing micro- and nanoparticles in a variety of applications, such as pharmaceutical compounds,1 superconductor precursors,2 and polymers.3,4 In this process, the solute is dissolved in an organic solvent, and the solution is sprayed into an excess of the supercritical antisolvent. By choosing substances and operating conditions such that the solvent and the antisolvent are either fully or partially miscible, and the solute and antisolvent have low mutual solubility, it is then possible to cause the solute to precipitate. Carbon dioxide is a common choice of antisolvent because it is inexpensive, nontoxic, and inert and its low critical temperature (Tc ) 31 °C) allows operation at moderate conditions. SAS is a complex process involving the interaction of jet hydrodynamics, droplet formation, mass transfer into and out of the droplets, phase equilibrium, nucleation, and growth. A complete description would have to take into account all of these processes. Such a model is not currently available. Recent theoretical models, although able to successfully explain aspects of the problem, incorporate only a limited number of the relevant processes. For instance, Kikic et al.5 do not address mass transport within the droplet, and Werling and Debenedetti consider neither hydrodynamics nor nucleation.6,7 As a preliminary step toward the development of a more complete model, in the present paper we investigate the characteristic time scales of jet breakup, mass transfer, and nucleation. * To whom correspondence should be addressed. Tel.: (609) 2585480. Fax: (609)2580211. E-mail: [email protected].

ethanol CO2 a

Tc (K)

Pc (bar)

ω

514.05 304.25

61.37 73.80

0.643 0.224

Values taken from ref 9.

Thermodynamics Our calculations are based on the carbon dioxideethanol system. Ethanol has been used as solvent for the precipitation of a variety of organic compounds, such as acetaminophen, ascorbic acid, chloramphenicol, insulin, and lecithin.3 We first calculated the liquid-vapor phase equilibrium of the system ethanol (solvent)carbon dioxide (antisolvent) by means of the SoaveRedlich-Kwong (SRK) equation of state.8 The values of the critical properties and acentric factors have been compiled in Table 1.9 The value of the binary interaction parameter was taken from the literature to be kij ) 0.0795.10 Using these values, we calculated the equilibrium compositions of the phases and their densities. These results are needed in order to estimate the equilibrium surface tension and the diffusion coefficients as detailed in the next sections. Here we consider conditions of temperature and pressure in which the solvent and antisolvent are partially miscible and there are two phases in equilibrium. The fully miscible case will be considered in a future publication. Figure 1 shows the calculated locus of critical points for the system ethanol-carbon dioxide. We verified that this curve is in excellent agreement with experimental results from the literature11 for temperatures below aproximately 360 K, which covers the vast majority of operation conditions typically used in SAS. We note however that, when trying to reproduce the whole

10.1021/ie021048j CCC: $25.00 © 2003 American Chemical Society Published on Web 05/30/2003

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the viscosity of the ambient gas. The functions in eq 2 are

F1 )

F3 )

I0() 2I1()

Jet Breakup Length The SAS process commonly involves feeding a liquid solution (solvent + solute) through a small orifice into the antisolvent continuum. The resulting jet eventually breaks up, and it is important to understand the mechanism and time scales associated with this process. The breakup of a liquid jet issuing from a nozzle and discharging into a stagnant gas has been studied for a long time. At low and moderate jet velocities, the growth of long wavelength, small amplitude disturbances is believed to initiate the liquid breakup process.12 Rayleigh was the first to show that the jet breakup is a consequence of hydrodynamic instability.13 In this classical result he neglected viscous effects in the liquid jet, and the density of the ambient gas. These factors were later taken into account by Weber, but his theoretical predictions did not agree well with experimental data.14 Sterling and Sleicher introduced a correction factor in the term controlling the aerodynamic effects and improved the agreement with experiments at low pressures.15 Recently, this model has been successfully applied to calculate jet lengths at operating conditions typical of the supercritical antisolvent process.16 This is also the model that we have used in this work to calculate the characteristic jet breakup time. The main idea is to assume that the interface, r ) a, of a cylindrical jet of radius a is perturbed by an axisymmetric wave with Fourier components of the form

η ) η0eβt+ikz

(1)

where η is the radial perturbation of the liquid surface, η0 is the amplitude of the initial perturbation, k ) 2π/λ is the wavenumber of the disturbance, λ is its wavelength, and β is its growth rate. A cylindrical coordinate system is used that moves in the axial direction, z, at the jet velocity, and whose origin z ) 0 is located at the nozzle exit at time t ) 0. The theory then analyzes the linear stability of the liquid surface to perturbations and arrives at the following equation,

β2F1 + βF3 ) F4 + csF5

(2)

where cs ) 0.175 is the correction factor, introduced in order to compensate for the effects of having neglected

2FLK1()

[

(

)]

σ (1 - 2)2 3 2FLa

F5 )

critical locus, the model tends to overestimate the critical pressure, in some cases by as much as 15 bar.

FVK0()

I0() I0() I0(1) 22 µ2 1 + - 1 2  2 2 I () 2 I () I FLa 1 -  1 1 1(1) F4 )

Figure 1. Mixture critical locus (dashed line) and pure component boiling curves (solid lines) for the system carbon dioxide-ethanol. Calculations performed using the Soave-Redlich-Kwong equation of state with binary parameter kij ) 0.0795. The filled circles represent the pure component critical points.

+

u2FV3K0() 2a2FLK1()

12 ) 2 +

βa2FL µ

(3)

where In and Kn are the nth-order modified Bessel functions of the first and second kind, respectively, u is the jet velocity, µ is the jet viscosity, FL and FV are the densities of the liquid and ambient gas, σ is the interfacial tension, and  is a dimensionless wavenumber defined by  ) ka. Solving numerically eq 2, it is possible to obtain the value of the growth rate associated with every wavenumber. It is reasonable to assume that the component with the fastest growth rate will eventually dominate the breakup. We denote the wavenumber of this component as * and the growth rate of this component as β*. The breakup of the jet occurs when the amplitude of the fastest growing wave is equal to the jet radius. Thus, from eq 1, the breakup time tbk and the jet length L are given by

tbk )

1 ln(a/η0) β*

(4)

and

L ) utbk

(5)

Most of the physical properties used in these equations are readily available, as they refer to pure components. The exception is the surface tension. If we assume that the interface between the liquid jet and the ambient gas achieves thermodynamic equilibrium sufficiently fast, then the surface tension to be used in eqs 2 and 3 is that between the phases at equilibrium. This is the approach used in this work, and the equilibrium surface tension, σeq, was calculated by means of the Parachor parameter equation:

σeq ) [

∑i (Pi)(CLxi - CVyi)]4

(6)

where Pi are the Parachor parameters of the pure components,17 CL and CV are the molar densities of the liquid and vapor phases, respectively, and xi and yi are the mole fractions of the components in the liquid and vapor phases, respectively. Figure 2 shows the dependence of the equilibrium surface tension on pressure and temperature. The equilibrium surface tension vanishes at the mixture

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Figure 4. Jet length as a function of pressure and temperature for We ) 10. Isobars are shown in increments of 10 bar, from 80 to 120 bar. D0 ) 200 µm. Ethanol jet in quiescent carbon dioxide. Figure 2. Equilibrium surface tension for the system ethanolcarbon dioxide for several conditions of temperature and pressure. Isobars are shown in intervals of 10 bar, between 80 and 120 bar.

Figure 3. Calculated (solid line) versus experimental (diamonds) jet length for liquid ethanol in quiescent carbon dioxide at T ) 308 K and P ) 57 bar. D0 ) 127 µm, cs ) 0.175, ln(a/η0) ) 12.

critical points. At any given pressure, it rises sharply near the lower critical temperature, goes through a maximum, and then decreases again toward the upper critical temperature. In the linear regime, where the Sterling-Sleicher model is appropriate, drops formed immediately after the breakup contain the same volume as a cylinder with the diameter of the jet and a length of λ. Thus, the droplet diameter can be estimated by18

Dd ) D0(6π/4*)1/3

(7)

where D0 is the nozzle diameter and * is the wavenumber of the fastest growing disturbance, that is to say, the one that generates the maximum β in eq 2. Figure 3 shows the results of model calculations for a stream of liquid ethanol in ambient carbon dioxide at T ) 308 K and P ) 57 bar. In the same figure we have included experimental data for comparison. The nozzle diameter used in these calculations and in the experiments was D0 ) 127 µm. Details of the experimental apparatus are given in the Appendix. It can be seen that the theoretical prediction agrees quite well with experimental observations. A few comments concerning the amplitude of the initial disturbance, η0 in eq 4, are appropriate at this stage. The usual approach, taken by many authors before (see, e.g., ref 15), has been to consider this quantity as an adjustable parameter, whose value is to be obtained by comparison with experiments. It has

Figure 5. Jet breakup time as a function of pressure and temperature for We ) 10. Isobars are shown in increments of 10 bar, from 80 to 120 bar. D0 ) 200 µm. Ethanol jet in quiescent carbon dioxide.

been suggested that the value of ln(a/η0) varies linearly with the Ohnesorge number,19 but at present a consistent theory or a reliable correlation is missing. Both our experimental results and results from the literature show that at low jet velocity the jet length is independent of pressure.16 This is the criterion followed in this work. We consider a base case with a value of η0 such that it best fits the experimental results; the value of η0 for other conditions of pressure and temperature is then calculated such that it produces the same jet length at low velocities. Figure 4 shows the calculated jet length for mixture subcritical conditions at a fixed Weber number, We ) D0u2FL/σL ) 10. The behavior of the breakup time under the same conditions is shown in Figure 5. Finally, the diameter of the droplets formed after breakup can be calculated by means of eq 7 for the same conditions. The results are shown in Figure 6. Similar calculations were performed for other values of the Weber number in the range 1 < We < 40, representative of practical SAS conditions, and it was confirmed that the behavior is qualitatively the same. The maximum order of magnitude of the breakup time is around 10-3 s in all cases. At constant temperature, jet length decreases with increasing pressure, as has been observed experimentally.16 Jet length, droplet diameter, and breakup time show similar behavior; they vanish at the lower critical point, then rise sharply with temperature, keeping the pressure constant, attain a maximum, and start approaching zero again near the upper critical point. The behavior of these quantities follows closely the behavior of the equilibrium surface

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Figure 6. Droplet diameter, normalized by orifice diameter, shown as a function of pressure and temperature for We ) 10. Isobars are shown in increments of 10 bar, from 80 to 120 bar. Ethanol jet in quiescent carbon dioxide.

tension, which vanishes at the critical points, as is shown in Figure 2. Mass Transfer The relevant mass transfer process is the diffusion of antisolvent into a solvent droplet.6,7 The characteristic time of molecular diffusion is

τD ) R2/DAB

(8)

where R is a characteristic diffusion length, here taken as R ) Dd/2, and DAB is the diffusion coefficient of the mixture (A refers to carbon dioxide and B refers to ethanol). At conditions far from the critical point, the diffusive flux can be assumed to be proportional to the concentration gradient as described by Fick’s law:

NA ) -FLDABxA + xAN

(9)

where DAB is the Fickian (hydrodynamic) diffusion coefficient. In reality, however, the flux is proportional to the chemical potential gradient, and we write20

NA ) -FLDABxA + xAN

(10)

[ (

(11)

DAB ) DAB 1 +

)]

∂ ln φˆ A ∂ ln xA

T,P

where φˆ A is the fugacity coefficient of component A in the mixture. The thermodynamic factor in brackets is not too different from unity in general, but it vanishes at the mixture critical points, where DAB goes to zero.21 This term, which plays a crucial role under the conditions considered here, can be calculated via the SRK equation of state. Since the droplets may contain a high concentration of carbon dioxide at saturation, an appropriate mixing rule is needed to approximate the diffusivity of the mixture. We used the following empirical relation, first proposed by Vignes22 and recommended by Reid et al.20 and Cussler:23

DAB ) (D°AB)(1-xA)(D°BA)xA

(12)

The liquid-phase diffusion coefficient at infinite dilution (D°AB) was calculated using the Tyn-Calus correlation.24 The gas-phase diffusion coefficient at infinite dilution (D°BA) was calculated using the Fuller correla-

Figure 7. Ratio of diffusion time to jet breakup time as a function of pressure and temperature. We ) 10 and D0 ) 200 µm. Isobars are shown in increments of 10 bar, from 80 to 120 bar.

tion,25 incorporating corrections for high pressures according to Takahashi.26 Although the mole fraction xA is a function of time at any location in the droplet, in our calculations we have used eqs 11 and 12 with the equilibrium compositions at the given temperature and pressure. This choice does not change the order of magnitude of the characteristic diffusion time. A comparison of the diffusion time, characteristic of the mass transfer process, and the breakup time, characteristic of the hydrodynamic process, is presented in Figure 7. Near both the upper and lower critical points, the term in brackets in eq 11 vanishes, and so, therefore, does the diffusion coefficient. In addition, from Figure 6, the breakup time approaches zero, resulting in the divergence of the ratio τD/tbk observed in Figure 7. It can be seen that, across the range of conditions investigated, the characteristic time of diffusion is at least 2 orders of magnitude larger than the jet breakup time. Nucleation Supersaturation is the driving force for solute precipitation. A natural definition for this driving force, which becomes exact under ideal mixture (dilute) conditions is s ) c/c0, where c and c0 are the actual and equilibrium solute concentrations, respectively. Classical nucleation theory then gives a well-known equation for the steady nucleation rate27

J0 ) ω*ZN(n*)

(13)

The rate expression can be written as the product of three terms. The first term, ω*, is the frequency of attachment of molecules to the nucleus of critical size. For the particular case of nucleation in solutions, it is given by28

ω* ) 4πr*2Cνλ0e-∆U/kBT

(14)

where r* is the radius of the critical nucleus, C is the number concentration of the solute, ν is a frequency factor, λ0 is the mean free path of particles in the liquid (which is approximately equal to the molecular diameter), kB is Boltzmann’s constant, T is the temperature, and ∆U is a characteristic energy that plays the role of a barrier to the incorporation of molecules into the critical nucleus. For single component nucleation from the melt, this term is normally associated with a

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viscous barrier; for solute nucleation from a binary mixture, the term “desolvation energy” is sometimes used.28 The second term, Z, is the Zeldovich nonequilibrium factor, given by

Z)

1 n*

x

∆G* 3πkBT

(15)

Table 2. Values of Parameters Used in the Calculations of Induction and Nucleation Times

σc C vs ν λ0

value

ref

0.015 N/m 3.98 × 1025 molecules/m3 1.94 × 10-28 m3/molecule 3 × 1013 s-1 4 × 10-10 m

30 30 30 28 28

where ∆G* is the reversible work of formation of a critical nucleus and n* is the number of molecules in the critical nucleus. The third term, N(n*), is the equilibrium concentration of critical nuclei and is given by

N(n*) ) Ce-∆G*/kBT

(16)

The exponential factor is responsible for the strong dependence of nucleation rate on supersaturation. In classical nucleation theory the work of formation of a critical nucleus is given by

∆G* )

16πvs2σc3 2

3kB T

2

1 (ln s)2

(17)

where vs is the volume per molecule in the embryo phase and σc is the interfacial tension between the metastable phase and the embryo. The above equations assume that a steady state has been established with respect to nucleation. In reality, some time must pass before the steady-state nucleation rate is established. A measure of this time is the socalled induction time, τI, given by29

τI )

4 Z2π2ω*

(18)

The key characteristic time for the nucleation process is the time it takes to nucleate one critical-sized nucleus in one droplet. If we let Vd denote the volume of the droplet, this nucleation time, τN, is defined by

Vd

∫0τ J(t) dt ) 1 N

(19)

where ∞

(-1)ie-i t/τ ] ∑ i)1

J(t) ) J0[1 + 2

2

I

(20)

This equation for the non-steady-state nucleation rate was first derived by Kaschiev.29 Upon substitution of eqs 14 and 15 in eq 18, an expression for the induction time is obtained. We also used eq 17 and expressions for r* and n* taken from classical nucleation theory.27 The resulting equation for the induction time is

τI )

σc 16 e∆U/kBT π k T(ln s)2Cνλ B 0

(21)

Equations 21 and 19, together with eq 20, allow the calculation of the characteristic time scales of the nucleation process. Additional difficulties arise from the strong dependence of the nucleation rate on supersaturation and desolvation energy. Relatively small changes

Figure 8. Logarithm of the induction time as a function of the supersaturation c/c0 for several values of the desolvation energy, ∆U ) 6, 8, 10, 12, and 14 kcal/mol.

in these parameters result in changes of orders of magnitude in the induction and nucleation times. To carry out the calculations, specific values for solute and solvent physical properties, such as desolvation energy, interfacial tension, and molecular diameter, are needed. From eq 21 we notice that the two magnitudes that have the largest effect on the induction time are the supersaturation, s, and the desolvation energy, ∆U. We proceeded to calculate the induction time as a function of supersaturation, for several values of the desolvation energy, using literature values for the rest of the parameters. In ref 30, the authors found experimentally the value for the dimensionless parameter A ) 16πσc3vs2/3kB3T3 ) 31.06 for the precipitation of paracetamol from ethanol solution at temperatures between 308 and 333 K. From this, a value of the crystal surface tension can be deduced, σc ) 0.015 N/m. The density of paracetamol and typical values of solute concentration were also taken from this source. It is important to stress that we are not interested in a particular solute but rather in the general effect of the different parameters on the orders of magnitude of the characteristic times. The values used in the calculations are compiled in Table 2. Calculation results for induction time are presented in Figure 8. When the supersaturation is small, the induction time is large, the steady-state nucleation rate, J0, is also very small, and consequently, from eq 19, the nucleation time is very large. At small values of supersaturation, it is found that τN . τI. For large values of supersaturation, the situation is reversed and τN < τI. This is shown in Figure 9 for a droplet of diameter D ) 200 µm. The calculations show that the ratio τN/τI is independent of ∆U. At very low values of supersaturation, both induction and nucleation times are extremely sensible to changes in supersaturation. This sensitivity is greatly reduced at large supersaturation values; for instance, when the supersaturation changes from 5 to 17, the induction time decreases only by a factor of 3. Figure 8 also illustrates the effect of the parameter ∆U. For the

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Figure 9. Ratio of nucleation to induction time as a function of supersaturation .

Figure 11. Ratio of diffusion to nucleation time as a function of droplet diameter for values of supersaturation s ) 2.2 and s ) 5. ∆U ) 14 kcal/mol.

interfacial tension (large energetic barrier to nucleation) and high desolvation energy (strong attractive forces between the solute and its solvent). Conclusions

Figure 10. Ratio of diffusion to nucleation time as a function of droplet diameter for values of supersaturation s ) 2.2 and s ) 5. ∆U ) 6 kcal/mol.

conditions considered here, a change in desolvation energy of only 1 kcal/mol leads to an increment in nucleation and induction time by a factor of approximately 20. Our calculations suggest that the most likely scenario is one in which the diffusion time is several orders of magnitude larger than the nucleation time. In this case, a diffusion-limited precipitation front is expected to occur. Figure 10 shows a comparison between nucleation and diffusion times as a function of droplet size for a value of ∆U ) 6 kcal/mol and two values of supersaturation. From the figure it can be seen that with a supersaturation value of s ) 5, even for very small droplets, the diffusion process is the limiting step. In fact, it can be seen that diffusion is the limiting process except for very low supersaturation and small droplets. Factors that favor the formation of a diffusion-limited precipitation front are small values of desolvation energy (low affinity between the solute and solvent) and low interfacial tension (small energetic barrier to creating the crystal phase). On the other hand, a nucleation-limited regime can also be found under the appropriate conditions. With smaller droplets, the diffusion time decreases rapidly and the nucleation time increases. Thus, finer atomization favors a nucleation-limited regime. Smaller droplets can be produced at conditions far from the critical points, for example by the use of atomization devices or by injecting the liquid at very high velocities. In Figure 11 it can be seen that, for supersaturation s ) 2.2, ∆U ) 14 kcal/mol, and droplet diameters smaller than approximately 250 µm, nucleation becomes the slowest process. This scenario is favored by high

The analysis of characteristic times presented in this paper provides insight into the relative importance of each of the basic processes in SAS over a range of operating conditions. It can also be used to validate certain approximations or assumptions made in particular models of the SAS process. Our calculations show that the characteristic time for mass transfer is at least 2 orders of magnitude larger than the characteristic time for the development of the hydrodynamic jet instability. Under these conditions it is reasonable to assume that, by the time the jet breaks up into droplets, little antisolvent has diffused into the liquid stream. This is consistent with the viewpoint adopted in recent modeling of mass transfer in the SAS process, where it was assumed the initial condition was an antisolventfree droplet of solution.6,7 In the case of the nucleation process, it is more difficult to draw conclusions, due to the large variability of the characteristic time. We also need to keep in mind that it is strongly coupled to the mass transfer process through the supersaturation. Nevertheless, on the basis of these calculations, two different regimes can be identified. The first one, characterized by a small nucleation to diffusion time ratio (fast nucleation process, slow diffusion), leads to a diffusion-limited precipitation front. Small values of desolvation energy, low interfacial tension, and high values of supersaturation all favor the occurrence of this regime. Also, factors that decrease the diffusivity, such as proximity to the critical points, favor the diffusion-controlled regime. In the second case, characterized by a large nucleation to diffusion time ratio (slow nucleation dynamics, fast diffusion), precipitation will occur homogeneously throughout the interior of the well-mixed droplet. In this paper we have focused on molecular diffusion, but a nucleation-limited scenario would be favored by the appearance of convective currents in the interior of the droplet. Other factors that tend to favor a nucleation-limited regime are large desolvation energy, large interfacial tension, low solute concentration, and low supersaturation. These considerations provide valuable guidelines to be used for the development and interpretation of complete models of the process or for applications to particular precipitating systems.

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Finally, we point out that in this work we have focused on conditions of temperature and pressure in which there are two phases in equilibrium. However, conditions for which the solvent-antisolvent mixture is fully miscible are also of interest in many SAS applications. This case will be considered in future work. Acknowledgment The authors gratefully acknowledge the support of the New Jersey Commission on Science and Technology through its R&D Excellence Award for the New Jersey Program for Engineered Particulates (NJPEP). Appendix The experimental apparatus consisted of a chamber with capacity 75 mL, cross-sectional area 12.7 mm × 12.7 mm, and a metering pump. The chamber is capable of operating at pressures up to 207 bar and temperatures up to 200 °C. Liquid solvent was injected through a sharp-edged stainless steel nozzle of 127 µm internal diameter. Carbon dioxide (Bone-dry, 99.9%) was used as antisolvent, and ethanol with a purity of 99.99% was used as the solvent. The chamber pressure was built up either by a high-pressure CO2 cylinder or by a CO2 pump and further regulated by a back-pressure regulator. The optical visualization system was comprised of a microscope lens, a high-speed CCD camera, and a computer with image capture software installed. The view field could be varied from 0.3 mm × 0.4 mm to 20 mm × 27 mm. Two different high-speed cameras were employed. The first one took pictures at the rate of 85 frames per second, with exposure time less than 100 µs and a pixel resolution of 648 × 484 per frame. The second one used a Nd:YAG dual cavity pulsed laser and could take double shots with an exposure time of 0.5 µs with a 15 µs delay and a pixel resolution of 1280 × 1024 per frame. The combined use of these cameras provided the flexibility of visualizing macrostructured jet phenomena, such as flow pattern, as well as microstructures. Literature Cited (1) Tan, H. S.; Borsadia, S. Particle Formation using Supercritical Fluids: Pharmaceutical Applications. Expert Opin. Ther. Pat. 2001, 11, 861. (2) Reverchon, E.; Della Porta, G.; Di Trolio, A.; Pace, S. Supercritical Antisolvent Precipitation of Nanoparticles of Superconductor Precursors. Ind. Eng. Chem. Res. 1998, 37, 952. (3) Jung, J.; Perrut, M. Particle Design Using Supercritical Fluids: Literature and Patent Survey. J. Supercrit. Fluids 2001, 20, 179. (4) Dixon, D. J.; Jonhston, K. P.; Bodmaier, R. A. Polymeric Materials Formed by Precipitation with a Compressed Fluid Antisolvent. AIChE J. 1993, 39, 127. (5) Kikic, I.; Lora, M.; Bertucco, A. Simulation of the Semicontinuous Supercritical Antisolvent Recrystallization Process. Ind. Eng. Chem. Res. 2000, 39, 1487. (6) Werling, J. O.; Debenedetti, P. G. Numerical Modeling of Mass Transfer in the Supercritical Antisolvent Process. J. Supercrit. Fluids 1999, 16, 167.

(7) Werling, J. O.; Debenedetti, P. G. Numerical Modeling of Mass Transfer in the Supercritical Antisolvent Process: Miscible Conditions. J. Supercrit. Fluids 2000, 18, 11. (8) Soave, G. Equilibrium Constants from a Modified RedlichKwong Equation of State. Chem. Eng. Sci. 1972, 27, 1197. (9) Perry, R. H.; Green, D. W.; Maloney, J. O. Perry’s Chemical Engineer’s Handbook; McGraw-Hill: New York, London, 1997. (10) Tanaka, H.; Kato, M. Vapor-Liquid Equilibrium Properties of Carbon Dioxide + Ethanol Mixture at High Pressures. J. Chem. Eng. Jpn. 1995, 19, 263. (11) Ziegler, J. W.; Chester, T. L.; Innis, D. P.; Page, S. H.; Dorsey, J. G. Supercritical Fluid Flow Injection Method for Mapping Liquid-Vapor Critical Loci of Binary Mixtures Containing CO2. In ACS Symposium Series; Hutchenson, K. W., Foster, N. R., Eds.; American Chemical Society: Washington, DC, 1995; Vol. 608, p 93. (12) Lin, S. P.; Reitz, R. D. Drop and Spray Formation from a Liquid Jet. Annu. Rev. Fluid Mech. 1998, 30, 85. (13) Rayleigh, L. On the Instability of Jets. Proc. London Math. Soc. 1879, 10, 4. (14) Weber, C. Z. Zum Zerfall eines Flussigkeitsstrahles. Math. Mech. 1931, 11, 136. (15) Sterling, A. M.; Sleicher, C. A. The Instability of Capillary Jets. J. Fluid Mech. 1975, 68, 477. (16) Lengsfeld, C. S.; Delplanque, J. P.; Barocas, V. H.; Randolph, T. W. Mechanism Governing Microparticle Morphology during Precipitation by a Compressed Antisolvent: Atomization vs Nucleation and Growth. J. Phys. Chem. 2000, 104, 2725. (17) Quayle, O. R. The Parachors of Organic Compounds. An Interpretation and Catalogue. Chem. Rev. 1953, 53, 439. (18) Brodkey, R. S. The Phenomena of Fluid Motion; AddisonWesley Pub. Co.: Reading, MA, 1967. (19) Grant, R. P.; Middleman, S. Newtonian Jet Instability. AIChE J. 1966, 12, 669. (20) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids, 4th ed.; McGraw-Hill: New York, 1987. (21) Levich, V. G. Physicochemical Hydrodynamics, 2nd ed.; Prentice Hall: Englewood Cliffs, NJ, 1962. (22) Vignes, A. Diffusion in Binary Solutions: Variation of Diffusion Coefficient with Composition. Ind. Eng. Chem. Fundam. 1966, 5, 189. (23) Cussler, E. L. Diffusion: Mass Transfer in Fluid Systems; Cambridge University Press: Cambridge, U.K., 1984. (24) Tyn, M. T.; Calus, W. F. Diffusion Coefficients in Dilute Binary Liquid Mixtures. J. Chem. Eng. Data 1975, 20, 106. (25) Fuller, E. N.; Ensley, K.; Giddings, J. C. Diffusion of Halogenated Hydrocarbons in Helium: The Effect of Structure on Collision Cross Sections. J. Phys. Chem. 1969, 73, 3679. (26) Takahashi, S. Preparation of a Generalized Chart for Diffusion Coefficients of Gases at High Pressures. J. Chem. Eng. Jpn. 1974, 7, 417. (27) Debenedetti, P. G. Metastable Liquids: Concepts and Principles; Princeton University Press: Princeton, NJ, 1996. (28) Markov, I. V. Crystal Growth for Beginners: Fundamentals of Nucleation, Crystal Growth, and Epitaxy; World Scientific: Singapore, River Edge, 1995. (29) Kashchiev, D. Solution of the Non-Steady-State Problem in Nucleation Kinetics. Surf. Sci. 1969, 14, 209. (30) Shekunov, B. Y.; Baldyga, J.; York, P. Particle Formation by Mixing with Supercritical Antisolvent at High Reynolds Numbers. Chem. Eng. Sci. 2001, 56, 2421.

Received for review December 30, 2002 Revised manuscript received April 23, 2003 Accepted April 28, 2003 IE021048J