Numerical analysis of the characteristic times controlling supercritical antisolvent micronization

Chemical Engineering Science 71 (2012) 39–45

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Numerical analysis of the characteristic times controlling supercritical antisolvent micronization Francesco Marra, Iolanda De Marco, Ernesto Reverchon n Department of Industrial Engineering, University of Salerno, Via Ponte Don Melillo, 1, 84084 Fisciano, SA, Italy

a r t i c l e i n f o

abstract

Article history: Received 30 July 2011 Received in revised form 9 December 2011 Accepted 10 December 2011 Available online 19 December 2011

In this work jet break-up time and of dynamic surface tension vanishing are considered as mechanisms in competition during supercritical antisolvent precipitation and a mathematical model, based on these two characteristic phenomena, is presented. Jet break-up time has been evaluated solving continuity and conservation of momentum equations; on the other hand, dynamic surface tension vanishing time has been evaluated according to the timeevolution model proposed by Cahn and Hilliard. Phase equilibria have also been taken into account, considering the Peng and Robinson equation of state and the related mixing rules. Calculations have been applied to yttrium acetate (YAc) as model solute, dimethylsulphoxide (DMSO) as liquid solvent and carbon dioxide (CO2) as antisolvent. The cross-over times, between jet break-up dominated and dynamic surface tension vanishing dominated regions, have been calculated at different pressures for pure DMSO and at different YAc concentrations in the liquid solution; a good agreement with previous experimental results has been obtained. The numerical results also correctly describe the influence of solute concentration on the pressure at which cross-over between the two regions is obtained. The characteristic times for acetone (AC) have also been evaluated, to perform a comparison with DMSO; in this case, cross-over has been observed in proximity of the mixture critical point pressure of the binary system. & 2011 Elsevier Ltd. All rights reserved.

Keywords: Jet break-up time Dynamic surface tension vanishing time Cross-over time Mathematical modeling Supercritical antisolvent precipitation Supercritical carbon dioxide

1. Introduction Supercritical antisolvent (SAS) precipitation is a process that has been successfully applied to produce microparticles, nanoparticles and crystals of several categories of compounds (Hakuta et al., 2003; Reverchon and De Marco, 2011; Shariati and Peters, 2003). It is based on two fundamental requisites: the solute to be micronized has to be soluble in the organic solvent and not soluble in the supercritical antisolvent (carbon dioxide, SC-CO2); organic solvent and SC-CO2 have to be completely miscible at the process operating conditions. Though many experimental evidences are available, the interpretation of the results demonstrated to be very complicated, due to the complex interactions among high pressure vapor liquid equilibria (VLEs), mass transfer, fluid dynamics, nucleation and growth processes, whose interlacing can produce different sizes and morphologies of SAS precipitates (Reverchon and De Marco, 2011).

n

Corresponding author. Fax: þ39 89 964057. E-mail addresses: [email protected] (I. De Marco), [email protected] (E. Reverchon). 0009-2509/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2011.12.019

Some papers have been dedicated to the modeling analysis of the SAS process or of part of it. (1) Mass transfer and its different evolutions at subcritical and supercritical conditions have been analyzed by Werling and Debenedetti (1999, 2000), that studied the evolution of the diameter of a single droplet of toluene at subcritical conditions and at supercritical miscible conditions; i.e., below and above the mixture critical point (MCP) of the binary system solvent– antisolvent. The mass transfer model was based on the diffusion approach. These works raised the problem of droplet definition at supercritical conditions, due to surface tension vanishing. Some authors (Dukhin et al., 2003; Lengsfeld et al., 2000) showed that dynamic surface tension (DST) does not vanish instantaneously when a liquid jet is fed in a supercritical fluid. Elvassore et al. (2004) added a fictitious solute, to investigate its interactions in the mass transfer modeling. Then, Fadli et al. (2010) introduced non-isothermal conditions in the description of the process; i.e., these authors considered a droplet surrounded by a fluid at a different temperature. (2) Hydrodynamics of the SAS process has also been the subject of some works. Several authors considered a jet of liquid

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F. Marra et al. / Chemical Engineering Science 71 (2012) 39–45

solvent injected into a gas, applying the classical jet break-up theory (Czerwonatis and Eggers, 2001; Kerst et al., 2000; Lengsfeld et al., 2000; Petit-Gas et al., 2009). (3) Some comprehensive approaches have also been attempted. For example, Martı´n and Cocero (2004) tried to model the whole SAS process above the MCP conditions, considering only gaseous mixing between the solution and SC-CO2. A computational fluid-dynamics (CFD) model was used and only qualitative results were obtained about particle size, particle size distribution and about the effect of the various SAS process parameters. A CFD approach was also attempted by Tavares Cardoso et al. (2008) in modeling minocycline hydrochloride micronization by SAS and by Ba"dyga et al. (2010) studying precipitation of paracetamol in the singlephase system and of nicotinic acid in the two-phase system. The limits of the currently available models are:

 No quantitative results have been obtained about particle size   

and their distributions; An explanation of the formation of the various particle morphologies that SAS can produce is still not available in most cases; Particle formation mechanisms require more investigations; The proposed models are not applicable to real systems or have been applied to very limited sets of experimental evidences.

Part of these limitations can be overcame, since a consistent explanation has been proposed by Reverchon and co-workers for several particle formation mechanisms, leading to different particles morphologies (Reverchon et al., 2007, 2008a,b, 2010; Reverchon and De Marco, 2011). These mechanisms can be applied to large sets of experimental evidences; i.e., to several solutes belonging to different classes of compounds (Reverchon et al., 2007, 2008a,b, 2010; Reverchon and De Marco, 2011). The proposed explanations are based on the competition between jet break-up time (JBU) and dynamic surface tension vanishing (DSTV) processes. This competition can take place near and far above the MCP of the system solvent–antisolvent, since it has been demonstrated (Dukhin et al., 2003; Lengsfeld et al., 2000; Reverchon et al., 2010) that DST vanishes in a finite period of time, even when SAS process is performed at supercritical conditions. The possibility of the formation of a gas plume (gas mixing) between the solvent and the antisolvent has to be taken into account. These two concepts are joined together through the definition of two characteristic times: a time of jet break-up (T) and a time of surface tension vanishing (t). When Tot, jet breakup prevails (always at subcritical conditions, sometimes also at supercritical conditions), droplets are formed and their subsequent drying produces perfectly spherical microparticles ranging from sub-micrometric to some micron diameters (Reverchon et al., 2008a). When T4t, gas mixing is obtained and the particles precipitate from a fluid phase: they are irregularly spherical nanoparticles (formed in the absence of surface tension), according to a gas-to-particle mechanism (Reverchon et al., 2007; Reverchon and De Marco, 2011). Spherical hollow microparticles are formed, instead, when jet break-up and DST vanishing operate in parallel. However, conceptually, these processes can be also partly in series, thus producing a solid precipitate formed by the coexistence of two different morphologies (Braeuer et al., 2011). Considering the practical and general relevance of these possible explanations and the fact that they have been confirmed for several compounds of industrial interest (Braeuer et al., 2011; De Marco and Reverchon, 2011; Reverchon et al., 2008b; Reverchon and De Marco, 2011), the aim of this work is to

propose a mathematical modeling of the processes producing JBU and DSTV, to evaluate from a theoretical point of view, the characteristic times and their evolution at different SAS process conditions, thus proposing an explanation on the particle morphologies that SAS can produce. The matching with the experimental results and modeling trends will also be attempted.

2. Modeling 2.1. Computation of jet break-up time (T) According to Cha´vez et al. (2003), the break-up of a liquid jet exiting from a nozzle can be calculated using the following expression:   1 a ð1Þ T ¼ n ln

Z0

b

n

where b is the fastest growth rate associated with wave-number that will dominate the jet break-up, Z0 is the amplitude of the initial jet perturbation, due to the presence of the dense gas, and a is the jet nozzle radius. According to the classical theory (Cha´vez et al., 2003), break-up of the jet occurs when the amplitude of the fastest growing wave is equal to the jet radius. n To know the value of b , it is necessary to solve continuity and conservation of momentum equations that describe the classic jet break-up, based on the Weber theory proposed by Sterling and Sleicher (1975). It results in a system of non linear equations consisting of modified Bessel functions of dimensionless wave numbers, of the first and the second kind, of order 0 and 1, as in the following equations (Cha´vez et al., 2003; Lengsfeld et al., 2000):

b2 F 1 þ bF 3 ¼ F 4 þ0:175F 5 F1 ¼

ð2Þ

eI0 ðeÞ rV eK 0 ðeÞ þ 2I1 ðeÞ 2rL K 1 ðeÞ "

ð3Þ 

me2 I0 ðeÞ 2e2 I0 ðeÞ I0 ðe1 Þ 1 þ 2 2 e e1 2e F3 ¼ I1 ðeÞ I1 ðeÞ I1 ðe1 Þ r L a2 e1 e F4 ¼

F5 ¼

s 2rL a3

ð1e2 Þe2

u2 rV e3 K 0 ðeÞ 2rL a2 K 1 ðeÞ

e21 ¼ e2 þ

ba2 rL

m

# ð4Þ

ð5Þ

ð6Þ

ð7Þ

where e ¼ ka is a dimensionless wavenumber, k ¼ 2p=l is the wavenumber of the disturbance, I0 and I1 are the order 0 and order 1 modified Bessel functions of the first order and K0 and K1 are the order 0 and order 1 modified Bessel functions of the second order. Eq. (2) already includes the so-called aerodynamic reduction factor that compensates the fact that the viscosity of the ambient gas has been neglected (not considered). According to Sterling and Sleicher (1975), it has been considered to be equal to 0.175. The solution of the system formed by Eqs. (2)–(7) allows to calculate b as a function of wave-numbers, of the jet velocity u, of the injected solution viscosity m, of the densities of liquid and dense phases, and of the surface tension. Physical data needed in the JBU time calculation (surface tension and kinematic viscosity, as function of solute concentration) were taken from Braeuer et al. (2011). The calculation procedure requires the value of the initial disturbance amplitude Z0, that has been determined by comparing the experimentally measured jet length (Braeuer et al., 2011) to the predicted jet length at low jet velocities.

F. Marra et al. / Chemical Engineering Science 71 (2012) 39–45

classical models proposed in the literature (Miqueu et al., 2003): in the conditions considered in this work they were:

2.2. Phase equilibria Phase equilibria for two binary systems (DMSO–CO2 and AC–CO2) considered in this work were taken into account by solving the Peng–Robinson equation of state (PR-EoS) and the relative mixing rules for the two considered cases. Peng–Robinson equation is: P¼

RT am  Vbm V 2 þ bm Vb2

ð8Þ

m

that, in terms of compressibility factor, Z, can be rewritten as ! l 1 þ b Z l l l l l l l Z ¼ b þ ðZ þ eUb ÞUðZ þ sUb ÞU ð9Þ l ql Ub

v

v

kii CO2–CO2 : 4.44e-21 J m5 mol  1 kij CO2–DMSO : 2.33e-20 J m5 mol  1 kjj DMSO–DMSO : 3.74e-19 J m5 mol  1 Eq. (18) can be further extended and rewritten in terms of total molecular density, being Ci ¼xiC and Cj ¼(1 xi)C, one obtains:    2   @C i @C j @C i @C i @x @C 2 @xi @C 2 ¼ ¼ C i þxi ¼ C þ xi j¼i @r @xi @r @r @r @r @r @r ð19aÞ j ai

for the liquid phase and Z v ¼ 1 þ b qv Ub U

v

Z v b v v ðZ þ eUb ÞUðZ v þ sUb Þ v

ð10Þ

for the vapor phase; where, in general terms, i¼j

p

bp 

b P RT

qp 

ap p b URUT

ðp ¼ l,vÞ ðp ¼ l,vÞ

ð11Þ

ð12Þ

are calculated on the basis of average values of coefficients b and a, according with following mixing rules: XX xi xj ð1wij Þðai aj Þ1=2 ð13Þ a¼ i

b¼

j

XX i

xi xj ðbi þ bj Þ

ð14Þ

j

where single contributions are given by ai ¼ C

aðT ri ; oÞR2 T 2ci P ci

2 aðT r ; oÞ ¼ ½1 þ ð0:37464þ 1:54226o0:26992o2 Þð1T 1=2 r Þ

bi ¼ O

RT ci P ci

41

ð15Þ ð16Þ ð17Þ

The above described equation systems have been solved using two Matlab routines: the first one solves the phases equilibrium and provides the values of dense phase density as function of temperature, pressure and composition; the second routine solves the set of Eqs. (2)–(7) using a recursive procedure to compute the values of the growth rate b. 2.3. Computation of dynamic surface tension vanishing time (t) In 2003, Cha´vez et al. compared the jet break-up time with the time needed by the anti-solvent to diffuse in a droplet of solvent; whereas, according to extended experimental observations performed by Reverchon and co-workers (Reverchon et al., 2008a,b, 2010; Reverchon and De Marco, 2011), it seems opportune to compare the classical jet break-up length with the time needed for the zeroing of the dynamic surface tension; the last consideration is in agreement with the time-evolution model already proposed by Cahn and Hilliard (1958): Z 1 XX @C @C j dr ð18Þ s¼ kij i @r @r 0 i j where kij is a contact fluid interaction parameter and where the molecular number density of the components are indicated with Ci and Cj. The fluid influence parameters were estimated using the

   @C i @C j @x @C @C @x ð1xi Þ C i ¼ C i þ xi @r @r @r @r @r @r  2    @xi @C @C ¼ C þ xi C ð1xi Þ @xi @xi @r

ð19bÞ

   @C j @C j @C i @C j @C @x @C @x C i C i ¼ ¼ ð1xi Þ ð1xi Þ @r @r @r @r @r @r @r @r  2  2 @xi @C ð1xi Þ C ð19cÞ ¼ @xi @r

Thus, substituting Eqs. (19a),(19b) and (19c), into Eq. (18): 2    Z 1"  @C @C @C s¼ kii xi þ C þ kij ð1xi Þ C xi þC @xi @xi @xi 0  2 # 2 @C @xi þkjj ð1xi Þ C dr ð20Þ @xi @r Again, the Peng–Robinson equation of state is needed to compute how the molecular concentration changes with mole fraction. The mole fraction gradient across the interface has to be determined. For this purpose, assuming that carbon dioxide simply diffuses into a solvent-rich environment, the mole fraction gradient will satisfy the one-dimensional, transient, diffusion equation accounting for the total molar flux as in the following expression: @C i @2 C @n ¼ Dij 2 i  i @t @r @r   @C @n1 @n2 ¼ þ @t @r @r

ð21Þ

ð22Þ

where ni is the total molar flux of i-component and Dij is the diffusion coefficient. As boundary conditions of Eqs. (21) and (22), equilibrium values of concentration are imposed for r ¼R and CO2 zero concentration at a distance of 1/5 of the nozzle radius. Eqs. (21) and (22) can be solved up to the time when the dynamic surface tension disappears, indicating that a hard-shell is formed and – thus – the final droplet size is reached, independently by the fact that inside the shell other mass transfer phenomena are going on. Eqs. (21) and (22) have been solved using Comsol Multiphysics 3.2, a commercial FEM based software, coupling the set of molar densities available according to the solution of Peng–Robinson equation of state as mentioned in the previous section. In the model based on diffusion, no adjustable parameters are present, therefore, the model is based on thermodynamic data. In the calculation of the DSTV time, numerical approximations on the zeroing of the dynamic surface tension have been done. Computational difficulties were related to the coupling of the equilibrium solvent–antisolvent in a dynamic situation (CO2 diffusion) and in the Z0 tuning of perturbations in the calculation of the jet break-up length.

42

F. Marra et al. / Chemical Engineering Science 71 (2012) 39–45

3. Results and discussion

11

As discussed in the Introduction, the explanation based on the competition between jet break-up and dynamic surface tension vanishing has been proposed to explain some relevant morphologies of the particles produced by SAS; namely: nanoparticles and microparticles. Now, it is relevant to attempt mathematical

50 mg/mL

10 9

25 mg/mL

d, mm

V, cm3

Tp, 1C

Q CO2 , kg/h

QLiq, mL/min

100

500

40

1.78

1

t, ms

8 Table 1 Experimental parameters: d is the injector diameter, V is the volume of the precipitator, Tp is the temperature inside the precipitator and Q CO2 and QLiq are the antisolvent and the liquid solution flow rates, respectively, used in this work.

7 6 5

10 mg/mL pure DMSO

4 3 80

100

120 p, bar

140

160

Fig. 2. Jet break-up time for the system YAc  DMSO  CO2 at different pressures and concentrations.

Table 3 Surface tension vanishing time for DMSO, ms. p, bar

c, mg/mL

80 120 160

0

10

25

50

22 5.1 2.9

23 5.7 3.3

26.3 7.2 3.7

34 8.9 4.3

35 30 50 mg/mL

t, ms

25 25 mg/mL

20 15

Fig. 1. YAc precipitated from DMSO: (a) microparticles obtained at 150 bar, 40 1C and 50 mg/mL; (b) nanoparticles obtained at 150 bar, 40 1C and 15 mg/mL.

Table 2 Jet break-up times for DMSO and DMSO þ YAc, ms. p, bar

80 120 160

10 5

10 mg/mL pure DMSO

c, mg/mL 0

10

25

50

7.8 5.0 3.2

8.4 5.3 3.6

9.2 5.7 4.0

10.3 6.0 4.2

0

80

100

120 p, bar

140

160

Fig. 3. Surface tension vanishing time for the system DMSO–YAc–CO2 at different pressures and concentrations.

F. Marra et al. / Chemical Engineering Science 71 (2012) 39–45

modeling of the mechanisms producing JBU and DSTV, to identify their characteristic times and their variations with SAS process relevant parameters, like pressure (P) and concentration of the liquid solution (c). To perform these operations, Yttrium acetate (YAc) has been considered as the reference solute and dimethylsulphoxide (DMSO) has been considered as organic solvent; the relevant experimental data have been summarized in Table 1. This choice of solute and solvent depended on the fact that thermodynamic vapor liquid equilibria of the system DMSO/CO2 is well known (Vega Gonzalez et al., 2002) and the morphology of YAc precipitates from DMSO, during SAS, has been the subject of some previous studies (Reverchon et al., 2007, 2008a,b, 2010). Examples of the experimentally observed morphologies at different concentrations are reported in Fig. 1a and b. SEM images in these figures propose examples of microparticles and nanoparticles (Reverchon et al., 2003, 2007, 2008b), together with the process conditions at which they have been produced. Using Eqs. (2)–(7) and the specific characteristics of YAc, DMSO, jet break-up times have been calculated at different pressures in the

43

range 80–160 bar and at different solute concentrations between 0 (solvent only) and 50 mg/mL. The calculated times are reported in Table 2 and the corresponding curves are reported in Fig. 2. Fig. 2 shows that jet break-up time ranges between about 11 and 3 ms and tends to shorten when the pressure is increased and to be longer when the initial concentration of YAc in DMSO is increased. The duration in milliseconds is congruent with the experimental observations made, using laser scattering techniques, by Braeuer et al. (2011). According to their calculations, the range 3–11 ms should correspond to a jet break-up length of 1.5–5.5 mm. These authors observed, for example, that in correspondence of a pressure of 160 bar, the jet break-up length is equal to 5 mm. The differences in times could be related to the different injector diameters used (in our calculation, we considered a 100 mm diameter nozzle, whereas Braeuer et al. (2011), used a 200 mm diameter nozzle). It is also expected that break-up conditions will be obtained earlier when the process pressure is increased: characteristic disturbances of the jet can develop faster at higher pressures.

30

30

pure DMSO

c=10 mg/mL

25 20 microparticles

15

t t, ms

t, ms

10

t

10

nanoparticles

T T 5

2

80

100

120 p, bar

140

80

160

30 25

40 35 30

c=25 mg/mL

20

120 p, bar

140

160

c=50 mg/mL

25

15

t

t

20 microparticles

microparticles

15

10

t, ms

t, ms

100

10 T

nanoparticles

5

nanoparticles

T 5 80

100

120 140 p, bar

160

180

80

100

120

140 p, bar

160

180

200

Fig. 4. Crossing of the jet break-up and dynamic surface tension curves for the system DMSO–YAc–CO2. T is the jet break-up time and t is the dynamic surface tension vanishing time.

44

F. Marra et al. / Chemical Engineering Science 71 (2012) 39–45

When solute concentration is increased, also the viscosity of the organic solution increases (thus the forces that oppose to jet break-up increase), produces a shift of the JBU curves to longer times; the corresponding curves also show that this effect varies not linearly with the concentration. This fact corresponds to an increase of viscosity that becomes relevant only at the higher YAc concentrations (Braeuer et al., 2011). Table 3 and Fig. 3 show the trend observed using Eqs. (20) and (22) for DSTV for pure DMSO and solutions DMSO þYAc at various concentrations. The family of curves generated by mathematical modeling shows similar characteristic times as jet break-up; i.e., they are in the same range of milliseconds; however, these curves show a

8 c=50

tcrossing, ms

7

6

c=25

5

more marked variation with pressure, covering times between about 35 and 3 ms. The increase of YAc concentration produces an increase of DSTV characteristic times; but, its influence is less marked with respect to jet break-up times. An increase of the DSTV times can be explained considering that, in the case of dynamic surface tension variation, interfacial propagation can be contrasted by the presence of the solute molecules. Considering together the results in Figs. 2 and 3, the diagrams reported in Fig. 4 are obtained. Fig. 4 shows the crossing points between JBU time and DSTV time; i.e., for different pressures and solute concentrations, the conditions at which the fluid dynamic behavior changes from the prevalence of JBU to the prevalence of dynamic surface tension vanishing. It means that these points, according to the discussion developed in the Introduction, mark the morphology crossing between droplets/microparticles formation to nanoparticles generation (cross-over point). It is possible to note that the conditions at which YAc precipitates change in morphology move to higher pressures when the starting concentration of the solute is increased; but, also longer cross times are shown. These results agree with the results obtained by Braeuer et al. (2011) on YAc precipitated at different pressures and concentrations of the liquid solution and with the results obtained by De Marco and Reverchon (2011) on gadolinium acetate, a compound that, processed by SAS, gave results similar to those obtained using YAc. This result could mean that the presence of solute can slower both JBU and DSTV processes and can be explained through the simultaneous increase of viscosity and slowering of DSTV. Thus,

15

c=10

pure AC c=0 4

JBU time DSTV time

120

130

140

150

10

Fig. 5. Cross-over points (T¼ t) at different pressures and concentrations, for the system YAc–DMSO–CO2.

t, ms

p, bar

5

0

80

100

120 p, bar

140

160

Fig. 7. Jet break  up time and surface tension vanishing time functions for AC.

Table 4 Physico-chemical data for DMSO and AC.

3

Density, g cm Viscosity, cp Surface tensionn, mN m  1 Fig. 6. Jet break-up and dynamic surface tension vanishing times as a function of pressure at different YAc concentrations.

n

DMSO

AC

1.104 1.996 43.54

0.793 0.3075 25.20

Data for surface tension are in air at 20 1C.

F. Marra et al. / Chemical Engineering Science 71 (2012) 39–45

operating at a constant concentration, a higher pressure is required to observe the transition from nanoparticles to microparticles, and operating at constant pressure, a higher concentration is required to observe the reverse transition; i.e., for micro to nanoparticles. The qualitative correspondence of this last numerical result with the SAS experiments has been verified in previous works (Reverchon et al., 2003; De Marco and Reverchon, 2008) and is quantitatively represented in Fig. 5. To summarize altogether these observations, it is possible to visualize in Fig. 6 the surfaces indicating the JBU time and the DSTV time in function of pressure and YAc concentration. The intersection between the two surfaces creates the locus of the cross-over times. To test the behavior of a different organic solvent, some ¨ calculations were made using AC (Pohler and Kiran, 1997). The jet break-up time and the dynamic surface tension vanishing time curves for this solvent are represented in Fig. 7. The transition from JBU controlled process to DSTV controlled process for AC is sharper and located very near the MCP pressure of the binary system, when compared with the same transition between the characteristic processes observed for DMSO. This different behavior could be related to the different physicochemical characteristics of the two solvents, reported in Table 4. These differences could also significantly affect the observable morphologies of the solutes precipitated by SAS. Indeed, in the case of AC, the crossover from the region dominated by JBU and the one dominated by DSTV is located in the surroundings of the MCP pressure; according with the previous discussion on Fig. 4, the region where microparticles formation is expected is very limited and, therefore, micro-morphology has been rarely observed.

4. Conclusions Mathematical modeling of atomization with jet break-up event and of the dynamic surface tension vanishing has been successfully performed and the cross-over condition between the two controlling processes have been obtained at different concentrations of a model solute. The comparison with the experimental results (Braeuer et al., 2011) showed fairly good correspondence with the experimentally measured characteristic times; the experimental trends with solute concentration have also been reproduced. The comparison between DMSO and AC behavior with respect to these characteristic times has been correctly reproduced. We expect similar trends not only for the model compound, but also for many other compounds showing similar experimental trends. References Ba"dyga, J., Kubicki, D., Shekunov, B.Y., Smith, K.B., 2010. Mixing effects on particle formation in supercritical fluids. Chem. Eng. Res. Des. 88, 1131–1141. Braeuer, A., Dowy, S., Torino, E., Rossmann, M., Luther, S.K., Schluecker, E., Leipertz, A., Reverchon, E., 2011. Analysis of the supercritical antisolvent mechanisms governing particles precipitation and morphology by in situ laser scattering techniques. Chem. Eng. J. 173, 258–266.

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